  In this paper, we identify many important properties and develop criteria for
the existence of subquasigroups in finite quasigroups. Based on these results,
we propose an effective method that concludes the nonexistence of subquasigroup
of a finite quasigroup, otherwise finds its all possible proper subquasigroups.
This has an important application in checking the cryptographic suitability of
a finite quasigroup. \par Further, we propose a binary operation using
arithmetic of finite fields to construct quasigroups of order $p^r$. We develop
the criteria under which these quasigroups have desirable cryptographic
properties, viz. polynomially completeness and possessing no proper
subquasigroups. Then a practical method is given to construct cryptographically
suitable quasigroups. We also illustrate these methods by some academic
examples and implement all proposed algorithms in the computer algebra system
{\sc{Singular}}.
