Finite Group DatabasebetaFinite Group DatabasebetaOrder 14400.
248 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | SL(2,𝔽3)×SL(2,𝔽5) | — | 5 | 2 | no | maximal |
| 2 | Dic5×SL(2,𝔽5) | — | 6 | 2 | no | maximal |
| 3 | Dic3×SL(2,𝔽5) | — | 10 | 2 | no | maximal |
| 4 | C2×C5×SL(2,𝔽5) | — | 6 | 2 | no | — |
| 5 | Q8×SL(2,𝔽5) | — | 5 | 2 | no | — |
| 6 | C2×C3×SL(2,𝔽5) | — | 10 | 2 | no | — |
| 7 | C5×SL(2,𝔽5) | — | 6 | 2 | no | — |
| 8 | SL(2,𝔽3)2 | — | 25 | 1 | no | Hall |
| 9 | Dic5×SL(2,𝔽3) | — | 30 | 2 | no | — |
| 10 | C4×SL(2,𝔽5) | — | 15 | 2 | no | — |
| 11 | Dic52 | — | 36 | 1 | no | — |
| 12 | C3×SL(2,𝔽5) | — | 10 | 2 | no | — |
| 13 | Dic3×SL(2,𝔽3) | — | 50 | 2 | no | — |
| 14 | C2×C5×SL(2,𝔽3) | — | 30 | 2 | no | — |
| 15 | C2×SL(2,𝔽5) | — | 60 | 2 | no | maximal |
| 16 | C2×SL(2,𝔽5) | A5 | 1 | 2 | yes | maximal normal |
| 17 | Dic3×Dic5 | — | 60 | 2 | no | — |
| 18 | C2×#1039 | — | 36 | 1 | no | — |
| 19 | C2×C5×Dic5 | — | 36 | 2 | no | — |
| 20 | Q8⋊SL(2,𝔽3) | — | 25 | 2 | no | — |
| 21 | Q8×SL(2,𝔽3) | — | 25 | 2 | no | — |
| 22 | Q8×Dic5 | — | 30 | 2 | no | — |
| 23 | C2×C3×SL(2,𝔽3) | — | 50 | 2 | no | — |
| 24 | Dic32 | — | 100 | 1 | no | — |
| 25 | C2×Dic15 | — | 60 | 2 | no | — |
| 26 | C2×C5×Dic3 | — | 60 | 2 | no | — |
| 27 | C2×C3×Dic5 | — | 60 | 2 | no | — |
| 28 | C5×SL(2,𝔽3) | — | 30 | 2 | no | — |
| 29 | SL(2,𝔽5) | — | 60 | 2 | no | — |
| 30 | SL(2,𝔽5) | SL(2,𝔽5) | 1 | 2 | yes | direct product factor |
| 31 | C22×C52 | — | 36 | 1 | no | — |
| 32 | #1039 | — | 36 | 1 | no | — |
| 33 | #1039 | — | 36 | 1 | no | — |
| 34 | C5×Dic5 | — | 36 | 2 | no | — |
| 35 | C5×Dic5 | — | 36 | 2 | no | — |
| 36 | Q8×Dic3 | — | 50 | 2 | no | — |
| 37 | C4×SL(2,𝔽3) | — | 75 | 2 | no | — |
| 38 | C2×C5×Q8 | — | 30 | 2 | no | — |
| 39 | C4⋊Dic5 | — | 90 | 2 | no | — |
| 40 | C4×Dic5 | — | 90 | 2 | no | — |
| 41 | C2×C3⋊Dic3 | — | 100 | 1 | no | — |
| 42 | C2×C3×Dic3 | — | 100 | 2 | no | — |
| 43 | C3×SL(2,𝔽3) | — | 50 | 2 | no | — |
| 44 | Q82 | — | 25 | 1 | no | Sylow |
| 45 | C22×C3×C5 | — | 60 | 2 | no | — |
| 46 | Dic15 | — | 60 | 4 | no | — |
| 47 | C3×Dic5 | — | 60 | 2 | no | — |
| 48 | C3×Dic5 | — | 60 | 2 | no | — |
| 49 | C5×Dic3 | — | 60 | 2 | no | — |
| 50 | C5×Dic3 | — | 60 | 2 | no | — |
| 51 | C2×C52 | — | 36 | 2 | no | — |
| 52 | C2×C52 | — | 36 | 1 | no | — |
| 53 | C2×C3×Q8 | — | 50 | 2 | no | — |
| 54 | C2×SL(2,𝔽3) | — | 5 | 2 | no | — |
| 55 | C2×SL(2,𝔽3) | — | 300 | 2 | no | — |
| 56 | C2×SL(2,𝔽3) | — | 100 | 2 | no | — |
| 57 | C4⋊Dic3 | — | 150 | 2 | no | — |
| 58 | C4×Dic3 | — | 150 | 2 | no | — |
| 59 | C5×Q8 | — | 30 | 2 | no | — |
| 60 | C5×Q8 | — | 90 | 2 | no | — |
| 61 | C2×C4×C5 | — | 90 | 2 | no | — |
| 62 | C2×Dic5 | — | 6 | 2 | no | — |
| 63 | C2×Dic5 | — | 90 | 2 | no | — |
| 64 | C2×Dic5 | — | 360 | 2 | no | — |
| 65 | C22×C32 | — | 100 | 1 | no | — |
