Subgroups — #22549833

Order 8128.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523231	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861166	C₂	1	1	yes	maximal, characteristic
4	#16861119	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
9	#14523231	C₄	1	4	yes	complemented
10	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₁₂₇	C₄	1	4	yes	complemented
13	Dic₁₂₇⋊C₄	C₄	1	2	yes	—
14	(C₂×D₁₂₇)⋊C₄	C₄	1	2	yes	—
15	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₁₂₇	C₂×C₄	1	2	yes	—
23	C₄×D₁₂₇	C₂×C₄	1	2	yes	—
24	C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₈×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₈×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₁₂₇	—	2	2	no	—
32	C₂×D₁₂₇	C₄²	1	2	yes	—
33	C₄×C₁₂₇	—	2	2	no	—
34	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	Dic₁₂₇	—	2	2	no	—
37	Dic₁₂₇	C₄²	1	2	yes	—
38	C₂×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₁₂₇	#131	1	2	yes	—
40	D₁₂₇	—	2	2	no	—
41	C₁₂₇	#405	1	1	yes	characteristic, complemented, Sylow
42	#405	—	127	1	no	maximal, Sylow
43	C₂×M₁₆	—	127	1	no	—
44	C₂²×C₈	—	127	1	no	—
45	#117	—	127	1	no	—
46	#97	—	127	1	no	—
47	#97	D₁₂₇	1	1	yes	characteristic, complemented
48	C₄×C₈	—	127	1	no	—
49	C₄×C₈	—	127	1	no	—
50	C₂²×C₄	—	127	1	no	—
51	M₁₆	—	127	4	no	—
52	C₂×C₈	—	127	1	no	—
53	C₂×C₈	—	127	1	no	—
54	C₂×C₈	—	127	4	no	—
55	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
57	C₄⋊C₄	—	127	2	no	—
58	C₂²⋊C₄	—	127	2	no	—
59	C₄²	—	127	1	no	—
60	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂³	—	127	1	no	—
62	C₂×C₄	—	127	1	no	—
63	C₂×C₄	—	127	2	no	—
64	C₂×C₄	—	127	2	no	—
65	C₂×C₄	—	127	2	no	—
66	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
67	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
68	C₈	—	127	2	no	—
69	C₈	—	127	2	no	—
70	C₈	C₄×D₁₂₇	1	2	yes	—
71	C₈	C₄×D₁₂₇	1	2	yes	—
72	C₂²	—	127	2	no	—
73	C₂²	—	254	2	no	—
74	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	127	2	no	—
77	C₄	—	254	2	no	—
78	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
79	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
80	C₂	—	254	2	no	—
81	C₂	C₄²×D₁₂₇	1	1	yes	characteristic
82	C₂	#16861217	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄²
Fitting	#97×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×C₄
hypercenter	#97	D₁₂₇
hypocenter	C₁₂₇	#405
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
4. #16861119	C₂	semidirect
5. C₄×C₁₂₇⋊C₈	C₂	semidirect
6. C₈×Dic₁₂₇	C₂	semidirect
7. #97×C₁₂₇	C₂	semidirect
9. #14523231	C₄	semidirect
12. C₈×D₁₂₇	C₄	semidirect
41. C₁₂₇	#405	semidirect
47. #97	D₁₂₇	semidirect