Subgroups — C₂²×C₅×#1044 #14522987

Order 2000.

340 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#1044	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅³	C₄	1	1	yes	characteristic, complemented
4	C₂×C₅×C₅⋊D₅	C₄	1	3	yes	complemented
5	C₂×C₅×C₅⋊D₅	C₂²	1	3	yes	—
6	C₅×#1044	C₂²	1	4	yes	direct product factor
7	C₂²×#1044	C₅	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₅×AGL(1,𝔽₅)	—	5	2	no	maximal
9	C₂×C₅³	C₂×C₄	1	3	yes	complemented
10	C₅×C₅⋊D₅	C₂×C₄	1	3	yes	complemented
11	C₅×C₅⋊D₅	C₂³	1	1	yes	characteristic
12	C₂²×C₅⋊D₅	C₂×C₅	1	1	yes	characteristic
13	C₂²×C₅×D₅	—	5	2	no	—
14	C₂²×C₅×D₅	—	10	2	no	—
15	C₂×#1044	C₂×C₅	1	6	yes	direct product factor
16	C₂×C₅×AGL(1,𝔽₅)	—	5	12	no	—
17	C₅³	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
18	C₂²×C₅²	—	2	2	no	—
19	C₂²×C₅²	—	4	4	no	—
20	C₂²×C₅²	—	4	2	no	—
21	C₂²×C₅²	C₄×C₅	1	1	yes	characteristic, complemented
22	C₂²×C₅²	AGL(1,𝔽₅)	1	2	yes	complemented
23	C₂×C₅⋊D₅	C₄×C₅	1	3	yes	complemented
24	C₂×C₅⋊D₅	C₂²×C₅	1	3	yes	—
25	C₂×C₅×D₅	—	5	6	no	—
26	C₂×C₅×D₅	—	5	6	no	—
27	C₂×C₅×D₅	—	10	6	no	—
28	C₂×C₅×D₅	—	10	6	no	—
29	#1044	C₂²×C₅	1	4	yes	direct product factor
30	C₅×AGL(1,𝔽₅)	—	5	8	no	—
31	C₂²×AGL(1,𝔽₅)	—	5	2	no	—
32	C₂²×C₄×C₅	—	25	1	no	—
33	C₂×C₅²	—	2	6	no	—
34	C₂×C₅²	—	4	12	no	—
35	C₂×C₅²	—	4	6	no	—
36	C₂×C₅²	C₂×C₄×C₅	1	3	yes	complemented
37	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	6	yes	complemented
38	C₅⋊D₅	C₂×C₄×C₅	1	3	yes	complemented
39	C₅⋊D₅	C₂³×C₅	1	1	yes	characteristic
40	C₅×D₅	—	5	2	no	—
41	C₅×D₅	—	5	6	no	—
42	C₅×D₅	—	10	6	no	—
43	C₅×D₅	—	10	2	no	—
44	C₂³×C₅	—	25	1	no	—
45	C₂²×D₅	—	5	2	no	—
46	C₂²×D₅	—	10	2	no	—
47	C₂×AGL(1,𝔽₅)	—	5	12	no	—
48	C₂×C₄×C₅	—	25	6	no	—
49	C₅²	—	2	2	no	—
50	C₅²	—	4	4	no	—
51	C₅²	—	4	2	no	—
52	C₅²	C₂²×C₄×C₅	1	1	yes	characteristic, complemented
53	C₅²	C₂²×AGL(1,𝔽₅)	1	2	yes	complemented
54	C₂²×C₅	—	2	2	no	—
55	C₂²×C₅	—	4	4	no	—
56	C₂²×C₅	—	4	2	no	—
57	C₂²×C₅	—	25	3	no	—
58	C₂²×C₅	—	25	3	no	—
59	C₂²×C₅	C₅×AGL(1,𝔽₅)	1	2	yes	complemented
60	C₂²×C₅	#1044	1	1	yes	characteristic, direct product factor
61	C₂×D₅	—	5	6	no	—
62	C₂×D₅	—	5	6	no	—
