Subgroups — #22550114

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	#16861185	C₂	1	2	yes	maximal
3	#16861143	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861133	C₂	1	2	yes	maximal, complemented
5	C₂×Q₁₆×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	—	2	2	no	—
9	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	Dic₁₂₇⋊C₄	—	2	2	no	—
11	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
12	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
13	Q₁₆×C₁₂₇	—	2	2	no	—
14	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₂₇	D₄	1	1	yes	characteristic
16	C₂×Dic₁₂₇	—	2	2	no	—
17	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
18	C₄×D₁₂₇	—	2	1	no	—
19	C₄×D₁₂₇	—	2	1	no	—
20	C₁₂₇⋊C₈	—	2	1	no	—
21	Q₈×C₁₂₇	—	2	2	no	—
22	Q₈×C₁₂₇	—	2	2	no	—
23	C₂×C₄×C₁₂₇	—	2	2	no	—
24	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₈×C₁₂₇	D₄	1	2	yes	—
26	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₂₇	—	2	1	no	—
28	C₂×D₁₂₇	—	2	1	no	—
29	C₂×D₁₂₇	Q₁₆	1	2	yes	complemented
30	C₄×C₁₂₇	—	4	2	no	—
31	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₂₇	#41	1	1	yes	characteristic
33	Dic₁₂₇	—	2	1	no	—
34	Dic₁₂₇	—	4	2	no	—
35	C₂×C₁₂₇	#121	1	1	yes	characteristic
36	C₂×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₂₇	#135	1	1	yes	characteristic
38	D₁₂₇	—	2	2	no	—
39	C₁₂₇	#467	1	1	yes	characteristic, complemented, Sylow
40	#467	—	127	1	no	maximal, Sylow
41	C₂×Q₁₆	D₁₂₇	1	1	yes	characteristic, complemented
42	C₂²×C₈	—	127	1	no	—
43	C₂²⋊Q₈	—	127	2	no	—
44	#107	—	127	1	no	—
45	#103	—	127	2	no	—
46	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
47	C₂²×C₄	—	127	1	no	—
48	Q₁₆	—	2	2	no	—
49	C₂×C₈	—	127	1	no	—
50	C₂×C₈	—	254	2	no	—
51	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
52	C₄⋊C₄	—	127	2	no	—
53	C₄⋊C₄	—	254	2	no	—
54	C₂²⋊C₄	—	254	2	no	—
55	C₂³	—	127	1	no	—
56	Q₈	—	2	2	no	—
57	Q₈	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	127	1	no	—
60	C₂×C₄	—	254	2	no	—
61	C₂×C₄	—	254	1	no	—
62	C₂×C₄	—	254	1	no	—
63	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
64	C₈	—	254	1	no	—
65	C₈	C₂²⋊D₁₂₇	1	2	yes	—
66	C₂²	—	127	2	no	—
67	C₂²	—	254	1	no	—
68	C₂²	—	254	1	no	—
69	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	508	2	no	—
72	C₄	—	254	1	no	—
73	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
74	C₄	#14523247	1	1	yes	characteristic
75	C₂	—	254	2	no	—
76	C₂	#16861180	1	1	yes	characteristic
77	C₂	Q₁₆×D₁₂₇	1	1	yes	characteristic
78	C₂	#16861232	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂×Q₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂×Q₁₆	D₁₂₇
hypocenter	C₁₂₇	#467
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #16861143	C₂	semidirect
4. #16861133	C₂	semidirect
5. C₂×Q₁₆×C₁₂₇	C₂	semidirect
29. C₂×D₁₂₇	Q₁₆	semidirect
39. C₁₂₇	#467	semidirect
41. C₂×Q₁₆	D₁₂₇	semidirect