  Fix an odd prime $p$ and let $X$ be the $p$-localization of a finite
suspended $CW$-complex. Given certain conditions on the reduced mod-$p$
homology $\bar H_*(X;\zmodp)$ of $X$, we use a decomposition of $\Omega\Sigma
X$ due to the second author and computations in modular representation theory
to show there are arbitrarily large integers $i$ such that $\Omega\Sigma^i X$
is a homotopy retract of $\Omega\Sigma X$. This implies the stable homotopy
groups of $\Sigma X$ are in a certain sense retracts of the unstable homotopy
groups, and by a result of Stanley, one can confirm the Moore conjecture for
$\Sigma X$. Under additional assumptions on $\bar H_*(X;\zmodp)$, we generalize
a result of Cohen and Neisendorfer to produce a homotopy decomposition of
$\Omega\Sigma X$ that has infinitely many finite $H$-spaces as factors.
