Subgroups — C₂²×C₅×S₄ #92122

Order 480.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×S₄	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×D₄	—	3	1	no	maximal, Hall
4	C₂²×C₅×D₃	—	4	1	no	maximal
5	C₂×C₅×A₄	C₂²	1	3	yes	complemented
6	C₅×S₄	C₂²	1	4	yes	direct product factor
7	C₂²×S₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂⁴×C₅	—	3	1	no	—
9	C₂⁴×C₅	D₃	1	1	yes	characteristic, complemented
10	C₂×C₅×D₄	—	3	6	no	—
11	C₂×C₅×D₄	—	3	6	no	—
12	C₂²×C₄×C₅	—	3	1	no	—
13	C₂²×C₃×C₅	—	4	1	no	—
14	C₂×C₅×D₃	—	4	6	no	—
15	C₅×A₄	C₂³	1	1	yes	characteristic, complemented
16	C₂²×A₄	C₂×C₅	1	1	yes	characteristic, complemented
17	C₂×S₄	C₂×C₅	1	6	yes	direct product factor
18	C₂³×C₅	—	3	1	no	—
19	C₂³×C₅	—	3	3	no	—
20	C₂³×C₅	—	3	6	no	—
21	C₂³×C₅	—	6	3	no	—
22	C₂³×C₅	—	6	1	no	—
23	C₂³×C₅	C₂×D₃	1	3	yes	complemented
24	C₅×D₄	—	3	12	no	—
25	C₅×D₄	—	3	4	no	—
26	C₂×C₄×C₅	—	3	6	no	—
27	C₂²×D₄	—	3	1	no	Sylow
28	C₂×C₃×C₅	—	4	3	no	—
29	C₅×D₃	—	4	4	no	—
30	C₂²×D₃	—	4	1	no	—
31	C₂×A₄	C₂²×C₅	1	3	yes	complemented
32	S₄	C₂²×C₅	1	4	yes	direct product factor
33	C₂²×C₅	—	3	3	no	—
34	C₂²×C₅	—	3	3	no	—
35	C₂²×C₅	—	3	3	no	—
36	C₂²×C₅	—	3	4	no	—
37	C₂²×C₅	—	6	1	no	—
38	C₂²×C₅	—	6	6	no	—
39	C₂²×C₅	—	6	6	no	—
40	C₂²×C₅	S₄	1	1	yes	characteristic, direct product factor
41	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic, complemented
42	C₄×C₅	—	3	4	no	—
43	C₂⁴	—	3	1	no	—
44	C₂⁴	C₅×D₃	1	1	yes	characteristic, complemented
45	C₂×D₄	—	3	6	no	—
46	C₂×D₄	—	3	6	no	—
47	C₂²×C₄	—	3	1	no	—
48	C₃×C₅	—	4	1	no	Hall
49	C₂²×C₃	—	4	1	no	—
50	C₂×D₃	—	4	6	no	—
51	A₄	C₂³×C₅	1	1	yes	characteristic, complemented
52	C₂×C₅	—	3	1	no	—
53	C₂×C₅	—	3	3	no	—
54	C₂×C₅	—	6	4	no	—
55	C₂×C₅	C₂×S₄	1	3	yes	direct product factor
56	C₂³	—	3	1	no	—
57	C₂³	—	3	3	no	—
58	C₂³	—	3	6	no	—
59	C₂³	—	6	3	no	—
60	C₂³	—	6	1	no	—
61	C₂³	C₂×C₅×D₃	1	3	yes	complemented
62	D₄	—	3	12	no	—
63	D₄	—	3	4	no	—
64	C₂×C₄	—	3	6	no	—
65	C₂×C₃	—	4	3	no	—
66	D₃	—	4	4	no	—
67	C₅	C₂²×S₄	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²	—	3	3	no	—
69	C₂²	—	3	3	no	—
70	C₂²	—	3	3	no	—
71	C₂²	—	3	4	no	—
72	C₂²	—	6	1	no	—
73	C₂²	—	6	6	no	—
74	C₂²	—	6	6	no	—
75	C₂²	C₅×S₄	1	1	yes	characteristic, direct product factor
76	C₂²	C₂²×C₅×D₃	1	1	yes	characteristic, complemented
77	C₄	—	3	4	no	—
78	C₃	—	4	1	no	Sylow
79	C₂	—	3	1	no	—
80	C₂	—	3	3	no	—
81	C₂	—	6	4	no	—
82	C₂	C₂×C₅×S₄	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅	S₄
commutator, cosocle, hypocenter	A₄	C₂³×C₅
Frattini	C₁	G
Fitting, socle	C₂⁴×C₅	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂²×C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×A₄	C₂	semidirect
2. C₂×C₅×S₄	C₂	direct (N × H)
5. C₂×C₅×A₄	C₂²	semidirect
6. C₅×S₄	C₂²	direct (N × H)
7. C₂²×S₄	C₅	direct (N × H)
9. C₂⁴×C₅	D₃	semidirect
15. C₅×A₄	C₂³	semidirect
16. C₂²×A₄	C₂×C₅	semidirect
17. C₂×S₄	C₂×C₅	direct (N × H)
23. C₂³×C₅	C₂×D₃	semidirect
31. C₂×A₄	C₂²×C₅	semidirect
32. S₄	C₂²×C₅	direct (N × H)
40. C₂²×C₅	S₄	direct (N × H)
41. C₂²×C₅	C₂²×D₃	semidirect
44. C₂⁴	C₅×D₃	semidirect
51. A₄	C₂³×C₅	semidirect
55. C₂×C₅	C₂×S₄	direct (N × H)
61. C₂³	C₂×C₅×D₃	semidirect
67. C₅	C₂²×S₄	direct (N × H)
75. C₂²	C₅×S₄	direct (N × H)
76. C₂²	C₂²×C₅×D₃	semidirect
82. C₂	C₂×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
C₂²×C₅×A₄	2
C₂×C₅×S₄	2
C₂²×C₅×D₄	3
C₂²×C₅×D₃	4
C₂²×S₄	5

Subgroups — C22×C5×S4 #92122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×S₄ #92122