Subgroups — C₃⋊(C₅⋊S₄) #67211

Order 360.

60 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	—	3	3	no	maximal
3	#1220	—	3	1	no	maximal
4	C₃⋊D₁₅	—	4	1	no	maximal
5	C₃⋊S₄	—	5	1	no	maximal, Hall
6	C₂²×C₃×C₅	D₃	1	1	yes	characteristic, complemented
7	C₂×D₁₅	—	3	1	no	—
8	C₅×A₄	D₃	1	3	yes	complemented
9	Dic₁₅	—	3	1	no	—
10	C₃²×C₅	—	4	1	no	Hall
11	C₂²⋊D₅	—	9	1	no	Hall
12	C₃×A₄	D₅	1	1	yes	characteristic, complemented
13	C₂×C₃×C₅	—	3	1	no	—
14	D₁₅	—	6	1	no	—
15	D₁₅	—	12	3	no	—
16	S₄	—	15	3	no	—
17	#67	—	15	1	no	—
18	C₂²×C₅	C₃⋊D₃	1	1	yes	characteristic, complemented
19	C₂×D₅	—	9	1	no	—
20	Dic₅	—	9	1	no	—
21	C₃⋊D₃	—	20	1	no	—
22	C₃×C₅	—	4	3	no	—
23	C₃×C₅	S₄	1	1	yes	characteristic, complemented
24	C₂²×C₃	D₁₅	1	1	yes	characteristic, complemented
25	C₂×D₃	—	15	1	no	—
26	A₄	D₁₅	1	3	yes	complemented
27	Dic₃	—	15	1	no	—
28	C₂×C₅	—	3	1	no	—
29	D₅	—	18	1	no	—
30	C₃²	—	4	1	no	Sylow
31	D₄	—	45	1	no	Sylow
32	C₂×C₃	—	3	1	no	—
33	D₃	—	30	1	no	—
34	D₃	—	60	3	no	—
35	C₅	C₃⋊S₄	1	1	yes	characteristic, complemented, Sylow
36	C₂²	—	45	1	no	—
37	C₂²	C₃⋊D₁₅	1	1	yes	characteristic, complemented
38	C₄	—	45	1	no	—
39	C₃	—	4	3	no	—
40	C₃	C₅⋊S₄	1	1	yes	characteristic, complemented
41	C₂	—	3	1	no	—
42	C₂	—	90	1	no	—

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

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

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