Subgroups — #14522708

Order 2000.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#92560	C₂	1	1	yes	maximal, characteristic
2	C₂×#92561	C₂	1	1	yes	maximal, characteristic
3	C₅×#6036	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₅⋊D₅	C₂²	1	1	yes	characteristic
5	#92561	—	2	1	no	—
6	#92560	—	2	1	no	—
7	C₅²×Dic₅	C₄	1	2	yes	complemented
8	C₄⋊#1042	—	5	2	no	maximal
9	#87832	—	5	1	no	maximal
10	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
11	C₅×C₅⋊D₅	D₄	1	1	yes	characteristic
12	C₅×C₅⋊D₅	Q₈	1	1	yes	characteristic
13	C₂×#1044	—	5	1	no	—
14	C₂×#1043	—	5	1	no	—
15	C₂×#1042	—	5	2	no	—
16	C₂×#1042	—	5	2	no	—
17	C₂×#1042	—	10	2	no	—
18	C₄×C₅×D₅	—	5	2	no	—
19	#6036	D₅	1	1	yes	characteristic
20	C₅³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₅⋊D₅	C₂×D₅	1	1	yes	characteristic
22	C₂×C₅×D₅	—	5	2	no	—
23	C₂×C₅×D₅	—	10	2	no	—
24	#1044	—	10	1	no	—
25	#1043	—	10	1	no	—
26	#1042	—	10	2	no	—
27	#1042	—	10	2	no	—
28	#1042	—	10	4	no	—
29	C₄×C₅²	—	5	2	no	—
30	C₅×Dic₅	—	4	2	no	—
31	C₅×Dic₅	Dic₅	1	2	yes	complemented
32	C₅×Dic₅	AGL(1,𝔽₅)	1	2	yes	complemented
33	C₄⋊AGL(1,𝔽₅)	—	25	2	no	—
34	C₄⋊Dic₅	—	25	1	no	—
35	C₂×C₅²	—	2	2	no	—
36	C₂×C₅²	—	4	2	no	—
37	C₂×C₅²	—	8	2	no	—
38	C₂×C₅²	C₂×Dic₅	1	1	yes	characteristic
39	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	2	yes	—
40	C₅⋊D₅	Dic₁₀	1	1	yes	characteristic
41	C₅⋊D₅	D₂₀	1	1	yes	characteristic
42	C₅×D₅	—	5	2	no	—
43	C₅×D₅	—	5	2	no	—
44	C₅×D₅	—	10	2	no	—
45	C₅×D₅	—	10	2	no	—
46	C₂×AGL(1,𝔽₅)	—	25	2	no	—
47	C₂×AGL(1,𝔽₅)	—	25	2	no	—
48	C₂×AGL(1,𝔽₅)	—	50	2	no	—
49	C₂×C₄×C₅	—	25	1	no	—
50	C₂×Dic₅	—	25	1	no	—
51	C₂×Dic₅	—	25	1	no	—
52	C₄×D₅	—	5	2	no	—
53	C₅²	—	2	2	no	—
54	C₅²	—	4	2	no	—
55	C₅²	—	8	2	no	—
56	C₅²	C₄⋊Dic₅	1	1	yes	characteristic, complemented
57	C₅²	C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
58	C₂²×C₅	—	25	1	no	—
59	C₂×D₅	—	5	2	no	—
60	C₂×D₅	—	10	2	no	—
61	AGL(1,𝔽₅)	—	50	2	no	—
62	AGL(1,𝔽₅)	—	50	2	no	—
63	AGL(1,𝔽₅)	—	50	4	no	—
64	C₄×C₅	—	5	2	no	—
65	C₄×C₅	—	5	2	no	—
66	C₄×C₅	—	20	2	no	—
67	Dic₅	—	50	1	no	—
68	Dic₅	—	50	1	no	—
69	Dic₅	#1042	1	2	yes	complemented
70	C₄⋊C₄	—	125	1	no	Sylow
71	C₂×C₅	—	2	2	no	—
72	C₂×C₅	—	4	2	no	—
73	C₂×C₅	—	8	2	no	—
74	C₂×C₅	—	25	1	no	—
75	C₂×C₅	—	25	1	no	—
76	C₂×C₅	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₂×C₅	C₂×#1042	1	2	yes	—
78	D₅	—	5	2	no	—
79	D₅	—	5	2	no	—
80	D₅	—	10	2	no	—
81	D₅	—	10	2	no	—
82	C₂×C₄	—	25	1	no	—
83	C₂×C₄	—	125	1	no	—
84	C₂×C₄	—	125	1	no	—
85	C₅	—	2	2	no	—
86	C₅	—	4	2	no	—
87	C₅	—	8	2	no	—
88	C₅	#87832	1	1	yes	characteristic, complemented
89	C₅	C₄⋊#1042	1	2	yes	complemented
90	C₂²	—	25	1	no	—
91	C₄	—	5	2	no	—
92	C₄	—	250	1	no	—
93	C₄	—	250	1	no	—
94	C₂	—	25	1	no	—
95	C₂	—	25	1	no	—
96	C₂	D₅⋊#1042	1	1	yes	characteristic

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

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
31. C₅×Dic₅	Dic₅	semidirect
32. C₅×Dic₅	AGL(1,𝔽₅)	semidirect
56. C₅²	C₄⋊Dic₅	semidirect
57. C₅²	C₄⋊AGL(1,𝔽₅)	semidirect
69. Dic₅	#1042	semidirect
88. C₅	#87832	semidirect
89. C₅	C₄⋊#1042	semidirect

Maximal subgroups

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

subgroup	index
C₂×#92560	2
C₂×#92561	2
C₅×#6036	2
C₄⋊#1042	5
#87832	5