Every positive divisor of the order of a finite cyclic group occurs as the order of a subgroup

Given: A finite cyclic group $G=⟨g⟩$ of order $n$, and positive integers $d,q$ satisfying $n=dq$.

The generator has $\mathrm{ord}(g)=\mathrm{\mid }G\mathrm{\mid }=n$, and $g^k=e$ exactly when $n\mid k$ (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity, If $\mathrm{ord}(g)=n$ then $g^k=e$ iff $k$ is an integer multiple of $n$, the powers $g^0,\dots ,g^{n-1}$ are distinct, and $⟨g⟩$ has exactly $n$ elements; if $g$ has infinite order then $g^j=g^k$ only for $j=k$).

Integer powers satisfy $(g^q{)}^k=g^{qk}$ (Powers $g^n$: natural exponents in a monoid and integer exponents in a group, with $g^0=e$, Exponent laws in a group: $g^{m+n}=g^m{g}^n$ and $(g^m{)}^n=g^{mn}$ for all $m,n\in \mathbb{Z}$, and $(gh{)}^n=g^n{h}^n$ when $g$ and $h$ commute).

The generated set $⟨g^q⟩$ is a subgroup, and its cardinality is the order of $g^q$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups, $⟨g⟩=\{g^n:n\in \mathbb{Z}\}$, and every cyclic group is abelian, If $\mathrm{ord}(g)=n$ then $g^k=e$ iff $k$ is an integer multiple of $n$, the powers $g^0,\dots ,g^{n-1}$ are distinct, and $⟨g⟩$ has exactly $n$ elements; if $g$ has infinite order then $g^j=g^k$ only for $j=k$).

The canonical embedding preserves products and order; $\mathbb{Z}$ is an ordered commutative ring and multiplication by a nonzero integer is cancellative (The naturals embed in the integers, The integers form a commutative ring, The integers form a totally ordered ring, The integers have no zero divisors; multiplicative cancellation, Divisibility in $\mathbb{Z}$: $d\mid a$ when $a=dq$ for some integer $q$).

If a positive natural exponent $r$ satisfies $x^r=e$, then $x$ has finite order, and its order is the least such positive exponent and is at most $r$; if there is no such exponent, its order is $\mathrm{\infty }$ (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).