_known_monster_maxes

# Solvable Subgroups of Maximal Order in Sporadic Simple Groups

### THOMAS BREUER
Lehrstuhl D für Mathematik
RWTH, 52056 Aachen, Germany

### May 14th, 2012

We determine the orders of solvable subgroups of maximal orders in sporadic simple groups and their automorphism groups, using the information in the ATLAS of Finite Groups [CCN+85] and the GAP system [GAP04], in particular its Character Table Library [Bre12] and its library of Tables of Marks [NMP11].
We also determine the conjugacy classes of these solvable subgroups in the big group, and the maximal overgroups.
A first version of this document, which was based on GAP 4.4.10, had been accessible in the web since August 2006. The differences to the current version are as follows.

- The format of the GAP output was adjusted to the changed behaviour of GAP 4.5.
- The (too wide) table of results was split into two tables, the first one lists the orders and indices of the subgroups, the second one lists the structure of subgroups and the maximal overgroups.
- The distribution of the solvable subgroups of maximal orders in the Baby Monster group and the Monster group to conjugacy classes is now proved.
- The sporadic simple Monster group has exactly one class of maximal subgroups of the type PSL(2, 41) (see [NW]), and has no maximal subgroups which have the socle PSL(2, 27) (see [Wil10]). This does not affect the arguments in Section 4.14, but some statements in this section had to be corrected.

# Contents

1 The Result
2 The Approach
2.1 Use the Table of Marks
2.2 Use Information from the Character Table Library
3 Cases where the Table of Marks is available in GAP
4 Cases where the Table of Marks is not available in GAP
4.1 G = Ru
4.2 G = Suz
4.3 G = ON
4.4 G = Co2
4.5 G = Fi22
4.6 G = HN
4.7 G = Ly
4.8 G = Th
4.9 G = Fi23
4.10 G = Co1
4.11 G = J4
4.12 G = Fi24′
4.13 G = B
4.14 G = M
5 Proof of the Corollary

## 1 The Result

The tables 1 and 2 list information about solvable subgroups of maximal order in sporadic simple groups and their automorphism groups. The first column in each table gives the names of the almost simple groups G, in alphabetical order. The remaining columns of Table 1 contain the order and the index of a solvable subgroup S of maximal order in G, the value log|G|(|S|), and the page number in the ATLAS [CCN+85] where the information about maximal subgroups of G is listed. The second and third columns of Table 2 a structure description of S and the structures of the maximal subgroups that contain S; the value "S" in the third column means that S is itself maximal in G. The fourth and fifth columns list the pages in the ATLAS with the information about the maximal subgroups of G and the section in this note with the proof of the table row, respectively. In the fourth column, page numbers in brackets refer to the ATLAS pages with information about the maximal subgroups of nonsolvable quotients of the maximal subgroups of G listed in the third column.
Note that in the case of nonmaximal subgroups S, we do not claim to describe the module structure of S in the third column of the table; we have kept the ATLAS description of the normal subgroups of the maximal overgroups of S. For example, the subgroup S listed for Co2 is contained in maximal subgroups of the types 21+8+:S6(2) and 24+10(S4 ×S3), so S has normal subgroups of the orders 2, 24, 29, 214, and 216; more ATLAS conformal notations would be 2[14](S4 ×S3) or 2[16](S3 ×S3).
Table 1: Solvable subgroups of maximal order - orders and indices

G | |S| | |G/S| | log|G|(|S|) | p.
M11 | 144 | 55 | 0.5536 | 18
M12 | 432 | 220 | 0.5294 | 33
M12.2 | 432 | 440 | 0.4992 | 33
J1 | 168 | 1 045 | 0.4243 | 36
M22 | 576 | 770 | 0.4888 | 39
M22.2 | 1 152 | 770 | 0.5147 | 39
J2 | 1 152 | 525 | 0.5295 | 42
J2.2 | 2 304 | 525 | 0.5527 | 42
M23 | 1 152 | 8 855 | 0.4368 | 71
HS | 2 000 | 22 176 | 0.4316 | 80
HS.2 | 4 000 | 22 176 | 0.4532 | 80
J3 | 1 944 | 25 840 | 0.4270 | 82
J3.2 | 3 888 | 25 840 | 0.4486 | 82
M24 | 13 824 | 17 710 | 0.4935 | 96
McL | 11 664 | 77 000 | 0.4542 | 100
McL.2 | 23 328 | 77 000 | 0.4719 | 100
He | 13 824 | 291 550 | 0.4310 | 104
He.2 | 18 432 | 437 325 | 0.4305 | 104
Ru | 49 152 | 2 968 875 | 0.4202 | 126
Suz | 139 968 | 3 203 200 | 0.4416 | 131
Suz.2 | 279 936 | 3 203 200 | 0.4557 | 131
O′N | 25 920 | 17 778 376 | 0.3784 | 132
O′N.2 | 51 840 | 17 778 376 | 0.3940 | 132
Co3 | 69 984 | 7 084 000 | 0.4142 | 134
Co2 | 2 359 296 | 17 931 375 | 0.4676 | 154
Fi22 | 5 038 848 | 12 812 800 | 0.4853 | 163
Fi22.2 | 10 077 696 | 12 812 800 | 0.4963 | 163
HN | 2 000 000 | 136 515 456 | 0.4364 | 166
HN.2 | 4 000 000 | 136 515 456 | 0.4479 | 166
Ly | 900 000 | 57 516 865 560 | 0.3562 | 174
Th | 944 784 | 96 049 408 000 | 0.3523 | 177
Fi23 | 3 265 173 504 | 1 252 451 200 | 0.5111 | 177
Co1 | 84 934 656 | 48 952 653 750 | 0.4258 | 183
J4 | 28 311 552 | 3 065 023 459 190 | 0.3737 | 190
Fi24′ | 29 386 561 536 | 42 713 595 724 800 | 0.4343 | 207
Fi24′.2 | 58 773 123 072 | 42 713 595 724 800 | 0.4413 | 207
B | 29 686 813 949 952 | 139 953 768 303 693 093 750 | 0.4007 | 217
M | 2 849 934 139 195 392 | 283 521 437 805 098 363 752 |  |
|  | 344 287 234 566 406 250 | 0.2866 | 234