63	C₂×D₅	—	10	6	no	—
64	C₂×D₅	—	10	6	no	—
65	AGL(1,𝔽₅)	—	5	8	no	—
66	C₄×C₅	—	25	4	no	—
67	C₂²×C₄	—	25	1	no	Sylow
68	C₂×C₅	—	2	6	no	—
69	C₂×C₅	—	4	12	no	—
70	C₂×C₅	—	4	6	no	—
71	C₂×C₅	—	25	1	no	—
72	C₂×C₅	—	25	3	no	—
73	C₂×C₅	C₂×C₅×AGL(1,𝔽₅)	1	6	yes	complemented
74	C₂×C₅	C₂×#1044	1	3	yes	direct product factor
75	D₅	—	5	6	no	—
76	D₅	—	5	2	no	—
77	D₅	—	10	6	no	—
78	D₅	—	10	2	no	—
79	C₂³	—	25	1	no	—
80	C₂×C₄	—	25	6	no	—
81	C₅	—	2	2	no	—
82	C₅	—	4	4	no	—
83	C₅	—	4	2	no	—
84	C₅	C₂²×C₅×AGL(1,𝔽₅)	1	2	yes	complemented
85	C₅	C₂²×#1044	1	1	yes	characteristic, direct product factor
86	C₂²	—	25	3	no	—
87	C₂²	—	25	3	no	—
88	C₂²	C₅×#1044	1	1	yes	characteristic, direct product factor
89	C₄	—	25	4	no	—
90	C₂	—	25	1	no	—
91	C₂	—	25	3	no	—
92	C₂	C₂×C₅×#1044	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅	#1044
commutator, hypocenter	C₅²	C₂²×C₄×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₅³	C₄
prime radical	C₂²×C₅×C₅⋊D₅	C₂
cosocle	C₅⋊D₅	C₂³×C₅
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂²×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#1044	C₂	direct (N × H)
3. C₂²×C₅³	C₄	semidirect
4. C₂×C₅×C₅⋊D₅	C₄	semidirect
6. C₅×#1044	C₂²	direct (N × H)
7. C₂²×#1044	C₅	direct (N × H)
9. C₂×C₅³	C₂×C₄	semidirect
10. C₅×C₅⋊D₅	C₂×C₄	semidirect
15. C₂×#1044	C₂×C₅	direct (N × H)
17. C₅³	C₂²×C₄	semidirect
21. C₂²×C₅²	C₄×C₅	semidirect
22. C₂²×C₅²	AGL(1,𝔽₅)	semidirect
23. C₂×C₅⋊D₅	C₄×C₅	semidirect
29. #1044	C₂²×C₅	direct (N × H)
36. C₂×C₅²	C₂×C₄×C₅	semidirect
37. C₂×C₅²	C₂×AGL(1,𝔽₅)	semidirect
38. C₅⋊D₅	C₂×C₄×C₅	semidirect
52. C₅²	C₂²×C₄×C₅	semidirect
53. C₅²	C₂²×AGL(1,𝔽₅)	semidirect
59. C₂²×C₅	C₅×AGL(1,𝔽₅)	semidirect
60. C₂²×C₅	#1044	direct (N × H)
73. C₂×C₅	C₂×C₅×AGL(1,𝔽₅)	semidirect
74. C₂×C₅	C₂×#1044	direct (N × H)
84. C₅	C₂²×C₅×AGL(1,𝔽₅)	semidirect
85. C₅	C₂²×#1044	direct (N × H)
88. C₂²	C₅×#1044	direct (N × H)
92. C₂	C₂×C₅×#1044	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×C₅×C₅⋊D₅	2
C₂×C₅×#1044	2
C₂²×#1044	5
C₂²×C₅×AGL(1,𝔽₅)	5

Subgroups — C22×C5×#1044 #14522987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#1044 #14522987