Subgroups — C₃×C₅×S₄ #67208

Order 360.

48 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	C₅×S₄	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅×D₄	—	3	1	no	maximal
4	C₃×C₅×D₃	—	4	1	no	maximal
5	C₃×S₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅	—	3	1	no	—
7	C₂²×C₃×C₅	D₃	1	1	yes	characteristic, complemented
8	C₅×A₄	—	2	1	no	—
9	C₅×A₄	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃×C₄×C₅	—	3	1	no	—
11	C₃²×C₅	—	4	1	no	Hall
12	C₅×D₄	—	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	—	6	1	no	—
16	C₅×D₃	—	4	1	no	—
17	S₄	C₃×C₅	1	1	yes	characteristic, direct product factor
18	C₃×D₄	—	3	1	no	—
19	C₂²×C₅	—	3	1	no	—
20	C₂²×C₅	C₃×D₃	1	1	yes	characteristic, complemented
21	C₄×C₅	—	3	1	no	—
22	C₃×D₃	—	4	1	no	—
23	C₃×C₅	—	4	1	no	—
24	C₃×C₅	—	8	1	no	—
25	C₃×C₅	S₄	1	1	yes	characteristic, direct product factor
26	C₂²×C₃	—	3	1	no	—
27	C₂²×C₃	C₅×D₃	1	1	yes	characteristic, complemented
28	A₄	—	2	1	no	—
29	A₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
30	C₃×C₄	—	3	1	no	—
31	C₂×C₅	—	3	1	no	—
32	C₂×C₅	—	6	1	no	—
33	C₃²	—	4	1	no	Sylow
34	D₄	—	3	1	no	Sylow
35	C₂×C₃	—	3	1	no	—
36	C₂×C₃	—	6	1	no	—
37	D₃	—	4	1	no	—
38	C₅	C₃×S₄	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²	—	3	1	no	—
40	C₂²	C₃×C₅×D₃	1	1	yes	characteristic, complemented
41	C₄	—	3	1	no	—
42	C₃	—	4	1	no	—
43	C₃	—	8	1	no	—
44	C₃	C₅×S₄	1	1	yes	characteristic, direct product factor
45	C₂	—	3	1	no	—
46	C₂	—	6	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅	S₄
commutator, cosocle, hypocenter	A₄	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×A₄	C₂	semidirect
2. C₅×S₄	C₃	direct (N × H)
5. C₃×S₄	C₅	direct (N × H)
7. C₂²×C₃×C₅	D₃	semidirect
9. C₅×A₄	C₂×C₃	semidirect
13. C₃×A₄	C₂×C₅	semidirect
17. S₄	C₃×C₅	direct (N × H)
20. C₂²×C₅	C₃×D₃	semidirect
25. C₃×C₅	S₄	direct (N × H)
27. C₂²×C₃	C₅×D₃	semidirect
29. A₄	C₂×C₃×C₅	semidirect
38. C₅	C₃×S₄	direct (N × H)
40. C₂²	C₃×C₅×D₃	semidirect
44. C₃	C₅×S₄	direct (N × H)

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
C₅×S₄	3
C₃×C₅×D₄	3
C₃×C₅×D₃	4
C₃×S₄	5

Subgroups — C3×C5×S4 #67208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×S₄ #67208