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C-projective structures are analogues of projective structures in the complexsetting. The maximal dimension of the Lie algebra of c-projective symmetries ofa complex connection on an almost complex manifold of C-dimension $n>1$ isclassically known to be $2n^2+4n$. We prove that the submaximal dimension isequal to $2n^2-2n+4+2\delta_{3,n}$. If the complex connection is minimal(encoded as a normal parabolic geometry), the harmonic curvature of thec-projective structure has three components and we specify the submaximalsymmetry dimensions and the corresponding geometric models for each of thesethree pure curvature types. If the connection is non-minimal, we introduce amodified normalization condition on the parabolic geometry and use this toresolve the symmetry gap problem. We prove that the submaximal symmetrydimension in the class of Levi-Civita connections for pseudo-K\"ahler metricsis $2n^2-2n+4$, and specializing to the K\"ahler case, we obtain $2n^2-2n+3$.This resolves the symmetry gap problem for metrizable c-projective structures.

Submaximally symmetric c-projective structures