Subgroups — #65207

Order 320.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4065	C₂	1	1	yes	maximal, characteristic
2	D₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#3961	C₂	1	1	yes	maximal, characteristic, complemented
5	#3960	C₂	1	1	yes	maximal, characteristic, complemented
6	#3932	C₂	1	1	yes	maximal, characteristic, complemented
7	#3930	C₂	1	1	yes	maximal, characteristic, complemented
8	#713	—	2	1	no	—
9	Q₈×D₅	—	2	1	no	—
10	C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₅×Q₁₆	C₂²	1	1	yes	characteristic
12	#689	—	2	1	no	—
13	#688	—	2	1	no	—
14	Dic₂₀	C₂²	1	1	yes	characteristic
15	D₄₀	C₂²	1	1	yes	characteristic
16	C₈×D₅	C₂²	1	1	yes	characteristic
17	C₅×C₁₆	C₂²	1	1	yes	characteristic
18	#672	C₂²	1	1	yes	characteristic
19	#510	—	5	1	no	maximal, Sylow
20	C₅×Q₈	—	2	1	no	—
21	C₅×Q₈	—	2	1	no	—
22	D₂₀	—	2	1	no	—
23	D₂₀	—	4	1	no	—
24	C₄×D₅	—	4	1	no	—
25	C₄×D₅	—	4	1	no	—
26	C₄×D₅	D₄	1	1	yes	characteristic
27	Dic₁₀	—	2	1	no	—
28	Dic₁₀	—	4	1	no	—
29	C₅×C₈	C₂³	1	1	yes	characteristic
30	#168	D₄	1	1	yes	characteristic
31	#135	—	5	1	no	—
32	C₂×Q₁₆	—	5	1	no	—
33	Q₃₂	—	5	1	no	—
34	Q₃₂	D₅	1	1	yes	characteristic, complemented
35	SD₃₂	—	5	1	no	—
36	SD₃₂	—	5	1	no	—
37	M₃₂	—	5	1	no	—
38	C₂×D₅	—	4	1	no	—
39	C₂×D₅	D₈	1	1	yes	characteristic
40	C₄×C₅	—	4	1	no	—
41	C₄×C₅	—	4	1	no	—
42	C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	Dic₅	—	4	1	no	—
44	Dic₅	D₈	1	1	yes	characteristic
45	#41	—	10	1	no	—
46	C₂×Q₈	—	10	1	no	—
47	Q₁₆	—	5	1	no	—
48	Q₁₆	—	10	1	no	—
49	Q₁₆	C₂×D₅	1	1	yes	characteristic
50	Q₁₆	C₂×D₅	1	1	yes	characteristic
51	SD₁₆	—	10	1	no	—
52	D₈	—	5	1	no	—
53	C₂×C₈	—	5	1	no	—
54	C₁₆	—	5	1	no	—
55	C₁₆	C₂×D₅	1	1	yes	characteristic
56	C₂×C₅	C₂×D₈	1	1	yes	characteristic
57	D₅	—	2	1	no	—
58	D₅	—	8	1	no	—
59	Q₈	—	2	1	no	—
60	Q₈	—	2	1	no	—
61	Q₈	—	10	1	no	—
62	Q₈	—	20	1	no	—
63	D₄	—	10	1	no	—
64	D₄	—	20	1	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	20	1	no	—
67	C₂×C₄	—	20	1	no	—
68	C₈	—	5	1	no	—
69	C₈	C₂²×D₅	1	1	yes	characteristic
70	C₅	#510	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	5	1	no	—
72	C₂²	—	20	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	5	1	no	—
76	C₄	—	20	1	no	—
77	C₄	D₄×D₅	1	1	yes	characteristic
78	C₂	—	10	1	no	—
79	C₂	—	40	1	no	—
80	C₂	D₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₅×D₈
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₅×Q₃₂	C₂
socle	C₂×C₅	C₂×D₈
prime radical	#4065	C₂
hypercenter	Q₃₂	D₅
hypocenter	C₅	#510
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. D₅×Q₁₆	C₂	semidirect
3. C₅×Q₃₂	C₂	semidirect
4. #3961	C₂	semidirect
5. #3960	C₂	semidirect
6. #3932	C₂	semidirect
7. #3930	C₂	semidirect
34. Q₃₂	D₅	semidirect
70. C₅	#510	semidirect

Maximal subgroups

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

subgroup	index
#4065	2
D₅×Q₁₆	2
C₅×Q₃₂	2
#3961	2
#3960	2
#3932	2
#3930	2
#510	5