Subgroups — #65309

Order 320.

96 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	#4015	C₂	1	1	yes	maximal, characteristic
3	#3967	C₂	1	2	yes	maximal
4	#3940	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	—	2	1	no	—
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	C₄⋊Dic₅	—	2	1	no	—
13	C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₂×#168	C₂²	1	2	yes	—
15	#475	—	5	1	no	maximal, Sylow
16	C₅×Q₈	—	2	1	no	—
17	C₅×Q₈	Q₈	1	2	yes	complemented
18	C₂×C₄×C₅	—	2	1	no	—
19	C₂×C₄×C₅	—	2	1	no	—
20	C₂×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×Dic₅	—	2	2	no	—
24	Dic₁₀	—	4	2	no	—
25	#168	—	2	2	no	—
26	C₄⋊Q₈	—	5	1	no	—
27	C₄×Q₈	D₅	1	1	yes	characteristic
28	#106	—	5	2	no	—
29	C₄⋊C₈	—	5	1	no	—
30	#103	—	5	2	no	—
31	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅	—	2	2	no	—
33	C₄×C₅	—	2	1	no	—
34	C₄×C₅	—	4	1	no	—
35	C₄×C₅	SD₁₆	1	2	yes	—
36	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
37	C₄×C₅	#41	1	1	yes	characteristic
38	Dic₅	—	4	2	no	—
39	C₂×Q₈	—	10	1	no	—
40	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
41	C₂×C₈	—	5	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	5	2	no	—
44	C₄⋊C₄	—	10	1	no	—
45	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
46	C₄²	—	2	1	no	—
47	C₄²	C₂×D₅	1	1	yes	characteristic
48	C₂×C₅	C₂²⋊Q₈	1	1	yes	characteristic
49	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
50	C₂×C₅	#137	1	1	yes	characteristic
51	Q₈	—	2	1	no	—
52	Q₈	—	20	2	no	—
53	Q₈	Dic₁₀	1	2	yes	complemented
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	10	2	no	—
57	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
60	C₈	—	10	2	no	—
61	C₅	#475	1	1	yes	characteristic, complemented, Sylow
62	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	20	2	no	—
67	C₄	#688	1	2	yes	—
68	C₄	C₂×Dic₁₀	1	1	yes	characteristic
69	C₄	#709	1	1	yes	characteristic
70	C₂	#4070	1	1	yes	characteristic
71	C₂	C₂×#688	1	1	yes	characteristic
72	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	#475
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
17. C₅×Q₈	Q₈	semidirect
53. Q₈	Dic₁₀	semidirect
61. C₅	#475	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
#4015	2
#3967	2
#3940	2
C₄⋊#168	2
#475	5