Subgroups — #10599853

Order 640.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₀	C₂	1	4	yes	maximal, complemented
2	#64984	C₂	1	2	yes	maximal, complemented
3	C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₀	—	2	2	no	—
5	C₂×D₄₀	C₂²	1	4	yes	—
6	#4020	—	2	2	no	—
7	C₂×C₅×C₁₆	C₂²	1	2	yes	—
8	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	D₈₀	—	2	8	no	—
10	#2246	—	5	1	no	maximal, Sylow
11	C₂×D₂₀	—	2	4	no	—
12	C₂×D₂₀	—	4	4	no	—
13	C₂×C₅×C₈	D₄	1	1	yes	characteristic
14	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
15	C₄²×C₅	D₄	1	1	yes	characteristic
16	D₄₀	—	2	8	no	—
17	D₄₀	—	4	4	no	—
18	C₅×C₁₆	D₄	1	4	yes	complemented
19	C₂×D₁₆	—	5	4	no	—
20	#493	—	5	2	no	—
21	C₄×C₁₆	D₅	1	1	yes	characteristic, complemented
22	C₂²×D₅	—	4	4	no	—
23	C₂×C₄×C₅	D₈	1	2	yes	—
24	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
25	D₂₀	—	4	4	no	—
26	D₂₀	—	4	4	no	—
27	D₂₀	—	8	8	no	—
28	C₅×C₈	D₈	1	2	yes	—
29	C₅×C₈	C₂×D₄	1	2	yes	—
30	C₂×D₈	—	5	4	no	—
31	C₂×D₈	—	10	2	no	—
32	#127	—	10	2	no	—
33	D₁₆	—	10	8	no	—
34	C₂×C₁₆	C₂×D₅	1	2	yes	—
35	C₄×C₈	C₂×D₅	1	1	yes	characteristic
36	C₂²×C₅	C₂×D₈	1	1	yes	characteristic
37	C₂×D₅	—	8	4	no	—
38	C₂×D₅	—	8	8	no	—
39	C₄×C₅	D₁₆	1	4	yes	complemented
40	C₄×C₅	#127	1	1	yes	characteristic
41	C₄×C₅	C₂×D₈	1	1	yes	characteristic
42	C₂×D₄	—	10	4	no	—
43	C₂×D₄	—	20	4	no	—
44	D₈	—	10	8	no	—
45	D₈	—	20	4	no	—
46	C₂×C₈	D₂₀	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
48	C₄²	D₂₀	1	1	yes	characteristic
49	C₁₆	D₂₀	1	4	yes	complemented
50	C₂×C₅	#493	1	1	yes	characteristic
51	C₂×C₅	C₂×D₁₆	1	2	yes	—
52	D₅	—	16	4	no	—
53	C₂³	—	20	4	no	—
54	D₄	—	20	4	no	—
55	D₄	—	20	4	no	—
56	D₄	—	40	8	no	—
57	C₂×C₄	D₄₀	1	2	yes	—
58	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
59	C₈	D₄₀	1	2	yes	—
60	C₈	C₂×D₂₀	1	2	yes	—
61	C₅	#2246	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	40	8	no	—
63	C₂²	—	40	4	no	—
64	C₂²	C₂×D₄₀	1	1	yes	characteristic
65	C₄	D₈₀	1	4	yes	complemented
66	C₄	#4020	1	1	yes	characteristic
67	C₄	C₂×D₄₀	1	1	yes	characteristic
68	C₂	—	80	4	no	—
69	C₂	#64984	1	1	yes	characteristic
70	C₂	C₂×D₈₀	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₀
commutator, cosocle	C₂×C₅×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₅
Fitting	C₄×C₅×C₁₆	C₂
socle	C₂²×C₅	C₂×D₈
hypercenter	C₄×C₁₆	D₅
hypocenter	C₅	#2246
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₈₀	C₂	semidirect
2. #64984	C₂	semidirect
3. C₄×C₅×C₁₆	C₂	semidirect
18. C₅×C₁₆	D₄	semidirect
21. C₄×C₁₆	D₅	semidirect
39. C₄×C₅	D₁₆	semidirect
49. C₁₆	D₂₀	semidirect
61. C₅	#2246	semidirect
65. C₄	D₈₀	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
#64984	2
C₄×C₅×C₁₆	2
#2246	5