Table 2: Solvable subgroups of maximal order - structures and overgroups

G | S | M(S) | [CCN+85] | see
M11 | 32:Q8.2 | S | 18 |  | 3
M12 | 32:2S4 | S | 33 |  | 3
| 32:2S4 | S | 33 |  | 3
M12.2 | 32:2S4 | M12 | 33 |  | 3
J1 | 23:7:3 | S | 36 |  | 3
M22 | 24:32:4 | 24:A6 | 39 | (4) | 3
M22.2 | 24:32:D8 | 24:S6 | 39 | (4) | 3
J2 | 22+4:(3 ×S3) | S | 42 |  | 3
J2.2 | 22+4:(S3 ×S3) | S | 42 |  | 3
M23 | 24:(3 ×A4):2 | 24:(3 ×A5):2, | 71 | (2) | 3
|  | 24:A7 |  | (10) |
HS | 51+2+:8:2 | U3(5).2 | 80 | (34) | 3
|  | U3(5).2 |  |  | 3
HS.2 | 51+2+:[25] | S | 80 | (34) | 3
J3 | 32.31+2+:8 | S | 82 |  | 3
J3.2 | 32.31+2+:QD16 | S | 82 |  | 3
M24 | 26:31+2+:D8 | 26:3.S6 | 96 | (4) | 3
McL | 31+4+:2S4 | 31+4+:2S5, | 100 | (2) | 3
|  | U4(3) | 52 |  | 3
McL.2 | 31+4+:4S4 | 31+4+:4S5, | 100 | (2) | 3
|  | U4(3).23 | 52 |  | 3
He | 26:31+2+:D8 | 26:3.S6 | 104 | (4) | 3
| 26:31+2+:D8 | 26:3.S6 | 104 | (4) | 3
He.2 | 24+4.(S3 ×S3).2 | S | 104 |  | 3
Ru | 2.24+6:S4 | 23+8:L3(2), | 126 | (3) | 4.1
|  | 2.24+6:S5 |  | (2) |
| 23+8:S4 | 23+8:L3(2), |  | (3) | 4.1
Suz | 32+4:2(A4 ×22).2 | S | 131 |  | 4.2
Suz.2 | 32+4:2(S4 ×D8) | S | 131 |  | 4.2
O′N | 34:21+4−D10 | S | 132 |  | 4.3
O′N.2 | 34:21+4−.(5:4) | S | 132 |  | 4.3
Co3 | 31+4+:4.32:D8 | 31+4+:4S6 | 134 | (4) | 3
|  | 35:(2 ×M11) |  | (18) |
Co2 | 24+10(S4 ×S3) | 21+8+:S6(2), | 154 | (46) | 4.4
|  | 24+10(S5 ×S3) |  | (2) |
Fi22 | 31+6+:23+4:32:2 | S | 163 |  | 4.5
Fi22.2 | 31+6+:23+4:(S3 ×S3) | S | 163 |  | 4.5
HN | 51+4+:21+4−.5.4 | S | 166 |  | 4.6
HN.2 | 51+4+:(4 Y21+4−.5.4) | S | 166 |  | 4.6
Ly | 51+4+:4.32:D8 | 51+4+:4S6 | 174 | (4) | 4.7
Th | [39].2S4 | S | 177 |  | 4.8
| 32.[37].2S4 | S |  |  |
Fi23 | 31+8+.21+6−.31+2+.2S4 | S | 177 |  | 4.9
Co1 | 24+12.(S3 ×31+2+:D8) | 24+12.(S3 ×3S6) | 183 |  | 4.10
J4 | 211:26:31+2+:D8 | 211:M24, | 190 | (96) | 4.11
|  | 21+12+.3M22:2 |  | (39) |
Fi24′ | 31+10+:21+6−:31+2+:2S4 | 31+10+:U5(2):2 | 207 | (73) | 4.12
Fi24′.2 | 31+10+:(2 ×21+6−:31+2+:2S4) | 31+10+:(2 ×U5(2):2) | 207 | (73) | 4.12
B | 22+10+20(24:32:D8 ×S3) | 22+10+20(M22:2 ×S3), | 217 | (39) | 4.13
|  | 29+16S8(2) |  | (123) |
M | 21+2+6+12+18.(S4 ×31+2+:D8) | 2[39].(L3(2) ×3S6), | 234 | (3, 4) | 4.14
|  | 21+24+.Co1 |  | (183) |
| 22+1+6+12+18.(S4 ×31+2+:D8) | 2[39].(L3(2) ×3S6), |  | (3, 4) | 4.14
|  | 22+11+22.(M24 ×S3) |  | (96) |

As a corollary (see Section 5), we read off the following.
Corollary 1 Exactly the following almost simple groups G with sporadic simple socle contain a solvable subgroup S with the property |S|2 ≥ |G|.

Fi23, J2, J2.2, M11, M12, M22.2.

The existence of the subgroups S of G with the structure and the order stated in Table 1 and 2 follows from the ATLAS: It is obvious in the cases where S is maximal in G, and in the other cases, the ATLAS information about a nonsolvable factor group of a maximal subgroup of G suffices.
In order to show that the table rows for the group G are correct, we have to show the following.

- G does not contain solvable subgroups of order larger than |S|.
- G contain exactly the conjugacy classes of solvable subgroups of order |S| that are listed in the second column of Table 2.
- S is contained exactly in the maximal subgroups listed in the third column of Table 2.

Remark 2

- The groups M12 and He contain two classes of isomorphic solvable subgroups of maximal order.
- The groups Ru, Th, and M contain two classes of nonisomorphic solvable subgroups of maximal order.
- The solvable subgroups of maximal order in McL.2 have the structure 31+4+:4S4, the subgroups are maximal in the maximal subgroups of the structures 31+4+:4S5 and U4(3).23 in McL.2. Note that the ATLAS claims another structure for these maximal subgroups of U4(3).23.

## 2 The Approach

We combine the information in the ATLAS [CCN+85] with explicit computations using the GAP system [GAP04], in particular its Character Table Library [Bre12] and its library of Tables of Marks [NMP11]. First we load these two packages.

        gap> LoadPackage( "CTblLib", "1.2" );
        true
        gap> LoadPackage( "TomLib" );
        true

The orders of solvable subgroups of maximal order are collected in a global record `MaxSolv`.

        gap> MaxSolv:= rec();;

### 2.1 Use the Table of Marks

If the GAP library of Tables of Marks [NMP11] contains the table of marks of a group G then we can easily inspect all conjugacy classes of subgroups of G. The following small GAP function can be used for that. It returns `false` if the table of marks of the group with the name `name` is not available, and the list `[ name, n, super ]` otherwise, where `n` is the maximal order of solvable subgroups of G, and `super` is a list of lists; for each conjugacy class of solvable subgroups S of order `n`, `super` contains the list of orders of representatives M of the classes of maximal subgroups of G such that M contains a conjugate of S.
Note that a subgroup in the i-th class of a table of marks contains a subgroup in the j-th class if and only if the entry in the position (i,j) of the table of marks is nonzero. For tables of marks objects in GAP, this is the case if and only if j is contained in the i-th row of the list that is stored as the value of the attribute `SubsTom` of the table of marks object; for this test, one need not unpack the matrix of marks.

        gap> MaximalSolvableSubgroupInfoFromTom:= function( name )
        >     local tom,          # table of marks for `name'
        >           n,            # maximal order of a solvable subgroup
        >           maxsubs,      # numbers of the classes of subgroups of order `n'
        >           orders,       # list of orders of the classes of subgroups
        >           i,            # loop over the classes of subgroups
        >           maxes,        # list of positions of the classes of max. subgroups
        >           subs,         # `SubsTom' value
        >           cont;         # list of list of positions of max. subgroups
        >
        >     tom:= TableOfMarks( name );
        >     if tom = fail then
        >       return false;
        >     fi;
        >     n:= 1;
        >     maxsubs:= [];
        >     orders:= OrdersTom( tom );
        >     for i in [ 1 .. Length( orders ) ] do
        >       if IsSolvableTom( tom, i ) then
        >         if orders[i] = n then
        >           Add( maxsubs, i );
        >         elif orders[i] > n then
        >           n:= orders[i];
        >           maxsubs:= [ i ];
        >         fi;
        >       fi;
        >     od;
        >     maxes:= MaximalSubgroupsTom( tom )[1];
        >     subs:= SubsTom( tom );
        >     cont:= List( maxsubs, j -> Filtered( maxes, i -> j in subs[i] ) );
        >
        >     return [ name, n, List( cont, l -> orders{ l } ) ];
        > end;;

### 2.2 Use Information from the Character Table Library

The GAP Character Table Library contains the character tables of all maximal subgroups of sporadic simple groups, except for the Monster group. This information can be used as follows.
We start, for a sporadic simple group G, with a known solvable subgroup of order n, say, in G. In order to show that G contains no solvable subgroup of larger order, it suffices to show that no maximal subgroup of G contains a larger solvable subgroup.
The point is that usually the orders of the maximal subgroups of G are not much larger than n, and that a maximal subgroup M contains a solvable subgroup of order n only if the factor group of M by its largest solvable normal subgroup N contains a solvable subgroup of order n/|N|. This reduces the question to relatively small groups.
What we can check automatically from the character table of M/N is whether M/N can contain subgroups (solvable or not) of indices between five and |M|/n, by computing possible permutation characters of these degrees. (Note that a solvable subgroup of a nonsolvable group has index at least five. This lower bound could be improved for example by considering the smallest degree of a nontrivial character, but this is not an issue here.)
Then we are left with a -hopefully short- list of maximal subgroups of G, together with upper bounds on the indices of possible solvable subgroups; excluding these possibilities then yields that the initially chosen solvable subgroup of G is indeed the largest one.
The following GAP function can be used to compute this information for the character table `tblM` of M and a given order `minorder`. It returns `false` if M cannot contain a solvable subgroup of order at least `minorder`, otherwise a list `[ tblM, m, k ]` where `m` is the maximal index of a subgroup that has order at least `minorder`, and `k` is the minimal index of a possible subgroup of M (a proper subgroup if M is nonsolvable), according to the GAP function `PermChars`.

