Subgroups — #14522113

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	C₂₅×#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#6014	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅⋊D₂₅	C₄	1	1	yes	characteristic
5	C₄×C₅×C₂₅	C₂²	1	1	yes	characteristic
6	#92524	C₄	1	1	yes	characteristic
7	#87769	—	5	1	no	maximal
8	#87712	—	5	1	no	maximal
9	C₂×C₅×C₂₅	C₂×C₄	1	1	yes	characteristic
10	C₅⋊D₂₅	—	2	1	no	—
11	C₄×C₅⋊D₅	—	5	1	no	—
12	C₅×#168	—	5	1	no	—
13	C₅×#168	D₅	1	1	yes	characteristic
14	C₄×D₂₅	—	5	1	no	—
15	C₄×D₂₅	—	10	2	no	—
16	C₈×C₂₅	—	5	1	no	—
17	#6014	D₅	1	1	yes	characteristic, complemented
18	C₅×C₂₅	M₁₆	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₅⋊D₅	—	5	1	no	—
20	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
21	#1039	—	5	1	no	—
22	C₂×D₂₅	—	5	1	no	—
23	C₂×D₂₅	—	10	2	no	—
24	C₄×C₂₅	—	2	2	no	—
25	C₄×C₂₅	C₂×D₅	1	1	yes	characteristic
26	Dic₂₅	—	5	1	no	—
27	Dic₂₅	—	10	2	no	—
28	#676	—	25	1	no	—
29	#676	—	25	1	no	—
30	C₂×C₅²	C₄×D₅	1	1	yes	characteristic
31	C₅⋊D₅	—	10	1	no	—
32	C₂×C₂₅	—	2	2	no	—
33	C₂×C₂₅	C₄×D₅	1	1	yes	characteristic
34	D₂₅	—	10	4	no	—
35	D₂₅	—	10	1	no	—
36	C₄×D₅	—	25	1	no	—
37	C₄×D₅	—	25	1	no	—
38	C₄×D₅	—	50	2	no	—
39	C₅×C₈	—	5	1	no	—
40	C₅×C₈	—	25	1	no	—
41	#168	—	5	1	no	—
42	#168	D₂₅	1	1	yes	characteristic, complemented
43	C₅²	#676	1	1	yes	characteristic
44	C₂₅	—	2	2	no	—
45	C₂₅	#676	1	1	yes	characteristic, complemented
46	C₂×D₅	—	25	1	no	—
47	C₂×D₅	—	25	1	no	—
48	C₂×D₅	—	50	2	no	—
49	C₄×C₅	—	2	2	no	—
50	C₄×C₅	C₂×D₂₅	1	1	yes	characteristic
51	C₄×C₅	D₅²	1	1	yes	characteristic
52	Dic₅	—	25	1	no	—
53	Dic₅	—	25	1	no	—
54	Dic₅	—	50	2	no	—
55	M₁₆	—	125	1	no	Sylow
56	C₂×C₅	—	2	2	no	—
57	C₂×C₅	C₄×D₂₅	1	1	yes	characteristic
58	C₂×C₅	#6036	1	1	yes	characteristic
59	D₅	—	50	4	no	—
60	D₅	—	50	1	no	—
61	D₅	—	50	1	no	—
62	C₂×C₄	—	125	1	no	—
63	C₈	—	5	1	no	—
64	C₈	—	25	1	no	—
65	C₅	—	2	2	no	—
66	C₅	#87712	1	1	yes	characteristic, complemented
67	C₅	#87769	1	1	yes	characteristic
68	C₂²	—	125	1	no	—
69	C₄	—	125	1	no	—
70	C₄	D₅×D₂₅	1	1	yes	characteristic
71	C₂	—	250	1	no	—
72	C₂	#11758652	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	D₅×D₂₅
commutator	C₂×C₅×C₂₅	C₂×C₄
Frattini	C₄×C₅	D₅²
Fitting, cosocle	C₄×C₅×C₂₅	C₂²
socle	C₂×C₅²	C₄×D₅
prime radical	C₂×C₅⋊D₂₅	C₄
hypocenter	C₅×C₂₅	M₁₆
LCS	G ▷ C₂×C₅×C₂₅ ▷ C₅×C₂₅
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₂₅×#168	C₂	semidirect
3. C₅×#6014	C₂	semidirect
17. #6014	D₅	semidirect
18. C₅×C₂₅	M₁₆	semidirect
42. #168	D₂₅	semidirect
45. C₂₅	#676	semidirect
66. C₅	#87712	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
C₂₅×#168	2
C₅×#6014	2
#87769	5
#87712	5