Subgroups — #20270010

Order 6752.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16809962	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₂₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₈₄₄	C₂	1	2	yes	maximal, complemented
4	#16809951	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₄₂₂	—	2	1	no	—
7	Dic₄₂₂	C₂²	1	2	yes	—
8	Dic₄₂₂	C₂²	1	1	yes	characteristic
9	D₈₄₄	C₂²	1	1	yes	characteristic
10	C₂²⋊D₂₁₁	—	2	1	no	—
11	C₂×Dic₂₁₁	—	2	1	no	—
12	C₄×D₂₁₁	—	2	1	no	—
13	C₂×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₈×C₂₁₁	C₂²	1	2	yes	—
15	C₂²×C₂₁₁	D₄	1	1	yes	characteristic
16	C₂×D₂₁₁	—	2	1	no	—
17	C₄×C₂₁₁	D₄	1	1	yes	characteristic
18	C₄×C₂₁₁	C₂³	1	1	yes	characteristic
19	Dic₂₁₁	—	2	1	no	—
20	Dic₂₁₁	—	2	2	no	—
21	C₂×C₂₁₁	—	2	1	no	—
22	C₂×C₂₁₁	C₂×D₄	1	1	yes	characteristic
23	D₂₁₁	—	4	1	no	—
24	C₂₁₁	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	211	1	no	maximal, Sylow
26	#41	—	211	1	no	—
27	C₂×Q₈	—	211	1	no	—
28	Q₁₆	—	211	2	no	—
29	SD₁₆	—	211	2	no	—
30	M₁₆	D₂₁₁	1	1	yes	characteristic, complemented
31	Q₈	—	422	1	no	—
32	Q₈	—	211	2	no	—
33	Q₈	—	211	1	no	—
34	D₄	—	422	1	no	—
35	D₄	—	211	1	no	—
36	C₂×C₄	—	422	1	no	—
37	C₂×C₄	—	422	1	no	—
38	C₂×C₄	C₂×D₂₁₁	1	1	yes	characteristic
39	C₈	C₂×D₂₁₁	1	2	yes	—
40	C₂²	—	422	1	no	—
41	C₂²	D₈₄₄	1	1	yes	characteristic
42	C₄	—	422	1	no	—
43	C₄	—	422	2	no	—
44	C₄	D₈₄₄	1	1	yes	characteristic
45	C₄	C₂²×D₂₁₁	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	844	1	no	—
48	C₂	C₂×D₈₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈₄₄
commutator	C₄×C₂₁₁	C₂³
Frattini	C₄	C₂²×D₂₁₁
Fitting	M₁₆×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₂×D₄
hypercenter	M₁₆	D₂₁₁
hypocenter	C₂₁₁	#137
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₄₂₂	C₂	semidirect
3. Dic₈₄₄	C₂	semidirect
4. #16809951	C₂	semidirect
5. M₁₆×C₂₁₁	C₂	semidirect
24. C₂₁₁	#137	semidirect
30. M₁₆	D₂₁₁	semidirect