Subgroups — #22549617

Order 8128.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861201	C₂	1	1	yes	maximal, characteristic
2	#16861120	C₂	1	1	yes	maximal, characteristic
3	#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₈	—	2	1	no	—
5	C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
6	C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
7	Dic₁₂₇⋊C₄	—	2	1	no	—
8	Dic₁₂₇⋊C₄	—	2	1	no	—
9	C₂×C₈×C₁₂₇	—	2	1	no	—
10	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₂₇	—	2	1	no	—
12	C₂×Dic₁₂₇	—	2	1	no	—
13	C₁₂₇⋊C₈	—	2	2	no	—
14	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
17	C₈×C₁₂₇	—	2	2	no	—
18	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₂₇	—	2	1	no	—
20	C₄×C₁₂₇	—	2	1	no	—
21	C₄×C₁₂₇	—	2	1	no	—
22	Dic₁₂₇	—	4	1	no	—
23	Dic₁₂₇	—	4	1	no	—
24	C₂×C₁₂₇	#101	1	1	yes	characteristic
25	C₂×C₁₂₇	#104	1	1	yes	characteristic
26	C₂×C₁₂₇	#104	1	1	yes	characteristic
27	C₁₂₇	#330	1	1	yes	characteristic, complemented, Sylow
28	#330	—	127	1	no	maximal, Sylow
29	#125	—	127	1	no	—
30	#97	—	127	1	no	—
31	#97	D₁₂₇	1	1	yes	characteristic
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	254	1	no	—
34	C₄⋊C₄	—	127	1	no	—
35	C₄⋊C₄	—	127	1	no	—
36	C₄⋊C₄	—	254	1	no	—
37	C₄⋊C₄	—	254	1	no	—
38	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
39	C₂×C₄	—	254	1	no	—
40	C₂×C₄	—	254	1	no	—
41	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
42	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
43	C₂×C₄	D₅₀₈	1	1	yes	characteristic
44	C₈	—	2	2	no	—
45	C₈	—	254	2	no	—
46	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	2	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	508	1	no	—
51	C₄	—	508	1	no	—
52	C₂	#16861127	1	1	yes	characteristic
53	C₂	#16861135	1	1	yes	characteristic
54	C₂	#16861137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₂₇)⋊C₄
commutator	C₂×C₄×C₁₂₇	C₂×C₄
Frattini	C₄²	C₂×D₁₂₇
Fitting	#97×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	#97	D₁₂₇
hypocenter	C₁₂₇	#330
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
27. C₁₂₇	#330	semidirect