Subgroups — #22550182

Order 8128.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861200	C₂	1	1	yes	maximal, characteristic
2	#16861174	C₂	1	1	yes	maximal, characteristic
3	#16861140	C₂	1	1	yes	maximal, characteristic
4	Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#16861134	C₂	1	1	yes	maximal, characteristic
6	#16861132	C₂	1	1	yes	maximal, characteristic
7	#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
9	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₂₇	—	2	1	no	—
11	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
12	Dic₁₂₇⋊C₄	—	2	1	no	—
13	Dic₁₂₇⋊C₄	—	2	1	no	—
14	C₄×Dic₁₂₇	—	2	1	no	—
15	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	Dic₂₅₄	—	2	1	no	—
20	Dic₂₅₄	Q₈	1	2	yes	—
21	C₂×Dic₁₂₇	—	2	1	no	—
22	C₂×Dic₁₂₇	—	2	1	no	—
23	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
24	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
25	C₁₂₇⋊C₈	—	2	1	no	—
26	C₂×C₄×C₁₂₇	—	2	1	no	—
27	C₂×C₄×C₁₂₇	—	2	1	no	—
28	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₈×C₁₂₇	—	2	1	no	—
30	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₂₇	—	4	1	no	—
32	C₄×C₁₂₇	—	4	1	no	—
33	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₂₇	#41	1	1	yes	characteristic
35	Dic₁₂₇	—	2	2	no	—
36	Dic₁₂₇	—	2	1	no	—
37	Dic₁₂₇	—	2	1	no	—
38	Dic₁₂₇	—	4	1	no	—
39	C₂×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂×C₁₂₇	#135	1	1	yes	characteristic
41	C₂×C₁₂₇	#137	1	1	yes	characteristic
42	C₁₂₇	#479	1	1	yes	characteristic, complemented, Sylow
43	#479	—	127	1	no	maximal, Sylow
44	#125	—	127	1	no	—
45	C₄×Q₈	—	127	1	no	—
46	#107	D₁₂₇	1	1	yes	characteristic
47	#106	—	127	1	no	—
48	C₄⋊C₈	—	127	1	no	—
49	#103	—	127	1	no	—
50	#103	—	127	1	no	—
51	C₂×Q₈	—	127	1	no	—
52	C₂×C₈	—	127	1	no	—
53	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
54	C₄⋊C₄	—	127	1	no	—
55	C₄⋊C₄	—	254	1	no	—
56	C₄⋊C₄	—	254	1	no	—
57	C₄⋊C₄	—	254	1	no	—
58	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
60	C₄²	—	127	1	no	—
61	C₄²	—	254	1	no	—
62	Q₈	—	127	2	no	—
63	Q₈	—	254	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	127	1	no	—
67	C₂×C₄	—	127	1	no	—
68	C₂×C₄	—	254	1	no	—
69	C₂×C₄	—	254	1	no	—
70	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	254	1	no	—
73	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
74	C₄	—	4	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	508	1	no	—
77	C₄	—	254	2	no	—
78	C₄	—	254	1	no	—
79	C₄	—	254	1	no	—
80	C₄	Q₈×D₁₂₇	1	1	yes	characteristic
81	C₄	#14523248	1	1	yes	characteristic
82	C₂	#16861187	1	1	yes	characteristic
83	C₂	#16861228	1	1	yes	characteristic
84	C₂	#16861241	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#107×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#107	D₁₂₇
hypocenter	C₁₂₇	#479
LCS	G ▷ C₂×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
42. C₁₂₇	#479	semidirect