Subgroups — #20289880

Order 6784.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811552	C₂	1	1	yes	maximal, characteristic
2	#16811550	C₂	1	1	yes	maximal, characteristic
3	#16810456	C₂	1	1	yes	maximal, characteristic
4	#13133288	—	2	1	no	—
5	#13133286	—	2	1	no	—
6	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
7	#13133123	C₄	1	1	yes	characteristic, complemented
8	C₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
9	C₄⋊#6157	—	4	1	no	—
10	#11719936	—	4	1	no	—
11	C₈×D₅₃	D₄	1	1	yes	characteristic
12	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	#11719908	Q₈	1	1	yes	characteristic
14	C₂×#6157	—	8	1	no	—
15	C₄×D₅₃	D₈	1	1	yes	characteristic
16	#88324	—	8	1	no	—
17	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	#88322	Q₁₆	1	1	yes	characteristic
19	C₂×D₅₃	SD₃₂	1	1	yes	characteristic
20	#6157	—	16	1	no	—
21	C₄×C₅₃	#107	1	1	yes	characteristic
22	Dic₅₃	SD₃₂	1	1	yes	characteristic
23	#1426	—	53	1	no	maximal, Sylow
24	C₂×C₅₃	#367	1	1	yes	characteristic
25	D₅₃	—	2	1	no	—
26	M₆₄	—	53	1	no	—
27	#368	—	53	1	no	—
28	#367	—	53	1	no	—
29	C₅₃	#1426	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₁₆	—	53	1	no	—
31	#108	—	106	1	no	—
32	#107	—	106	1	no	—
33	C₃₂	—	53	1	no	—
34	C₃₂	#6157	1	1	yes	characteristic, complemented
35	M₁₆	—	212	1	no	—
36	C₂×C₈	—	53	1	no	—
37	C₄⋊C₄	—	212	1	no	—
38	C₁₆	—	53	1	no	—
39	C₁₆	C₂×#6157	1	1	yes	characteristic
40	C₂×C₄	—	53	1	no	—
41	C₂×C₄	—	424	1	no	—
42	C₈	—	53	1	no	—
43	C₈	—	424	1	no	—
44	C₈	C₄⋊#6157	1	1	yes	characteristic
45	C₂²	—	53	1	no	—
46	C₄	—	53	1	no	—
47	C₄	—	848	1	no	—
48	C₄	#13133286	1	1	yes	characteristic
49	C₂	—	106	1	no	—
50	C₂	#16811550	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811550
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₁₆	C₂×#6157
Fitting	C₃₂×C₅₃	C₄
socle	C₂×C₅₃	#367
hypercenter	C₃₂	#6157
hypocenter	C₅₃	#1426
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
7. #13133123	C₄	semidirect
8. C₃₂×C₅₃	C₄	semidirect
29. C₅₃	#1426	semidirect
34. C₃₂	#6157	semidirect