Subgroups — C₅²×#688 #14522619

Order 2000.

304 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, complemented
2	C₅²×D₂₀	C₂	1	1	yes	maximal, characteristic
3	C₅²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅²×D₅	—	2	1	no	—
5	C₄×C₅³	—	2	1	no	—
6	C₄×C₅³	C₂²	1	1	yes	characteristic
7	C₅²×SD₁₆	—	5	1	no	maximal
8	C₅×#688	C₅	1	6	yes	maximal, direct product factor
9	C₂×C₅³	D₄	1	1	yes	characteristic
10	C₅²×D₅	—	4	1	no	—
11	C₅²×Q₈	—	2	12	no	—
12	C₅²×Q₈	D₅	1	1	yes	characteristic, complemented
13	C₅²×Q₈	C₂×C₅	1	6	yes	complemented
14	C₅²×D₄	—	5	1	no	—
15	C₅×D₂₀	C₂×C₅	1	6	yes	—
16	C₅²×C₈	—	5	1	no	—
17	C₅×#168	C₂×C₅	1	6	yes	complemented
18	C₅³	SD₁₆	1	1	yes	characteristic, complemented, Sylow
19	C₂²×C₅²	—	10	1	no	—
20	C₂×C₅×D₅	—	2	6	no	—
21	C₄×C₅²	—	2	24	no	—
22	C₄×C₅²	—	2	12	no	—
23	C₄×C₅²	—	2	1	no	—
24	C₄×C₅²	—	2	6	no	—
25	C₄×C₅²	C₂×D₅	1	1	yes	characteristic
26	C₄×C₅²	C₂²×C₅	1	6	yes	—
27	C₅×SD₁₆	—	5	6	no	—
28	#688	C₅²	1	1	yes	characteristic, direct product factor
29	C₂×C₅²	—	2	12	no	—
30	C₂×C₅²	—	20	1	no	—
31	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
32	C₂×C₅²	C₅×D₄	1	6	yes	—
33	C₅×D₅	—	4	6	no	—
34	C₅×Q₈	—	2	12	no	—
35	C₅×Q₈	C₅×D₅	1	6	yes	complemented
36	C₅×Q₈	C₂×C₅²	1	1	yes	characteristic, complemented
37	C₅×D₄	—	5	6	no	—
38	D₂₀	C₂×C₅²	1	1	yes	characteristic
39	C₅×C₈	—	5	6	no	—
40	#168	C₂×C₅²	1	1	yes	characteristic, complemented
41	C₅²	—	2	12	no	—
42	C₅²	#688	1	1	yes	characteristic, direct product factor
43	C₅²	C₅×SD₁₆	1	6	yes	complemented
44	C₂²×C₅	—	10	6	no	—
45	C₂×D₅	—	2	1	no	—
46	C₄×C₅	—	2	24	no	—
47	C₄×C₅	—	2	6	no	—
48	C₄×C₅	—	2	1	no	—
49	C₄×C₅	—	2	12	no	—
50	C₄×C₅	C₂×C₅×D₅	1	6	yes	—
51	C₄×C₅	C₂²×C₅²	1	1	yes	characteristic
52	SD₁₆	—	5	1	no	Sylow
53	C₂×C₅	—	2	12	no	—
54	C₂×C₅	—	20	6	no	—
55	C₂×C₅	C₅×C₂²⋊D₅	1	6	yes	—
56	C₂×C₅	C₅²×D₄	1	1	yes	characteristic
57	D₅	—	4	1	no	—
58	Q₈	C₅²×D₅	1	1	yes	characteristic, complemented
59	D₄	—	5	1	no	—
60	C₈	—	5	1	no	—
61	C₅	—	2	12	no	—
62	C₅	C₅×#688	1	6	yes	direct product factor
63	C₅	C₅²×SD₁₆	1	1	yes	characteristic, complemented
64	C₂²	—	10	1	no	—
65	C₄	—	2	1	no	—
66	C₄	C₂×C₅²×D₅	1	1	yes	characteristic
67	C₂	—	20	1	no	—
68	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	C₂×C₅³	D₄
prime radical	C₅²×D₂₀	C₂
hypercenter	C₅²×Q₈	D₅
hypocenter	C₅	C₅²×SD₁₆
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
1. C₅³×Q₈	C₂	semidirect
3. C₅²×#168	C₂	semidirect
8. C₅×#688	C₅	direct (N × H)
12. C₅²×Q₈	D₅	semidirect
13. C₅²×Q₈	C₂×C₅	semidirect
17. C₅×#168	C₂×C₅	semidirect
18. C₅³	SD₁₆	semidirect
28. #688	C₅²	direct (N × H)
35. C₅×Q₈	C₅×D₅	semidirect
36. C₅×Q₈	C₂×C₅²	semidirect
40. #168	C₂×C₅²	semidirect
42. C₅²	#688	direct (N × H)
43. C₅²	C₅×SD₁₆	semidirect
58. Q₈	C₅²×D₅	semidirect
62. C₅	C₅×#688	direct (N × H)
63. C₅	C₅²×SD₁₆	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×D₂₀	2
C₅²×#168	2
C₅²×SD₁₆	5
C₅×#688	5

Subgroups — C52×#688 #14522619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#688 #14522619