Subgroups — #14522727

Order 2000.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11758751	C₂	1	1	yes	maximal, characteristic
2	#11758739	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×#170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅⋊(C₅⋊D₅)	C₄	1	1	yes	characteristic
5	C₅×#1039	C₄	1	1	yes	characteristic
6	C₅²×Dic₅	C₂²	1	1	yes	characteristic
7	#87829	—	5	6	no	maximal
8	#87799	—	5	1	no	maximal
9	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
10	C₅⋊(C₅⋊D₅)	—	2	1	no	—
11	C₄×C₅⋊D₅	—	5	1	no	—
12	#6036	—	5	6	no	—
13	#6032	—	5	6	no	—
14	C₅×#170	D₅	1	6	yes	complemented
15	C₅²×C₈	—	5	1	no	—
16	#6029	—	5	1	no	—
17	C₅³	M₁₆	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₅⋊D₅	—	5	6	no	—
19	C₂×C₅⋊D₅	—	5	1	no	—
20	C₂×C₅⋊D₅	—	20	6	no	—
21	C₄×C₅²	—	5	1	no	—
22	#1039	AGL(1,𝔽₅)	1	1	yes	characteristic
23	C₅×Dic₅	—	5	6	no	—
24	C₅×Dic₅	C₂×D₅	1	6	yes	—
25	#700	—	25	1	no	—
26	#676	—	25	6	no	—
27	C₂×C₅²	—	4	6	no	—
28	C₂×C₅²	C₄×D₅	1	6	yes	—
29	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
30	C₅⋊D₅	—	10	1	no	—
31	C₅⋊D₅	—	10	6	no	—
32	C₅⋊D₅	—	20	12	no	—
33	C₄×D₅	—	25	6	no	—
34	C₄×D₅	—	25	1	no	—
35	#170	—	25	1	no	—
36	#170	C₅⋊D₅	1	1	yes	characteristic, complemented
37	C₅×C₈	—	5	6	no	—
38	#168	—	25	6	no	—
39	C₅²	—	4	6	no	—
40	C₅²	#676	1	6	yes	complemented
41	C₅²	#700	1	1	yes	characteristic, complemented
42	C₂×D₅	—	25	1	no	—
43	C₂×D₅	—	25	6	no	—
44	C₂×D₅	—	100	6	no	—
45	C₄×C₅	—	5	6	no	—
46	C₄×C₅	—	25	1	no	—
47	Dic₅	—	5	6	no	—
48	Dic₅	C₂×C₅⋊D₅	1	1	yes	characteristic
49	M₁₆	—	125	1	no	Sylow
50	C₂×C₅	—	4	6	no	—
51	C₂×C₅	C₄×C₅⋊D₅	1	1	yes	characteristic
52	C₂×C₅	D₅×AGL(1,𝔽₅)	1	6	yes	—
53	D₅	—	50	6	no	—
54	D₅	—	50	1	no	—
55	D₅	—	100	12	no	—
56	C₂×C₄	—	125	1	no	—
57	C₈	—	5	1	no	—
58	C₈	—	125	1	no	—
59	C₅	—	4	6	no	—
60	C₅	#87799	1	1	yes	characteristic, complemented
61	C₅	#87829	1	6	yes	complemented
62	C₂²	—	125	1	no	—
63	C₄	—	5	1	no	—
64	C₄	—	25	1	no	—
65	C₂	—	250	1	no	—
66	C₂	AGL(1,𝔽₅)×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₅)×C₅⋊D₅
commutator, Fitting, socle	C₂×C₅³	C₂×C₄
prime radical	C₂×C₅⋊(C₅⋊D₅)	C₄
cosocle	C₅²×Dic₅	C₂²
hypocenter	C₅³	M₁₆
LCS	G ▷ C₂×C₅³ ▷ C₅³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. #11758739	C₂	semidirect
3. C₅²×#170	C₂	semidirect
14. C₅×#170	D₅	semidirect
17. C₅³	M₁₆	semidirect
36. #170	C₅⋊D₅	semidirect
40. C₅²	#676	semidirect
41. C₅²	#700	semidirect
60. C₅	#87799	semidirect
61. C₅	#87829	semidirect

Maximal subgroups

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

subgroup	index
#11758751	2
#11758739	2
C₅²×#170	2
#87829	5
#87799	5