Finite Group DatabasebetaFinite Group DatabasebetaOrder 23765.
40 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C72×C97 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 2 | C5×C7×C97 | C7 | 1 | 8 | yes | maximal, direct product factor |
| 3 | C7×C97 | C5×C7 | 1 | 8 | yes | direct product factor |
| 4 | C5×C97 | C72 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 5 | C5×C72 | C97 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 6 | C97 | C5×C72 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 7 | C72 | C5×C97 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 8 | C5×C7 | C7×C97 | 1 | 8 | yes | direct product factor |
| 9 | C7 | C5×C7×C97 | 1 | 8 | yes | direct product factor |
| 10 | C5 | C72×C97 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C72×C97 | C5 | direct (N × H) |
| 2. C5×C7×C97 | C7 | direct (N × H) |
| 3. C7×C97 | C5×C7 | direct (N × H) |
| 4. C5×C97 | C72 | direct (N × H) |
| 5. C5×C72 | C97 | direct (N × H) |
| 6. C97 | C5×C72 | direct (N × H) |
| 7. C72 | C5×C97 | direct (N × H) |
| 8. C5×C7 | C7×C97 | direct (N × H) |
| 9. C7 | C5×C7×C97 | direct (N × H) |
| 10. C5 | C72×C97 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C72×C97 | 5 |
| C5×C7×C97 | 7 |
| C5×C72 | 97 |