Subgroups — C₅²×#700 #14522782

Order 2000.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₅²×#170	C₂	1	2	yes	maximal, complemented
3	C₂×C₅²×D₅	C₄	1	1	yes	characteristic
4	C₄×C₅³	C₄	1	1	yes	characteristic
5	C₅²×Dic₅	C₂²	1	1	yes	characteristic
6	C₅×#700	C₅	1	6	yes	maximal, direct product factor
7	C₅²×M₁₆	—	5	1	no	maximal
8	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
9	C₅²×D₅	—	2	1	no	—
10	C₂×C₄×C₅²	—	5	1	no	—
11	C₄×C₅×D₅	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	—	5	1	no	—
16	C₂×C₅×D₅	C₄×C₅	1	6	yes	—
17	C₄×C₅²	—	4	6	no	—
18	C₄×C₅²	—	5	1	no	—
19	C₄×C₅²	C₄×C₅	1	6	yes	—
20	C₄×C₅²	AGL(1,𝔽₅)	1	1	yes	characteristic
21	C₅×Dic₅	C₂²×C₅	1	6	yes	—
22	#700	C₅²	1	1	yes	characteristic, direct product factor
23	C₅×M₁₆	—	5	6	no	—
24	C₂×C₅²	—	4	6	no	—
25	C₂×C₅²	—	10	1	no	—
26	C₂×C₅²	C₂×C₄×C₅	1	6	yes	—
27	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₅×D₅	—	2	6	no	—
29	C₂×C₄×C₅	—	5	6	no	—
30	C₄×D₅	C₂×C₅²	1	1	yes	characteristic
31	#170	C₂×C₅²	1	2	yes	complemented
32	C₅×C₈	—	5	12	no	—
33	C₅²	—	4	6	no	—
34	C₅²	C₅×M₁₆	1	6	yes	complemented
35	C₅²	#700	1	1	yes	characteristic, direct product factor
36	C₂²×C₅	—	5	6	no	—
37	C₂×D₅	C₄×C₅²	1	1	yes	characteristic
38	C₄×C₅	—	4	6	no	—
39	C₄×C₅	—	5	6	no	—
40	C₄×C₅	C₄×C₅²	1	1	yes	characteristic
41	C₄×C₅	C₅×AGL(1,𝔽₅)	1	6	yes	—
42	Dic₅	C₂²×C₅²	1	1	yes	characteristic
43	M₁₆	—	5	1	no	Sylow
44	C₂×C₅	—	4	6	no	—
45	C₂×C₅	—	10	6	no	—
46	C₂×C₅	C₂×C₄×C₅²	1	1	yes	characteristic
47	C₂×C₅	C₂×C₅×AGL(1,𝔽₅)	1	6	yes	—
48	D₅	—	2	1	no	—
49	C₂×C₄	—	5	1	no	—
50	C₈	—	5	2	no	—
51	C₅	—	4	6	no	—
52	C₅	C₅²×M₁₆	1	1	yes	characteristic, complemented
53	C₅	C₅×#700	1	6	yes	direct product factor
54	C₂²	—	5	1	no	—
55	C₄	—	5	1	no	—
56	C₄	C₅²×AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂	—	10	1	no	—
58	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	C₄×C₅³	C₄
socle	C₂×C₅³	C₂×C₄
prime radical	C₂×C₅²×D₅	C₄
cosocle	Dic₅	C₂²×C₅²
hypercenter	C₄×C₅²	AGL(1,𝔽₅)
hypocenter	C₅	C₅²×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅² ◁ C₄×C₅²

Complements

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

N	H ≅ G/N	type
2. C₅²×#170	C₂	semidirect
6. C₅×#700	C₅	direct (N × H)
12. C₅×#170	C₂×C₅	semidirect
14. C₅³	M₁₆	semidirect
22. #700	C₅²	direct (N × H)
31. #170	C₂×C₅²	semidirect
34. C₅²	C₅×M₁₆	semidirect
35. C₅²	#700	direct (N × H)
52. C₅	C₅²×M₁₆	semidirect
53. C₅	C₅×#700	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅²×D₅	2
C₅²×#170	2
C₅×#700	5
C₅²×M₁₆	5

Subgroups — C52×#700 #14522782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#700 #14522782