Subgroups — C₅×#87806 #14522630

Order 2000.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#1039	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#1039	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅³	C₂²	1	1	yes	characteristic
5	C₄×C₅³	—	2	1	no	—
6	C₅×#1039	—	2	1	no	—
7	C₅×#1039	C₄	1	2	yes	complemented
8	#87806	C₅	1	1	yes	maximal, characteristic, direct product factor
9	C₅×#683	—	5	6	no	maximal
10	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅³	D₄	1	1	yes	characteristic
12	C₂×C₅³	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₅²	—	2	12	no	—
14	C₂×C₄×C₅²	D₅	1	6	yes	—
15	C₂×C₄×C₅²	C₂×C₅	1	1	yes	characteristic
16	C₂×#1039	C₂×C₅	1	1	yes	characteristic
17	C₂×#1039	C₂×C₅	1	1	yes	characteristic
18	C₂×C₅×Dic₅	—	5	6	no	—
19	C₂×C₅×Dic₅	—	5	6	no	—
20	C₅³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₅²	—	2	12	no	—
22	C₂²×C₅²	C₂×D₅	1	6	yes	—
23	C₂²×C₅²	C₂²×C₅	1	1	yes	characteristic
24	C₄×C₅²	—	2	24	no	—
25	C₄×C₅²	—	2	6	no	—
26	C₄×C₅²	—	2	1	no	—
27	#1039	—	2	1	no	—
28	#1039	C₄×C₅	1	2	yes	complemented
29	C₅×Dic₅	—	5	12	no	—
30	C₅×Dic₅	—	10	6	no	—
31	C₅×C₄⋊C₄	—	25	1	no	—
32	#683	—	5	6	no	—
33	C₂×C₅²	—	2	12	no	—
34	C₂×C₅²	—	2	12	no	—
35	C₂×C₅²	—	2	12	no	—
36	C₂×C₅²	Dic₁₀	1	6	yes	—
37	C₂×C₅²	C₄×D₅	1	6	yes	—
38	C₂×C₅²	C₂²⋊D₅	1	6	yes	—
39	C₂×C₅²	C₂×C₄×C₅	1	1	yes	characteristic
40	C₂×C₅²	C₅×D₄	1	1	yes	characteristic
41	C₂×C₅²	C₅×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₅	—	2	12	no	—
43	C₂×C₄×C₅	—	25	1	no	—
44	C₂×C₄×C₅	—	25	1	no	—
45	C₂×C₄×C₅	C₅×D₅	1	6	yes	—
46	C₂×C₄×C₅	C₅⋊D₅	1	1	yes	characteristic
47	C₂×Dic₅	—	5	6	no	—
48	C₂×Dic₅	—	5	6	no	—
49	C₅²	—	2	12	no	—
50	C₅²	#683	1	6	yes	complemented
51	C₅²	C₅×C₄⋊C₄	1	1	yes	characteristic, complemented
52	C₂²×C₅	—	2	12	no	—
53	C₂²×C₅	C₂×C₅×D₅	1	6	yes	—
54	C₂²×C₅	C₂×C₅⋊D₅	1	1	yes	characteristic
55	C₄×C₅	—	2	24	no	—
56	C₄×C₅	—	2	6	no	—
57	C₄×C₅	—	2	1	no	—
58	C₄×C₅	—	25	2	no	—
59	C₄×C₅	—	50	1	no	—
60	Dic₅	—	5	12	no	—
61	Dic₅	—	10	6	no	—
62	C₄⋊C₄	—	25	1	no	Sylow
63	C₂×C₅	—	2	12	no	—
64	C₂×C₅	—	2	12	no	—
65	C₂×C₅	—	2	12	no	—
66	C₂×C₅	C₅×Dic₁₀	1	6	yes	—
67	C₂×C₅	C₄×C₅×D₅	1	6	yes	—
68	C₂×C₅	C₅×C₂²⋊D₅	1	6	yes	—
69	C₂×C₅	#6045	1	1	yes	characteristic
70	C₂×C₅	C₄×C₅⋊D₅	1	1	yes	characteristic
71	C₂×C₅	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
72	C₂×C₄	—	25	1	no	—
73	C₂×C₄	—	25	1	no	—
74	C₂×C₄	C₅×C₅⋊D₅	1	1	yes	characteristic
75	C₅	—	2	12	no	—
76	C₅	C₅×#683	1	6	yes	complemented
77	C₅	#87806	1	1	yes	characteristic, direct product factor
78	C₂²	C₂×C₅×C₅⋊D₅	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	25	2	no	—
81	C₄	—	50	1	no	—
82	C₂	C₅×#6045	1	1	yes	characteristic
83	C₂	C₄×C₅×C₅⋊D₅	1	1	yes	characteristic
84	C₂	C₅×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₅×#1039	C₄	semidirect
8. #87806	C₅	direct (N × H)
20. C₅³	C₄⋊C₄	semidirect
28. #1039	C₄×C₅	semidirect
50. C₅²	#683	semidirect
51. C₅²	C₅×C₄⋊C₄	semidirect
76. C₅	C₅×#683	semidirect
77. C₅	#87806	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×C₄×C₅³	2
C₂×C₅×#1039	2
#87806	5
C₅×#683	5

Subgroups — C5×#87806 #14522630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#87806 #14522630