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Label | Name | Order | Exponent | $\card{\mathrm{conj}(G)}$ | Center | Type - length |
---|---|---|---|---|---|---|
1.1 | $C_1$ | $1$ | $1$ | $1$ | $C_1$ | Trivial |
2.1 | $C_2$ | $2$ | $2$ | $2$ | $C_2$ | Cyclic |
3.1 | $C_3$ | $3$ | $3$ | $3$ | $C_3$ | Cyclic |
4.1 | $C_4$ | $2^{2}$ | $2^{2}$ | $4$ | $C_4$ | Cyclic |
4.2 | $C_2^2$ | $2^{2}$ | $2$ | $4$ | $C_2^2$ | Abelian - 2 |
5.1 | $C_5$ | $5$ | $5$ | $5$ | $C_5$ | Cyclic |
6.1 | $S_3$ | $2 \cdot 3$ | $2 \cdot 3$ | $3$ | $C_1$ | Solvable - 2 |
6.2 | $C_6$ | $2 \cdot 3$ | $2 \cdot 3$ | $6$ | $C_6$ | Cyclic |
7.1 | $C_7$ | $7$ | $7$ | $7$ | $C_7$ | Cyclic |
8.1 | $C_8$ | $2^{3}$ | $2^{3}$ | $8$ | $C_8$ | Cyclic |
8.2 | $C_2\times C_4$ | $2^{3}$ | $2^{2}$ | $8$ | $C_2\times C_4$ | Abelian - 2 |
8.3 | $D_4$ | $2^{3}$ | $2^{2}$ | $5$ | $C_2$ | Nilpotent - 2 |
8.4 | $Q_8$ | $2^{3}$ | $2^{2}$ | $5$ | $C_2$ | Nilpotent - 2 |
8.5 | $C_2^3$ | $2^{3}$ | $2$ | $8$ | $C_2^3$ | Abelian - 3 |
9.1 | $C_9$ | $3^{2}$ | $3^{2}$ | $9$ | $C_9$ | Cyclic |
9.2 | $C_3^2$ | $3^{2}$ | $3$ | $9$ | $C_3^2$ | Abelian - 2 |
10.1 | $D_5$ | $2 \cdot 5$ | $2 \cdot 5$ | $4$ | $C_1$ | Solvable - 2 |
10.2 | $C_{10}$ | $2 \cdot 5$ | $2 \cdot 5$ | $10$ | $C_{10}$ | Cyclic |
11.1 | $C_{11}$ | $11$ | $11$ | $11$ | $C_{11}$ | Cyclic |
12.1 | $C_3:C_4$ | $2^{2} \cdot 3$ | $2^{2} \cdot 3$ | $6$ | $C_2$ | Solvable - 2 |
12.2 | $C_{12}$ | $2^{2} \cdot 3$ | $2^{2} \cdot 3$ | $12$ | $C_{12}$ | Cyclic |
12.3 | $A_4$ | $2^{2} \cdot 3$ | $2 \cdot 3$ | $4$ | $C_1$ | Solvable - 2 |
12.4 | $D_6$ | $2^{2} \cdot 3$ | $2 \cdot 3$ | $6$ | $C_2$ | Solvable - 2 |
12.5 | $C_2\times C_6$ | $2^{2} \cdot 3$ | $2 \cdot 3$ | $12$ | $C_2\times C_6$ | Abelian - 2 |
13.1 | $C_{13}$ | $13$ | $13$ | $13$ | $C_{13}$ | Cyclic |
14.1 | $D_7$ | $2 \cdot 7$ | $2 \cdot 7$ | $5$ | $C_1$ | Solvable - 2 |
14.2 | $C_{14}$ | $2 \cdot 7$ | $2 \cdot 7$ | $14$ | $C_{14}$ | Cyclic |
15.1 | $C_{15}$ | $3 \cdot 5$ | $3 \cdot 5$ | $15$ | $C_{15}$ | Cyclic |
16.1 | $C_{16}$ | $2^{4}$ | $2^{4}$ | $16$ | $C_{16}$ | Cyclic |
16.2 | $C_4^2$ | $2^{4}$ | $2^{2}$ | $16$ | $C_4^2$ | Abelian - 2 |
16.3 | $C_2^2:C_4$ | $2^{4}$ | $2^{2}$ | $10$ | $C_2^2$ | Nilpotent - 2 |
16.4 | $C_4:C_4$ | $2^{4}$ | $2^{2}$ | $10$ | $C_2^2$ | Nilpotent - 2 |
16.5 | $C_2\times C_8$ | $2^{4}$ | $2^{3}$ | $16$ | $C_2\times C_8$ | Abelian - 2 |
16.6 | $\OD_{16}$ | $2^{4}$ | $2^{3}$ | $10$ | $C_4$ | Nilpotent - 2 |
16.7 | $D_8$ | $2^{4}$ | $2^{3}$ | $7$ | $C_2$ | Nilpotent - 3 |
16.8 | $\SD_{16}$ | $2^{4}$ | $2^{3}$ | $7$ | $C_2$ | Nilpotent - 3 |
16.9 | $Q_{16}$ | $2^{4}$ | $2^{3}$ | $7$ | $C_2$ | Nilpotent - 3 |
16.10 | $C_2^2\times C_4$ | $2^{4}$ | $2^{2}$ | $16$ | $C_2^2\times C_4$ | Abelian - 3 |
16.11 | $C_2\times D_4$ | $2^{4}$ | $2^{2}$ | $10$ | $C_2^2$ | Nilpotent - 2 |
16.12 | $C_2\times Q_8$ | $2^{4}$ | $2^{2}$ | $10$ | $C_2^2$ | Nilpotent - 2 |
16.13 | $D_4:C_2$ | $2^{4}$ | $2^{2}$ | $10$ | $C_4$ | Nilpotent - 2 |
16.14 | $C_2^4$ | $2^{4}$ | $2$ | $16$ | $C_2^4$ | Abelian - 4 |
17.1 | $C_{17}$ | $17$ | $17$ | $17$ | $C_{17}$ | Cyclic |
18.1 | $D_9$ | $2 \cdot 3^{2}$ | $2 \cdot 3^{2}$ | $6$ | $C_1$ | Solvable - 2 |
18.2 | $C_{18}$ | $2 \cdot 3^{2}$ | $2 \cdot 3^{2}$ | $18$ | $C_{18}$ | Cyclic |
18.3 | $C_3\times S_3$ | $2 \cdot 3^{2}$ | $2 \cdot 3$ | $9$ | $C_3$ | Solvable - 2 |
18.4 | $C_3:S_3$ | $2 \cdot 3^{2}$ | $2 \cdot 3$ | $6$ | $C_1$ | Solvable - 2 |
18.5 | $C_3\times C_6$ | $2 \cdot 3^{2}$ | $2 \cdot 3$ | $18$ | $C_3\times C_6$ | Abelian - 2 |
19.1 | $C_{19}$ | $19$ | $19$ | $19$ | $C_{19}$ | Cyclic |
20.1 | $C_5:C_4$ | $2^{2} \cdot 5$ | $2^{2} \cdot 5$ | $8$ | $C_2$ | Solvable - 2 |