Subgroup structure of symmetric group:S7
This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S7.
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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|| 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)|