        gap> SolvableSubgroupInfoFromCharacterTable:= function( tblM, minorder )
        >     local maxindex,  # index of subgroups of order `minorder'
        >           N,         # class positions describing a solvable normal subgroup
        >           fact,      # character table of the factor by `N'
        >           classes,   # class sizes in `fact'
        >           nsg,       # list of class positions of normal subgroups
        >           i;         # loop over the possible indices
        >
        >     maxindex:= Int( Size( tblM ) / minorder );
        >     if   maxindex = 0 then
        >       return false;
        >     elif IsSolvableCharacterTable( tblM ) then
        >       return [ tblM, maxindex, 1 ];
        >     elif maxindex < 5 then
        >       return false;
        >     fi;
        >
        >     N:= [ 1 ];
        >     fact:= tblM;
        >     repeat
        >       fact:= fact / N;
        >       classes:= SizesConjugacyClasses( fact );
        >       nsg:= Difference( ClassPositionsOfNormalSubgroups( fact ), [ [ 1 ] ] );
        >       N:= First( nsg, x -> IsPrimePowerInt( Sum( classes{ x } ) ) );
        >     until N = fail;
        >
        >     for i in [ 5 .. maxindex ] do
        >       if Length( PermChars( fact, rec( torso:= [ i ] ) ) ) > 0 then
        >         return [ tblM, maxindex, i ];
        >       fi;
        >     od;
        >
        >     return false;
        > end;;

## 3 Cases where the Table of Marks is available in GAP

For twelve sporadic simple groups, the GAP library of Tables of Marks knows the tables of marks, so we can use `MaximalSolvableSubgroupInfoFromTom`.

        gap> solvinfo:= Filtered( List(
        >         AllCharacterTableNames( IsSporadicSimple, true,
        >                                 IsDuplicateTable, false ),
        >         MaximalSolvableSubgroupInfoFromTom ), x -> x <> false );;
        gap> for entry in solvinfo do
        >      MaxSolv.( entry[1] ):= entry[2];
        >    od;
        gap> for entry in solvinfo do
        >      Print( String( entry[1], 5 ), String( entry[2], 7 ),
        >             String( entry[3], 28 ), "\n" );
        >    od;
          Co3  69984     [ [ 3849120, 699840 ] ]
           HS   2000      [ [ 252000, 252000 ] ]
           He  13824  [ [ 138240 ], [ 138240 ] ]
           J1    168                 [ [ 168 ] ]
           J2   1152                [ [ 1152 ] ]
           J3   1944                [ [ 1944 ] ]
          M11    144                 [ [ 144 ] ]
          M12    432        [ [ 432 ], [ 432 ] ]
          M22    576                [ [ 5760 ] ]
          M23   1152         [ [ 40320, 5760 ] ]
          M24  13824              [ [ 138240 ] ]
          McL  11664      [ [ 3265920, 58320 ] ]

We see that for J1, J2, J3, M11, and M12, the subgroup S is maximal. For M12 and He, there are two classes of subgroups S. For the other groups, the class of subgroups S is unique, and there are one or two classes of maximal subgroups of G that contain S. From the shown orders of these maximal subgroups, their structures can be read off from the ATLAS, on the pages listed in Table 2.
Similarly, the ATLAS tells us about the extensions of the subgroups S in Aut(G). In particular,

- the order 2 000 subgroups of HS are contained in maximal subgroups of the type U3(5).2 (two classes) which do not extend to HS.2, but there are novelties of the type 51+2+:[25] and of the order 4 000, so the solvable subgroups of maximal order in HS do in fact extend to HS.2.
- the order 13 824 subgroups of He are contained in maximal subgroups of the type 26:3S6 (two classes) which do not extend to He.2, but there are novelties of the type 24+4.(S3 ×S3).2 and of the order 18 432. (So the solvable subgroups S of maximal order in He do not extend to He.2 but there are larger solvable subgroups in He.2.)
We inspect the maximal subgroups of He.2 in order to show that these are in fact the solvable subgroups of maximal order (see [CCN+85,p. 104]): Any other solvable subgroup of order at least n in He.2 must be contained in a subgroup of one of the types S4(4).4 (of index at most 212), 22.L3(4).D12 (of index at most 52), or 21+6+.L3(2).2 (of index at most 2). By [CCN+85,pp. 44, 23, 3], this is not the case.
- the maximal subgroups of order 1 152 in J2 extend to subgroups of order 2 304 in J2.2.
- the maximal subgroups of order 1 944 in J3 extend to subgroups of the type 32.31+2+:8.2 and of order 3888 in J3.2. (The structure stated in [CCN+85,p. 82] is not correct, see [BN95].)
- the maximal subgroups of order 432 in M12 (two classes) do not extend in M12.2, and we see from the table of marks of M12.2 that there are no larger solvable subgroups in this group, i. e., the solvable subgroups of maximal order in M12.2 lie in M12.
- the order 576 subgroups of M22 are contained in maximal subgroups of the type 24:A6 which extend to subgroups of the type 24:S6 in M22.2, so the solvable subgroups of maximal order in M22.2 have the type 24:32:D8 and the order 1 152. In fact the structure is S4 \wr S2.
- the order 11 664 subgroups of McL are contained in maximal subgroups of the type 31+4+:2S5 which extend to subgroups of the type 31+4:4S5 in McL.2, so the solvable subgroups of maximal order in McL.2 have the type 31+4:4S4 and the order 23 328.

        gap> MaxSolv.( "HS.2" ):= 2 * MaxSolv.( "HS" );;
        gap> n:= 2^(4+4) * ( 6 * 6 ) * 2;  MaxSolv.( "He.2" ):= n;;
        18432
        gap> List( [ Size( CharacterTable( "S4(4).4" ) ),
        >            Factorial( 5 )^2 * 2,
        >            Size( CharacterTable( "2^2.L3(4).D12" ) ),
        >            2^7 * Size( CharacterTable( "L3(2)" ) ) * 2,
        >            7^2 * 2 * Size( CharacterTable( "L2(7)" ) ) * 2,
        >            3 * Factorial( 7 ) * 2 ], i -> Int( i / n ) );
        [ 212, 1, 52, 2, 1, 1 ]
        gap> MaxSolv.( "J2.2" ):= 2 * MaxSolv.( "J2" );;
        gap> MaxSolv.( "J3.2" ):= 2 * MaxSolv.( "J3" );;
        gap> info:= MaximalSolvableSubgroupInfoFromTom( "M12.2" );
        [ "M12.2", 432, [ [ 95040 ] ] ]
        gap> MaxSolv.( "M12.2" ):= info[2];;
        gap> MaxSolv.( "M22.2" ):= 2 * MaxSolv.( "M22" );;
        gap> MaxSolv.( "McL.2" ):= 2 * MaxSolv.( "McL" );;

## 4 Cases where the Table of Marks is not available in GAP

We use the GAP function `SolvableSubgroupInfoFromCharacterTable`, and individual arguments. In several cases, information about smaller sporadic simple groups is needed, so we deal with the groups in increasing order.

