Subgroups — #64960

Order 320.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₀	C₂	1	3	yes	maximal, complemented
2	C₂×#685	C₂	1	3	yes	maximal, complemented
3	C₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₅	C₂²	1	3	yes	—
5	C₂²×C₄×C₅	C₂²	1	3	yes	—
6	C₂²×Dic₅	C₂²	1	1	yes	characteristic
7	C₂×D₂₀	—	2	12	no	—
8	C₂×D₂₀	—	2	6	no	—
9	#685	—	2	6	no	—
10	#392	—	5	1	no	maximal, Sylow
11	C₂³×C₅	C₂³	1	1	yes	characteristic
12	C₂²×D₅	—	2	3	no	—
13	C₂²×D₅	—	2	6	no	—
14	C₂²×D₅	—	2	6	no	—
15	C₂²×D₅	—	2	3	no	—
16	C₂²×D₅	D₄	1	6	yes	complemented
17	C₂×C₄×C₅	—	2	6	no	—
18	C₂×C₄×C₅	D₄	1	6	yes	complemented
19	C₂×Dic₅	—	2	3	no	—
20	D₂₀	—	2	24	no	—
21	C₂²×D₄	—	5	3	no	—
22	C₂×C₂²⋊C₄	—	5	3	no	—
23	#95	D₅	1	1	yes	characteristic, complemented
24	C₂²×C₅	C₂×D₄	1	3	yes	—
25	C₂²×C₅	C₂×D₄	1	3	yes	—
26	C₂²×C₅	#41	1	1	yes	characteristic
27	C₂×D₅	—	2	12	no	—
28	C₂×D₅	—	2	6	no	—
29	C₂×D₅	—	4	3	no	—
30	C₂×D₅	—	4	6	no	—
31	C₂×D₅	—	4	3	no	—
32	C₄×C₅	—	2	6	no	—
33	Dic₅	—	4	1	no	—
34	C₂⁴	—	5	3	no	—
35	C₂×D₄	—	10	12	no	—
36	C₂×D₄	—	10	6	no	—
37	C₂²×C₄	—	5	1	no	—
38	C₂²×C₄	C₂×D₅	1	3	yes	—
39	C₂²⋊C₄	—	10	6	no	—
40	C₂×C₅	#120	1	3	yes	—
41	C₂×C₅	#121	1	3	yes	—
42	C₂×C₅	#127	1	1	yes	characteristic
43	D₅	—	4	6	no	—
44	C₂³	—	5	6	no	—
45	C₂³	—	10	3	no	—
46	C₂³	—	10	6	no	—
47	C₂³	—	10	6	no	—
48	C₂³	—	10	3	no	—
49	C₂³	C₂²×D₅	1	1	yes	characteristic
50	D₄	—	10	24	no	—
51	C₂×C₄	—	2	6	no	—
52	C₂×C₄	—	10	3	no	—
53	C₂×C₄	D₂₀	1	6	yes	complemented
54	C₅	#392	1	1	yes	characteristic, complemented, Sylow
55	C₂²	—	10	12	no	—
56	C₂²	—	10	6	no	—
57	C₂²	—	20	3	no	—
58	C₂²	—	20	3	no	—
59	C₂²	—	20	6	no	—
60	C₂²	C₂×D₂₀	1	3	yes	—
61	C₂²	D₄×D₅	1	3	yes	—
62	C₂²	#713	1	1	yes	characteristic
63	C₄	—	2	6	no	—
64	C₄	—	20	1	no	—
65	C₂	—	20	6	no	—
66	C₂	#4020	1	1	yes	characteristic
67	C₂	#4028	1	3	yes	—
68	C₂	#4041	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₅
commutator, socle, cosocle	C₂³×C₅	C₂³
Fitting	C₅×#95	C₂
hypercenter	#95	D₅
hypocenter	C₅	#392
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂²×D₂₀	C₂	semidirect
2. C₂×#685	C₂	semidirect
3. C₅×#95	C₂	semidirect
16. C₂²×D₅	D₄	semidirect
18. C₂×C₄×C₅	D₄	semidirect
23. #95	D₅	semidirect
53. C₂×C₄	D₂₀	semidirect
54. C₅	#392	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₂₀	2
C₂×#685	2
C₅×#95	2
#392	5