I am following J.B. Fraleigh: A first course in abstract algebra. In the text page $101$ the author supposes $H,K$ are subgroups of $G$ then uses the notation $H \leq K \leq G$. Does he intend to write, $H \subseteq K \subseteq G$, $|H| \leq |K| \leq |G|$, or something different? I cannot recall that such notation for sets has been defined in my prior maths courses.
As the other answer makes clear people will prefer different things. But it is completely standard to write $$ H\leq G $$ when we want to say that $H$ is a subgroup of $G$. So saying that $$ H\leq K\leq G $$ says that $H$ is a subgroup of $K$ and $K$ is a subgroup of $G$. (As a side note, $\leq$ is a transitive, so this would also mean that $H$ is a subgroup of $G$.)
I looked in a couple of my abstract algebra books, and the following books use $\leq$ for subgroups
- Gallian's Contemporary Abstract Algebra book
- Fraleigh's A First Course in Abstract Algebra
- Rotman's Advanced Modern Algebra
- Herstein's Topics in Aglebra
- Dummit and Foote's Abstract Algebra
Even Wikipedia's article on subgroups uses the notation. Hungerford uses $<$.
One advantage of using $\leq$ over $\subseteq$ is that it distinguishes between being a subset and a subgroup. In a proof you might first show that $H$ is a subset of $G$ and then later conclude that $H$ is a subgroup. So having different notations can be helpful.