Subgroups — #65928

Order 320.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	#4099	C₂	1	2	yes	maximal, complemented
3	#4083	C₂	1	3	yes	maximal, complemented
4	#4082	C₂	1	3	yes	maximal, complemented
5	#4031	C₂	1	6	yes	maximal, complemented
6	C₂⁴×C₅	C₂²	1	1	yes	characteristic, complemented
7	C₂³×D₅	C₂²	1	1	yes	characteristic, complemented
8	C₂×C₅×D₄	C₂²	1	3	yes	—
9	C₂×C₂²⋊D₅	C₂²	1	6	yes	—
10	C₂×Dic₁₀	C₂²	1	3	yes	complemented
11	C₅×C₂²⋊C₄	C₂²	1	3	yes	complemented
12	C₂²⋊Dic₅	C₂²	1	6	yes	complemented
13	C₂²⋊Dic₅	C₂²	1	3	yes	complemented
14	#685	C₂²	1	6	yes	complemented
15	C₄×Dic₅	C₂²	1	3	yes	complemented
16	H(3,𝔽₄)	—	5	1	no	maximal, Sylow
17	C₂³×C₅	—	4	3	no	—
18	C₂³×C₅	C₂³	1	1	yes	characteristic
19	C₂³×C₅	C₂³	1	3	yes	—
20	C₂²×D₅	—	4	3	no	—
21	C₂²×D₅	C₂³	1	2	yes	—
22	C₅×D₄	—	4	3	no	—
23	C₂×C₄×C₅	C₂³	1	3	yes	—
24	C₂²⋊D₅	—	4	6	no	—
25	C₂×Dic₅	C₂³	1	6	yes	—
26	Dic₁₀	—	4	3	no	—
27	#124	—	5	3	no	—
28	#124	—	5	6	no	—
29	#120	—	5	2	no	—
30	#120	—	5	3	no	—
31	#120	D₅	1	1	yes	characteristic, complemented
32	C₂²×C₅	—	2	6	no	—
33	C₂²×C₅	—	2	3	no	—
34	C₂²×C₅	—	2	3	no	—
35	C₂²×C₅	—	4	4	no	—
36	C₂²×C₅	C₂⁴	1	1	yes	characteristic
37	C₂×D₅	—	2	6	no	—
38	C₂×D₅	—	4	4	no	—
39	C₄×C₅	—	2	3	no	—
40	Dic₅	—	2	6	no	—
41	C₂⁴	—	5	1	no	—
42	C₂⁴	C₂×D₅	1	1	yes	characteristic, complemented
43	C₂×Q₈	—	5	3	no	—
44	C₂×D₄	—	5	6	no	—
45	C₂×D₄	C₂×D₅	1	3	yes	—
46	C₂²⋊C₄	—	5	3	no	—
47	C₂²⋊C₄	—	5	6	no	—
48	C₂²⋊C₄	—	5	6	no	—
49	C₂²⋊C₄	C₂×D₅	1	3	yes	complemented
50	C₄²	—	5	3	no	—
51	C₂×C₅	—	4	1	no	—
52	C₂×C₅	—	4	3	no	—
53	C₂×C₅	H(5,𝔽₂)	1	3	yes	—
54	D₅	—	4	2	no	—
55	C₂³	—	4	3	no	—
56	C₂³	—	5	2	no	—
57	C₂³	—	20	3	no	—
58	C₂³	C₂²×D₅	1	1	yes	characteristic
59	C₂³	C₂²×D₅	1	3	yes	—
60	Q₈	—	20	3	no	—
61	D₄	—	4	3	no	—
62	D₄	—	20	6	no	—
63	C₂×C₄	—	5	6	no	—
64	C₂×C₄	C₂²×D₅	1	3	yes	—
65	C₅	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	10	6	no	—
71	C₂²	—	20	4	no	—
72	C₂²	C₂³×D₅	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	10	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	20	2	no	—
78	C₂	#4144	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#120	C₂
hypercenter	#120	D₅
hypocenter	C₅	H(3,𝔽₄)
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. C₅×#120	C₂	semidirect
2. #4099	C₂	semidirect
3. #4083	C₂	semidirect
4. #4082	C₂	semidirect
5. #4031	C₂	semidirect
6. C₂⁴×C₅	C₂²	semidirect
7. C₂³×D₅	C₂²	semidirect
10. C₂×Dic₁₀	C₂²	semidirect
11. C₅×C₂²⋊C₄	C₂²	semidirect
12. C₂²⋊Dic₅	C₂²	semidirect
13. C₂²⋊Dic₅	C₂²	semidirect
14. #685	C₂²	semidirect
15. C₄×Dic₅	C₂²	semidirect
31. #120	D₅	semidirect
42. C₂⁴	C₂×D₅	semidirect
49. C₂²⋊C₄	C₂×D₅	semidirect
65. C₅	H(3,𝔽₄)	semidirect

Maximal subgroups

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

subgroup	index
C₅×#120	2
#4099	2
#4083	2
#4082	2
#4031	2
H(3,𝔽₄)	5