Subgroups — #65520

Order 320.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#685	C₂	1	3	yes	maximal
3	#3963	C₂	1	3	yes	maximal, complemented
4	C₂³×D₅	C₂²	1	1	yes	characteristic
5	C₂×C₅×Q₈	—	2	6	no	—
6	C₂²×C₄×C₅	C₂²	1	3	yes	—
7	C₂²×Dic₅	C₂²	1	3	yes	—
8	#685	—	2	6	no	—
9	#393	—	5	1	no	maximal, Sylow
10	C₂³×C₅	C₂³	1	1	yes	characteristic
11	C₂²×D₅	—	2	3	no	—
12	C₂²×D₅	—	2	3	no	—
13	C₂²×D₅	Q₈	1	2	yes	complemented
14	C₅×Q₈	—	2	8	no	—
15	C₂×C₄×C₅	—	2	6	no	—
16	C₂×C₄×C₅	D₄	1	6	yes	—
17	C₂×Dic₅	—	2	3	no	—
18	C₂×Dic₅	—	2	6	no	—
19	C₂²×Q₈	D₅	1	1	yes	characteristic, complemented
20	C₂×C₂²⋊C₄	—	5	3	no	—
21	#95	—	5	3	no	—
22	C₂²×C₅	C₂×D₄	1	3	yes	—
23	C₂²×C₅	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₅	#41	1	3	yes	—
25	C₂×D₅	—	2	6	no	—
26	C₂×D₅	—	4	1	no	—
27	C₂×D₅	—	4	3	no	—
28	C₄×C₅	—	2	6	no	—
29	Dic₅	—	4	3	no	—
30	C₂⁴	—	5	1	no	—
31	C₂×Q₈	—	2	6	no	—
32	C₂²×C₄	—	5	3	no	—
33	C₂²×C₄	C₂×D₅	1	3	yes	—
34	C₂²⋊C₄	—	10	6	no	—
35	C₂×C₅	#120	1	1	yes	characteristic
36	C₂×C₅	C₂²⋊Q₈	1	3	yes	—
37	C₂×C₅	#124	1	3	yes	—
38	D₅	—	4	2	no	—
39	C₂³	—	5	2	no	—
40	C₂³	—	10	3	no	—
41	C₂³	—	10	3	no	—
42	C₂³	C₂²×D₅	1	1	yes	characteristic
43	Q₈	—	2	8	no	—
44	C₂×C₄	—	2	6	no	—
45	C₂×C₄	—	10	6	no	—
46	C₂×C₄	—	10	3	no	—
47	C₂×C₄	C₂²⋊D₅	1	6	yes	—
48	C₅	#393	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	10	6	no	—
50	C₂²	—	20	1	no	—
51	C₂²	—	20	3	no	—
52	C₂²	Q₈×D₅	1	1	yes	characteristic
53	C₂²	#713	1	3	yes	—
54	C₂²	C₂×C₂²⋊D₅	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	20	3	no	—
57	C₂	—	20	2	no	—
58	C₂	#4092	1	3	yes	—
59	C₂	#4093	1	3	yes	—
60	C₂	#4099	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₅
commutator, socle, cosocle	C₂³×C₅	C₂³
Fitting	C₂²×C₅×Q₈	C₂
prime radical	C₂³×D₅	C₂²
hypercenter	C₂²×Q₈	D₅
hypocenter	C₅	#393
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×Q₈	C₂	semidirect
3. #3963	C₂	semidirect
13. C₂²×D₅	Q₈	semidirect
19. C₂²×Q₈	D₅	semidirect
48. C₅	#393	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×Q₈	2
C₂×#685	2
#3963	2
#393	5