Subgroups — #22549661

Order 8128.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861215	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14523248	C₄	1	1	yes	characteristic
5	#14523232	C₄	1	1	yes	characteristic
6	#14523231	—	2	1	no	—
7	C₈×D₁₂₇	—	2	1	no	—
8	C₁₂₇⋊C₁₆	—	2	1	no	—
9	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₂₇	—	2	1	no	—
11	Dic₂₅₄	C₈	1	1	yes	characteristic
12	D₅₀₈	C₈	1	1	yes	characteristic
13	C₂²⋊D₁₂₇	—	2	1	no	—
14	C₄×D₁₂₇	—	2	1	no	—
15	C₁₂₇⋊C₈	—	2	1	no	—
16	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂²×C₁₂₇	M₁₆	1	1	yes	characteristic
20	C₂×D₁₂₇	—	2	1	no	—
21	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
23	Dic₁₂₇	—	2	1	no	—
24	C₂×C₁₂₇	—	2	1	no	—
25	C₂×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
26	D₁₂₇	—	4	1	no	—
27	C₁₂₇	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	127	1	no	maximal, Sylow
29	#131	—	127	1	no	—
30	M₃₂	D₁₂₇	1	1	yes	characteristic, complemented
31	C₂×C₁₆	—	127	1	no	—
32	#41	—	127	1	no	—
33	M₁₆	—	127	1	no	—
34	M₁₆	—	254	1	no	—
35	C₂×C₈	—	254	1	no	—
36	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	254	1	no	—
39	Q₈	—	127	1	no	—
40	D₄	—	127	1	no	—
41	D₄	—	254	1	no	—
42	C₂×C₄	—	254	1	no	—
43	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
44	C₈	—	254	1	no	—
45	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
46	C₈	D₅₀₈	1	1	yes	characteristic
47	C₂²	—	254	1	no	—
48	C₂²	#14523231	1	1	yes	characteristic
49	C₄	—	254	1	no	—
50	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
51	C₄	C₈×D₁₂₇	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	508	1	no	—
54	C₂	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	M₃₂×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²⋊C₈
hypercenter	M₃₂	D₁₂₇
hypocenter	C₁₂₇	#350
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₂₇⋊C₁₆	C₂	semidirect
3. M₃₂×C₁₂₇	C₂	semidirect
27. C₁₂₇	#350	semidirect
30. M₃₂	D₁₂₇	semidirect