Subgroups — #65359

Order 320.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#127	C₂	1	1	yes	maximal, characteristic
2	#4015	C₂	1	1	yes	maximal, characteristic, complemented
3	#3964	C₂	1	4	yes	maximal, complemented
4	C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×D₄	—	2	1	no	—
6	C₂×C₅×D₄	—	2	1	no	—
7	C₂×C₅×D₄	C₂²	1	2	yes	—
8	C₂×Dic₁₀	—	2	1	no	—
9	C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₅	—	2	1	no	—
11	C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂×#168	C₂²	1	2	yes	—
13	#486	—	5	1	no	maximal, Sylow
14	C₂³×C₅	—	2	2	no	—
15	C₅×D₄	—	2	4	no	—
16	C₅×D₄	—	4	2	no	—
17	C₅×D₄	—	4	2	no	—
18	C₂×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×Dic₅	—	2	2	no	—
22	Dic₁₀	—	4	2	no	—
23	#168	—	2	2	no	—
24	C₄⋊Q₈	—	5	1	no	—
25	#127	D₅	1	1	yes	characteristic
26	#102	—	5	4	no	—
27	C₄×C₈	—	5	1	no	—
28	C₂²×C₅	—	4	2	no	—
29	C₂²×C₅	—	4	2	no	—
30	C₂²×C₅	—	4	2	no	—
31	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅	D₈	1	2	yes	—
33	C₄×C₅	SD₁₆	1	2	yes	—
34	C₄×C₅	#41	1	2	yes	—
35	Dic₅	—	4	2	no	—
36	C₂×Q₈	—	10	1	no	—
37	C₂×D₄	—	2	1	no	—
38	C₂×D₄	—	2	1	no	—
39	C₂×D₄	C₂×D₅	1	2	yes	—
40	C₂×C₈	—	5	2	no	—
41	C₄⋊C₄	—	5	2	no	—
42	C₄⋊C₄	—	10	1	no	—
43	C₄²	C₂×D₅	1	1	yes	characteristic
44	C₂×C₅	—	8	2	no	—
45	C₂×C₅	#124	1	1	yes	characteristic
46	C₂×C₅	C₂×D₈	1	1	yes	characteristic
47	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
48	C₂³	—	2	2	no	—
49	Q₈	—	20	2	no	—
50	D₄	—	2	4	no	—
51	D₄	—	4	2	no	—
52	D₄	—	4	2	no	—
53	C₂×C₄	—	10	2	no	—
54	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
57	C₈	—	10	2	no	—
58	C₅	#486	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
63	C₄	—	20	2	no	—
64	C₄	#686	1	2	yes	—
65	C₄	#687	1	2	yes	—
66	C₄	#711	1	2	yes	—
67	C₂	—	8	2	no	—
68	C₂	C₂×#686	1	1	yes	characteristic
69	C₂	C₂×#687	1	1	yes	characteristic
70	C₂	#4082	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting, prime radical	C₅×#127	C₂
socle	C₂²×C₅	C₂×D₄
hypercenter	#127	D₅
hypocenter	C₅	#486
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. #4015	C₂	semidirect
3. #3964	C₂	semidirect
4. C₄×#168	C₂	semidirect
58. C₅	#486	semidirect

Maximal subgroups

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

subgroup	index
C₅×#127	2
#4015	2
#3964	2
C₄×#168	2
#486	5