Subgroups — #14522595

Order 2000.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#92553	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#1039	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅²×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅³	C₂²	1	1	yes	characteristic
5	#92553	—	2	1	no	—
6	C₅×#1039	—	2	1	no	—
7	C₅²×Dic₅	C₄	1	2	yes	complemented
8	C₄⋊#1039	—	5	1	no	maximal
9	#87780	—	5	6	no	maximal
10	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅³	D₄	1	1	yes	characteristic
12	C₂×C₅³	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₅²	—	5	1	no	—
14	C₂×#1039	—	5	1	no	—
15	C₂×#1039	—	5	6	no	—
16	C₂×#1039	—	10	12	no	—
17	C₂×#1039	D₅	1	1	yes	characteristic
18	C₂×C₅×Dic₅	—	5	6	no	—
19	C₂×C₅×Dic₅	D₅	1	6	yes	—
20	C₅³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₅²	—	2	12	no	—
22	C₂²×C₅²	C₂×D₅	1	1	yes	characteristic
23	C₂²×C₅²	C₂×D₅	1	6	yes	—
24	C₄×C₅²	—	5	2	no	—
25	#1039	—	2	1	no	—
26	#1039	—	10	24	no	—
27	#1039	—	10	6	no	—
28	#1039	—	10	1	no	—
29	C₅×Dic₅	—	10	6	no	—
30	C₅×Dic₅	Dic₅	1	12	yes	complemented
31	C₄⋊Dic₅	—	25	6	no	—
32	#683	—	25	1	no	—
33	C₂×C₅²	—	2	12	no	—
34	C₂×C₅²	—	2	12	no	—
35	C₂×C₅²	—	2	12	no	—
36	C₂×C₅²	Dic₁₀	1	6	yes	—
37	C₂×C₅²	Dic₁₀	1	1	yes	characteristic
38	C₂×C₅²	C₄×D₅	1	1	yes	characteristic
39	C₂×C₅²	D₂₀	1	6	yes	—
40	C₂×C₅²	C₂×Dic₅	1	6	yes	—
41	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
42	C₂×C₄×C₅	—	5	6	no	—
43	C₂×C₄×C₅	—	25	1	no	—
44	C₂×Dic₅	—	5	6	no	—
45	C₂×Dic₅	—	25	6	no	—
46	C₂×Dic₅	—	25	1	no	—
47	C₂×Dic₅	—	50	12	no	—
48	C₂×Dic₅	C₅⋊D₅	1	1	yes	characteristic
49	C₅²	—	2	12	no	—
50	C₅²	#683	1	1	yes	characteristic, complemented
51	C₅²	C₄⋊Dic₅	1	6	yes	complemented
52	C₂²×C₅	—	2	12	no	—
53	C₂²×C₅	D₅²	1	6	yes	—
54	C₂²×C₅	C₂×C₅⋊D₅	1	1	yes	characteristic
55	C₄×C₅	—	5	12	no	—
56	C₄×C₅	—	50	1	no	—
57	Dic₅	—	10	6	no	—
58	Dic₅	—	50	24	no	—
59	Dic₅	—	50	6	no	—
60	Dic₅	—	50	1	no	—
61	Dic₅	#1039	1	2	yes	complemented
62	C₄⋊C₄	—	125	1	no	Sylow
63	C₂×C₅	—	2	12	no	—
64	C₂×C₅	—	2	12	no	—
65	C₂×C₅	—	2	12	no	—
66	C₂×C₅	D₅×Dic₅	1	6	yes	—
67	C₂×C₅	#6038	1	6	yes	—
68	C₂×C₅	#6039	1	6	yes	—
69	C₂×C₅	#6045	1	1	yes	characteristic
70	C₂×C₅	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
71	C₂×C₅	C₂×#1039	1	1	yes	characteristic
72	C₂×C₄	—	5	1	no	—
73	C₂×C₄	—	25	1	no	—
74	C₂×C₄	—	125	1	no	—
75	C₅	—	2	12	no	—
76	C₅	#87780	1	6	yes	complemented
77	C₅	C₄⋊#1039	1	1	yes	characteristic, complemented
78	C₂²	D₅×C₅⋊D₅	1	1	yes	characteristic
79	C₄	—	5	2	no	—
80	C₄	—	50	1	no	—
81	C₄	—	250	1	no	—
82	C₂	D₅×#1039	1	1	yes	characteristic
83	C₂	#11758754	1	1	yes	characteristic
84	C₂	#11758755	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₅²×Dic₅	C₄	semidirect
20. C₅³	C₄⋊C₄	semidirect
30. C₅×Dic₅	Dic₅	semidirect
50. C₅²	#683	semidirect
51. C₅²	C₄⋊Dic₅	semidirect
61. Dic₅	#1039	semidirect
76. C₅	#87780	semidirect
77. C₅	C₄⋊#1039	semidirect

Maximal subgroups

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

subgroup	index
C₂×#92553	2
C₂×C₅×#1039	2
C₂×C₅²×Dic₅	2
C₄⋊#1039	5
#87780	5