Subgroups — C₅²×#704 #14522786

Order 2000.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₅²×#170	C₂	1	2	yes	maximal, complemented
3	C₂²×C₅³	C₄	1	1	yes	characteristic
4	C₅²×Dic₅	C₄	1	1	yes	characteristic
5	C₅²×Dic₅	C₂²	1	1	yes	characteristic
6	C₅×#704	C₅	1	6	yes	maximal, direct product factor
7	C₅²×M₁₆	—	5	1	no	maximal
8	C₂×C₅³	—	2	1	no	—
9	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₅²	—	5	1	no	—
11	C₂×C₅×Dic₅	C₂×C₅	1	6	yes	—
12	C₅×#170	C₂×C₅	1	12	yes	complemented
13	C₅²×C₈	—	5	2	no	—
14	C₅³	M₁₆	1	1	yes	characteristic, complemented, Sylow
15	C₂²×C₅²	—	4	6	no	—
16	C₂²×C₅²	C₄×C₅	1	6	yes	—
17	C₂²×C₅²	AGL(1,𝔽₅)	1	1	yes	characteristic
18	C₄×C₅²	—	5	1	no	—
19	C₄×C₅²	—	5	1	no	—
20	C₅×Dic₅	C₄×C₅	1	6	yes	—
21	C₅×Dic₅	C₂²×C₅	1	6	yes	—
22	#704	C₅²	1	1	yes	characteristic, direct product factor
23	C₅×M₁₆	—	5	6	no	—
24	C₂×C₅²	—	2	1	no	—
25	C₂×C₅²	—	2	6	no	—
26	C₂×C₅²	—	4	12	no	—
27	C₂×C₅²	—	4	6	no	—
28	C₂×C₅²	C₂×C₄×C₅	1	6	yes	—
29	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
30	C₂×C₄×C₅	—	5	6	no	—
31	C₂×Dic₅	C₂×C₅²	1	1	yes	characteristic
32	#170	C₂×C₅²	1	2	yes	complemented
33	C₅×C₈	—	5	12	no	—
34	C₅²	—	4	6	no	—
35	C₅²	C₅×M₁₆	1	6	yes	complemented
36	C₅²	#704	1	1	yes	characteristic, direct product factor
37	C₂²×C₅	—	4	6	no	—
38	C₂²×C₅	C₄×C₅²	1	1	yes	characteristic
39	C₂²×C₅	C₅×AGL(1,𝔽₅)	1	6	yes	—
40	C₄×C₅	—	5	6	no	—
41	C₄×C₅	—	5	6	no	—
42	Dic₅	C₄×C₅²	1	1	yes	characteristic
43	Dic₅	C₂²×C₅²	1	1	yes	characteristic
44	M₁₆	—	5	1	no	Sylow
45	C₂×C₅	—	2	1	no	—
46	C₂×C₅	—	2	6	no	—
47	C₂×C₅	—	4	12	no	—
48	C₂×C₅	—	4	6	no	—
49	C₂×C₅	C₂×C₄×C₅²	1	1	yes	characteristic
50	C₂×C₅	C₂×C₅×AGL(1,𝔽₅)	1	6	yes	—
51	C₂×C₄	—	5	1	no	—
52	C₈	—	5	2	no	—
53	C₅	—	4	6	no	—
54	C₅	C₅²×M₁₆	1	1	yes	characteristic, complemented
55	C₅	C₅×#704	1	6	yes	direct product factor
56	C₂²	C₅²×AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₄	—	5	1	no	—
58	C₄	—	5	1	no	—
59	C₂	—	2	1	no	—
60	C₂	C₂×C₅²×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅²	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄×C₅²
Frattini	C₂	C₂×C₅²×AGL(1,𝔽₅)
Fitting, prime radical	C₂²×C₅³	C₄
socle	C₂×C₅³	C₂×C₄
cosocle	Dic₅	C₂²×C₅²
hypercenter	C₂²×C₅²	AGL(1,𝔽₅)
hypocenter	C₅	C₅²×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅² ◁ C₂²×C₅²

Complements

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

N	H ≅ G/N	type
2. C₅²×#170	C₂	semidirect
6. C₅×#704	C₅	direct (N × H)
12. C₅×#170	C₂×C₅	semidirect
14. C₅³	M₁₆	semidirect
22. #704	C₅²	direct (N × H)
32. #170	C₂×C₅²	semidirect
35. C₅²	C₅×M₁₆	semidirect
36. C₅²	#704	direct (N × H)
54. C₅	C₅²×M₁₆	semidirect
55. C₅	C₅×#704	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²×Dic₅	2
C₅²×#170	2
C₅×#704	5
C₅²×M₁₆	5

Subgroups — C52×#704 #14522786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#704 #14522786