Subgroups — #14522716

Order 2000.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#92561	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#1044	C₂	1	1	yes	maximal, characteristic
3	#11758756	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₅⋊D₅	C₂²	1	1	yes	characteristic
5	#92561	C₄	1	2	yes	complemented
6	C₅×#1044	C₄	1	2	yes	complemented
7	C₅×#1039	C₄	1	2	yes	complemented
8	#87830	—	5	1	no	maximal
9	Dic₅×AGL(1,𝔽₅)	—	5	2	no	maximal
10	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
11	C₅×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
12	C₅×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
13	C₂×#1044	D₅	1	1	yes	characteristic
14	C₂×#1043	—	5	1	no	—
15	C₂×#1042	—	5	2	no	—
16	C₂×#1042	—	10	2	no	—
17	C₂×C₅×AGL(1,𝔽₅)	—	5	2	no	—
18	#6036	—	5	1	no	—
19	D₅×Dic₅	—	5	2	no	—
20	C₅³	C₄²	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	Dic₅	1	2	yes	complemented
25	#1043	—	5	2	no	—
26	#1042	—	5	4	no	—
27	#1042	—	10	4	no	—
28	C₅×AGL(1,𝔽₅)	—	5	4	no	—
29	#1039	AGL(1,𝔽₅)	1	2	yes	complemented
30	C₅×Dic₅	—	5	2	no	—
31	C₅×Dic₅	—	5	2	no	—
32	C₅×Dic₅	—	20	2	no	—
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	C₄×D₅	1	1	yes	characteristic
39	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	2	yes	—
40	C₅⋊D₅	C₄×D₅	1	1	yes	characteristic
41	C₅⋊D₅	C₂×Dic₅	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)	—	5	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	—	25	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)	—	5	4	no	—
62	AGL(1,𝔽₅)	—	25	4	no	—
63	AGL(1,𝔽₅)	—	50	4	no	—
64	C₄×C₅	—	25	2	no	—
65	C₄×C₅	—	25	2	no	—
66	Dic₅	—	5	2	no	—
67	Dic₅	—	5	2	no	—
68	Dic₅	—	20	2	no	—
69	Dic₅	—	25	2	no	—
70	C₄²	—	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	2	yes	—
77	C₂×C₅	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
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	Dic₅×AGL(1,𝔽₅)	1	2	yes	complemented
89	C₅	#87830	1	1	yes	characteristic, complemented
90	C₂²	—	25	1	no	—
91	C₄	—	25	2	no	—
92	C₄	—	25	2	no	—
93	C₄	—	125	2	no	—
94	C₂	—	25	1	no	—
95	C₂	—	25	1	no	—
96	C₂	#11758789	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#11758789
commutator, hypocenter	C₅³	C₄²
Fitting, socle	C₂×C₅³	C₂×C₄
prime radical, cosocle	C₂×C₅×C₅⋊D₅	C₂²
LCS	G ▷ C₅³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
5. #92561	C₄	semidirect
6. C₅×#1044	C₄	semidirect
7. C₅×#1039	C₄	semidirect
20. C₅³	C₄²	semidirect
24. #1044	Dic₅	semidirect
29. #1039	AGL(1,𝔽₅)	semidirect
56. C₅²	C₄×Dic₅	semidirect
57. C₅²	C₄×AGL(1,𝔽₅)	semidirect
88. C₅	Dic₅×AGL(1,𝔽₅)	semidirect
89. C₅	#87830	semidirect

Maximal subgroups

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

subgroup	index
C₂×#92561	2
C₂×C₅×#1044	2
#11758756	2
#87830	5
Dic₅×AGL(1,𝔽₅)	5