Subgroups — #66285

Order 320.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#4150	C₂	1	6	yes	maximal, complemented
3	D₅×#41	C₂	1	6	yes	maximal, complemented
4	#4144	C₂	1	9	yes	maximal, complemented
5	C₂×#711	C₂	1	9	yes	maximal, complemented
6	C₅×#41	C₂²	1	6	yes	complemented
7	C₂×C₅×D₄	C₂²	1	9	yes	complemented
8	C₂×C₂²⋊D₅	C₂²	1	18	yes	complemented
9	C₂²×Dic₅	C₂²	1	6	yes	complemented
10	#713	C₂²	1	2	yes	complemented
11	Q₈×D₅	C₂²	1	6	yes	complemented
12	#711	C₂²	1	18	yes	complemented
13	#711	C₂²	1	36	yes	complemented
14	D₄×D₅	C₂²	1	18	yes	complemented
15	#709	C₂²	1	18	yes	complemented
16	C₂×C₄×D₅	C₂²	1	9	yes	complemented
17	C₂×Dic₁₀	C₂²	1	9	yes	complemented
18	#585	—	5	1	no	maximal, Sylow
19	C₂³×C₅	C₂³	1	6	yes	—
20	C₂²×D₅	C₂³	1	9	yes	—
21	C₅×Q₈	C₂³	1	2	yes	—
22	C₅×D₄	C₂³	1	18	yes	—
23	C₂×C₄×C₅	C₂³	1	9	yes	—
24	C₂²⋊D₅	C₂³	1	36	yes	—
25	C₂×Dic₅	C₂³	1	18	yes	—
26	C₂×Dic₅	C₂³	1	9	yes	—
27	D₂₀	C₂³	1	6	yes	—
28	C₄×D₅	C₂³	1	6	yes	—
29	C₄×D₅	C₂³	1	18	yes	—
30	Dic₁₀	C₂³	1	18	yes	—
31	2₋¹⁺⁴	—	5	6	no	—
32	H(5,𝔽₂)	—	5	9	no	—
33	H(5,𝔽₂)	D₅	1	1	yes	characteristic, complemented
34	C₂×#41	—	5	6	no	—
35	C₂×#41	—	5	9	no	—
36	C₂²×C₅	—	4	6	no	—
37	C₂²×C₅	C₂⁴	1	9	yes	—
38	C₂×D₅	—	4	9	no	—
39	C₂×D₅	C₂⁴	1	6	yes	—
40	C₄×C₅	C₂⁴	1	6	yes	—
41	Dic₅	C₂⁴	1	9	yes	—
42	Dic₅	C₂⁴	1	1	yes	characteristic
43	#41	—	5	2	no	—
44	#41	—	5	18	no	—
45	#41	—	5	36	no	—
46	#41	—	5	18	no	—
47	#41	C₂×D₅	1	6	yes	complemented
48	C₂×Q₈	—	5	6	no	—
49	C₂×Q₈	—	5	9	no	—
50	C₂×D₄	—	5	18	no	—
51	C₂×D₄	—	5	18	no	—
52	C₂×D₄	C₂×D₅	1	9	yes	complemented
53	C₂²×C₄	—	5	6	no	—
54	C₂²×C₄	—	5	9	no	—
55	C₂×C₅	—	2	9	no	—
56	C₂×C₅	C₂⁵	1	1	yes	characteristic
57	D₅	—	2	6	no	—
58	C₂³	—	5	9	no	—
59	C₂³	C₂²×D₅	1	6	yes	—
60	Q₈	—	5	18	no	—
61	Q₈	C₂²×D₅	1	2	yes	—
62	D₄	—	5	36	no	—
63	D₄	—	5	6	no	—
64	D₄	C₂²×D₅	1	18	yes	—
65	C₂×C₄	—	5	9	no	—
66	C₂×C₄	—	5	18	no	—
67	C₂×C₄	—	5	6	no	—
68	C₂×C₄	—	5	18	no	—
69	C₂×C₄	C₂²×D₅	1	9	yes	—
70	C₅	#585	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	4	6	no	—
72	C₂²	—	5	6	no	—
73	C₂²	—	20	9	no	—
74	C₂²	C₂³×D₅	1	9	yes	—
75	C₄	—	5	1	no	—
76	C₄	—	5	9	no	—
77	C₄	C₂³×D₅	1	6	yes	—
78	C₂	—	2	9	no	—
79	C₂	—	10	6	no	—
80	C₂	C₂⁴×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₅
commutator, socle, cosocle	C₂×C₅	C₂⁵
Fitting	C₅×H(5,𝔽₂)	C₂
hypercenter	H(5,𝔽₂)	D₅
hypocenter	C₅	#585
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₅×H(5,𝔽₂)	C₂	semidirect
2. #4150	C₂	semidirect
3. D₅×#41	C₂	semidirect
4. #4144	C₂	semidirect
5. C₂×#711	C₂	semidirect
6. C₅×#41	C₂²	semidirect
7. C₂×C₅×D₄	C₂²	semidirect
8. C₂×C₂²⋊D₅	C₂²	semidirect
9. C₂²×Dic₅	C₂²	semidirect
10. #713	C₂²	semidirect
11. Q₈×D₅	C₂²	semidirect
12. #711	C₂²	semidirect
13. #711	C₂²	semidirect
14. D₄×D₅	C₂²	semidirect
15. #709	C₂²	semidirect
16. C₂×C₄×D₅	C₂²	semidirect
17. C₂×Dic₁₀	C₂²	semidirect
33. H(5,𝔽₂)	D₅	semidirect
47. #41	C₂×D₅	semidirect
52. C₂×D₄	C₂×D₅	semidirect
70. C₅	#585	semidirect

Maximal subgroups

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

subgroup	index
C₅×H(5,𝔽₂)	2
#4150	2
D₅×#41	2
#4144	2
C₂×#711	2
#585	5