Subgroups — #22549688

Order 8128.

62 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	#16861149	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×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	—
7	C₁₂₇⋊C₁₆	—	2	1	no	—
8	M₁₆×C₁₂₇	—	2	1	no	—
9	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×D₁₂₇	—	4	1	no	—
11	D₅₀₈	—	2	2	no	—
12	D₅₀₈	—	4	1	no	—
13	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
14	C₈×C₁₂₇	—	4	1	no	—
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	C₂×D₁₂₇	—	4	2	no	—
19	C₂×D₁₂₇	—	8	2	no	—
20	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₂₇	D₈	1	1	yes	characteristic
22	C₂×C₁₂₇	—	2	1	no	—
23	C₂×C₁₂₇	#102	1	1	yes	characteristic
24	D₁₂₇	—	8	2	no	—
25	C₁₂₇	#361	1	1	yes	characteristic, complemented, Sylow
26	#361	—	127	1	no	maximal, Sylow
27	C₂×D₈	—	127	1	no	—
28	M₃₂	—	127	1	no	—
29	#108	D₁₂₇	1	1	yes	characteristic, complemented
30	C₂×D₄	—	254	1	no	—
31	D₈	—	127	2	no	—
32	D₈	—	254	1	no	—
33	M₁₆	—	2	1	no	—
34	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
35	C₁₆	—	254	1	no	—
36	C₂³	—	508	1	no	—
37	D₄	—	508	1	no	—
38	D₄	—	254	2	no	—
39	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
40	C₈	—	4	1	no	—
41	C₈	C₄×D₁₂₇	1	1	yes	characteristic
42	C₈	D₅₀₈	1	1	yes	characteristic
43	C₂²	—	1016	2	no	—
44	C₂²	—	508	2	no	—
45	C₂²	#14523235	1	1	yes	characteristic
46	C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
47	C₄	#14523233	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	1016	2	no	—
50	C₂	#16861129	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16861129
commutator	C₈×C₁₂₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	#108×C₁₂₇	C₂
socle	C₂×C₁₂₇	#102
hypercenter	#108	D₁₂₇
hypocenter	C₁₂₇	#361
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #16861149	C₂	semidirect
3. #108×C₁₂₇	C₂	semidirect
25. C₁₂₇	#361	semidirect
29. #108	D₁₂₇	semidirect