Subgroups — #14522115

Order 2000.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₅⋊D₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₅×Dic₅₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₅×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅⋊D₂₅	—	2	1	no	—
5	C₄×C₅×C₂₅	C₂²	1	1	yes	characteristic
6	C₅×Dic₂₅	—	2	1	no	—
7	#87774	—	5	1	no	maximal
8	#87713	—	5	1	no	maximal
9	C₂×C₅×C₂₅	D₄	1	1	yes	characteristic
10	C₅⋊D₂₅	—	4	1	no	—
11	C₄⋊(C₅⋊D₅)	—	5	1	no	—
12	C₅×Dic₁₀	—	5	1	no	—
13	C₅×#168	D₅	1	1	yes	characteristic
14	D₁₀₀	—	5	1	no	—
15	D₁₀₀	—	10	2	no	—
16	Dic₅₀	D₅	1	1	yes	characteristic, complemented
17	C₈×C₂₅	—	5	1	no	—
18	C₅×C₂₅	SD₁₆	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₅⋊D₅	—	10	1	no	—
20	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
21	C₅×Dic₅	—	10	1	no	—
22	C₂×D₂₅	—	10	4	no	—
23	C₂×D₂₅	—	10	1	no	—
24	C₄×C₂₅	—	2	2	no	—
25	C₄×C₂₅	C₂×D₅	1	1	yes	characteristic
26	Dic₂₅	—	2	1	no	—
27	#688	—	25	1	no	—
28	#677	—	25	1	no	—
29	C₂×C₅²	D₂₀	1	1	yes	characteristic
30	C₅⋊D₅	—	20	1	no	—
31	C₂×C₂₅	—	2	2	no	—
32	C₂×C₂₅	C₂²⋊D₅	1	1	yes	characteristic
33	D₂₅	—	20	4	no	—
34	D₂₅	—	20	1	no	—
35	C₅×Q₈	—	25	1	no	—
36	D₂₀	—	25	1	no	—
37	D₂₀	—	25	1	no	—
38	D₂₀	—	50	2	no	—
39	Dic₁₀	—	5	1	no	—
40	C₅×C₈	—	5	1	no	—
41	#168	D₂₅	1	1	yes	characteristic, complemented
42	C₅²	#677	1	1	yes	characteristic
43	C₂₅	—	2	2	no	—
44	C₂₅	#688	1	1	yes	characteristic, complemented
45	C₂×D₅	—	50	4	no	—
46	C₂×D₅	—	50	1	no	—
47	C₂×D₅	—	50	1	no	—
48	C₄×C₅	—	2	2	no	—
49	C₄×C₅	—	50	1	no	—
50	C₄×C₅	C₂×D₂₅	1	1	yes	characteristic
51	C₄×C₅	D₅²	1	1	yes	characteristic
52	Dic₅	—	10	1	no	—
53	SD₁₆	—	125	1	no	Sylow
54	C₂×C₅	—	2	2	no	—
55	C₂×C₅	D₁₀₀	1	1	yes	characteristic
56	C₂×C₅	#6038	1	1	yes	characteristic
57	D₅	—	100	4	no	—
58	D₅	—	100	1	no	—
59	D₅	—	100	1	no	—
60	Q₈	—	25	1	no	—
61	D₄	—	125	1	no	—
62	C₈	—	5	1	no	—
63	C₅	—	2	2	no	—
64	C₅	#87713	1	1	yes	characteristic, complemented
65	C₅	#87774	1	1	yes	characteristic
66	C₂²	—	250	1	no	—
67	C₄	—	50	1	no	—
68	C₄	D₅×D₂₅	1	1	yes	characteristic
69	C₂	—	500	1	no	—
70	C₂	#11758653	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11758653
commutator, Fitting, cosocle	C₄×C₅×C₂₅	C₂²
Frattini	C₄×C₅	D₅²
socle	C₂×C₅²	D₂₀
prime radical	C₄⋊(C₅⋊D₂₅)	C₂
hypercenter	C₄	D₅×D₂₅
hypocenter	C₅×C₂₅	SD₁₆
LCS	G ▷ C₄×C₅×C₂₅ ▷ C₂×C₅×C₂₅ ▷ C₅×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₅×Dic₅₀	C₂	semidirect
3. C₂₅×#168	C₂	semidirect
16. Dic₅₀	D₅	semidirect
18. C₅×C₂₅	SD₁₆	semidirect
41. #168	D₂₅	semidirect
44. C₂₅	#688	semidirect
64. C₅	#87713	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₅⋊D₂₅)	2
C₅×Dic₅₀	2
C₂₅×#168	2
#87774	5
#87713	5