Non-soluble groups

See soluble groups.

Groups of order 60

A5Alternating group on 5 letters; = SL2(𝔽4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simple53+A560,5

Groups of order 120

S5Symmetric group on 5 letters; = PGL2(𝔽5) = Aut(A5) = 5-cell symmetries; almost simple54+S5120,34
SL2(𝔽5)Special linear group on 𝔽52; = C2.A5 = 2I = <2,3,5>242-SL(2,5)120,5
C2×A5Direct product of C2 and A5; = icosahedron/dodecahedron symmetries103+C2xA5120,35

Groups of order 168

GL3(𝔽2)General linear group on 𝔽23; = Aut(C23) = L3(2) = L2(7); 2nd non-abelian simple73GL(3,2)168,42

Groups of order 180

C3×A5Direct product of C3 and A5; = GL2(𝔽4)153C3xA5180,19

Groups of order 240

CSU2(𝔽5)Conformal special unitary group on 𝔽52; = C2.2S5484-CSU(2,5)240,89
A5⋊C4The semidirect product of A5 and C4 acting via C4/C2=C2124A5:C4240,91
C4.A5The central extension by C4 of A5242C4.A5240,93
C2.S52nd central stem extension by C2 of S5404-C2.S5240,90
C2×S5Direct product of C2 and S5; = O3(𝔽5)104+C2xS5240,189
C4×A5Direct product of C4 and A5203C4xA5240,92
C22×A5Direct product of C22 and A520C2^2xA5240,190
C2×SL2(𝔽5)Direct product of C2 and SL2(𝔽5)48C2xSL(2,5)240,94

Groups of order 300

C5×A5Direct product of C5 and A5; = U2(𝔽4)253C5xA5300,22

Groups of order 336

SL2(𝔽7)Special linear group on 𝔽72; = C2.GL3(𝔽2)164SL(2,7)336,114
PGL2(𝔽7)Projective linear group on 𝔽72; = GL3(𝔽2)C2 = Aut(GL3(𝔽2)); almost simple86+PGL(2,7)336,208
C2×GL3(𝔽2)Direct product of C2 and GL3(𝔽2)143C2xGL(3,2)336,209

Groups of order 360

A6Alternating group on 6 letters; = PSL2(𝔽9) = L2(9); 3rd non-abelian simple65+A6360,118
ΓL2(𝔽4)Semilinear group on 𝔽42; = C3S5156GammaL(2,4)360,120
C3×S5Direct product of C3 and S5154C3xS5360,119
S3×A5Direct product of S3 and A5156+S3xA5360,121
C6×A5Direct product of C6 and A5303C6xA5360,122
C3×SL2(𝔽5)Direct product of C3 and SL2(𝔽5)722C3xSL(2,5)360,51

Groups of order 420

C7×A5Direct product of C7 and A5353C7xA5420,13

Groups of order 480

GL2(𝔽5)General linear group on 𝔽52; = SL2(𝔽5)1C4 = Aut(C52)244GL(2,5)480,218
C4⋊S5The semidirect product of C4 and S5 acting via S5/A5=C2206C4:S5480,944
C22⋊S5The semidirect product of C22 and S5 acting via S5/A5=C2206+C2^2:S5480,951
A5⋊Q8The semidirect product of A5 and Q8 acting via Q8/C4=C2246A5:Q8480,945
A5⋊C8The semidirect product of A5 and C8 acting via C8/C4=C2404A5:C8480,217
C4.3S53rd non-split extension by C4 of S5 acting via S5/A5=C2404C4.3S5480,948
C8.A5The central extension by C8 of A5482C8.A5480,221
D4.A5The non-split extension by D4 of A5 acting through Inn(D4)484-D4.A5480,957
Q8.A5The non-split extension by Q8 of A5 acting through Inn(Q8)484+Q8.A5480,959
C4.6S53rd central extension by C4 of S5484C4.6S5480,946
C4.S52nd non-split extension by C4 of S5 acting via S5/A5=C2484C4.S5480,947
C22.S5The non-split extension by C22 of S5 acting via S5/A5=C2484-C2^2.S5480,953
C22.2S51st central extension by C22 of S596C2^2.2S5480,219
C4×S5Direct product of C4 and S5; = CO3(𝔽5)204C4xS5480,943
D4×A5Direct product of D4 and A5206+D4xA5480,956
C22×S5Direct product of C22 and S520C2^2xS5480,1186
C8×A5Direct product of C8 and A5403C8xA5480,220
Q8×A5Direct product of Q8 and A5406-Q8xA5480,958
C23×A5Direct product of C23 and A540C2^3xA5480,1187
C4×SL2(𝔽5)Direct product of C4 and SL2(𝔽5)96C4xSL(2,5)480,222
C2×CSU2(𝔽5)Direct product of C2 and CSU2(𝔽5)96C2xCSU(2,5)480,949
C22×SL2(𝔽5)Direct product of C22 and SL2(𝔽5)96C2^2xSL(2,5)480,960
C2×A5⋊C4Direct product of C2 and A5⋊C424C2xA5:C4480,952
C2×C4×A5Direct product of C2×C4 and A540C2xC4xA5480,954
C2×C4.A5Direct product of C2 and C4.A548C2xC4.A5480,955
C2×C2.S5Direct product of C2 and C2.S580C2xC2.S5480,950