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Subgroup label Subgroup order Cyclic Abelian Solvable
Normal Characteristic Perfect Maximal Central
Ambient label Ambient order Direct product Semidirect product
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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
46080.bx.1.a1 $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_1$ $1$ ✓
46080.bx.2.A $C_2^3\times S_5\times S_4$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.B $C_2^3.(S_4\times S_5)$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.C $(C_2\times S_5).\GL(2,\mathbb{Z}/4)$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.D $(C_2\times \GL(2,\mathbb{Z}/4)).S_5$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.E $(A_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.F $C_2^5.(C_6\times S_5)$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.G $(C_2^5\times A_5):D_6$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.H $C_2^4.D_6\times S_5$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.I $(C_2^4\times S_5):D_6$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.J $C_2^3:D_{12}\times S_5$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.K $(C_2^4\times S_5):D_6$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.2.L $D_4\times S_4\times S_5$ $2^{9} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2$ $2$ ✓
46080.bx.3.a1 $C_2\times S_5\times D_4^2$ $2^{10} \cdot 3 \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $3$ $-$
46080.bx.4.A $A_5.(C_2^3\times S_4)$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.B $(C_2^3\times A_4).S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.C $C_2\times A_5\times \GL(2,\mathbb{Z}/4)$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.D $(C_2^3\times A_4).S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.E $C_2^3\times A_4:S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.F $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.G $C_2^5:C_6\times A_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.H $C_2^2\times S_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.I $C_2^2\times S_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.J $C_2^4.D_6\times A_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.K $C_2^2.S_5\times S_4$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.L $C_2^2\times S_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.M $C_2^3:C_{12}\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.N $C_2^2.S_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.O $C_2^2\times S_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.P $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.Q $(C_2^2\times S_4):S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.R $C_2^3:D_{12}\times A_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.S $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.T $(C_2^4\times S_5):C_6$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.U $(C_2^4\times A_5).D_6$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.V $C_2^2.(S_4\times S_5)$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.W $(C_2^3\times A_5):D_{12}$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.X $C_2^2\times S_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.Y $S_4\times C_2^2:S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.Z $S_5\times \GL(2,\mathbb{Z}/4)$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.BA $\GL(2,\mathbb{Z}/4):S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.BB $D_4\times S_4\times A_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.BC $D_4\times A_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.BD $D_4\times A_4:S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.BE $C_4\times S_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.BF $S_4\times C_4:S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.BG $C_4:S_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.BH $C_4:(S_4\times S_5)$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $C_2^2$ $2^{2}$ ✓
46080.bx.4.BI $D_4\times D_6\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ ✓ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $2^{2}$ $-$
46080.bx.4.BJ $C_2^2\times S_4\times S_5$ $2^{8} \cdot 3^{2} \cdot 5$ $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ $2^{10} \cdot 3^{2} \cdot 5$ $2^{2}$ $-$
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