Subgroups — A₄×D₁₅ #67214

Order 360.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₅	C₃	1	1	yes	maximal, characteristic, complemented
3	D₅×A₄	—	3	1	no	maximal
4	C₃×D₁₅	—	4	1	no	maximal
5	D₃×A₄	—	5	1	no	maximal, Hall
6	C₂²×C₃×C₅	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×D₁₅	—	3	1	no	—
8	C₂×D₁₅	—	3	1	no	—
9	C₅×A₄	—	2	1	no	—
10	C₅×A₄	D₃	1	1	yes	characteristic, complemented
11	C₃²×C₅	—	4	1	no	Hall
12	C₂²×D₅	—	3	1	no	Hall
13	C₃×A₄	D₅	1	1	yes	characteristic, complemented
14	C₂×C₃×C₅	—	3	1	no	—
15	D₁₅	—	3	1	no	—
16	D₁₅	A₄	1	1	yes	characteristic, direct product factor
17	C₃×D₅	—	12	1	no	—
18	C₂²×D₃	—	5	1	no	—
19	C₂×A₄	—	15	1	no	—
20	C₂²×C₅	C₃×D₃	1	1	yes	characteristic, complemented
21	C₂×D₅	—	9	1	no	—
22	C₂×D₅	—	9	1	no	—
23	C₃×D₃	—	20	1	no	—
24	C₃×C₅	—	4	1	no	—
25	C₃×C₅	—	8	1	no	—
26	C₃×C₅	C₂×A₄	1	1	yes	characteristic, complemented
27	C₂²×C₃	C₃×D₅	1	1	yes	characteristic, complemented
28	C₂×D₃	—	15	1	no	—
29	C₂×D₃	—	15	1	no	—
30	A₄	—	2	1	no	—
31	A₄	D₁₅	1	1	yes	characteristic, direct product factor
32	C₂×C₅	—	3	1	no	—
33	D₅	—	3	1	no	—
34	D₅	—	9	1	no	—
35	C₃²	—	4	1	no	Sylow
36	C₂³	—	15	1	no	Sylow
37	C₂×C₃	—	3	1	no	—
38	C₂×C₃	—	60	1	no	—
39	D₃	—	5	1	no	—
40	D₃	—	15	1	no	—
41	C₅	D₃×A₄	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	45	1	no	—
43	C₂²	—	45	1	no	—
44	C₂²	C₃×D₁₅	1	1	yes	characteristic, complemented
45	C₃	—	4	1	no	—
46	C₃	—	8	1	no	—
47	C₃	D₅×A₄	1	1	yes	characteristic, complemented
48	C₂	—	3	1	no	—
49	C₂	—	15	1	no	—
50	C₂	—	45	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₃×C₅	C₂×C₃
LCS	G ▷ C₂²×C₃×C₅
UCS	C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×C₅×A₄	C₂	semidirect
2. C₂²×D₁₅	C₃	semidirect
6. C₂²×C₃×C₅	C₂×C₃	semidirect
10. C₅×A₄	D₃	semidirect
13. C₃×A₄	D₅	semidirect
16. D₁₅	A₄	direct (N × H)
20. C₂²×C₅	C₃×D₃	semidirect
26. C₃×C₅	C₂×A₄	semidirect
27. C₂²×C₃	C₃×D₅	semidirect
31. A₄	D₁₅	direct (N × H)
41. C₅	D₃×A₄	semidirect
44. C₂²	C₃×D₁₅	semidirect
47. C₃	D₅×A₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×A₄	2
C₂²×D₁₅	3
D₅×A₄	3
C₃×D₁₅	4
D₃×A₄	5

Subgroups — A4×D15 #67214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₄×D₁₅ #67214