Subgroups — #65164

Order 320.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4034	C₂	1	1	yes	maximal, characteristic
2	#4033	C₂	1	1	yes	maximal, characteristic
3	C₅×#107	C₂	1	1	yes	maximal, characteristic
4	#3948	C₂	1	1	yes	maximal, characteristic
5	#3946	C₂	1	1	yes	maximal, characteristic
6	#3942	C₂	1	1	yes	maximal, characteristic
7	#3939	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀	—	2	1	no	—
9	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	#683	—	2	1	no	—
15	#683	—	2	1	no	—
16	C₄×Dic₅	—	2	1	no	—
17	C₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×#168	C₂²	1	1	yes	characteristic
19	#477	—	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	—	2	1	no	—
28	Dic₁₀	—	4	2	no	—
29	Dic₁₀	Q₈	1	2	yes	complemented
30	C₅×C₈	—	2	1	no	—
31	#168	—	2	1	no	—
32	C₄⋊Q₈	—	5	1	no	—
33	C₄×Q₈	—	5	1	no	—
34	#107	—	5	1	no	—
35	#107	D₅	1	1	yes	characteristic
36	C₄⋊C₈	—	5	1	no	—
37	#103	—	5	1	no	—
38	#103	—	5	1	no	—
39	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
40	C₄×C₅	—	4	1	no	—
41	C₄×C₅	—	4	1	no	—
42	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
43	C₄×C₅	#41	1	1	yes	characteristic
44	Dic₅	—	2	2	no	—
45	Dic₅	—	2	1	no	—
46	Dic₅	—	4	1	no	—
47	Dic₅	Q₁₆	1	2	yes	—
48	C₂×Q₈	—	5	1	no	—
49	C₂×Q₈	—	10	1	no	—
50	C₂×C₈	—	5	1	no	—
51	C₂×C₈	C₂×D₅	1	1	yes	characteristic
52	C₄⋊C₄	—	5	1	no	—
53	C₄⋊C₄	—	10	1	no	—
54	C₄⋊C₄	—	10	1	no	—
55	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
56	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
57	C₄²	—	5	1	no	—
58	C₄²	—	10	1	no	—
59	C₂×C₅	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂×C₅	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Q₈	—	5	2	no	—
63	Q₈	—	10	1	no	—
64	Q₈	—	20	2	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	5	1	no	—
68	C₂×C₄	—	5	1	no	—
69	C₂×C₄	—	10	1	no	—
70	C₂×C₄	—	10	1	no	—
71	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	10	1	no	—
74	C₅	#477	1	1	yes	characteristic, complemented, Sylow
75	C₂²	D₄×D₅	1	1	yes	characteristic
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	5	2	no	—
79	C₄	—	10	2	no	—
80	C₄	—	10	1	no	—
81	C₄	—	20	1	no	—
82	C₄	#709	1	1	yes	characteristic
83	C₄	Q₈×D₅	1	1	yes	characteristic
84	C₂	#4042	1	1	yes	characteristic
85	C₂	#4057	1	1	yes	characteristic
86	C₂	D₅×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#107	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	#107	D₅
hypocenter	C₅	#477
LCS	G ▷ C₂×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
29. Dic₁₀	Q₈	semidirect
74. C₅	#477	semidirect

Maximal subgroups

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

subgroup	index
#4034	2
#4033	2
C₅×#107	2
#3948	2
#3946	2
#3942	2
#3939	2
#477	5