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 | 2p12 + 2p11 + 6p10 + 8p9 + 12p8 + 12p7 + 18p6 + 16p5 + 16p4 + 12p3 + 10p2 + 6p + 8 |
| Classes of subgroups | 6 |
| Automorphisms | p49 − p48 − p47 + p44 + p42 + p41 − p39 − p38 − 2p37 + 2p33 + p32 + p31 − p29 − p28 − p26 + p23 + p22 − p21 |
| Classes with a complement | 6 |
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)6 | p6 + p5 + p4 + p3 + p2 + p + 1 |
| (Cp)2 (direct factor) | (Cp)5 | p10 + p9 + 2p8 + 2p7 + 3p6 + 3p5 + 3p4 + 2p3 + 2p2 + p + 1 |
| (Cp)3 (direct factor) | (Cp)4 | p12 + p11 + 2p10 + 3p9 + 4p8 + 4p7 + 5p6 + 4p5 + 4p4 + 3p3 + 2p2 + p + 1 |
| (Cp)4 (direct factor) | (Cp)3 | p12 + p11 + 2p10 + 3p9 + 4p8 + 4p7 + 5p6 + 4p5 + 4p4 + 3p3 + 2p2 + p + 1 |
| (Cp)5 (direct factor) | (Cp)2 | p10 + p9 + 2p8 + 2p7 + 3p6 + 3p5 + 3p4 + 2p3 + 2p2 + p + 1 |
| (Cp)6 (direct factor) | Cp | p6 + p5 + p4 + p3 + p2 + p + 1 |