  We develop a theory of right group-like projections in Hopf algebras linking
them with the theory of left coideal subalgebras with two sided counital
integrals. Every right group-like projection is associated with a left coideal
subalgebra, maximal among the ones containing the given group-like projection
as an integral, and we show that that subalgebra is finite dimensional. We
observe that in a semisimple Hopf algebra $H$ every left coideal subalgebra has
an integral and we prove a 1-1 correspondence between right group-like
projections and left coideal subalgebras of $H$. We provide a number of
equivalent conditions for a right group-like projections to be left group-like
projection and prove a 1-1 correspondence between semisimple left coideal
subalgebras preserved by the squared antipode and two sided group-like
projections.
  We also classify left coideal subalgebras in Taft Hopf algebras $H_{n^2}$
over a field $\mathbb{k}$, showing that the automorphism group splits them into
  - a class of cardinality $|\mathbb{k}|-1$ of semisimple ones which correspond
to right group-like projections which are not two sided;
  - finitely many semisimple singletons, each corresponding to two sided
group-like projection; the number of those singletons for $H_{n^2}$ is equal to
the number of divisors of $n$;
  - finitely many singletons, each non-semisimple and admitting no right
group-like projection; the number of those singletons for $H_{n^2}$ is equal to
the number of divisors of $n$.
  In particular we answer the question of Landstad and Van Daele showing that
there do exist right group-like projections which are not left group-like
projections.
