Quiz:Subgroup structure of alternating group:A4

See subgroup structure of alternating group:A4 for background information.

Basic stuff

Summary table on the structure of subgroups: [SHOW MORE]

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	--	--	--

View the lattice of subgroups as a picture: [SHOW MORE]

1 Which of the following factors of 12 does not occur as the order of a subgroup of the alternating group of degree four?

	1
	2
	3
	4
	6

2 Which of the following groups is not isomorphic to any subgroup of the alternating group of degree four?

	trivial group
	cyclic group:Z2, the cyclic group of order two
	cyclic group:Z3, the cyclic group of order three
	cyclic group:Z4, the cyclic group of order four
	Klein four-group, the non-cyclic group of order four