Subgroups — #64977

Order 320.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#676	C₂	1	2	yes	maximal, complemented
2	C₄×D₂₀	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3952	C₂	1	2	yes	maximal, complemented
5	C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀	C₄	1	1	yes	characteristic
7	C₂×C₄×D₅	C₂²	1	2	yes	—
8	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₄²×C₅	C₂²	1	1	yes	characteristic
11	#685	C₄	1	2	yes	—
12	C₄⋊Dic₅	C₄	1	1	yes	characteristic
13	C₂×#168	C₂²	1	2	yes	—
14	#676	—	2	4	no	—
15	#436	—	5	1	no	maximal, Sylow
16	C₂²×D₅	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂×Dic₅	C₂×C₄	1	2	yes	—
21	D₂₀	—	2	2	no	—
22	C₄×D₅	—	2	2	no	—
23	C₄×D₅	—	2	2	no	—
24	C₅×C₈	—	2	1	no	—
25	C₅×C₈	D₄	1	2	yes	complemented
26	#168	—	2	2	no	—
27	C₂×M₁₆	—	5	2	no	—
28	C₄×D₄	—	5	1	no	—
29	C₄⋊C₈	—	5	1	no	—
30	C₂²⋊C₈	—	5	2	no	—
31	C₄×C₈	D₅	1	1	yes	characteristic, complemented
32	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₅	—	2	2	no	—
34	C₂×D₅	—	2	2	no	—
35	C₂×D₅	—	2	2	no	—
36	C₄×C₅	—	2	1	no	—
37	C₄×C₅	M₁₆	1	2	yes	complemented
38	C₄×C₅	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅	#41	1	1	yes	characteristic
40	Dic₅	—	2	2	no	—
41	C₂×D₄	—	5	1	no	—
42	C₂²×C₄	—	5	2	no	—
43	M₁₆	—	10	4	no	—
44	C₂×C₈	—	5	2	no	—
45	C₂×C₈	C₂×D₅	1	1	yes	characteristic
46	C₂×C₈	C₂×D₅	1	1	yes	characteristic
47	C₄⋊C₄	—	5	1	no	—
48	C₂²⋊C₄	—	5	2	no	—
49	C₄²	C₂×D₅	1	1	yes	characteristic
50	C₂×C₅	C₄×D₄	1	1	yes	characteristic
51	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₅	#131	1	1	yes	characteristic
53	D₅	—	4	2	no	—
54	C₂³	—	5	2	no	—
55	D₄	—	10	2	no	—
56	C₂×C₄	—	5	2	no	—
57	C₂×C₄	—	10	2	no	—
58	C₂×C₄	—	10	2	no	—
59	C₂×C₄	C₄×D₅	1	1	yes	characteristic
60	C₂×C₄	C₄×D₅	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	—	10	2	no	—
64	C₈	D₂₀	1	2	yes	complemented
65	C₅	#436	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	10	2	no	—
67	C₂²	—	10	2	no	—
68	C₂²	—	10	2	no	—
69	C₂²	C₂×C₄×D₅	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	10	2	no	—
72	C₄	#676	1	2	yes	complemented
73	C₄	C₂×D₂₀	1	1	yes	characteristic
74	C₄	#709	1	1	yes	characteristic
75	C₂	—	20	2	no	—
76	C₂	C₄×D₂₀	1	1	yes	characteristic
77	C₂	C₂×#676	1	1	yes	characteristic
78	C₂	#4047	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₄×C₅×C₈	C₂
prime radical	C₂×D₂₀	C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₄×C₈	D₅
hypocenter	C₅	#436
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#676	C₂	semidirect
3. C₄×C₅×C₈	C₂	semidirect
4. #3952	C₂	semidirect
5. C₄⋊#168	C₂	semidirect
25. C₅×C₈	D₄	semidirect
31. C₄×C₈	D₅	semidirect
37. C₄×C₅	M₁₆	semidirect
64. C₈	D₂₀	semidirect
65. C₅	#436	semidirect
72. C₄	#676	semidirect

Maximal subgroups

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

subgroup	index
C₂×#676	2
C₄×D₂₀	2
C₄×C₅×C₈	2
#3952	2
C₄⋊#168	2
#436	5