### 4.1 G = Ru

The group Ru contains exactly two conjugacy classes of nonisomorphic solvable subgroups of order n = 49 152, and no larger solvable subgroups.

        gap> t:= CharacterTable( "Ru" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 49152;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "2^3+8:L3(2)" ), 7, 7 ],
          [ CharacterTable( "2.2^4+6:S5" ), 5, 5 ] ]

The maximal subgroups of the structure 2.24+6:S5 in Ru contain one class of solvable subgroups of order n and with the structure 2.24+6:S4, see [CCN+85,p. 126, p. 2].
The maximal subgroups of the structure 23+8:L3(2) in Ru contain two classes of solvable subgroups of order n and with the structure 23+8:S4, see [CCN+85,p. 126, p. 3]. These groups are the stabilizers of vectors and two-dimensional subspaces, respectively, in the three-dimensional submodule; note that each 23+8:L3(2) type subgroup H of Ru is the normalizer of an elementary abelian group of order eight all of whose involutions are in the Ru-class `2A` and are conjugate in H. Since the 2.24+6:S5 type subgroups of Ru are the normalizers of `2A`-elements in Ru, the groups in one of the two classes in question coincide with the largest solvable subgroups in the 2.24+6:S5 type subgroups. The groups in the other class do not centralize a `2A`-element in Ru and are therefore not isomorphic with the 2.24+6:S4 type groups.

        gap> MaxSolv.( "Ru" ):= n;;
        gap> s:= info[1][1];;
        gap> cls:= SizesConjugacyClasses( s );;
        gap> nsg:= Filtered( ClassPositionsOfNormalSubgroups( s ),
        >                    x -> Sum( cls{ x } ) = 2^3 );
        [ [ 1, 2 ] ]
        gap> cls{ nsg[1] };
        [ 1, 7 ]
        gap> GetFusionMap( s, t ){ nsg[1] };
        [ 1, 2 ]

### 4.2 G = Suz

The group Suz contains a unique conjugacy class of solvable subgroups of order n = 139 968, and no larger solvable subgroups.

        gap> t:= CharacterTable( "Suz" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 139968;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "G2(4)" ), 1797, 416 ],
          [ CharacterTable( "3_2.U4(3).2_3'" ), 140, 72 ],
          [ CharacterTable( "3^5:M11" ), 13, 11 ],
          [ CharacterTable( "2^4+6:3a6" ), 7, 6 ],
          [ CharacterTable( "3^2+4:2(2^2xa4)2" ), 1, 1 ] ]

The maximal subgroups S of the structure 32+4:2(A4 ×22).2 in Suz are solvable and have order n, see [CCN+85,p. 131].
In order to show that Suz contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in G2(4) of index at most 1 797 (see p. 97), in U4(3).23′ of index at most 140 (see p. 52), in M11 of index at most 13 (see p. 18), and in A6 of index at most 7 (see p. 4).
The group S extends to a group of the structure 32+4:2(S4 ×D8) in the automorphism group Suz.2.

        gap> MaxSolv.( "Suz" ):= n;;
        gap> MaxSolv.( "Suz.2" ):= 2 * n;;

### 4.3 G = ON

The group ON contains a unique conjugacy class of solvable subgroups of order 25 920, and no larger solvable subgroups.

        gap> t:= CharacterTable( "ON" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 25920;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "L3(7).2" ), 144, 114 ],
          [ CharacterTable( "ONM2" ), 144, 114 ],
          [ CharacterTable( "3^4:2^(1+4)D10" ), 1, 1 ] ]

The maximal subgroups S of the structure 34:21+4−D10 in ON are solvable and have order n, see [CCN+85,p. 132].
In order to show that ON contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in L3(7).2 of index at most 144 (see p. 50); note that the groups in the second class of maximal subgroups of ON are isomorphic with L3(7).2.
The group S extends to a group of order |S.2| in the automorphism group ON.2.

        gap> MaxSolv.( "ON" ):= n;;
        gap> MaxSolv.( "ON.2" ):= 2 * n;;

### 4.4 G = Co2

The group Co2 contains a unique conjugacy class of solvable subgroups of order 2 359 296, and no larger solvable subgroups.

        gap> t:= CharacterTable( "Co2" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 2359296;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "U6(2).2" ), 7796, 672 ],
          [ CharacterTable( "2^10:m22:2" ), 385, 22 ],
          [ CharacterTable( "McL" ), 380, 275 ],
          [ CharacterTable( "2^1+8:s6f2" ), 315, 28 ],
          [ CharacterTable( "2^1+4+6.a8" ), 17, 8 ],
          [ CharacterTable( "U4(3).D8" ), 11, 8 ],
          [ CharacterTable( "2^(4+10)(S5xS3)" ), 5, 5 ] ]

The maximal subgroups of the structure 24+10(S5 ×S3) in Co2 contain solvable subgroups S of order n and with the structure 24+10(S4 ×S3), see [CCN+85,p. 154].
The subgroups S are contained also in the maximal subgroups of the type 21+8+:S6(2); note that the 21+8+:S6(2) type subgroups are described as normalizers of elements in the Co2-class `2A`, and S normalizes an elementary abelian group of order 16 containing an S-class of length five that is contained in the Co2-class `2A`.

        gap> s:= info[7][1];
        CharacterTable( "2^(4+10)(S5xS3)" )
        gap> cls:= SizesConjugacyClasses( s );;
        gap> nsg:= Filtered( ClassPositionsOfNormalSubgroups( s ),
        >                    x -> Sum( cls{ x } ) = 2^4 );
        [ [ 1, 2, 3 ] ]
        gap> cls{ nsg[1] };
        [ 1, 5, 10 ]
        gap> GetFusionMap( s, t ){ nsg[1] };
        [ 1, 2, 3 ]

The stabilizers of these involutions in 24+10(S5 ×S3) have index five, they are solvable, and they are contained in 21+8+:S6(2) type subgroups, so they are Co2-conjugates of S. (The corresponding subgroups of S6(2) are maximal and have the type 2.[26]:(S3 ×S3).)
In order to show that G contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in U6(2) of index at most 7 796 (see p. 115), in M22.2 of index at most 385 (see p. 39 or Section 3), in McL of index at most 380 (see p. 100 or Section 3), in A8 of index at most 17 (see p. 20), and in U4(3).D8 of index at most 11 (see p. 52).

        gap> MaxSolv.( "Co2" ):= n;;

### 4.5 G = Fi22

The group Fi22 contains a unique conjugacy class of solvable subgroups of order 5 038 848, and no larger solvable subgroups.

        gap> t:= CharacterTable( "Fi22" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 5038848;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "2.U6(2)" ), 3650, 672 ],
          [ CharacterTable( "O7(3)" ), 910, 351 ],
          [ CharacterTable( "Fi22M3" ), 910, 351 ],
          [ CharacterTable( "O8+(2).3.2" ), 207, 6 ],
          [ CharacterTable( "2^10:m22" ), 90, 22 ],
          [ CharacterTable( "3^(1+6):2^(3+4):3^2:2" ), 1, 1 ] ]

The maximal subgroups S of the structure 31+6:23+4:32:2 in Fi22 are solvable and have order n, see [CCN+85,p. 163].
In order to show that Fi22 contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in U6(2) of index at most 3 650 (see p. 115), in O7(3) of index at most 910 (see p. 109), in O8+(2).S3 of index at most 207 (see p. 85), and in M22.2 of index at most 90 (see p. 39 or Section 3); note that the groups in the third class of maximal subgroups of Fi22 are isomorphic with O7(3).
The group S extends to a group of order |S.2| in the automorphism group Fi22.2.

        gap> MaxSolv.( "Fi22" ):= n;;
        gap> MaxSolv.( "Fi22.2" ):= 2 * n;;

### 4.6 G = HN

The group HN contains a unique conjugacy class of solvable subgroups of order 2 000 000, and no larger solvable subgroups.

        gap> t:= CharacterTable( "HN" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 2000000;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "A12" ), 119, 12 ],
          [ CharacterTable( "5^(1+4):2^(1+4).5.4" ), 1, 1 ] ]

The maximal subgroups S of the structure 51+4:21+4.5.4 in HN are solvable and have order n, see [CCN+85,p. 166].
In order to show that HN contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in A12 of index at most 119 (see p. 91).
The group S extends to a group of order |S.2| in the automorphism group HN.2.

        gap> MaxSolv.( "HN" ):= n;;
        gap> MaxSolv.( "HN.2" ):= 2 * n;;

### 4.7 G = Ly

The group Ly contains a unique conjugacy class of solvable subgroups of order 900 000, and no larger solvable subgroups.

        gap> t:= CharacterTable( "Ly" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 900000;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "G2(5)" ), 6510, 3906 ],
          [ CharacterTable( "3.McL.2" ), 5987, 275 ],
          [ CharacterTable( "5^3.psl(3,5)" ), 51, 31 ],
          [ CharacterTable( "2.A11" ), 44, 11 ],
          [ CharacterTable( "5^(1+4):4S6" ), 10, 6 ] ]

