Subgroups — #22549828

Order 8128.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523231	C₂	1	2	yes	maximal
2	C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861120	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
5	#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₂₇	C₄	1	2	yes	—
7	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
8	#14523231	C₄	1	8	yes	complemented
9	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₁₂₇	C₄	1	2	yes	—
11	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
17	C₄×D₁₂₇	—	2	4	no	—
18	C₄×D₁₂₇	C₂×C₄	1	4	yes	—
19	C₁₂₇⋊C₈	C₂×C₄	1	4	yes	complemented
20	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
23	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×D₁₂₇	—	2	2	no	—
25	C₂×D₁₂₇	C₄²	1	2	yes	—
26	C₄×C₁₂₇	—	2	2	no	—
27	C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
28	Dic₁₂₇	C₄²	1	2	yes	—
29	Dic₁₂₇	M₁₆	1	4	yes	complemented
30	C₂×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₁₂₇	C₂×M₁₆	1	2	yes	—
32	D₁₂₇	—	2	2	no	—
33	C₁₂₇	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×M₁₆	—	127	1	no	maximal, Sylow
35	C₂×M₁₆	—	127	2	no	—
36	C₂×C₄²	—	127	1	no	—
37	#97	—	127	1	no	—
38	#97	D₁₂₇	1	1	yes	characteristic, complemented
39	C₄×C₈	—	127	2	no	—
40	C₂²×C₄	—	127	1	no	—
41	C₂²×C₄	—	127	2	no	—
42	M₁₆	—	127	8	no	—
43	C₂×C₈	—	127	2	no	—
44	C₂×C₈	C₂×D₁₂₇	1	2	yes	—
45	C₄²	—	127	2	no	—
46	C₄²	—	127	1	no	—
47	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
48	C₂³	—	127	1	no	—
49	C₂×C₄	—	127	2	no	—
50	C₂×C₄	—	127	1	no	—
51	C₂×C₄	—	127	4	no	—
52	C₂×C₄	—	254	4	no	—
53	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
54	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
55	C₈	—	127	4	no	—
56	C₈	C₄×D₁₂₇	1	4	yes	complemented
57	C₂²	—	127	2	no	—
58	C₂²	—	254	2	no	—
59	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	127	4	no	—
62	C₄	—	127	2	no	—
63	C₄	C₂×C₄×D₁₂₇	1	2	yes	—
64	C₂	—	254	2	no	—
65	C₂	C₄²×D₁₂₇	1	1	yes	characteristic
66	C₂	M₁₆×D₁₂₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄²
Fitting	#97×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×C₄
hypercenter	#97	D₁₂₇
hypocenter	C₁₂₇	C₄×M₁₆
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
3. #16861120	C₂	semidirect
4. C₈×Dic₁₂₇	C₂	semidirect
5. #97×C₁₂₇	C₂	semidirect
8. #14523231	C₄	semidirect
19. C₁₂₇⋊C₈	C₂×C₄	semidirect
22. C₈×C₁₂₇	C₂×C₄	semidirect
29. Dic₁₂₇	M₁₆	semidirect
33. C₁₂₇	C₄×M₁₆	semidirect
38. #97	D₁₂₇	semidirect
56. C₈	C₄×D₁₂₇	semidirect