Let K = Q(\sqrt{d}) be a real quadratic number field with discriminant d and K_{w-ur} its maximal weakly unramified extension. We denote by Γ the Galois group Gal(K_{w-ur}/K) and l_{w} be the length of the narrow Hilbert class field tower of K: K = K_{0,nar} ⫅ K_{1,nar} ⫅ K_{2,nar} ⫅ ⋯ (K_{i+1,nar} is the Hilbert class field of K_{i,nar}), that is, l_{w} is the smallest number with K_{l_{w}+1,nar} = K_{l_{w},nar}.

Under GRH we have determined the structure of Γ for d ≦ 800 except for d = 641, 661, 673, 677, 701, 709, 757, 769, 773, 776, 797. We give a table of Γ for K for which we suceed in concluding K_{w-ur} ≠ K_{1,nar} below, however we omit K for which we expect K_{w-ur} = K but we have not suceeded in proving yet.

Table of real quadratic number fields K = Q(\sqrt{d}), d ≦ 800 with K_{w-ur} ≠ K_{1,nar}

\begin{matrix}d & Cl(K) & l_{w} & Γ & d & Cl(K) & l_{w} & Γ \\ 69 & C_{2} & 2 & D_{3} & 357 & V_{4} & 2 & D_{10} \\ 92 & C_{2} & 2 & D_{3} & 376 & C_{2} & 2 & D_{5} \\ 93 & C_{2} & 2 & D_{3} & 380 & V_{4} & 2 & D_{8} \\ 124 & C_{2} & 2 & D_{3} & 381 & C_{2} & 2 & D_{5} \\ 141 & C_{2} & 2 & D_{5} & 385 & V_{4} & 2 & D_{4} \\ 156 & V_{4} & 2 & D_{4} & 393 & C_{2} & 2 & D_{5} \\ 161 & C_{2} & 2 & D_{3} & 408 & V_{4} & 2 & D_{4} \\ 165 & V_{4} & 2 & D_{4} & 412 & C_{2} & 2 & D_{5} \\ 168 & V_{4} & 2 & D_{4} & 413 & C_{2} & 2 & D_{3} \\ 177 & C_{2} & 2 & D_{3} & 417 & C_{2} & 2 & D_{3} \\ 184 & C_{2} & 2 & D_{3} & 428 & C_{2} & 2 & D_{3} \\ 188 & C_{2} & 2 & D_{5} & 429 & V_{4} & 2 & D_{4} ⋏ D_{5} \\ 204 & V_{4} & 2 & D_{4} & 437 & C_{2} & 3 & S_{4} \\ 213 & C_{2} & 2 & D_{7} & 440 & V_{4} & 2 & D_{4} \\ 217 & C_{2} & 2 & D_{3} & 444 & V_{4} & 2 & D_{8} \\ 220 & V_{4} & 2 & D_{4} & 453 & C_{2} & 2 & D_{7} \\ 229 & C_{3} & 2 & A_{4} & 456 & V_{4} & 2 & D_{6} \\ 236 & C_{2} & 2 & D_{3} & 460 & V_{4} & 2 & D_{6} \\ 237 & C_{2} & 2 & D_{5} & 465 & V_{4} & 2 & D_{4} ⋏ D_{6} \\ 248 & C_{2} & 2 & D_{3} & 472 & C_{2} & 3 & S_{4} \\ 249 & C_{2} & 2 & D_{3} & 476 & V_{4} & 2 & D_{4} ⋏ D_{10} \\ 253 & C_{2} & 2 & D_{3} & 492 & V_{4} & 2 & D_{8} \\ 257 & C_{3} & 2 & A_{4} & 497 & C_{2} & 2 & D_{7} \\ 273 & V_{4} & 2 & D_{4} & 501 & C_{2} & 2 & D_{11} \\ 284 & C_{2} & 2 & D_{7} & 508 & C_{2} & 2 & D_{5} \\ 285 & V_{4} & 2 & D_{8} & 517 & C_{2} & 2 & D_{5} \\ 309 & C_{2} & 2 & D_{5} & 520 & V_{4} & 2 & Q_{8} \\ 312 & V_{4} & 2 & D_{12} & 524 & C_{2} & 2 & D_{5} \\ 316 & C_{6} & 2 & D_{5} × C_{3} & 537 & C_{2} & 2 & D_{5} \\ 321 & C_{6} & 2 & D_{3} × C_{3} & 552 & V_{4} & 2 & Q_{12} \\ 329 & C_{2} & 2 & D_{5} & 553 & C_{2} & 2 & D_{5} \\ 332 & C_{2} & 3 & S_{4} & 556 & C_{2} & 2 & D_{3} \\ 341 & C_{2} & 3 & S_{4} & 568 & C_{6} & 2 & D_{7} × C_{3} \\ 345 & V_{4} & 2 & D_{6} & 572 & V_{4} & 2 & D_{10} \\ 348 & V_{4} & 2 & D_{3} × D_{3} & 573 & C_{2} & 2 & D_{13} \end{matrix}

