The results below are complete, since the LMFDB contains all groups of order at most 2000 except orders larger than 500 that are multiples of 128

Refine search

Order Exponent Nilpotency class Nilpotent
Center Commutator Central quotient Abelianization
Abelian Cyclic Solvable Simple
Perfect Direct product Semidirect product Wreath product
Automorphism group Automorphisms Min. transitive degree Min. permutation degree
Number of subgroups Num. of normal subs Num. of conj. classes
Order statistics Family Order factorization Commutator length
Min. degree of $\C$-irrep Min. degree of $\Q$-irrep Min. degree of $\C$-rep Min. degree of $\Q$-rep
Outer aut. group Outer automorphisms Metabelian Metacyclic
Almost simple Quasisimple A-group Z-group
Frattini subgroup Derived length Rank Schur multiplier
Supersolvable Monomial Rational
Num. of char. subgroups Num. of aut. classes Number of divisions
Sort order Select

Results (1-50 of 5211 matches)

Next   displayed columns for results
Label Name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
385.1 $C_{11}:C_{35}$ $5 \cdot 7 \cdot 11$ $5 \cdot 7 \cdot 11$ 49 $C_7$ $C_{11}:C_5$ $C_{11}$ $C_{35}$ Solvable - 2
385.2 $C_{385}$ $5 \cdot 7 \cdot 11$ $5 \cdot 7 \cdot 11$ 385 $C_{385}$ $C_1$ $C_1$ $C_{385}$ Cyclic
386.1 $D_{193}$ $2 \cdot 193$ $2 \cdot 193$ 98 $C_1$ $D_{193}$ $C_{193}$ $C_{2}$ Solvable - 2
386.2 $C_{386}$ $2 \cdot 193$ $2 \cdot 193$ 386 $C_{386}$ $C_1$ $C_1$ $C_{386}$ Cyclic
387.1 $C_{43}:C_9$ $3^{2} \cdot 43$ $3^{2} \cdot 43$ 51 $C_3$ $C_{43}:C_3$ $C_{43}$ $C_{9}$ Solvable - 2
387.2 $C_{387}$ $3^{2} \cdot 43$ $3^{2} \cdot 43$ 387 $C_{387}$ $C_1$ $C_1$ $C_{387}$ Cyclic
387.3 $C_{129}:C_3$ $3^{2} \cdot 43$ $3 \cdot 43$ 51 $C_3$ $C_{43}:C_3$ $C_{43}$ $C_{3}^{2}$ Solvable - 2
387.4 $C_3\times C_{129}$ $3^{2} \cdot 43$ $3 \cdot 43$ 387 $C_3\times C_{129}$ $C_1$ $C_1$ $C_{3} \times C_{129}$ Abelian - 2
388.1 $C_{97}:C_4$ $2^{2} \cdot 97$ $2^{2} \cdot 97$ 100 $C_2$ $D_{97}$ $C_{97}$ $C_{4}$ Solvable - 2
388.2 $C_{388}$ $2^{2} \cdot 97$ $2^{2} \cdot 97$ 388 $C_{388}$ $C_1$ $C_1$ $C_{388}$ Cyclic
388.3 $C_{97}:C_4$ $2^{2} \cdot 97$ $2^{2} \cdot 97$ 28 $C_1$ $C_{97}:C_4$ $C_{97}$ $C_{4}$ Solvable - 2
388.4 $D_{194}$ $2^{2} \cdot 97$ $2 \cdot 97$ 100 $C_2$ $D_{97}$ $C_{97}$ $C_{2}^{2}$ Solvable - 2
388.5 $C_2\times C_{194}$ $2^{2} \cdot 97$ $2 \cdot 97$ 388 $C_2\times C_{194}$ $C_1$ $C_1$ $C_{2} \times C_{194}$ Abelian - 2
389.1 $C_{389}$ $389$ $389$ 389 $C_{389}$ $C_1$ $C_1$ $C_{389}$ Cyclic
390.1 $C_{13}:C_{30}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 40 $C_5$ $C_{13}:C_6$ $C_{13}$ $C_{30}$ Solvable - 2
390.2 $C_{65}:C_6$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 28 $C_1$ $C_{65}:C_6$ $C_{65}$ $C_{6}$ Solvable - 2
390.3 $C_{65}:C_6$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 22 $C_1$ $C_{65}:C_6$ $C_{65}$ $C_{6}$ Solvable - 2
390.4 $C_{13}:C_{30}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 70 $C_{10}$ $C_{13}:C_3$ $C_{13}$ $C_{30}$ Solvable - 2
390.5 $C_{15}\times D_{13}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 120 $C_{15}$ $D_{13}$ $C_{13}$ $C_{30}$ Solvable - 2
390.6 $D_5\times C_{39}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 156 $C_{39}$ $D_5$ $C_5$ $C_{78}$ Solvable - 2
390.7 $C_3\times D_{65}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 102 $C_3$ $D_{65}$ $C_{65}$ $C_{6}$ Solvable - 2
390.8 $S_3\times C_{65}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 195 $C_{65}$ $S_3$ $C_3$ $C_{130}$ Solvable - 2
