View source for Subgroup structure of symmetric group:S4

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{{group-specific information|
information type = subgroup structure|
group = symmetric group:S4|
connective = of}}

The [[symmetric group:S4|symmetric group of degree four]] has many subgroups.

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]].

==Tables for quick information==

{{finite solvable group subgroup structure facts to check against}}

===Table classifying subgroups up to automorphisms===

<section begin="summary"/>

{| 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 !! Size of each conjugacy class !! Number of subgroups !! 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 in S4]] || <math>\{ (), (1,2,3), (1,3,2) \}</math>  || [[cyclic group:Z3]] || 3 || 8 || 1 || 4 || 4 || -- || -- || 3-Sylow
|-
| [[Z4 in S4]] || <math>\langle (1,2,3,4) \rangle</math> || [[cyclic group:Z4]] || 4 || 6 ||  1 || 3 || 3 || -- || -- ||
|-
| [[normal Klein four-subgroup of S4]] || <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}</math> || [[Klein four-group]] || 4 || 6 || 1 || 1 || 1 || [[symmetric group:S3]] || 1 || 2-core
|-
| [[non-normal Klein four-subgroups of S4]] || <math>\langle (1,2), (3,4) \rangle</math> || [[Klein four-group]] || 4 || 6 || 1 || 3 || 3 || -- || -- ||
|-
| [[S3 in S4]] || <math>\langle (1,2,3), (1,2) \rangle</math> ||[[symmetric group:S3]] || 6 || 4 || 1 || 4 || 4 || -- || -- ||
|-
| [[D8 in S4]] || <math>\langle (1,2,3,4), (1,3) \rangle</math> || [[dihedral group:D8]] || 8 || 3 || 1 || 3 || 3 || -- || -- || 2-Sylow
|-
| [[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 || -- || -- ||
|}

<section end="summary"/>

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

{| class="sortable" border="1"
! Group name !! Order !! Second part of GAP ID (first part is order) !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
|-
| [[Trivial group]] || 1 || 1 || 1 || 1 || 1 || 1
|-
| [[Cyclic group:Z2]] || 2 || 1 || 9 || 2 || 0 || 0
|-
| [[Cyclic group:Z3]] || 3 || 1 || 4 || 1 || 0 || 0
|-
| [[Cyclic group:Z4]] || 4 || 1 || 3 || 1 || 0 || 0
|-
| [[Klein four-group]] || 4 || 2 || 4 || 2 || 1 || 1
|-
| [[Symmetric group:S3]] || 6 || 1 || 4 || 1 || 0 || 0
|-
| [[Dihedral group:D8]] || 8 || 3 || 3 || 1 || 0 || 0
|-
| [[Alternating group:A4]] || 12 || 3 || 1 || 1 || 1 || 1
|-
| [[Symmetric group:S4]] || 24 || 12 || 1 || 1 || 1 || 1
|-
| Total || -- || -- || 30 || 11 || 4 || 4
|}

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

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>p</math>. For <math>p = 2</math>, 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"
! Group order !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
|-
| 1 || 1 || 1 || 1 || 1
|-
| 2 || 9 || 2 || 0 || 0
|-
| 3 || 4 || 1 || 0 || 0
|-
| 4 || 7 || 3 || 1 || 1
|-
| 6 || 4 || 1 || 0 || 0
|-
| 8 || 3 || 1 || 0 || 0
|-
| 12 || 1 || 1 || 1 || 1
|-
| 24 || 1 || 1 || 1 || 1
|-
| Total || 30 || 11 || 4 || 4
|}

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

{| class="sortable" border="1"
! Group property !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
|-
|[[Cyclic group]] || 17 || 5 || 1 || 1
|-
|[[Abelian group]] || 21 || 7 || 2 || 2
|-
|[[Nilpotent group]] || 24 || 8 || 2 || 2
|-
|[[Solvable group]] || 30 || 11 || 4 || 4
|}

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

{| class="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]] || 4 || 4 || 4
|-
|[[Characteristic subgroup]] || 4 || 4 || 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
|-
| trivial subgroup || 1 || 1 + 1 + 1 + 1 || Yes || The subgroup fixes each point, so the orbits are singleton subsets.
|-
| [[S2 in S4]] || 6 || 2 + 1 + 1 || Yes || <math>\{ (), (1,2) \}</math> has orbits <math>\{ 1,2 \}, \{ 3 \}, \{ 4 \}</math>
|-
| [[subgroup generated by double transposition in S4]] || 3 || 2 + 2 || No || <math>\{ (), (1,2)(3,4) \}</math> has orbits <math>\{ 1,2 \}, \{ 3, 4 \}</math>
|-
| [[A3 in S4]] || 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.
|}