  For each submanifold of a stratified group, we find a number and a measure
only depending on its tangent bundle, the grading and the fixed Riemannian
metric. In two step stratified groups, we show that such number and measure
coincide with the Hausdorff dimension and with the spherical Hausdorff measure
of the submanifold with respect to the Carnot-Caratheodory distance,
respectively. Our main technical tool is an intrinsic blow-up at points of
maximum degree. We also show that the intrinsic tangent cone to the submanifold
at these points is always a subgroup. Finally, by direct computations in the
Engel group, we show how our results can be extended to higher step stratified
groups, provided the submanifold is sufficiently regular.
