A series of new flow observables mixed harmonic multiparticle cumulants, which allow for the first time to quantify the strength of the correlation between different orders of flow coefficients with various moments, was investigated using the hydrodynamic model. These new observables are constructed based on multiparticle cumulants and, thus, by design, will be less sensitive to the nonflow contaminations. In addition to the previous study of correlation involving two flow coefficients with their second moments both correlations of three flow coefficients and the correlations of higher-order moments of v(2) and v(3) are systematically investigated using the iEBE-VISHNU hybrid model with two different initial conditions, AMPT and TRENTo, respectively. These systematic studies using hydrodynamic models will significantly improve the understanding of the correlations between different orders (and moments) of flow coefficients. Furthermore, the hydrodynamic predictions shown in this paper and the future comparisons to experimental measurements will provide more constraints on theoretical models and extract more information about the transport properties of the quark-gluon plasma created in heavy-ion collisions.