The processes X(3872)→D^{*0}D[over ¯]^{0}+c.c.,γJ/ψ,γψ(2S), and γD^{+}D^{-} are searched for in a 9.0 fb^{-1} data sample collected at center-of-mass energies between 4.178 and 4.278 GeV with the BESIII detector. We observe X(3872)→D^{*0}D^{0}[over ¯]+c.c. and find evidence for X(3872)→γJ/ψ with statistical significances of 7.4σ and 3.5σ, respectively. No evident signals for X(3872)→γψ(2S) and γD^{+}D^{-} are found, and the upper limit on the relative branching ratio R_{γψ}≡{B[X(3872)→γψ(2S)]}/{B[X(3872)→γJ/ψ]}<0.59 is set at 90% confidence level. Measurements of branching ratios relative to decay X(3872)→π^{+}π^{-}J/ψ are also reported for decays X(3872)→D^{*0}D^{0}[over ¯]+c.c.,γψ(2S),γJ/ψ, and γD^{+}D^{-}, as well as the non-D^{*0}D^{0}[over ¯] three-body decays π^{0}D^{0}D^{0}[over ¯] and γD^{0}D^{0}[over ¯].