Subgroups — #20272783

Order 6784.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810444	C₂	1	2	yes	maximal
2	C₅₃×#346	C₂	1	1	yes	maximal, characteristic
3	#13133157	—	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	D₄	1	1	yes	characteristic
9	C₄²×C₅₃	Q₈	1	1	yes	characteristic
10	C₁₆×C₅₃	—	2	1	no	—
11	C₁₆×C₅₃	C₈	1	2	yes	—
12	#11719908	—	4	2	no	—
13	C₂×C₄×C₅₃	—	2	1	no	—
14	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
15	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₈×C₅₃	M₁₆	1	1	yes	characteristic
17	C₈×C₅₃	SD₁₆	1	2	yes	—
18	C₂²×C₅₃	#108	1	1	yes	characteristic
19	C₄×C₅₃	—	4	1	no	—
20	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₅₃	#106	1	1	yes	characteristic
22	#1369	—	53	1	no	maximal, Sylow
23	C₂×C₅₃	—	2	1	no	—
24	C₂×C₅₃	#334	1	1	yes	characteristic
25	#364	—	53	2	no	—
26	#346	D₅₃	1	1	yes	characteristic
27	C₅₃	#1369	1	1	yes	characteristic, complemented, Sylow
28	M₃₂	—	106	2	no	—
29	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
30	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
31	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
32	C₂×C₈	D₂₁₂	1	1	yes	characteristic
33	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
34	C₄²	Dic₁₀₆	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₁₆	—	212	2	no	—
37	C₁₆	#88322	1	2	yes	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
40	C₈	#11719913	1	2	yes	—
41	C₈	C₂×#88322	1	1	yes	characteristic
42	C₈	#11719917	1	1	yes	characteristic
43	C₂²	#13133153	1	1	yes	characteristic
44	C₄	—	4	1	no	—
45	C₄	C₄⋊#88322	1	1	yes	characteristic
46	C₄	#13133149	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#16810374	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133149
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂×C₅₃	#334
hypercenter	#346	D₅₃
hypocenter	C₅₃	#1369
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
27. C₅₃	#1369	semidirect