The maximal subgroups of the structure 5(1+4):4S6 in Ly contain solvable subgroups S of order n and with the structure 51+4:4.32.D8, see [CCN+85,p. 174].
In order to show that Ly contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in G2(5) of index at most 6 510 (see p. 114), in McL.2 of index at most 5 987 (see p. 100 or Section 3), in L3(5) of index at most 51 (see p. 38), and in A11 of index at most 44 (see p. 75).

        gap> MaxSolv.( "Ly" ):= n;;

### 4.8 G = Th

The group Th contains exactly two conjugacy classes of nonisomorphic solvable subgroups of order n = 944 784, and no larger solvable subgroups.

        gap> t:= CharacterTable( "Th" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 944784;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "2^5.psl(5,2)" ), 338, 31 ],
          [ CharacterTable( "2^1+8.a9" ), 98, 9 ],
          [ CharacterTable( "U3(8).6" ), 35, 6 ], [ CharacterTable( "ThN3B" ), 1, 1 ],
          [ CharacterTable( "ThM7" ), 1, 1 ] ]

The maximal subgroups S of the structures [39].2S4 and 32.[37].2S4 in Th are solvable and have order n, see [CCN+85,p. 177].
In order to show that Th contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in L5(2) of index at most 338 (see p. 70), in A9 of index at most 98 (see p. 37), and in U3(8).6 of index at most 35 (see p. 66).

        gap> MaxSolv.( "Th" ):= n;;

### 4.9 G = Fi23

The group Fi23 contains a unique conjugacy class of solvable subgroups of order n = 3 265 173 504, and no larger solvable subgroups.

        gap> t:= CharacterTable( "Fi23" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 3265173504;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "2.Fi22" ), 39545, 3510 ],
          [ CharacterTable( "O8+(3).3.2" ), 9100, 6 ],
          [ CharacterTable( "3^(1+8).2^(1+6).3^(1+2).2S4" ), 1, 1 ] ]

The maximal subgroups S of the structure 31+8+.21+6−.31+2+.2S4 in Fi23 are solvable and have order n, see [CCN+85,p. 177].
In order to show that Fi23 contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in Fi22 of index at most 39 545 (see Section 4.5) and in O8+(3).S3 of index at most 9 100 (see p. 140).

        gap> MaxSolv.( "Fi23" ):= n;;

### 4.10 G = Co1

The group Co1 contains a unique conjugacy class of solvable subgroups of order n = 84 934 656, and no larger solvable subgroups.

        gap> t:= CharacterTable( "Co1" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 84934656;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "Co2" ), 498093, 2300 ],
          [ CharacterTable( "3.Suz.2" ), 31672, 1782 ],
          [ CharacterTable( "2^11:M24" ), 5903, 24 ],
          [ CharacterTable( "Co3" ), 5837, 276 ],
          [ CharacterTable( "2^(1+8)+.O8+(2)" ), 1050, 120 ],
          [ CharacterTable( "U6(2).3.2" ), 649, 6 ],
          [ CharacterTable( "2^(2+12):(A8xS3)" ), 23, 8 ],
          [ CharacterTable( "2^(4+12).(S3x3S6)" ), 10, 6 ] ]

The maximal subgroups of the structure 24+12.(S3 ×3S6) in Co1 contain solvable subgroups S of order n and with the structure 24+12.(S3 ×31+2+:D8), see [CCN+85,p. 183].
In order to show that Co1 contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in Co2 of index at most 498 093 (see Section 4.4), in Suz.2 of index at most 31 672 (see Section 4.2), in M24 of index at most 5 903 (see Section 3), in Co3 of index at most 5 837 (see p. 134 or Section 3), in O8+(2) of index at most 1 050 (see p. 185), in U6(2).S3 of index at most 649 (see p. 115), and in A8 of index at most 23 (see p. 22).

        gap> MaxSolv.( "Co1" ):= n;;

### 4.11 G = J4

The group J4 contains a unique conjugacy class of solvable subgroups of order 28 311 552, and no larger solvable subgroups.

        gap> t:= CharacterTable( "J4" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 28311552;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "mx1j4" ), 17710, 24 ],
          [ CharacterTable( "c2aj4" ), 770, 22 ],
          [ CharacterTable( "2^10:L5(2)" ), 361, 31 ],
          [ CharacterTable( "J4M4" ), 23, 5 ] ]

The maximal subgroups of the structure 211:M24 in J4 contain solvable subgroups S of order n and with the structure 211:26:31+2+:D8, see Section 3 and [CCN+85,p. 190].
(The subgroups in the first four classes of maximal subgroups of J4 have the structures 211:M24, 21+12+.3M22:2, 210:L5(2), and 23+12.(S5 ×L3(2)), in this order.)
The subgroups S are contained also in the maximal subgroups of the type 21+12+.3M22:2; note that these subgroups are described as normalizers of elements in the J4-class `2A`, and S normalizes an elementary abelian group of order 211 containing an S-class of length 1 771 that is contained in the J4-class `2A`.

        gap> s:= info[1][1];
        CharacterTable( "mx1j4" )
        gap> cls:= SizesConjugacyClasses( s );;
        gap> nsg:= Filtered( ClassPositionsOfNormalSubgroups( s ),
        >                    x -> Sum( cls{ x } ) = 2^11 );
        [ [ 1, 2, 3 ] ]
        gap> cls{ nsg[1] };
        [ 1, 276, 1771 ]
        gap> GetFusionMap( s, t ){ nsg[1] };
        [ 1, 3, 2 ]

The stabilizers of these involutions in 211:M24 have index 1 771, they have the structure 211:26:3.S6, and they are contained in 21+12+.3M22:2 type subgroups; so also S, which has index 10 in 211:26:3.S6, is contained in 21+12+.3M22:2. (The corresponding subgroups of M22:2 are of course the solvable groups of maximal order described in Section 3.)
In order to show that G contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in L5(2) of index at most 361 (see p. 70) and in S5 ×L3(2) of index at most 23 (see pp. 2, 3).

        gap> MaxSolv.( "J4" ):= n;;

### 4.12 G = Fi24′

The group Fi24′ contains a unique conjugacy class of solvable subgroups of order 29 386 561 536, and no larger solvable subgroups.

        gap> t:= CharacterTable( "Fi24'" );;
        gap> mx:= List( Maxes( t ), CharacterTable );;
        gap> n:= 29386561536;;
        gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
        gap> info:= Filtered( info, IsList );
        [ [ CharacterTable( "Fi23" ), 139161244, 31671 ],
          [ CharacterTable( "2.Fi22.2" ), 8787, 3510 ],
          [ CharacterTable( "(3xO8+(3):3):2" ), 3033, 6 ],
          [ CharacterTable( "O10-(2)" ), 851, 495 ],
          [ CharacterTable( "3^(1+10):U5(2):2" ), 165, 165 ],
          [ CharacterTable( "2^2.U6(2).3.2" ), 7, 6 ] ]

The maximal subgroups of the structure 31+10+:U5(2):2 in Fi24′ contain solvable subgroups S of order n and with the structure 31+10+:21+6−:31+2+:2S4, see [CCN+85,p. 73, p. 207].
In order to show that G contains no other solvable subgroups of order larger than or equal to |S|, we check that there are no solvable subgroups in Fi23 of order at least n (see Section 4.9), in Fi22.2 of order at least n (see Section 4.5), in O8+(3).S3 of index at most 3 033 (see p. 140), in O10−(2) of index at most 851 (see p. 147), and in U6(2).S3 of index at most 7 (see p. 115).
The group S extends to a group of order |S.2| in the automorphism group Fi24.

        gap> MaxSolv.( "Fi24'" ):= n;;
        gap> MaxSolv.( "Fi24'.2" ):= 2 * n;;

### 4.13 G = B

The group B contains a unique conjugacy class of solvable subgroups of order n = 29 686 813 949 952, and no larger solvable subgroups.
The maximal subgroups of the structure 22+10+20(M22:2 ×S3) in B contain solvable subgroups S of order n and with the structure 22+10+20(24:32:D8 ×S3), see [CCN+85,p. 217] and Section 3.

        gap> n:= 29686813949952;;
        gap> n = 2^(2+10+20) * 2^4 * 3^2 * 8 * 6;
        true
        gap> n = 2^(2+10+20) * MaxSolv.( "M22.2" ) * 6;
        true

By [Wil99,Table 1], the only maximal subgroups of B of order bigger than |S| have the following structures.

