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Order Exponent Nilpotency class Nilpotent
Center Commutator Central quotient Abelianization
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Number of subgroups Num. of normal subs Num. of conj. classes
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Outer aut. group Outer automorphisms Metabelian Metacyclic
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Results (1-50 of 2969 matches)

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Label Name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
6.1 $S_3$ $2 \cdot 3$ $2 \cdot 3$ 3 $C_1$ $S_3$ $C_3$ $C_{2}$ Solvable - 2
20.3 $F_5$ $2^{2} \cdot 5$ $2^{2} \cdot 5$ 5 $C_1$ $F_5$ $C_5$ $C_{4}$ Solvable - 2
24.12 $S_4$ $2^{3} \cdot 3$ $2^{2} \cdot 3$ 5 $C_1$ $S_4$ $A_4$ $C_{2}$ Solvable - 3
42.1 $F_7$ $2 \cdot 3 \cdot 7$ $2 \cdot 3 \cdot 7$ 7 $C_1$ $F_7$ $C_7$ $C_{6}$ Solvable - 2
54.6 $C_9:C_6$ $2 \cdot 3^{3}$ $2 \cdot 3^{2}$ 10 $C_1$ $C_9:C_6$ $C_9$ $C_{6}$ Solvable - 2
110.1 $F_{11}$ $2 \cdot 5 \cdot 11$ $2 \cdot 5 \cdot 11$ 11 $C_1$ $F_{11}$ $C_{11}$ $C_{10}$ Solvable - 2
120.34 $S_5$ $2^{3} \cdot 3 \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 7 $C_1$ $S_5$ $A_5$ $C_{2}$ Non-Solvable - 2
120.36 $S_3\times F_5$ $2^{3} \cdot 3 \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 15 $C_1$ $S_3\times F_5$ $C_{15}$ $C_{2} \times C_{4}$ Solvable - 2
144.182 $F_9:C_2$ $2^{4} \cdot 3^{2}$ $2^{3} \cdot 3$ 9 $C_1$ $F_9:C_2$ $C_3^2:C_4$ $C_{2}^{2}$ Solvable - 3
144.183 $S_3\times S_4$ $2^{4} \cdot 3^{2}$ $2^{2} \cdot 3$ 15 $C_1$ $S_3\times S_4$ $C_3\times A_4$ $C_{2}^{2}$ Solvable - 3
156.7 $F_{13}$ $2^{2} \cdot 3 \cdot 13$ $2^{2} \cdot 3 \cdot 13$ 13 $C_1$ $F_{13}$ $C_{13}$ $C_{12}$ Solvable - 2
168.43 $F_8:C_3$ $2^{3} \cdot 3 \cdot 7$ $2 \cdot 3 \cdot 7$ 8 $C_1$ $F_8:C_3$ $F_8$ $C_{3}$ Solvable - 3
216.90 $C_3^2.S_4$ $2^{3} \cdot 3^{3}$ $2^{2} \cdot 3^{2}$ 19 $C_1$ $C_3^2.S_4$ $C_2^2:C_9$ $C_{6}$ Solvable - 3
252.26 $S_3\times F_7$ $2^{2} \cdot 3^{2} \cdot 7$ $2 \cdot 3 \cdot 7$ 21 $C_1$ $S_3\times F_7$ $C_{21}$ $C_{2} \times C_{6}$ Solvable - 2
272.50 $F_{17}$ $2^{4} \cdot 17$ $2^{4} \cdot 17$ 17 $C_1$ $F_{17}$ $C_{17}$ $C_{16}$ Solvable - 2
320.1635 $C_2^4:F_5$ $2^{6} \cdot 5$ $2^{3} \cdot 5$ 11 $C_1$ $C_2^4:F_5$ $C_2^4:C_5$ $C_{4}$ Solvable - 3
