Finding quotient groups

Question

Let $G = U_{44}, N = \langle |3| \rangle, K = \langle |9| \rangle$.

a) Find $G/ K, G/ N, N/ K, (G/ K)/ (N/ K)$

b) Explain why $(G/ K)/ (N/ K) \cong G/ N $

Answer 1

$U_{44}$ has order $\phi(44)=20$. Actually, $U_{44} \cong U_4 \times U_{11} \cong C_2 \times C_{10}$.

$3$ has order $10$ mod $11$ and so has order $10$ mod $44$. So, $G/N$ has order $2$ and is $C_2$.

$9$ has order $5$ mod $44$ and mod $11$. So, $G/K \cong C_2 \times C_2$.

$N/K$ has order $2$ and so is $C_2$.