390.9 $C_5\times D_{39}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 105 $C_5$ $D_{39}$ $C_{39}$ $C_{10}$ Solvable - 2
390.10 $C_{13}\times D_{15}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 117 $C_{13}$ $D_{15}$ $C_{15}$ $C_{26}$ Solvable - 2
390.11 $D_{195}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 99 $C_1$ $D_{195}$ $C_{195}$ $C_{2}$ Solvable - 2
390.12 $C_{390}$ $2 \cdot 3 \cdot 5 \cdot 13$ $2 \cdot 3 \cdot 5 \cdot 13$ 390 $C_{390}$ $C_1$ $C_1$ $C_{390}$ Cyclic
391.1 $C_{391}$ $17 \cdot 23$ $17 \cdot 23$ 391 $C_{391}$ $C_1$ $C_1$ $C_{391}$ Cyclic
392.1 $C_{49}:C_8$ $2^{3} \cdot 7^{2}$ $2^{3} \cdot 7^{2}$ 104 $C_4$ $D_{49}$ $C_{49}$ $C_{8}$ Solvable - 2
392.2 $C_{392}$ $2^{3} \cdot 7^{2}$ $2^{3} \cdot 7^{2}$ 392 $C_{392}$ $C_1$ $C_1$ $C_{392}$ Cyclic
392.3 $C_{49}:Q_8$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7^{2}$ 101 $C_2$ $D_{98}$ $C_{98}$ $C_{2}^{2}$ Solvable - 2
392.4 $C_4\times D_{49}$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7^{2}$ 104 $C_4$ $D_{49}$ $C_{49}$ $C_{2} \times C_{4}$ Solvable - 2
392.5 $D_{196}$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7^{2}$ 101 $C_2$ $D_{98}$ $C_{98}$ $C_{2}^{2}$ Solvable - 2
392.6 $C_{98}:C_4$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7^{2}$ 104 $C_2^2$ $D_{49}$ $C_{49}$ $C_{2} \times C_{4}$ Solvable - 2
392.7 $C_{49}:D_4$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7^{2}$ 101 $C_2$ $D_{98}$ $C_{98}$ $C_{2}^{2}$ Solvable - 2
392.8 $C_2\times C_{196}$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7^{2}$ 392 $C_2\times C_{196}$ $C_1$ $C_1$ $C_{2} \times C_{196}$ Abelian - 2
392.9 $D_4\times C_{49}$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7^{2}$ 245 $C_{98}$ $C_2^2$ $C_2$ $C_{2} \times C_{98}$ Nilpotent - 2
392.10 $Q_8\times C_{49}$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7^{2}$ 245 $C_{98}$ $C_2^2$ $C_2$ $C_{2} \times C_{98}$ Nilpotent - 2
392.11 $C_2^3:C_{49}$ $2^{3} \cdot 7^{2}$ $2 \cdot 7^{2}$ 56 $C_7$ $F_8$ $C_2^3$ $C_{49}$ Solvable - 2
392.12 $C_2\times D_{98}$ $2^{3} \cdot 7^{2}$ $2 \cdot 7^{2}$ 104 $C_2^2$ $D_{49}$ $C_{49}$ $C_{2}^{3}$ Solvable - 2
392.13 $C_2^2\times C_{98}$ $2^{3} \cdot 7^{2}$ $2 \cdot 7^{2}$ 392 $C_2^2\times C_{98}$ $C_1$ $C_1$ $C_{2}^{2} \times C_{98}$ Abelian - 3
392.14 $C_7:C_{56}$ $2^{3} \cdot 7^{2}$ $2^{3} \cdot 7$ 140 $C_{28}$ $D_7$ $C_7$ $C_{56}$ Solvable - 2
392.15 $C_7^2:C_8$ $2^{3} \cdot 7^{2}$ $2^{3} \cdot 7$ 104 $C_4$ $C_7:D_7$ $C_7^2$ $C_{8}$ Solvable - 2
392.16 $C_7\times C_{56}$ $2^{3} \cdot 7^{2}$ $2^{3} \cdot 7$ 392 $C_7\times C_{56}$ $C_1$ $C_1$ $C_{7} \times C_{56}$ Abelian - 2
392.17 $C_7^2:C_8$ $2^{3} \cdot 7^{2}$ $2^{3} \cdot 7$ 32 $C_2$ $C_7^2:C_4$ $C_7^2$ $C_{8}$ Solvable - 2
392.18 $C_{14}.D_{14}$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7$ 50 $C_2$ $D_7^2$ $C_7^2$ $C_{2} \times C_{4}$ Solvable - 2
392.19 $C_{14}.D_{14}$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7$ 50 $C_2$ $D_7^2$ $C_7^2$ $C_{2} \times C_{4}$ Solvable - 2
392.20 $D_{14}:D_7$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7$ 47 $C_2$ $D_7^2$ $C_7\times C_{14}$ $C_{2}^{2}$ Solvable - 2
392.21 $C_7:D_{28}$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7$ 47 $C_2$ $D_7^2$ $C_7\times C_{14}$ $C_{2}^{2}$ Solvable - 2
392.22 $C_7^2:Q_8$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7$ 47 $C_2$ $D_7^2$ $C_7\times C_{14}$ $C_{2}^{2}$ Solvable - 2
392.23 $C_7^2:Q_8$ $2^{3} \cdot 7^{2}$ $2^{2} \cdot 7$ 119 $C_{14}$ $D_{14}$ $C_{14}$ $C_{2} \times C_{14}$ Solvable - 2
Next   displayed columns for results