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Subgroup structure of symmetric group:S4

2,283 bytes added, 03:48, 18 February 2014
Table classifying subgroups up to automorphisms
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{{group-specific information|
information type = subgroup structure|
{{finite solvable group subgroup structure facts to check against}}
 
<section begin="summary"/>
===Quick summary===
{| class="sortable" border="1"
! Item !! Value
|-
| [[Number of subgroups]] || 30<br>Compared with <math>S_n, n = 1,2,3,4,5,\dots</math>: 1,2,6,'''30''',156,1455,11300, 151221
|-
| [[Number of conjugacy classes of subgroups]] || 11<br>Compared with <math>S_n, n = 1,2,3,4,5,\dots</math>: 1,2,4,'''11''',19,56,96,296,554,1593
|-
| [[Number of automorphism classes of subgroups]] || 11<br>Compared with <math>S_n, n = 1,2,3,4,5,\dots</math>: 1,2,4,'''11''',19,37,96,296,554,1593
|-
| 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===
{{subgroup order sorting note}}
 
<small>
{| class="sortable" border="1"
! Automorphism class of subgroups !! Representative !! Isomorphism class !! [[Order of a group|Order]] of subgroups !! [[Index of a subgroup|Index]] of subgroups !! Number of conjugacy classes (=1 iff [[automorph-conjugate subgroup]]) !! Size of each conjugacy class (=1 iff [[normal subgroup]]) !! Number of subgroups (=1 iff [[characteristic subgroup]])!! Isomorphism class of quotient (if exists) !! [[Subnormal depth]] (if subnormal)!! Note
|-
| trivial subgroup || <math>\{ () \}</math> || [[trivial group]] || 1 || 24 || 1 || 1 || 1 || [[symmetric group:S4]] || 1||
|-
| [[S2 in S4]] || <math>\{ (), (1,2) \}</math> || [[cyclic group:Z2]] || 2 || 12 || 1 || 6 || 6 || -- || --||
|-
| [[subgroup generated by double transposition in S4]] || <math>\{ (), (1,2)(3,4) \}</math> || [[cyclic group:Z2]] || 2 || 12 || 1 || 3 || 3 || -- || 2||
|-
| [[A3 Z4 in S4]] || <math>\{ (), langle (1,2,3), (1,3,24) \}rangle</math> || [[cyclic group:Z3Z4]] || 4 || 6 || 1 || 4 3 || 3 || -- || --||
|-
| [[Z4 in normal Klein four-subgroup of S4]] || <math>\langle { (), (1,2)(3,4), </math><br><math>(1,3)(2,4), (1,4)(2,3) \rangle}</math> || [[cyclic Klein four-group:Z4]] || 4 || 6 || 1 || 1 || 1 || 3 [[symmetric group:S3]] || -- 1 || 2--core
|-
| [[non-normal Klein four-subgroup subgroups of S4]] || <math>\{ (), langle (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}rangle</math> || [[Klein four-group]] || 4 || 6 || 1 || 1 3 || 3 || -- || [[symmetric group:S3]] -- || 1
|-
| [[non-normal Klein four-subgroups of D8 in S4]] || <math>\langle (1,2,3,4), (1,3,4) \rangle</math> || [[Klein four-dihedral group:D8]] || 8 || 3 || 1 || 3 || 3 || -- || --|| 2-Sylow, fusion system is [[non-inner non-simple fusion system for dihedral group:D8]]
|-
| [[S3 A3 in S4]] || <math>\langle { (), (1,2,3), (1,3,2) \rangle}</math> ||[[symmetric cyclic group:S3Z3]] || 3 || 8 || 1 || 4 || 4 || -- || --|| 3-Sylow, fusion system is [[non-inner fusion system for cyclic group:Z3]]
|-
| [[D8 S3 in S4]] || <math>\langle (1,2,3,4), (1,32) \rangle</math> || [[dihedral symmetric group:D8S3]] || 6 || 4 || 1 || 3 4 || 4 || -- || --||
|-
| [[A4 in S4]] || <math>\langle (1,2,3), (1,2)(3,4) \rangle</math> || [[alternating group:A4]] || 12 || 2 || 1 || 1 || 1 || [[cyclic group:Z2]] || 1||
|-
| whole group || <math>\langle (1,2,3,4), (1,2) \rangle</math> || [[symmetric group:S4]] || 24 || 1 || 1 || 1 || 1 || [[trivial group]] || 0|||-! Total (11 rows) !! -- !! -- !! -- !! -- !! 11 !! -- !! 30 !! -- !! -- !! --
|}
</small>
<section end="summary"/>
===Table classifying isomorphism types of subgroups===
| [[Symmetric group:S4]] || 24 || 12 || 1 || 1 || 1 || 1
|-
| ! Total || -- || -- || 30 || 11 || 4 || 4
|}
===Table listing number of subgroups by order===
Note that these orders These numbers satisfy the [[congruence condition on number of subgroups of given prime power order]]: the number of subgroups of order <math>p^r</math> for a fixed nonnegative integer <math>r</math> is congruent to 1 mod <math>1p</math>. For <math>p = 2</math> modulo , this means the number is odd, and for <math>p= 3</math>, this means the number is congruent to 1 mod 3 (so it is among 1,4,7,...)
{| class="sortable" border="1"
| 24 || 1 || 1 || 1 || 1
|-
| ! Total || 30 || 11 || 4 || 4
|}
 
===Table listing numbers of subgroups by group property===
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