Subgroups — #14522696

Order 2000.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₅×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	#11758743	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×C₅⋊D₅	C₄	1	1	yes	characteristic
4	#92553	C₂²	1	1	yes	characteristic
5	C₅²×Dic₅	C₄	1	1	yes	characteristic
6	#87862	—	5	1	no	maximal
7	D₅⋊#170	—	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	—	5	1	no	—
11	D₅×Dic₅	—	5	2	no	—
12	D₅×Dic₅	—	10	2	no	—
13	#6034	—	5	2	no	—
14	#6034	—	5	2	no	—
15	#6034	—	10	4	no	—
16	#6033	—	5	2	no	—
17	C₅³	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₅⋊D₅	AGL(1,𝔽₅)	1	1	yes	characteristic
19	C₂×C₅×D₅	—	5	2	no	—
20	C₂×C₅×D₅	—	10	2	no	—
21	C₄×C₅²	—	5	1	no	—
22	#1039	—	5	2	no	—
23	#1039	—	5	1	no	—
24	#1039	—	10	4	no	—
25	#1039	—	10	2	no	—
26	#1039	—	20	4	no	—
27	C₅×Dic₅	—	2	2	no	—
28	C₅×Dic₅	AGL(1,𝔽₅)	1	2	yes	—
29	C₂×#170	—	25	1	no	—
30	D₅⋊C₈	—	25	2	no	—
31	C₂×C₅²	—	2	4	no	—
32	C₂×C₅²	—	2	2	no	—
33	C₂×C₅²	—	4	4	no	—
34	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	2	yes	—
36	C₅⋊D₅	#170	1	2	yes	complemented
37	C₅×D₅	—	5	4	no	—
38	C₅×D₅	—	10	4	no	—
39	C₂×Dic₅	—	25	1	no	—
40	C₄×D₅	—	25	2	no	—
41	C₄×D₅	—	50	2	no	—
42	#170	—	25	2	no	—
43	#170	—	25	2	no	—
44	#170	—	25	2	no	—
45	#170	—	50	4	no	—
46	C₅²	—	2	4	no	—
47	C₅²	—	2	2	no	—
48	C₅²	—	4	4	no	—
49	C₅²	D₅⋊C₈	1	2	yes	complemented
50	C₅²	C₂×#170	1	1	yes	characteristic, complemented
51	C₂²×C₅	—	25	1	no	—
52	C₂×D₅	—	5	2	no	—
53	C₂×D₅	—	10	2	no	—
54	C₄×C₅	—	5	2	no	—
55	C₄×C₅	—	10	2	no	—
56	Dic₅	—	25	2	no	—
57	Dic₅	—	25	1	no	—
58	Dic₅	—	50	4	no	—
59	Dic₅	—	50	2	no	—
60	Dic₅	—	100	4	no	—
61	Dic₅	#1044	1	1	yes	characteristic
62	C₂×C₈	—	125	1	no	Sylow
63	C₂×C₅	—	2	4	no	—
64	C₂×C₅	—	2	2	no	—
65	C₂×C₅	—	4	4	no	—
66	C₂×C₅	—	25	2	no	—
67	C₂×C₅	D₅⋊AGL(1,𝔽₅)	1	2	yes	—
68	C₂×C₅	C₂×#1044	1	1	yes	characteristic
69	D₅	—	5	4	no	—
70	D₅	—	10	4	no	—
71	C₂×C₄	—	125	1	no	—
72	C₈	—	125	2	no	—
73	C₅	—	2	2	no	—
74	C₅	—	2	4	no	—
75	C₅	—	4	4	no	—
76	C₅	D₅⋊#170	1	2	yes	complemented
77	C₅	#87862	1	1	yes	characteristic, complemented
78	C₂²	—	25	1	no	—
79	C₄	—	5	1	no	—
80	C₄	—	125	1	no	—
81	C₂	—	25	2	no	—
82	C₂	D₅⋊#1044	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅⋊#1044
commutator, hypocenter	C₅³	C₂×C₈
Fitting, socle	C₂×C₅³	C₂×C₄
prime radical	C₂×C₅×C₅⋊D₅	C₄
cosocle	#92553	C₂²
LCS	G ▷ C₅³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. #11758743	C₂	semidirect
9. C₅×C₅⋊D₅	C₈	semidirect
17. C₅³	C₂×C₈	semidirect
36. C₅⋊D₅	#170	semidirect
49. C₅²	D₅⋊C₈	semidirect
50. C₅²	C₂×#170	semidirect
76. C₅	D₅⋊#170	semidirect
77. C₅	#87862	semidirect

Maximal subgroups

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

subgroup	index
Dic₅×C₅⋊D₅	2
#11758743	2
#87862	5
D₅⋊#170	5