Subgroups — #65375

Order 320.

124 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	C₂×#689	C₂	1	2	yes	maximal, complemented
3	C₂×#688	C₂	1	2	yes	maximal
4	#4021	C₂	1	1	yes	maximal, characteristic
5	#3935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×Q₈	C₂²	1	2	yes	—
7	C₂×D₂₀	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
9	C₅×C₄⋊C₄	—	2	2	no	—
10	C₄²×C₅	C₂²	1	1	yes	characteristic
11	#689	—	2	4	no	—
12	#688	—	2	4	no	—
13	#685	—	2	2	no	—
14	C₂×#168	C₂²	1	2	yes	—
15	#497	—	5	1	no	maximal, Sylow
16	C₂²×D₅	—	2	1	no	—
17	C₅×Q₈	—	2	2	no	—
18	C₅×Q₈	—	2	2	no	—
19	C₂×C₄×C₅	—	2	2	no	—
20	C₂×C₄×C₅	D₄	1	2	yes	—
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	D₄	1	4	yes	—
26	C₂×Q₁₆	—	5	2	no	—
27	C₂×SD₁₆	—	5	2	no	—
28	C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
29	#124	—	5	1	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	2	no	—
34	C₄×C₅	—	2	2	no	—
35	C₄×C₅	—	4	2	no	—
36	C₄×C₅	C₂×D₄	1	2	yes	—
37	Dic₅	—	4	1	no	—
38	C₂×Q₈	—	5	1	no	—
39	C₂×Q₈	C₂×D₅	1	2	yes	—
40	C₂×D₄	—	5	1	no	—
41	Q₁₆	—	10	4	no	—
42	SD₁₆	—	10	4	no	—
43	C₂×C₈	—	5	2	no	—
44	C₄⋊C₄	—	2	2	no	—
45	C₂²⋊C₄	—	10	2	no	—
46	C₄²	C₂×D₅	1	1	yes	characteristic
47	C₂×C₅	#127	1	1	yes	characteristic
48	C₂×C₅	#137	1	2	yes	—
49	D₅	—	8	1	no	—
50	C₂³	—	10	1	no	—
51	Q₈	—	2	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	10	2	no	—
54	D₄	—	10	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	10	1	no	—
57	C₂×C₄	C₂²⋊D₅	1	2	yes	—
58	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
59	C₈	—	5	4	no	—
60	C₅	#497	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	20	1	no	—
62	C₂²	—	20	2	no	—
63	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	2	no	—
66	C₄	—	20	1	no	—
67	C₄	D₄×D₅	1	2	yes	—
68	C₂	—	40	1	no	—
69	C₂	#4086	1	1	yes	characteristic
70	C₂	#4088	1	2	yes	—

	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	#497
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
2. C₂×#689	C₂	semidirect
5. #3935	C₂	semidirect
28. C₄⋊Q₈	D₅	semidirect
60. C₅	#497	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
C₂×#689	2
C₂×#688	2
#4021	2
#3935	2
#497	5