Subgroups — #22549676

Order 8128.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₀₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₅₀₈	—	2	1	no	—
5	D₁₀₁₆	—	2	1	no	—
6	D₁₀₁₆	C₄	1	2	yes	complemented
7	C₁₂₇⋊C₁₆	—	2	1	no	—
8	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₁₂₇	—	2	1	no	—
10	C₂²×D₁₂₇	—	4	1	no	—
11	D₅₀₈	—	2	2	no	—
12	D₅₀₈	—	4	1	no	—
13	C₂×C₄×C₁₂₇	—	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	C₂×D₁₂₇	—	4	2	no	—
19	C₂×D₁₂₇	—	8	1	no	—
20	C₂×D₁₂₇	—	8	1	no	—
21	C₄×C₁₂₇	—	8	1	no	—
22	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
24	C₂×C₁₂₇	#102	1	1	yes	characteristic
25	C₂×C₁₂₇	D₁₆	1	1	yes	characteristic
26	C₂×C₁₂₇	SD₃₂	1	1	yes	characteristic
27	D₁₂₇	—	8	2	no	—
28	C₁₂₇	#357	1	1	yes	characteristic, complemented, Sylow
29	#357	—	127	1	no	maximal, Sylow
30	C₂×D₈	—	127	1	no	—
31	C₂×C₁₆	—	127	1	no	—
32	#107	D₁₂₇	1	1	yes	characteristic, complemented
33	C₂×D₄	—	254	1	no	—
34	D₈	—	127	2	no	—
35	D₈	—	254	1	no	—
36	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
37	C₄⋊C₄	—	2	1	no	—
38	C₁₆	—	254	1	no	—
39	C₂³	—	508	1	no	—
40	D₄	—	508	1	no	—
41	D₄	—	254	2	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
44	C₈	C₄×D₁₂₇	1	1	yes	characteristic
45	C₈	D₅₀₈	1	1	yes	characteristic
46	C₂²	—	1016	1	no	—
47	C₂²	—	1016	1	no	—
48	C₂²	—	508	2	no	—
49	C₂²	#14523233	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
52	C₄	#14523235	1	1	yes	characteristic
53	C₂	—	1016	2	no	—
54	C₂	#16861129	1	1	yes	characteristic
55	C₂	#16861150	1	1	yes	characteristic
56	C₂	#16861152	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14523233
commutator	C₈×C₁₂₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	#107×C₁₂₇	C₂
socle	C₂²×C₁₂₇	D₈
hypercenter	#107	D₁₂₇
hypocenter	C₁₂₇	#357
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₂₇⋊C₁₆	C₂	semidirect
3. #107×C₁₂₇	C₂	semidirect
6. D₁₀₁₆	C₄	semidirect
28. C₁₂₇	#357	semidirect
32. #107	D₁₂₇	semidirect