Subgroup structure of alternating group:A7
From Groupprops
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
This article discusses the subgroup structure of alternating group:A7, which is the alternating group on the set . The group has order 2520.
Family contexts
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
Quick summary
Item | Value |
---|---|
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) |