Subgroups — #20272921

Order 6784.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810453	C₂	1	1	yes	maximal, characteristic
2	#16810451	C₂	1	1	yes	maximal, characteristic
3	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
4	#13133153	—	2	1	no	—
5	#13133151	—	2	1	no	—
6	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₃₂×C₅₃	C₄	1	2	yes	complemented
8	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
9	C₄⋊Dic₅₃	—	4	1	no	—
10	#11719917	—	4	1	no	—
11	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₅₃	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
14	C₂×Dic₅₃	—	8	1	no	—
15	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
16	C₈×C₅₃	Q₁₆	1	1	yes	characteristic
17	#88322	—	8	1	no	—
18	C₂²×C₅₃	SD₃₂	1	1	yes	characteristic
19	C₄×C₅₃	#107	1	1	yes	characteristic
20	C₄×C₅₃	SD₃₂	1	1	yes	characteristic
21	Dic₅₃	—	16	1	no	—
22	#1426	—	53	1	no	maximal, Sylow
23	C₂×C₅₃	—	2	1	no	—
24	C₂×C₅₃	#367	1	1	yes	characteristic
25	M₆₄	D₅₃	1	1	yes	characteristic
26	#368	—	53	1	no	—
27	#367	—	53	1	no	—
28	C₅₃	#1426	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
30	#108	—	106	1	no	—
31	#107	—	106	1	no	—
32	C₃₂	Dic₅₃	1	2	yes	complemented
33	M₁₆	—	212	1	no	—
34	C₂×C₈	D₂₁₂	1	1	yes	characteristic
35	C₄⋊C₄	—	212	1	no	—
36	C₁₆	Dic₁₀₆	1	1	yes	characteristic
37	C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
38	C₂×C₄	—	424	1	no	—
39	C₂×C₄	D₄₂₄	1	1	yes	characteristic
40	C₈	—	424	1	no	—
41	C₈	Dic₂₁₂	1	1	yes	characteristic
42	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
43	C₂²	#13133162	1	1	yes	characteristic
44	C₄	—	848	1	no	—
45	C₄	#13133151	1	1	yes	characteristic
46	C₄	#13133162	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#16810451	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810451
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₅₃×M₆₄	C₂
socle	C₂×C₅₃	#367
hypercenter	M₆₄	D₅₃
hypocenter	C₅₃	#1426
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₅₃	C₄	semidirect
28. C₅₃	#1426	semidirect
32. C₃₂	Dic₅₃	semidirect