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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
256.53563.1.a1.a1 $C_4^2.C_2^4$ $2^{8}$ $2$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_1$ $1$ ✓
256.53563.2.a1.a1 $(C_2\times D_4):D_4$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.b1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.c1.a1 $(C_2\times Q_{16}):C_2^2$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.d1.a1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.e1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.f1.a1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.g1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.h1.a1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.i1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.j1.a1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.k1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.l1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.m1.a1 $(C_2\times D_4).D_4$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.n1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.o1.a1 $(C_2\times D_4).D_4$ $2^{7}$ ✓ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.p1.a1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.p1.b1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.q1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.q1.b1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.r1.a1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.r1.b1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.s1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.s1.b1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.t1.a1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.t1.b1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.u1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.u1.b1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.v1.a1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.v1.b1 $C_4^2.D_4$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.w1.a1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.2.w1.b1 $C_4^2.C_2^3$ $2^{7}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2$ $2$ ✓
256.53563.4.a1.a1 $D_4:C_2^3$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.b1.a1 $D_8:C_2^2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.c1.a1 $C_4^2.C_2^2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.d1.a1 $C_4^2.C_2^2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.e1.a1 $C_2^3.D_4$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.f1.a1 $C_2^3.D_4$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.g1.a1 $Q_{16}:C_2^2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.h1.a1 $C_8:C_2^3$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.i1.a1 $C_4^2:C_2^2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.j1.a1 $C_4^2:C_2^2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.k1.a1 $C_2^2.D_8$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.l1.a1 $C_2^3.D_4$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.m1.a1 $C_2^2\times \SD_{16}$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.n1.a1 $C_2^2.C_2^4$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.o1.a1 $C_4^2.C_2^2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.p1.a1 $C_4^2.C_2^2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.q1.a1 $C_4^2.C_2^2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
256.53563.4.r1.a1 $(C_2^2\times C_8):C_2$ $2^{6}$ ✓ ✓ $C_4^2.C_2^4$ $2^{8}$ $C_2^2$ $2^{2}$ ✓
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