Subgroups — C₄⋊(C₅⋊S₄) #91950

Order 480.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅⋊S₄	C₂	1	2	yes	maximal, complemented
2	C₄×C₅×A₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#4076	—	3	1	no	maximal, Hall
4	C₂×C₅×A₄	C₂²	1	1	yes	characteristic
5	C₅⋊S₄	—	2	2	no	—
6	D₆₀	—	4	1	no	maximal
7	C₄⋊S₄	—	5	1	no	maximal, Hall
8	C₂²×C₄×C₅	D₃	1	1	yes	characteristic, complemented
9	C₂×C₂²⋊D₅	—	3	2	no	—
10	C₂×D₂₀	—	3	1	no	—
11	#685	—	3	2	no	—
12	C₄⋊Dic₅	—	3	1	no	—
13	C₂×D₁₅	—	4	2	no	—
14	C₅×A₄	D₄	1	1	yes	characteristic, complemented
15	C₃×C₄×C₅	—	4	1	no	—
16	C₂×S₄	—	5	2	no	—
17	C₄×A₄	D₅	1	1	yes	characteristic, complemented
18	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
19	C₂²×D₅	—	3	2	no	—
20	C₂×C₄×C₅	—	3	1	no	—
21	C₂×C₄×C₅	—	3	1	no	—
22	C₂²⋊D₅	—	6	2	no	—
23	C₂²⋊D₅	—	6	2	no	—
24	C₂×Dic₅	—	3	2	no	—
25	D₂₀	—	6	1	no	—
26	D₂₀	—	6	1	no	—
27	#121	—	15	1	no	Sylow
28	C₂×C₃×C₅	—	4	1	no	—
29	D₁₅	—	8	2	no	—
30	C₂×A₄	C₂×D₅	1	1	yes	characteristic
31	S₄	—	10	2	no	—
32	D₁₂	—	20	1	no	—
33	C₂²×C₅	—	3	1	no	—
34	C₂²×C₅	—	3	1	no	—
35	C₂²×C₅	D₁₂	1	1	yes	characteristic, complemented
36	C₂×D₅	—	6	2	no	—
37	C₂×D₅	—	6	2	no	—
38	C₂×D₅	—	6	2	no	—
39	C₄×C₅	—	3	1	no	—
40	C₄×C₅	S₄	1	1	yes	characteristic, complemented
41	Dic₅	—	6	2	no	—
42	C₂×D₄	—	15	1	no	—
43	C₂×D₄	—	15	2	no	—
44	C₂²×C₄	D₁₅	1	1	yes	characteristic, complemented
45	C₄⋊C₄	—	15	1	no	—
46	C₂²⋊C₄	—	15	2	no	—
47	C₃×C₅	—	4	1	no	Hall
48	C₂×D₃	—	20	2	no	—
49	A₄	D₂₀	1	1	yes	characteristic, complemented
50	C₃×C₄	—	4	1	no	—
51	C₂×C₅	—	3	1	no	—
52	C₂×C₅	—	3	1	no	—
53	C₂×C₅	C₂×S₄	1	1	yes	characteristic
54	D₅	—	12	2	no	—
55	C₂³	—	15	2	no	—
56	C₂³	C₂×D₁₅	1	1	yes	characteristic
57	D₄	—	30	1	no	—
58	D₄	—	30	2	no	—
59	D₄	—	30	2	no	—
60	D₄	—	30	1	no	—
61	C₂×C₄	—	3	1	no	—
62	C₂×C₄	—	3	1	no	—
63	C₂×C₄	—	15	2	no	—
64	C₂×C₃	—	4	1	no	—
65	D₃	—	40	2	no	—
66	C₅	C₄⋊S₄	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	3	1	no	—
68	C₂²	—	3	1	no	—
69	C₂²	—	30	2	no	—
70	C₂²	—	30	2	no	—
71	C₂²	—	30	2	no	—
72	C₂²	D₆₀	1	1	yes	characteristic, complemented
73	C₄	—	3	1	no	—
74	C₄	—	30	2	no	—
75	C₄	C₅⋊S₄	1	1	yes	characteristic, complemented
76	C₃	—	4	1	no	Sylow
77	C₂	—	3	1	no	—
78	C₂	—	3	1	no	—
79	C₂	—	60	2	no	—
80	C₂	C₂×C₅⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₅⋊S₄
commutator, cosocle	C₂×C₅×A₄	C₂²
Fitting	C₂²×C₄×C₅	D₃
socle	C₂³×C₅	C₂×D₃
hypercenter	C₄	C₅⋊S₄
hypocenter	C₅×A₄	D₄
LCS	G ▷ C₂×C₅×A₄ ▷ C₅×A₄
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₅⋊S₄	C₂	semidirect
2. C₄×C₅×A₄	C₂	semidirect
8. C₂²×C₄×C₅	D₃	semidirect
14. C₅×A₄	D₄	semidirect
17. C₄×A₄	D₅	semidirect
35. C₂²×C₅	D₁₂	semidirect
40. C₄×C₅	S₄	semidirect
44. C₂²×C₄	D₁₅	semidirect
49. A₄	D₂₀	semidirect
66. C₅	C₄⋊S₄	semidirect
72. C₂²	D₆₀	semidirect
75. C₄	C₅⋊S₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅⋊S₄	2
C₄×C₅×A₄	2
#4076	3
D₆₀	4
C₄⋊S₄	5

Subgroups — C4⋊(C5⋊S4) #91950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(C₅⋊S₄) #91950