Subgroups — #22550278

Order 8128.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861201	C₂	1	2	yes	maximal
2	#16861143	C₂	1	2	yes	maximal
3	#16861141	C₂	1	2	yes	maximal
4	C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
6	Dic₁₂₇⋊C₄	—	2	4	no	—
7	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₂₇	—	2	4	no	—
11	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
12	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
13	C₈×C₁₂₇	Q₈	1	2	yes	—
14	C₈×C₁₂₇	Q₈	1	2	yes	—
15	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
16	C₄×C₁₂₇	—	2	2	no	—
17	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
18	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
19	Dic₁₂₇	—	4	4	no	—
20	C₂×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₁₂₇	#135	1	2	yes	—
22	C₁₂₇	#499	1	1	yes	characteristic, complemented, Sylow
23	#499	—	127	1	no	maximal, Sylow
24	#125	—	127	2	no	—
25	#107	—	127	2	no	—
26	#106	—	127	2	no	—
27	C₄×C₈	D₁₂₇	1	1	yes	characteristic
28	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
29	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
30	C₄⋊C₄	—	127	4	no	—
31	C₄⋊C₄	—	254	4	no	—
32	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
33	C₂×C₄	—	254	4	no	—
34	C₂×C₄	D₅₀₈	1	2	yes	—
35	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
36	C₈	Dic₂₅₄	1	2	yes	—
37	C₈	Dic₂₅₄	1	2	yes	—
38	C₂²	C₂×D₅₀₈	1	1	yes	characteristic
39	C₄	—	2	2	no	—
40	C₄	—	508	4	no	—
41	C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
42	C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
43	C₂	#16861209	1	1	yes	characteristic
44	C₂	#16861231	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₄×C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#499
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
22. C₁₂₇	#499	semidirect