Subgroups — C₂²×#92559 #14522993

Order 2000.

492 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#92559	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅³	C₄	1	1	yes	characteristic, complemented
4	C₂×C₅²×D₅	C₄	1	3	yes	complemented
5	C₂×C₅²×D₅	C₂²	1	3	yes	—
6	#92559	C₂²	1	4	yes	direct product factor
7	C₂²×#1042	—	5	6	no	maximal
8	C₂²×#1039	—	5	1	no	maximal
9	C₂×C₅³	C₂×C₄	1	3	yes	complemented
10	C₅²×D₅	C₂×C₄	1	3	yes	complemented
11	C₅²×D₅	C₂³	1	1	yes	characteristic
12	C₂³×C₅²	—	5	1	no	—
13	C₂²×C₅×D₅	D₅	1	6	yes	—
14	C₂×#1042	—	5	36	no	—
15	C₂×#1039	—	5	6	no	—
16	C₅³	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
17	C₂²×C₅²	—	4	6	no	—
18	C₂²×C₅²	—	5	3	no	—
19	C₂²×C₅²	—	5	3	no	—
20	C₂²×C₅²	Dic₅	1	6	yes	complemented
21	C₂²×C₅²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
22	C₂×C₅×D₅	Dic₅	1	18	yes	complemented
23	C₂×C₅×D₅	C₂×D₅	1	18	yes	—
24	#1042	—	5	24	no	—
25	#1039	—	5	4	no	—
26	C₂²×AGL(1,𝔽₅)	—	25	1	no	—
27	C₂²×Dic₅	—	25	6	no	—
28	C₂×C₅²	—	4	18	no	—
29	C₂×C₅²	—	5	1	no	—
30	C₂×C₅²	—	5	3	no	—
31	C₂×C₅²	C₂×Dic₅	1	18	yes	complemented
32	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
33	C₅×D₅	C₂×Dic₅	1	18	yes	complemented
34	C₅×D₅	C₂²×D₅	1	6	yes	—
35	C₂³×C₅	—	5	6	no	—
36	C₂²×D₅	C₅⋊D₅	1	1	yes	characteristic
37	C₂×AGL(1,𝔽₅)	—	25	6	no	—
38	C₂×Dic₅	—	25	36	no	—
39	C₅²	—	4	6	no	—
40	C₅²	C₂²×Dic₅	1	6	yes	complemented
41	C₅²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
42	C₂²×C₅	—	4	6	no	—
43	C₂²×C₅	—	5	18	no	—
44	C₂²×C₅	—	5	18	no	—
45	C₂²×C₅	#1039	1	1	yes	characteristic, complemented
46	C₂²×C₅	#1042	1	6	yes	complemented
47	C₂×D₅	#1039	1	3	yes	complemented
48	C₂×D₅	C₂×C₅⋊D₅	1	3	yes	—
49	AGL(1,𝔽₅)	—	25	4	no	—
50	Dic₅	—	25	24	no	—
51	C₂²×C₄	—	125	1	no	Sylow
52	C₂×C₅	—	4	18	no	—
53	C₂×C₅	—	5	6	no	—
54	C₂×C₅	—	5	18	no	—
55	C₂×C₅	C₂×#1039	1	3	yes	complemented
56	C₂×C₅	C₂×#1042	1	18	yes	complemented
57	D₅	C₂×#1039	1	3	yes	complemented
58	D₅	C₂²×C₅⋊D₅	1	1	yes	characteristic
59	C₂³	—	5	1	no	—
60	C₂×C₄	—	125	6	no	—
61	C₅	—	4	6	no	—
62	C₅	C₂²×#1039	1	1	yes	characteristic, complemented
63	C₅	C₂²×#1042	1	6	yes	complemented
64	C₂²	—	5	3	no	—
65	C₂²	—	5	3	no	—
66	C₂²	#92559	1	1	yes	characteristic, direct product factor
67	C₄	—	125	4	no	—
68	C₂	—	5	1	no	—
69	C₂	—	5	3	no	—
70	C₂	C₂×#92559	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	#92559
commutator, hypocenter	C₅³	C₂²×C₄
Frattini	C₁	G
Fitting, socle	C₂²×C₅³	C₄
prime radical	C₂²×C₅²×D₅	C₂
cosocle	C₅²×D₅	C₂³
LCS	G ▷ C₅³
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×#92559	C₂	direct (N × H)
3. C₂²×C₅³	C₄	semidirect
4. C₂×C₅²×D₅	C₄	semidirect
6. #92559	C₂²	direct (N × H)
9. C₂×C₅³	C₂×C₄	semidirect
10. C₅²×D₅	C₂×C₄	semidirect
16. C₅³	C₂²×C₄	semidirect
20. C₂²×C₅²	Dic₅	semidirect
21. C₂²×C₅²	AGL(1,𝔽₅)	semidirect
22. C₂×C₅×D₅	Dic₅	semidirect
31. C₂×C₅²	C₂×Dic₅	semidirect
32. C₂×C₅²	C₂×AGL(1,𝔽₅)	semidirect
33. C₅×D₅	C₂×Dic₅	semidirect
40. C₅²	C₂²×Dic₅	semidirect
41. C₅²	C₂²×AGL(1,𝔽₅)	semidirect
45. C₂²×C₅	#1039	semidirect
46. C₂²×C₅	#1042	semidirect
47. C₂×D₅	#1039	semidirect
55. C₂×C₅	C₂×#1039	semidirect
56. C₂×C₅	C₂×#1042	semidirect
57. D₅	C₂×#1039	semidirect
62. C₅	C₂²×#1039	semidirect
63. C₅	C₂²×#1042	semidirect
66. C₂²	#92559	direct (N × H)
70. C₂	C₂×#92559	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅²×D₅	2
C₂×#92559	2
C₂²×#1042	5
C₂²×#1039	5

Subgroups — C22×#92559 #14522993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#92559 #14522993