Subgroups — #22550309

Order 8128.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861232	C₂	1	2	yes	maximal, complemented
2	#16861152	C₂	1	2	yes	maximal, complemented
3	D₂₀₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523249	—	2	2	no	—
7	#14523235	—	2	2	no	—
8	D₁₀₁₆	C₂²	1	2	yes	—
9	C₈×D₁₂₇	C₂²	1	1	yes	characteristic
10	C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	Q₁₆×C₁₂₇	C₂²	1	2	yes	—
12	C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	D₅₀₈	—	2	2	no	—
14	D₅₀₈	—	4	2	no	—
15	C₄×D₁₂₇	—	4	2	no	—
16	C₄×D₁₂₇	D₄	1	1	yes	characteristic
17	C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
18	Q₈×C₁₂₇	—	2	2	no	—
19	C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₂×D₁₂₇	—	4	2	no	—
21	C₂×D₁₂₇	D₈	1	1	yes	characteristic
22	C₄×C₁₂₇	—	4	2	no	—
23	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
24	Dic₁₂₇	D₈	1	1	yes	characteristic
25	C₂×C₁₂₇	C₂×D₈	1	1	yes	characteristic
26	D₁₂₇	—	2	1	no	—
27	D₁₂₇	—	8	2	no	—
28	C₁₂₇	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	127	1	no	maximal, Sylow
30	#135	—	127	2	no	—
31	Q₃₂	D₁₂₇	1	1	yes	characteristic, complemented
32	SD₃₂	—	127	2	no	—
33	D₁₆	—	127	1	no	—
34	C₂×C₁₆	—	127	1	no	—
35	#41	—	254	2	no	—
36	Q₁₆	C₂×D₁₂₇	1	2	yes	—
37	SD₁₆	—	254	2	no	—
38	D₈	—	127	2	no	—
39	C₂×C₈	—	127	1	no	—
40	C₁₆	—	127	1	no	—
41	C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
42	Q₈	—	2	2	no	—
43	D₄	—	508	2	no	—
44	D₄	—	254	2	no	—
45	C₂×C₄	—	127	1	no	—
46	C₂×C₄	—	508	2	no	—
47	C₈	—	127	1	no	—
48	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
49	C₂²	—	127	1	no	—
50	C₂²	—	508	2	no	—
51	C₄	—	4	2	no	—
52	C₄	—	127	1	no	—
53	C₄	D₄×D₁₂₇	1	1	yes	characteristic
54	C₂	—	1016	2	no	—
55	C₂	—	254	1	no	—
56	C₂	D₈×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₁₂₇
commutator	C₈×C₁₂₇	C₂³
Frattini	C₈	C₂²×D₁₂₇
Fitting	Q₃₂×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₈
hypercenter	Q₃₂	D₁₂₇
hypocenter	C₁₂₇	#508
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #16861232	C₂	semidirect
2. #16861152	C₂	semidirect
3. D₂₀₃₂	C₂	semidirect
4. C₁₆×D₁₂₇	C₂	semidirect
5. Q₃₂×C₁₂₇	C₂	semidirect
28. C₁₂₇	#508	semidirect
31. Q₃₂	D₁₂₇	semidirect