Subgroups — #14522723

Order 2000.

244 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#92559	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	D₅×#1039	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅²×D₅	C₂²	1	1	yes	characteristic
5	#92559	—	2	1	no	—
6	C₅²×AGL(1,𝔽₅)	—	2	1	no	—
7	#92553	C₄	1	1	yes	characteristic, complemented
8	C₅×#1039	C₄	1	1	yes	characteristic, complemented
9	#87825	—	5	6	no	maximal
10	#87806	—	5	1	no	maximal
11	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
12	C₅²×D₅	D₄	1	1	yes	characteristic
13	C₅²×D₅	Q₈	1	1	yes	characteristic
14	C₂×#1042	—	5	6	no	—
15	C₂×C₅×AGL(1,𝔽₅)	D₅	1	6	yes	—
16	C₂×C₄×C₅²	—	5	1	no	—
17	C₂×#1039	—	5	1	no	—
18	C₂×#1039	—	5	1	no	—
19	D₅×Dic₅	—	5	6	no	—
20	C₅³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₅²	—	5	1	no	—
22	C₂×C₅×D₅	C₂×D₅	1	6	yes	—
23	#1042	—	10	6	no	—
24	C₅×AGL(1,𝔽₅)	—	2	6	no	—
25	C₄×C₅²	—	10	1	no	—
26	#1039	—	5	6	no	—
27	#1039	—	5	1	no	—
28	#1039	—	10	1	no	—
29	#1039	—	20	6	no	—
30	#1039	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
31	C₅×Dic₅	—	5	6	no	—
32	C₄⋊AGL(1,𝔽₅)	—	25	1	no	—
33	#683	—	25	6	no	—
34	C₂×C₅²	—	4	6	no	—
35	C₂×C₅²	—	5	1	no	—
36	C₂×C₅²	—	5	1	no	—
37	C₂×C₅²	C₄×D₅	1	6	yes	—
38	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₅×D₅	Dic₁₀	1	6	yes	—
40	C₅×D₅	C₂²⋊D₅	1	6	yes	—
41	C₂×AGL(1,𝔽₅)	—	25	1	no	—
42	C₂×AGL(1,𝔽₅)	C₅⋊D₅	1	1	yes	characteristic
43	C₂×C₄×C₅	—	5	6	no	—
44	C₂×Dic₅	—	25	6	no	—
45	C₂×Dic₅	—	25	6	no	—
46	C₄×D₅	—	25	1	no	—
47	C₅²	—	4	6	no	—
48	C₅²	#683	1	6	yes	complemented
49	C₅²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
50	C₂²×C₅	—	5	6	no	—
51	C₂×D₅	C₂×C₅⋊D₅	1	1	yes	characteristic
52	AGL(1,𝔽₅)	—	2	1	no	—
53	AGL(1,𝔽₅)	—	50	1	no	—
54	C₄×C₅	—	10	6	no	—
55	C₄×C₅	—	25	1	no	—
56	Dic₅	—	5	6	no	—
57	Dic₅	—	25	1	no	—
58	Dic₅	—	25	6	no	—
59	Dic₅	—	50	6	no	—
60	Dic₅	—	100	6	no	—
61	C₄⋊C₄	—	125	1	no	Sylow
62	C₂×C₅	—	4	6	no	—
63	C₂×C₅	—	5	6	no	—
64	C₂×C₅	—	5	6	no	—
65	C₂×C₅	C₄×C₅⋊D₅	1	1	yes	characteristic
66	C₂×C₅	D₅×AGL(1,𝔽₅)	1	6	yes	—
67	D₅	#6045	1	1	yes	characteristic
68	D₅	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
69	C₂×C₄	—	5	1	no	—
70	C₂×C₄	—	125	1	no	—
71	C₂×C₄	—	125	1	no	—
72	C₅	—	4	6	no	—
73	C₅	#87806	1	1	yes	characteristic, complemented
74	C₅	#87825	1	6	yes	complemented
75	C₂²	—	5	1	no	—
76	C₄	—	10	1	no	—
77	C₄	—	25	1	no	—
78	C₄	—	125	1	no	—
79	C₄	—	250	1	no	—
80	C₂	—	5	1	no	—
81	C₂	—	5	1	no	—
82	C₂	AGL(1,𝔽₅)×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₅)×C₅⋊D₅
commutator, Fitting, socle	C₂×C₅³	C₂×C₄
prime radical, cosocle	C₂×C₅²×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. #92553	C₄	semidirect
8. C₅×#1039	C₄	semidirect
20. C₅³	C₄⋊C₄	semidirect
30. #1039	AGL(1,𝔽₅)	semidirect
48. C₅²	#683	semidirect
49. C₅²	C₄⋊AGL(1,𝔽₅)	semidirect
73. C₅	#87806	semidirect
74. C₅	#87825	semidirect

Maximal subgroups

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

subgroup	index
C₂×#92559	2
C₂×C₅²×AGL(1,𝔽₅)	2
D₅×#1039	2
#87825	5
#87806	5