Subgroups — #22549630

Order 8128.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₂₇	C₄	1	2	yes	—
5	C₂×C₁₂₇⋊C₈	—	2	1	no	—
6	C₂×C₁₂₇⋊C₈	—	2	1	no	—
7	C₂×C₁₂₇⋊C₈	C₄	1	2	yes	complemented
8	C₄×Dic₁₂₇	—	2	1	no	—
9	C₄×Dic₁₂₇	—	2	1	no	—
10	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₂₇	—	2	1	no	—
12	C₂×C₈×C₁₂₇	—	2	1	no	—
13	C₂×C₈×C₁₂₇	C₄	1	2	yes	complemented
14	C₂×Dic₁₂₇	—	2	2	no	—
15	C₂×Dic₁₂₇	—	2	2	no	—
16	C₂×Dic₁₂₇	C₈	1	4	yes	complemented
17	C₁₂₇⋊C₈	—	2	2	no	—
18	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
25	C₈×C₁₂₇	—	2	2	no	—
26	C₂²×C₁₂₇	C₄²	1	1	yes	characteristic
27	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₂₇	C₂×C₈	1	2	yes	—
30	C₂²×C₁₂₇	M₁₆	1	2	yes	—
31	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
34	C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
35	Dic₁₂₇	—	2	4	no	—
36	C₂×C₁₂₇	#95	1	1	yes	characteristic
37	C₂×C₁₂₇	C₄×C₈	1	1	yes	characteristic
38	C₂×C₁₂₇	#97	1	1	yes	characteristic
39	C₂×C₁₂₇	C₂²⋊C₈	1	2	yes	—
40	C₂×C₁₂₇	C₄⋊C₈	1	2	yes	—
41	C₁₂₇	#336	1	1	yes	characteristic, complemented, Sylow
42	#336	—	127	1	no	maximal, Sylow
43	C₂²×C₈	—	127	1	no	—
44	C₂²×C₈	D₁₂₇	1	1	yes	characteristic
45	C₂×C₄²	—	127	1	no	—
46	C₂²×C₄	—	127	2	no	—
47	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	—	127	2	no	—
51	C₂×C₈	—	254	1	no	—
52	C₂×C₈	—	254	1	no	—
53	C₂×C₈	Dic₁₂₇	1	2	yes	complemented
54	C₄²	—	254	1	no	—
55	C₄²	—	254	1	no	—
56	C₂³	C₄×D₁₂₇	1	1	yes	characteristic
57	C₂×C₄	—	127	4	no	—
58	C₂×C₄	—	254	2	no	—
59	C₂×C₄	—	254	2	no	—
60	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
61	C₂×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
64	C₂×C₄	D₅₀₈	1	1	yes	characteristic
65	C₂×C₄	Dic₂₅₄	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	254	2	no	—
68	C₂²	C₄×Dic₁₂₇	1	1	yes	characteristic
69	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
70	C₂²	Dic₁₂₇⋊C₄	1	1	yes	characteristic
71	C₂²	C₈×D₁₂₇	1	2	yes	—
72	C₂²	#14523231	1	2	yes	—
73	C₄	—	254	4	no	—
74	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
75	C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
76	C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
77	C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
78	C₂	C₁₂₇⋊#95	1	1	yes	characteristic
79	C₂	C₈×Dic₁₂₇	1	1	yes	characteristic
80	C₂	#16861119	1	1	yes	characteristic
81	C₂	(C₂×D₁₂₇)⋊C₈	1	2	yes	—
82	C₂	Dic₁₂₇⋊C₈	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₄×C₈
Fitting	C₂²×C₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×C₄
hypercenter	C₂²×C₈	D₁₂₇
hypocenter	C₁₂₇	#336
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
7. C₂×C₁₂₇⋊C₈	C₄	semidirect
13. C₂×C₈×C₁₂₇	C₄	semidirect
16. C₂×Dic₁₂₇	C₈	semidirect
41. C₁₂₇	#336	semidirect
53. C₂×C₈	Dic₁₂₇	semidirect