Subgroups — #22550190

Order 8128.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861183	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861143	C₂	1	2	yes	maximal, complemented
4	#16861131	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
8	C₂²×Dic₁₂₇	—	2	1	no	—
9	C₄⋊Dic₁₂₇	—	2	1	no	—
10	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
12	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
13	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂²×D₁₂₇	—	2	1	no	—
16	D₅₀₈	—	2	2	no	—
17	C₂²⋊D₁₂₇	—	4	2	no	—
18	C₂×Dic₁₂₇	—	2	1	no	—
19	C₂×Dic₁₂₇	—	2	2	no	—
20	C₂×Dic₁₂₇	—	4	2	no	—
21	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	—	2	2	no	—
23	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₈×C₁₂₇	—	2	2	no	—
26	C₂²×C₁₂₇	—	2	1	no	—
27	C₂²×C₁₂₇	—	2	1	no	—
28	C₂²×C₁₂₇	D₈	1	2	yes	—
29	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
30	C₂×D₁₂₇	—	4	1	no	—
31	C₂×D₁₂₇	—	4	1	no	—
32	C₂×D₁₂₇	—	4	1	no	—
33	C₄×C₁₂₇	—	2	1	no	—
34	C₄×C₁₂₇	#41	1	2	yes	—
35	Dic₁₂₇	—	4	2	no	—
36	Dic₁₂₇	—	4	1	no	—
37	C₂×C₁₂₇	—	2	2	no	—
38	C₂×C₁₂₇	#123	1	1	yes	characteristic
39	C₂×C₁₂₇	C₂×D₈	1	1	yes	characteristic
40	C₂×C₁₂₇	#137	1	1	yes	characteristic
41	D₁₂₇	—	8	1	no	—
42	C₁₂₇	#480	1	1	yes	characteristic, complemented, Sylow
43	#480	—	127	1	no	maximal, Sylow
44	#121	—	127	1	no	—
45	C₂×C₄⋊C₄	—	127	1	no	—
46	#107	—	127	2	no	—
47	#102	—	127	2	no	—
48	C₂²⋊C₈	D₁₂₇	1	1	yes	characteristic, complemented
49	C₂×D₄	—	127	1	no	—
50	C₂×D₄	—	254	1	no	—
51	C₂²×C₄	—	254	1	no	—
52	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
53	C₂×C₈	C₂×D₁₂₇	1	2	yes	—
54	C₄⋊C₄	—	127	2	no	—
55	C₄⋊C₄	—	127	1	no	—
56	C₄⋊C₄	—	254	1	no	—
57	C₂²⋊C₄	—	254	1	no	—
58	C₂³	—	254	1	no	—
59	C₂³	D₅₀₈	1	1	yes	characteristic
60	D₄	—	508	2	no	—
61	D₄	—	254	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	508	2	no	—
64	C₂×C₄	—	254	1	no	—
65	C₂×C₄	—	254	2	no	—
66	C₂×C₄	D₅₀₈	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	508	1	no	—
72	C₂²	—	508	1	no	—
73	C₂²	—	508	1	no	—
74	C₂²	D₁₀₁₆	1	2	yes	—
75	C₂²	C₂×D₅₀₈	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	508	2	no	—
78	C₄	—	508	1	no	—
79	C₄	#14523247	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	1016	1	no	—
82	C₂	#16861193	1	1	yes	characteristic
83	C₂	C₂×D₁₀₁₆	1	1	yes	characteristic
84	C₂	#16861242	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂²⋊C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂²⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#480
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
3. #16861143	C₂	semidirect
4. #16861131	C₂	semidirect
5. C₂²⋊C₈×C₁₂₇	C₂	semidirect
42. C₁₂₇	#480	semidirect
48. C₂²⋊C₈	D₁₂₇	semidirect