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Subgroup label Subgroup order Cyclic Abelian Solvable
Normal Characteristic Perfect Maximal Central
Ambient label Ambient order Direct product Semidirect product
Quotient label Subgroup index Cyclic quotient Abelian quotient Solvable quotient
Maximal quotient Non-trivial proper Hall subgroup Sylow subgroup
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Label Subgroup Ambient Quotient
Name Order Sylow norm char max cent ab Name Order Name Size max ab
324000.bm.1.a1 $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $C_1$ $1$ ✓
324000.bm.2.a1 $C_3^3:D_5\wr S_3$ $2^{4} \cdot 3^{4} \cdot 5^{3}$ ✓ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $C_2$ $2$ ✓
324000.bm.2.b1 $(C_{15}^3.A_4):C_4$ $2^{4} \cdot 3^{4} \cdot 5^{3}$ ✓ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $C_2$ $2$ ✓
324000.bm.2.c1 $(C_{15}^3.A_4):C_4$ $2^{4} \cdot 3^{4} \cdot 5^{3}$ ✓ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $C_2$ $2$ ✓
324000.bm.3.a1 $C_{15}^3.C_4^2.C_2$ $2^{5} \cdot 3^{3} \cdot 5^{3}$ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $3$ $-$
324000.bm.4.a1 $C_{15}^3.S_4$ $2^{3} \cdot 3^{4} \cdot 5^{3}$ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $C_4$ $2^{2}$ ✓
324000.bm.4.b1 $C_3^3:D_5\wr C_3$ $2^{3} \cdot 3^{4} \cdot 5^{3}$ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $C_2^2$ $2^{2}$ ✓
324000.bm.4.c1 $C_{15}^3.S_4$ $2^{3} \cdot 3^{4} \cdot 5^{3}$ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $C_4$ $2^{2}$ ✓
324000.bm.4.d1 $C_{15}\wr S_3:C_4$ $2^{3} \cdot 3^{4} \cdot 5^{3}$ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2}$ $-$
324000.bm.6.a1 $C_{15}^3.C_4.C_2^2$ $2^{4} \cdot 3^{3} \cdot 5^{3}$ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $S_3$ $2 \cdot 3$
324000.bm.6.b1 $C_{15}^3.C_2.C_2^3$ $2^{4} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3$ $-$
324000.bm.6.c1 $C_5^3.(C_6\times S_3^2).C_2$ $2^{4} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3$ $-$
324000.bm.6.d1 $C_{15}^3.C_4.C_2^2$ $2^{4} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3$ $-$
324000.bm.6.e1 $C_{15}^3.C_4^2$ $2^{4} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3$ $-$
324000.bm.6.f1 $C_{15}^3.C_2.D_4$ $2^{4} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3$ $-$
324000.bm.6.g1 $C_{15}^3.C_2.D_4$ $2^{4} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3$ $-$
324000.bm.8.a1 $C_{15}^3.A_4$ $2^{2} \cdot 3^{4} \cdot 5^{3}$ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $C_2\times C_4$ $2^{3}$ ✓
324000.bm.8.b1 $C_{15}\wr S_3:C_2$ $2^{2} \cdot 3^{4} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{3}$ $-$
324000.bm.8.c1 $C_{15}^3.C_{12}$ $2^{2} \cdot 3^{4} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{3}$ $-$
324000.bm.8.d1 $C_{15}\wr C_3:C_4$ $2^{2} \cdot 3^{4} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{3}$ $-$
324000.bm.9.a1 $C_5^3.(C_4\times S_3\wr C_2)$ $2^{5} \cdot 3^{2} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $3^{2}$ $-$
324000.bm.12.a1 $C_{15}^3.C_2^3$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $D_6$ $2^{2} \cdot 3$
324000.bm.12.b1 $C_3\times C_5^3.(C_6\times S_3).C_2$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.c1 $C_{15}^3.C_2^2.C_2$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.d1 $C_{15}^3.C_2^3$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.e1 $C_{15}^2.C_{30}.C_2^2$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.f1 $C_{15}^3.D_4$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.g1 $C_{15}^2:(S_3\times F_5)$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.h1 $C_{15}^3.C_4.C_2$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.i1 $(C_5\times C_{15}^2).C_{12}.C_2$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.j1 $C_5^3.C_3:S_3.C_6.C_2$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.k1 $C_{15}^3.C_4.C_2$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.l1 $C_{15}^3.C_4.C_2$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.m1 $C_{15}^3.D_4$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.12.n1 $C_{15}^3.D_4$ $2^{3} \cdot 3^{3} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 3$ $-$
324000.bm.15.a1 $(C_3\times C_{15}^2).C_4^2.C_2$ $2^{5} \cdot 3^{3} \cdot 5^{2}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $3 \cdot 5$ $-$
324000.bm.16.a1 $C_{15}\wr S_3$ $2 \cdot 3^{4} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{4}$ $-$
324000.bm.16.b1 $C_{15}\wr C_3:C_2$ $2 \cdot 3^{4} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{4}$ $-$
324000.bm.16.c1 $(C_3\times C_{15}^2):D_{15}$ $2 \cdot 3^{4} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{4}$ $-$
324000.bm.18.a1 $C_5^3.(C_6\times S_3).C_2^2$ $2^{4} \cdot 3^{2} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3^{2}$ $-$
324000.bm.18.b1 $C_5^3.(C_4\times S_3^2)$ $2^{4} \cdot 3^{2} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3^{2}$ $-$
324000.bm.18.c1 $C_5^3.S_3^2:C_4$ $2^{4} \cdot 3^{2} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3^{2}$ $-$
324000.bm.18.d1 $C_5^3.C_3:S_3.C_2^3$ $2^{4} \cdot 3^{2} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3^{2}$ $-$
324000.bm.18.e1 $(C_5\times C_{15}^2).C_4^2$ $2^{4} \cdot 3^{2} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3^{2}$ $-$
324000.bm.18.f1 $C_5^3.(C_3\times C_6).D_4$ $2^{4} \cdot 3^{2} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3^{2}$ $-$
324000.bm.18.g1 $C_5^3.(C_4\times S_3^2)$ $2^{4} \cdot 3^{2} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3^{2}$ $-$
324000.bm.18.h1 $C_5^3.S_3^2:C_4$ $2^{4} \cdot 3^{2} \cdot 5^{3}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2 \cdot 3^{2}$ $-$
324000.bm.20.a1 $C_{15}^2:C_6.D_6$ $2^{3} \cdot 3^{4} \cdot 5^{2}$ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $2^{2} \cdot 5$ $-$
324000.bm.24.a1 $C_{15}^3.C_4$ $2^{2} \cdot 3^{3} \cdot 5^{3}$ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $S_4$ $2^{3} \cdot 3$
324000.bm.24.b1 $C_{15}^3.C_2^2$ $2^{2} \cdot 3^{3} \cdot 5^{3}$ ✓ ✓ $C_{15}^3.(C_4\times S_4)$ $2^{5} \cdot 3^{4} \cdot 5^{3}$ $C_4\times S_3$ $2^{3} \cdot 3$
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