Cyclic subgroup of a cyclic group

Is there any non trivial example of subgroup of a cyclic group?

Every subgroup of a cyclic group is cyclic. I couldnt find any non trivial examples of it since every subgroup would have the generator and if the generator is there then the entire group is there.

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$\{0 + 4\mathbb Z, 2 + 4\mathbb Z\} \subseteq Z/4\mathbb Z$ for a finite example, or think of $n\mathbb Z \subseteq \mathbb Z$. –  martini Dec 10 '13 at 12:43
Wow i totally missed that thanks. –  vinothkr Dec 10 '13 at 12:44
And from the answer below, you conclude that if the order is prime, there is no proper subgroup at all. –  Jean-Claude Arbaut Dec 10 '13 at 12:46
That was going to be my next question. You read my mind :) –  vinothkr Dec 10 '13 at 12:47

Not every subgroup has to contain the generator. Consider the cyclic group of order $4$, which we denote as $\{1,a,a^2,a^3\}$. Then $\{1,a^2\}$ is a proper subgroup.
More generally, if $\langle a \rangle$ is a cyclic group of order $x$ and $d$ divides $x$, we have a cyclic subgroup $\langle a^d \rangle$.
If your group is $G=(\mathbb{Z},+)$, your generator is $1$. If your subgroup is $H=(2\mathbb{Z},+)$, the generator is $2$ which is not the generator for $G$.