Subgroups — #22549989

Order 8128.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₀₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅₀₈	C₂	1	2	yes	maximal, complemented
3	#16861141	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861131	C₂	1	2	yes	maximal, complemented
5	#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₀₈	C₂²	1	2	yes	—
7	C₂×C₄×D₁₂₇	—	2	2	no	—
8	D₁₀₁₆	C₄	1	4	yes	complemented
9	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
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	C₂²×D₁₂₇	—	2	2	no	—
15	D₅₀₈	—	2	2	no	—
16	D₅₀₈	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₁₂₇	—	2	2	no	—
18	C₄×D₁₂₇	—	4	4	no	—
19	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
20	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₈×C₁₂₇	—	2	1	no	—
22	C₈×C₁₂₇	C₂×C₄	1	2	yes	complemented
23	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₁₂₇	—	2	4	no	—
25	C₂×D₁₂₇	—	4	2	no	—
26	C₂×D₁₂₇	—	4	2	no	—
27	C₄×C₁₂₇	—	2	2	no	—
28	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₁₂₇	#41	1	1	yes	characteristic
30	Dic₁₂₇	—	4	2	no	—
31	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
32	C₂×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
33	D₁₂₇	—	4	4	no	—
34	C₁₂₇	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	127	1	no	maximal, Sylow
36	C₂×D₈	—	127	1	no	—
37	C₄×D₄	—	127	2	no	—
38	#106	—	127	1	no	—
39	#102	—	127	2	no	—
40	#97	D₁₂₇	1	1	yes	characteristic, complemented
41	C₂×D₄	—	127	2	no	—
42	C₂²×C₄	—	254	2	no	—
43	D₈	—	127	4	no	—
44	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
45	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
46	C₄⋊C₄	—	127	2	no	—
47	C₂²⋊C₄	—	254	2	no	—
48	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
49	C₂³	—	254	2	no	—
50	D₄	—	127	4	no	—
51	D₄	—	254	2	no	—
52	C₂×C₄	—	508	4	no	—
53	C₂×C₄	—	254	2	no	—
54	C₂×C₄	D₅₀₈	1	2	yes	—
55	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄×D₁₂₇	1	2	yes	complemented
58	C₂²	—	508	2	no	—
59	C₂²	—	508	2	no	—
60	C₂²	—	254	4	no	—
61	C₂²	C₂×D₅₀₈	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	508	2	no	—
64	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
65	C₄	#14523248	1	1	yes	characteristic
66	C₂	—	508	4	no	—
67	C₂	C₄×D₅₀₈	1	1	yes	characteristic
68	C₂	#16861237	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₀₈
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#97×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#97	D₁₂₇
hypocenter	C₁₂₇	#442
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. C₄×D₅₀₈	C₂	semidirect
3. #16861141	C₂	semidirect
4. #16861131	C₂	semidirect
5. #97×C₁₂₇	C₂	semidirect
8. D₁₀₁₆	C₄	semidirect
16. D₅₀₈	C₂×C₄	semidirect
22. C₈×C₁₂₇	C₂×C₄	semidirect
34. C₁₂₇	#442	semidirect
40. #97	D₁₂₇	semidirect
57. C₈	C₄×D₁₂₇	semidirect