Subgroups — #20272897

Order 6784.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#16810449	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₁₂	—	2	1	no	—
5	Dic₄₂₄	—	2	1	no	—
6	Dic₄₂₄	C₄	1	2	yes	complemented
7	#13133151	—	2	1	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₃₂×C₅₃	—	2	1	no	—
10	C₂×Dic₁₀₆	—	4	1	no	—
11	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
12	C₄⋊Dic₅₃	—	4	1	no	—
13	Dic₂₁₂	—	2	2	no	—
14	Dic₂₁₂	—	4	1	no	—
15	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
18	C₂×Dic₅₃	—	8	1	no	—
19	C₂×Dic₅₃	—	8	1	no	—
20	Dic₁₀₆	—	4	2	no	—
21	Dic₁₀₆	—	8	1	no	—
22	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
25	C₄×C₅₃	#102	1	1	yes	characteristic
26	C₄×C₅₃	SD₃₂	1	1	yes	characteristic
27	Dic₅₃	—	8	2	no	—
28	Dic₅₃	—	16	1	no	—
29	#1416	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	#357	1	1	yes	characteristic
31	C₂×C₅₃	SD₆₄	1	1	yes	characteristic
32	C₂×C₅₃	Q₆₄	1	1	yes	characteristic
33	C₂×Q₃₂	—	53	1	no	—
34	C₂×C₃₂	D₅₃	1	1	yes	characteristic
35	#366	—	53	1	no	—
36	C₅₃	#1416	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₁₆	—	106	1	no	—
38	Q₃₂	—	53	2	no	—
39	Q₃₂	—	106	1	no	—
40	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
41	#107	—	106	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×Q₈	—	212	1	no	—
44	Q₁₆	—	106	2	no	—
45	Q₁₆	—	212	1	no	—
46	C₂×C₈	D₂₁₂	1	1	yes	characteristic
47	C₄⋊C₄	—	212	1	no	—
48	C₁₆	C₄×D₅₃	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
50	Q₈	—	424	1	no	—
51	Q₈	—	212	2	no	—
52	C₂×C₄	—	424	1	no	—
53	C₂×C₄	—	424	1	no	—
54	C₂×C₄	D₄₂₄	1	1	yes	characteristic
55	C₈	#11719913	1	1	yes	characteristic
56	C₈	#11719921	1	1	yes	characteristic
57	C₂²	D₈₄₈	1	1	yes	characteristic
58	C₄	—	848	1	no	—
59	C₄	—	424	2	no	—
60	C₄	#13133139	1	1	yes	characteristic
61	C₄	#13133162	1	1	yes	characteristic
62	C₂	#16810425	1	1	yes	characteristic
63	C₂	#16810462	1	1	yes	characteristic
64	C₂	Dic₈₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈₄₈
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₂×C₃₂×C₅₃	C₂
socle	C₂²×C₅₃	D₁₆
hypercenter	C₂×C₃₂	D₅₃
hypocenter	C₅₃	#1416
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
6. Dic₄₂₄	C₄	semidirect
36. C₅₃	#1416	semidirect