Subgroups — #14522599

Order 2000.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₅×#6029	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×C₅⋊D₅	C₄	1	1	yes	characteristic
4	C₄×C₅³	C₂²	1	1	yes	characteristic
5	C₅×#1039	C₄	1	1	yes	characteristic
6	#87769	—	5	1	no	maximal
7	#87768	—	5	2	no	maximal
8	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
9	C₅×C₅⋊D₅	—	2	1	no	—
10	C₄×C₅⋊D₅	D₅	1	1	yes	characteristic
11	C₄×C₅×D₅	—	5	2	no	—
12	C₄×C₅×D₅	—	10	2	no	—
13	#6029	D₅	1	2	yes	complemented
14	C₅×#168	—	5	2	no	—
15	C₅×#168	—	5	2	no	—
16	C₅×#168	—	10	4	no	—
17	C₅³	M₁₆	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₅⋊D₅	Dic₅	1	1	yes	characteristic
19	C₂×C₅×D₅	—	5	2	no	—
20	C₂×C₅×D₅	—	10	2	no	—
21	C₄×C₅²	—	2	4	no	—
22	C₄×C₅²	—	2	2	no	—
23	C₄×C₅²	—	4	4	no	—
24	C₄×C₅²	C₂×D₅	1	2	yes	—
25	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
26	#1039	Dic₅	1	1	yes	characteristic
27	C₅×Dic₅	—	5	2	no	—
28	C₅×Dic₅	—	10	2	no	—
29	#681	—	25	1	no	—
30	#676	—	25	2	no	—
31	C₂×C₅²	—	2	4	no	—
32	C₂×C₅²	—	2	2	no	—
33	C₂×C₅²	—	4	4	no	—
34	C₂×C₅²	C₄×D₅	1	2	yes	—
35	C₂×C₅²	C₂×Dic₅	1	1	yes	characteristic
36	C₅⋊D₅	—	2	1	no	—
37	C₅×D₅	—	10	4	no	—
38	C₅×D₅	—	10	2	no	—
39	C₂×C₄×C₅	—	25	1	no	—
40	C₄×D₅	—	5	2	no	—
41	C₄×D₅	—	10	2	no	—
42	C₅×C₈	—	25	2	no	—
43	#168	—	5	2	no	—
44	#168	—	5	2	no	—
45	#168	—	10	4	no	—
46	C₅²	—	2	4	no	—
47	C₅²	—	2	2	no	—
48	C₅²	—	4	4	no	—
49	C₅²	#676	1	2	yes	complemented
50	C₅²	#681	1	1	yes	characteristic, complemented
51	C₂²×C₅	—	25	1	no	—
52	C₂×D₅	—	5	2	no	—
53	C₂×D₅	—	10	2	no	—
54	C₄×C₅	—	2	4	no	—
55	C₄×C₅	—	2	2	no	—
56	C₄×C₅	—	4	4	no	—
57	C₄×C₅	—	25	1	no	—
58	C₄×C₅	D₅²	1	2	yes	—
59	C₄×C₅	D₅²	1	1	yes	characteristic
60	Dic₅	—	5	2	no	—
61	Dic₅	—	10	2	no	—
62	M₁₆	—	125	1	no	Sylow
63	C₂×C₅	—	2	4	no	—
64	C₂×C₅	—	2	2	no	—
65	C₂×C₅	—	4	4	no	—
66	C₂×C₅	—	50	1	no	—
67	C₂×C₅	D₅×Dic₅	1	2	yes	—
68	C₂×C₅	#6036	1	1	yes	characteristic
69	D₅	—	10	4	no	—
70	D₅	—	10	2	no	—
71	C₂×C₄	—	25	1	no	—
72	C₈	—	25	2	no	—
73	C₅	—	2	2	no	—
74	C₅	—	2	4	no	—
75	C₅	—	4	4	no	—
76	C₅	#87768	1	2	yes	complemented
77	C₅	#87769	1	1	yes	characteristic, complemented
78	C₂²	—	25	1	no	—
79	C₄	—	25	1	no	—
80	C₄	#92566	1	1	yes	characteristic
81	C₂	—	50	1	no	—
82	C₂	#11758756	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	#92566
commutator, socle	C₂×C₅³	C₂×C₄
Fitting, cosocle	C₄×C₅³	C₂²
prime radical	C₂×C₅×C₅⋊D₅	C₄
hypocenter	C₅³	M₁₆
LCS	G ▷ C₂×C₅³ ▷ C₅³
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₅×#6029	C₂	semidirect
13. #6029	D₅	semidirect
17. C₅³	M₁₆	semidirect
49. C₅²	#676	semidirect
50. C₅²	#681	semidirect
76. C₅	#87768	semidirect
77. C₅	#87769	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₅⋊D₅	2
C₅×#6029	2
#87769	5
#87768	5