324.118 $C_3^2.S_3^2$ $2^{2} \cdot 3^{4}$ $2 \cdot 3^{2}$ 30 $C_1$ $C_3^2.S_3^2$ $C_3\times C_9$ $C_{2} \times C_{6}$ Solvable - 2
336.208 $\PGL(2,7)$ $2^{4} \cdot 3 \cdot 7$ $2^{3} \cdot 3 \cdot 7$ 9 $C_1$ $\PGL(2,7)$ $\PSL(2,7)$ $C_{2}$ Non-Solvable - 2
336.210 $F_8:C_6$ $2^{4} \cdot 3 \cdot 7$ $2 \cdot 3 \cdot 7$ 16 $C_2$ $F_8:C_3$ $F_8$ $C_{6}$ Solvable - 3
342.7 $F_{19}$ $2 \cdot 3^{2} \cdot 19$ $2 \cdot 3^{2} \cdot 19$ 19 $C_1$ $F_{19}$ $C_{19}$ $C_{18}$ Solvable - 2
384.5677 $C_2^2.\GL(2,\mathbb{Z}/4)$ $2^{7} \cdot 3$ $2^{3} \cdot 3$ 16 $C_1$ $C_2^2.\GL(2,\mathbb{Z}/4)$ $C_4^2:C_6$ $C_{2}^{2}$ Solvable - 3
384.5678 $C_2^4:S_4$ $2^{7} \cdot 3$ $2^{3} \cdot 3$ 16 $C_1$ $C_2^4:S_4$ $C_2^3:A_4$ $C_{2}^{2}$ Solvable - 3
432.520 $\He_3:\SD_{16}$ $2^{4} \cdot 3^{3}$ $2^{3} \cdot 3$ 14 $C_1$ $\He_3:\SD_{16}$ $\He_3:C_4$ $C_{2}^{2}$ Solvable - 4
432.523 $C_6^2:D_6$ $2^{4} \cdot 3^{3}$ $2^{2} \cdot 3$ 20 $C_1$ $C_6^2:D_6$ $C_3^2:A_4$ $C_{2}^{2}$ Solvable - 3
432.734 $C_3^2:\GL(2,3)$ $2^{4} \cdot 3^{3}$ $2^{3} \cdot 3$ 11 $C_1$ $C_3^2:\GL(2,3)$ $\PU(3,2)$ $C_{2}$ Solvable - 5
480.1189 $F_5\times S_4$ $2^{5} \cdot 3 \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 25 $C_1$ $F_5\times S_4$ $C_5\times A_4$ $C_{2} \times C_{4}$ Solvable - 3
486.31 $C_{27}:C_{18}$ $2 \cdot 3^{5}$ $2 \cdot 3^{3}$ 31 $C_1$ $C_{27}:C_{18}$ $C_{27}$ $C_{18}$ Solvable - 2
500.18 $C_{25}:C_{20}$ $2^{2} \cdot 5^{3}$ $2^{2} \cdot 5^{2}$ 26 $C_1$ $C_{25}:C_{20}$ $C_{25}$ $C_{20}$ Solvable - 2
506.1 $F_{23}$ $2 \cdot 11 \cdot 23$ $2 \cdot 11 \cdot 23$ 23 $C_1$ $F_{23}$ $C_{23}$ $C_{22}$ Solvable - 2
660.15 $S_3\times F_{11}$ $2^{2} \cdot 3 \cdot 5 \cdot 11$ $2 \cdot 3 \cdot 5 \cdot 11$ 33 $C_1$ $S_3\times F_{11}$ $C_{33}$ $C_{2} \times C_{10}$ Solvable - 2
720.767 $S_3\times S_5$ $2^{4} \cdot 3^{2} \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 21 $C_1$ $S_3\times S_5$ $\GL(2,4)$ $C_{2}^{2}$ Non-Solvable - 4
800.1191 $F_5\wr C_2$ $2^{5} \cdot 5^{2}$ $2^{3} \cdot 5$ 20 $C_1$ $F_5\wr C_2$ $C_5:F_5$ $C_{2} \times C_{4}$ Solvable - 3
812.7 $F_{29}$ $2^{2} \cdot 7 \cdot 29$ $2^{2} \cdot 7 \cdot 29$ 29 $C_1$ $F_{29}$ $C_{29}$ $C_{28}$ Solvable - 2
