Subgroups — C₅×C₄⋊S₄ #91939

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	C₅×#121	—	3	1	no	maximal, Hall
4	C₂×C₅×A₄	C₂²	1	1	yes	characteristic
5	C₅×S₄	—	2	2	no	—
6	C₅×D₁₂	—	4	1	no	maximal
7	C₄⋊S₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₅×D₄	—	3	1	no	—
9	C₂×C₅×D₄	—	3	2	no	—
10	C₂²×C₄×C₅	D₃	1	1	yes	characteristic, complemented
11	C₅×C₄⋊C₄	—	3	1	no	—
12	C₅×C₂²⋊C₄	—	3	2	no	—
13	C₂×C₅×D₃	—	4	2	no	—
14	C₅×A₄	D₄	1	1	yes	characteristic, complemented
15	C₃×C₄×C₅	—	4	1	no	—
16	C₂×S₄	C₂×C₅	1	2	yes	complemented
17	C₄×A₄	C₂×C₅	1	1	yes	characteristic, complemented
18	C₂³×C₅	—	3	2	no	—
19	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
20	C₅×D₄	—	6	1	no	—
21	C₅×D₄	—	6	2	no	—
22	C₅×D₄	—	6	2	no	—
23	C₅×D₄	—	6	1	no	—
24	C₂×C₄×C₅	—	3	1	no	—
25	C₂×C₄×C₅	—	3	1	no	—
26	C₂×C₄×C₅	—	3	2	no	—
27	#121	—	3	1	no	Sylow
28	C₂×C₃×C₅	—	4	1	no	—
29	C₅×D₃	—	8	2	no	—
30	C₂×A₄	C₂²×C₅	1	1	yes	characteristic
31	S₄	—	2	2	no	—
32	D₁₂	—	4	1	no	—
33	C₂²×C₅	—	3	1	no	—
34	C₂²×C₅	—	3	1	no	—
35	C₂²×C₅	—	6	2	no	—
36	C₂²×C₅	—	6	2	no	—
37	C₂²×C₅	—	6	2	no	—
38	C₂²×C₅	D₁₂	1	1	yes	characteristic, complemented
39	C₄×C₅	—	3	1	no	—
40	C₄×C₅	—	6	2	no	—
41	C₄×C₅	S₄	1	1	yes	characteristic, complemented
42	C₂×D₄	—	3	1	no	—
43	C₂×D₄	—	3	2	no	—
44	C₂²×C₄	C₅×D₃	1	1	yes	characteristic, complemented
45	C₄⋊C₄	—	3	1	no	—
46	C₂²⋊C₄	—	3	2	no	—
47	C₃×C₅	—	4	1	no	Hall
48	C₂×D₃	—	4	2	no	—
49	A₄	C₅×D₄	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	—	12	2	no	—
54	C₂×C₅	C₂×S₄	1	1	yes	characteristic
55	C₂³	—	3	2	no	—
56	C₂³	C₂×C₅×D₃	1	1	yes	characteristic
57	D₄	—	6	1	no	—
58	D₄	—	6	2	no	—
59	D₄	—	6	2	no	—
60	D₄	—	6	1	no	—
61	C₂×C₄	—	3	1	no	—
62	C₂×C₄	—	3	1	no	—
63	C₂×C₄	—	3	2	no	—
64	C₂×C₃	—	4	1	no	—
65	D₃	—	8	2	no	—
66	C₅	C₄⋊S₄	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²	—	3	1	no	—
68	C₂²	—	3	1	no	—
69	C₂²	—	6	2	no	—
70	C₂²	—	6	2	no	—
71	C₂²	—	6	2	no	—
72	C₂²	C₅×D₁₂	1	1	yes	characteristic, complemented
73	C₄	—	3	1	no	—
74	C₄	—	6	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	—	12	2	no	—
80	C₂	C₂×C₅×S₄	1	1	yes	characteristic

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

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
7. C₄⋊S₄	C₅	direct (N × H)
10. C₂²×C₄×C₅	D₃	semidirect
14. C₅×A₄	D₄	semidirect
16. C₂×S₄	C₂×C₅	semidirect
17. C₄×A₄	C₂×C₅	semidirect
38. C₂²×C₅	D₁₂	semidirect
41. C₄×C₅	S₄	semidirect
44. C₂²×C₄	C₅×D₃	semidirect
49. A₄	C₅×D₄	semidirect
66. C₅	C₄⋊S₄	direct (N × H)
72. C₂²	C₅×D₁₂	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
C₅×#121	3
C₅×D₁₂	4
C₄⋊S₄	5

Subgroups — C5×C4⋊S4 #91939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₄⋊S₄ #91939