Subgroups — #64985

Order 320.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄₀	C₂	1	1	yes	maximal, characteristic
3	C₂×#677	C₂	1	2	yes	maximal, complemented
4	#4021	C₂	1	2	yes	maximal, complemented
5	C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀	C₂²	1	2	yes	—
7	C₂×Dic₁₀	C₂²	1	2	yes	—
8	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₄²×C₅	C₂²	1	1	yes	characteristic
11	#685	—	2	4	no	—
12	Dic₂₀	—	2	2	no	—
13	D₄₀	—	2	2	no	—
14	#677	—	2	4	no	—
15	#495	—	5	1	no	maximal, Sylow
16	C₂²×D₅	—	2	2	no	—
17	C₂×C₄×C₅	D₄	1	2	yes	—
18	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×Dic₅	—	2	2	no	—
20	D₂₀	—	2	2	no	—
21	D₂₀	—	2	2	no	—
22	Dic₁₀	—	2	2	no	—
23	Dic₁₀	—	2	2	no	—
24	C₅×C₈	D₄	1	2	yes	—
25	C₅×C₈	D₄	1	2	yes	—
26	C₂×Q₁₆	—	5	1	no	—
27	C₂×SD₁₆	—	5	2	no	—
28	C₂×D₈	—	5	1	no	—
29	#124	—	5	2	no	—
30	C₄×C₈	D₅	1	1	yes	characteristic, complemented
31	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
32	C₂×D₅	—	4	2	no	—
33	C₂×D₅	—	4	4	no	—
34	C₄×C₅	—	2	2	no	—
35	C₄×C₅	C₂×D₄	1	1	yes	characteristic
36	C₄×C₅	C₂×D₄	1	1	yes	characteristic
37	Dic₅	—	4	2	no	—
38	C₂×Q₈	—	5	2	no	—
39	C₂×D₄	—	5	2	no	—
40	Q₁₆	—	10	2	no	—
41	SD₁₆	—	10	4	no	—
42	D₈	—	10	2	no	—
43	C₂×C₈	C₂×D₅	1	1	yes	characteristic
44	C₂×C₈	C₂×D₅	1	1	yes	characteristic
45	C₂²⋊C₄	—	10	4	no	—
46	C₄²	C₂×D₅	1	1	yes	characteristic
47	C₂×C₅	#127	1	1	yes	characteristic
48	C₂×C₅	#135	1	2	yes	—
49	D₅	—	8	2	no	—
50	C₂³	—	10	2	no	—
51	Q₈	—	10	2	no	—
52	Q₈	—	10	2	no	—
53	D₄	—	10	2	no	—
54	D₄	—	10	2	no	—
55	C₂×C₄	—	10	2	no	—
56	C₂×C₄	D₂₀	1	2	yes	—
57	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
58	C₈	D₂₀	1	2	yes	—
59	C₈	D₂₀	1	2	yes	—
60	C₅	#495	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	20	2	no	—
62	C₂²	—	20	4	no	—
63	C₂²	C₂×D₂₀	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	20	2	no	—
66	C₄	C₂×D₂₀	1	1	yes	characteristic
67	C₄	C₂×D₂₀	1	1	yes	characteristic
68	C₂	—	40	2	no	—
69	C₂	#4020	1	1	yes	characteristic
70	C₂	#4050	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×C₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×D₄₀	C₂
hypercenter	C₄×C₈	D₅
hypocenter	C₅	#495
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₂₀	C₂	semidirect
3. C₂×#677	C₂	semidirect
4. #4021	C₂	semidirect
5. C₄×C₅×C₈	C₂	semidirect
30. C₄×C₈	D₅	semidirect
60. C₅	#495	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀	2
C₂×D₄₀	2
C₂×#677	2
#4021	2
C₄×C₅×C₈	2
#495	5