Subgroups — #22549685

Order 8128.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861231	C₂	1	1	yes	maximal, characteristic
2	#16861149	C₂	1	1	yes	maximal, characteristic, complemented
3	#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14523248	—	2	1	no	—
5	Dic₅₀₈	C₄	1	1	yes	characteristic, complemented
6	#14523237	—	2	1	no	—
7	D₁₀₁₆	C₄	1	1	yes	characteristic, complemented
8	C₁₂₇⋊C₁₆	—	2	1	no	—
9	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₁₂₇	—	2	1	no	—
11	Dic₂₅₄	—	2	1	no	—
12	D₅₀₈	—	2	1	no	—
13	C₂²⋊D₁₂₇	—	4	1	no	—
14	C₄×D₁₂₇	—	4	1	no	—
15	C₂×C₄×C₁₂₇	—	4	1	no	—
16	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
17	C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂²×C₁₂₇	D₈	1	1	yes	characteristic
20	C₂×D₁₂₇	—	4	1	no	—
21	C₄×C₁₂₇	—	8	1	no	—
22	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
24	Dic₁₂₇	—	4	1	no	—
25	C₂×C₁₂₇	—	2	1	no	—
26	C₂×C₁₂₇	#102	1	1	yes	characteristic
27	D₁₂₇	—	8	1	no	—
28	C₁₂₇	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	127	1	no	maximal, Sylow
30	#135	—	127	1	no	—
31	M₃₂	—	127	1	no	—
32	#106	D₁₂₇	1	1	yes	characteristic, complemented
33	#41	—	254	1	no	—
34	Q₁₆	—	127	1	no	—
35	SD₁₆	—	254	1	no	—
36	D₈	—	127	1	no	—
37	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
38	C₄⋊C₄	—	2	1	no	—
39	C₁₆	—	254	1	no	—
40	Q₈	—	254	1	no	—
41	D₄	—	508	1	no	—
42	D₄	—	254	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	508	1	no	—
45	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
46	C₈	C₄×D₁₂₇	1	1	yes	characteristic
47	C₈	D₅₀₈	1	1	yes	characteristic
48	C₂²	—	508	1	no	—
49	C₂²	#14523233	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	—	508	1	no	—
52	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
53	C₄	#14523235	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	1016	1	no	—
56	C₂	#16861129	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16861129
commutator	C₈×C₁₂₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	#106×C₁₂₇	C₂
socle	C₂×C₁₂₇	#102
hypercenter	#106	D₁₂₇
hypocenter	C₁₂₇	#360
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

Complements

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

N	H ≅ G/N	type
2. #16861149	C₂	semidirect
3. #106×C₁₂₇	C₂	semidirect
5. Dic₅₀₈	C₄	semidirect
7. D₁₀₁₆	C₄	semidirect
28. C₁₂₇	#360	semidirect
32. #106	D₁₂₇	semidirect