2.2E6(2).2
21+22.Co2
Fi23
29+16S8(2)
Th
(22 ×F4(2)):2
22+10+20(M22:2 ×S3)
25+5+10+10L5(2)
S3 ×Fi22:2
2[35](S5 ×L3(2))
HN:2
O8+(3):S4

(The character tables of the maximal subgroups of B are meanwhile available in GAP. Note that we cannot apply the function `SolvableSubgroupInfoFromCharacterTable` to these character tables because some indices are too large for forming ranges.)

        gap> b:= CharacterTable( "B" );;
        gap> mx:= List( Maxes( b ), CharacterTable );;
        gap> Filtered( mx, x -> Size( x ) >= n );
        [ CharacterTable( "2.2E6(2).2" ), CharacterTable( "2^(1+22).Co2" ),
          CharacterTable( "Fi23" ), CharacterTable( "2^(9+16).S8(2)" ),
          CharacterTable( "Th" ), CharacterTable( "(2^2xF4(2)):2" ),
          CharacterTable( "2^(2+10+20).(M22.2xS3)" ), CharacterTable( "[2^30].L5(2)" )
            , CharacterTable( "S3xFi22.2" ), CharacterTable( "[2^35].(S5xL3(2))" ),
          CharacterTable( "HN.2" ), CharacterTable( "O8+(3).S4" ) ]

For the subgroups 21+22.Co2, Fi23, Th, S3 ×Fi22:2, and HN:2, the solvable subgroups of maximal order are known from the previous sections or can be derived from known values, and are smaller than n.

        gap> List( [ 2^(1+22) * MaxSolv.( "Co2" ),
        >            MaxSolv.( "Fi23" ),
        >            MaxSolv.( "Th" ),
        >            6 * MaxSolv.( "Fi22.2" ),
        >            MaxSolv.( "HN.2" ) ], i -> Int( i / n ) );
        [ 0, 0, 0, 0, 0 ]

If one of the remaining maximal groups U from the above list has a solvable subgroup of order at least n then the index of this subgroup in U is bounded as follows.

        gap> List( [ Size( CharacterTable( "2.2E6(2).2" ) ),
        >            2^(9+16) * Size( CharacterTable( "S8(2)" ) ),
        >            2^3 * Size( CharacterTable( "F4(2)" ) ),
        >            2^(2+10+20) * Size( CharacterTable( "M22.2" ) ) * 6,
        >            2^30 * Size( CharacterTable( "L5(2)" ) ),
        >            2^35 * Factorial(5) * Size( CharacterTable( "L3(2)" ) ),
        >            Size( CharacterTable( "O8+(3)" ) ) * 24 ],
        >          i -> Int( i / n ) );
        [ 10311982931, 53550, 892, 770, 361, 23, 4 ]

The group O8+(3):S4 is nonsolvable, and its order is less than 5 n, thus its solvable subgroups have orders less than n.
The largest solvable subgroup of S5 ×L3(2) has index 35, thus the solvable subgroups of 2[35](S5 ×L3(2)) have orders less than n.
The groups of type 25+5+10+10L5(2) cannot contain solvable subgroups of order at least n because L5(2) has no solvable subgroup of index up to 361 -such a subgroup would be contained in 24:L4(2), of index at most 361/31  = 11 (see [CCN+85,p. 70]), and L4(2) ≅ A8 does not have such subgroups (see [CCN+85,p. 22]).
The largest proper subgroup of F4(2) has index 69 615 (see [CCN+85,p. 170]), which excludes solvable subgroups of order at least n in (22 ×F4(2)):2.
Ruling out the group 2.2E6(2).2 is more involved. We consider the list of maximal subgroups of 2E6(2) in [CCN+85,p. 191] (which is complete, see [BN95]), and compute the maximal index of a group of order n/4; the possible subgroups of 2E6(2) to consider are the following

21+20:U6(2)
28+16:O8−(2)
F4(2)
22.29.218:(L3(4) ×S3)
Fi22
O10−(2)
23.212.215:(S5 ×L3(2))

(The order of S3 ×U6(2) is already smaller than n/4.)

        gap> List( [ 2^(1+20) * Size( CharacterTable( "U6(2)" ) ),
        >            2^(8+16) * Size( CharacterTable( "O8-(2)" ) ),
        >            Size( CharacterTable( "F4(2)" ) ),
        >            2^(2+9+18) * Size( CharacterTable( "L3(4)" ) ) * 6,
        >            Size( CharacterTable( "Fi22" ) ),
        >            Size( CharacterTable( "O10-(2)" ) ),
        >            2^(3+12+15) * 120 * Size( CharacterTable( "L3(2)" ) ),
        >            6 * Size( CharacterTable( "U6(2)" ) ) ],
        >          i -> Int( i / ( n / 4 ) ) );
        [ 2598, 446, 446, 8, 8, 3, 2, 0 ]

The indices of the solvable groups of maximal orders in the groups U6(2), O8−(2), F4(2), L3(4), and Fi22 are larger than the bounds we get for n, see [CCN+85,pp. 115, 89, 170, 23, 163].
It remains to consider the subgroups of the type 29+16S8(2). The group S8(2) contains maximal subgroups of the type 23+8:(S3 ×S6) and of index 5 355 (see [CCN+85,p. 123]), which contain solvable subgroups S′ of index 10. This yields solvable subgroups of order 29+16+3+8 ·6 ·72 = n.

        gap> 2^(9+16+3+8) * 6 * 72 = n;
        true

There are no other solvable subgroups of larger or equal order in S8(2): We would need solvable subgroups of index at most 446 in O8−(2):2, 393 in O8+(2):2, 210 in S6(2), or 23 in A8, which is not the case by [CCN+85,pp. 89, 85, 46, 22].

        gap> index:= Int( 2^(9+16) * Size( CharacterTable( "S8(2)" ) ) / n );
        53550
        gap> List( [ 120, 136, 255, 2295 ], i -> Int( index / i ) );
        [ 446, 393, 210, 23 ]
        gap> MaxSolv.( "B" ):= n;;

So the 29+16S8(2) type subgroups of B yield solvable subgroups S′ of the type 29+16.23+8:(S3 ×32:D8), and of order n.
We want to show that S′ is a B-conjugate of S. For that, we first show the following:
Lemma 1 The group B contains exactly two conjugacy classes of Klein four groups whose involutions lie in the class `2B`. (We will call these Klein four groups `2B`-pure.) Their normalizers in B have the orders 22 858 846 741 463 040 and 292 229 574 819 840, respectively.
PROOF. Let V be a `2B`-pure Klein four group in B, and set N = NB(V). Let x ∈ V be an involution and set H = CB(x), then H is maximal in B and has the structure 21+22.Co2. The index of C = CB(V) = CH(V) in N divides 6, and C stabilizes the central involution in H and another `2B` involution. The group H contains exactly four conjugacy classes of `2B` elements.

        gap> h:= mx[2];
        CharacterTable( "2^(1+22).Co2" )
        gap> pos:= Positions( GetFusionMap( h, b ), 3 );
        [ 2, 4, 11, 20 ]

The B-classes of `2B`-pure Klein four groups arise from those of these classes yH ⊂ H such that x ≠ y holds and x y is a `2B` element. We compute this subset.

        gap> pos:= Filtered( Difference( pos, [ 2 ] ), i -> ForAny( pos,
        >             j -> NrPolyhedralSubgroups( h, 2, i, j ).number <> 0 ) );
        [ 4, 11 ]

The two classes have lengths 93 150 and 7 286 400, thus the index of C in H is one of these numbers.

        gap> SizesConjugacyClasses( h ){ pos };
        [ 93150, 7286400 ]

Next we compute the number n0 of `2B`-pure Klein four groups in B.

