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Lie symmetry analysis is one of the powerful tools to analyze nonlinearordinary differential equations. We review the effectiveness of this method interms of various symmetries. We present the method of deriving Lie pointsymmetries, contact symmetries, hidden symmetries, nonlocal symmetries,$\lambda$-symmetries, adjoint symmetries and telescopic vector fields of asecond-order ordinary differential equation. We also illustrate the algorithminvolved in each method by considering a nonlinear oscillator equation as anexample. The connections between (i) symmetries and integrating factors and(ii) symmetries and integrals are also discussed and illustrated through thesame example. The interconnections between some of the above symmetries, thatis (i) Lie point symmetries and $\lambda$-symmetries and (ii) exponentialnonlocal symmetries and $\lambda$-symmetries are also discussed. The orderreduction procedure is invoked to derive the general solution of thesecond-order equation.

Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study