Subgroups — #22550276

Order 8128.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861200	C₂	1	2	yes	maximal
2	#16861142	C₂	1	2	yes	maximal
3	#16861141	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₁₂₇	—	2	2	no	—
8	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	Dic₁₂₇⋊C₄	—	2	2	no	—
10	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂×Dic₁₂₇	—	2	2	no	—
14	C₂×Dic₁₂₇	D₄	1	2	yes	—
15	C₁₂₇⋊C₈	Q₈	1	2	yes	—
16	C₂×C₄×C₁₂₇	—	2	2	no	—
17	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₈×C₁₂₇	Q₈	1	2	yes	—
19	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₂₇	—	4	2	no	—
21	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
22	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
23	Dic₁₂₇	—	2	2	no	—
24	Dic₁₂₇	—	4	2	no	—
25	C₂×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₁₂₇	#135	1	2	yes	—
27	C₁₂₇	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	127	1	no	maximal, Sylow
29	#125	—	127	2	no	—
30	#107	—	127	2	no	—
31	#106	—	127	1	no	—
32	#106	D₁₂₇	1	1	yes	characteristic
33	C₄×C₈	—	127	1	no	—
34	C₂×C₈	—	127	1	no	—
35	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
36	C₄⋊C₄	—	127	2	no	—
37	C₄⋊C₄	—	254	2	no	—
38	C₄⋊C₄	—	254	2	no	—
39	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
40	C₄²	—	127	1	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	127	2	no	—
43	C₂×C₄	—	254	2	no	—
44	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
45	C₈	—	127	2	no	—
46	C₈	Dic₂₅₄	1	2	yes	—
47	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
48	C₄	—	4	2	no	—
49	C₄	—	508	2	no	—
50	C₄	—	254	2	no	—
51	C₄	Q₈×D₁₂₇	1	1	yes	characteristic
52	C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
53	C₂	#16861208	1	1	yes	characteristic
54	C₂	#16861230	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#106×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#106	D₁₂₇
hypocenter	C₁₂₇	#499
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

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