840.139 $F_5\times F_7$ $2^{3} \cdot 3 \cdot 5 \cdot 7$ $2^{2} \cdot 3 \cdot 5 \cdot 7$ 35 $C_1$ $F_5\times F_7$ $C_{35}$ $C_{2} \times C_{12}$ Solvable - 2
864.4662 $F_9:D_6$ $2^{5} \cdot 3^{3}$ $2^{3} \cdot 3$ 27 $C_1$ $F_9:D_6$ $C_3^2:C_{12}$ $C_{2}^{3}$ Solvable - 3
896.564 $C_2^4.F_8$ $2^{7} \cdot 7$ $2^{2} \cdot 7$ 17 $C_2$ $C_2^3.F_8$ $C_2^4.C_2^3$ $C_{7}$ Solvable - 3
930.1 $F_{31}$ $2 \cdot 3 \cdot 5 \cdot 31$ $2 \cdot 3 \cdot 5 \cdot 31$ 31 $C_1$ $F_{31}$ $C_{31}$ $C_{30}$ Solvable - 2
936.167 $S_3\times F_{13}$ $2^{3} \cdot 3^{2} \cdot 13$ $2^{2} \cdot 3 \cdot 13$ 39 $C_1$ $S_3\times F_{13}$ $C_{39}$ $C_{2} \times C_{12}$ Solvable - 2
960.11359 $F_{16}:C_4$ $2^{6} \cdot 3 \cdot 5$ $2^{3} \cdot 3 \cdot 5$ 13 $C_1$ $F_{16}:C_4$ $F_{16}$ $C_{4}$ Solvable - 3
972.429 $C_3^3.S_3^2$ $2^{2} \cdot 3^{5}$ $2 \cdot 3^{2}$ 39 $C_1$ $C_3^3.S_3^2$ $C_3^2:C_9$ $C_{2} \times C_{6}$ Solvable - 3
972.445 $C_3^3.S_3^2$ $2^{2} \cdot 3^{5}$ $2 \cdot 3^{2}$ 35 $C_1$ $C_3^3.S_3^2$ $C_3\wr C_3$ $C_{2} \times C_{6}$ Solvable - 3
972.446 $C_3^3.S_3^2$ $2^{2} \cdot 3^{5}$ $2 \cdot 3^{2}$ 35 $C_1$ $C_3^3.S_3^2$ $C_3.\He_3$ $C_{2} \times C_{6}$ Solvable - 3
972.447 $C_3^3.S_3^2$ $2^{2} \cdot 3^{5}$ $2 \cdot 3^{2}$ 35 $C_1$ $C_3^3.S_3^2$ $\He_3:C_3$ $C_{2} \times C_{6}$ Solvable - 3
1008.885 $F_8:C_3\times S_3$ $2^{4} \cdot 3^{2} \cdot 7$ $2 \cdot 3 \cdot 7$ 24 $C_1$ $F_8:C_3\times S_3$ $C_3\times F_8$ $C_{6}$ Solvable - 3
1008.886 $S_4\times F_7$ $2^{4} \cdot 3^{2} \cdot 7$ $2^{2} \cdot 3 \cdot 7$ 35 $C_1$ $S_4\times F_7$ $C_7\times A_4$ $C_{2} \times C_{6}$ Solvable - 3
1080.271 $D_{45}:C_{12}$ $2^{3} \cdot 3^{3} \cdot 5$ $2^{2} \cdot 3^{2} \cdot 5$ 50 $C_1$ $D_{45}:C_{12}$ $C_{45}$ $C_{2} \times C_{12}$ Solvable - 2
1152.157849 $S_4\wr C_2$ $2^{7} \cdot 3^{2}$ $2^{3} \cdot 3$ 20 $C_1$ $S_4\wr C_2$ $\PSOPlus(4,3)$ $C_{2}^{2}$ Solvable - 4
1200.946 $F_{25}:C_2$ $2^{4} \cdot 3 \cdot 5^{2}$ $2^{3} \cdot 3 \cdot 5$ 20 $C_1$ $F_{25}:C_2$ $C_5^2:C_6$ $C_{2} \times C_{4}$ Solvable - 3
1296.2920 $C_6^2.S_3^2$ $2^{4} \cdot 3^{4}$ $2^{2} \cdot 3^{2}$ 50 $C_1$ $C_6^2.S_3^2$ $C_9\times A_4$ $C_{2} \times C_{6}$ Solvable - 3
1296.2922 $C_6^2.S_3^2$ $2^{4} \cdot 3^{4}$ $2^{2} \cdot 3^{2}$ 57 $C_1$ $C_6^2.S_3^2$ $C_3^2.A_4$ $C_{2} \times C_{6}$ Solvable - 3
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