Subgroups — #22550159

Order 8128.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861208	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861141	C₂	1	1	yes	maximal, characteristic
4	#16861140	C₂	1	1	yes	maximal, characteristic
5	Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#16861133	C₂	1	1	yes	maximal, characteristic
7	#103×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₅₄	—	2	1	no	—
9	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₂₇	—	2	1	no	—
11	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
13	Dic₁₂₇⋊C₄	—	2	1	no	—
14	C₄×Dic₁₂₇	—	2	1	no	—
15	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	Dic₂₅₄	—	4	2	no	—
20	C₂×Dic₁₂₇	—	2	1	no	—
21	C₂×Dic₁₂₇	—	2	1	no	—
22	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
23	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
24	C₁₂₇⋊C₈	—	2	1	no	—
25	Q₈×C₁₂₇	—	2	1	no	—
26	Q₈×C₁₂₇	Q₈	1	2	yes	complemented
27	C₂×C₄×C₁₂₇	—	2	1	no	—
28	C₂×C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₈×C₁₂₇	—	2	1	no	—
31	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₂₇	—	2	2	no	—
33	C₄×C₁₂₇	—	4	1	no	—
34	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₁₂₇	#41	1	1	yes	characteristic
36	Dic₁₂₇	—	2	1	no	—
37	Dic₁₂₇	—	4	1	no	—
38	Dic₁₂₇	—	4	1	no	—
39	Dic₁₂₇	SD₁₆	1	2	yes	—
40	C₂×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₁₂₇	#137	1	1	yes	characteristic
43	C₁₂₇	#475	1	1	yes	characteristic, complemented, Sylow
44	#475	—	127	1	no	maximal, Sylow
45	C₄⋊Q₈	—	127	1	no	—
46	C₄×Q₈	—	127	1	no	—
47	#106	—	127	1	no	—
48	#106	—	127	1	no	—
49	C₄⋊C₈	—	127	1	no	—
50	#103	—	127	1	no	—
51	#103	D₁₂₇	1	1	yes	characteristic
52	C₂×Q₈	—	254	1	no	—
53	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
54	C₂×C₈	—	127	1	no	—
55	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	C₄⋊C₄	—	127	1	no	—
57	C₄⋊C₄	—	127	1	no	—
58	C₄⋊C₄	—	254	1	no	—
59	C₄⋊C₄	—	254	1	no	—
60	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
61	C₄²	—	127	1	no	—
62	C₄²	—	254	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	508	2	no	—
65	Q₈	Dic₂₅₄	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	127	1	no	—
69	C₂×C₄	—	127	1	no	—
70	C₂×C₄	—	254	1	no	—
71	C₂×C₄	—	254	1	no	—
72	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	254	1	no	—
75	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	127	2	no	—
79	C₄	—	508	1	no	—
80	C₄	—	508	1	no	—
81	C₄	—	254	1	no	—
82	C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
83	C₄	#14523247	1	1	yes	characteristic
84	C₂	#16861186	1	1	yes	characteristic
85	C₂	SD₁₆×D₁₂₇	1	1	yes	characteristic
86	C₂	#16861241	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#103×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#103	D₁₂₇
hypocenter	C₁₂₇	#475
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₁₂₇	Q₈	semidirect
43. C₁₂₇	#475	semidirect
65. Q₈	Dic₂₅₄	semidirect