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→Table classifying subgroups up to automorphisms

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{{group-specific information|

information type = subgroup structure|

Note that since <math>S_4</math> is a [[complete group]], every automorphism is inner, so the classification of subgroups upto conjugacy is equivalent to the classification of subgroups upto automorphism. In other words, every subgroup is an [[automorph-conjugate subgroup]].

<section begin="summary"/>=~~Tables for quick information~~==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 ||

|-

| ~~trivial subgroup ~~[[S2 in S4]] || <math>\{ (), (1,2) \}</math> || [[~~trivial ~~cyclic group:Z2]] || 2 || 12 || 1 || ~~1 ~~6 || 6 || -- || -- || ~~1~~

|-

| [[~~S2 ~~subgroup generated by double transposition in S4]] || <math>\{ (), (1,2)(3,4) \}</math> || [[cyclic group:Z2]] || 2 || 12 || 1 || ~~6 ~~3 || 3 || -- || ~~--~~2 ||

|-

| [[~~subgroup generated by double transposition ~~Z4 in S4]] || <math>\langle (1,2,3,4) \rangle</math> || [[cyclic group:~~Z2~~Z4]] || 4 || 6 || 1 || 3 || 3 || -- || -- || ~~2~~

|-

| [[~~A3 in ~~normal Klein four-subgroup of S4]] || <math>\{ (), (1,2)(3,4), </math><br><math>(1,3)(2,4), (1,4)(2,3) \}</math> || [[~~cyclic ~~Klein four-group~~:Z3~~]] || 4 || 6 || 1 || ~~4 ~~1 || 1 || [[symmetric group:S3]] || ~~-- ~~1 || 2-~~-~~core

|-

| [[~~Z4 in ~~non-normal Klein four-subgroups of S4]] || <math>\langle (1,2), (3,4) \rangle</math> || [[~~cyclic ~~Klein four-group~~:Z4~~]] || 4 || 6 || 1 || 3 || 3 || -- || --||

|-

| [[~~normal Klein four-subgroup of ~~D8 in S4]] || <math>\langle (1,2,3,4), (1,3) \rangle</math> || [[~~Klein four-~~dihedral group:D8]] || 8 || 3 || 1 || ~~1 ~~3 || 3 || -- || -- || 2-Sylow, fusion system is [[~~symmetric ~~non-inner non-simple fusion system for dihedral group:~~S3~~D8]] ~~|| 1~~

|-

| [[~~non-normal Klein four-subgroups of ~~A3 in S4]] || <math>\{ (), (1,2,3), (1,3,2) \}</math> || [[~~Klein four-~~cyclic group:Z3]] || 3 || 8 || 1 || ~~3 ~~4 || 4 || -- || -- || 3-Sylow, fusion system is [[non-inner fusion system for cyclic group:Z3]]

|-

| [[S3 in S4]] || <math>\langle (1,2,3), (1,2) \rangle</math> || [[symmetric group:S3]] || 6 || 4 || 1 || 4 || 4 || -- || --||

|-

| [[~~D8 ~~A4 in S4]] || <math>\langle (1,2,3), (1,2)(3,4) \rangle</math> || [[~~dihedral ~~alternating group:~~D8~~A4]] || 12 || 2 || 1 || 1 || 1 || ~~3 ~~[[cyclic group:Z2]] || ~~-- ~~1 || ~~--~~

|-

| ~~[[A4 in S4]] ~~whole group || <math>\langle (1,2,3,4), (1,2) \rangle</math> || [[~~alternating ~~symmetric group:~~A4~~S4]] || 24 || 1 || 1 || 1 || 1 || [[~~cyclic ~~trivial group~~:Z2~~]] || ~~1~~0 ||

|-

|}

</small>

<section end="summary"/>

===Table classifying isomorphism types of subgroups===

| [[Symmetric group:S4]] || 24 || 12 || 1 || 1 || 1 || 1

|-

|}

===Table listing number of subgroups by order===

{| class="~~wikitable~~sortable" border="1"

! Group order !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup

|-

| 24 || 1 || 1 || 1 || 1

|-

|}

===Table listing numbers of subgroups by group property===

{| class="~~wikitable~~sortable" border="1"

! Group property !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup

|-

|[[Nilpotent group]] || 24 || 8 || 2 || 2

|-

|[[Solvable group]] || 30 || 11 || ~~5 ~~4 || ~~5~~4

|}

===Table listing numbers of subgroups by subgroup property===

{| class="~~wikitable~~sortable" border="1"

! Subgroup property !! Occurences as subgroup !! Conjugacy classes of occurrences as subgroup !! Automorphism classes of occurrences as subgroup

|-

|[[Subgroup]] || 30 || 11 || 11

|-

|[[Normal subgroup]] || ~~5 ~~4 || ~~5 ~~4 || ~~5~~4

|-

|[[Characteristic subgroup]] || ~~5 ~~4 || ~~5 ~~4 || ~~5~~4|} ==Subgroup structure viewed as symmetric group== ===Classification based on partition given by orbit sizes=== For any subgroup of <math>S_4</math>, the natural action on <math>\{ 1,2,3,4 \}</math> induces a partition of the set <math>\{ 1,2,3 \}</math> into orbits, which in turn induces an unordered integer partition of the number 4. Below, we classify this information for the subgroups. {| class="sortable" border="1"! Conjugacy class of subgroups !! Size of conjugacy class !! Induced partition of 4 !! Direct product of transitive subgroups on each orbit? !! Illustration with representative

|-

|~~[[Fully characteristic ~~trivial subgroup~~]] ~~|| ~~5 ~~1 || ~~5 ~~1 + 1 + 1 + 1 || ~~5~~Yes || The subgroup fixes each point, so the orbits are singleton subsets.

|-

|[[~~Self-centralizing subgroup~~S2 in S4]] || ~~20 ~~6 || ~~8 ~~2 + 1 + 1 || ~~8~~Yes || <math>\{ (), (1,2) \}</math> has orbits <math>\{ 1,2 \}, \{ 3 \}, \{ 4 \}</math>

|-

|[[~~Pronormal ~~subgroupgenerated by double transposition in S4]] || ~~21 ~~3 || ~~8 ~~2 + 2 || ~~8~~No || <math>\{ (), (1,2)(3,4) \}</math> has orbits <math>\{ 1,2 \}, \{ 3, 4 \}</math>

|-

|[[~~Retract~~A3 in S4]] || ~~ 12 ~~4 || 3 + 1 || Yes || <math>\{ (), (1,2,3), (1,3,2) \}</math> has orbits <math>\{ 1,2,3 \}, \{ 4 \}</math>|-| [[Z4 in S4]] || 3 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.|-| [[normal Klein four-subgroup of S4]] || 1 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.|-| [[non-normal Klein four-subgroups of S4]] || 3 || 2 + 2 || Yes || <math>\langle (1,2), (3,4) \rangle</math> has orbits <math>\{ 1,2 \}, \{ 3,4 \}</math>|-| [[S3 in S4]] || 4 || 3 + 1 || Yes || <math>\langle (1,2,3), (1,2) \rangle</math> has orbits <math>\{ 1,2,3 \}, \{ 4 \}</math>|-| [[D8 in S4]] || 3 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.|-| [[A4 in S4]] || 1 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.|-| whole group || 1 || 4 || Yes || The action is a [[transitive group action]], so only one orbit.

|}

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