Subgroups — #22551621

Order 8224.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861617	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₀₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861608	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861606	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861601	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₅₁₄	—	2	1	no	—
9	Dic₅₁₄	C₂²	1	1	yes	characteristic
10	Dic₅₁₄	C₂²	1	1	yes	characteristic
11	C₂²⋊D₂₅₇	—	2	1	no	—
12	C₂×Dic₂₅₇	—	2	1	no	—
13	C₄×D₂₅₇	—	2	1	no	—
14	C₄×D₂₅₇	C₂²	1	1	yes	characteristic
15	#14523339	C₂²	1	1	yes	characteristic
16	Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
17	D₄×C₂₅₇	C₂²	1	1	yes	characteristic
18	C₈×C₂₅₇	C₂²	1	1	yes	characteristic
19	C₂²×C₂₅₇	—	2	1	no	—
20	C₂×D₂₅₇	D₄	1	1	yes	characteristic
21	C₄×C₂₅₇	—	2	1	no	—
22	C₄×C₂₅₇	C₂³	1	1	yes	characteristic
23	Dic₂₅₇	—	2	1	no	—
24	Dic₂₅₇	—	2	1	no	—
25	Dic₂₅₇	D₄	1	1	yes	characteristic
26	C₂×C₂₅₇	—	4	1	no	—
27	C₂×C₂₅₇	C₂×D₄	1	1	yes	characteristic
28	D₂₅₇	—	2	1	no	—
29	C₂₅₇	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	257	1	no	maximal, Sylow
31	#41	—	257	1	no	—
32	C₂×Q₈	—	257	1	no	—
33	Q₁₆	—	257	1	no	—
34	Q₁₆	—	257	1	no	—
35	SD₁₆	—	257	1	no	—
36	SD₁₆	D₂₅₇	1	1	yes	characteristic, complemented
37	M₁₆	—	257	1	no	—
38	Q₈	—	257	1	no	—
39	Q₈	—	257	1	no	—
40	Q₈	—	514	1	no	—
41	Q₈	C₂×D₂₅₇	1	1	yes	characteristic
42	D₄	—	514	1	no	—
43	D₄	C₂×D₂₅₇	1	1	yes	characteristic
44	C₂×C₄	—	257	1	no	—
45	C₂×C₄	—	514	1	no	—
46	C₂×C₄	—	514	1	no	—
47	C₈	—	257	1	no	—
48	C₈	C₂×D₂₅₇	1	1	yes	characteristic
49	C₂²	—	257	1	no	—
50	C₂²	—	2	1	no	—
51	C₄	—	257	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	514	1	no	—
54	C₄	—	514	1	no	—
55	C₄	C₂²×D₂₅₇	1	1	yes	characteristic
56	C₂	—	514	1	no	—
57	C₂	—	4	1	no	—
58	C₂	D₄×D₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₅₇
commutator	C₄×C₂₅₇	C₂³
Frattini	C₄	C₂²×D₂₅₇
Fitting	SD₁₆×C₂₅₇	C₂
socle	C₂×C₂₅₇	C₂×D₄
hypercenter	SD₁₆	D₂₅₇
hypocenter	C₂₅₇	#137
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₂₅₇	C₂	semidirect
3. Dic₁₀₂₈	C₂	semidirect
4. #16861608	C₂	semidirect
5. #16861606	C₂	semidirect
6. #16861601	C₂	semidirect
7. SD₁₆×C₂₅₇	C₂	semidirect
29. C₂₅₇	#137	semidirect
36. SD₁₆	D₂₅₇	semidirect