Subgroups — #22550308

Order 8128.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861231	C₂	1	2	yes	maximal, complemented
2	Dic₁₀₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861154	C₂	1	2	yes	maximal, complemented
4	D₂₀₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523248	—	2	2	no	—
7	Dic₅₀₈	C₂²	1	2	yes	—
8	#14523237	—	2	2	no	—
9	D₁₀₁₆	C₂²	1	2	yes	—
10	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	Dic₂₅₄	—	2	2	no	—
14	D₅₀₈	—	2	2	no	—
15	C₂²⋊D₁₂₇	—	4	2	no	—
16	C₄×D₁₂₇	—	4	2	no	—
17	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
18	C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₂²×C₁₂₇	D₈	1	1	yes	characteristic
21	C₂×D₁₂₇	—	4	2	no	—
22	C₄×C₁₂₇	D₈	1	1	yes	characteristic
23	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
24	Dic₁₂₇	—	4	2	no	—
25	C₂×C₁₂₇	—	2	1	no	—
26	C₂×C₁₂₇	C₂×D₈	1	1	yes	characteristic
27	D₁₂₇	—	8	2	no	—
28	C₁₂₇	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	127	1	no	maximal, Sylow
30	#135	—	127	2	no	—
31	Q₃₂	—	127	1	no	—
32	SD₃₂	—	127	2	no	—
33	D₁₆	—	127	1	no	—
34	C₂×C₁₆	D₁₂₇	1	1	yes	characteristic, complemented
35	#41	—	254	2	no	—
36	Q₁₆	—	127	2	no	—
37	SD₁₆	—	254	2	no	—
38	D₈	—	127	2	no	—
39	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
40	C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
41	C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
42	Q₈	—	254	2	no	—
43	D₄	—	508	2	no	—
44	D₄	—	254	2	no	—
45	C₂×C₄	—	508	2	no	—
46	C₂×C₄	D₅₀₈	1	1	yes	characteristic
47	C₈	D₅₀₈	1	1	yes	characteristic
48	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
49	C₂²	—	508	2	no	—
50	C₂²	D₁₀₁₆	1	1	yes	characteristic
51	C₄	—	508	2	no	—
52	C₄	D₁₀₁₆	1	1	yes	characteristic
53	C₄	C₂×D₅₀₈	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	1016	2	no	—
56	C₂	C₂×D₁₀₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₀₁₆
commutator	C₈×C₁₂₇	C₂³
Frattini	C₈	C₂²×D₁₂₇
Fitting	C₂×C₁₆×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₈
hypercenter	C₂×C₁₆	D₁₂₇
hypocenter	C₁₂₇	#508
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #16861231	C₂	semidirect
2. Dic₁₀₁₆	C₂	semidirect
3. #16861154	C₂	semidirect
4. D₂₀₃₂	C₂	semidirect
5. C₂×C₁₆×C₁₂₇	C₂	semidirect
28. C₁₂₇	#508	semidirect
34. C₂×C₁₆	D₁₂₇	semidirect