Subgroups — #22550354

Order 8128.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₂₇	C₂	1	8	yes	maximal, complemented
2	#16861168	C₂	1	6	yes	maximal
3	C₂²×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₁₂₇	C₂²	1	12	yes	—
5	C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
6	C₄×Dic₁₂₇	C₂²	1	12	yes	—
7	C₂×Q₈×C₁₂₇	C₄	1	8	yes	complemented
8	C₂×Q₈×C₁₂₇	C₂²	1	4	yes	—
9	C₂²×C₄×C₁₂₇	C₂²	1	3	yes	—
10	C₂×Dic₁₂₇	C₂³	1	8	yes	—
11	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
12	Q₈×C₁₂₇	C₂×C₄	1	16	yes	—
13	C₂×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
14	C₂×C₄×C₁₂₇	C₂³	1	6	yes	—
15	C₂²×C₁₂₇	—	2	2	no	—
16	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
17	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
18	C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
19	Dic₁₂₇	—	2	8	no	—
20	C₂×C₁₂₇	—	2	2	no	—
21	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
22	C₂×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
23	C₁₂₇	#519	1	1	yes	characteristic, complemented, Sylow
24	#519	—	127	1	no	maximal, Sylow
25	C₂²×Q₈	D₁₂₇	1	1	yes	characteristic
26	C₄×Q₈	—	127	8	no	—
27	#117	—	127	6	no	—
28	C₂×Q₈	Dic₁₂₇	1	8	yes	complemented
29	C₂×Q₈	C₂×D₁₂₇	1	4	yes	—
30	C₂²×C₄	C₂×D₁₂₇	1	3	yes	—
31	C₄⋊C₄	—	127	12	no	—
32	C₂²⋊C₄	—	127	4	no	—
33	C₄²	—	127	12	no	—
34	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
35	Q₈	C₂×Dic₁₂₇	1	16	yes	—
36	C₂×C₄	—	127	8	no	—
37	C₂×C₄	C₂×Dic₁₂₇	1	12	yes	—
38	C₂×C₄	C₂²×D₁₂₇	1	6	yes	—
39	C₂²	—	2	2	no	—
40	C₂²	C₂²×Dic₁₂₇	1	2	yes	—
41	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
42	C₄	—	254	8	no	—
43	C₄	C₂²×Dic₁₂₇	1	12	yes	—
44	C₂	—	2	2	no	—
45	C₂	C₂³×Dic₁₂₇	1	1	yes	characteristic
46	C₂	#16861257	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂³×C₄
Fitting	C₂²×Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂⁴
hypercenter	C₂²×Q₈	D₁₂₇
hypocenter	C₁₂₇	#519
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₂₇	C₂	semidirect
7. C₂×Q₈×C₁₂₇	C₄	semidirect
23. C₁₂₇	#519	semidirect
28. C₂×Q₈	Dic₁₂₇	semidirect