Subgroups — #20289739

Order 6784.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811484	C₂	1	2	yes	maximal
2	#16810398	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#88324	—	2	2	no	—
4	C₂×#11719908	C₄	1	1	yes	characteristic
5	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#88324	—	4	2	no	—
8	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₄×Dic₅₃	D₄	1	1	yes	characteristic
10	C₂×#88322	Q₈	1	1	yes	characteristic
11	C₁₆×C₅₃	C₈	1	2	yes	complemented
12	#11719908	—	2	1	no	—
13	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
14	C₂×Dic₅₃	SD₁₆	1	2	yes	—
15	#88324	—	8	2	no	—
16	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₈×C₅₃	M₁₆	1	1	yes	characteristic
18	#88322	—	2	1	no	—
19	C₂²×C₅₃	#106	1	1	yes	characteristic
20	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₅₃	#108	1	1	yes	characteristic
22	Dic₅₃	—	2	2	no	—
23	#1368	—	53	1	no	maximal, Sylow
24	C₂×C₅₃	#334	1	1	yes	characteristic
25	C₂×C₅₃	#365	1	2	yes	—
26	#346	—	53	1	no	—
27	#334	—	53	2	no	—
28	C₅₃	#1368	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₁₆	—	53	1	no	—
30	C₂×C₁₆	#6157	1	1	yes	characteristic
31	C₄⋊C₈	—	106	2	no	—
32	C₄×C₈	—	53	1	no	—
33	C₂×C₈	—	53	1	no	—
34	C₂×C₈	—	212	2	no	—
35	C₂×C₈	C₂×#6157	1	1	yes	characteristic
36	C₄²	—	53	1	no	—
37	C₁₆	—	106	1	no	—
38	C₁₆	#88324	1	2	yes	complemented
39	C₂×C₄	—	53	2	no	—
40	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
41	C₈	—	106	1	no	—
42	C₈	—	424	2	no	—
43	C₈	#11719936	1	1	yes	characteristic
44	C₈	C₂×#88324	1	1	yes	characteristic
45	C₂²	#13133285	1	1	yes	characteristic
46	C₄	—	106	2	no	—
47	C₄	C₄⋊#88324	1	1	yes	characteristic
48	C₄	#13133287	1	1	yes	characteristic
49	C₂	#16811484	1	1	yes	characteristic
50	C₂	#16811547	1	2	yes	—

	subgroup	G/H
center	C₂²	#13133285
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₂×C₈	C₂×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#106
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	#1368
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
11. C₁₆×C₅₃	C₈	semidirect
28. C₅₃	#1368	semidirect
38. C₁₆	#88324	semidirect