Subgroups — #22548629

Order 8032.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₂₅₁	C₄	1	1	yes	characteristic, complemented
5	C₂²⋊D₂₅₁	—	2	1	no	—
6	C₂²⋊D₂₅₁	—	2	1	no	—
7	C₂×Dic₂₅₁	—	2	1	no	—
8	C₂×Dic₂₅₁	C₄	1	1	yes	characteristic, complemented
9	C₂³×C₂₅₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₂₅₁	—	2	1	no	—
11	C₂²×C₂₅₁	—	4	1	no	—
12	C₂²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₂₅₁	D₄	1	1	yes	characteristic
14	C₂²×C₂₅₁	D₄	1	1	yes	characteristic
15	C₂×D₂₅₁	—	2	1	no	—
16	C₂×D₂₅₁	—	4	1	no	—
17	C₄×C₂₅₁	—	4	1	no	—
18	Dic₂₅₁	—	2	1	no	—
19	Dic₂₅₁	—	4	1	no	—
20	C₂×C₂₅₁	—	2	1	no	—
21	C₂×C₂₅₁	—	2	1	no	—
22	C₂×C₂₅₁	—	2	1	no	—
23	C₂×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
24	D₂₅₁	—	4	1	no	—
25	C₂₅₁	#99	1	1	yes	characteristic, complemented, Sylow
26	#99	—	251	1	no	maximal, Sylow
27	C₂×D₄	—	251	1	no	—
28	C₂²⋊C₄	—	251	1	no	—
29	C₂²⋊C₄	D₂₅₁	1	1	yes	characteristic, complemented
30	C₂³	—	251	1	no	—
31	C₂³	C₂×D₂₅₁	1	1	yes	characteristic
32	D₄	—	502	1	no	—
33	D₄	—	502	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	502	1	no	—
36	C₂×C₄	—	251	1	no	—
37	C₂²	—	4	1	no	—
38	C₂²	—	1004	1	no	—
39	C₂²	—	502	1	no	—
40	C₂²	C₄×D₂₅₁	1	1	yes	characteristic
41	C₂²	C₂²⋊D₂₅₁	1	1	yes	characteristic
42	C₂²	D₁₀₀₄	1	1	yes	characteristic
43	C₄	—	4	1	no	—
44	C₄	—	1004	1	no	—
45	C₄	—	502	1	no	—
46	C₂	—	2	1	no	—
47	C₂	—	2	1	no	—
48	C₂	—	2	1	no	—
49	C₂	—	1004	1	no	—
50	C₂	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×D₂₅₁)⋊C₄
commutator	C₂²×C₂₅₁	C₂×C₄
Frattini	C₂³	C₂×D₂₅₁
Fitting	C₂²⋊C₄×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂²⋊C₄
hypercenter	C₂²⋊C₄	D₂₅₁
hypocenter	C₂₅₁	#99
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²⋊Dic₂₅₁	C₂	semidirect
3. C₂²⋊C₄×C₂₅₁	C₂	semidirect
4. C₂²×D₂₅₁	C₄	semidirect
8. C₂×Dic₂₅₁	C₄	semidirect
25. C₂₅₁	#99	semidirect
29. C₂²⋊C₄	D₂₅₁	semidirect