Subgroups — #65362

Order 320.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#686	C₂	1	4	yes	maximal, complemented
3	#4020	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×D₄	—	2	2	no	—
6	C₂×C₅×D₄	C₂²	1	2	yes	—
7	C₂×D₂₀	—	2	2	no	—
8	C₂×D₂₀	C₂²	1	2	yes	—
9	C₄²×C₅	C₂²	1	1	yes	characteristic
10	#686	—	2	8	no	—
11	C₂×#168	C₂²	1	2	yes	—
12	#493	—	5	1	no	maximal, Sylow
13	C₂³×C₅	—	2	2	no	—
14	C₂²×D₅	—	2	2	no	—
15	C₅×D₄	—	2	4	no	—
16	C₅×D₄	—	4	4	no	—
17	C₂×C₄×C₅	D₄	1	2	yes	—
18	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
19	D₂₀	—	2	4	no	—
20	D₂₀	—	4	4	no	—
21	#168	D₄	1	4	yes	complemented
22	C₂×D₈	—	5	4	no	—
23	#127	—	5	1	no	—
24	#127	D₅	1	1	yes	characteristic, complemented
25	C₄×C₈	—	5	1	no	—
26	C₂²×C₅	—	4	2	no	—
27	C₂²×C₅	—	4	4	no	—
28	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
29	C₂×D₅	—	4	2	no	—
30	C₂×D₅	—	4	4	no	—
31	C₄×C₅	D₈	1	4	yes	complemented
32	C₄×C₅	C₂×D₄	1	2	yes	—
33	C₂×D₄	—	2	2	no	—
34	C₂×D₄	—	5	2	no	—
35	C₂×D₄	—	10	2	no	—
36	C₂×D₄	C₂×D₅	1	2	yes	—
37	D₈	—	10	8	no	—
38	C₂×C₈	—	5	2	no	—
39	C₄²	C₂×D₅	1	1	yes	characteristic
40	C₂×C₅	—	8	2	no	—
41	C₂×C₅	#127	1	1	yes	characteristic
42	C₂×C₅	C₂×D₈	1	2	yes	—
43	D₅	—	8	2	no	—
44	C₂³	—	2	2	no	—
45	C₂³	—	10	2	no	—
46	D₄	—	2	4	no	—
47	D₄	—	4	4	no	—
48	D₄	—	10	4	no	—
49	D₄	—	20	4	no	—
50	C₂×C₄	C₂²⋊D₅	1	2	yes	—
51	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
52	C₈	—	5	4	no	—
53	C₅	#493	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	4	4	no	—
55	C₂²	—	4	2	no	—
56	C₂²	—	20	4	no	—
57	C₂²	—	20	2	no	—
58	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
59	C₄	#686	1	4	yes	complemented
60	C₄	D₄×D₅	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	40	2	no	—
63	C₂	C₂×#686	1	2	yes	—
64	C₂	#4086	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#127	C₂
socle	C₂²×C₅	C₂×D₄
hypercenter	#127	D₅
hypocenter	C₅	#493
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₅×#127	C₂	semidirect
2. C₂×#686	C₂	semidirect
3. #4020	C₂	semidirect
4. C₄×#168	C₂	semidirect
21. #168	D₄	semidirect
24. #127	D₅	semidirect
31. C₄×C₅	D₈	semidirect
53. C₅	#493	semidirect
59. C₄	#686	semidirect

Maximal subgroups

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

subgroup	index
C₅×#127	2
C₂×#686	2
#4020	2
C₄×#168	2
#493	5