Subgroups — #14522140

Order 2000.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#92524	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅×C₂₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×Dic₂₅	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅×C₂₅	C₂²	1	1	yes	characteristic
5	#92524	—	2	1	no	—
6	Dic₅×C₂₅	—	2	1	no	—
7	C₅×Dic₂₅	C₄	1	2	yes	complemented
8	#87780	—	5	1	no	maximal
9	#87718	—	5	1	no	maximal
10	C₂×C₅×C₂₅	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅×C₂₅	D₄	1	1	yes	characteristic
12	C₂×C₅×C₂₅	Q₈	1	1	yes	characteristic
13	C₂×#1039	—	5	1	no	—
14	C₂×C₅×Dic₅	—	5	1	no	—
15	C₂×C₅×Dic₅	D₅	1	1	yes	characteristic
16	C₂×C₄×C₂₅	—	5	1	no	—
17	C₂×Dic₂₅	—	5	1	no	—
18	C₂×Dic₂₅	—	10	2	no	—
19	C₂×Dic₂₅	D₅	1	1	yes	characteristic
20	C₅×C₂₅	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₅²	C₂×D₅	1	1	yes	characteristic
22	#1039	—	10	1	no	—
23	C₅×Dic₅	—	2	1	no	—
24	C₅×Dic₅	—	5	2	no	—
25	C₂²×C₂₅	—	2	2	no	—
26	C₂²×C₂₅	C₂×D₅	1	1	yes	characteristic
27	C₄×C₂₅	—	10	1	no	—
28	Dic₂₅	—	10	4	no	—
29	Dic₂₅	—	10	1	no	—
30	Dic₂₅	Dic₅	1	2	yes	complemented
31	C₄⋊Dic₅	—	25	1	no	—
32	#683	—	25	1	no	—
33	C₂×C₅²	Dic₁₀	1	1	yes	characteristic
34	C₂×C₅²	C₄×D₅	1	1	yes	characteristic
35	C₂×C₅²	C₂²⋊D₅	1	1	yes	characteristic
36	C₂×C₂₅	—	2	2	no	—
37	C₂×C₂₅	—	2	2	no	—
38	C₂×C₂₅	—	2	2	no	—
39	C₂×C₂₅	Dic₁₀	1	1	yes	characteristic
40	C₂×C₂₅	D₂₀	1	1	yes	characteristic
41	C₂×C₂₅	C₂×Dic₅	1	1	yes	characteristic
42	C₂×C₄×C₅	—	5	1	no	—
43	C₂×C₄×C₅	—	25	1	no	—
44	C₂×Dic₅	—	5	1	no	—
45	C₂×Dic₅	—	25	1	no	—
46	C₂×Dic₅	—	25	1	no	—
47	C₂×Dic₅	—	50	2	no	—
48	C₂×Dic₅	D₂₅	1	1	yes	characteristic
49	C₅²	#683	1	1	yes	characteristic
50	C₂₅	—	2	2	no	—
51	C₂₅	C₄⋊Dic₅	1	1	yes	characteristic, complemented
52	C₂²×C₅	—	2	2	no	—
53	C₂²×C₅	C₂×D₂₅	1	1	yes	characteristic
54	C₂²×C₅	D₅²	1	1	yes	characteristic
55	C₄×C₅	—	10	1	no	—
56	C₄×C₅	—	25	2	no	—
57	Dic₅	—	2	1	no	—
58	Dic₅	—	5	2	no	—
59	Dic₅	—	50	4	no	—
60	Dic₅	—	50	1	no	—
61	Dic₅	—	50	1	no	—
62	C₄⋊C₄	—	125	1	no	Sylow
63	C₂×C₅	—	2	2	no	—
64	C₂×C₅	—	2	2	no	—
65	C₂×C₅	—	2	2	no	—
66	C₂×C₅	Dic₅₀	1	1	yes	characteristic
67	C₂×C₅	C₄×D₂₅	1	1	yes	characteristic
68	C₂×C₅	C₂²⋊D₂₅	1	1	yes	characteristic
69	C₂×C₅	D₅×Dic₅	1	1	yes	characteristic
70	C₂×C₅	#6038	1	1	yes	characteristic
71	C₂×C₅	#6039	1	1	yes	characteristic
72	C₂×C₄	—	5	1	no	—
73	C₂×C₄	—	25	1	no	—
74	C₂×C₄	—	125	1	no	—
75	C₅	—	2	2	no	—
76	C₅	#87718	1	1	yes	characteristic, complemented
77	C₅	#87780	1	1	yes	characteristic
78	C₂²	D₅×D₂₅	1	1	yes	characteristic
79	C₄	—	10	1	no	—
80	C₄	—	25	2	no	—
81	C₄	—	250	1	no	—
82	C₂	#11758650	1	1	yes	characteristic
83	C₂	Dic₅×D₂₅	1	1	yes	characteristic
84	C₂	#11758656	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	D₅×D₂₅
commutator	C₂×C₅×C₂₅	C₂×C₄
Frattini	C₂²×C₅	D₅²
Fitting, cosocle	C₂²×C₅×C₂₅	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₅
hypocenter	C₅×C₂₅	C₄⋊C₄
LCS	G ▷ C₂×C₅×C₂₅ ▷ C₅×C₂₅
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
7. C₅×Dic₂₅	C₄	semidirect
20. C₅×C₂₅	C₄⋊C₄	semidirect
30. Dic₂₅	Dic₅	semidirect
51. C₂₅	C₄⋊Dic₅	semidirect
76. C₅	#87718	semidirect

Maximal subgroups

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

subgroup	index
C₂×#92524	2
C₂×Dic₅×C₂₅	2
C₂×C₅×Dic₂₅	2
#87780	5
#87718	5