Subgroups — #22550273

Order 8128.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861208	C₂	1	2	yes	maximal
2	#16861141	C₂	1	1	yes	maximal, characteristic
3	#16861140	C₂	1	2	yes	maximal
4	C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₅₄	—	2	2	no	—
7	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
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	Dic₂₅₄	—	4	4	no	—
14	C₂×Dic₁₂₇	—	2	2	no	—
15	C₂×Dic₁₂₇	D₄	1	2	yes	—
16	C₁₂₇⋊C₈	Q₈	1	2	yes	complemented
17	C₂×C₄×C₁₂₇	—	2	2	no	—
18	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₈×C₁₂₇	Q₈	1	2	yes	complemented
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	—	4	2	no	—
25	Dic₁₂₇	SD₁₆	1	4	yes	—
26	C₂×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₁₂₇	C₂×SD₁₆	1	2	yes	—
28	C₁₂₇	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	127	1	no	maximal, Sylow
30	C₄⋊Q₈	—	127	2	no	—
31	#106	—	127	1	no	—
32	#106	—	127	2	no	—
33	#106	D₁₂₇	1	1	yes	characteristic
34	C₄×C₈	—	127	1	no	—
35	C₂×Q₈	—	254	2	no	—
36	C₂×C₈	—	127	1	no	—
37	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
38	C₄⋊C₄	—	127	2	no	—
39	C₄⋊C₄	—	254	2	no	—
40	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
41	C₄²	—	127	1	no	—
42	Q₈	—	508	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	127	2	no	—
45	C₂×C₄	—	254	2	no	—
46	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
47	C₈	—	127	2	no	—
48	C₈	Dic₂₅₄	1	2	yes	complemented
49	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	127	4	no	—
52	C₄	—	508	2	no	—
53	C₄	Q₈×D₁₂₇	1	1	yes	characteristic
54	C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
55	C₂	#16861208	1	1	yes	characteristic
56	C₂	SD₁₆×D₁₂₇	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	#498
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
16. C₁₂₇⋊C₈	Q₈	semidirect
19. C₈×C₁₂₇	Q₈	semidirect
28. C₁₂₇	#498	semidirect
48. C₈	Dic₂₅₄	semidirect