Subgroups — #65161

Order 320.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₀	C₂	1	1	yes	maximal, characteristic
2	#4039	C₂	1	2	yes	maximal, complemented
3	C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#3941	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₀	C₂²	1	2	yes	—
7	C₂×C₄×D₅	—	2	2	no	—
8	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	#685	—	2	2	no	—
11	C₄×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×#168	C₂²	1	1	yes	characteristic
13	D₄₀	C₄	1	4	yes	complemented
14	C₄×D₈	—	5	1	no	maximal, Sylow
15	C₂²×D₅	—	2	2	no	—
16	C₂×C₄×C₅	—	2	2	no	—
17	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₂×Dic₅	D₄	1	1	yes	characteristic
19	C₂×Dic₅	D₄	1	1	yes	characteristic
20	D₂₀	—	2	2	no	—
21	D₂₀	C₂×C₄	1	4	yes	complemented
22	C₄×D₅	—	4	4	no	—
23	C₅×C₈	C₂×C₄	1	2	yes	complemented
24	#168	—	2	1	no	—
25	C₂×D₈	—	5	1	no	—
26	C₄×D₄	—	5	2	no	—
27	#107	D₅	1	1	yes	characteristic, complemented
28	#102	—	5	2	no	—
29	C₄×C₈	—	5	1	no	—
30	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
31	C₂×D₅	—	2	4	no	—
32	C₂×D₅	—	4	2	no	—
33	C₂×D₅	—	4	2	no	—
34	C₄×C₅	—	4	2	no	—
35	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₅	#41	1	1	yes	characteristic
37	Dic₅	—	2	1	no	—
38	Dic₅	D₈	1	2	yes	complemented
39	C₂×D₄	—	5	2	no	—
40	C₂²×C₄	—	10	2	no	—
41	D₈	—	5	4	no	—
42	C₂×C₈	—	5	1	no	—
43	C₂×C₈	C₂×D₅	1	1	yes	characteristic
44	C₄⋊C₄	C₂×D₅	1	2	yes	—
45	C₂²⋊C₄	—	10	2	no	—
46	C₄²	—	5	1	no	—
47	C₂×C₅	C₄×D₄	1	1	yes	characteristic
48	C₂×C₅	C₂×D₈	1	1	yes	characteristic
49	C₂×C₅	#135	1	1	yes	characteristic
50	D₅	—	4	4	no	—
51	C₂³	—	10	2	no	—
52	D₄	—	5	4	no	—
53	D₄	—	10	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	5	1	no	—
56	C₂×C₄	—	5	1	no	—
57	C₂×C₄	—	20	4	no	—
58	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
59	C₈	—	10	1	no	—
60	C₈	C₄×D₅	1	2	yes	complemented
61	C₅	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	10	4	no	—
63	C₂²	—	20	2	no	—
64	C₂²	—	20	2	no	—
65	C₂²	D₄×D₅	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	5	2	no	—
68	C₄	—	10	1	no	—
69	C₄	C₂×C₄×D₅	1	1	yes	characteristic
70	C₄	#713	1	1	yes	characteristic
71	C₂	—	20	4	no	—
72	C₂	#4039	1	1	yes	characteristic
73	C₂	D₅×D₈	1	1	yes	characteristic
74	C₂	#4065	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#107	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×D₄₀	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#107	D₅
hypocenter	C₅	C₄×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
2. #4039	C₂	semidirect
3. C₅×#107	C₂	semidirect
4. C₈×Dic₅	C₂	semidirect
5. #3941	C₂	semidirect
13. D₄₀	C₄	semidirect
21. D₂₀	C₂×C₄	semidirect
23. C₅×C₈	C₂×C₄	semidirect
27. #107	D₅	semidirect
38. Dic₅	D₈	semidirect
60. C₈	C₄×D₅	semidirect
61. C₅	C₄×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₀	2
#4039	2
C₅×#107	2
C₈×Dic₅	2
#3941	2
C₄×D₈	5