Finite Group DatabasebetaFinite Group DatabasebetaOrder 16320.
69 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | GO−(4,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | SL(2,𝔽16) | C4 | 1 | 1 | yes | characteristic, complemented |
| 3 | AΓL(1,𝔽16) | — | 17 | 1 | no | maximal, Hall |
| 4 | ((C24⋊C5)⋊C2)⋊C3 | — | 17 | 1 | no | — |
| 5 | #66297 | — | 51 | 1 | no | Hall |
| 6 | AGL(1,𝔽16) | — | 17 | 1 | no | — |
| 7 | A5⋊C4 | — | 68 | 1 | no | maximal |
| 8 | #4625 | — | 85 | 1 | no | Hall |
| 9 | (C24⋊C5)⋊C2 | — | 17 | 1 | no | — |
| 10 | #3633 | — | 120 | 1 | no | maximal |
| 11 | D3×AGL(1,𝔽5) | — | 136 | 1 | no | maximal |
| 12 | C2×A5 | — | 68 | 1 | no | — |
| 13 | #120⋊C3 | — | 85 | 1 | no | — |
| 14 | C24⋊C5 | — | 17 | 1 | no | — |
| 15 | #595 | — | 120 | 1 | no | — |
| 16 | AΣL(1,𝔽16) | — | 255 | 1 | no | Sylow |
| 17 | D3×D5 | — | 136 | 1 | no | — |
| 18 | ΓL(1,𝔽16) | — | 136 | 1 | no | — |
| 19 | C3×AGL(1,𝔽5) | — | 136 | 1 | no | — |
| 20 | A5 | — | 68 | 1 | no | — |
| 21 | C22⋊A4 | — | 17 | 1 | no | — |
| 22 | SL(2,ℤ/4) | — | 340 | 1 | no | — |
| 23 | C2×AGL(1,𝔽5) | — | 408 | 1 | no | — |
| 24 | D17 | — | 120 | 1 | no | — |
| 25 | #120 | — | 85 | 1 | no | — |
| 26 | #100 | — | 255 | 1 | no | — |
| 27 | #99 | — | 255 | 1 | no | — |
| 28 | D15 | — | 136 | 1 | no | — |
| 29 | C3×D5 | — | 136 | 1 | no | — |
| 30 | C5×D3 | — | 136 | 1 | no | — |
| 31 | C2×A4 | — | 340 | 1 | no | — |
| 32 | C4×D3 | — | 680 | 1 | no | — |
| 33 | C2×D5 | — | 408 | 1 | no | — |
| 34 | AGL(1,𝔽5) | — | 136 | 1 | no | — |
| 35 | AGL(1,𝔽5) | — | 408 | 1 | no | — |
| 36 | C17 | — | 120 | 1 | no | Sylow |
| 37 | C24 | — | 17 | 1 | no | — |
| 38 | C2×D4 | — | 255 | 1 | no | — |
| 39 | M16 | — | 510 | 1 | no | — |
| 40 | C22⋊C4 | — | 510 | 1 | no | — |
| 41 | C22⋊C4 | — | 255 | 1 | no | — |
| 42 | C3×C5 | — | 136 | 1 | no | Hall |
| 43 | C2×D3 | — | 680 | 1 | no | — |
| 44 | A4 | — | 340 | 1 | no | — |
| 45 | C3×C4 | — | 680 | 1 | no | — |
| 46 | Dic3 | — | 680 | 1 | no | — |
| 47 | C2×C5 | — | 408 | 1 | no | — |
| 48 | D5 | — | 136 | 1 | no | — |
| 49 | D5 | — | 408 | 1 | no | — |
| 50 | C23 | — | 85 | 1 | no | — |
| 51 | C23 | — | 255 | 1 | no | — |
| 52 | D4 | — | 1020 | 1 | no | — |
| 53 | C2×C4 | — | 1020 | 1 | no | — |
| 54 | C2×C4 | — | 255 | 1 | no | — |
| 55 | C8 | — | 1020 | 1 | no | — |
| 56 | C2×C3 | — | 680 | 1 | no | — |
| 57 | D3 | — | 136 | 1 | no | — |
| 58 | D3 | — | 680 | 1 | no | — |
| 59 | C5 | — | 136 | 1 | no | Sylow |
| 60 | C22 | — | 85 | 1 | no | — |
| 61 | C22 | — | 510 | 1 | no | — |
| 62 | C22 | — | 510 | 1 | no | — |
| 63 | C4 | — | 680 | 1 | no | — |
| 64 | C4 | — | 510 | 1 | no | — |
| 65 | C3 | — | 136 | 1 | no | Sylow |
| 66 | C2 | — | 68 | 1 | no | — |
| 67 | C2 | — | 255 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | SL(2,𝔽16) | C4 |
| prime radical, cosocle | GO−(4,𝔽4) | C2 |
| LCS | G ▷ SL(2,𝔽16) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. SL(2,𝔽16) | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| GO−(4,𝔽4) | 2 |
| AΓL(1,𝔽16) | 17 |
| A5⋊C4 | 68 |
| #3633 | 120 |
| D3×AGL(1,𝔽5) | 136 |