Subgroups — #65145

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4036	C₂	1	1	yes	maximal, characteristic
2	#4034	C₂	1	1	yes	maximal, characteristic
3	C₅×#106	C₂	1	1	yes	maximal, characteristic
4	#3950	C₂	1	1	yes	maximal, characteristic
5	#3947	C₂	1	1	yes	maximal, characteristic
6	#3940	C₂	1	1	yes	maximal, characteristic
7	#3939	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀	—	2	1	no	—
9	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₅	—	2	1	no	—
13	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	#683	—	2	1	no	—
16	#683	—	2	1	no	—
17	C₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×#168	C₂²	1	1	yes	characteristic
19	#501	—	5	1	no	maximal, Sylow
20	C₂×C₄×C₅	—	2	1	no	—
21	C₂×C₄×C₅	—	2	1	no	—
22	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×Dic₅	—	2	1	no	—
24	C₂×Dic₅	—	2	1	no	—
25	C₂×Dic₅	D₄	1	1	yes	characteristic
26	C₂×Dic₅	D₄	1	1	yes	characteristic
27	Dic₁₀	—	4	2	no	—
28	C₅×C₈	Q₈	1	2	yes	complemented
29	#168	Q₈	1	2	yes	complemented
30	C₄⋊Q₈	—	5	1	no	—
31	#125	—	5	1	no	—
32	#107	—	5	1	no	—
33	#107	—	5	1	no	—
34	#106	—	5	1	no	—
35	#106	D₅	1	1	yes	characteristic
36	#97	—	5	1	no	—
37	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
38	C₄×C₅	—	4	1	no	—
39	C₄×C₅	—	4	1	no	—
40	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
41	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
42	Dic₅	—	2	1	no	—
43	Dic₅	—	2	1	no	—
44	Dic₅	—	4	1	no	—
45	Dic₅	—	4	1	no	—
46	C₂×Q₈	—	10	1	no	—
47	C₂×C₈	—	5	1	no	—
48	C₂×C₈	C₂×D₅	1	1	yes	characteristic
49	C₄⋊C₄	—	5	1	no	—
50	C₄⋊C₄	—	5	1	no	—
51	C₄⋊C₄	—	10	1	no	—
52	C₄⋊C₄	—	10	1	no	—
53	C₄⋊C₄	—	10	1	no	—
54	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
55	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
56	C₄²	—	5	1	no	—
57	C₂×C₅	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂×C₅	#137	1	1	yes	characteristic
60	Q₈	—	20	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	5	1	no	—
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	10	1	no	—
66	C₂×C₄	—	10	1	no	—
67	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
68	C₈	—	5	2	no	—
69	C₈	Dic₁₀	1	2	yes	complemented
70	C₅	#501	1	1	yes	characteristic, complemented, Sylow
71	C₂²	D₄×D₅	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	10	1	no	—
75	C₄	—	10	1	no	—
76	C₄	—	20	1	no	—
77	C₄	—	20	1	no	—
78	C₄	C₂×Dic₁₀	1	1	yes	characteristic
79	C₄	Q₈×D₅	1	1	yes	characteristic
80	C₂	#4034	1	1	yes	characteristic
81	C₂	#4060	1	1	yes	characteristic
82	C₂	#4061	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#106	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	#106	D₅
hypocenter	C₅	#501
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
28. C₅×C₈	Q₈	semidirect
29. #168	Q₈	semidirect
69. C₈	Dic₁₀	semidirect
70. C₅	#501	semidirect

Maximal subgroups

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

subgroup	index
#4036	2
#4034	2
C₅×#106	2
#3950	2
#3947	2
#3940	2
#3939	2
#501	5