        gap> nr:= NrPolyhedralSubgroups( b, 3, 3, 3 );
        rec( number := 14399283809600746875, type := "V4" )
        gap> n0:= nr.number;;

The B-conjugacy class of V has length [B:N] = [B:H] ·[H:C] / [N:C], where [N:C] divides 6. We see that [N:C] = 6 in both cases.

        gap> cand:= List( pos, i -> Size( b ) / SizesCentralizers( h )[i] / 6 );
        [ 181758140654146875, 14217525668946600000 ]
        gap> Sum( cand ) = n0;
        true

The orders of the normalizers of the two classes of `2B`-pure Klein four groups are as claimed.

        gap> List( cand, x -> Size( b ) / x );
        [ 22858846741463040, 292229574819840 ]

[¯]
The subgroup S of order n is contained in a maximal subgroup M of the type 22+10+20(M22:2 ×S3) in B. The group M is the normalizer of a `2B`-pure Klein four group in B, and the other class of normalizers of `2B`-pure Klein four groups does not contain subgroups of order n. Thus the conjugates of S are uniquely determined by |S| and the property that they normalize `2B`-pure Klein four groups.

        gap> m:= mx[7];
        CharacterTable( "2^(2+10+20).(M22.2xS3)" )
        gap> Size( m );
        22858846741463040
        gap> nsg:= ClassPositionsOfMinimalNormalSubgroups( m );
        [ [ 1, 2 ] ]
        gap> SizesConjugacyClasses( m ){ nsg[1] };
        [ 1, 3 ]
        gap> GetFusionMap( m, b ){ nsg[1] };
        [ 1, 3 ]
        gap> List( cand, x -> Size( b ) / ( n * x ) );
        [ 770, 315/32 ]

Now consider the subgroup S′ of order n, which is contained in a maximal subgroup of the type 29+16S8(2) in B. In order to prove that S′ is B-conjugate to S, it is enough to show that S′ normalizes a `2B`-pure Klein four group.
The unique minimal normal subgroup V of 29+16S8(2) has order 28. Its involutions lie in the class `2B` of B.

        gap> m:= mx[4];
        CharacterTable( "2^(9+16).S8(2)" )
        gap> nsg:= ClassPositionsOfMinimalNormalSubgroups( m );
        [ [ 1, 2 ] ]
        gap> SizesConjugacyClasses( m ){ nsg[1] };
        [ 1, 255 ]
        gap> GetFusionMap( m, b ){ nsg[1] };
        [ 1, 3 ]

The group V is central in the normal subgroup W = 29+16, since all nonidentity elements of V lie in one conjugacy class of odd length. As a module for S8(2), V is the unique irreducible eight-dimensional module in characteristic two.

        gap> CharacterDegrees( CharacterTable( "S8(2)" ) mod 2 );
        [ [ 1, 1 ], [ 8, 1 ], [ 16, 1 ], [ 26, 1 ], [ 48, 1 ], [ 128, 1 ],
          [ 160, 1 ], [ 246, 1 ], [ 416, 1 ], [ 768, 1 ], [ 784, 1 ], [ 2560, 1 ],
          [ 3936, 1 ], [ 4096, 1 ], [ 12544, 1 ], [ 65536, 1 ] ]

Hence we are done if the restriction of the S8(2)-action on V to S′/W leaves a two-dimensional subspace of V invariant. In fact we show that already the restriction of the S8(2)-action on V to the maximal subgroups of the structure 23+8:(S3 ×S6) has a two-dimensional submodule.
These maximal subgroups have index 5 355 in S8(2). The primitive permutation representation of degree 5 355 of S8(2) and the irreducible eight-dimensional matrix representation of S8(2) over the field with two elements are available via the GAP package AtlasRep, see [WPN+11]. We compute generators for an index 5 355 subgroup in the matrix group via an isomorphism to the permutation group.

        gap> permg:= AtlasGroup( "S8(2)", NrMovedPoints, 5355 );
        <permutation group of size 47377612800 with 2 generators>
        gap> matg:= AtlasGroup( "S8(2)", Dimension, 8 );
        <matrix group of size 47377612800 with 2 generators>
        gap> hom:= GroupHomomorphismByImagesNC( matg, permg,
        >              GeneratorsOfGroup( matg ), GeneratorsOfGroup( permg ) );;
        gap> max:= PreImages( hom, Stabilizer( permg, 1 ) );;

These generators define the action of the index 5 355 subgroup of S8(2) on the eight-dimensional module. We compute the dimensions of the factors of an ascending composition series of this module.

        gap> m:= GModuleByMats( GeneratorsOfGroup( max ), GF(2) );;
        gap> comp:= MTX.CompositionFactors( m );;
        gap> List( comp, r -> r.dimension );
        [ 2, 4, 2 ]

### 4.14 G = M

The group M contains exactly two conjugacy classes of solvable subgroups of order n = 2 849 934 139 195 392, and no larger solvable subgroups.
The maximal subgroups of the structure 21+24+.Co1 in the group M contain solvable subgroups S of order n and with the structure 21+24+.24+12.(S3 ×31+2+:D8), see [CCN+85,p. 234] and Section 4.10.

        gap> n:= 2^25 * MaxSolv.( "Co1" );
        2849934139195392

The solvable subgroups of maximal order in groups of the types 22+11+22.(M24 ×S3) and 2[39].(L3(2) ×3S6) have order n.

        gap> 2^(2+11+22) * MaxSolv.( "M24" ) * 6 = n;
        true
        gap> 2^39 * 24 * 3 * 72 = n;
        true

For inspecting the other maximal subgroups of M, we use the description from [NW]. Currently 44_known_monster_maxes classes of maximal subgroups are listed there, and any possible other maximal subgroup of G has socle isomorphic to one of L2(13), Sz(8), U3(4), U3(8); so these maximal subgroups are isomorphic to subgroups of the automorphism groups of these groups - the maximum of these group orders is smaller than n, hence we may ignore these possible subgroups.

        gap> cand:= [ "L2(13)", "Sz(8)", "U3(4)", "U3(8)" ];;
        gap> List( cand, nam -> ExtensionInfoCharacterTable(
        > CharacterTable( nam ) ) );
        [ [ "2", "2" ], [ "2^2", "3" ], [ "", "4" ], [ "3", "(3xS3)" ] ]
        gap> ll:= List( cand, x -> Size( CharacterTable( x ) ) );
        [ 1092, 29120, 62400, 5515776 ]
        gap> 18* ll[4];
        99283968
        gap> 2^39 * 24 * 3 * 72;
        2849934139195392

Thus only the following maximal subgroups of M have order bigger than |S|.

2.B
21+24+.Co1
3.Fi24
22.2E6(2):S3
210+16.O10+(2)
22+11+22.(M24 ×S3)
31+12+.2Suz.2
25+10+20.(S3 ×L5(2))
S3 ×Th
2[39].(L3(2) ×3S6)
38.O8−(3).23
(D10 ×HN).2

For the subgroups 2.B, 3.Fi24, 31+12+.2Suz.2, S3 ×Th, and (D10 ×HN).2, the solvable subgroups of maximal order are smaller than n.

        gap> List( [ 2 * MaxSolv.( "B" ),
        >            6 * MaxSolv.( "Fi24'" ),
        >            3^13 * 2 * MaxSolv.( "Suz" ) * 2,
        >            6 * MaxSolv.( "Th" ),
        >            10 * MaxSolv.( "HN" ) * 2 ], i -> Int( i / n ) );
        [ 0, 0, 0, 0, 0 ]

The subgroup 22.2E6(2):S3 can be excluded by the fact that this group is only six times larger than the subgroup 2.2E6(2):2 of B, but n is 96 times larger than the maximal solvable subgroup in B.

        gap> n / MaxSolv.( "B" );
        96

The group 38.O8−(3).23 can be excluded by the fact that a solvable subgroup of order at least n would imply the existence of a solvable subgroup of index at most 46 in O8−(3).23, which is not the case (see [CCN+85,p. 141]).

        gap> Int( 3^8 * Size( CharacterTable( "O8-(3)" ) ) * 2 / n );
        46

Similarly, the existence of a solvable subgroup of order at least n in 25+10+20.(S3 ×L5(2)) would imply the existence of a solvable subgroup of index at most 723 in L5(2) and in turn of a solvable subgroup of index at most 23 in L4(2), which is not the case (see [CCN+85,p. 70]).

