Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
121 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C3×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×S5 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×S5 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 4 | C3×A5 | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×C3×S4 | — | 5 | 1 | no | maximal, Hall |
| 6 | C2×C3×AGL(1,𝔽5) | — | 6 | 1 | no | maximal |
| 7 | C2×A5 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | S5 | C2×C3 | 1 | 2 | yes | direct product factor |
| 9 | C22×C3×D3 | — | 10 | 1 | no | maximal |
| 10 | C2×C3×A4 | — | 5 | 1 | no | — |
| 11 | C3×S4 | — | 5 | 2 | no | — |
| 12 | C2×C3×D5 | — | 6 | 1 | no | — |
| 13 | C3×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 14 | A5 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C2×S4 | — | 5 | 1 | no | — |
| 16 | C2×C3×D4 | — | 15 | 1 | no | — |
| 17 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 18 | C22×C32 | — | 10 | 1 | no | — |
| 19 | C2×C3×D3 | — | 10 | 1 | no | — |
| 20 | C2×C3×D3 | — | 10 | 2 | no | — |
| 21 | C2×C3×D3 | — | 10 | 1 | no | — |
| 22 | C2×C3×D3 | — | 10 | 2 | no | — |
| 23 | C3×A4 | — | 5 | 1 | no | — |
| 24 | C2×C3×C5 | — | 6 | 1 | no | — |
| 25 | C3×D5 | — | 6 | 1 | no | — |
| 26 | C3×D5 | — | 6 | 1 | no | — |
| 27 | C23×C3 | — | 5 | 1 | no | — |
| 28 | C23×C3 | — | 15 | 1 | no | — |
| 29 | C22×D3 | — | 10 | 1 | no | — |
| 30 | C2×A4 | — | 5 | 1 | no | — |
| 31 | C2×A4 | — | 10 | 1 | no | — |
| 32 | S4 | — | 5 | 2 | no | — |
| 33 | C3×D4 | — | 15 | 2 | no | — |
| 34 | C3×D4 | — | 15 | 2 | no | — |
| 35 | C2×C3×C4 | — | 15 | 1 | no | — |
| 36 | C2×D5 | — | 6 | 1 | no | — |
| 37 | AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 38 | C2×C32 | — | 10 | 2 | no | — |
| 39 | C2×C32 | — | 10 | 1 | no | — |
| 40 | C3×D3 | — | 10 | 2 | no | — |
| 41 | C3×D3 | — | 10 | 1 | no | — |
| 42 | C3×D3 | — | 10 | 1 | no | — |
| 43 | C2×D4 | — | 15 | 1 | no | Sylow |
| 44 | C3×C5 | — | 6 | 1 | no | — |
| 45 | C22×C3 | — | 5 | 1 | no | — |
| 46 | C22×C3 | — | 10 | 1 | no | — |
| 47 | C22×C3 | — | 10 | 1 | no | — |
| 48 | C22×C3 | — | 15 | 2 | no | — |
| 49 | C22×C3 | — | 15 | 1 | no | — |
| 50 | C22×C3 | — | 15 | 1 | no | — |
| 51 | C22×C3 | — | 20 | 1 | no | — |
| 52 | C22×C3 | — | 30 | 1 | no | — |
| 53 | C2×D3 | — | 10 | 1 | no | — |
| 54 | C2×D3 | — | 10 | 2 | no | — |
| 55 | C2×D3 | — | 10 | 1 | no | — |
| 56 | C2×D3 | — | 10 | 2 | no | — |
| 57 | A4 | — | 5 | 1 | no | — |
| 58 | A4 | — | 10 | 1 | no | — |
| 59 | C3×C4 | — | 15 | 2 | no | — |
| 60 | C2×C5 | — | 6 | 1 | no | — |
| 61 | D5 | — | 6 | 1 | no | — |
| 62 | D5 | — | 6 | 1 | no | — |
| 63 | C32 | — | 10 | 1 | no | Sylow |
| 64 | C23 | — | 5 | 1 | no | — |
| 65 | C23 | — | 15 | 1 | no | — |
| 66 | D4 | — | 15 | 2 | no | — |
| 67 | D4 | — | 15 | 2 | no | — |
| 68 | C2×C4 | — | 15 | 1 | no | — |
| 69 | C2×C3 | — | 10 | 2 | no | — |
| 70 | C2×C3 | — | 10 | 2 | no | — |
| 71 | C2×C3 | — | 10 | 1 | no | — |
| 72 | C2×C3 | — | 15 | 1 | no | — |
| 73 | C2×C3 | — | 15 | 1 | no | — |
| 74 | C2×C3 | — | 20 | 2 | no | — |
| 75 | C2×C3 | — | 20 | 1 | no | — |
| 76 | C2×C3 | S5 | 1 | 1 | yes | characteristic, direct product factor |
| 77 | D3 | — | 10 | 2 | no | — |
| 78 | D3 | — | 10 | 1 | no | — |
| 79 | D3 | — | 10 | 1 | no | — |
| 80 | C5 | — | 6 | 1 | no | Sylow |
| 81 | C22 | — | 5 | 1 | no | — |
| 82 | C22 | — | 10 | 1 | no | — |
| 83 | C22 | — | 15 | 2 | no | — |
| 84 | C22 | — | 15 | 1 | no | — |
| 85 | C22 | — | 15 | 1 | no | — |
| 86 | C22 | — | 30 | 1 | no | — |
| 87 | C4 | — | 15 | 2 | no | — |
| 88 | C3 | — | 10 | 1 | no | — |
| 89 | C3 | — | 20 | 1 | no | — |
| 90 | C3 | C2×S5 | 1 | 1 | yes | characteristic, direct product factor |
| 91 | C2 | — | 10 | 2 | no | — |
| 92 | C2 | — | 15 | 1 | no | — |
| 93 | C2 | — | 15 | 1 | no | — |
| 94 | C2 | C3×S5 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C3 | S5 |
| commutator, cosocle, hypocenter | A5 | C22×C3 |
| Frattini | C1 | G |
| socle | C2×C3×A5 | C2 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×A5 | C2 | semidirect |
| 2. C3×S5 | C2 | direct (N × H) |
| 3. C2×S5 | C3 | direct (N × H) |
| 4. C3×A5 | C22 | semidirect |
| 7. C2×A5 | C2×C3 | semidirect |
| 8. S5 | C2×C3 | direct (N × H) |
| 14. A5 | C22×C3 | semidirect |
| 76. C2×C3 | S5 | direct (N × H) |
| 90. C3 | C2×S5 | direct (N × H) |
| 94. C2 | C3×S5 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×A5 | 2 |
| C3×S5 | 2 |
| C2×S5 | 3 |
| C2×C3×S4 | 5 |
| C2×C3×AGL(1,𝔽5) | 6 |
| C22×C3×D3 | 10 |