Subgroups — #14522583

Order 2000.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic
2	C₅²×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	#11758733	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₅⋊D₅	—	2	1	no	—
5	C₄×C₅³	C₂²	1	1	yes	characteristic
6	C₅²×Dic₅	—	2	1	no	—
7	#87811	—	5	1	no	maximal
8	#87772	—	5	6	no	maximal
9	C₂×C₅³	D₄	1	1	yes	characteristic
10	C₅×C₅⋊D₅	—	4	1	no	—
11	C₅²×Q₈	—	5	1	no	—
12	C₄⋊(C₅⋊D₅)	D₅	1	1	yes	characteristic
13	C₅×D₂₀	—	5	6	no	—
14	C₅×Dic₁₀	D₅	1	6	yes	complemented
15	#6029	—	5	1	no	—
16	#6029	—	5	6	no	—
17	#6029	—	10	12	no	—
18	C₅³	SD₁₆	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₅⋊D₅	—	2	1	no	—
20	C₂×C₅×D₅	—	10	6	no	—
21	C₄×C₅²	—	2	12	no	—
22	C₄×C₅²	—	10	1	no	—
23	C₄×C₅²	C₂×D₅	1	6	yes	—
24	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
25	C₅×Dic₅	—	2	6	no	—
26	#688	—	25	6	no	—
27	#687	—	25	1	no	—
28	C₂×C₅²	—	2	12	no	—
29	C₂×C₅²	C₂²⋊D₅	1	6	yes	—
30	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
31	C₅⋊D₅	—	4	1	no	—
32	C₅×D₅	—	20	6	no	—
33	C₅×Q₈	—	5	6	no	—
34	C₅×D₄	—	25	1	no	—
35	D₂₀	—	5	6	no	—
36	Dic₁₀	C₅⋊D₅	1	1	yes	characteristic, complemented
37	#168	—	25	6	no	—
38	#168	—	25	1	no	—
39	#168	—	50	12	no	—
40	C₅²	—	2	12	no	—
41	C₅²	#687	1	1	yes	characteristic, complemented
42	C₅²	#688	1	6	yes	complemented
43	C₂²×C₅	—	50	1	no	—
44	C₂×D₅	—	10	6	no	—
45	C₄×C₅	—	2	12	no	—
46	C₄×C₅	—	10	6	no	—
47	C₄×C₅	D₅²	1	6	yes	—
48	C₄×C₅	C₂×C₅⋊D₅	1	1	yes	characteristic
49	Dic₅	—	2	1	no	—
50	SD₁₆	—	125	1	no	Sylow
51	C₂×C₅	—	2	12	no	—
52	C₂×C₅	—	100	1	no	—
53	C₂×C₅	#6037	1	6	yes	—
54	C₂×C₅	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
55	D₅	—	20	6	no	—
56	Q₈	—	5	1	no	—
57	D₄	—	25	1	no	—
58	C₈	—	125	1	no	—
59	C₅	—	2	12	no	—
60	C₅	#87772	1	6	yes	complemented
61	C₅	#87811	1	1	yes	characteristic, complemented
62	C₂²	—	50	1	no	—
63	C₄	—	10	1	no	—
64	C₄	D₅×C₅⋊D₅	1	1	yes	characteristic
65	C₂	—	100	1	no	—
66	C₂	#11758752	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11758752
commutator, Fitting, cosocle	C₄×C₅³	C₂²
Frattini, hypercenter	C₄	D₅×C₅⋊D₅
socle	C₂×C₅³	D₄
prime radical	C₅×C₄⋊(C₅⋊D₅)	C₂
hypocenter	C₅³	SD₁₆
LCS	G ▷ C₄×C₅³ ▷ C₂×C₅³ ▷ C₅³
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₅²×Dic₁₀	C₂	semidirect
3. #11758733	C₂	semidirect
14. C₅×Dic₁₀	D₅	semidirect
18. C₅³	SD₁₆	semidirect
36. Dic₁₀	C₅⋊D₅	semidirect
41. C₅²	#687	semidirect
42. C₅²	#688	semidirect
60. C₅	#87772	semidirect
61. C₅	#87811	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₄⋊(C₅⋊D₅)	2
C₅²×Dic₁₀	2
#11758733	2
#87811	5
#87772	5