Subgroups — #65131

Order 320.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₂₀	C₂	1	1	yes	maximal, characteristic
2	#4015	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3953	C₂	1	2	yes	maximal
5	#3949	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₀	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅	—	2	1	no	—
8	C₂×Dic₁₀	—	2	1	no	—
9	C₂×C₅×C₈	C₂²	1	2	yes	—
10	C₄²×C₅	C₂²	1	1	yes	characteristic
11	#685	—	2	1	no	—
12	C₄⋊Dic₅	—	2	1	no	—
13	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₅	C₂²	1	2	yes	—
15	#476	—	5	1	no	maximal, Sylow
16	C₂²×D₅	—	2	1	no	—
17	C₂×C₄×C₅	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂×Dic₅	—	2	2	no	—
21	C₂×Dic₅	—	2	1	no	—
22	D₂₀	—	2	1	no	—
23	D₂₀	Q₈	1	2	yes	complemented
24	C₄×D₅	—	4	2	no	—
25	Dic₁₀	—	4	2	no	—
26	C₅×C₈	—	2	2	no	—
27	C₄⋊Q₈	—	5	1	no	—
28	C₄×D₄	—	5	1	no	—
29	#106	—	5	2	no	—
30	C₄⋊C₈	D₅	1	1	yes	characteristic, complemented
31	#102	—	5	2	no	—
32	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
33	C₂×D₅	—	2	2	no	—
34	C₂×D₅	—	4	1	no	—
35	C₂×D₅	—	4	1	no	—
36	C₄×C₅	—	2	1	no	—
37	C₄×C₅	SD₁₆	1	2	yes	—
38	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
39	C₄×C₅	#41	1	1	yes	characteristic
40	Dic₅	—	4	1	no	—
41	Dic₅	—	4	2	no	—
42	C₂×Q₈	—	10	1	no	—
43	C₂×D₄	—	5	1	no	—
44	C₂²×C₄	—	10	1	no	—
45	C₂×C₈	C₂×D₅	1	2	yes	—
46	C₄⋊C₄	—	5	2	no	—
47	C₄⋊C₄	—	5	1	no	—
48	C₄⋊C₄	—	10	1	no	—
49	C₂²⋊C₄	—	10	1	no	—
50	C₄²	C₂×D₅	1	1	yes	characteristic
51	C₂×C₅	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
53	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
54	D₅	—	4	2	no	—
55	C₂³	—	10	1	no	—
56	Q₈	—	20	2	no	—
57	D₄	—	5	2	no	—
58	D₄	—	10	1	no	—
59	C₂×C₄	—	10	1	no	—
60	C₂×C₄	—	10	2	no	—
61	C₂×C₄	—	20	2	no	—
62	C₂×C₄	D₂₀	1	1	yes	characteristic
63	C₂×C₄	D₂₀	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₅	#476	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	10	2	no	—
68	C₂²	—	20	1	no	—
69	C₂²	—	20	1	no	—
70	C₂²	C₂×D₂₀	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	20	1	no	—
73	C₄	—	20	2	no	—
74	C₄	#677	1	2	yes	—
75	C₄	#711	1	1	yes	characteristic
76	C₄	Q₈×D₅	1	1	yes	characteristic
77	C₂	—	20	2	no	—
78	C₂	#4043	1	1	yes	characteristic
79	C₂	C₂×#677	1	1	yes	characteristic
80	C₂	#4054	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊C₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×D₂₀	C₂²
hypercenter	C₄⋊C₈	D₅
hypocenter	C₅	#476
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
2. #4015	C₂	semidirect
3. C₅×C₄⋊C₈	C₂	semidirect
5. #3949	C₂	semidirect
23. D₂₀	Q₈	semidirect
30. C₄⋊C₈	D₅	semidirect
66. C₅	#476	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₂₀	2
#4015	2
C₅×C₄⋊C₈	2
#3953	2
#3949	2
#476	5