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 | p7 |
| Subgroups | 2p7 + 8p6 + 10p5 + 16p4 + 12p3 + 10p2 + 6p + 8 |
| Classes of subgroups | 22 |
| Automorphisms | p29 − 2p28 − p27 + 3p26 + p25 − p24 − 3p23 + p22 + 2p21 − p20 |
| Classes with a complement | 10 |
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) | (Cp2)2 × (Cp)2 | p4 + p3 + p2 |
| Cp | Cp2 × (Cp)4 | p + 1 |
| Cp2 (direct factor) | Cp2 × (Cp)3 | p5 + p4 |
| (Cp)2 (direct factor) | (Cp2)2 × Cp | p6 + p5 + p4 |
| (Cp)2 | Cp2 × (Cp)3 | p4 + 2p3 + 2p2 + p |
| (Cp)2 (characteristic) | (Cp)5 | 1 |
| Cp2 × Cp (direct factor) | Cp2 × (Cp)2 | p7 + 2p6 + 2p5 + p4 |
| Cp2 × Cp | (Cp)4 | p4 + p3 |
| (Cp)3 (direct factor) | (Cp2)2 | p6 |
| (Cp)3 | Cp2 × (Cp)2 | p5 + 2p4 + 2p3 + p2 |
| (Cp)3 | (Cp)4 | p2 + p + 1 |
| (Cp2)2 (direct factor) | (Cp)3 | p6 |
| Cp2 × (Cp)2 (direct factor) | Cp2 × Cp | p7 + 2p6 + 2p5 + p4 |
| Cp2 × (Cp)2 | (Cp)3 | p5 + 2p4 + 2p3 + p2 |
| (Cp)4 | Cp2 × Cp | p4 + p3 |
| (Cp)4 | (Cp)3 | p2 + p + 1 |
| (Cp2)2 × Cp (direct factor) | (Cp)2 | p6 + p5 + p4 |
| Cp2 × (Cp)3 (direct factor) | Cp2 | p5 + p4 |
| Cp2 × (Cp)3 | (Cp)2 | p4 + 2p3 + 2p2 + p |
| (Cp)5 (characteristic) | (Cp)2 | 1 |
| (Cp2)2 × (Cp)2 (direct factor) | Cp | p4 + p3 + p2 |
| Cp2 × (Cp)4 | Cp | p + 1 |