Subgroups — #65193

Order 320.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4050	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₄₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#3932	C₂	1	2	yes	maximal, complemented
5	D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
6	#709	—	2	2	no	—
7	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
8	Dic₂₀	C₂²	1	2	yes	—
9	D₄₀	C₂²	1	2	yes	—
10	#677	—	2	2	no	—
11	C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₅×C₁₆	C₂²	1	1	yes	characteristic
13	#508	—	5	1	no	maximal, Sylow
14	C₂×C₄×C₅	D₄	1	1	yes	characteristic
15	C₂²⋊D₅	—	4	2	no	—
16	D₂₀	—	2	2	no	—
17	C₄×D₅	—	4	2	no	—
18	Dic₁₀	—	2	2	no	—
19	C₅×C₈	D₄	1	1	yes	characteristic
20	C₅×C₈	C₂³	1	1	yes	characteristic
21	#135	—	5	2	no	—
22	Q₃₂	—	5	1	no	—
23	SD₃₂	—	5	2	no	—
24	D₁₆	—	5	1	no	—
25	C₂×C₁₆	D₅	1	1	yes	characteristic, complemented
26	C₂²×C₅	D₈	1	1	yes	characteristic
27	C₂×D₅	—	4	2	no	—
28	C₄×C₅	D₈	1	1	yes	characteristic
29	C₄×C₅	C₂×D₄	1	1	yes	characteristic
30	Dic₅	—	4	2	no	—
31	#41	—	10	2	no	—
32	Q₁₆	—	5	2	no	—
33	SD₁₆	—	10	2	no	—
34	D₈	—	5	2	no	—
35	C₂×C₈	C₂×D₅	1	1	yes	characteristic
36	C₁₆	C₂×D₅	1	1	yes	characteristic
37	C₁₆	C₂×D₅	1	1	yes	characteristic
38	C₂×C₅	—	2	1	no	—
39	C₂×C₅	C₂×D₈	1	1	yes	characteristic
40	D₅	—	8	2	no	—
41	Q₈	—	10	2	no	—
42	D₄	—	10	2	no	—
43	D₄	—	20	2	no	—
44	C₂×C₄	—	20	2	no	—
45	C₂×C₄	D₂₀	1	1	yes	characteristic
46	C₈	D₂₀	1	1	yes	characteristic
47	C₈	C₂²×D₅	1	1	yes	characteristic
48	C₅	#508	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	20	2	no	—
50	C₂²	D₄₀	1	1	yes	characteristic
51	C₄	—	20	2	no	—
52	C₄	D₄₀	1	1	yes	characteristic
53	C₄	C₂×D₂₀	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	40	2	no	—
56	C₂	C₂×D₄₀	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₀
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₂×C₅×C₁₆	C₂
socle	C₂×C₅	C₂×D₈
hypercenter	C₂×C₁₆	D₅
hypocenter	C₅	#508
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #4050	C₂	semidirect
2. C₂×C₅×C₁₆	C₂	semidirect
3. Dic₄₀	C₂	semidirect
4. #3932	C₂	semidirect
5. D₈₀	C₂	semidirect
25. C₂×C₁₆	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
#4050	2
C₂×C₅×C₁₆	2
Dic₄₀	2
#3932	2
D₈₀	2
#508	5