Subgroups — #65376

Order 320.

108 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	C₄×D₂₀	C₂	1	1	yes	maximal, characteristic
3	#3941	C₂	1	2	yes	maximal
4	#3939	C₂	1	2	yes	maximal, complemented
5	C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×Q₈	—	2	1	no	—
7	C₂×D₂₀	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₅	—	2	1	no	—
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	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₂×#168	C₂²	1	2	yes	—
15	#474	—	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	1	no	—
24	D₂₀	Q₈	1	2	yes	complemented
25	C₄×D₅	—	4	2	no	—
26	#168	—	2	2	no	—
27	C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
28	C₄×D₄	—	5	1	no	—
29	#107	—	5	2	no	—
30	C₄⋊C₈	—	5	1	no	—
31	#102	—	5	2	no	—
32	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
33	C₂×D₅	—	2	2	no	—
34	C₂×D₅	—	4	1	no	—
35	C₂×D₅	—	4	1	no	—
36	C₄×C₅	—	2	1	no	—
37	C₄×C₅	—	4	2	no	—
38	C₄×C₅	D₈	1	2	yes	—
39	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
40	C₄×C₅	#41	1	1	yes	characteristic
41	Dic₅	—	4	1	no	—
42	C₂×Q₈	—	2	1	no	—
43	C₂×D₄	—	5	1	no	—
44	C₂²×C₄	—	10	1	no	—
45	C₂×C₈	—	5	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	5	1	no	—
48	C₄⋊C₄	C₂×D₅	1	2	yes	—
49	C₂²⋊C₄	—	10	1	no	—
50	C₄²	C₂×D₅	1	1	yes	characteristic
51	C₂×C₅	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂×C₅	C₂×D₈	1	1	yes	characteristic
53	C₂×C₅	#137	1	1	yes	characteristic
54	D₅	—	4	2	no	—
55	C₂³	—	10	1	no	—
56	Q₈	—	4	2	no	—
57	D₄	—	5	2	no	—
58	D₄	—	10	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	10	1	no	—
61	C₂×C₄	—	20	2	no	—
62	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
65	C₈	—	10	2	no	—
66	C₅	#474	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	10	2	no	—
68	C₂²	—	20	1	no	—
69	C₂²	—	20	1	no	—
70	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	20	1	no	—
74	C₄	#686	1	2	yes	—
75	C₄	Q₈×D₅	1	1	yes	characteristic
76	C₄	#713	1	1	yes	characteristic
77	C₂	—	20	2	no	—
78	C₂	C₂×#686	1	1	yes	characteristic
79	C₂	#4088	1	1	yes	characteristic
80	C₂	#4092	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	#474
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
4. #3939	C₂	semidirect
5. C₄⋊#168	C₂	semidirect
24. D₂₀	Q₈	semidirect
27. C₄⋊Q₈	D₅	semidirect
66. C₅	#474	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
C₄×D₂₀	2
#3941	2
#3939	2
C₄⋊#168	2
#474	5