Let G be a connected simply connected nilpotent Lie group with discrete uniform subgroup \(\Gamma \). A connected closed subgroup H of G is called \(\Gamma \)-rational if \(H\cap \Gamma \) is a discrete uniform subgroup of H. For a closed connected subgroup H of G, let \(\mathcal {I}(H, \Gamma )\) denote the identity component of the closure of the subgroup generated by H and \(\Gamma \). In this paper, we prove that \(\mathcal {I}(H, \Gamma )\) is the smallest normal \(\Gamma \)-rational connected closed subgroup containing H. As an immediate consequence, we obtain that \(\mathcal {I}(H, \Gamma )\) depends only on the commensurability class of \(\Gamma \). As applications, we give two results. In the first, we determine explicitly the smallest \(\Gamma \)-rational connected closed subgroup containing H. The second is a characterization of ergodicity of nilflow \( (G/\Gamma , H)\) in terms of \(\mathcal {I}(H, \Gamma )\). Furthermore, a characterization of the irreducible unitary representations of G for which the restriction to \(\Gamma \) remain irreducible is given.