Subgroups — #20276990

Order 6784.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811028	C₂	1	2	yes	maximal
2	#16810449	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133217	—	2	2	no	—
5	#13133151	—	2	2	no	—
6	#13133151	C₂²	1	4	yes	—
7	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀₆	—	4	2	no	—
10	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
11	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
12	C₄²×C₅₃	D₄	1	1	yes	characteristic
13	C₄⋊Dic₅₃	—	2	4	no	—
14	C₄⋊Dic₅₃	—	4	2	no	—
15	C₁₆×C₅₃	Q₈	1	4	yes	complemented
16	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
17	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
18	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
19	C₂×Dic₅₃	—	4	4	no	—
20	Dic₁₀₆	—	8	4	no	—
21	C₈×C₅₃	Q₁₆	1	2	yes	—
22	C₈×C₅₃	C₂×Q₈	1	2	yes	—
23	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
24	C₄×C₅₃	D₁₆	1	2	yes	—
25	C₄×C₅₃	Q₃₂	1	2	yes	—
26	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₅₃	—	8	4	no	—
29	#2252	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	#500	1	1	yes	characteristic
31	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
32	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
33	#500	—	53	2	no	—
34	#366	—	53	4	no	—
35	C₄×C₁₆	D₅₃	1	1	yes	characteristic
36	C₅₃	#2252	1	1	yes	characteristic, complemented, Sylow
37	C₄⋊Q₈	—	106	2	no	—
38	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
39	#107	—	53	4	no	—
40	#107	—	106	2	no	—
41	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
42	C₂×Q₈	—	212	2	no	—
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	2	no	—
47	C₄²	D₂₁₂	1	1	yes	characteristic
48	C₁₆	Dic₁₀₆	1	4	yes	complemented
49	Q₈	—	424	4	no	—
50	C₂×C₄	—	212	4	no	—
51	C₂×C₄	D₄₂₄	1	1	yes	characteristic
52	C₂×C₄	D₄₂₄	1	1	yes	characteristic
53	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
54	C₈	Dic₂₁₂	1	2	yes	—
55	C₈	C₂×Dic₁₀₆	1	2	yes	—
56	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
57	C₄	—	424	4	no	—
58	C₄	D₈₄₈	1	2	yes	—
59	C₄	Dic₄₂₄	1	2	yes	—
60	C₄	#13133217	1	1	yes	characteristic
61	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
62	C₂	#16811028	1	1	yes	characteristic
63	C₂	C₂×D₈₄₈	1	1	yes	characteristic
64	C₂	C₂×Dic₄₂₄	1	1	yes	characteristic

	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	#2252
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
15. C₁₆×C₅₃	Q₈	semidirect
36. C₅₃	#2252	semidirect
48. C₁₆	Dic₁₀₆	semidirect