Finite Group DatabasebetaFinite Group DatabasebetaOrder 120.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C3×D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | ΓL(1,𝔽16) | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×AGL(1,𝔽5) | — | 3 | 1 | no | maximal, Hall |
| 4 | C2×C3×C5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C3×D5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C3×D5 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×Dic3 | — | 5 | 1 | no | maximal, Hall |
| 8 | C2×D5 | D3 | 1 | 1 | yes | characteristic |
| 9 | AGL(1,𝔽5) | — | 3 | 2 | no | — |
| 10 | C3×C5 | C2×C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 11 | C22×C3 | — | 5 | 1 | no | — |
| 12 | Dic3 | — | 5 | 2 | no | — |
| 13 | C2×C5 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 14 | D5 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 15 | D5 | C2×D3 | 1 | 1 | yes | characteristic |
| 16 | C2×C4 | — | 15 | 1 | no | Sylow |
| 17 | C2×C3 | — | 5 | 1 | no | — |
| 18 | C2×C3 | — | 5 | 1 | no | — |
| 19 | C2×C3 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 20 | C5 | C2×Dic3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | C22 | — | 5 | 1 | no | — |
| 22 | C4 | — | 15 | 2 | no | — |
| 23 | C3 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 24 | C2 | — | 5 | 1 | no | — |
| 25 | C2 | — | 5 | 1 | no | — |
| 26 | C2 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C2×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5 | C4 |
| prime radical | C2×C3×D5 | C2 |
| cosocle | C3×D5 | C22 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. ΓL(1,𝔽16) | C2 | direct (N × H) |
| 4. C2×C3×C5 | C4 | semidirect |
| 5. C3×D5 | C4 | semidirect |
| 10. C3×C5 | C2×C4 | semidirect |
| 13. C2×C5 | Dic3 | semidirect |
| 14. D5 | Dic3 | semidirect |
| 19. C2×C3 | AGL(1,𝔽5) | semidirect |
| 20. C5 | C2×Dic3 | semidirect |
| 23. C3 | C2×AGL(1,𝔽5) | semidirect |
| 26. C2 | ΓL(1,𝔽16) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D5 | 2 |
| ΓL(1,𝔽16) | 2 |
| C2×AGL(1,𝔽5) | 3 |
| C2×Dic3 | 5 |