Subgroup structure of symmetric group:S7
From Groupprops
This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S7.
View subgroup structure of particular groups | View other specific information about symmetric group:S7
This article discusses the subgroup structure of symmetric group:S7, which is the symmetric group on the set . The group has order 5040.
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 | 11300 Compared with : 1,2,6,30,156,1455,11300,151221 |
Number of conjugacy classes of subgroups | 96 Compared with : 1,2,4,11,19,56,96,296,554,1593,... |
Number of automorphism classes of subgroups | 96 Compared with : 1,2,4,11,19,37,96,296,554,1593,... |
Isomorphism classes of Sylow subgroups | 2-Sylow: direct product of D8 and Z2 (order 16) 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 144) and -Hall subgroups (of order 720), the latter being S6 in S7. Note that the -Hall subgroups are not contained in -Hall subgroups. |
maximal subgroups | maximal subgroups have orders 42, 144, 240, 720, 2520 |
normal subgroups | the whole group, trivial subgroup, and alternating group:A7 (embedded as A7 in S7) |
subgroups that are simple non-abelian groups | 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), alternating group:A7 (order 2520) |