Partial table of results (sorted by $\sqrt{a+b+c}$) :
$\small \begin{array} {r|lllllll} 49&[97, 192, 2112] & [376, 465, 1560] & [720, 801, 880]\\ 71&[*280, 945, 3816] & [*280, 1320, 3441] & [417, 1792, 2832]\\ 79&[312, *1057, 4872] & [616, 2520, 3105] & [*1057, 1752, 3432]\\ 85&[336, 1600, 5289] & [*2041, 2184, 3000] & [984, *2041, 4200]\\ 87&[344, *680, 6545]& [513, 1008, 6048]& [*1169, 1640, 4760]& [*680, *1169, 5720]\\ 91&[360, 3240, 4681]& [1392, 3097, 3792]& [1720, 2505, 4056]\\ 95&[561, 2464, 6000]& [744, 1281, 7000]& [2136, 2625, 4264]\\ 97&[193, 1488, 7728]& [*1128, 2353, 5928]& [*1128, 3633, 4648]\\ 98&[388, 768, 8448]& [1504, 1860, 6240]& [2880, 3204, 3520]\\ 103&[808, *2328, 7473] &[1584, 3040, 5985]& [2688, 3553, 4368]& [1393, *2328, 6888]\\ 105&[1224, 2376, 7425] &[209, *2000, 8816] &[416, *4625, 5984] &[*824, 3800, 6401]& [1616, *4625, 4784]& [*824, *1025, 9176]&[*1025, *2000, 8000]\\ 111&[1296, 5265, 5760]& [1712, 2912, 7697]& [2520, 3105, 6696]\\ 113&[225, 3744, 8800]& [448, 1953, 10368]& [1320, 4305, 7144]& [2568, 4488, 5713]& [2769, 4120, 5880]\\ 115&[*681, 1344, 11200]& [2200, 4200, 6825]& [*681, 4944, 7600]\\ 117&[233, 3488, 9968]& [1145, *3080, 9464]& [*2240, 3689, 7760]& [*3080, 4664, 5945]&[464, *2240, 10985]\\ 119&[472, 6417, 7272]& [*2280, 3345, 8536] &[*2280, 4945, 6936]\\ 121&[480, 3616, 10545]& [*1185, 1416, 12040]& [*1185, 4440, 9016]\\ 123&[2360, 6104, 6665]& [1904, 2585, 10640]& [3465, 4104, 7560]\\ 125&[249, 6976, 8400]& [*1225, 5016, 9384]& [4176, 5625, 5824] &[4600, 4809, 6216]&[984, *1225, 13416]\\ 127&[504, 4680, 10945]& [1968, 3808, 10353]& [3360, 5104, 7665]\\ 129&[512, 7232, 8897]& [1016, 4760, 10865]& [2241, 4320, 10080]\\ 131&[1785, 6136, 9240] &[*2032, 5712, 9417]& [3936, 5280, 7945]&[777, *2032, 14352]\\ 133&[1048, 6873, 9768]& [2313, 7488, 7888]& [4000, 4464, 9225]\\ 135&[*536, 1064, 16625]& [*2600, 5681, 9944]& [*536, *2600, 15089]\\ 137&[273, 2128, 16368]& [*1345, 3144, 14280]& [*1345, 5544, 11880]\\ 141&[3497, 7112, 9272]& [*281, 560, 19040]& [1656, 7560, 10665]& [2457, 6192, 11232]& [2720, 4505, 12656]& [*281, 8000, 11600]\\ \end{array} $