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Subgroup structure of symmetric group:S4

54 bytes removed, 01:41, 13 November 2010
Tables for quick information
===Table classifying subgroups up to automorphisms===
{| class="wikitablesortable" border="1"! Automorphism class of subgroups !! Isomorphism class !! Number of conjugacy classes !! Size of each conjugacy class !! Isomorphism class of quotient (if exists) !! [[Subnormal depth ]] (if subnormal)
|-
| trivial subgroup || [[trivial group]] || 1 || 1 || -- || 1
===Table classifying isomorphism types of subgroups===
{| class="wikitablesortable" 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]] || <math>(1,|| 1)</math> || 1 || 1 || 1 || 1
|-
| [[Cyclic group:Z2]] || <math>(2,|| 1)</math> || 9 || 2 || 0 || 0
|-
| [[Cyclic group:Z3]] || <math>(3,|| 1)</math> || 4 || 1 || 0 || 0
|-
| [[Cyclic group:Z4]] || <math>(4,|| 1)</math> || 3 || 1 || 0 || 0
|-
| [[Klein four-group]] || <math>(4,|| 2)</math> || 4 || 2 || 1 || 1
|-
| [[Symmetric group:S3]] || <math>(6,|| 1)</math> || 4 || 1 || 0 || 0
|-
| [[Dihedral group:D8]] || <math>(8,|| 3)</math> || 3 || 1 || 0 || 0
|-
| [[Alternating group:A4]] || <math>(12,|| 3)</math> || 1 || 1 || 1 || 1
|-
| [[Symmetric group:S4]] || <math>(24,|| 12)</math> || 1 || 1 || 1 || 1
|-
| Total || -- || -- || 30 || 11 || 4 || 4
|}
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