diff --git a/src/dense.jl b/src/dense.jl
index e68f7e94..21017c7e 100644
--- a/src/dense.jl
+++ b/src/dense.jl
@@ -1587,24 +1587,25 @@ factorize(A::Transpose) = transpose(factorize(parent(A)))
 factorize(a::Number)    = a # same as how factorize behaves on Diagonal types
 
 function getstructure(A::StridedMatrix)
+    require_one_based_indexing(A)
     m, n = size(A)
     if m == 1 return A[1] end
     utri    = true
     utri1   = true
     herm    = true
     sym     = true
-    for j = 1:n-1, i = j+1:m
-        if utri1
+    for j = 1:n, i = j:m
+        if (j < n) && (i > j) && utri1 # indices are off-diagonal
             if A[i,j] != 0
                 utri1 = i == j + 1
                 utri = false
             end
         end
         if sym
-            sym &= A[i,j] == A[j,i]
+            sym &= A[i,j] == transpose(A[j,i])
         end
         if herm
-            herm &= A[i,j] == conj(A[j,i])
+            herm &= A[i,j] == adjoint(A[j,i])
         end
         if !(utri1|herm|sym) break end
     end
@@ -1617,10 +1618,12 @@ function getstructure(A::StridedMatrix)
     if ltri1
         for i = 1:n-1
             if A[i,i+1] != 0
-                ltri &= false
+                ltri = false
                 break
             end
         end
+    else
+        ltri = false
     end
     return (utri, utri1, ltri, ltri1, sym, herm)
 end
diff --git a/test/dense.jl b/test/dense.jl
index 236587df..79492bdb 100644
--- a/test/dense.jl
+++ b/test/dense.jl
@@ -1347,4 +1347,11 @@ end
     end
 end
 
+@testset "structure of dense matrices" begin
+    # A is neither triangular nor symmetric/Hermitian
+    A = [1 im 2; -im 0  3; 2 3 im]
+    @test factorize(A) isa LU{ComplexF64, Matrix{ComplexF64}, Vector{Int}}
+    @test !any(LinearAlgebra.getstructure(A))
+end
+
 end # module TestDense