Subgroups — #14522741

Order 2000.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11758751	C₂	1	1	yes	maximal, characteristic
2	C₅×#6032	C₂	1	2	yes	maximal, complemented
3	C₂×C₅⋊(C₅⋊D₅)	C₄	1	1	yes	characteristic
4	C₅×#1039	C₄	1	1	yes	characteristic
5	C₅²×Dic₅	C₂²	1	1	yes	characteristic
6	#87829	—	5	2	no	maximal
7	#87769	—	5	1	no	maximal
8	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
9	C₅⋊(C₅⋊D₅)	—	2	1	no	—
10	C₄×C₅⋊D₅	—	5	1	no	—
11	#6036	—	5	2	no	—
12	#6036	—	10	2	no	—
13	#6032	D₅	1	2	yes	complemented
14	C₅×#170	—	5	2	no	—
15	C₅×#168	—	5	2	no	—
16	C₅³	M₁₆	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₅⋊D₅	—	5	2	no	—
18	C₂×C₅⋊D₅	—	5	1	no	—
19	C₂×C₅⋊D₅	—	10	2	no	—
20	C₂×C₅⋊D₅	—	20	2	no	—
21	C₂×C₅⋊D₅	—	20	4	no	—
22	C₄×C₅²	—	5	1	no	—
23	#1039	AGL(1,𝔽₅)	1	1	yes	characteristic
24	C₅×Dic₅	—	2	2	no	—
25	C₅×Dic₅	—	5	2	no	—
26	C₅×Dic₅	—	10	2	no	—
27	C₅×Dic₅	C₂×D₅	1	2	yes	—
28	#700	—	25	1	no	—
29	#676	—	25	2	no	—
30	C₂×C₅²	—	2	2	no	—
31	C₂×C₅²	—	4	4	no	—
32	C₂×C₅²	—	4	2	no	—
33	C₂×C₅²	C₄×D₅	1	2	yes	—
34	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₅⋊D₅	—	10	1	no	—
36	C₅⋊D₅	—	10	4	no	—
37	C₅⋊D₅	—	10	2	no	—
38	C₅⋊D₅	—	20	8	no	—
39	C₅⋊D₅	—	20	4	no	—
40	C₄×D₅	—	25	2	no	—
41	C₄×D₅	—	25	1	no	—
42	C₄×D₅	—	50	2	no	—
43	#170	—	5	2	no	—
44	C₅×C₈	—	25	2	no	—
45	#168	—	5	2	no	—
46	C₅²	—	2	2	no	—
47	C₅²	—	4	4	no	—
48	C₅²	—	4	2	no	—
49	C₅²	#676	1	2	yes	complemented
50	C₅²	#700	1	1	yes	characteristic, complemented
51	C₂×D₅	—	25	1	no	—
52	C₂×D₅	—	25	2	no	—
53	C₂×D₅	—	50	2	no	—
54	C₂×D₅	—	100	2	no	—
55	C₂×D₅	—	100	4	no	—
56	C₄×C₅	—	5	2	no	—
57	C₄×C₅	—	10	2	no	—
58	C₄×C₅	—	25	1	no	—
59	Dic₅	—	5	2	no	—
60	Dic₅	—	10	2	no	—
61	Dic₅	D₅²	1	1	yes	characteristic
62	M₁₆	—	125	1	no	Sylow
63	C₂×C₅	—	2	2	no	—
64	C₂×C₅	—	4	4	no	—
65	C₂×C₅	—	4	2	no	—
66	C₂×C₅	#6036	1	1	yes	characteristic
67	C₂×C₅	D₅×AGL(1,𝔽₅)	1	2	yes	—
68	D₅	—	50	4	no	—
69	D₅	—	50	2	no	—
70	D₅	—	50	1	no	—
71	D₅	—	100	8	no	—
72	D₅	—	100	4	no	—
73	C₂×C₄	—	125	1	no	—
74	C₈	—	25	2	no	—
75	C₅	—	2	2	no	—
76	C₅	—	4	4	no	—
77	C₅	—	4	2	no	—
78	C₅	#87769	1	1	yes	characteristic, complemented
79	C₅	#87829	1	2	yes	complemented
80	C₂²	—	125	1	no	—
81	C₄	—	5	1	no	—
82	C₄	—	25	1	no	—
83	C₂	—	250	1	no	—
84	C₂	#1042⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#1042⋊D₅
commutator, Fitting, socle	C₂×C₅³	C₂×C₄
prime radical	C₂×C₅⋊(C₅⋊D₅)	C₄
cosocle	C₅²×Dic₅	C₂²
hypocenter	C₅³	M₁₆
LCS	G ▷ C₂×C₅³ ▷ C₅³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₅×#6032	C₂	semidirect
13. #6032	D₅	semidirect
16. C₅³	M₁₆	semidirect
49. C₅²	#676	semidirect
50. C₅²	#700	semidirect
78. C₅	#87769	semidirect
79. C₅	#87829	semidirect

Maximal subgroups

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

subgroup	index
#11758751	2
C₅×#6032	2
#87829	5
#87769	5