Finite Group DatabasebetaFinite Group DatabasebetaValid at every prime. The subgroup lattice of this shape is one object: each count below is a polynomial in p, and substituting a prime gives the lattice of that group.
| Order | p5 |
| Subgroups | 2p6 + 2p5 + 6p4 + 6p3 + 6p2 + 4p + 6 |
| Classes of subgroups | 4 |
| Automorphisms | p25 − p24 − p23 + p20 + p19 + p18 − p17 − p16 − p15 + p12 + p11 − p10 |
| Classes with a complement | 4 |
Each circle is one class of subgroups, labeled by the exponents of its shape — 3,2 is Cp3 × Cp2, and 2²,1 is (Cp2)2 × Cp. The badge is how many subgroups the class holds. Hover a row of the table to light its circle, or click a circle to jump to its row.
| Subgroup | Quotient | Subgroups |
|---|---|---|
| Cp (direct factor) | (Cp)4 | p4 + p3 + p2 + p + 1 |
| (Cp)2 (direct factor) | (Cp)3 | p6 + p5 + 2p4 + 2p3 + 2p2 + p + 1 |
| (Cp)3 (direct factor) | (Cp)2 | p6 + p5 + 2p4 + 2p3 + 2p2 + p + 1 |
| (Cp)4 (direct factor) | Cp | p4 + p3 + p2 + p + 1 |