Subgroups — C₅²×#689 #14522620

Order 2000.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅³×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅²×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	C₅²×#168	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅³	—	2	1	no	—
5	C₄×C₅³	C₂²	1	1	yes	characteristic
6	C₅²×Dic₅	—	2	1	no	—
7	C₅²×Q₁₆	—	5	1	no	maximal
8	C₅×#689	C₅	1	6	yes	maximal, direct product factor
9	C₂×C₅³	D₄	1	1	yes	characteristic
10	C₅²×Q₈	—	2	12	no	—
11	C₅²×Q₈	—	5	1	no	—
12	C₅²×Q₈	D₅	1	1	yes	characteristic
13	C₅²×Q₈	C₂×C₅	1	6	yes	—
14	C₅×Dic₁₀	C₂×C₅	1	6	yes	—
15	C₅²×C₈	—	5	1	no	—
16	C₅×#168	C₂×C₅	1	6	yes	—
17	C₅³	Q₁₆	1	1	yes	characteristic, complemented, Sylow
18	C₄×C₅²	—	2	24	no	—
19	C₄×C₅²	—	2	12	no	—
20	C₄×C₅²	—	2	1	no	—
21	C₄×C₅²	—	2	6	no	—
22	C₄×C₅²	—	10	1	no	—
23	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
24	C₄×C₅²	C₂²×C₅	1	6	yes	—
25	C₅×Dic₅	—	2	6	no	—
26	C₅×Q₁₆	—	5	6	no	—
27	#689	C₅²	1	1	yes	characteristic, direct product factor
28	C₂×C₅²	—	2	12	no	—
29	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
30	C₂×C₅²	C₅×D₄	1	6	yes	—
31	C₅×Q₈	—	2	12	no	—
32	C₅×Q₈	—	5	6	no	—
33	C₅×Q₈	C₅×D₅	1	6	yes	—
34	C₅×Q₈	C₂×C₅²	1	1	yes	characteristic
35	Dic₁₀	C₂×C₅²	1	1	yes	characteristic
36	C₅×C₈	—	5	6	no	—
37	#168	C₂×C₅²	1	1	yes	characteristic
38	C₅²	—	2	12	no	—
39	C₅²	#689	1	1	yes	characteristic, direct product factor
40	C₅²	C₅×Q₁₆	1	6	yes	complemented
41	C₄×C₅	—	2	24	no	—
42	C₄×C₅	—	2	6	no	—
43	C₄×C₅	—	2	1	no	—
44	C₄×C₅	—	2	12	no	—
45	C₄×C₅	—	10	6	no	—
46	C₄×C₅	C₂×C₅×D₅	1	6	yes	—
47	C₄×C₅	C₂²×C₅²	1	1	yes	characteristic
48	Dic₅	—	2	1	no	—
49	Q₁₆	—	5	1	no	Sylow
50	C₂×C₅	—	2	12	no	—
51	C₂×C₅	C₅×C₂²⋊D₅	1	6	yes	—
52	C₂×C₅	C₅²×D₄	1	1	yes	characteristic
53	Q₈	—	5	1	no	—
54	Q₈	C₅²×D₅	1	1	yes	characteristic
55	C₈	—	5	1	no	—
56	C₅	—	2	12	no	—
57	C₅	C₅×#689	1	6	yes	direct product factor
58	C₅	C₅²×Q₁₆	1	1	yes	characteristic, complemented
59	C₄	—	2	1	no	—
60	C₄	—	10	1	no	—
61	C₄	C₂×C₅²×D₅	1	1	yes	characteristic
62	C₂	C₅²×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅²	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂²×C₅²
Frattini	C₄	C₂×C₅²×D₅
Fitting	C₅³×Q₈	C₂
socle, prime radical	C₂×C₅³	D₄
hypercenter	C₅²×Q₈	D₅
hypocenter	C₅	C₅²×Q₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅² ◁ C₄×C₅² ◁ C₅²×Q₈

Complements

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

N	H ≅ G/N	type
8. C₅×#689	C₅	direct (N × H)
17. C₅³	Q₁₆	semidirect
27. #689	C₅²	direct (N × H)
39. C₅²	#689	direct (N × H)
40. C₅²	C₅×Q₁₆	semidirect
57. C₅	C₅×#689	direct (N × H)
58. C₅	C₅²×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₅³×Q₈	2
C₅²×Dic₁₀	2
C₅²×#168	2
C₅²×Q₁₆	5
C₅×#689	5

Subgroups — C52×#689 #14522620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#689 #14522620