        gap> Int( 2^(10+16) * Size( CharacterTable( "O10+(2)" ) ) / n );
        553350
        gap> Int( 2^(5+10+20) * 6 * Size( CharacterTable( "L5(2)" ) ) / n );
        723
        gap> Int( 723 / 31 );
        23

It remains to exclude the subgroup 210+16.O10+(2), which means to show that O10+(2) does not contain a solvable subgroup of index at most 553 350. If such a subgroup would exist then it would be contained in one of the following maximal subgroups of O10+(2) (see [CCN+85,p. 146]): in S8(2) (of index at most 1 115), in 28:O8+(2) (of index at most 1 050), in 210:L5(2) (of index at most 241), in (3 ×O8−(2)):2 (of index at most 27), in (21+12+:(S3 ×A8) (of index at most 23), or in 23+12:(S3 ×S3 ×L3(2)) (of index at most 4). By [CCN+85,pp. 123, 85, 70, 89, 22], this is not the case.

        gap> index:= Int( 2^(10+16) * Size( CharacterTable( "O10+(2)" ) ) / n );
        553350
        gap> List( [ 496, 527, 2295, 19840, 23715, 118575 ], i -> Int( index / i ) );
        [ 1115, 1050, 241, 27, 23, 4 ]

As a consequence, we have shown that the largest solvable subgroups of M have order n.

        gap> MaxSolv.( "M" ):= n;;

In order to prove the statement about the conjugacy of subgroups of order n in M, we first show the following.
Lemma 2 The group M contains exactly three conjugacy classes of `2B`-pure Klein four groups. Their normalizers in M have the orders 50 472 333 605 150 392 320, 259 759 622 062 080, and 9 567 039 651 840, respectively.
PROOF. The idea is the same as for the Baby Monster group, see Section 4.13. Let V be a `2B`-pure Klein four group in M, and set N = NM(V). Let x ∈ V be an involution and set H = CM(x), then H is maximal in M and has the structure 21+24+.Co1. The index of C = CM(V) = CH(V) in N divides 6, and C stabilizes the central involution in H and another `2B` involution.
The group H contains exactly five conjugacy classes of `2B` elements, three of them consist of elements that generate a `2B`-pure Klein four group together with x.

        gap> m:= CharacterTable( "M" );;
        gap> h:= CharacterTable( "2^1+24.Co1" );
        CharacterTable( "2^1+24.Co1" )
        gap> pos:= Positions( GetFusionMap( h, m ), 3 );
        [ 2, 4, 7, 9, 16 ]
        gap> pos:= Filtered( Difference( pos, [ 2 ] ), i -> ForAny( pos,
        >             j -> NrPolyhedralSubgroups( h, 2, i, j ).number <> 0 ) );
        [ 4, 9, 16 ]

The two classes have lengths 93 150 and 7 286 400, thus the index of C in H is one of these numbers.

        gap> SizesConjugacyClasses( h ){ pos };
        [ 16584750, 3222483264000, 87495303168000 ]

Next we compute the number n0 of `2B`-pure Klein four groups in M.

        gap> nr:= NrPolyhedralSubgroups( m, 3, 3, 3 );
        rec( number := 87569110066985387357550925521828244921875, type := "V4" )
        gap> n0:= nr.number;;

The M-conjugacy class of V has length [M:N] = [M:H] ·[H:C] / [N:C], where [N:C] divides 6. We see that [N:C] = 6 in both cases.

        gap> cand:= List( pos, i -> Size( m ) / SizesCentralizers( h )[i] / 6 );
        [ 16009115629875684006343550944921875,
          3110635203347364905168577322802100000000,
          84458458854522392576698341855475200000000 ]
        gap> Sum( cand ) = n0;
        true

The orders of the normalizers of the three classes of `2B`-pure Klein four groups are as claimed.

        gap> List( cand, x -> Size( m ) / x );
        [ 50472333605150392320, 259759622062080, 9567039651840 ]

[¯]
As we have seen above, the group M contains exactly the following (solvable) subgroups of order n.

1. One class in 21+24+.Co1 type subgroups,
2. one class in 22+11+22.(M24 ×S3) type subgroups, and
3. two classes in 2[39].(L3(2) ×3S6) type subgroups.

Note that 2[39].(L3(2) ×3S6) contains an elementary abelian normal subgroup of order eight whose involutions lie in the class `2B`, see [CCN+85,p. 234]. As a module for the group L3(2), this normal subgroup is irreducible, and the restriction of the action to the two classes of S4 type subgroups fixes a one- and a two-dimensional subspace, respectively. Hence we have one class of subgroups of order n that centralize a `2B` element and one class of subgroups of order n that normalize a `2B`-pure Klein four group. Clearly the subgroups in the first class coincide with the subgroups of order n in 21+24+.Co1 type subgroups. By the above classification of `2B`-pure Klein four groups in M, the subgroups in the second class coincide with the subgroups of order n in 22+11+22.(M24 ×S3) type subgroups.
It remains to show that the subgroups of order n do not stabilize both a `2B` element and a `2B`-pure Klein four group. We do this by direct computations with a 22+11+22.(M24 ×S3) type group, which is available via the AtlasRep package, see [WPN+11].
First we fetch the group, and factor out the largest solvable normal subgroup, by suitable actions on blocks.

        gap> g:= AtlasGroup( "2^(2+11+22).(M24xS3)" );
        <permutation group of size 50472333605150392320 with 2 generators>
        gap> NrMovedPoints( g );
        294912
        gap> bl:= Blocks( g, MovedPoints( g ) );;
        gap> Length( bl );
        147456
        gap> hom1:= ActionHomomorphism( g, bl, OnSets );;
        gap> act1:= Image( hom1 );;
        gap> Size( g ) / Size( act1 );
        8192
        gap> bl2:= Blocks( act1, MovedPoints( act1 ) );;
        gap> Length( bl2 );
        72
        gap> hom2:= ActionHomomorphism( act1, bl2, OnSets );;
        gap> act2:= Image( hom2 );;
        gap> Size( act2 );
        1468938240
        gap> Size( MathieuGroup( 24 ) ) * 6;
        1468938240
        gap> bl3:= AllBlocks( act2 );;
        gap> List( bl3, Length );
        [ 24, 3 ]
        gap> bl3:= Orbit( act2, bl3[2], OnSets );;
        gap> hom3:= ActionHomomorphism( act2, bl3, OnSets );;
        gap> act3:= Image( hom3 );;

Now we compute an isomorphism from the factor group of type M24 to the group that belongs to GAP's table of marks. Then we use the information from the table of marks to compute a solvable subgroup of maximal order in M24 (which is 13 824), and take the preimage under the isomorphism. Finally, we take the preimage of this group in te original group.

        gap> tom:= TableOfMarks( "M24" );;
        gap> tomgroup:= UnderlyingGroup( tom );;
        gap> iso:= IsomorphismGroups( act3, tomgroup );;
        gap> pos:= Positions( OrdersTom( tom ), 13824 );
        [ 1508 ]
        gap> sub:= RepresentativeTom( tom, pos[1] );;
        gap> pre:= PreImages( iso, sub );;
        gap> pre:= PreImages( hom3, pre );;
        gap> pre:= PreImages( hom2, pre );;
        gap> pre:= PreImages( hom1, pre );;
        gap> Size( pre ) = n;
        true

The subgroups stabilizes a Klein four group. It does not stabilize a `2B` element because its centre is trivial.

        gap> pciso:= IsomorphismPcGroup( pre );;
        gap> Size( Centre( Image( pciso ) ) );
        1

## 5 Proof of the Corollary

With the computations in the previous sections, we have collected the information that is needed to show the corollary stated in Section 1.

        gap> Filtered( Set( RecNames( MaxSolv ) ),
        >              x -> MaxSolv.( x )^2 >= Size( CharacterTable( x ) ) );
        [ "Fi23", "J2", "J2.2", "M11", "M12", "M22.2" ]

## References

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