Subgroups — #22549657

Order 8128.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523231	C₂	1	1	yes	maximal, characteristic
2	#16861149	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₁₂₇	C₄	1	1	yes	characteristic
5	#14523231	—	2	1	no	—
6	#14523231	—	2	1	no	—
7	C₂×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
8	C₁₂₇⋊C₁₆	—	2	1	no	—
9	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₂₇	—	2	1	no	—
11	C₂²×D₁₂₇	C₈	1	1	yes	characteristic
12	C₂×Dic₁₂₇	C₈	1	1	yes	characteristic
13	C₄×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	C₂×C₈	1	1	yes	characteristic
20	C₂×D₁₂₇	—	2	1	no	—
21	C₂×D₁₂₇	—	4	1	no	—
22	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
24	Dic₁₂₇	—	2	1	no	—
25	C₂×C₁₂₇	—	2	1	no	—
26	C₂×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
27	D₁₂₇	—	4	1	no	—
28	C₁₂₇	#349	1	1	yes	characteristic, complemented, Sylow
29	#349	—	127	1	no	maximal, Sylow
30	C₂×M₁₆	—	127	1	no	—
31	M₃₂	—	127	1	no	—
32	M₃₂	D₁₂₇	1	1	yes	characteristic, complemented
33	C₂²×C₄	—	127	1	no	—
34	M₁₆	—	254	1	no	—
35	M₁₆	—	254	1	no	—
36	C₂×C₈	—	127	1	no	—
37	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	C₁₆	—	254	1	no	—
40	C₂³	—	127	1	no	—
41	C₂×C₄	—	127	1	no	—
42	C₂×C₄	—	254	1	no	—
43	C₂×C₄	—	254	1	no	—
44	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
45	C₈	—	254	1	no	—
46	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
47	C₈	D₅₀₈	1	1	yes	characteristic
48	C₂²	—	508	1	no	—
49	C₂²	—	254	1	no	—
50	C₂²	C₈×D₁₂₇	1	1	yes	characteristic
51	C₄	—	254	1	no	—
52	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
53	C₄	#14523231	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	508	1	no	—
56	C₂	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×D₁₂₇)⋊C₄
commutator	C₂²×C₁₂₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	M₃₂×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²⋊C₈
hypercenter	M₃₂	D₁₂₇
hypocenter	C₁₂₇	#349
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #16861149	C₂	semidirect
3. M₃₂×C₁₂₇	C₂	semidirect
28. C₁₂₇	#349	semidirect
32. M₃₂	D₁₂₇	semidirect