Subgroups — #22549608

Order 8128.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₅₄	C₄	1	1	yes	characteristic
5	C₂×C₁₂₇⋊C₈	—	2	1	no	—
6	C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
7	Dic₁₂₇⋊C₄	—	2	1	no	—
8	C₄×Dic₁₂₇	—	2	1	no	—
9	C₂×C₈×C₁₂₇	—	2	1	no	—
10	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	Dic₂₅₄	—	2	1	no	—
12	Dic₂₅₄	C₈	1	2	yes	complemented
13	C₂×Dic₁₂₇	—	2	1	no	—
14	C₂×Dic₁₂₇	—	2	1	no	—
15	C₁₂₇⋊C₈	—	4	1	no	—
16	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
19	C₈×C₁₂₇	—	2	2	no	—
20	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₂₇	—	2	1	no	—
22	C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
23	C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
24	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
25	C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
26	Dic₁₂₇	—	2	2	no	—
27	Dic₁₂₇	—	4	1	no	—
28	C₂×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₁₂₇	#103	1	1	yes	characteristic
30	C₂×C₁₂₇	#104	1	1	yes	characteristic
31	C₁₂₇	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	127	1	no	maximal, Sylow
33	C₄×Q₈	—	127	1	no	—
34	C₄⋊C₈	—	127	1	no	—
35	C₄×C₈	D₁₂₇	1	1	yes	characteristic
36	C₂×Q₈	—	127	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	254	1	no	—
39	C₄⋊C₄	—	127	1	no	—
40	C₄⋊C₄	—	254	1	no	—
41	C₄²	—	254	1	no	—
42	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
43	Q₈	—	127	2	no	—
44	Q₈	—	254	1	no	—
45	C₂×C₄	—	254	1	no	—
46	C₂×C₄	—	254	1	no	—
47	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
49	C₂×C₄	D₅₀₈	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	508	1	no	—
52	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	508	1	no	—
55	C₄	—	254	2	no	—
56	C₄	C₈×D₁₂₇	1	1	yes	characteristic
57	C₄	#14523231	1	1	yes	characteristic
58	C₄	#14523237	1	1	yes	characteristic
59	C₄	Dic₅₀₈	1	1	yes	characteristic
60	C₂	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
61	C₂	#16861134	1	1	yes	characteristic
62	C₂	#16861137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅₀₈
commutator	C₄×C₁₂₇	C₂×C₈
Frattini	C₄²	C₂×D₁₂₇
Fitting	C₄×C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	Q₈⋊C₈
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. Dic₂₅₄	C₈	semidirect
31. C₁₂₇	Q₈⋊C₈	semidirect