Subgroups — #22549913

Order 8128.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#16861167	C₂	1	2	yes	maximal
3	#16861143	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861140	C₂	1	2	yes	maximal, complemented
5	#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₈×D₁₂₇	C₄	1	4	yes	complemented
9	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
11	C₄×Dic₁₂₇	—	2	2	no	—
12	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×D₁₂₇	D₄	1	1	yes	characteristic
15	C₂×Dic₁₂₇	—	2	2	no	—
16	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
17	C₄×D₁₂₇	C₂×C₄	1	2	yes	—
18	C₄×D₁₂₇	Q₈	1	2	yes	—
19	C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₁₂₇	—	2	2	no	—
21	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₁₂₇	—	2	1	no	—
25	C₂×D₁₂₇	—	2	1	no	—
26	C₂×D₁₂₇	C₄⋊C₄	1	2	yes	—
27	C₄×C₁₂₇	—	4	2	no	—
28	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
30	Dic₁₂₇	—	4	2	no	—
31	Dic₁₂₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₁₂₇	#135	1	1	yes	characteristic
34	C₂×C₁₂₇	#135	1	1	yes	characteristic
35	D₁₂₇	—	2	2	no	—
36	C₁₂₇	#427	1	1	yes	characteristic, complemented, Sylow
37	#427	—	127	1	no	maximal, Sylow
38	C₂²×C₈	—	127	1	no	—
39	#117	—	127	2	no	—
40	#107	—	127	1	no	—
41	#107	D₁₂₇	1	1	yes	characteristic, complemented
42	#106	—	127	2	no	—
43	C₂²×C₄	—	127	1	no	—
44	C₂×C₈	—	127	1	no	—
45	C₂×C₈	—	127	4	no	—
46	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
47	C₄⋊C₄	—	127	2	no	—
48	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
49	C₂²⋊C₄	—	254	2	no	—
50	C₄²	—	254	2	no	—
51	C₂³	—	127	1	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	127	1	no	—
54	C₂×C₄	—	127	2	no	—
55	C₂×C₄	—	127	2	no	—
56	C₂×C₄	—	254	2	no	—
57	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
58	C₈	—	127	2	no	—
59	C₈	C₄×D₁₂₇	1	2	yes	—
60	C₂²	—	127	2	no	—
61	C₂²	—	254	1	no	—
62	C₂²	—	254	1	no	—
63	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	127	2	no	—
66	C₄	—	508	2	no	—
67	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
68	C₄	Q₈×D₁₂₇	1	1	yes	characteristic
69	C₂	—	254	2	no	—
70	C₂	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
71	C₂	#16861228	1	1	yes	characteristic
72	C₂	#16861232	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#107×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#107	D₁₂₇
hypocenter	C₁₂₇	#427
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
3. #16861143	C₂	semidirect
4. #16861140	C₂	semidirect
5. #107×C₁₂₇	C₂	semidirect
8. C₈×D₁₂₇	C₄	semidirect
36. C₁₂₇	#427	semidirect
41. #107	D₁₂₇	semidirect