Subgroups — #22549693

Order 8128.

54 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	#16861145	C₂	1	1	yes	maximal, characteristic
3	M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₅₄	—	2	1	no	—
5	Dic₅₀₈	—	2	1	no	—
6	Dic₅₀₈	C₄	1	2	yes	—
7	#14523232	—	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	C₁₂₇⋊C₈	—	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	SD₁₆	1	1	yes	characteristic
18	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₂₇	D₈	1	1	yes	characteristic
20	Dic₁₂₇	—	4	2	no	—
21	C₂×C₁₂₇	—	2	1	no	—
22	C₂×C₁₂₇	#102	1	1	yes	characteristic
23	C₁₂₇	#362	1	1	yes	characteristic, complemented, Sylow
24	#362	—	127	1	no	maximal, Sylow
25	C₂×Q₁₆	—	127	1	no	—
26	M₃₂	D₁₂₇	1	1	yes	characteristic
27	#108	—	127	1	no	—
28	C₂×Q₈	—	254	1	no	—
29	Q₁₆	—	127	2	no	—
30	Q₁₆	—	254	1	no	—
31	M₁₆	—	254	1	no	—
32	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
33	C₁₆	—	2	1	no	—
34	Q₈	—	508	1	no	—
35	Q₈	—	254	2	no	—
36	C₂×C₄	—	508	1	no	—
37	C₂×C₄	D₅₀₈	1	1	yes	characteristic
38	C₈	—	508	1	no	—
39	C₈	C₄×D₁₂₇	1	1	yes	characteristic
40	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
41	C₂²	#14523237	1	1	yes	characteristic
42	C₄	—	508	2	no	—
43	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
44	C₄	D₁₀₁₆	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#16861131	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16861131
commutator	C₈×C₁₂₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	M₃₂×C₁₂₇	C₂
socle	C₂×C₁₂₇	#102
hypercenter	M₃₂	D₁₂₇
hypocenter	C₁₂₇	#362
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
23. C₁₂₇	#362	semidirect