Get Max Value of overlapping times in a list of Start and End Times

I am trying to get a count of the max value of times where the times intersect.

The expected result that I want need from the code example below should be 4.

There are a total of 8 times, there are 6 values that intersect in total, a group of 4, and a group of 2.

What I am trying to get is the max value of intersections but just cant get it to work.

This is the code as it stands at the moment.

void Main()
{
var times = new List<Times> {
new Times
        {
            Start = DateTime.Now,
            End = DateTime.Now.AddMinutes(10)
        },
new Times
        {
            Start = DateTime.Now,
            End = DateTime.Now.AddMinutes(10)
        },
new Times
        {
            Start = DateTime.Now.AddMinutes(2),
            End = DateTime.Now.AddMinutes(10)
        },

new Times
        {
            Start = DateTime.Now.AddMinutes(15),
            End = DateTime.Now.AddMinutes(35)
        },
new Times
        {
            Start = DateTime.Now.AddMinutes(25),
            End = DateTime.Now.AddMinutes(42)
        },
new Times
        {
            Start = DateTime.Now.AddMinutes(43),
            End = DateTime.Now.AddMinutes(50)
        },
new Times
        {
            Start = DateTime.Now.AddMinutes(55),
            End = DateTime.Now.AddMinutes(89)
        },
new Times
        {
            Start = DateTime.Now.AddMinutes(2),
            End = DateTime.Now.AddMinutes(12)
        }
};


times.OrderBy(x => x.Start);

var overlappingEvents =
                        (
                        from e1 in times
                        where times
                            .Where(e2 => e1 != e2)
                            .Where(e2 => e1.Start <= e2.End)
                            .Where(e2 => e1.End >= e2.Start)
                            .Any()
                        select e1).ToList();

overlappingEvents.OrderBy(x => x.Start);
overlappingEvents.Distinct().OrderBy(x => x.Start);
overlappingEvents.Distinct().OrderBy(x => x.Start).Count();

}

public class Times
{
public DateTime Start { get; set; }
public DateTime End { get; set; }
}

Select allOpen in new window

This is the output of the times object

Start               |  End

05/04/2017 08:38:57 |  05/04/2017 08:48:57 

05/04/2017 08:38:57 |  05/04/2017 08:48:57 

05/04/2017 08:40:57 |  05/04/2017 08:48:57 

05/04/2017 08:40:57 |  05/04/2017 08:50:57 

05/04/2017 08:53:57 |  05/04/2017 09:13:57 

05/04/2017 09:03:57 |  05/04/2017 09:20:57 

05/04/2017 09:21:57 |  05/04/2017 09:28:57 

05/04/2017 09:33:57 |  05/04/2017 10:07:57 

This is the output of the overlapping object ( I have added the line where the intersect stops)

Start               |  End

05/04/2017 08:38:57 | 05/04/2017 08:48:57 

05/04/2017 08:38:57 | 05/04/2017 08:48:57 

05/04/2017 08:40:57 |  05/04/2017 08:48:57 

05/04/2017 08:40:57 |  05/04/2017 08:50:57 


---------------------------------------

05/04/2017 08:53:57 |  05/04/2017 09:13:57

05/04/2017 09:03:57 |  05/04/2017 09:20:57

Select allOpen in new window

I would be very grateful if someone can help me get the Max value of the intersections.

Thanks