Subgroups — #20272851

Order 6784.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810457	C₂	1	2	yes	maximal
2	C₅₃×#346	C₂	1	1	yes	maximal, characteristic
3	#13133123	C₄	1	4	yes	complemented
4	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₈×C₅₃	C₈	1	1	yes	characteristic
8	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₅₃	C₈	1	1	yes	characteristic
10	C₁₆×C₅₃	C₈	1	2	yes	—
11	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₅₃	—	2	1	no	—
13	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
14	C₈×C₅₃	—	2	1	no	—
15	C₈×C₅₃	C₄²	1	1	yes	characteristic
16	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₂²×C₅₃	M₃₂	1	1	yes	characteristic
18	C₄×C₅₃	—	4	1	no	—
19	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
20	C₄×C₅₃	M₃₂	1	1	yes	characteristic
21	#1398	—	53	1	no	maximal, Sylow
22	C₂×C₅₃	—	2	1	no	—
23	C₂×C₅₃	#346	1	1	yes	characteristic
24	M₆₄	—	53	2	no	—
25	#346	D₅₃	1	1	yes	characteristic
26	C₅₃	#1398	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
29	C₄×C₈	Dic₅₃	1	1	yes	characteristic
30	C₃₂	—	53	4	no	—
31	C₂×C₈	#88322	1	1	yes	characteristic
32	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
33	C₄²	#88322	1	1	yes	characteristic
34	C₁₆	#88322	1	2	yes	—
35	C₁₆	C₄×D₅₃	1	2	yes	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₂×#88322	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	C₂×#88322	1	1	yes	characteristic
40	C₈	C₄×Dic₅₃	1	1	yes	characteristic
41	C₂²	#13133157	1	1	yes	characteristic
42	C₄	—	4	1	no	—
43	C₄	C₄×#88322	1	1	yes	characteristic
44	C₄	#13133157	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#16810399	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂×C₅₃	#346
hypercenter	#346	D₅₃
hypocenter	C₅₃	#1398
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
3. #13133123	C₄	semidirect
26. C₅₃	#1398	semidirect