Subgroup structure of alternating group:A4

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This article gives specific information, namely, subgroup structure, about a particular group, namely: alternating group:A4.
View subgroup structure of particular groups | View other specific information about alternating group:A4
A4latticeofsubgroups.png

The alternating group on \{ 1,2,3,4 \} is a group of order 12.

There is no subgroup of order 6. The alternating group of degree four is the group of smallest possible order (in this case 12) not having subgroups of all orders dividing the group order.

Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group


Quick summary

Item Value
Number of subgroups 10
Number of conjugacy classes of subgroups 5
Number of automorphism classes of subgroups 5

Table classifying subgroups up to automorphism

Automorphism class of subgroups List of subgroups Isomorphism class Order of subgroups Index of subgroups Number of conjugacy classes Size of each conjugacy class Total number of subgroups Isomorphism class of quotient (if exists) Subnormal depth (if subnormal) Note
trivial subgroup \{ () \} trivial group 1 12 1 1 1 alternating group:A4 1 trivial
subgroup generated by double transposition in A4 \{ (), (1,2)(3,4) \}, \{ (), (1,3)(2,4) \}, \{ (), (1,4)(2,3) \} cyclic group:Z2 2 6 1 3 3 -- 2
V4 in A4 \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \} Klein four-group 4 3 1 1 1 cyclic group:Z3 1 2-Sylow, minimal normal, maximal
A3 in A4 \{ (), (2,3,4), (2,4,3) \}, \{ (), (1,3,4), (1,4,3)\}, \{ (), (1,2,4), (1,4,2)\}, \{ (), (1,2,3), (1,3,2) \} cyclic group:Z3 3 4 1 4 4 -- -- 3-Sylow, maximal
whole group all elements alternating group:A4 12 1 1 1 1 trivial group 1 whole
Total (5 rows) -- -- -- -- 5 -- 10 -- -- --


Table classifying isomorphism types of subgroups

Group name Order Second part of GAP ID (first part is order) Occurrences as subgroup Conjugacy classes of occurrence as subgroup Automorphism classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
trivial group 1 1 1 1 1 1 1
cyclic group:Z2 2 1 3 1 1 0 0
cyclic group:Z3 3 1 4 1 1 0 0
Klein four-group 4 2 1 1 1 1 1
alternating group:A4 12 3 1 1 1 1 1
Total -- -- 10 5 5 3 3

Table listing numbers of subgroups by group property

Group property Occurrences as subgroup Conjugacy classes of occurrence as subgroup Automorphism classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
cyclic group 8 3 3 1 1
abelian group 9 4 4 2 2
nilpotent group 9 4 4 2 2
solvable group 10 5 5 3 3

A4latticeofsubgroups.png