Subgroups — #22549680

Order 8128.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#16861143	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₁₂₇	—	2	1	no	—
5	C₁₂₇⋊C₁₆	—	2	1	no	—
6	C₂×Q₈×C₁₂₇	—	2	1	no	—
7	Q₁₆×C₁₂₇	—	2	1	no	—
8	Q₁₆×C₁₂₇	C₄	1	2	yes	complemented
9	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₂₇	—	4	1	no	—
11	Q₈×C₁₂₇	—	2	2	no	—
12	Q₈×C₁₂₇	—	4	1	no	—
13	C₂×C₄×C₁₂₇	—	4	1	no	—
14	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
15	C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂²×C₁₂₇	D₈	1	1	yes	characteristic
18	C₄×C₁₂₇	—	4	2	no	—
19	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
21	Dic₁₂₇	—	8	1	no	—
22	C₂×C₁₂₇	#102	1	1	yes	characteristic
23	C₂×C₁₂₇	SD₃₂	1	1	yes	characteristic
24	C₂×C₁₂₇	Q₃₂	1	1	yes	characteristic
25	C₁₂₇	#358	1	1	yes	characteristic, complemented, Sylow
26	#358	—	127	1	no	maximal, Sylow
27	C₂×Q₁₆	D₁₂₇	1	1	yes	characteristic
28	C₂×C₁₆	—	127	1	no	—
29	#107	—	127	1	no	—
30	C₂×Q₈	—	2	1	no	—
31	Q₁₆	—	2	1	no	—
32	Q₁₆	Dic₁₂₇	1	2	yes	complemented
33	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
34	C₄⋊C₄	—	254	1	no	—
35	C₁₆	—	254	1	no	—
36	Q₈	—	2	2	no	—
37	Q₈	—	4	1	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	508	1	no	—
40	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
41	C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
42	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
43	C₂²	#14523233	1	1	yes	characteristic
44	C₄	—	4	2	no	—
45	C₄	—	1016	1	no	—
46	C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
47	C₄	#14523236	1	1	yes	characteristic
48	C₂	#16861130	1	1	yes	characteristic
49	C₂	#16861152	1	1	yes	characteristic
50	C₂	#16861155	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14523233
commutator	C₈×C₁₂₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	C₂×Q₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	D₈
hypercenter	C₂×Q₁₆	D₁₂₇
hypocenter	C₁₂₇	#358
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
8. Q₁₆×C₁₂₇	C₄	semidirect
25. C₁₂₇	#358	semidirect
32. Q₁₆	Dic₁₂₇	semidirect