Subgroups — #20276994

Order 6784.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811025	C₂	1	2	yes	maximal
2	#16810451	C₂	1	2	yes	maximal
3	#16810449	C₂	1	2	yes	maximal
4	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#13133209	—	2	2	no	—
6	#13133151	C₂²	1	4	yes	—
7	#13133149	—	2	2	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
13	C₄²×C₅₃	D₄	1	1	yes	characteristic
14	C₄⋊Dic₅₃	—	2	4	no	—
15	#11719919	—	4	4	no	—
16	C₁₆×C₅₃	Q₈	1	2	yes	—
17	C₁₆×C₅₃	Q₈	1	2	yes	—
18	C₂×C₄×C₅₃	D₈	1	2	yes	—
19	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
20	C₂×Dic₅₃	—	4	4	no	—
21	C₈×C₅₃	Q₁₆	1	2	yes	—
22	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
23	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
25	C₄×C₅₃	—	2	2	no	—
26	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₅₃	—	8	4	no	—
29	#2253	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	#500	1	1	yes	characteristic
31	C₂×C₅₃	#508	1	2	yes	—
32	#499	—	53	2	no	—
33	#367	—	53	2	no	—
34	#366	—	53	2	no	—
35	C₄×C₁₆	D₅₃	1	1	yes	characteristic
36	C₅₃	#2253	1	1	yes	characteristic, complemented, Sylow
37	#125	—	106	2	no	—
38	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
39	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
40	#107	—	53	4	no	—
41	#106	—	106	2	no	—
42	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
43	C₂×C₈	D₂₁₂	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
45	C₄⋊C₄	—	106	4	no	—
46	C₄⋊C₄	—	212	4	no	—
47	C₄²	D₂₁₂	1	1	yes	characteristic
48	C₁₆	Dic₁₀₆	1	2	yes	—
49	C₁₆	Dic₁₀₆	1	2	yes	—
50	C₂×C₄	—	212	4	no	—
51	C₂×C₄	D₄₂₄	1	2	yes	—
52	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
53	C₈	Dic₂₁₂	1	2	yes	—
54	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
55	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
56	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	424	4	no	—
59	C₄	#13133217	1	1	yes	characteristic
60	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
61	C₂	#16811028	1	1	yes	characteristic
62	C₂	#16811055	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#2253
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
36. C₅₃	#2253	semidirect