Subgroups — #22550277

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	#16861143	C₂	1	1	yes	maximal, characteristic
3	#16861140	C₂	1	2	yes	maximal
4	C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#107×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	1	yes	characteristic
15	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
16	C₁₂₇⋊C₈	Q₈	1	2	yes	—
17	C₂×C₄×C₁₂₇	—	2	2	no	—
18	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₈×C₁₂₇	Q₈	1	2	yes	—
20	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
21	C₄×C₁₂₇	—	4	2	no	—
22	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₂₇	—	2	1	no	—
25	Dic₁₂₇	—	2	1	no	—
26	Dic₁₂₇	—	4	2	no	—
27	C₂×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₁₂₇	#135	1	1	yes	characteristic
29	C₂×C₁₂₇	#135	1	1	yes	characteristic
30	C₁₂₇	#499	1	1	yes	characteristic, complemented, Sylow
31	#499	—	127	1	no	maximal, Sylow
32	#125	—	127	2	no	—
33	#107	—	127	1	no	—
34	#107	D₁₂₇	1	1	yes	characteristic
35	#106	—	127	2	no	—
36	C₄×C₈	—	127	1	no	—
37	C₂×C₈	—	127	1	no	—
38	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
39	C₄⋊C₄	—	127	2	no	—
40	C₄⋊C₄	—	254	2	no	—
41	C₄⋊C₄	—	254	2	no	—
42	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
43	C₄²	—	127	1	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	127	1	no	—
46	C₂×C₄	—	127	1	no	—
47	C₂×C₄	—	254	2	no	—
48	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
49	C₈	—	127	2	no	—
50	C₈	Dic₂₅₄	1	2	yes	—
51	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
52	C₄	—	4	2	no	—
53	C₄	—	508	2	no	—
54	C₄	—	254	1	no	—
55	C₄	—	254	1	no	—
56	C₄	Q₈×D₁₂₇	1	1	yes	characteristic
57	C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
58	C₂	#16861208	1	1	yes	characteristic
59	C₂	#16861228	1	1	yes	characteristic
60	C₂	#16861232	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#107×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#107	D₁₂₇
hypocenter	C₁₂₇	#499
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

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