Subgroups — #20272781

Order 6784.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810374	C₂	1	2	yes	maximal
2	C₅₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#88322	—	2	2	no	—
4	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
8	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
9	C₄²×C₅₃	D₄	1	1	yes	characteristic
10	C₂×#88322	—	4	2	no	—
11	C₁₆×C₅₃	—	2	1	no	—
12	C₁₆×C₅₃	C₈	1	2	yes	complemented
13	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
15	C₈×C₅₃	—	2	1	no	—
16	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₈×C₅₃	M₁₆	1	1	yes	characteristic
18	#88322	—	8	2	no	—
19	C₂²×C₅₃	#106	1	1	yes	characteristic
20	C₄×C₅₃	—	2	2	no	—
21	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₅₃	#108	1	1	yes	characteristic
23	#1368	—	53	1	no	maximal, Sylow
24	C₂×C₅₃	#334	1	1	yes	characteristic
25	C₂×C₅₃	#365	1	2	yes	—
26	#346	D₅₃	1	1	yes	characteristic
27	#334	—	53	2	no	—
28	C₅₃	#1368	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
30	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
31	C₄⋊C₈	—	106	2	no	—
32	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
33	C₂×C₈	—	212	2	no	—
34	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
35	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
36	C₄²	D₂₁₂	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	#88322	1	2	yes	complemented
39	C₂×C₄	#11719913	1	2	yes	—
40	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
41	C₈	—	2	1	no	—
42	C₈	—	424	2	no	—
43	C₈	C₂×#88322	1	1	yes	characteristic
44	C₈	#11719917	1	1	yes	characteristic
45	C₂²	#13133149	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄⋊#88322	1	1	yes	characteristic
48	C₄	#13133153	1	1	yes	characteristic
49	C₂	#16810374	1	1	yes	characteristic
50	C₂	#16810446	1	2	yes	—

	subgroup	G/H
center	C₂²	#13133149
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂²×C₅₃	#106
hypercenter	#346	D₅₃
hypocenter	C₅₃	#1368
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₅₃	C₈	semidirect
28. C₅₃	#1368	semidirect
38. C₁₆	#88322	semidirect