Subgroups — #65154

Order 320.

130 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, complemented
2	C₂×#676	C₂	1	1	yes	maximal, characteristic, complemented
3	#4041	C₂	1	2	yes	maximal, complemented
4	C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	#3941	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₀	—	2	2	no	—
7	C₂×D₂₀	C₂²	1	2	yes	—
8	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	#685	—	2	2	no	—
12	C₂×#168	C₂²	1	1	yes	characteristic
13	D₄₀	—	2	2	no	—
14	#676	—	2	2	no	—
15	#469	—	5	1	no	maximal, Sylow
16	C₂²×D₅	—	2	2	no	—
17	C₂²×D₅	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅	—	2	2	no	—
19	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂×Dic₅	D₄	1	1	yes	characteristic
21	D₂₀	—	2	2	no	—
22	D₂₀	—	2	2	no	—
23	D₂₀	—	4	4	no	—
24	C₄×D₅	—	2	1	no	—
25	C₄×D₅	—	2	1	no	—
26	C₅×C₈	D₄	1	2	yes	complemented
27	#168	—	2	1	no	—
28	C₂×D₈	—	5	1	no	—
29	C₂×M₁₆	—	5	1	no	—
30	#121	—	5	2	no	—
31	#106	D₅	1	1	yes	characteristic, complemented
32	#102	—	5	2	no	—
33	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
34	C₂×D₅	—	2	1	no	—
35	C₂×D₅	—	2	2	no	—
36	C₂×D₅	—	4	2	no	—
37	C₂×D₅	—	4	2	no	—
38	C₂×D₅	—	4	2	no	—
39	C₄×C₅	—	4	2	no	—
40	C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₄×C₅	#41	1	1	yes	characteristic
42	Dic₅	—	2	1	no	—
43	C₂×D₄	—	5	2	no	—
44	C₂×D₄	—	10	2	no	—
45	C₂²×C₄	—	5	1	no	—
46	D₈	—	10	2	no	—
47	M₁₆	—	10	2	no	—
48	C₂×C₈	—	5	1	no	—
49	C₂×C₈	C₂×D₅	1	1	yes	characteristic
50	C₄⋊C₄	C₂×D₅	1	2	yes	—
51	C₂²⋊C₄	—	10	2	no	—
52	C₂×C₅	#121	1	1	yes	characteristic
53	C₂×C₅	AGL(1,ℤ/8)	1	2	yes	—
54	D₅	—	4	1	no	—
55	D₅	—	8	2	no	—
56	C₂³	—	5	1	no	—
57	C₂³	—	10	2	no	—
58	D₄	—	10	2	no	—
59	D₄	—	10	2	no	—
60	D₄	—	20	4	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	5	1	no	—
63	C₂×C₄	—	10	1	no	—
64	C₂×C₄	—	10	1	no	—
65	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
66	C₈	—	10	1	no	—
67	C₈	D₂₀	1	2	yes	complemented
68	C₅	#469	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	10	1	no	—
70	C₂²	—	10	2	no	—
71	C₂²	—	20	2	no	—
72	C₂²	—	20	2	no	—
73	C₂²	—	20	2	no	—
74	C₂²	D₄×D₅	1	1	yes	characteristic
75	C₄	—	4	2	no	—
76	C₄	—	10	1	no	—
77	C₄	C₂×D₂₀	1	1	yes	characteristic
78	C₄	#713	1	1	yes	characteristic
79	C₂	—	20	1	no	—
80	C₂	—	40	2	no	—
81	C₂	#4041	1	1	yes	characteristic
82	C₂	#4060	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#106	C₂
socle	C₂²×C₅	C₂×D₄
hypercenter	#106	D₅
hypocenter	C₅	#469
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
1. C₂×D₄₀	C₂	semidirect
2. C₂×#676	C₂	semidirect
3. #4041	C₂	semidirect
4. C₅×#106	C₂	semidirect
5. #3941	C₂	semidirect
26. C₅×C₈	D₄	semidirect
31. #106	D₅	semidirect
67. C₈	D₂₀	semidirect
68. C₅	#469	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
C₂×#676	2
#4041	2
C₅×#106	2
#3941	2
#469	5