Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
98 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×#168 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#674 | C2 | 1 | 6 | yes | maximal, direct product factor |
| 3 | C22×C4×C5 | C4 | 1 | 1 | yes | characteristic |
| 4 | C2×#168 | C4 | 1 | 3 | yes | — |
| 5 | C2×#168 | C22 | 1 | 3 | yes | — |
| 6 | #674 | C22 | 1 | 4 | yes | direct product factor |
| 7 | C22×C16 | — | 5 | 1 | no | maximal, Sylow |
| 8 | C23×C5 | C8 | 1 | 1 | yes | characteristic |
| 9 | C2×C4×C5 | C8 | 1 | 3 | yes | — |
| 10 | C2×C4×C5 | C2×C4 | 1 | 3 | yes | — |
| 11 | #168 | C2×C4 | 1 | 3 | yes | — |
| 12 | #168 | C23 | 1 | 1 | yes | characteristic |
| 13 | C22×C8 | — | 5 | 1 | no | — |
| 14 | C2×C16 | — | 5 | 6 | no | — |
| 15 | C22×C5 | C16 | 1 | 4 | yes | complemented |
| 16 | C22×C5 | C2×C8 | 1 | 3 | yes | — |
| 17 | C4×C5 | C2×C8 | 1 | 3 | yes | — |
| 18 | C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 19 | C22×C4 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 20 | C2×C8 | — | 5 | 3 | no | — |
| 21 | C2×C8 | — | 5 | 3 | no | — |
| 22 | C16 | — | 5 | 4 | no | — |
| 23 | C2×C5 | C2×C16 | 1 | 6 | yes | complemented |
| 24 | C2×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 25 | C23 | #170 | 1 | 1 | yes | characteristic |
| 26 | C2×C4 | #170 | 1 | 3 | yes | — |
| 27 | C2×C4 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 28 | C8 | — | 5 | 1 | no | — |
| 29 | C8 | — | 5 | 3 | no | — |
| 30 | C5 | C22×C16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 31 | C22 | #674 | 1 | 4 | yes | direct product factor |
| 32 | C22 | C2×#170 | 1 | 3 | yes | — |
| 33 | C4 | C2×#170 | 1 | 3 | yes | — |
| 34 | C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 35 | C2 | C2×#674 | 1 | 6 | yes | direct product factor |
| 36 | C2 | C22×#170 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C22×C16 |
| Frattini | C4 | C22×AGL(1,𝔽5) |
| Fitting | C22×C4×C5 | C4 |
| socle, prime radical | C23×C5 | C8 |
| cosocle | #168 | C23 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#674 | C2 | direct (N × H) |
| 6. #674 | C22 | direct (N × H) |
| 15. C22×C5 | C16 | semidirect |
| 23. C2×C5 | C2×C16 | semidirect |
| 30. C5 | C22×C16 | semidirect |
| 31. C22 | #674 | direct (N × H) |
| 35. C2 | C2×#674 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#168 | 2 |
| C2×#674 | 2 |
| C22×C16 | 5 |