Subgroups — #22550613

Order 8128.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861203	C₂	1	2	yes	maximal, complemented
2	#16861190	C₂	1	3	yes	maximal, complemented
3	#16861181	C₂	1	6	yes	maximal, complemented
4	#16861177	C₂	1	3	yes	maximal, complemented
5	#127×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₁₂₇	C₂²	1	6	yes	—
8	C₂²×Dic₁₂₇	C₂²	1	3	yes	complemented
9	Dic₁₂₇⋊C₄	C₂²	1	6	yes	complemented
10	C₂²⋊Dic₁₂₇	C₂²	1	3	yes	—
11	C₂²⋊Dic₁₂₇	C₂²	1	3	yes	complemented
12	(C₂×D₁₂₇)⋊C₄	C₂²	1	6	yes	—
13	C₂×D₄×C₁₂₇	C₂²	1	3	yes	—
14	C₂×D₄×C₁₂₇	C₂²	1	3	yes	complemented
15	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²×D₁₂₇	—	4	3	no	—
17	C₂²×D₁₂₇	C₂³	1	2	yes	—
18	C₂²⋊D₁₂₇	—	4	6	no	—
19	C₂×Dic₁₂₇	—	4	3	no	—
20	C₂×Dic₁₂₇	C₂³	1	6	yes	—
21	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂³×C₁₂₇	C₂³	1	3	yes	—
23	D₄×C₁₂₇	—	4	3	no	—
24	D₄×C₁₂₇	—	4	3	no	—
25	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
26	C₂²×C₁₂₇	—	2	3	no	—
27	C₂²×C₁₂₇	—	2	6	no	—
28	C₂²×C₁₂₇	—	2	3	no	—
29	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
30	C₂×D₁₂₇	—	2	6	no	—
31	C₂×D₁₂₇	—	4	4	no	—
32	C₄×C₁₂₇	—	2	3	no	—
33	Dic₁₂₇	—	2	6	no	—
34	C₂×C₁₂₇	—	4	1	no	—
35	C₂×C₁₂₇	—	4	3	no	—
36	C₂×C₁₂₇	H(5,𝔽₂)	1	3	yes	—
37	D₁₂₇	—	4	2	no	—
38	C₁₂₇	#560	1	1	yes	characteristic, complemented, Sylow
39	#560	—	127	1	no	maximal, Sylow
40	#127	D₁₂₇	1	1	yes	characteristic, complemented
41	#126	—	127	2	no	—
42	#123	—	127	3	no	—
43	#121	—	127	6	no	—
44	#120	—	127	3	no	—
45	C₂⁴	—	127	1	no	—
46	C₂×D₄	—	127	6	no	—
47	C₂×D₄	C₂×D₁₂₇	1	3	yes	—
48	C₂×D₄	C₂×D₁₂₇	1	3	yes	complemented
49	C₂²×C₄	—	127	3	no	—
50	C₄⋊C₄	—	127	6	no	—
51	C₂²⋊C₄	—	127	3	no	—
52	C₂²⋊C₄	—	127	3	no	—
53	C₂²⋊C₄	—	127	6	no	—
54	C₄²	C₂×D₁₂₇	1	1	yes	characteristic, complemented
55	C₂³	—	127	2	no	—
56	C₂³	—	508	3	no	—
57	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
58	C₂³	C₂²×D₁₂₇	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	508	6	no	—
62	C₂×C₄	—	127	6	no	—
63	C₂×C₄	—	508	3	no	—
64	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	508	4	no	—
69	C₂²	—	254	6	no	—
70	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	254	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	508	2	no	—
76	C₂	#16861254	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	#127×C₁₂₇	C₂
hypercenter	#127	D₁₂₇
hypocenter	C₁₂₇	#560
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #16861203	C₂	semidirect
2. #16861190	C₂	semidirect
3. #16861181	C₂	semidirect
4. #16861177	C₂	semidirect
5. #127×C₁₂₇	C₂	semidirect
8. C₂²×Dic₁₂₇	C₂²	semidirect
9. Dic₁₂₇⋊C₄	C₂²	semidirect
11. C₂²⋊Dic₁₂₇	C₂²	semidirect
14. C₂×D₄×C₁₂₇	C₂²	semidirect
15. C₄²×C₁₂₇	C₂²	semidirect
38. C₁₂₇	#560	semidirect
40. #127	D₁₂₇	semidirect
48. C₂×D₄	C₂×D₁₂₇	semidirect
54. C₄²	C₂×D₁₂₇	semidirect