Subgroups — #14522951

Order 2000.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₅⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	#11758780	C₂	1	2	yes	maximal, complemented
3	C₅×D₅²	C₄	1	1	yes	characteristic
4	C₅×#1044	C₄	1	1	yes	characteristic
5	C₅×#1043	C₂²	1	1	yes	characteristic
6	#6053⋊C₂	—	5	1	no	maximal
7	C₅×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
8	C₅²×D₅	—	2	1	no	—
9	C₂×C₅×AGL(1,𝔽₅)	—	10	1	no	—
10	D₅⋊AGL(1,𝔽₅)	D₅	1	1	yes	characteristic
11	#6053	—	5	2	no	—
12	C₅³	M₁₆	1	1	yes	characteristic, complemented, Sylow
13	C₂×C₅×D₅	—	10	1	no	—
14	D₅²	Dic₅	1	1	yes	characteristic
15	#1044	Dic₅	1	1	yes	characteristic
16	#1043	C₂×D₅	1	1	yes	characteristic
17	C₅×AGL(1,𝔽₅)	—	10	1	no	—
18	C₅×AGL(1,𝔽₅)	—	10	1	no	—
19	C₅×AGL(1,𝔽₅)	—	20	1	no	—
20	#681	—	25	1	no	maximal
21	C₂×C₅²	—	10	1	no	—
22	C₅⋊D₅	C₂×Dic₅	1	1	yes	characteristic
23	C₅×D₅	—	2	1	no	—
24	C₅×D₅	—	2	1	no	—
25	C₅×D₅	—	8	1	no	—
26	C₅×D₅	—	10	1	no	—
27	C₅×D₅	—	20	1	no	—
28	C₂×AGL(1,𝔽₅)	—	10	1	no	—
29	C₂×C₄×C₅	—	25	1	no	—
30	#168	—	25	2	no	—
31	C₅²	—	2	1	no	—
32	C₅²	—	4	1	no	—
33	C₅²	—	8	1	no	—
34	C₅²	—	16	1	no	—
35	C₅²	#681	1	1	yes	characteristic, complemented
36	C₂²×C₅	—	25	1	no	—
37	C₂×D₅	—	10	1	no	—
38	AGL(1,𝔽₅)	—	10	1	no	—
39	AGL(1,𝔽₅)	—	10	1	no	—
40	AGL(1,𝔽₅)	—	20	1	no	—
41	C₄×C₅	—	25	1	no	—
42	C₄×C₅	—	25	1	no	—
43	M₁₆	—	125	1	no	Sylow
44	C₂×C₅	—	10	1	no	—
45	C₂×C₅	—	10	1	no	—
46	C₂×C₅	—	25	1	no	—
47	C₂×C₅	—	40	1	no	—
48	D₅	—	2	1	no	—
49	D₅	—	10	1	no	—
50	D₅	—	20	1	no	—
51	C₂×C₄	—	25	1	no	—
52	C₈	—	125	2	no	—
53	C₅	—	2	1	no	—
54	C₅	—	4	1	no	—
55	C₅	—	8	1	no	—
56	C₅	—	16	1	no	—
57	C₅	#6053⋊C₂	1	1	yes	characteristic, complemented
58	C₂²	—	25	1	no	—
59	C₄	—	25	1	no	—
60	C₄	—	25	1	no	—
61	C₂	—	10	1	no	—
62	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator	C₅×C₅⋊D₅	C₂×C₄
Fitting, socle, hypocenter	C₅³	M₁₆
prime radical	C₅×D₅²	C₄
cosocle	C₅×#1043	C₂²
LCS	G ▷ C₅×C₅⋊D₅ ▷ C₅³
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. #11758780	C₂	semidirect
12. C₅³	M₁₆	semidirect
35. C₅²	#681	semidirect
57. C₅	#6053⋊C₂	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₅⋊AGL(1,𝔽₅)	2
#11758780	2
#6053⋊C₂	5
#681	25