Subgroups — #14522590

Order 2000.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11758769	C₂	1	1	yes	maximal, characteristic
2	C₅×#6045	C₂	1	1	yes	maximal, characteristic
3	C₅²×#168	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅³	C₂²	1	1	yes	characteristic
5	#92553	—	2	1	no	—
6	C₅×#1039	—	2	1	no	—
7	#87802	—	5	1	no	maximal
8	#87776	—	5	6	no	maximal
9	C₂×C₅³	D₄	1	1	yes	characteristic
10	#6045	—	5	1	no	—
11	#6045	—	5	6	no	—
12	#6045	—	10	12	no	—
13	#6045	D₅	1	1	yes	characteristic
14	C₅×Dic₁₀	—	5	6	no	—
15	C₅²×C₈	—	5	1	no	—
16	C₅×#168	D₅	1	6	yes	—
17	C₅³	Q₁₆	1	1	yes	characteristic, complemented, Sylow
18	C₄×C₅²	—	2	12	no	—
19	C₄×C₅²	C₂×D₅	1	6	yes	—
20	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
21	#1039	—	2	1	no	—
22	#1039	—	10	24	no	—
23	#1039	—	10	6	no	—
24	#1039	—	10	1	no	—
25	C₅×Dic₅	—	10	6	no	—
26	#689	—	25	1	no	—
27	Dic₂₀	—	25	6	no	—
28	C₂×C₅²	—	2	12	no	—
29	C₂×C₅²	D₂₀	1	6	yes	—
30	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
31	C₅×Q₈	—	25	1	no	—
32	Dic₁₀	—	5	6	no	—
33	Dic₁₀	—	25	6	no	—
34	Dic₁₀	—	25	1	no	—
35	Dic₁₀	—	50	12	no	—
36	C₅×C₈	—	5	6	no	—
37	#168	C₅⋊D₅	1	1	yes	characteristic
38	C₅²	—	2	12	no	—
39	C₅²	Dic₂₀	1	6	yes	complemented
40	C₅²	#689	1	1	yes	characteristic, complemented
41	C₄×C₅	—	2	12	no	—
42	C₄×C₅	—	50	1	no	—
43	C₄×C₅	D₅²	1	6	yes	—
44	C₄×C₅	C₂×C₅⋊D₅	1	1	yes	characteristic
45	Dic₅	—	10	6	no	—
46	Dic₅	—	50	24	no	—
47	Dic₅	—	50	6	no	—
48	Dic₅	—	50	1	no	—
49	Q₁₆	—	125	1	no	Sylow
50	C₂×C₅	—	2	12	no	—
51	C₂×C₅	#6038	1	6	yes	—
52	C₂×C₅	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
53	Q₈	—	25	1	no	—
54	Q₈	—	125	1	no	—
55	C₈	—	5	1	no	—
56	C₅	—	2	12	no	—
57	C₅	#87776	1	6	yes	complemented
58	C₅	#87802	1	1	yes	characteristic, complemented
59	C₄	—	50	1	no	—
60	C₄	—	250	1	no	—
61	C₄	D₅×C₅⋊D₅	1	1	yes	characteristic
62	C₂	#11758754	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11758754
commutator, Fitting, cosocle	C₄×C₅³	C₂²
Frattini, hypercenter	C₄	D₅×C₅⋊D₅
socle, prime radical	C₂×C₅³	D₄
hypocenter	C₅³	Q₁₆
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
17. C₅³	Q₁₆	semidirect
39. C₅²	Dic₂₀	semidirect
40. C₅²	#689	semidirect
57. C₅	#87776	semidirect
58. C₅	#87802	semidirect

Maximal subgroups

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

subgroup	index
#11758769	2
C₅×#6045	2
C₅²×#168	2
#87802	5
#87776	5