Let $\{X(t)\}_{t\geqslant0}$ be the generalized fractional Brownian motion introduced by Pang and Taqqu (2019): \begin{align*} \{X(t)\}_{t\ge0}\overset{d}{=}&\left\{ \int_{\mathbb R} \left((t-u)_+^α-(-u)_+^α \right) |u|^{-γ/2} B(du) \right\}_{t\ge0}, \end{align*} where $ γ\in [0,1), \ \ α\in \left(-\frac12+\fracγ{2}, \ \frac12+\fracγ{2} \right)$ are constants. For any $θ>0$, let \begin{align*} Y(t)=\frac{1}{Γ(θ)}\int_0^t (t-u)^{θ-1} X(u)du, \quad t\ge 0. \end{align*} Building upon the arguments of Talagrand (1996), we give integral criteria for the lower classes of $Y$ at $t=0$ and at infinity, respectively. As a consequence, we derive its Chung-type laws of the iterated logarithm. In the proofs, the small ball probability estimates play important roles.