Subgroups — #22550319

Order 8128.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861231	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₀₁₆	C₂	1	2	yes	maximal, complemented
4	#16861154	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523248	—	2	1	no	—
7	C₂×Dic₂₅₄	—	2	1	no	—
8	Dic₅₀₈	—	2	1	no	—
9	Dic₅₀₈	C₂²	1	2	yes	—
10	Dic₅₀₈	C₂²	1	1	yes	characteristic
11	#14523237	—	2	1	no	—
12	D₁₀₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₁₆×C₁₂₇	C₂²	1	2	yes	—
15	Dic₂₅₄	—	2	1	no	—
16	Dic₂₅₄	—	2	2	no	—
17	Dic₂₅₄	—	4	1	no	—
18	D₅₀₈	—	2	1	no	—
19	C₂²⋊D₁₂₇	—	4	1	no	—
20	C₂×Dic₁₂₇	—	4	1	no	—
21	C₄×D₁₂₇	—	4	1	no	—
22	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
23	C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₈×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₂₇	D₈	1	1	yes	characteristic
26	C₂×D₁₂₇	—	4	1	no	—
27	C₄×C₁₂₇	D₈	1	1	yes	characteristic
28	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
29	Dic₁₂₇	—	4	1	no	—
30	Dic₁₂₇	—	4	2	no	—
31	C₂×C₁₂₇	—	2	1	no	—
32	C₂×C₁₂₇	C₂×D₈	1	1	yes	characteristic
33	D₁₂₇	—	8	1	no	—
34	C₁₂₇	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	127	1	no	maximal, Sylow
36	#135	—	127	1	no	—
37	C₂×Q₁₆	—	127	1	no	—
38	Q₃₂	—	127	2	no	—
39	SD₃₂	—	127	2	no	—
40	M₃₂	D₁₂₇	1	1	yes	characteristic, complemented
41	#41	—	254	1	no	—
42	C₂×Q₈	—	254	1	no	—
43	Q₁₆	—	127	2	no	—
44	Q₁₆	—	127	1	no	—
45	Q₁₆	—	254	1	no	—
46	SD₁₆	—	254	1	no	—
47	D₈	—	127	1	no	—
48	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
49	C₁₆	C₂×D₁₂₇	1	2	yes	—
50	Q₈	—	508	1	no	—
51	Q₈	—	254	1	no	—
52	Q₈	—	254	2	no	—
53	D₄	—	508	1	no	—
54	D₄	—	254	1	no	—
55	C₂×C₄	—	508	1	no	—
56	C₂×C₄	—	508	1	no	—
57	C₂×C₄	D₅₀₈	1	1	yes	characteristic
58	C₈	D₅₀₈	1	1	yes	characteristic
59	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
60	C₂²	—	508	1	no	—
61	C₂²	D₁₀₁₆	1	1	yes	characteristic
62	C₄	—	508	1	no	—
63	C₄	—	508	2	no	—
64	C₄	D₁₀₁₆	1	1	yes	characteristic
65	C₄	C₂×D₅₀₈	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	1016	1	no	—
68	C₂	C₂×D₁₀₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₀₁₆
commutator	C₈×C₁₂₇	C₂³
Frattini	C₈	C₂²×D₁₂₇
Fitting	M₃₂×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₈
hypercenter	M₃₂	D₁₂₇
hypocenter	C₁₂₇	#510
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₅₀₈	C₂	semidirect
3. Dic₁₀₁₆	C₂	semidirect
4. #16861154	C₂	semidirect
5. M₃₂×C₁₂₇	C₂	semidirect
34. C₁₂₇	#510	semidirect
40. M₃₂	D₁₂₇	semidirect