Subgroups — #20289749

Order 6784.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811486	C₂	1	2	yes	maximal
2	C₁₆×Dic₅₃	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	—	2	1	no	—
12	#11719908	C₈	1	2	yes	complemented
13	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
14	C₂×Dic₅₃	D₈	1	1	yes	characteristic
15	C₂×Dic₅₃	Q₁₆	1	1	yes	characteristic
16	#88324	—	8	2	no	—
17	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₈×C₅₃	M₁₆	1	1	yes	characteristic
19	#88322	—	2	1	no	—
20	C₂²×C₅₃	#107	1	1	yes	characteristic
21	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₅₃	#108	1	1	yes	characteristic
23	Dic₅₃	—	2	1	no	—
24	Dic₅₃	SD₃₂	1	2	yes	—
25	#1372	—	53	1	no	maximal, Sylow
26	C₂×C₅₃	#335	1	1	yes	characteristic
27	C₂×C₅₃	#367	1	1	yes	characteristic
28	C₂×C₅₃	#368	1	1	yes	characteristic
29	C₄×C₁₆	—	53	1	no	—
30	#335	—	53	2	no	—
31	C₅₃	#1372	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₁₆	—	53	1	no	—
33	C₂×C₁₆	#6157	1	1	yes	characteristic
34	C₄⋊C₈	—	106	2	no	—
35	C₄×C₈	—	53	1	no	—
36	C₂×C₈	—	53	1	no	—
37	C₂×C₈	—	212	2	no	—
38	C₂×C₈	C₂×#6157	1	1	yes	characteristic
39	C₄²	—	53	1	no	—
40	C₁₆	—	2	1	no	—
41	C₁₆	—	53	2	no	—
42	C₂×C₄	—	53	1	no	—
43	C₂×C₄	—	53	1	no	—
44	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
45	C₈	—	106	1	no	—
46	C₈	—	424	2	no	—
47	C₈	D₅₃⋊C₈	1	1	yes	characteristic
48	C₈	#11719940	1	1	yes	characteristic
49	C₂²	#13133286	1	1	yes	characteristic
50	C₄	—	53	2	no	—
51	C₄	—	106	1	no	—
52	C₄	#13133284	1	1	yes	characteristic
53	C₄	#13133287	1	1	yes	characteristic
54	C₂	#16811487	1	1	yes	characteristic
55	C₂	#16811550	1	1	yes	characteristic
56	C₂	#16811551	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133286
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₂×C₈	C₂×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#107
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	#1372
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
12. #11719908	C₈	semidirect
31. C₅₃	#1372	semidirect