Subgroups — C₅×D₃×A₄ #67213

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×C₅×D₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×A₄	—	3	1	no	maximal
4	C₃×C₅×D₃	—	4	1	no	maximal
5	D₃×A₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₅×D₃	—	3	1	no	—
8	C₂×C₅×D₃	—	3	1	no	—
9	C₅×A₄	—	2	1	no	—
10	C₅×A₄	D₃	1	1	yes	characteristic, direct product factor
11	C₃²×C₅	—	4	1	no	Hall
12	C₂³×C₅	—	3	1	no	Hall
13	C₃×A₄	C₂×C₅	1	1	yes	characteristic, complemented
14	C₂×C₃×C₅	—	3	1	no	—
15	C₂×C₃×C₅	—	12	1	no	—
16	C₅×D₃	—	3	1	no	—
17	C₅×D₃	A₄	1	1	yes	characteristic, direct product factor
18	C₂²×D₃	C₃×C₅	1	1	yes	characteristic, complemented
19	C₂×A₄	—	3	1	no	—
20	C₂²×C₅	—	9	1	no	—
21	C₂²×C₅	—	9	1	no	—
22	C₂²×C₅	C₃×D₃	1	1	yes	characteristic, complemented
23	C₃×D₃	—	4	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	C₂×C₃×C₅	1	1	yes	characteristic, complemented
28	C₂×D₃	—	3	1	no	—
29	C₂×D₃	—	3	1	no	—
30	A₄	—	2	1	no	—
31	A₄	C₅×D₃	1	1	yes	characteristic, direct product factor
32	C₂×C₅	—	3	1	no	—
33	C₂×C₅	—	3	1	no	—
34	C₂×C₅	—	9	1	no	—
35	C₃²	—	4	1	no	Sylow
36	C₂³	—	3	1	no	Sylow
37	C₂×C₃	—	3	1	no	—
38	C₂×C₃	—	12	1	no	—
39	D₃	—	3	1	no	—
40	D₃	C₅×A₄	1	1	yes	characteristic, direct product factor
41	C₅	D₃×A₄	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	—	9	1	no	—
43	C₂²	—	9	1	no	—
44	C₂²	C₃×C₅×D₃	1	1	yes	characteristic, complemented
45	C₃	—	4	1	no	—
46	C₃	—	8	1	no	—
47	C₃	C₂×C₅×A₄	1	1	yes	characteristic, complemented
48	C₂	—	3	1	no	—
49	C₂	—	3	1	no	—
50	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₅	D₃×A₄
commutator, cosocle, hypocenter	C₂²×C₃	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅	C₂×C₃
LCS	G ▷ C₂²×C₃
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×A₄	C₂	semidirect
2. C₂²×C₅×D₃	C₃	semidirect
5. D₃×A₄	C₅	direct (N × H)
6. C₂²×C₃×C₅	C₂×C₃	semidirect
10. C₅×A₄	D₃	direct (N × H)
13. C₃×A₄	C₂×C₅	semidirect
17. C₅×D₃	A₄	direct (N × H)
18. C₂²×D₃	C₃×C₅	semidirect
22. C₂²×C₅	C₃×D₃	semidirect
26. C₃×C₅	C₂×A₄	semidirect
27. C₂²×C₃	C₂×C₃×C₅	semidirect
31. A₄	C₅×D₃	direct (N × H)
40. D₃	C₅×A₄	direct (N × H)
41. C₅	D₃×A₄	direct (N × H)
44. C₂²	C₃×C₅×D₃	semidirect
47. C₃	C₂×C₅×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×C₅×D₃	3
C₂×C₅×A₄	3
C₃×C₅×D₃	4
D₃×A₄	5

Subgroups — C5×D3×A4 #67213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×A₄ #67213