Subgroups — #22549727

Order 8128.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₂₇⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₂₇	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic
5	#95×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
7	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
9	Dic₁₂₇⋊C₄	C₄	1	8	yes	complemented
10	C₄×Dic₁₂₇	—	2	4	no	—
11	C₄×Dic₁₂₇	C₄	1	4	yes	complemented
12	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂×Dic₁₂₇	—	2	2	no	—
15	C₂×Dic₁₂₇	—	2	2	no	—
16	C₂×Dic₁₂₇	C₂×C₄	1	4	yes	—
17	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
18	C₂×Dic₁₂₇	C₂×C₄	1	4	yes	—
19	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
20	C₂×Dic₁₂₇	D₄	1	2	yes	—
21	C₂×Dic₁₂₇	Q₈	1	2	yes	—
22	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	—	2	2	no	—
24	C₂×C₄×C₁₂₇	—	2	2	no	—
25	C₂×C₄×C₁₂₇	—	2	2	no	—
26	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
28	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₁₂₇	C₂²×C₄	1	2	yes	—
30	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
32	C₂²×C₁₂₇	#41	1	2	yes	—
33	C₄×C₁₂₇	—	2	2	no	—
34	C₄×C₁₂₇	—	2	4	no	—
35	Dic₁₂₇	—	2	4	no	—
36	Dic₁₂₇	C₄²	1	4	yes	complemented
37	Dic₁₂₇	C₄⋊C₄	1	4	yes	complemented
38	C₂×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₁₂₇	#117	1	1	yes	characteristic
41	C₂×C₁₂₇	C₄×D₄	1	2	yes	—
42	C₂×C₁₂₇	C₄×Q₈	1	2	yes	—
43	C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
44	C₄×C₄⋊C₄	—	127	1	no	maximal, Sylow
45	C₂×C₄⋊C₄	—	127	2	no	—
46	C₂×C₄²	—	127	1	no	—
47	C₂×C₄²	—	127	2	no	—
48	#95	—	127	1	no	—
49	#95	D₁₂₇	1	1	yes	characteristic
50	C₂²×C₄	—	127	1	no	—
51	C₂²×C₄	—	127	1	no	—
52	C₂²×C₄	—	127	2	no	—
53	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
54	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
55	C₄⋊C₄	—	127	8	no	—
56	C₄²	—	127	4	no	—
57	C₄²	—	254	4	no	—
58	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	127	2	no	—
63	C₂×C₄	—	127	2	no	—
64	C₂×C₄	—	127	4	no	—
65	C₂×C₄	—	127	2	no	—
66	C₂×C₄	—	127	2	no	—
67	C₂×C₄	—	127	4	no	—
68	C₂×C₄	—	254	2	no	—
69	C₂×C₄	—	254	2	no	—
70	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
71	C₂×C₄	C₄×D₁₂₇	1	4	yes	—
72	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
73	C₂²	C₂×C₄×D₁₂₇	1	2	yes	—
74	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
75	C₂²	Q₈×D₁₂₇	1	1	yes	characteristic
76	C₂²	#14523247	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	—	2	4	no	—
79	C₄	—	127	4	no	—
80	C₄	—	127	4	no	—
81	C₄	—	254	4	no	—
82	C₂	C₄²×D₁₂₇	1	1	yes	characteristic
83	C₂	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
84	C₂	#16861165	1	1	yes	characteristic
85	C₂	(C₂²⋊D₁₂₇)⋊C₄	1	2	yes	—
86	C₂	#16861174	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄²
Fitting	#95×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	#95	D₁₂₇
hypocenter	C₁₂₇	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
9. Dic₁₂₇⋊C₄	C₄	semidirect
11. C₄×Dic₁₂₇	C₄	semidirect
36. Dic₁₂₇	C₄²	semidirect
37. Dic₁₂₇	C₄⋊C₄	semidirect
43. C₁₂₇	C₄×C₄⋊C₄	semidirect