## Catalog of Sporadic Finite Simple Groups

The sporadic finite simple groups are as follows:

Name Abbreviation Order Date discovered Discovered by
Mathieu group $\mathrm M_{11}$ $2^4 \times 3^2 \times 5 \times 11$ $1861$
Mathieu group $\mathrm M_{12}$ $2^6 \times 3^3 \times 5 \times 11$ $1861$
Mathieu group $\mathrm M_{22}$ $2^7 \times 3^2 \times 5 \times 7 \times 11$ $1873$
Mathieu group $\mathrm M_{23}$ $2^7 \times 3^2 \times 5 \times 7 \times 11 \times 23$ $1873$
Mathieu group $\mathrm M_{24}$ $2^{10} \times 3^3 \times 5 \times 7 \times 11 \times 23$ $1873$
Janko group $\mathrm J_1$ $2^3 \times 3 \times 5 \times 7 \times 11 \times 19$ $1966$
Janko group $\mathrm J_2$ $2^7 \times 3^3 \times 5^2 \times 7$ $1968$
Janko group $\mathrm J_3$ $2^7 \times 3^5 \times 5 \times 17 \times 19$ $1968$
Janko group $\mathrm J_4$ $2^{21} \times 3^3 \times 5 \times 7 \times 11^3 \times 23 \times 29 \times 31 \times 37 \times 43$ $1976$
$\mathrm {HS}$ $2^9 \times 3^2 \times 5^3 \times 7 \times 11$ $1968$
$\mathrm {McL}$ $2^7 \times 3^6 \times 5^3 \times 7 \times 11$ $1969$
$\mathrm {Suz}$ $2^{13} \times 3^7 \times 5^2 \times 7 \times 11 \times 13$ $1969$
$\mathrm {Ru}$ $2^{14} \times 3^3 \times 5^3 \times 7 \times 13 \times 29$ $1973$
$\mathrm {He}$ $2^{10} \times 3^2 \times 5^2 \times 7^3 \times 17$ $1969$
$\mathrm {Ly}$ $2^8 \times 3^7 \times 5^6 \times 7 \times 11 \times 31 \times 37 \times 67$ $1972$
$\text {O'N}$ $2^9 \times 3^4 \times 5 \times 7^3 \times 11 \times 19 \times 31$ $1976$
$\mathrm {Co3}$ $2^{10} \times 3^7 \times 5^3 \times 7 \times 11 \times 23$ $1969$
$\mathrm {Co2}$ $2^{18} \times 3^6 \times 5^3 \times 7 \times 11 \times 23$ $1969$
$\mathrm {Co1}$ $2^{21} \times 3^9 \times 5^4 \times 7^2 \times 11 \times 13 \times 23$ $1969$
$\mathrm {Fi} \left({22}\right)$ $2^{17} \times 3^9 \times 5^2 \times 7 \times 11 \times 13$ $1971$
$\mathrm {Fi} \left({23}\right)$ $2^{18} \times 3^{13} \times 5^2 \times 7 \times 11 \times 13 \times 17 \times 23$ $1971$
$\mathrm {Fi} \left({24}\right)'$ $2^{21} \times 3^{16} \times 5^2 \times 7^3 \times 11 \times 13 \times 17 \times 23 \times 29$ $1971$
$\mathrm {HN}$ $2^{14} \times 3^6 \times 5^6 \times 7 \times 11 \times 19$ $1976$
$\mathrm {Th}$ $2^{15} \times 3^{10} \times 5^3 \times 7^2 \times 13 \times 19 \times 31$ $1976$
$\mathrm {BM}$ $2^{41} \times 3^{13} \times 5^6 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 23 \times 31 \times 47$ $1976$
Fischer-Griess Monster $\mathrm {M}$ $2^{46} \times 3^{20} \times 5^9 \times 7^6 \times 11^2 \times 13^3 \times 17 \times 19 \times 23 \times 29 \times 31 \times 41 \times 47 \times 59 \times 71$ $1981$