Subgroups — #14522773

Order 2000.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11758758	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758751	C₂	1	3	yes	maximal
3	C₅×#6039	C₂	1	3	yes	maximal, complemented
4	C₂×C₅⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
5	C₅×#1039	C₂²	1	3	yes	—
6	C₅²×Dic₅	C₂²	1	3	yes	—
7	#87872	—	5	3	no	maximal
8	C₂×C₅³	C₂³	1	1	yes	characteristic
9	C₅⋊(C₅⋊D₅)	Q₈	1	2	yes	complemented
10	C₄×C₅⋊D₅	—	5	3	no	—
11	C₅×Dic₁₀	—	5	6	no	—
12	#6039	—	5	3	no	—
13	#6039	D₅	1	3	yes	complemented
14	#6036	—	5	6	no	—
15	#6036	—	10	6	no	—
16	C₅³	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₅⋊D₅	—	5	3	no	—
18	C₂×C₅⋊D₅	—	10	6	no	—
19	C₂×C₅⋊D₅	—	20	4	no	—
20	C₄×C₅²	—	5	3	no	—
21	#1039	C₂×D₅	1	3	yes	—
22	C₅×Dic₅	—	2	6	no	—
23	C₅×Dic₅	—	5	6	no	—
24	C₅×Dic₅	—	10	6	no	—
25	C₅×Dic₅	C₂×D₅	1	6	yes	—
26	Q₈×D₅	—	25	3	no	—
27	C₂×C₅²	—	2	6	no	—
28	C₂×C₅²	—	4	4	no	—
29	C₂×C₅²	C₂²×D₅	1	3	yes	—
30	C₅⋊D₅	—	5	6	no	—
31	C₅⋊D₅	—	10	12	no	—
32	C₅⋊D₅	—	20	8	no	—
33	C₅×Q₈	—	25	3	no	—
34	C₄×D₅	—	25	6	no	—
35	C₄×D₅	—	25	3	no	—
36	C₄×D₅	—	50	6	no	—
37	Dic₁₀	—	5	6	no	—
38	Dic₁₀	—	25	3	no	—
39	C₅²	—	2	6	no	—
40	C₅²	—	4	4	no	—
41	C₅²	Q₈×D₅	1	3	yes	complemented
42	C₂×D₅	—	25	3	no	—
43	C₂×D₅	—	50	6	no	—
44	C₂×D₅	—	100	4	no	—
45	C₄×C₅	—	5	6	no	—
46	C₄×C₅	—	10	6	no	—
47	C₄×C₅	—	25	3	no	—
48	Dic₅	—	5	6	no	—
49	Dic₅	—	10	6	no	—
50	Dic₅	D₅²	1	3	yes	—
51	C₂×Q₈	—	125	1	no	Sylow
52	C₂×C₅	—	2	6	no	—
53	C₂×C₅	—	4	4	no	—
54	C₂×C₅	C₂×D₅²	1	3	yes	—
55	D₅	—	25	6	no	—
56	D₅	—	50	12	no	—
57	D₅	—	100	8	no	—
58	Q₈	—	25	3	no	—
59	Q₈	—	125	1	no	—
60	C₂×C₄	—	125	3	no	—
61	C₅	—	2	6	no	—
62	C₅	—	4	4	no	—
63	C₅	#87872	1	3	yes	complemented
64	C₂²	—	125	1	no	—
65	C₄	—	5	3	no	—
66	C₄	—	25	3	no	—
67	C₂	—	125	2	no	—
68	C₂	D₅³	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅³
commutator, Fitting, socle, cosocle	C₂×C₅³	C₂³
prime radical	C₂×C₅⋊(C₅⋊D₅)	C₂²
hypocenter	C₅³	C₂×Q₈
LCS	G ▷ C₂×C₅³ ▷ C₅³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. #11758758	C₂	semidirect
3. C₅×#6039	C₂	semidirect
9. C₅⋊(C₅⋊D₅)	Q₈	semidirect
13. #6039	D₅	semidirect
16. C₅³	C₂×Q₈	semidirect
41. C₅²	Q₈×D₅	semidirect
63. C₅	#87872	semidirect

Maximal subgroups

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

subgroup	index
#11758758	2
#11758751	2
C₅×#6039	2
#87872	5