Subgroup structure of alternating group:A7
This article gives specific information, namely, subgroup structure, about a particular group, namely: alternating group:A7.
View subgroup structure of particular groups | View other specific information about alternating group:A7
|Family name||Parameter values||General discussion of subgroup structure of family|
|alternating group||degree , i.e., the group||subgroup structure of alternating groups|
Tables for quick information
FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate
|Number of subgroups|| 3786|
Compared with : 2, 10, 59, 501, 3786, 48337, ...
|Number of conjugacy classes of subgroups|| 40|
Compared with : 2, 5, 9, 22, 40, 137, ...
|Number of automorphism classes of subgroups|| 37|
Compared with : 2, 5, 9, 16, 37, 112, ...
|Isomorphism classes of Sylow subgroups and the corresponding fusion systems|| 2-Sylow: dihedral group:D8 (order 8) as D8 in A7 (with its non-inner fusion system -- see fusion systems for dihedral group:D8). Sylow number is 315.|
3-Sylow: elementary abelian group:E9 (order 9) as E9 in A7. Sylow number is 70.
5-Sylow: cyclic group:Z5 (order 5) as Z5 in A7. Sylow number is 126.
7-Sylow: cyclic group:Z7 (order 7) as Z7 in A7. Sylow number is 120.
|Hall subgroups||Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are -Hall subgroups (of order 72) and -Hall subgroups (of order 360), the latter being A6 in A7. Note that the -Hall subgroups are not contained in -Hall subgroups.|
|maximal subgroups||maximal subgroups have orders 72, 120, 168, 360.|
|normal subgroups||only the whole group and the trivial subgroup, because the group is simple. See alternating groups are simple.|
|subgroups that are simple non-abelian groups (apart from the whole group itself)||alternating group:A5 (order 60), projective special linear group:PSL(3,2) (order 168, the embedding arises via the natural permutation representation on the two-dimensional projective space over field:F2, which has size ), alternating group:A6 (order 360)|