| 66 | C3⋊Dic3 | — | 100 | 2 | no | — |
| 67 | C3×Dic3 | — | 100 | 2 | no | — |
| 68 | C3×Dic3 | — | 100 | 2 | no | — |
| 69 | C4⋊Q8 | — | 225 | 1 | no | — |
| 70 | C4×Q8 | — | 75 | 2 | no | — |
| 71 | C2×C3×C5 | — | 60 | 2 | no | — |
| 72 | C2×C3×C5 | — | 60 | 2 | no | — |
| 73 | C2×C3×C5 | — | 60 | 2 | no | — |
| 74 | C52 | — | 36 | 1 | no | Sylow |
| 75 | C3×Q8 | — | 50 | 2 | no | — |
| 76 | C3×Q8 | — | 150 | 2 | no | — |
| 77 | C2×C3×C4 | — | 150 | 2 | no | — |
| 78 | C2×Dic3 | — | 10 | 2 | no | — |
| 79 | C2×Dic3 | — | 600 | 1 | no | — |
| 80 | C2×Dic3 | — | 150 | 2 | no | — |
| 81 | SL(2,𝔽3) | — | 5 | 2 | no | — |
| 82 | SL(2,𝔽3) | — | 300 | 2 | no | — |
| 83 | SL(2,𝔽3) | — | 100 | 2 | no | — |
| 84 | C22×C5 | — | 6 | 2 | no | — |
| 85 | C22×C5 | — | 72 | 2 | no | — |
| 86 | C4×C5 | — | 90 | 2 | no | — |
| 87 | C4×C5 | — | 90 | 2 | no | — |
| 88 | Dic5 | — | 6 | 2 | no | — |
| 89 | Dic5 | — | 6 | 2 | no | — |
| 90 | Dic5 | — | 360 | 2 | no | — |
| 91 | Dic5 | — | 360 | 2 | no | — |
| 92 | Dic5 | — | 180 | 2 | no | — |
| 93 | C2×C32 | — | 100 | 2 | no | — |
| 94 | C2×C32 | — | 100 | 1 | no | — |
| 95 | C2×Q8 | — | 5 | 2 | no | — |
| 96 | C2×Q8 | — | 75 | 2 | no | — |
| 97 | C4⋊C4 | — | 225 | 2 | no | — |
| 98 | C42 | — | 225 | 1 | no | — |
| 99 | C3×C5 | — | 60 | 2 | no | — |
| 100 | C22×C3 | — | 10 | 2 | no | — |
| 101 | C22×C3 | — | 200 | 1 | no | — |
| 102 | C3×C4 | — | 150 | 2 | no | — |
| 103 | C3×C4 | — | 150 | 2 | no | — |
| 104 | Dic3 | — | 10 | 2 | no | — |
| 105 | Dic3 | — | 10 | 2 | no | — |
| 106 | Dic3 | — | 300 | 2 | no | — |
| 107 | Dic3 | — | 600 | 2 | no | — |
| 108 | C2×C5 | — | 6 | 2 | no | — |
| 109 | C2×C5 | — | 6 | 2 | no | — |
| 110 | C2×C5 | — | 6 | 2 | no | — |
| 111 | C2×C5 | — | 72 | 4 | no | — |
| 112 | C2×C5 | — | 72 | 2 | no | — |
| 113 | C32 | — | 100 | 1 | no | Sylow |
| 114 | Q8 | — | 5 | 2 | no | — |
| 115 | Q8 | — | 15 | 2 | no | — |
| 116 | Q8 | — | 300 | 2 | no | — |
| 117 | C2×C4 | — | 15 | 2 | no | — |
| 118 | C2×C4 | — | 225 | 1 | no | — |
| 119 | C2×C3 | — | 10 | 2 | no | — |
| 120 | C2×C3 | — | 10 | 2 | no | — |
| 121 | C2×C3 | — | 10 | 2 | no | — |
| 122 | C2×C3 | — | 200 | 2 | no | — |
| 123 | C2×C3 | — | 200 | 1 | no | — |
| 124 | C5 | — | 6 | 2 | no | — |
| 125 | C5 | — | 72 | 2 | no | — |
| 126 | C22 | A52 | 1 | 1 | yes | characteristic |
| 127 | C4 | — | 15 | 2 | no | — |
| 128 | C4 | — | 15 | 2 | no | — |
| 129 | C4 | — | 450 | 1 | no | — |
| 130 | C3 | — | 10 | 2 | no | — |
| 131 | C3 | — | 200 | 1 | no | — |
| 132 | C2 | A5×SL(2,𝔽5) | 1 | 2 | yes | — |
| 133 | C2 | Ω+(4,𝔽5) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C22 | A52 |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 30. SL(2,𝔽5) | SL(2,𝔽5) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽3)×SL(2,𝔽5) | 5 |
| Dic5×SL(2,𝔽5) | 6 |
| Dic3×SL(2,𝔽5) | 10 |
| C2×SL(2,𝔽5) | 60 |