Irreducible constituents of a faithful representation I am new to representation theory and have a doubt:

Are all the irreducible constituents of a faithful representation of a
finite group also faithful?

My thoughts:
If the Group action is faithful in the whole vector space, it must be so in any subspace. So the answer should be yes. Am I missing something?
 A: You can easily generate a counterexample by taking two faithful irreducible representations $(\pi_i, V_i)$ of groups $G_i$ ($i=1,2$) and letting $(\pi, V_1 \oplus V_2)$ be the faithful representation of $G = G_1 \times G_2)$ by
$$\pi(g_1, g_2)\Big( v_1 + v_2 \Big) = \pi_1(g_1)v_1 + \pi_2(g_2)v_2.$$
Here $V_1$ and $V_2$ are the irreducible constituents of the faithful representation $\pi$, but they are clearly not faithful representations of $G$, because the $G_2$ component acts trivially on $V_1$, and the $G_1$ component acts trivially on $V_2$.
A: $C_2\times C_3$ acts faithfully on $\Bbb C^2$ as $(m,n)\mapsto\operatorname{diag}(e^{im\pi},e^{2in/3})$. This representation decomposes into the two non-faithful irreducible representations $$C_2\times C_3\stackrel{\pi_1}{\longrightarrow}C_2\stackrel{-1}{\longrightarrow}\Bbb C^*\\ C_2\times C_3\stackrel{\pi_2}{\longrightarrow}C_3\stackrel{e^{2i\pi/3}}{\longrightarrow}\Bbb C^*$$
A: Possibly you have the other thing as below in your mind: If $V$ is a faithful representation of $G$ then for any subgroup $H$(non-trivial)  $V$ is a faithful representation: this is obvious as as an injective homomorphism $\rho$ defined with $G$ as domain remains injective when restricted to $H$.
However here the question is different. All the maps $\rho(g)$ are non-identity on a vector space $V,\ g\neq e$. (meaning of V being faithful)   But on a subspace $W$, (irred subrepresentation)  some of  of the $\rho(g)$  could become identity maps and so faithfulness is no more valid.
Take the natural representation $S_n$ on the vector space of linear homogeneous polynomials in $n$ variables, $x_i$, by permuting the variables. Every permutation disturbs at east on non-zero polynomial.  But the 1-dimensional subspace generated by polynomials where all the coefficients are same, $c\sum_i x_i,\ c\in \mathbf{C}$ is an irred subrepresentation, is not faithful.
