Subgroups — C₅×#87862 #14522795

Order 2000.

156 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×#6034	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	#87862	C₅	1	1	yes	maximal, characteristic, direct product factor
7	C₅×D₅⋊C₈	—	5	2	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	2	no	—
12	C₄×C₅×D₅	—	10	2	no	—
13	#6034	C₂×C₅	1	2	yes	complemented
14	C₅×#170	—	5	4	no	—
15	C₅³	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₅⋊D₅	C₄×C₅	1	1	yes	characteristic
17	C₂×C₅×D₅	—	5	2	no	—
18	C₂×C₅×D₅	—	10	2	no	—
19	C₄×C₅²	—	2	2	no	—
20	C₄×C₅²	—	4	4	no	—
21	C₄×C₅²	—	4	2	no	—
22	C₄×C₅²	C₄×C₅	1	1	yes	characteristic
23	C₄×C₅²	AGL(1,𝔽₅)	1	2	yes	—
24	#1039	C₂²×C₅	1	1	yes	characteristic
25	C₅×Dic₅	—	5	2	no	—
26	C₅×Dic₅	—	10	2	no	—
27	D₅⋊C₈	—	5	2	no	—
28	C₂×C₅×C₈	—	25	1	no	—
29	C₂×C₅²	—	2	2	no	—
30	C₂×C₅²	—	4	4	no	—
31	C₂×C₅²	—	4	2	no	—
32	C₂×C₅²	C₂×C₄×C₅	1	1	yes	characteristic
33	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	2	yes	—
34	C₅⋊D₅	C₅×C₈	1	2	yes	complemented
35	C₅×D₅	—	5	4	no	—
36	C₅×D₅	—	10	4	no	—
37	C₂×C₄×C₅	—	25	1	no	—
38	C₄×D₅	—	5	2	no	—
39	C₄×D₅	—	10	2	no	—
40	#170	—	5	4	no	—
41	C₅×C₈	—	25	2	no	—
42	C₅²	—	2	2	no	—
43	C₅²	—	4	4	no	—
44	C₅²	—	4	2	no	—
45	C₅²	C₂×C₅×C₈	1	1	yes	characteristic, complemented
46	C₅²	D₅⋊C₈	1	2	yes	complemented
47	C₂²×C₅	—	25	1	no	—
48	C₂×D₅	—	5	2	no	—
49	C₂×D₅	—	10	2	no	—
50	C₄×C₅	—	2	2	no	—
51	C₄×C₅	—	4	4	no	—
52	C₄×C₅	—	4	2	no	—
53	C₄×C₅	—	25	1	no	—
54	C₄×C₅	C₅×AGL(1,𝔽₅)	1	2	yes	—
55	C₄×C₅	#1044	1	1	yes	characteristic
56	Dic₅	—	5	2	no	—
57	Dic₅	—	10	2	no	—
58	C₂×C₈	—	25	1	no	Sylow
59	C₂×C₅	—	2	2	no	—
60	C₂×C₅	—	4	4	no	—
61	C₂×C₅	—	4	2	no	—
62	C₂×C₅	—	25	2	no	—
63	C₂×C₅	C₂×C₅×AGL(1,𝔽₅)	1	2	yes	—
64	C₂×C₅	C₂×#1044	1	1	yes	characteristic
65	D₅	—	5	4	no	—
66	D₅	—	10	4	no	—
67	C₂×C₄	—	25	1	no	—
68	C₈	—	25	2	no	—
69	C₅	—	2	2	no	—
70	C₅	—	4	4	no	—
71	C₅	—	4	2	no	—
72	C₅	C₅×D₅⋊C₈	1	2	yes	complemented
73	C₅	#87862	1	1	yes	characteristic, direct product factor
74	C₂²	—	25	1	no	—
75	C₄	—	25	1	no	—
76	C₄	C₅×#1044	1	1	yes	characteristic
77	C₂	—	25	2	no	—
78	C₂	C₂×C₅×#1044	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄×C₅	#1044
commutator, hypocenter	C₅²	C₂×C₅×C₈
Frattini	C₂	C₂×C₅×#1044
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×#6034	C₂	semidirect
6. #87862	C₅	direct (N × H)
9. C₅×C₅⋊D₅	C₈	semidirect
13. #6034	C₂×C₅	semidirect
15. C₅³	C₂×C₈	semidirect
34. C₅⋊D₅	C₅×C₈	semidirect
45. C₅²	C₂×C₅×C₈	semidirect
46. C₅²	D₅⋊C₈	semidirect
72. C₅	C₅×D₅⋊C₈	semidirect
73. C₅	#87862	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×#6034	2
#87862	5
C₅×D₅⋊C₈	5

Subgroups — C5×#87862 #14522795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#87862 #14522795