Subgroups — #22550238

Order 8128.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861197	C₂	1	1	yes	maximal, characteristic
2	#16861132	C₂	1	2	yes	maximal, complemented
3	#16861129	C₂	1	2	yes	maximal
4	#16861120	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
7	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
10	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₁₂₇	—	2	1	no	—
12	C₄⋊C₄×C₁₂₇	—	2	1	no	—
13	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₂₇	—	2	1	no	—
16	Dic₂₅₄	—	2	2	no	—
17	D₅₀₈	—	2	2	no	—
18	C₂×Dic₁₂₇	—	2	1	no	—
19	C₁₂₇⋊C₈	—	2	2	no	—
20	Q₈×C₁₂₇	—	4	2	no	—
21	C₂×C₄×C₁₂₇	—	2	2	no	—
22	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
26	C₂×D₁₂₇	—	4	1	no	—
27	C₂×D₁₂₇	—	4	1	no	—
28	C₂×D₁₂₇	—	4	1	no	—
29	C₄×C₁₂₇	—	2	1	no	—
30	C₄×C₁₂₇	—	2	1	no	—
31	C₄×C₁₂₇	—	4	2	no	—
32	C₄×C₁₂₇	#41	1	2	yes	—
33	Dic₁₂₇	—	4	1	no	—
34	C₂×C₁₂₇	#124	1	1	yes	characteristic
35	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₁₂₇	#137	1	1	yes	characteristic
37	D₁₂₇	—	8	1	no	—
38	C₁₂₇	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	127	1	no	maximal, Sylow
40	C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
41	#124	—	127	1	no	—
42	#103	—	127	2	no	—
43	#102	—	127	2	no	—
44	#97	—	127	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	127	1	no	—
47	C₂×D₄	—	127	1	no	—
48	C₂×C₈	—	127	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
51	C₂²⋊C₄	—	254	1	no	—
52	C₂²⋊C₄	—	254	1	no	—
53	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
54	C₂³	—	254	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	254	2	no	—
57	D₄	—	254	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	254	1	no	—
60	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
63	C₈	—	254	2	no	—
64	C₂²	—	508	1	no	—
65	C₂²	—	508	1	no	—
66	C₂²	—	508	1	no	—
67	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	508	1	no	—
72	C₄	#14523249	1	2	yes	—
73	C₂	—	1016	1	no	—
74	C₂	#16861198	1	1	yes	characteristic
75	C₂	#16861235	1	1	yes	characteristic
76	C₂	#16861240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₄⋊Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#489
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #16861132	C₂	semidirect
4. #16861120	C₂	semidirect
5. C₄⋊Q₈×C₁₂₇	C₂	semidirect
38. C₁₂₇	#489	semidirect
40. C₄⋊Q₈	D₁₂₇	semidirect