Finite Group DatabasebetaFinite Group DatabasebetaOrder 23125.
64 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C53×C37 | C5 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C37×H(3,𝔽5) | C5 | 1 | 5 | yes | maximal, complemented |
| 3 | C52×C37 | — | 5 | 1 | no | — |
| 4 | C52×C37 | — | 5 | 5 | no | — |
| 5 | C52×C37 | — | 5 | 5 | no | — |
| 6 | C52×C37 | C52 | 1 | 1 | yes | characteristic |
| 7 | H(3,𝔽5)⋊C5 | C37 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 8 | C5×C37 | — | 5 | 5 | no | — |
| 9 | C5×C37 | — | 5 | 1 | no | — |
| 10 | C5×C37 | — | 25 | 5 | no | — |
| 11 | C5×C37 | H(3,𝔽5) | 1 | 1 | yes | characteristic |
| 12 | C53 | C5×C37 | 1 | 1 | yes | characteristic, complemented |
| 13 | H(3,𝔽5) | C5×C37 | 1 | 5 | yes | complemented |
| 14 | C37 | H(3,𝔽5)⋊C5 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 15 | C52 | — | 5 | 5 | no | — |
| 16 | C52 | — | 5 | 1 | no | — |
| 17 | C52 | — | 5 | 5 | no | — |
| 18 | C52 | C52×C37 | 1 | 1 | yes | characteristic |
| 19 | C5 | — | 5 | 5 | no | — |
| 20 | C5 | — | 5 | 1 | no | — |
| 21 | C5 | — | 25 | 5 | no | — |
| 22 | C5 | C37×H(3,𝔽5) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C37 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C52 | C52×C37 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C37 ◁ C52×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×C37 | C5 | semidirect |
| 2. C37×H(3,𝔽5) | C5 | semidirect |
| 7. H(3,𝔽5)⋊C5 | C37 | direct (N × H) |
| 12. C53 | C5×C37 | semidirect |
| 13. H(3,𝔽5) | C5×C37 | semidirect |
| 14. C37 | H(3,𝔽5)⋊C5 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×C37 | 5 |
| C37×H(3,𝔽5) | 5 |
| H(3,𝔽5)⋊C5 | 37 |