Subgroups — #65208

Order 320.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4065	C₂	1	2	yes	maximal, complemented
2	C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#3960	C₂	1	2	yes	maximal, complemented
4	D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#713	—	2	2	no	—
7	C₅×Q₁₆	C₂²	1	2	yes	—
8	#688	—	2	2	no	—
9	D₄₀	C₂²	1	2	yes	—
10	C₈×D₅	C₂²	1	1	yes	characteristic
11	C₅×C₁₆	C₂²	1	1	yes	characteristic
12	#672	C₂²	1	1	yes	characteristic
13	#508	—	5	1	no	maximal, Sylow
14	C₅×Q₈	—	2	2	no	—
15	D₂₀	—	2	2	no	—
16	D₂₀	—	4	2	no	—
17	C₄×D₅	—	4	2	no	—
18	C₄×D₅	D₄	1	1	yes	characteristic
19	C₅×C₈	C₂³	1	1	yes	characteristic
20	#168	D₄	1	1	yes	characteristic
21	#135	—	5	2	no	—
22	Q₃₂	D₅	1	1	yes	characteristic, complemented
23	SD₃₂	—	5	2	no	—
24	D₁₆	—	5	1	no	—
25	C₂×C₁₆	—	5	1	no	—
26	C₂×D₅	—	4	2	no	—
27	C₂×D₅	D₈	1	1	yes	characteristic
28	C₄×C₅	—	4	2	no	—
29	C₄×C₅	C₂×D₄	1	1	yes	characteristic
30	Dic₅	D₈	1	1	yes	characteristic
31	#41	—	10	2	no	—
32	Q₁₆	C₂×D₅	1	2	yes	—
33	SD₁₆	—	10	2	no	—
34	D₈	—	5	2	no	—
35	C₂×C₈	—	5	1	no	—
36	C₁₆	—	5	1	no	—
37	C₁₆	C₂×D₅	1	1	yes	characteristic
38	C₂×C₅	C₂×D₈	1	1	yes	characteristic
39	D₅	—	2	1	no	—
40	D₅	—	8	2	no	—
41	Q₈	—	2	2	no	—
42	D₄	—	10	2	no	—
43	D₄	—	20	2	no	—
44	C₂×C₄	—	5	1	no	—
45	C₂×C₄	—	20	2	no	—
46	C₈	—	5	1	no	—
47	C₈	C₂²×D₅	1	1	yes	characteristic
48	C₅	#508	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	5	1	no	—
50	C₂²	—	20	2	no	—
51	C₄	—	4	2	no	—
52	C₄	—	5	1	no	—
53	C₄	D₄×D₅	1	1	yes	characteristic
54	C₂	—	10	1	no	—
55	C₂	—	40	2	no	—
56	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
hypercenter	Q₃₂	D₅
hypocenter	C₅	#508
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
1. #4065	C₂	semidirect
2. C₅×Q₃₂	C₂	semidirect
3. #3960	C₂	semidirect
4. D₈₀	C₂	semidirect
5. D₅×C₁₆	C₂	semidirect
22. Q₃₂	D₅	semidirect
48. C₅	#508	semidirect

Maximal subgroups

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

subgroup	index
#4065	2
C₅×Q₃₂	2
#3960	2
D₈₀	2
D₅×C₁₆	2
#508	5