Subgroups — C₅×#87855 #14522788

Order 2000.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₅×#6033	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×C₅⋊D₅	C₄	1	1	yes	characteristic
4	C₄×C₅³	C₄	1	1	yes	characteristic
5	C₅×#1039	C₂²	1	1	yes	characteristic
6	#87855	C₅	1	1	yes	maximal, characteristic, direct product factor
7	C₅×D₅⋊C₈	—	5	6	no	maximal
8	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
9	C₅×C₅⋊D₅	C₈	1	2	yes	complemented
10	C₄×C₅⋊D₅	C₂×C₅	1	1	yes	characteristic
11	C₄×C₅×D₅	—	5	6	no	—
12	#6033	C₂×C₅	1	2	yes	complemented
13	C₅×#170	—	5	12	no	—
14	C₅³	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
15	C₂×C₅⋊D₅	C₄×C₅	1	1	yes	characteristic
16	C₂×C₅×D₅	—	5	6	no	—
17	C₄×C₅²	—	4	6	no	—
18	C₄×C₅²	C₄×C₅	1	1	yes	characteristic
19	C₄×C₅²	AGL(1,𝔽₅)	1	6	yes	—
20	#1039	C₂²×C₅	1	1	yes	characteristic
21	C₅×Dic₅	—	5	6	no	—
22	D₅⋊C₈	—	5	6	no	—
23	C₂×C₅×C₈	—	25	1	no	—
24	C₂×C₅²	—	4	6	no	—
25	C₂×C₅²	C₂×C₄×C₅	1	1	yes	characteristic
26	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	6	yes	—
27	C₅⋊D₅	C₅×C₈	1	2	yes	complemented
28	C₅×D₅	—	5	12	no	—
29	C₂×C₄×C₅	—	25	1	no	—
30	C₄×D₅	—	5	6	no	—
31	#170	—	5	12	no	—
32	C₅×C₈	—	25	2	no	—
33	C₅²	—	4	6	no	—
34	C₅²	C₂×C₅×C₈	1	1	yes	characteristic, complemented
35	C₅²	D₅⋊C₈	1	6	yes	complemented
36	C₂²×C₅	—	25	1	no	—
37	C₂×D₅	—	5	6	no	—
38	C₄×C₅	—	4	6	no	—
39	C₄×C₅	—	25	1	no	—
40	C₄×C₅	C₅×AGL(1,𝔽₅)	1	6	yes	—
41	C₄×C₅	#1043	1	1	yes	characteristic
42	Dic₅	—	5	6	no	—
43	C₂×C₈	—	25	1	no	Sylow
44	C₂×C₅	—	4	6	no	—
45	C₂×C₅	—	25	2	no	—
46	C₂×C₅	C₂×C₅×AGL(1,𝔽₅)	1	6	yes	—
47	C₂×C₅	C₂×#1043	1	1	yes	characteristic
48	D₅	—	5	12	no	—
49	C₂×C₄	—	25	1	no	—
50	C₈	—	25	2	no	—
51	C₅	—	4	6	no	—
52	C₅	C₅×D₅⋊C₈	1	6	yes	complemented
53	C₅	#87855	1	1	yes	characteristic, direct product factor
54	C₂²	—	25	1	no	—
55	C₄	—	25	1	no	—
56	C₄	C₅×#1043	1	1	yes	characteristic
57	C₂	—	25	2	no	—
58	C₂	C₂×C₅×#1043	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄×C₅	#1043
commutator, hypocenter	C₅²	C₂×C₅×C₈
Frattini	C₂	C₂×C₅×#1043
Fitting	C₄×C₅³	C₄
socle	C₂×C₅³	C₂×C₄
prime radical	C₂×C₅×C₅⋊D₅	C₄
cosocle	#1039	C₂²×C₅
LCS	G ▷ C₅²
UCS	C₁ ◁ C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×#6033	C₂	semidirect
6. #87855	C₅	direct (N × H)
9. C₅×C₅⋊D₅	C₈	semidirect
12. #6033	C₂×C₅	semidirect
14. C₅³	C₂×C₈	semidirect
27. C₅⋊D₅	C₅×C₈	semidirect
34. C₅²	C₂×C₅×C₈	semidirect
35. C₅²	D₅⋊C₈	semidirect
52. C₅	C₅×D₅⋊C₈	semidirect
53. C₅	#87855	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×C₅×C₅⋊D₅	2
C₅×#6033	2
#87855	5
C₅×D₅⋊C₈	5

Subgroups — C5×#87855 #14522788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#87855 #14522788