Subgroups — #22549994

Order 8128.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861231	C₂	1	1	yes	maximal, characteristic
2	#16861215	C₂	1	2	yes	maximal, complemented
3	#16861145	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861137	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523248	C₂²	1	2	yes	—
7	Dic₅₀₈	C₄	1	1	yes	characteristic, complemented
8	#14523237	C₄	1	2	yes	complemented
9	D₁₀₁₆	C₄	1	1	yes	characteristic, complemented
10	#14523232	C₂²	1	2	yes	—
11	#14523231	—	2	2	no	—
12	C₈×D₁₂₇	—	2	2	no	—
13	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	Dic₂₅₄	C₂×C₄	1	2	yes	—
17	D₅₀₈	C₂×C₄	1	2	yes	—
18	C₂²⋊D₁₂₇	—	2	2	no	—
19	C₄×D₁₂₇	—	2	2	no	—
20	C₁₂₇⋊C₈	—	2	2	no	—
21	C₂×C₄×C₁₂₇	—	2	1	no	—
22	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₂₇	D₄	1	1	yes	characteristic
26	C₈×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂²×C₁₂₇	#41	1	1	yes	characteristic
28	C₂×D₁₂₇	—	2	2	no	—
29	C₄×C₁₂₇	—	2	2	no	—
30	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	Dic₁₂₇	—	2	2	no	—
33	C₂×C₁₂₇	—	2	1	no	—
34	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
35	D₁₂₇	—	4	2	no	—
36	C₁₂₇	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	127	1	no	maximal, Sylow
38	#135	—	127	1	no	—
39	#131	—	127	2	no	—
40	#108	—	127	1	no	—
41	#104	—	127	2	no	—
42	C₄×C₈	D₁₂₇	1	1	yes	characteristic, complemented
43	#41	—	127	2	no	—
44	Q₁₆	—	127	1	no	—
45	SD₁₆	—	127	2	no	—
46	D₈	—	127	1	no	—
47	M₁₆	—	127	2	no	—
48	M₁₆	—	254	2	no	—
49	C₂×C₈	—	254	2	no	—
50	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
52	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
53	Q₈	—	127	2	no	—
54	D₄	—	127	2	no	—
55	D₄	—	254	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	254	2	no	—
58	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
59	C₈	—	254	2	no	—
60	C₈	C₄×D₁₂₇	1	1	yes	characteristic
61	C₈	C₄×D₁₂₇	1	1	yes	characteristic
62	C₈	D₅₀₈	1	1	yes	characteristic
63	C₈	D₅₀₈	1	1	yes	characteristic
64	C₂²	—	254	2	no	—
65	C₂²	#14523248	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	254	2	no	—
68	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
69	C₄	C₂×D₅₀₈	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	508	2	no	—
72	C₂	C₄×D₅₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₅₀₈
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₄×C₈×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₄×D₄
hypercenter	C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#443
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #16861215	C₂	semidirect
3. #16861145	C₂	semidirect
4. #16861137	C₂	semidirect
5. C₄×C₈×C₁₂₇	C₂	semidirect
7. Dic₅₀₈	C₄	semidirect
8. #14523237	C₄	semidirect
9. D₁₀₁₆	C₄	semidirect
36. C₁₂₇	#443	semidirect
42. C₄×C₈	D₁₂₇	semidirect