Finite Group DatabasebetaFinite Group DatabasebetaGroups sandwiched between a non-abelian simple group and its automorphism group.
| order | group | socle | G/socle |
|---|---|---|---|
| 60 | A5 #304 | simple | |
| 120 | S5 #1224 | A5 | C2 |
| 168 | GL(3,𝔽2) #4272 | simple | |
| 336 | SO(3,𝔽7) #66739 | GL(3,𝔽2) | C2 |
| 360 | A6 #67188 | simple | |
| 504 | SL(2,𝔽8) #92730 | simple | |
| 660 | PSL(2,𝔽11) #10620343 | simple | |
| 720 | S6 #10624294 | A6 | C2 |
| 720 | PGL(2,𝔽9) #10624295 | A6 | C2 |
| 720 | #10624296 | A6 | C2 |
| 1092 | PSL(2,𝔽13) #11763977 | simple | |
| 1320 | PGL(2,𝔽11) #13050429 | PSL(2,𝔽11) | C2 |
| 1440 | PΓL(2,𝔽9) #13089524 | A6 | C22 |
| 1512 | ΣL(2,𝔽8) #13094882 | SL(2,𝔽8) | C3 |
| 2184 | PGL(2,𝔽13) #72210356 | PSL(2,𝔽13) | C2 |
| 2448 | PSL(2,𝔽17) #72209593 | simple | |
| 2520 | A7 #72209594 | simple | |
| 3420 | PSL(2,𝔽19) #16811951 | simple | |
| 4080 | SL(2,𝔽16) #72209607 | simple | |
| 4896 | PGL(2,𝔽17) #72210361 | PSL(2,𝔽17) | C2 |
| 5040 | S7 #72210209 | A7 | C2 |
| 5616 | SL(3,𝔽3) #72209613 | simple | |
| 6048 | SU(3,𝔽3) #72209615 | simple | |
| 6072 | PSL(2,𝔽23) #72209616 | simple | |
| 6840 | PGL(2,𝔽19) #72210366 | PSL(2,𝔽19) | C2 |
| 7800 | PSL(2,𝔽25) #72209633 | simple | |
| 7920 | M11 #72209634 | simple | |
| 8160 | GO−(4,𝔽4) #72219522 | SL(2,𝔽16) | C2 |
| 9828 | PSL(2,𝔽27) #72209638 | simple | |
| 12144 | PGL(2,𝔽23) #72210372 | PSL(2,𝔽23) | C2 |
| 12180 | PSL(2,𝔽29) #27131832 | simple | |
| 14880 | PSL(2,𝔽31) #72209679 | simple | |
| 15600 | PGL(2,𝔽25) #72210377 | PSL(2,𝔽25) | C2 |
| 15600 | PΣL(2,𝔽25) #72219556 | PSL(2,𝔽25) | C2 |
| 15600 | #72224461 | PSL(2,𝔽25) | C2 |
| 16320 | ΣL(2,𝔽16) #72224462 | SL(2,𝔽16) | C4 |
| 19656 | PGL(2,𝔽27) #72210380 | PSL(2,𝔽27) | C2 |
| 20160 | A8 #72209754 | simple | |
| 20160 | PSL(3,𝔽4) #72209755 | simple | |
| 24360 | PGL(2,𝔽29) #72210386 | PSL(2,𝔽29) | C2 |
| 25308 | PSL(2,𝔽37) #38761268 | simple | |
| 25920 | SU(4,𝔽2) #72209939 | simple | |
| 29120 | #72209943 | simple | |
| 29484 | PΣL(2,𝔽27) #72224464 | PSL(2,𝔽27) | C3 |
| 29760 | PGL(2,𝔽31) #72210389 | PSL(2,𝔽31) | C2 |
| 31200 | PΓL(2,𝔽25) #72224463 | PSL(2,𝔽25) | C22 |
| 32736 | SL(2,𝔽32) #72210239 | simple | |
| 34440 | PSL(2,𝔽41) #72210533 | simple | |
| 39732 | PSL(2,𝔽43) #51432461 | simple | |
| 40320 | S8 #72210210 | A8 | C2 |
| 40320 | PΣL(3,𝔽4) #72224470 | PSL(3,𝔽4) | C2 |
| 50616 | PGL(2,𝔽37) #72210400 | PSL(2,𝔽37) | C2 |
| 51840 | SO(5,𝔽3) #72225614 | SU(4,𝔽2) | C2 |
| 51888 | PSL(2,𝔽47) #72210538 | simple | |
| 58800 | PSL(2,𝔽49) #72210558 | simple | |
| 58968 | PΓL(2,𝔽27) #72224466 | PSL(2,𝔽27) | C2×C3 |
| 60480 | PGL(3,𝔽4) #72210415 | PSL(3,𝔽4) | C3 |
| 62400 | #72268924 | simple | |