Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
338 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×Dic5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×#170 | C2 | 1 | 14 | yes | maximal, direct product factor |
| 3 | C24×C5 | C4 | 1 | 1 | yes | characteristic |
| 4 | C22×Dic5 | C4 | 1 | 7 | yes | — |
| 5 | C22×Dic5 | C22 | 1 | 7 | yes | — |
| 6 | C2×#170 | C22 | 1 | 28 | yes | direct product factor |
| 7 | C23×C8 | — | 5 | 1 | no | maximal, Sylow |
| 8 | C23×C5 | C8 | 1 | 8 | yes | complemented |
| 9 | C23×C5 | C2×C4 | 1 | 7 | yes | — |
| 10 | C2×Dic5 | C2×C4 | 1 | 21 | yes | — |
| 11 | C2×Dic5 | C23 | 1 | 7 | yes | — |
| 12 | #170 | C23 | 1 | 8 | yes | direct product factor |
| 13 | C23×C4 | — | 5 | 1 | no | — |
| 14 | C22×C8 | — | 5 | 14 | no | — |
| 15 | C22×C5 | C2×C8 | 1 | 28 | yes | complemented |
| 16 | C22×C5 | C22×C4 | 1 | 7 | yes | — |
| 17 | Dic5 | C22×C4 | 1 | 7 | yes | — |
| 18 | Dic5 | C24 | 1 | 1 | yes | characteristic |
| 19 | C24 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 20 | C22×C4 | — | 5 | 7 | no | — |
| 21 | C22×C4 | — | 5 | 7 | no | — |
| 22 | C2×C8 | — | 5 | 28 | no | — |
| 23 | C2×C5 | C22×C8 | 1 | 14 | yes | complemented |
| 24 | C2×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 25 | C23 | #170 | 1 | 8 | yes | direct product factor |
| 26 | C23 | C2×AGL(1,𝔽5) | 1 | 7 | yes | — |
| 27 | C2×C4 | — | 5 | 7 | no | — |
| 28 | C2×C4 | — | 5 | 21 | no | — |
| 29 | C8 | — | 5 | 8 | no | — |
| 30 | C5 | C23×C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 31 | C22 | C2×#170 | 1 | 28 | yes | direct product factor |
| 32 | C22 | C22×AGL(1,𝔽5) | 1 | 7 | yes | — |
| 33 | C4 | — | 5 | 1 | no | — |
| 34 | C4 | — | 5 | 7 | no | — |
| 35 | C2 | C22×#170 | 1 | 14 | yes | direct product factor |
| 36 | C2 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C23×C8 |
| Frattini | C2 | C23×AGL(1,𝔽5) |
| Fitting, socle, prime radical | C24×C5 | C4 |
| cosocle | Dic5 | C24 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#170 | C2 | direct (N × H) |
| 6. C2×#170 | C22 | direct (N × H) |
| 8. C23×C5 | C8 | semidirect |
| 12. #170 | C23 | direct (N × H) |
| 15. C22×C5 | C2×C8 | semidirect |
| 23. C2×C5 | C22×C8 | semidirect |
| 25. C23 | #170 | direct (N × H) |
| 30. C5 | C23×C8 | semidirect |
| 31. C22 | C2×#170 | direct (N × H) |
| 35. C2 | C22×#170 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Dic5 | 2 |
| C22×#170 | 2 |
| C23×C8 | 5 |