Subgroups — #22548747

Order 8032.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16860811	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#16860800	C₂	1	1	yes	maximal, characteristic, complemented
4	#16860799	C₂	1	1	yes	maximal, characteristic, complemented
5	#16860797	C₂	1	1	yes	maximal, characteristic, complemented
6	#16860793	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₅₀₂	—	2	1	no	—
9	Dic₅₀₂	C₂²	1	1	yes	characteristic
10	D₁₀₀₄	—	2	1	no	—
11	D₁₀₀₄	C₂²	1	1	yes	characteristic
12	C₄×D₂₅₁	—	2	1	no	—
13	C₄×D₂₅₁	—	2	1	no	—
14	C₄×D₂₅₁	C₂²	1	1	yes	characteristic
15	C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
17	Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
18	C₈×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×D₂₅₁	—	2	1	no	—
20	C₂×D₂₅₁	D₄	1	1	yes	characteristic
21	C₄×C₂₅₁	—	2	1	no	—
22	C₄×C₂₅₁	—	2	1	no	—
23	C₄×C₂₅₁	C₂³	1	1	yes	characteristic
24	Dic₂₅₁	—	2	1	no	—
25	Dic₂₅₁	D₄	1	1	yes	characteristic
26	C₂×C₂₅₁	C₂×D₄	1	1	yes	characteristic
27	D₂₅₁	—	2	1	no	—
28	D₂₅₁	—	4	1	no	—
29	C₂₅₁	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	251	1	no	maximal, Sylow
31	#41	—	251	1	no	—
32	C₂×Q₈	—	251	1	no	—
33	Q₁₆	—	251	1	no	—
34	Q₁₆	D₂₅₁	1	1	yes	characteristic, complemented
35	SD₁₆	—	251	1	no	—
36	SD₁₆	—	251	1	no	—
37	M₁₆	—	251	1	no	—
38	Q₈	—	502	1	no	—
39	Q₈	—	251	1	no	—
40	Q₈	C₂×D₂₅₁	1	1	yes	characteristic
41	Q₈	C₂×D₂₅₁	1	1	yes	characteristic
42	D₄	—	502	1	no	—
43	D₄	—	251	1	no	—
44	C₂×C₄	—	502	1	no	—
45	C₂×C₄	—	502	1	no	—
46	C₂×C₄	—	251	1	no	—
47	C₈	—	251	1	no	—
48	C₈	C₂×D₂₅₁	1	1	yes	characteristic
49	C₂²	—	502	1	no	—
50	C₂²	—	251	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	502	1	no	—
54	C₄	—	251	1	no	—
55	C₄	C₂²×D₂₅₁	1	1	yes	characteristic
56	C₂	—	1004	1	no	—
57	C₂	—	502	1	no	—
58	C₂	D₄×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³
Frattini	C₄	C₂²×D₂₅₁
Fitting	Q₁₆×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂×D₄
hypercenter	Q₁₆	D₂₅₁
hypocenter	C₂₅₁	#137
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₂₅₁	C₂	semidirect
3. #16860800	C₂	semidirect
4. #16860799	C₂	semidirect
5. #16860797	C₂	semidirect
6. #16860793	C₂	semidirect
7. Q₁₆×C₂₅₁	C₂	semidirect
29. C₂₅₁	#137	semidirect
34. Q₁₆	D₂₅₁	semidirect