Subgroups — C₅×#87904 #14522916

Order 2000.

464 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	C₅×C₄⋊(C₅⋊D₅)	C₂	1	3	yes	maximal, complemented
3	C₄×C₅×C₅⋊D₅	C₂	1	3	yes	maximal, complemented
4	C₂×C₅×C₅⋊D₅	C₂²	1	3	yes	—
5	C₄×C₅³	C₂²	1	3	yes	—
6	C₅×#1039	C₂²	1	1	yes	characteristic
7	#87904	C₅	1	1	yes	maximal, characteristic, direct product factor
8	C₅×#713	—	5	6	no	maximal
9	C₂×C₅³	C₂³	1	1	yes	characteristic
10	C₅×C₅⋊D₅	—	2	3	no	—
11	C₅²×Q₈	—	2	12	no	—
12	C₅²×Q₈	D₅	1	6	yes	complemented
13	C₅²×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
14	C₄⋊(C₅⋊D₅)	C₂×C₅	1	3	yes	complemented
15	C₄×C₅⋊D₅	C₂×C₅	1	3	yes	complemented
16	C₅×D₂₀	—	5	18	no	—
17	C₄×C₅×D₅	—	5	18	no	—
18	C₅³	#41	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₅⋊D₅	C₂²×C₅	1	3	yes	—
20	C₂×C₅×D₅	—	5	18	no	—
21	C₄×C₅²	—	2	36	no	—
22	C₄×C₅²	C₂×D₅	1	18	yes	—
23	C₄×C₅²	C₂²×C₅	1	3	yes	—
24	#1039	C₂²×C₅	1	1	yes	characteristic
25	C₅×Dic₅	—	5	6	no	—
26	C₅×#41	—	25	1	no	—
27	#713	—	5	6	no	—
28	C₂×C₅²	—	2	12	no	—
29	C₂×C₅²	C₂²×D₅	1	6	yes	—
30	C₂×C₅²	C₂³×C₅	1	1	yes	characteristic
31	C₅⋊D₅	—	2	3	no	—
32	C₅×D₅	—	10	18	no	—
33	C₅×Q₈	—	2	12	no	—
34	C₅×Q₈	C₅×D₅	1	6	yes	complemented
35	C₅×Q₈	C₅⋊D₅	1	1	yes	characteristic, complemented
36	C₅×D₄	—	25	3	no	—
37	C₂×C₄×C₅	—	25	3	no	—
38	D₂₀	—	5	18	no	—
39	C₄×D₅	—	5	18	no	—
40	C₅²	—	2	12	no	—
41	C₅²	#713	1	6	yes	complemented
42	C₅²	C₅×#41	1	1	yes	characteristic, complemented
43	C₂²×C₅	—	25	3	no	—
44	C₂×D₅	—	5	18	no	—
45	C₄×C₅	—	2	36	no	—
46	C₄×C₅	—	25	1	no	—
47	C₄×C₅	C₂×C₅×D₅	1	18	yes	—
48	C₄×C₅	C₂×C₅⋊D₅	1	3	yes	—
49	Dic₅	—	5	6	no	—
50	#41	—	25	1	no	Sylow
51	C₂×C₅	—	2	12	no	—
52	C₂×C₅	—	50	3	no	—
53	C₂×C₅	C₂²×C₅×D₅	1	6	yes	—
54	C₂×C₅	C₂²×C₅⋊D₅	1	1	yes	characteristic
55	D₅	—	10	18	no	—
56	Q₈	C₅×C₅⋊D₅	1	1	yes	characteristic, complemented
57	D₄	—	25	3	no	—
58	C₂×C₄	—	25	3	no	—
59	C₅	—	2	12	no	—
60	C₅	C₅×#713	1	6	yes	complemented
61	C₅	#87904	1	1	yes	characteristic, direct product factor
62	C₂²	—	25	3	no	—
63	C₄	—	25	1	no	—
64	C₄	C₂×C₅×C₅⋊D₅	1	3	yes	—
65	C₂	—	50	3	no	—
66	C₂	C₂²×C₅×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂²×C₅⋊D₅
commutator, cosocle	C₂×C₅²	C₂³×C₅
Frattini	C₂	C₂²×C₅×C₅⋊D₅
Fitting	C₅³×Q₈	C₂
socle	C₂×C₅³	C₂³
hypercenter	C₅×Q₈	C₅⋊D₅
hypocenter	C₅²	C₅×#41
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₅ ◁ C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₅³×Q₈	C₂	semidirect
2. C₅×C₄⋊(C₅⋊D₅)	C₂	semidirect
3. C₄×C₅×C₅⋊D₅	C₂	semidirect
7. #87904	C₅	direct (N × H)
12. C₅²×Q₈	D₅	semidirect
13. C₅²×Q₈	C₂×C₅	semidirect
14. C₄⋊(C₅⋊D₅)	C₂×C₅	semidirect
15. C₄×C₅⋊D₅	C₂×C₅	semidirect
18. C₅³	#41	semidirect
34. C₅×Q₈	C₅×D₅	semidirect
35. C₅×Q₈	C₅⋊D₅	semidirect
41. C₅²	#713	semidirect
42. C₅²	C₅×#41	semidirect
56. Q₈	C₅×C₅⋊D₅	semidirect
60. C₅	C₅×#713	semidirect
61. C₅	#87904	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅³×Q₈	2
C₅×C₄⋊(C₅⋊D₅)	2
C₄×C₅×C₅⋊D₅	2
#87904	5
C₅×#713	5

Subgroups — C5×#87904 #14522916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#87904 #14522916