Subgroups — #14522342

Order 2000.

66 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	#11758644	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×D₂₅	C₄	1	1	yes	characteristic
4	C₄×C₅×C₂₅	C₄	1	1	yes	characteristic
5	C₅×Dic₂₅	C₂²	1	1	yes	characteristic
6	#87849	—	5	1	no	maximal
7	#87735	—	5	1	no	maximal
8	C₂×C₅×C₂₅	C₂×C₄	1	1	yes	characteristic
9	C₅×D₂₅	—	2	1	no	—
10	C₄×C₅×D₅	—	5	1	no	—
11	#6032	—	5	2	no	—
12	C₄×D₂₅	D₅	1	1	yes	characteristic
13	#6016	—	5	2	no	—
14	C₅×C₂₅	M₁₆	1	1	yes	characteristic, complemented, Sylow
15	C₂×C₅×D₅	—	5	1	no	—
16	C₄×C₅²	AGL(1,𝔽₅)	1	1	yes	characteristic
17	C₅×Dic₅	—	5	1	no	—
18	C₂×D₂₅	Dic₅	1	1	yes	characteristic
19	C₄×C₂₅	—	4	1	no	—
20	C₄×C₂₅	Dic₅	1	1	yes	characteristic
21	Dic₂₅	C₂×D₅	1	1	yes	characteristic
22	#700	—	25	1	no	—
23	#681	—	25	1	no	—
24	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₅×D₅	—	10	1	no	—
26	C₂×C₂₅	—	4	1	no	—
27	C₂×C₂₅	C₂×Dic₅	1	1	yes	characteristic
28	D₂₅	—	2	1	no	—
29	C₂×C₄×C₅	—	25	1	no	—
30	C₄×D₅	—	5	1	no	—
31	#170	—	25	2	no	—
32	#168	—	25	2	no	—
33	C₅²	#700	1	1	yes	characteristic
34	C₂₅	—	4	1	no	—
35	C₂₅	#681	1	1	yes	characteristic, complemented
36	C₂²×C₅	—	25	1	no	—
37	C₂×D₅	—	5	1	no	—
38	C₄×C₅	—	4	1	no	—
39	C₄×C₅	—	25	1	no	—
40	C₄×C₅	#1035	1	1	yes	characteristic
41	C₄×C₅	#1042	1	1	yes	characteristic
42	Dic₅	—	5	1	no	—
43	M₁₆	—	125	1	no	Sylow
44	C₂×C₅	—	4	1	no	—
45	C₂×C₅	—	50	1	no	—
46	C₂×C₅	C₂×#1035	1	1	yes	characteristic
47	C₂×C₅	C₂×#1042	1	1	yes	characteristic
48	D₅	—	10	1	no	—
49	C₂×C₄	—	25	1	no	—
50	C₈	—	125	2	no	—
51	C₅	—	4	1	no	—
52	C₅	#87735	1	1	yes	characteristic, complemented
53	C₅	#87849	1	1	yes	characteristic
54	C₂²	—	25	1	no	—
55	C₄	—	25	1	no	—
56	C₄	#92534	1	1	yes	characteristic
57	C₂	—	50	1	no	—
58	C₂	C₂×#92534	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#92534
commutator	C₂×C₅×C₂₅	C₂×C₄
Frattini	C₂×C₅	C₂×#1042
Fitting	C₄×C₅×C₂₅	C₄
socle	C₂×C₅²	C₂×AGL(1,𝔽₅)
prime radical	C₂×C₅×D₂₅	C₄
cosocle	C₅×Dic₂₅	C₂²
hypercenter	C₄	#92534
hypocenter	C₅×C₂₅	M₁₆
LCS	G ▷ 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. #11758644	C₂	semidirect
14. C₅×C₂₅	M₁₆	semidirect
35. C₂₅	#681	semidirect
52. C₅	#87735	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
#11758644	2
#87849	5
#87735	5