Subgroups — #14522588

Order 2000.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#6045	C₂	1	1	yes	maximal, characteristic
2	C₅²×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	#11758733	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅³	C₂²	1	1	yes	characteristic
5	C₅×#1039	—	2	1	no	—
6	C₅²×Dic₅	—	2	1	no	—
7	#87812	—	5	1	no	maximal
8	#87775	—	5	6	no	maximal
9	C₂×C₅³	D₄	1	1	yes	characteristic
10	C₅²×Q₈	—	5	1	no	—
11	#6045	D₅	1	1	yes	characteristic
12	C₅×Dic₁₀	—	5	6	no	—
13	C₅×Dic₁₀	D₅	1	6	yes	—
14	#6029	—	5	1	no	—
15	#6029	—	5	6	no	—
16	#6029	—	10	12	no	—
17	C₅³	Q₁₆	1	1	yes	characteristic, complemented, Sylow
18	C₄×C₅²	—	2	12	no	—
19	C₄×C₅²	—	10	1	no	—
20	C₄×C₅²	C₂×D₅	1	6	yes	—
21	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
22	#1039	—	2	1	no	—
23	C₅×Dic₅	—	2	6	no	—
24	C₅×Dic₅	—	10	6	no	—
25	#689	—	25	6	no	—
26	#689	—	25	1	no	—
27	C₂×C₅²	—	2	12	no	—
28	C₂×C₅²	C₂²⋊D₅	1	6	yes	—
29	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
30	C₅×Q₈	—	5	6	no	—
31	C₅×Q₈	—	25	1	no	—
32	Dic₁₀	—	5	6	no	—
33	Dic₁₀	C₅⋊D₅	1	1	yes	characteristic
34	#168	—	25	6	no	—
35	#168	—	25	1	no	—
36	#168	—	50	12	no	—
37	C₅²	—	2	12	no	—
38	C₅²	#689	1	6	yes	complemented
39	C₅²	#689	1	1	yes	characteristic, complemented
40	C₄×C₅	—	2	12	no	—
41	C₄×C₅	—	10	6	no	—
42	C₄×C₅	—	50	1	no	—
43	C₄×C₅	D₅²	1	6	yes	—
44	C₄×C₅	C₂×C₅⋊D₅	1	1	yes	characteristic
45	Dic₅	—	2	1	no	—
46	Dic₅	—	10	6	no	—
47	Q₁₆	—	125	1	no	Sylow
48	C₂×C₅	—	2	12	no	—
49	C₂×C₅	#6037	1	6	yes	—
50	C₂×C₅	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
51	Q₈	—	5	1	no	—
52	Q₈	—	25	1	no	—
53	C₈	—	125	1	no	—
54	C₅	—	2	12	no	—
55	C₅	#87775	1	6	yes	complemented
56	C₅	#87812	1	1	yes	characteristic, complemented
57	C₄	—	10	1	no	—
58	C₄	—	50	1	no	—
59	C₄	D₅×C₅⋊D₅	1	1	yes	characteristic
60	C₂	#11758752	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11758752
commutator, Fitting, cosocle	C₄×C₅³	C₂²
Frattini, hypercenter	C₄	D₅×C₅⋊D₅
socle, prime radical	C₂×C₅³	D₄
hypocenter	C₅³	Q₁₆
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
17. C₅³	Q₁₆	semidirect
38. C₅²	#689	semidirect
39. C₅²	#689	semidirect
55. C₅	#87775	semidirect
56. C₅	#87812	semidirect

Maximal subgroups

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

subgroup	index
C₅×#6045	2
C₅²×Dic₁₀	2
#11758733	2
#87812	5
#87775	5