Finite Group DatabasebetaFinite Group DatabasebetaOrder 2400.
182 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 16 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 | C2×C5×SL(2,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5⋊#6732 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D5×SL(2,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5×SL(2,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 5 | #4147⋊C3 | — | 5 | 1 | no | maximal |
| 6 | #91881 | — | 5 | 1 | no | maximal |
| 7 | #87879 | — | 6 | 1 | no | maximal |
| 8 | C2×C5×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 9 | #6779 | — | 10 | 1 | no | maximal |
| 10 | D5×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 11 | C5⋊(#41⋊C3) | — | 5 | 1 | no | — |
| 12 | C2×SL(2,𝔽5) | — | 5 | 1 | no | — |
| 13 | C2×SL(2,𝔽5) | D5 | 1 | 1 | yes | characteristic, complemented |
| 14 | #6732 | — | 5 | 1 | no | — |
| 15 | C22⋊(C5⋊D5) | — | 6 | 1 | no | — |
| 16 | C5×C22⋊D5 | — | 6 | 1 | no | — |
| 17 | C2×C5×Dic5 | — | 6 | 1 | no | — |
| 18 | #6039 | — | 6 | 1 | no | — |
| 19 | #6038 | — | 6 | 1 | no | — |
| 20 | #6036 | — | 6 | 1 | no | — |
| 21 | D5×Dic5 | — | 6 | 1 | no | — |
| 22 | #4147 | — | 5 | 1 | no | — |
| 23 | #1220 | — | 10 | 1 | no | — |
| 24 | C2×C5×Dic3 | — | 10 | 1 | no | — |
| 25 | C3×C22⋊D5 | — | 10 | 1 | no | — |
| 26 | C5×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 27 | #1204 | — | 10 | 1 | no | — |
| 28 | #1202 | — | 10 | 1 | no | — |
| 29 | #1200 | — | 10 | 1 | no | — |
| 30 | D5×Dic3 | — | 10 | 1 | no | — |
| 31 | SL(2,𝔽5) | C2×D5 | 1 | 1 | yes | characteristic |
| 32 | C22×C52 | — | 6 | 1 | no | — |
| 33 | C2×C5⋊D5 | — | 6 | 1 | no | — |
| 34 | C2×C5×D5 | — | 6 | 1 | no | — |
| 35 | #1039 | — | 6 | 1 | no | — |
| 36 | C5×Dic5 | — | 6 | 1 | no | — |
| 37 | C5×Dic5 | — | 6 | 1 | no | — |
| 38 | C5×Dic5 | — | 6 | 1 | no | — |
| 39 | 2−1+4⋊C3 | — | 25 | 1 | no | Hall |
| 40 | C2×C5×Q8 | — | 5 | 1 | no | — |
| 41 | #713 | — | 5 | 1 | no | — |
| 42 | #713 | — | 15 | 1 | no | — |
| 43 | Q8×D5 | — | 5 | 1 | no | — |
| 44 | Q8×D5 | — | 15 | 1 | no | — |
| 45 | #711 | — | 30 | 1 | no | — |
| 46 | #709 | — | 15 | 1 | no | — |
| 47 | #709 | — | 15 | 1 | no | — |
| 48 | C22×C3×C5 | — | 10 | 1 | no | — |
| 49 | C2×D15 | — | 10 | 1 | no | — |
| 50 | C2×C3×D5 | — | 10 | 1 | no | — |
| 51 | Dic15 | — | 10 | 1 | no | — |
| 52 | C3×Dic5 | — | 10 | 1 | no | — |
| 53 | C5×Dic3 | — | 10 | 1 | no | — |
