Letting
$p$
vary over all primes and
$E$
vary over all elliptic curves over the finite field
${{\mathbb{F}}_{p}}$
, we study the frequency to which a given group
$G$
arises as a group of points
$E\left( {{\mathbb{F}}_{p}} \right)$
. It is well known that the only permissible groups are of the form
${{G}_{m,\,k}}\,:=\,\mathbb{Z}\,/m\mathbb{Z}\,\times \,\mathbb{Z}/mk\mathbb{Z}$
. Given such a candidate group, we let
$M\left( {{G}_{m,\,k}} \right)$
be the frequency to which the group
${{G}_{m,\,k}}$
arises in this way. Previously, C.David and E. Smith determined an asymptotic formula for
$M\left( {{G}_{m,\,k}} \right)$
assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for
$M\left( {{G}_{m,\,k}} \right)$
, pointwise and on average. In particular, we show that
$M\left( {{G}_{m,\,k}} \right)$
is bounded above by a constant multiple of the expected quantity when
$m\,\le \,{{k}^{A}}$
and that the conjectured asymptotic for
$M\left( {{G}_{m,\,k}} \right)$
holds for almost all groups
${{G}_{m,\,k}}$
when
$m\,\le \,{{k}^{1/4-\in }}$
. We also apply our methods to study the frequency to which a given integer
$N$
arises as a group order
$\#E\left( {{\mathbb{F}}_{p}} \right)$
.