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Subgroup structure of symmetric group:S4

1,013 bytes added, 21:35, 28 April 2012
Tables for quick information
{{finite solvable group subgroup structure facts to check against}}
 
<section begin="summary"/>
===Quick summary===
{| class="sortable" border="1"
! Item !! Value
|-
| [[Number of subgroups]] || 30
|-
| [[Number of conjugacy classes of subgroups]] || 11
|-
| [[Number of automorphism classes of subgroups]] || 11
|-
| Isomorphism classes of [[Sylow subgroup]]s and the corresponding [[Sylow number]]s and [[fusion system]]s || 2-Sylow: [[dihedral group:D8]] (order 8), Sylow number is 3, fusion system is [[non-inner non-simple fusion system for dihedral group:D8]]<br>3-Sylow: [[cyclic group:Z3]], Sylow number is 4, fusion system is [[non-inner fusion system for cyclic group:Z3]]
|-
| [[Hall subgroup]]s || Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups
|-
| [[maximal subgroup]]s || maximal subgroups have order 6 ([[S3 in S4]]), 8 ([[D8 in S4]]), and 12 ([[A4 in S4]]).
|-
| [[normal subgroup]]s || There are four normal subgroups: the whole group, the trivial subgroup, [[A4 in S4]], and [[normal V4 in S4]].
|}
===Table classifying subgroups up to automorphisms===
<section begin="summary"/>
<small>
{| class="sortable" border="1"
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