Subgroups — #20289741

Order 6784.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811484	C₂	1	1	yes	maximal, characteristic
2	#16811484	C₂	1	1	yes	maximal, characteristic
3	#16810398	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88324	—	2	1	no	—
5	C₄⋊#88324	—	2	1	no	—
6	C₂×#11719908	C₄	1	1	yes	characteristic
7	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂×#88324	—	4	1	no	—
10	C₂×#88324	—	4	1	no	—
11	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₄×Dic₅₃	D₄	1	1	yes	characteristic
13	C₂×#88322	Q₈	1	1	yes	characteristic
14	C₁₆×C₅₃	—	2	1	no	—
15	#11719908	C₈	1	2	yes	complemented
16	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
17	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
18	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
19	#88324	—	8	1	no	—
20	#88324	—	8	1	no	—
21	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
22	C₈×C₅₃	M₁₆	1	1	yes	characteristic
23	#88322	—	2	1	no	—
24	C₂²×C₅₃	#106	1	1	yes	characteristic
25	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
26	C₄×C₅₃	#108	1	1	yes	characteristic
27	Dic₅₃	—	2	1	no	—
28	Dic₅₃	—	2	1	no	—
29	#1368	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	#334	1	1	yes	characteristic
31	C₂×C₅₃	#365	1	1	yes	characteristic
32	C₂×C₅₃	#365	1	1	yes	characteristic
33	#346	—	53	1	no	—
34	#334	—	53	1	no	—
35	#334	—	53	1	no	—
36	C₅₃	#1368	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₁₆	—	53	1	no	—
38	C₂×C₁₆	#6157	1	1	yes	characteristic
39	C₄⋊C₈	—	106	1	no	—
40	C₄⋊C₈	—	106	1	no	—
41	C₄×C₈	—	53	1	no	—
42	C₂×C₈	—	53	1	no	—
43	C₂×C₈	—	212	1	no	—
44	C₂×C₈	—	212	1	no	—
45	C₂×C₈	C₂×#6157	1	1	yes	characteristic
46	C₄²	—	53	1	no	—
47	C₁₆	—	2	1	no	—
48	C₁₆	—	53	2	no	—
49	C₂×C₄	—	53	1	no	—
50	C₂×C₄	—	53	1	no	—
51	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
52	C₈	—	106	1	no	—
53	C₈	—	424	1	no	—
54	C₈	—	424	1	no	—
55	C₈	D₅₃⋊C₈	1	1	yes	characteristic
56	C₈	#11719940	1	1	yes	characteristic
57	C₂²	#13133285	1	1	yes	characteristic
58	C₄	—	106	1	no	—
59	C₄	—	106	1	no	—
60	C₄	#13133284	1	1	yes	characteristic
61	C₄	#13133288	1	1	yes	characteristic
62	C₂	#16811485	1	1	yes	characteristic
63	C₂	#16811547	1	1	yes	characteristic
64	C₂	#16811548	1	1	yes	characteristic

	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
15. #11719908	C₈	semidirect
36. C₅₃	#1368	semidirect