This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a consequence, many Gromov-Pyatetski-Shapiro and Agol-Belolipetsky-Thomson non-arithmetic manifolds embed geodesically. Moreover, we show that the number of commensurability classes of hyperbolic manifolds with a representative of volume ≤ v that bounds geometrically is at least v Cv , for v large enough.A. K. and S. R. were supported by the SNSF project no.