Subgroups — #14522587

Order 2000.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₅⋊(C₅⋊D₅))	C₂	1	1	yes	maximal, characteristic
2	C₅×#6045	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅⋊(C₅⋊D₅)	—	2	1	no	—
5	C₄×C₅³	C₂²	1	1	yes	characteristic
6	C₅×#1039	—	2	1	no	—
7	#87800	—	5	1	no	maximal
8	#87774	—	5	6	no	maximal
9	C₂×C₅³	D₄	1	1	yes	characteristic
10	C₅⋊(C₅⋊D₅)	—	4	1	no	—
11	C₄⋊(C₅⋊D₅)	—	5	1	no	—
12	C₄⋊(C₅⋊D₅)	—	5	6	no	—
13	C₄⋊(C₅⋊D₅)	—	10	12	no	—
14	#6045	D₅	1	1	yes	characteristic, complemented
15	C₅×Dic₁₀	—	5	6	no	—
16	C₅²×C₈	—	5	1	no	—
17	C₅×#168	D₅	1	6	yes	complemented
18	C₅³	SD₁₆	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₅⋊D₅	—	10	24	no	—
20	C₂×C₅⋊D₅	—	10	6	no	—
21	C₂×C₅⋊D₅	—	10	1	no	—
22	C₄×C₅²	—	2	12	no	—
23	C₄×C₅²	C₂×D₅	1	6	yes	—
24	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
25	#1039	—	2	1	no	—
26	C₅×Dic₅	—	10	6	no	—
27	#688	—	25	1	no	—
28	#677	—	25	6	no	—
29	C₂×C₅²	—	2	12	no	—
30	C₂×C₅²	D₂₀	1	6	yes	—
31	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
32	C₅⋊D₅	—	20	24	no	—
33	C₅⋊D₅	—	20	6	no	—
34	C₅⋊D₅	—	20	1	no	—
35	C₅×Q₈	—	25	1	no	—
36	D₂₀	—	25	6	no	—
37	D₂₀	—	25	1	no	—
38	D₂₀	—	50	12	no	—
39	Dic₁₀	—	5	6	no	—
40	C₅×C₈	—	5	6	no	—
41	#168	C₅⋊D₅	1	1	yes	characteristic, complemented
42	C₅²	—	2	12	no	—
43	C₅²	#677	1	6	yes	complemented
44	C₅²	#688	1	1	yes	characteristic, complemented
45	C₂×D₅	—	50	24	no	—
46	C₂×D₅	—	50	6	no	—
47	C₂×D₅	—	50	1	no	—
48	C₄×C₅	—	2	12	no	—
49	C₄×C₅	—	50	1	no	—
50	C₄×C₅	D₅²	1	6	yes	—
51	C₄×C₅	C₂×C₅⋊D₅	1	1	yes	characteristic
52	Dic₅	—	10	6	no	—
53	SD₁₆	—	125	1	no	Sylow
54	C₂×C₅	—	2	12	no	—
55	C₂×C₅	#6038	1	6	yes	—
56	C₂×C₅	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
57	D₅	—	100	24	no	—
58	D₅	—	100	6	no	—
59	D₅	—	100	1	no	—
60	Q₈	—	25	1	no	—
61	D₄	—	125	1	no	—
62	C₈	—	5	1	no	—
63	C₅	—	2	12	no	—
64	C₅	#87774	1	6	yes	complemented
65	C₅	#87800	1	1	yes	characteristic, complemented
66	C₂²	—	250	1	no	—
67	C₄	—	50	1	no	—
68	C₄	D₅×C₅⋊D₅	1	1	yes	characteristic
69	C₂	—	500	1	no	—
70	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	C₂×C₅³	D₄
prime radical	C₄⋊(C₅⋊(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. C₅×#6045	C₂	semidirect
3. C₅²×#168	C₂	semidirect
14. #6045	D₅	semidirect
17. C₅×#168	D₅	semidirect
18. C₅³	SD₁₆	semidirect
41. #168	C₅⋊D₅	semidirect
43. C₅²	#677	semidirect
44. C₅²	#688	semidirect
64. C₅	#87774	semidirect
65. C₅	#87800	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₅⋊(C₅⋊D₅))	2
C₅×#6045	2
C₅²×#168	2
#87800	5
#87774	5