Subgroups — #22550681

Order 8128.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861255	C₂	1	2	yes	maximal, complemented
2	#16861237	C₂	1	6	yes	maximal, complemented
3	#16861231	C₂	1	3	yes	maximal, complemented
4	C₂×D₁₀₁₆	C₂	1	3	yes	maximal, complemented
5	#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523249	—	2	2	no	—
7	#14523248	C₂²	1	6	yes	complemented
8	C₂×D₅₀₈	C₂²	1	6	yes	complemented
9	D₄×D₁₂₇	—	2	6	no	—
10	Dic₅₀₈	C₂²	1	1	yes	characteristic, complemented
11	#14523237	C₂²	1	6	yes	complemented
12	D₁₀₁₆	C₂²	1	3	yes	complemented
13	D₁₀₁₆	C₂²	1	6	yes	complemented
14	#41×C₁₂₇	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₂₇	C₂²	1	3	yes	complemented
16	C₂×C₈×C₁₂₇	C₂²	1	3	yes	complemented
17	C₂²×D₁₂₇	—	2	6	no	—
18	Dic₂₅₄	C₂³	1	2	yes	—
19	D₅₀₈	—	2	6	no	—
20	D₅₀₈	C₂³	1	6	yes	—
21	C₂²⋊D₁₂₇	—	2	6	no	—
22	C₄×D₁₂₇	—	2	6	no	—
23	Q₈×C₁₂₇	D₄	1	1	yes	characteristic
24	D₄×C₁₂₇	D₄	1	3	yes	—
25	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
26	C₈×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₈×C₁₂₇	C₂³	1	3	yes	—
28	C₂²×C₁₂₇	C₂×D₄	1	3	yes	—
29	C₂×D₁₂₇	—	2	6	no	—
30	C₂×D₁₂₇	—	8	6	no	—
31	C₄×C₁₂₇	C₂×D₄	1	3	yes	—
32	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
33	Dic₁₂₇	—	2	2	no	—
34	C₂×C₁₂₇	—	2	3	no	—
35	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
36	D₁₂₇	—	4	6	no	—
37	C₁₂₇	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	127	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	127	2	no	—
40	AGL(1,ℤ/8)	—	127	6	no	—
41	#135	—	127	3	no	—
42	C₂×D₈	—	127	3	no	—
43	#131	D₁₂₇	1	1	yes	characteristic, complemented
44	#41	—	127	6	no	—
45	#41	—	254	2	no	—
46	#41	C₂×D₁₂₇	1	1	yes	characteristic
47	C₂×D₄	—	127	6	no	—
48	C₂×D₄	—	254	6	no	—
49	Q₁₆	—	127	1	no	—
50	SD₁₆	—	127	6	no	—
51	D₈	—	127	3	no	—
52	D₈	—	127	6	no	—
53	M₁₆	C₂×D₁₂₇	1	3	yes	complemented
54	C₂×C₈	C₂×D₁₂₇	1	3	yes	complemented
55	C₂³	—	254	6	no	—
56	Q₈	—	127	2	no	—
57	Q₈	D₅₀₈	1	1	yes	characteristic
58	D₄	—	127	6	no	—
59	D₄	—	254	6	no	—
60	D₄	—	254	6	no	—
61	D₄	D₅₀₈	1	3	yes	—
62	C₂×C₄	—	254	6	no	—
63	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
64	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
65	C₈	C₂²×D₁₂₇	1	3	yes	—
66	C₂²	—	1016	6	no	—
67	C₂²	—	254	6	no	—
68	C₂²	C₂×D₅₀₈	1	3	yes	—
69	C₄	—	254	2	no	—
70	C₄	C₂×D₅₀₈	1	3	yes	—
71	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	508	6	no	—
74	C₂	C₂²×D₅₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₅₀₈
commutator	C₄×C₁₂₇	C₂⁴
Frattini	C₄	C₂³×D₁₂₇
Fitting	#131×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²×D₄
hypercenter	#131	D₁₂₇
hypocenter	C₁₂₇	#576
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #16861255	C₂	semidirect
2. #16861237	C₂	semidirect
3. #16861231	C₂	semidirect
4. C₂×D₁₀₁₆	C₂	semidirect
5. #131×C₁₂₇	C₂	semidirect
7. #14523248	C₂²	semidirect
8. C₂×D₅₀₈	C₂²	semidirect
10. Dic₅₀₈	C₂²	semidirect
11. #14523237	C₂²	semidirect
12. D₁₀₁₆	C₂²	semidirect
13. D₁₀₁₆	C₂²	semidirect
15. M₁₆×C₁₂₇	C₂²	semidirect
16. C₂×C₈×C₁₂₇	C₂²	semidirect
37. C₁₂₇	#576	semidirect
43. #131	D₁₂₇	semidirect
53. M₁₆	C₂×D₁₂₇	semidirect
54. C₂×C₈	C₂×D₁₂₇	semidirect