Subgroups — #22549614

Order 8128.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861188	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₅₄	C₄	1	1	yes	characteristic, complemented
5	C₂²×Dic₁₂₇	—	2	1	no	—
6	C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic, complemented
7	Dic₁₂₇⋊C₄	—	2	1	no	—
8	C₂²⋊Dic₁₂₇	—	2	1	no	—
9	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₁₂₇	—	2	1	no	—
11	Dic₂₅₄	—	4	1	no	—
12	C₂×Dic₁₂₇	—	2	2	no	—
13	C₂×Dic₁₂₇	—	2	1	no	—
14	C₂×Dic₁₂₇	—	2	1	no	—
15	C₂×Dic₁₂₇	—	4	1	no	—
16	C₂×Dic₁₂₇	—	4	1	no	—
17	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	—	4	1	no	—
19	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
21	C₈×C₁₂₇	—	4	1	no	—
22	C₂²×C₁₂₇	—	2	1	no	—
23	C₂²×C₁₂₇	—	2	1	no	—
24	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₁₂₇	SD₁₆	1	1	yes	characteristic
26	C₂²×C₁₂₇	Q₁₆	1	1	yes	characteristic
27	C₄×C₁₂₇	—	2	1	no	—
28	C₄×C₁₂₇	—	2	1	no	—
29	Dic₁₂₇	—	4	2	no	—
30	Dic₁₂₇	—	4	1	no	—
31	Dic₁₂₇	—	4	1	no	—
32	C₂×C₁₂₇	—	2	1	no	—
33	C₂×C₁₂₇	—	2	1	no	—
34	C₂×C₁₂₇	#99	1	1	yes	characteristic
35	C₂×C₁₂₇	#103	1	1	yes	characteristic
36	C₂×C₁₂₇	#104	1	1	yes	characteristic
37	C₁₂₇	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	127	1	no	maximal, Sylow
39	C₂²⋊Q₈	—	127	1	no	—
40	C₂²⋊C₈	D₁₂₇	1	1	yes	characteristic
41	#95	—	127	1	no	—
42	C₂×Q₈	—	127	1	no	—
43	C₂²×C₄	—	254	1	no	—
44	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
45	C₂×C₈	—	2	1	no	—
46	C₄⋊C₄	—	127	1	no	—
47	C₄⋊C₄	—	254	1	no	—
48	C₂²⋊C₄	—	254	1	no	—
49	C₂³	D₅₀₈	1	1	yes	characteristic
50	Q₈	—	508	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	—	508	1	no	—
53	C₂×C₄	—	508	1	no	—
54	C₂×C₄	—	254	2	no	—
55	C₂×C₄	—	254	1	no	—
56	C₂×C₄	—	254	1	no	—
57	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
59	C₈	—	4	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
63	C₂²	#14523237	1	1	yes	characteristic
64	C₂²	Dic₅₀₈	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	508	2	no	—
68	C₄	—	508	1	no	—
69	C₄	—	508	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	1	no	—
72	C₂	#16861123	1	1	yes	characteristic
73	C₂	#16861134	1	1	yes	characteristic
74	C₂	#16861135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₂₇)⋊C₄
commutator	C₂×C₄×C₁₂₇	C₂×C₄
Frattini	C₂²×C₄	C₂×D₁₂₇
Fitting	C₂²⋊C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	C₂²⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#328
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×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
4. C₂×Dic₂₅₄	C₄	semidirect
6. C₄⋊Dic₁₂₇	C₄	semidirect
37. C₁₂₇	#328	semidirect