Subgroups — #65198

Order 320.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
2	#4050	C₂	1	1	yes	maximal, characteristic
3	C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₄₀	C₂	1	2	yes	maximal, complemented
5	#3932	C₂	1	2	yes	maximal, complemented
6	#709	—	2	1	no	—
7	C₂×Dic₁₀	—	2	1	no	—
8	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
9	Dic₂₀	—	2	1	no	—
10	Dic₂₀	C₂²	1	2	yes	—
11	Dic₂₀	C₂²	1	1	yes	characteristic
12	D₄₀	C₂²	1	1	yes	characteristic
13	#677	—	2	1	no	—
14	C₅×C₁₆	C₂²	1	2	yes	—
15	#510	—	5	1	no	maximal, Sylow
16	C₂×C₄×C₅	D₄	1	1	yes	characteristic
17	C₂²⋊D₅	—	4	1	no	—
18	C₂×Dic₅	—	4	1	no	—
19	D₂₀	—	2	1	no	—
20	C₄×D₅	—	4	1	no	—
21	Dic₁₀	—	2	1	no	—
22	Dic₁₀	—	2	2	no	—
23	Dic₁₀	—	4	1	no	—
24	C₅×C₈	D₄	1	1	yes	characteristic
25	C₅×C₈	C₂³	1	1	yes	characteristic
26	#135	—	5	1	no	—
27	C₂×Q₁₆	—	5	1	no	—
28	Q₃₂	—	5	2	no	—
29	SD₃₂	—	5	2	no	—
30	M₃₂	D₅	1	1	yes	characteristic, complemented
31	C₂²×C₅	D₈	1	1	yes	characteristic
32	C₂×D₅	—	4	1	no	—
33	C₄×C₅	D₈	1	1	yes	characteristic
34	C₄×C₅	C₂×D₄	1	1	yes	characteristic
35	Dic₅	—	4	2	no	—
36	Dic₅	—	4	1	no	—
37	#41	—	10	1	no	—
38	C₂×Q₈	—	10	1	no	—
39	Q₁₆	—	5	2	no	—
40	Q₁₆	—	5	1	no	—
41	Q₁₆	—	10	1	no	—
42	SD₁₆	—	10	1	no	—
43	D₈	—	5	1	no	—
44	C₂×C₈	C₂×D₅	1	1	yes	characteristic
45	C₁₆	C₂×D₅	1	2	yes	—
46	C₂×C₅	—	2	1	no	—
47	C₂×C₅	C₂×D₈	1	1	yes	characteristic
48	D₅	—	8	1	no	—
49	Q₈	—	10	1	no	—
50	Q₈	—	10	2	no	—
51	Q₈	—	20	1	no	—
52	D₄	—	10	1	no	—
53	D₄	—	20	1	no	—
54	C₂×C₄	—	20	1	no	—
55	C₂×C₄	—	20	1	no	—
56	C₂×C₄	D₂₀	1	1	yes	characteristic
57	C₈	D₂₀	1	1	yes	characteristic
58	C₈	C₂²×D₅	1	1	yes	characteristic
59	C₅	#510	1	1	yes	characteristic, complemented, Sylow
60	C₂²	—	20	1	no	—
61	C₂²	D₄₀	1	1	yes	characteristic
62	C₄	—	20	1	no	—
63	C₄	—	20	2	no	—
64	C₄	D₄₀	1	1	yes	characteristic
65	C₄	C₂×D₂₀	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	40	1	no	—
68	C₂	C₂×D₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄₀
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₅×M₃₂	C₂
socle	C₂×C₅	C₂×D₈
prime radical	#4050	C₂
hypercenter	M₃₂	D₅
hypocenter	C₅	#510
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₂₀	C₂	semidirect
3. C₅×M₃₂	C₂	semidirect
4. Dic₄₀	C₂	semidirect
5. #3932	C₂	semidirect
30. M₃₂	D₅	semidirect
59. C₅	#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀	2
#4050	2
C₅×M₃₂	2
Dic₄₀	2
#3932	2
#510	5