Finite Group DatabasebetaFinite Group DatabasebetaOrder 24198.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #27130719 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D37×C109 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C37×C109 | C2×C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C2×C109⋊C3 | — | 37 | 1 | no | maximal, Hall |
| 5 | C109⋊C3 | — | 37 | 1 | no | Hall |
| 6 | D37⋊C3 | — | 109 | 1 | no | maximal, Hall |
| 7 | C2×C109 | — | 111 | 1 | no | Hall |
| 8 | C37⋊C3 | — | 109 | 1 | no | Hall |
| 9 | C109 | D37⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | D37 | C109⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 11 | C37 | C2×C109⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×C3 | — | 4033 | 1 | no | Hall |
| 13 | C3 | — | 4033 | 1 | no | Sylow |
| 14 | C2 | — | 111 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C37×C109 | C2×C3 |
| LCS | G ▷ C37×C109 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27130719 | C2 | semidirect |
| 2. D37×C109 | C3 | semidirect |
| 3. C37×C109 | C2×C3 | semidirect |
| 9. C109 | D37⋊C3 | semidirect |
| 10. D37 | C109⋊C3 | semidirect |
| 11. C37 | C2×C109⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27130719 | 2 |
| D37×C109 | 3 |
| C2×C109⋊C3 | 37 |
| D37⋊C3 | 109 |