Is this job posting legal?

Brigham Young University is looking for a professor in plant biology (aka botany). I honestly wonder if this is legal:

Plant Biology - Brigham Young University

The Department of Biology seeks to fill a full time, continuing status position in plant biology. Qualified applicants with a PhD, postdoctoral experience, and expertise in evolutionary or organismal biology (including, but not limited to, modern applications such as molecular ecology, systematics, genomics, evolutionary development, and so forth) are encouraged to apply. The successful candidate is expected to maintain an externally funded research program involving both undergraduate and graduate students. Excellence in teaching is required; teaching responsibilities will include general biology, plant diversity, and a graduate course in the candidate's area of expertise. The department offers competitive start-up packages and reduced teaching loads for new faculty. Interested persons should complete a BYU faculty application form at https://yjobs.byu.edu and attach a current CV and statements of teaching and research interests. Questions can be directed to: Dr. Clint Whipple, Plant Biology Search Committee Chair, 401 WIDB, Department of Biology, BYU, Provo, UT 84602.

The review process will begin September 15th, 2011 for this position and continue until the position is filled. Additional department and college information is available at website: http://biology.byu.edu/home. Brigham Young University, an equal opportunity employer, does not discriminate on the basis of race, color, gender, age, national origin, veteran status, or against qualified individuals with disabilities. All faculty are required to abide by the university's honor code and dress and grooming standards. Preference is given to qualified candidates who are members in good standing of the affiliated church, The Church of Jesus Christ of Latter-day Saints. Successful candidates are expected to support and contribute to the academic and religious missions of the university within the context of the principles and doctrine of the affiliated church.

Are they allowed to not include religion in the list of things they don't discriminate against? Are they allowed to give preferential treatment to Mormons?

Other fun stuff at BYU:

Dress and Grooming Standards

Men
A clean and well-cared-for appearance should be maintained. Clothing is inappropriate when it is sleeveless, revealing, or form fitting. Shorts must be knee-length or longer. Hairstyles should be clean and neat, avoiding extreme styles or colors, and trimmed above the collar, leaving the ear uncovered. Sideburns should not extend below the earlobe or onto the cheek. If worn, moustaches should be neatly trimmed and may not extend beyond or below the corners of the mouth. Men are expected to be clean-shaven; beards are not acceptable. Earrings and other body piercing are not acceptable. Shoes should be worn in all public campus areas.

Women
A clean and well-cared-for appearance should be maintained. Clothing is inappropriate when it is sleeveless, strapless, backless, or revealing; has slits above the knee; or is form fitting. Dresses, skirts, and shorts must be knee-length or longer. Hairstyles should be clean and neat, avoiding extremes in styles or colors. Excessive ear piercing (more than one per ear) and all other body piercing are not acceptable. Shoes should be worn in all public campus areas.

Clones, in other words.