Subgroups — C₅²×#686 #14522617

Order 2000.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₅³	—	2	1	no	—
5	C₂×C₅²×D₅	—	2	1	no	—
6	C₄×C₅³	C₂²	1	1	yes	characteristic
7	C₅²×D₈	—	5	1	no	maximal
8	C₅×#686	C₅	1	6	yes	maximal, direct product factor
9	C₂×C₅³	—	4	1	no	—
10	C₂×C₅³	D₄	1	1	yes	characteristic
11	C₅²×D₅	—	4	1	no	—
12	C₅²×D₄	—	2	12	no	—
13	C₅²×D₄	—	5	1	no	—
14	C₅²×D₄	D₅	1	1	yes	characteristic, complemented
15	C₅²×D₄	C₂×C₅	1	6	yes	complemented
16	C₅×D₂₀	C₂×C₅	1	6	yes	complemented
17	C₅²×C₈	—	5	1	no	—
18	C₅×#168	C₂×C₅	1	6	yes	complemented
19	C₅³	D₈	1	1	yes	characteristic, complemented, Sylow
20	C₂²×C₅²	—	2	24	no	—
21	C₂²×C₅²	—	2	1	no	—
22	C₂²×C₅²	—	2	6	no	—
23	C₂²×C₅²	—	10	1	no	—
24	C₂×C₅×D₅	—	2	6	no	—
25	C₄×C₅²	—	2	12	no	—
26	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
27	C₄×C₅²	C₂²×C₅	1	6	yes	—
28	C₅×D₈	—	5	6	no	—
29	#686	C₅²	1	1	yes	characteristic, direct product factor
30	C₂×C₅²	—	2	12	no	—
31	C₂×C₅²	—	4	24	no	—
32	C₂×C₅²	—	4	1	no	—
33	C₂×C₅²	—	4	6	no	—
34	C₂×C₅²	—	20	1	no	—
35	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
36	C₂×C₅²	C₅×D₄	1	6	yes	—
37	C₅×D₅	—	4	6	no	—
38	C₅×D₄	—	2	12	no	—
39	C₅×D₄	—	5	6	no	—
40	C₅×D₄	C₅×D₅	1	6	yes	complemented
41	C₅×D₄	C₂×C₅²	1	1	yes	characteristic, complemented
42	D₂₀	C₂×C₅²	1	1	yes	characteristic, complemented
43	C₅×C₈	—	5	6	no	—
44	#168	C₂×C₅²	1	1	yes	characteristic, complemented
45	C₅²	—	2	12	no	—
46	C₅²	#686	1	1	yes	characteristic, direct product factor
47	C₅²	C₅×D₈	1	6	yes	complemented
48	C₂²×C₅	—	2	24	no	—
49	C₂²×C₅	—	2	6	no	—
50	C₂²×C₅	—	2	1	no	—
51	C₂²×C₅	—	10	6	no	—
52	C₂×D₅	—	2	1	no	—
53	C₄×C₅	—	2	12	no	—
54	C₄×C₅	C₂×C₅×D₅	1	6	yes	—
55	C₄×C₅	C₂²×C₅²	1	1	yes	characteristic
56	D₈	—	5	1	no	Sylow
57	C₂×C₅	—	2	12	no	—
58	C₂×C₅	—	4	24	no	—
59	C₂×C₅	—	4	1	no	—
60	C₂×C₅	—	4	6	no	—
61	C₂×C₅	—	20	6	no	—
62	C₂×C₅	C₅×C₂²⋊D₅	1	6	yes	—
63	C₂×C₅	C₅²×D₄	1	1	yes	characteristic
64	D₅	—	4	1	no	—
65	D₄	—	5	1	no	—
66	D₄	C₅²×D₅	1	1	yes	characteristic, complemented
67	C₈	—	5	1	no	—
68	C₅	—	2	12	no	—
69	C₅	C₅×#686	1	6	yes	direct product factor
70	C₅	C₅²×D₈	1	1	yes	characteristic, complemented
71	C₂²	—	2	1	no	—
72	C₂²	—	10	1	no	—
73	C₄	C₂×C₅²×D₅	1	1	yes	characteristic
74	C₂	—	4	1	no	—
75	C₂	—	20	1	no	—
76	C₂	C₅²×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅²	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂²×C₅²
Frattini	C₄	C₂×C₅²×D₅
Fitting	C₅³×D₄	C₂
socle	C₂×C₅³	D₄
hypercenter	C₅²×D₄	D₅
hypocenter	C₅	C₅²×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅² ◁ C₄×C₅² ◁ C₅²×D₄

Complements

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

N	H ≅ G/N	type
1. C₅³×D₄	C₂	semidirect
2. C₅²×D₂₀	C₂	semidirect
3. C₅²×#168	C₂	semidirect
8. C₅×#686	C₅	direct (N × H)
14. C₅²×D₄	D₅	semidirect
15. C₅²×D₄	C₂×C₅	semidirect
16. C₅×D₂₀	C₂×C₅	semidirect
18. C₅×#168	C₂×C₅	semidirect
19. C₅³	D₈	semidirect
29. #686	C₅²	direct (N × H)
40. C₅×D₄	C₅×D₅	semidirect
41. C₅×D₄	C₂×C₅²	semidirect
42. D₂₀	C₂×C₅²	semidirect
44. #168	C₂×C₅²	semidirect
46. C₅²	#686	direct (N × H)
47. C₅²	C₅×D₈	semidirect
66. D₄	C₅²×D₅	semidirect
69. C₅	C₅×#686	direct (N × H)
70. C₅	C₅²×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₅³×D₄	2
C₅²×D₂₀	2
C₅²×#168	2
C₅²×D₈	5
C₅×#686	5

Subgroups — C52×#686 #14522617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#686 #14522617