Subgroups — #22549626

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	1	yes	maximal, characteristic
2	C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₂₇⋊C₈	—	2	1	no	—
5	C₂×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₂₇	—	2	1	no	—
8	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₁₂₇⋊C₈	—	2	1	no	—
10	C₁₂₇⋊C₈	—	4	1	no	—
11	C₁₂₇⋊C₈	C₈	1	2	yes	complemented
12	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
14	C₂×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
15	C₈×C₁₂₇	—	4	1	no	—
16	C₂²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₁₂₇	—	2	1	no	—
18	C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
19	C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
20	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
21	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
22	C₂×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₁₂₇	#106	1	1	yes	characteristic
24	C₂×C₁₂₇	#108	1	1	yes	characteristic
25	C₁₂₇	#334	1	1	yes	characteristic, complemented, Sylow
26	#334	—	127	1	no	maximal, Sylow
27	C₄⋊C₈	—	127	1	no	—
28	C₄⋊C₈	D₁₂₇	1	1	yes	characteristic
29	C₄×C₈	—	127	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	127	1	no	—
32	C₂×C₈	—	127	1	no	—
33	C₂×C₈	—	254	1	no	—
34	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
35	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
37	C₂×C₄	Dic₂₅₄	1	1	yes	characteristic
38	C₈	—	4	1	no	—
39	C₈	—	127	2	no	—
40	C₈	—	508	1	no	—
41	C₈	—	254	1	no	—
42	C₂²	Dic₁₂₇⋊C₄	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₁₂₇	1	1	yes	characteristic
45	C₄	#14523231	1	1	yes	characteristic
46	C₄	#14523235	1	1	yes	characteristic
47	C₄	#14523236	1	1	yes	characteristic
48	C₂	Dic₁₂₇⋊C₈	1	1	yes	characteristic
49	C₂	#16861140	1	1	yes	characteristic
50	C₂	#16861144	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₂₇
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	#334
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
11. C₁₂₇⋊C₈	C₈	semidirect
25. C₁₂₇	#334	semidirect