Subgroups — #22550006

Order 8128.

94 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	Dic₁₂₇⋊C₈	C₂	1	2	yes	maximal
4	#16861119	C₂	1	2	yes	maximal
5	C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₅₄	C₄	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
9	Dic₁₂₇⋊C₄	C₄	1	2	yes	—
10	C₄×Dic₁₂₇	C₂²	1	2	yes	—
11	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	Dic₂₅₄	—	2	2	no	—
15	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
16	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
17	C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₈×C₁₂₇	—	2	1	no	—
22	C₈×C₁₂₇	Q₈	1	2	yes	complemented
23	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₁₂₇	—	2	1	no	—
25	C₄×C₁₂₇	M₁₆	1	2	yes	—
26	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₁₂₇	#41	1	1	yes	characteristic
28	Dic₁₂₇	—	2	2	no	—
29	Dic₁₂₇	—	2	2	no	—
30	C₂×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₁₂₇	#131	1	1	yes	characteristic
33	C₁₂₇	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	127	1	no	maximal, Sylow
35	C₄×Q₈	—	127	1	no	—
36	C₄⋊C₈	—	127	2	no	—
37	C₄⋊C₈	—	127	1	no	—
38	#97	—	127	2	no	—
39	C₄×C₈	D₁₂₇	1	1	yes	characteristic
40	C₂×Q₈	—	127	1	no	—
41	C₂×C₈	—	127	2	no	—
42	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
43	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
44	C₄⋊C₄	—	127	2	no	—
45	C₄⋊C₄	—	127	1	no	—
46	C₄²	—	127	2	no	—
47	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
48	Q₈	—	254	2	no	—
49	C₂×C₄	—	127	2	no	—
50	C₂×C₄	—	127	2	no	—
51	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
52	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
54	C₈	—	2	1	no	—
55	C₈	—	254	2	no	—
56	C₈	Dic₂₅₄	1	2	yes	complemented
57	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	254	2	no	—
60	C₄	—	254	2	no	—
61	C₄	#14523231	1	2	yes	—
62	C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
63	C₄	#14523248	1	1	yes	characteristic
64	C₂	C₄×Dic₂₅₄	1	1	yes	characteristic
65	C₂	C₂×#14523231	1	1	yes	characteristic
66	C₂	#16861215	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₄
Fitting	C₄×C₈×C₁₂₇	C₂
hypercenter	C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#446
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
22. C₈×C₁₂₇	Q₈	semidirect
33. C₁₂₇	#446	semidirect
56. C₈	Dic₂₅₄	semidirect