The methods described above have been developed with a particular interest in the tables of marks of simple and almost simple groups. While it should be possible to adapt them to larger classes of groups, it should be noted that arguments involving Sylow subgroups (in particular Corollary 4.8) loose their meaning when the table of marks of a p-group is to be determined.
Table 4 in the appendix lists the numbers of subgroups and the
numbers of conjugacy classes of subgroups for the following groups: all
projective special linear groups
of order
; the projective
special linear groups
for n = 3, 4, 5; the alternating groups
for
; the symmetric groups
for
; the unitary groups
for n = 3, 4, 5 and
; the Suzuki
group Sz(8); the symplectic group
; the Mathieu groups
,
,
and
plus their automorphism groups
and
; the Janko groups
,
and
; and the McLaughlin group
McL.
The complete tables of marks of these groups have been determined by the methods described above. These tables form a library which is part of GAP.
In [Buekenhout and Rees 1988] the poset structure of the Mathieu group
is
determined. The poset structure of the sporadic simple Janko group
is
determined in [Pahlings 1987]. The poset structure of the sporadic simple
Janko group
is determined in [Pfeiffer 1991]. Informations about
parts of the subgroup lattice of the sporadic simple McLaughlin group McL
can be found in [Diawara 1987].
A rather incomplete collection of facts about subgroups of the Mathieu groups is given in [Greenberg 1973]. In his review (MR 50#4731), P. Fong points out that ``this can not remain the last work on the subject.''