Subgroups — #22549641

Order 8128.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523232	C₂	1	2	yes	maximal
2	C₂²×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#14523232	—	2	2	no	—
4	#14523232	C₄	1	4	yes	—
5	C₂×C₁₂₇⋊C₈	—	2	2	no	—
6	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₁₂₇	—	2	2	no	—
8	C₂×C₈×C₁₂₇	C₄	1	2	yes	—
9	C₁₂₇⋊C₈	—	2	4	no	—
10	C₂³×C₁₂₇	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
14	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
15	C₈×C₁₂₇	—	2	2	no	—
16	C₂²×C₁₂₇	—	2	2	no	—
17	C₂²×C₁₂₇	C₄⋊C₄	1	2	yes	—
18	C₂²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₁₂₇	C₄²	1	1	yes	characteristic
20	C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
21	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₁₂₇	—	2	2	no	—
23	C₂×C₁₂₇	#95	1	1	yes	characteristic
24	C₂×C₁₂₇	#108	1	2	yes	—
25	C₁₂₇	#341	1	1	yes	characteristic, complemented, Sylow
26	#341	—	127	1	no	maximal, Sylow
27	C₂×M₁₆	—	127	2	no	—
28	C₂²×C₈	D₁₂₇	1	1	yes	characteristic
29	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
30	M₁₆	—	127	4	no	—
31	M₁₆	—	254	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	254	2	no	—
34	C₂×C₈	Dic₁₂₇	1	2	yes	—
35	C₂³	Dic₂₅₄	1	1	yes	characteristic
36	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
37	C₂×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
38	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
39	C₂×C₄	D₅₀₈	1	1	yes	characteristic
40	C₈	—	2	2	no	—
41	C₈	—	254	4	no	—
42	C₂²	—	2	2	no	—
43	C₂²	Dic₁₂₇⋊C₄	1	2	yes	—
44	C₂²	C₄⋊Dic₁₂₇	1	1	yes	characteristic
45	C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
46	C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
47	C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
48	C₂	—	2	2	no	—
49	C₂	C₁₂₇⋊#95	1	1	yes	characteristic
50	C₂	#16861145	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₅₀₈
commutator	C₄×C₁₂₇	C₄²
Frattini	C₂²×C₄	C₂×D₁₂₇
Fitting	C₂²×C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₄⋊C₄
hypercenter	C₂²×C₈	D₁₂₇
hypocenter	C₁₂₇	#341
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
25. C₁₂₇	#341	semidirect