$\textbf{Cyclic groups}$ are groups in which every element is a power of some fixed element.
$\textbf{Definition:}$ Let $\text{G}$ be a group, $g \in \text{G}.$ Let $1$ be the identity element. The order of $g$ is the smallest positive integer $n$ such that $g^{n} = 1.$ If there is no positive integer $n$ such that $g^{n} = 1,$ then $g$ has infinite order.
$\textbf{Definition:}$ If $\text{G}$ is a group and $g \in \text{G},$ then the subgroup generated by $g$ is $\langle g \rangle = \{g^{n} | n \in \mathbf{Z}\}.$
$\textbf{Definition:}$ A group $\text{G}$ is cyclic if $\text{G} = \langle g \rangle$ for some $g \in \text{G}.\; g$ is a generator of $\langle g \rangle.$ If a generator $g$ has order $n, \text{G} = \langle g \rangle$ is cyclic of order $n.$ If a generator $g$ has infinite order, $\text{G} = \langle g \rangle$ is infinite cyclic.
Since, $\text{G}$ is a cyclic group with generator ‘$a$’, where Order of ‘$a$’ is $29.$ So, Order of $\text{G}$ is $29.$ By Lagrange theorem, Order of every subgroup of $\text{G}$ must divide the order of $\text{G}.$ So, only two possible orders of subgroups of $\text{G}$ are $1,29,$ as $29$ is prime number.
