Subgroups — #65378

Order 320.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#4021	C₂	1	1	yes	maximal, characteristic
3	#3942	C₂	1	2	yes	maximal, complemented
4	#3941	C₂	1	2	yes	maximal
5	#3935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×Q₈	—	2	1	no	—
7	C₂×D₂₀	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
9	C₅×C₄⋊C₄	—	2	1	no	—
10	C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₄²×C₅	C₂²	1	1	yes	characteristic
12	#685	—	2	1	no	—
13	#685	—	2	1	no	—
14	C₂×#168	C₂²	1	2	yes	—
15	#489	—	5	1	no	maximal, Sylow
16	C₂²×D₅	—	2	1	no	—
17	C₅×Q₈	—	4	2	no	—
18	C₂×C₄×C₅	—	2	2	no	—
19	C₂×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×Dic₅	—	2	1	no	—
23	D₂₀	—	2	2	no	—
24	Dic₁₀	—	2	2	no	—
25	#168	—	2	2	no	—
26	C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
27	#124	—	5	1	no	—
28	#103	—	5	2	no	—
29	#102	—	5	2	no	—
30	#97	—	5	1	no	—
31	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
32	C₂×D₅	—	4	1	no	—
33	C₂×D₅	—	4	1	no	—
34	C₂×D₅	—	4	1	no	—
35	C₄×C₅	—	2	1	no	—
36	C₄×C₅	—	2	1	no	—
37	C₄×C₅	—	4	2	no	—
38	C₄×C₅	#41	1	2	yes	—
39	Dic₅	—	4	1	no	—
40	C₂×Q₈	—	2	1	no	—
41	C₂×Q₈	—	5	1	no	—
42	C₂×D₄	—	5	1	no	—
43	C₂×C₈	—	5	2	no	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	C₂×D₅	1	2	yes	—
46	C₂²⋊C₄	—	10	1	no	—
47	C₂²⋊C₄	—	10	1	no	—
48	C₄²	C₂×D₅	1	1	yes	characteristic
49	C₂×C₅	#124	1	1	yes	characteristic
50	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₂×C₅	#137	1	1	yes	characteristic
52	D₅	—	8	1	no	—
53	C₂³	—	10	1	no	—
54	Q₈	—	4	2	no	—
55	Q₈	—	10	2	no	—
56	D₄	—	10	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	10	1	no	—
59	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
60	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
62	C₈	—	10	2	no	—
63	C₅	#489	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	20	1	no	—
65	C₂²	—	20	1	no	—
66	C₂²	—	20	1	no	—
67	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	20	1	no	—
72	C₄	#713	1	2	yes	—
73	C₂	—	40	1	no	—
74	C₂	#4078	1	1	yes	characteristic
75	C₂	#4088	1	1	yes	characteristic
76	C₂	#4093	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊Q₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×D₂₀	C₂²
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	#489
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
1. C₅×C₄⋊Q₈	C₂	semidirect
3. #3942	C₂	semidirect
5. #3935	C₂	semidirect
26. C₄⋊Q₈	D₅	semidirect
63. C₅	#489	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₄⋊Q₈	2
#4021	2
#3942	2
#3941	2
#3935	2
#489	5