In order to understand the unusual magnetic phase of multiferroic hexagonal manganites, we have investigated the doping effects of hexagonal manganites ${\text{YMn}}_{0.9}{X}_{0.1}{\text{O}}_3$ ($X=\text{Al}$, Ru, and Zn) by using bulk measurements as well as neutron-diffraction experiments. We have also re-examined the magnetic structure of ${\text{YMnO}}_3$ to find that it has mainly the ${\Gamma }_3$ representation with an about 18% mixing of the ${\Gamma }_4$ representation. Upon doping nonmagnetic atoms such as Al, Ru, and Zn at the Mn site, the mixing of the two representations gets increased with an angle growing between the magnetic moment and the crystallographic $a$ axis $\left(\varphi \right)$; for example, $\varphi$ changes from $10°$ for ${\text{YMnO}}_3$ to $45°$ for ${\text{YMn}}_{0.9}{\text{Zn}}_{0.1}{\text{O}}_3$. At the same time, the ordered moment becomes significantly reduced with doping, while the transition temperature is suppressed slightly.