Subgroups — #65320

Order 320.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#689	C₂	1	2	yes	maximal
3	#4015	C₂	1	1	yes	maximal, characteristic
4	#3942	C₂	1	2	yes	maximal
5	C₄⋊#168	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₀	C₂²	1	2	yes	—
8	C₅×C₄⋊C₄	—	2	1	no	—
9	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₄²×C₅	—	2	1	no	—
11	C₄²×C₅	C₂²	1	1	yes	characteristic
12	#689	—	2	4	no	—
13	C₄⋊Dic₅	—	2	1	no	—
14	C₄⋊Dic₅	—	2	1	no	—
15	C₂×#168	C₂²	1	2	yes	—
16	#462	—	5	1	no	maximal, Sylow
17	C₅×Q₈	—	2	1	no	—
18	C₅×Q₈	D₄	1	2	yes	—
19	C₂×C₄×C₅	—	2	1	no	—
20	C₂×C₄×C₅	—	2	1	no	—
21	C₂×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×Dic₅	—	2	2	no	—
25	Dic₁₀	—	2	2	no	—
26	Dic₁₀	—	2	2	no	—
27	#168	—	2	2	no	—
28	C₂×Q₁₆	—	5	2	no	—
29	C₄⋊Q₈	—	5	1	no	—
30	C₄×Q₈	D₅	1	1	yes	characteristic
31	C₄⋊C₈	—	5	1	no	—
32	#103	—	5	2	no	—
33	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅	—	2	2	no	—
35	C₄×C₅	—	2	1	no	—
36	C₄×C₅	—	4	1	no	—
37	C₄×C₅	Q₁₆	1	2	yes	complemented
38	C₄×C₅	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅	#41	1	1	yes	characteristic
40	Dic₅	—	4	2	no	—
41	C₂×Q₈	—	5	2	no	—
42	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
43	Q₁₆	—	10	4	no	—
44	C₂×C₈	—	5	2	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	10	1	no	—
47	C₄⋊C₄	—	10	1	no	—
48	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
49	C₄²	—	2	1	no	—
50	C₄²	C₂×D₅	1	1	yes	characteristic
51	C₂×C₅	#121	1	1	yes	characteristic
52	C₂×C₅	C₂×Q₁₆	1	1	yes	characteristic
53	C₂×C₅	#137	1	1	yes	characteristic
54	Q₈	—	2	1	no	—
55	Q₈	—	10	2	no	—
56	Q₈	—	10	2	no	—
57	Q₈	D₂₀	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	10	2	no	—
61	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
63	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
64	C₈	—	10	2	no	—
65	C₅	#462	1	1	yes	characteristic, complemented, Sylow
66	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	2	no	—
69	C₄	—	4	1	no	—
70	C₄	—	20	2	no	—
71	C₄	#689	1	2	yes	complemented
72	C₄	C₂×D₂₀	1	1	yes	characteristic
73	C₄	#709	1	1	yes	characteristic
74	C₂	#4076	1	1	yes	characteristic
75	C₂	C₂×#689	1	1	yes	characteristic
76	C₂	#4097	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×Q₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	#462
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
37. C₄×C₅	Q₁₆	semidirect
65. C₅	#462	semidirect
71. C₄	#689	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×Q₈	2
C₂×#689	2
#4015	2
#3942	2
C₄⋊#168	2
#462	5