  In the paper "The Steenrod algebra and its dual", J.Milnor determined the
structure of the dual Steenrod algebra which is a graded commutative Hopf
algebra of finite type. We consider the affine group scheme $G_p$ represented
by the dual Hopf algebra of the mod $p$ Steenrod algebra. Then, $G_p$ assigns a
graded commutative algebra $A_*$ over a prime field of finite characteristic
$p$ to a set of isomorphisms of the additive formal group law over $A_*$, whose
group structure is given by the composition of formal power series. The aim of
this paper is to show some group theoretic properties of $G_p$ by making use of
this presentation of $G_p(A_*)$. We give a decreasing filtration of subgroup
schemes of $G_p$ which we use for estimating the length of the lower central
series of finite subgroup schemes of $G_p$. We also give a successive quotient
maps
$G_p\xrightarrow{\rho_0}G_p^{\langle1\rangle}\xrightarrow{\rho_1}G_p^{\langle2\rangle}\xrightarrow{\rho_2}\cdots\xrightarrow{\rho_{k-1}}
G_p^{\langle k\rangle}\xrightarrow{\rho_k}G_p^{\langle
k+1\rangle}\xrightarrow{\rho_{k+1}}\cdots$ of affine group schemes over a prime
field ${\boldsymbol F}_p$ such that the kernel of $\rho_k$ is a maximal abelian
subgroup.
