Subgroups — #10598095

Order 640.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₀	C₂	1	1	yes	maximal, characteristic
2	C₅×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#64726	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₀	—	2	1	no	—
5	C₂×C₅×C₁₆	C₂²	1	1	yes	characteristic
6	#3951	—	2	1	no	—
7	D₈₀	—	2	1	no	—
8	D₈₀	C₄	1	2	yes	—
9	C₅×C₃₂	—	2	1	no	—
10	#1419	—	5	1	no	maximal, Sylow
11	C₂×D₂₀	—	4	1	no	—
12	C₂×C₅×C₈	D₄	1	1	yes	characteristic
13	#681	—	4	1	no	—
14	D₄₀	—	2	2	no	—
15	D₄₀	—	4	1	no	—
16	C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₅×C₁₆	D₄	1	1	yes	characteristic
18	C₂×D₁₆	—	5	1	no	—
19	M₆₄	D₅	1	1	yes	characteristic, complemented
20	#368	—	5	1	no	—
21	C₂²×D₅	—	8	1	no	—
22	C₂×C₄×C₅	D₈	1	1	yes	characteristic
23	D₂₀	—	4	2	no	—
24	D₂₀	—	8	1	no	—
25	C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
26	C₅×C₈	SD₁₆	1	1	yes	characteristic
27	#168	—	8	1	no	—
28	C₂×D₈	—	10	1	no	—
29	D₁₆	—	5	2	no	—
30	D₁₆	—	10	1	no	—
31	C₂×C₁₆	C₂×D₅	1	1	yes	characteristic
32	#108	—	10	1	no	—
33	C₃₂	—	2	1	no	—
34	C₂²×C₅	SD₃₂	1	1	yes	characteristic
35	C₂×D₅	—	8	2	no	—
36	C₂×D₅	—	16	2	no	—
37	C₄×C₅	#102	1	1	yes	characteristic
38	C₄×C₅	D₁₆	1	1	yes	characteristic
39	C₂×D₄	—	20	1	no	—
40	D₈	—	10	2	no	—
41	D₈	—	20	1	no	—
42	M₁₆	—	20	1	no	—
43	C₂×C₈	D₂₀	1	1	yes	characteristic
44	C₁₆	C₄×D₅	1	1	yes	characteristic
45	C₁₆	C₂²⋊D₅	1	1	yes	characteristic
46	C₂×C₅	—	2	1	no	—
47	C₂×C₅	#357	1	1	yes	characteristic
48	D₅	—	16	2	no	—
49	C₂³	—	40	1	no	—
50	D₄	—	20	2	no	—
51	D₄	—	40	1	no	—
52	C₂×C₄	D₄₀	1	1	yes	characteristic
53	C₈	—	40	1	no	—
54	C₈	#677	1	1	yes	characteristic
55	C₈	#685	1	1	yes	characteristic
56	C₅	#1419	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	40	2	no	—
58	C₂²	—	80	2	no	—
59	C₂²	#3932	1	1	yes	characteristic
60	C₄	D₈₀	1	1	yes	characteristic
61	C₄	#3953	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	80	2	no	—
64	C₂	#64729	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#64729
commutator	C₅×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₅
Fitting	C₅×M₆₄	C₂
socle	C₂×C₅	#357
prime radical	C₂×D₈₀	C₂
cosocle	C₂×C₅×C₁₆	C₂²
hypercenter	M₆₄	D₅
hypocenter	C₅	#1419
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₅×M₆₄	C₂	semidirect
3. #64726	C₂	semidirect
19. M₆₄	D₅	semidirect
56. C₅	#1419	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈₀	2
C₅×M₆₄	2
#64726	2
#1419	5