Continued (under GRH)

d Cl(K) l_{w}  Γ  d Cl(K) l_{w}  Γ
581 C_{2} 2 D_{3}  705  V_{4}  2  D_{10}
589 C_{2} 2 D_{3}  713  C_{2}  2  D_{3} ⋏ D_{3}
597 C_{2} 3 S_{4} ⋏ D_{9} 716  C_{2}  2  D_{5}
604 C_{2} 2 D_{7}  717  C_{2}  3  S_{4} ⋏ D_{5}
609 V_{4}  2 D_{4} ⋏ D_{6} 721  C_{2}  2  D_{5}
616 V_{4}  2 D_{4}  728  V_{4}  2  D_{6}
620 V_{4}  2 D_{4} ⋏ D_{6} 732  V_{4}  2  D_{8} ⋏ D_{6}
632 C_{2} 2 D_{5}  741  V_{4}  2  D_{4} ⋏ D_{6}
633 C_{2} 2 D_{3}  744  V_{4}  2  D_{8} ⋉ C_{3}
636 V_{4}  2 D_{5} × D_{3} 748  V_{4}  2  D_{4}
645 V_{4}  2 D_{14}  749  C_{2}  4  S̃_{4}
649 C_{2} 2 D_{3}  753  C_{2}  2  D_{7}
653 {1} 0 A_{5}  760  V_{4}  2  D_{8}
664 C_{2} 2 D_{3}  761  C_{3}  2  A_{4}
665 V_{4}  2 D_{8}  764  C_{2}  2  D_{13}
668 C_{2} 2 D_{11}  776  C_{4} ≧ 2  C_{4} ⋉ C_{3}^{2}?
669 C_{2} 2 D_{7}  777 C_{4} × C_{2} 2 (C_{4} × C_{2}) ⋉ C_{8}
680 V_{4}  2 D_{4}  780  C_{2}^{3}  2  C_{2}^{3} ⋉ C_{4}^{2}
681 C_{2} 2 D_{5}  781  C_{2}  2  D_{7}
696 V_{4}  2 D_{6}  789  C_{2}  2  D_{13}
697 C_{6} 2 A_{4} × C_{2} 796  C_{2}  2  D_{9}

As a remarkable fact, we note the following. For K = Q(\sqrt{653}), we have Gal(K_{w-ur}/K) ≅ A_{5}, a nonabelian simple group. For this K we have Gal(K_{w-ur}/Q) ≅ A_{5} × C_{2}. Though this is conditional, this is only example of a number field K for which Gal(K_{w-ur}/K) is nonabelian simple. In the range d ≦ 2001, for which we have [K_{w-ur} : K] < ∞ under GRH, another fields have weakly unramified nonsolvable Galois extension: Q(\sqrt{1609}) has a weakly unramified A_{5}-extension that is an S_{5}-extension of Q. Q(\sqrt{1417}) has a weakly unramified PSL(2, 7)-extension that is an PSL(2, 7) × C_{2}-extension of Q. Q(\sqrt{1709}) has a weakly unramified PSL(2, 7)- extension that is an PGL(2, 7)-extension of Q.