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→Table classifying subgroups up to automorphisms

{{quiz ad}}{{group-specific information|information type = subgroup structure|group = symmetric group:S4|connective = of}} The [[symmetric group:S4|symmetric group ~~on ~~of degree four ~~letters~~]] has many ~~sugbroups~~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]].

===~~Subgroup-defining functions yielding this subgroup~~Table classifying subgroups up to automorphisms===

<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 (=~~Subgroup properties satisfied by this ~~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 |||-| [[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), </math><br><math>(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 || -- || -- |||-| [[D8 in S4]] || <math>\langle (1,2,3,4), (1,3) \rangle</math> || [[dihedral group:D8]] || 8 || 3 || 1 || 3 || 3 || -- || -- || 2-Sylow, fusion system is [[non-inner non-simple fusion system for dihedral group:D8]]|-| [[A3 in S4]] || <math>\{ (), (1,2,3), (1,3,2) \}</math> || [[cyclic group:Z3]] || 3 || 8 || 1 || 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 || -- || -- |||-| [[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"/>

{| class="sortable" border=~~=Subgroup properties not satisfied by this ~~"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|}

{| class="sortable" border=~~The two~~"1"! Group property !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup|-~~element subgroup generated by a transposition (type (~~|[[Cyclic group]] || 17 || 5 || 1 || 1|-|[[Abelian group]] || 21 || 7 || 2 || 2|-|[[Nilpotent group]] || 24 || 8 || 2 || 2~~))==~~|-|[[Solvable group]] || 30 || 11 || 4 || 4|}

===~~Subgroup properties satisfied ~~Table listing numbers of subgroups by ~~these subgroups~~subgroup property===

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