Subgroups — #22549746

Order 8128.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₁₂₇	C₂	1	6	yes	maximal
2	C₂×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₁₂₇	C₂²	1	4	yes	—
4	C₄⋊Dic₁₂₇	—	2	12	no	—
5	C₂²×C₄×C₁₂₇	C₂²	1	3	yes	—
6	C₄²×C₁₂₇	C₄	1	4	yes	complemented
7	C₂×Dic₁₂₇	—	2	12	no	—
8	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
9	C₂×C₄×C₁₂₇	C₂×C₄	1	6	yes	—
10	C₂×C₄×C₁₂₇	D₄	1	6	yes	—
11	C₂×C₄×C₁₂₇	Q₈	1	6	yes	—
12	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
13	C₂²×C₁₂₇	C₂×D₄	1	3	yes	—
14	C₂²×C₁₂₇	C₂×Q₈	1	3	yes	—
15	C₄×C₁₂₇	C₄⋊C₄	1	12	yes	complemented
16	Dic₁₂₇	—	4	4	no	—
17	C₂×C₁₂₇	C₂×C₄⋊C₄	1	3	yes	—
18	C₂×C₁₂₇	#127	1	1	yes	characteristic
19	C₂×C₁₂₇	C₄⋊Q₈	1	3	yes	—
20	C₁₂₇	#384	1	1	yes	characteristic, complemented, Sylow
21	#384	—	127	1	no	maximal, Sylow
22	C₂×C₄⋊C₄	—	127	6	no	—
23	C₂×C₄²	D₁₂₇	1	1	yes	characteristic
24	C₂²×C₄	—	127	4	no	—
25	C₂²×C₄	C₂×D₁₂₇	1	3	yes	—
26	C₄⋊C₄	—	254	12	no	—
27	C₄²	Dic₁₂₇	1	4	yes	complemented
28	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
29	C₂×C₄	—	254	12	no	—
30	C₂×C₄	C₂×Dic₁₂₇	1	6	yes	—
31	C₂×C₄	D₅₀₈	1	6	yes	—
32	C₂×C₄	Dic₂₅₄	1	6	yes	—
33	C₂²	C₂²×Dic₁₂₇	1	1	yes	characteristic
34	C₂²	C₂×D₅₀₈	1	3	yes	—
35	C₂²	C₂×Dic₂₅₄	1	3	yes	—
36	C₄	—	508	4	no	—
37	C₄	C₄⋊Dic₁₂₇	1	12	yes	complemented
38	C₂	C₂×C₄⋊Dic₁₂₇	1	3	yes	—
39	C₂	#16861206	1	1	yes	characteristic
40	C₂	#16861209	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₄
Fitting	C₂×C₄²×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂×C₄²	D₁₂₇
hypocenter	C₁₂₇	#384
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. C₄²×C₁₂₇	C₄	semidirect
15. C₄×C₁₂₇	C₄⋊C₄	semidirect
20. C₁₂₇	#384	semidirect
27. C₄²	Dic₁₂₇	semidirect
37. C₄	C₄⋊Dic₁₂₇	semidirect