Subgroups — #22549681

Order 8128.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
2	#16861143	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₅₄	—	2	1	no	—
5	Dic₅₀₈	—	2	1	no	—
6	Dic₅₀₈	C₄	1	2	yes	complemented
7	C₄⋊Dic₁₂₇	—	2	1	no	—
8	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₁₆×C₁₂₇	—	2	1	no	—
10	Dic₂₅₄	—	2	2	no	—
11	Dic₂₅₄	—	4	1	no	—
12	C₂×Dic₁₂₇	—	4	1	no	—
13	C₂×Dic₁₂₇	—	4	1	no	—
14	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
15	C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂²×C₁₂₇	D₈	1	1	yes	characteristic
18	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
20	Dic₁₂₇	—	4	2	no	—
21	Dic₁₂₇	—	8	1	no	—
22	C₂×C₁₂₇	#102	1	1	yes	characteristic
23	C₂×C₁₂₇	SD₃₂	1	1	yes	characteristic
24	C₂×C₁₂₇	Q₃₂	1	1	yes	characteristic
25	C₁₂₇	#358	1	1	yes	characteristic, complemented, Sylow
26	#358	—	127	1	no	maximal, Sylow
27	C₂×Q₁₆	—	127	1	no	—
28	C₂×C₁₆	D₁₂₇	1	1	yes	characteristic
29	#107	—	127	1	no	—
30	C₂×Q₈	—	254	1	no	—
31	Q₁₆	—	127	2	no	—
32	Q₁₆	—	254	1	no	—
33	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
34	C₄⋊C₄	—	254	1	no	—
35	C₁₆	—	2	1	no	—
36	Q₈	—	508	1	no	—
37	Q₈	—	254	2	no	—
38	C₂×C₄	—	508	1	no	—
39	C₂×C₄	—	508	1	no	—
40	C₂×C₄	D₅₀₈	1	1	yes	characteristic
41	C₈	C₄×D₁₂₇	1	1	yes	characteristic
42	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
43	C₂²	D₁₀₁₆	1	1	yes	characteristic
44	C₄	—	1016	1	no	—
45	C₄	—	508	2	no	—
46	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
47	C₄	#14523237	1	1	yes	characteristic
48	C₂	#16861131	1	1	yes	characteristic
49	C₂	#16861154	1	1	yes	characteristic
50	C₂	Dic₁₀₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₀₁₆
commutator	C₈×C₁₂₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	C₂×C₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	D₈
hypercenter	C₂×C₁₆	D₁₂₇
hypocenter	C₁₂₇	#358
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
6. Dic₅₀₈	C₄	semidirect
25. C₁₂₇	#358	semidirect