  A well-known classical result states that $c_0$ is linearly embeddable in a
Banach lattice if and only if it is lattice embeddable. Improving results of
H.P.~Lotz, H.P.~Rosenthal and N.~Ghoussoub, we prove that $C[0,1]$ shares this
property with $c_0$. Furthermore, we show that any infinite-dimensional
sublattice of $C[0,1]$ is either lattice isomorphic to $c_0$ or contains a
sublattice isomorphic to $C[0,1]$. As a consequence, it is proved that for a
separable Banach lattice $X$ the following conditions are equivalent:
  (1) $X$ is linearly embeddable in a Banach lattice if and only if it is
lattice embeddable;
  (2) $X$ is lattice embeddable into $C[0,1]$.
