  Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a
subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for
every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in
G$, there exists $c_1\in G$ satisfying that $c_{1|[1,i]}=c_{|[1,i]}$, $supp
(c_1)\subseteq [1,n_i]$, and order$(c_1)$ divides order$(c_{|[1,n_i]})$. In
this paper we investigate the structure of order controllable group codes. It
is proved that if $G$ is an order controllable, shift invariant, group code
over a finite abelian group $H$, then $G$ possesses a finite canonical
generating set. Furthermore, our construction also yields that $G$ is
algebraically conjugate to a full group shift.
