<p><b>Abstract</b>—The purpose of this paper is to investigate combinatorial properties of the hypernet network. The hypernet network owns two structural advantages: expansibility and equal degree. Besides, it was shown efficient in both communication and computation. Since the number of nodes contained in the hypernet network increases very rapidly with expansion level, we emphasize the hypernet network of two levels (denoted by HN<tmath>$(d, 2)$</tmath>) with a practical view. Recently, combinatorial properties such as container (i.e., node-disjoint paths), wide diameter, and fault diameter have received much attention due to their increasing importance and applications in networks. In this paper, the following results are obtained for HN<tmath>$(d, 2)$</tmath>: 1) best containers with width <tmath>$d-1$</tmath>, 2) containers with (maximum) width <tmath>$d$</tmath>, 3) the (<tmath>$d-1$</tmath>)-wide diameter, 4) the <tmath>$d$</tmath>-wide diameter, 5) the <tmath>$(d - 2)$</tmath>-fault diameter, and 6) the <tmath>$(d-1)$</tmath>-fault diameter. More specifically, between every two nodes of HN<tmath>$(d, 2)$</tmath>, <tmath>$d$</tmath> (or <tmath>$d-1$</tmath>) packets can be transmitted simultaneously with at most <tmath>$D + 2$</tmath> (or <tmath>$D + 1$</tmath>) parallel steps, where <tmath>$D=2d+1$</tmath> is the diameter of HN<tmath>$(d, 2)$</tmath>. Besides, the diameter of HN<tmath>$(d, 2)$</tmath> will increase by at most two (or one), if there are at most <tmath>$d-1$</tmath> (or <tmath>$d-2$</tmath>) node faults. Our results reveal that HN<tmath>$(d, 2)$</tmath> is not only efficient in parallel transmission, but robust in fault tolerance.</p>