Subgroups — #22549686

Order 8128.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861149	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861141	C₂	1	1	yes	maximal, characteristic, complemented
3	#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₁₂₇	—	2	1	no	—
5	C₁₂₇⋊C₁₆	—	2	1	no	—
6	#41×C₁₂₇	—	2	1	no	—
7	Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic, complemented
8	SD₁₆×C₁₂₇	—	2	1	no	—
9	D₈×C₁₂₇	C₄	1	1	yes	characteristic, complemented
10	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₂₇	—	4	1	no	—
12	Q₈×C₁₂₇	—	2	1	no	—
13	D₄×C₁₂₇	—	2	1	no	—
14	D₄×C₁₂₇	—	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	—	4	1	no	—
20	C₂²×C₁₂₇	D₈	1	1	yes	characteristic
21	C₄×C₁₂₇	—	4	1	no	—
22	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
24	Dic₁₂₇	—	8	1	no	—
25	C₂×C₁₂₇	—	2	1	no	—
26	C₂×C₁₂₇	—	8	1	no	—
27	C₂×C₁₂₇	#102	1	1	yes	characteristic
28	C₁₂₇	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	127	1	no	maximal, Sylow
30	#135	D₁₂₇	1	1	yes	characteristic
31	M₃₂	—	127	1	no	—
32	#106	—	127	1	no	—
33	#41	—	2	1	no	—
34	Q₁₆	Dic₁₂₇	1	1	yes	characteristic, complemented
35	SD₁₆	—	2	1	no	—
36	D₈	Dic₁₂₇	1	1	yes	characteristic, complemented
37	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
38	C₄⋊C₄	—	254	1	no	—
39	C₁₆	—	254	1	no	—
40	Q₈	—	2	1	no	—
41	D₄	—	2	1	no	—
42	D₄	—	4	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	C₂×Dic₁₂₇	1	1	yes	characteristic
47	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	#14523233	1	1	yes	characteristic
50	C₄	—	4	1	no	—
51	C₄	—	1016	1	no	—
52	C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
53	C₄	#14523236	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	8	1	no	—
56	C₂	#16861130	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16861130
commutator	C₈×C₁₂₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	#135×C₁₂₇	C₂
socle	C₂×C₁₂₇	#102
hypercenter	#135	D₁₂₇
hypocenter	C₁₂₇	#360
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #16861149	C₂	semidirect
2. #16861141	C₂	semidirect
7. Q₁₆×C₁₂₇	C₄	semidirect
9. D₈×C₁₂₇	C₄	semidirect
28. C₁₂₇	#360	semidirect
34. Q₁₆	Dic₁₂₇	semidirect
36. D₈	Dic₁₂₇	semidirect