Subgroup structure of alternating group:A8
From Groupprops
This article gives specific information, namely, subgroup structure, about a particular group, namely: alternating group:A8.
View subgroup structure of particular groups | View other specific information about alternating group:A8
This article discusses the subgroup structure of alternating group:A8, which is the alternating group on the set . The group has order 20160.
Family contexts
Family name | Parameter values | General information on subgroup structure of family |
---|---|---|
alternating group | degree , i.e., the group | subgroup structure of alternating groups |
projective special linear group of degree four over a finite field | field size , so field:F2, i.e., the group is . Note that because of isomorphism between linear groups over field:F2, this is also , , and . | ? |
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 | 48337 Compared with : 2, 10, 59, 501, 3786, 48337, ... |
Number of conjugacy classes of subgroups | 137 Compared with : 2, 5, 9, 22, 40, 137, ... |
Number of automorphism classes of subgroups | 112 Compared with : 2, 5, 9, 16, 37, 112, ... |
Isomorphism classes of Sylow subgroups | 2-Sylow: unitriangular matrix group:UT(4,2) (order 64). 3-Sylow: elementary abelian group:E9 (order 9). 5-Sylow: cyclic group:Z5 (order 5) 7-Sylow: cyclic group:Z7 (order 7) |
Hall subgroups | Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are -Hall subgroups of order 576. |
maximal subgroups | maximal subgroups have orders 360, 576, 720, 1344, 2520 |
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 (other than the whole group) | 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 , or via embedding it as in the alternating group on a set of size ), alternating group:A6 (order 360), alternating group:A7 (order 2520) |