Subgroups — #22549629

Order 8128.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal
2	C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₂₇⋊C₈	—	2	2	no	—
4	C₂×C₈×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂×C₈×C₁₂₇	C₄	1	1	yes	characteristic
6	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₁₂₇⋊C₈	—	4	2	no	—
8	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
10	C₂×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
11	C₈×C₁₂₇	—	2	1	no	—
12	C₈×C₁₂₇	C₈	1	2	yes	complemented
13	C₂²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₁₂₇	—	2	1	no	—
15	C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
16	C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
17	C₄×C₁₂₇	D₈	1	1	yes	characteristic
18	C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
19	C₂×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
20	C₂×C₁₂₇	#107	1	1	yes	characteristic
21	C₂×C₁₂₇	#108	1	1	yes	characteristic
22	C₁₂₇	#335	1	1	yes	characteristic, complemented, Sylow
23	#335	—	127	1	no	maximal, Sylow
24	C₄⋊C₈	—	127	2	no	—
25	C₄×C₈	D₁₂₇	1	1	yes	characteristic
26	C₂×C₈	—	254	2	no	—
27	C₂×C₈	Dic₁₂₇	1	1	yes	characteristic
28	C₂×C₈	Dic₁₂₇	1	1	yes	characteristic
29	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
30	C₂×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
31	C₂×C₄	D₅₀₈	1	1	yes	characteristic
32	C₂×C₄	Dic₂₅₄	1	1	yes	characteristic
33	C₈	—	2	1	no	—
34	C₈	—	508	2	no	—
35	C₈	C₁₂₇⋊C₈	1	2	yes	complemented
36	C₂²	C₄⋊Dic₁₂₇	1	1	yes	characteristic
37	C₄	—	2	1	no	—
38	C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
39	C₄	#14523232	1	1	yes	characteristic
40	C₄	D₁₀₁₆	1	1	yes	characteristic
41	C₄	Dic₅₀₈	1	1	yes	characteristic
42	C₂	C₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
43	C₂	#16861143	1	1	yes	characteristic
44	C₂	#16861145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅₀₈
commutator	C₄×C₁₂₇	C₂×C₈
Frattini	C₄²	C₂×D₁₂₇
Fitting	C₄×C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₄⋊C₄
hypercenter	C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#335
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₁₂₇	C₈	semidirect
22. C₁₂₇	#335	semidirect
35. C₈	C₁₂₇⋊C₈	semidirect