Subgroups — C₅×#87836 #14522743

Order 2000.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#1044	C₂	1	1	yes	maximal, characteristic
2	C₅×#6036	C₂	1	2	yes	maximal
3	C₂×C₅×C₅⋊D₅	C₂²	1	1	yes	characteristic
4	C₅×#1044	C₄	1	2	yes	complemented
5	C₅²×Dic₅	—	2	2	no	—
6	#87836	C₅	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₅³	D₄	1	1	yes	characteristic
8	C₅×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
9	C₅×C₅⋊D₅	Q₈	1	1	yes	characteristic
10	C₂×#1044	C₂×C₅	1	1	yes	characteristic
11	C₂×C₅×AGL(1,𝔽₅)	—	10	1	no	—
12	C₄×C₅×D₅	—	10	2	no	—
13	#6036	C₂×C₅	1	2	yes	—
14	C₅³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
15	C₂×C₅⋊D₅	C₂²×C₅	1	1	yes	characteristic
16	C₂×C₅×D₅	—	10	2	no	—
17	C₂×C₅×D₅	—	10	1	no	—
18	#1044	C₄×C₅	1	2	yes	complemented
19	C₅×AGL(1,𝔽₅)	—	10	2	no	—
20	C₄×C₅²	—	10	2	no	—
21	C₅×Dic₅	—	2	2	no	—
22	C₅×Dic₅	—	2	2	no	—
23	C₅×Dic₅	—	4	4	no	—
24	C₅×C₄⋊C₄	—	25	1	no	maximal
25	C₂×C₅²	—	2	2	no	—
26	C₂×C₅²	—	2	1	no	—
27	C₂×C₅²	—	4	4	no	—
28	C₂×C₅²	—	8	1	no	—
29	C₂×C₅²	C₅×D₄	1	1	yes	characteristic
30	C₅⋊D₅	C₂×C₄×C₅	1	1	yes	characteristic
31	C₅⋊D₅	C₅×Q₈	1	1	yes	characteristic
32	C₅×D₅	—	10	1	no	—
33	C₅×D₅	—	10	2	no	—
34	C₅×D₅	—	10	2	no	—
35	C₅×D₅	—	10	1	no	—
36	C₂×AGL(1,𝔽₅)	—	10	1	no	—
37	C₂×C₄×C₅	—	25	2	no	—
38	C₂×C₄×C₅	—	25	1	no	—
39	C₄×D₅	—	10	2	no	—
40	C₅²	—	2	2	no	—
41	C₅²	—	2	1	no	—
42	C₅²	—	4	4	no	—
43	C₅²	—	8	1	no	—
44	C₅²	C₅×C₄⋊C₄	1	1	yes	characteristic, complemented
45	C₂²×C₅	—	25	1	no	—
46	C₂×D₅	—	10	2	no	—
47	C₂×D₅	—	10	1	no	—
48	AGL(1,𝔽₅)	—	10	2	no	—
49	C₄×C₅	—	10	2	no	—
50	C₄×C₅	—	10	2	no	—
51	C₄×C₅	—	20	4	no	—
52	C₄×C₅	—	25	2	no	—
53	Dic₅	—	2	2	no	—
54	C₄⋊C₄	—	25	1	no	Sylow
55	C₂×C₅	—	2	2	no	—
56	C₂×C₅	—	2	1	no	—
57	C₂×C₅	—	4	4	no	—
58	C₂×C₅	—	8	1	no	—
59	C₂×C₅	—	25	1	no	—
60	C₂×C₅	—	25	1	no	—
61	C₂×C₅	D₅²⋊C₂	1	1	yes	characteristic
62	D₅	—	10	1	no	—
63	D₅	—	10	2	no	—
64	D₅	—	10	2	no	—
65	D₅	—	10	1	no	—
66	C₂×C₄	—	25	2	no	—
67	C₂×C₄	—	25	1	no	—
68	C₅	—	2	2	no	—
69	C₅	—	2	1	no	—
70	C₅	—	4	4	no	—
71	C₅	—	8	1	no	—
72	C₅	#87836	1	1	yes	characteristic, direct product factor
73	C₂²	—	25	1	no	—
74	C₄	—	10	2	no	—
75	C₄	—	25	2	no	—
76	C₂	—	25	1	no	—
77	C₂	—	25	1	no	—
78	C₂	C₅×D₅²⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	D₅²⋊C₂
commutator	C₅⋊D₅	C₂×C₄×C₅
Frattini	C₂	C₅×D₅²⋊C₂
Fitting, socle	C₂×C₅³	D₄
prime radical	C₂×C₅×C₅⋊D₅	C₂²
cosocle	C₂×C₅⋊D₅	C₂²×C₅
hypocenter	C₅²	C₅×C₄⋊C₄
LCS	G ▷ C₅⋊D₅ ▷ C₅²
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
4. C₅×#1044	C₄	semidirect
6. #87836	C₅	direct (N × H)
14. C₅³	C₄⋊C₄	semidirect
18. #1044	C₄×C₅	semidirect
44. C₅²	C₅×C₄⋊C₄	semidirect
72. C₅	#87836	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#1044	2
C₅×#6036	2
#87836	5
C₅×C₄⋊C₄	25

Subgroups — C5×#87836 #14522743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#87836 #14522743