Q:

Are normal subgroups subsets of each other?

I mean, let $G$ be a group and has normal subgroups $N_1,N_2,N_3,...$
Let $N:max(N_1,N_2,N_3,...)$
Is the following true :
$$\bigcup_{k=1}^{\infty} N_k = N$$
I'm asking this because I'm concerned with the definition of Maximal normal subgroup.
A proper normal subgroup of $G$ is called maximal if I can't find a proper normal subgroup of $G$ that contains it? However this doesn't mean I can't find another normal subgroup which has order larger than.

A:

In a general partially ordered set $(S,\leq)$, an element is called $x$ maximal if there does not exist $y$ such that $x < y$. For instance, if $S = \{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9\}$ and the order is the divisor order ($x_i \leq x_j$ if and only if $i$ divides $j$), then the elements $x_5, x_6, x_7, x_8, x_9$ are all maximal. This is much weaker than saying that for every $y$ we have $y\leq x$. Such an $x$ might be called maximum or greatest. In finite sets this is the same as saying $x$ is the unique maximal element. For example if we add the element $x_{5\cdot 7 \cdot 8 \cdot 9}$ to our set then this is the greatest element.
In group theory, the relation of inclusion $M\subseteq N$ is far more natural and useful than the weaker order relation $|M|\leq |N|$, and a subgroup is called maximal if it is maximal with respect to inclusion. As you say this is the same as not being contained in a larger subgroup. 