| 54 | C5×Dic3 | — | 10 | 1 | no | — |
| 55 | C2×C52 | — | 6 | 1 | no | — |
| 56 | C2×C52 | — | 12 | 1 | no | — |
| 57 | C5⋊D5 | — | 12 | 1 | no | — |
| 58 | C5×D5 | — | 12 | 1 | no | — |
| 59 | #237 | — | 50 | 1 | no | — |
| 60 | #41⋊C3 | — | 25 | 1 | no | — |
| 61 | C2×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 62 | C2×SL(2,𝔽3) | — | 25 | 1 | no | — |
| 63 | C5×Q8 | — | 5 | 1 | no | — |
| 64 | C5×Q8 | — | 15 | 1 | no | — |
| 65 | C5×D4 | — | 30 | 1 | no | — |
| 66 | C2×C4×C5 | — | 15 | 1 | no | — |
| 67 | C22⋊D5 | — | 15 | 1 | no | — |
| 68 | C22⋊D5 | — | 30 | 1 | no | — |
| 69 | C22⋊D5 | — | 30 | 1 | no | — |
| 70 | C22⋊D5 | — | 60 | 2 | no | — |
| 71 | C22⋊D5 | A5 | 1 | 1 | yes | maximal normal, characteristic |
| 72 | C2×Dic5 | — | 6 | 1 | no | — |
| 73 | C2×Dic5 | — | 30 | 1 | no | — |
| 74 | D20 | — | 15 | 1 | no | — |
| 75 | D20 | — | 15 | 1 | no | — |
| 76 | C4×D5 | — | 15 | 2 | no | — |
| 77 | C4×D5 | — | 15 | 1 | no | — |
| 78 | C4×D5 | — | 15 | 1 | no | — |
| 79 | C4×D5 | — | 30 | 1 | no | — |
| 80 | Dic10 | — | 15 | 1 | no | — |
| 81 | Dic10 | — | 15 | 1 | no | — |
| 82 | Dic10 | — | 30 | 1 | no | — |
| 83 | 2−1+4 | — | 25 | 1 | no | Sylow |
| 84 | C2×C3×C5 | — | 10 | 1 | no | — |
| 85 | C2×C3×C5 | — | 20 | 1 | no | — |
| 86 | D15 | — | 20 | 1 | no | — |
| 87 | C3×D5 | — | 20 | 1 | no | — |
| 88 | C52 | — | 6 | 1 | no | Sylow |
| 89 | C3×D4 | — | 50 | 1 | no | — |
| 90 | #67 | — | 50 | 1 | no | — |
| 91 | #67 | — | 50 | 1 | no | — |
| 92 | C2×Dic3 | — | 10 | 1 | no | — |
| 93 | C2×Dic3 | — | 50 | 1 | no | — |
| 94 | C4×D3 | — | 50 | 1 | no | — |
| 95 | Dic6 | — | 50 | 1 | no | — |
| 96 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 97 | C22×C5 | — | 6 | 1 | no | — |
| 98 | C22×C5 | — | 12 | 2 | no | — |
| 99 | C22×C5 | — | 30 | 1 | no | — |
| 100 | C22×C5 | C2×A5 | 1 | 1 | yes | characteristic |
| 101 | C2×D5 | — | 15 | 1 | no | — |
| 102 | C2×D5 | — | 30 | 1 | no | — |
| 103 | C2×D5 | — | 60 | 2 | no | — |
| 104 | C2×D5 | C2×A5 | 1 | 1 | yes | characteristic |
| 105 | C4×C5 | — | 15 | 1 | no | — |
| 106 | C4×C5 | — | 15 | 1 | no | — |
| 107 | C4×C5 | — | 30 | 1 | no | — |
| 108 | Dic5 | — | 6 | 1 | no | — |
| 109 | Dic5 | — | 6 | 1 | no | — |
| 110 | Dic5 | — | 15 | 1 | no | — |
| 111 | Dic5 | — | 30 | 1 | no | — |
| 112 | Dic5 | — | 60 | 2 | no | — |
| 113 | Dic5 | C2×A5 | 1 | 1 | yes | characteristic |
| 114 | #41 | — | 25 | 1 | no | — |
| 115 | #41 | — | 75 | 1 | no | — |
| 116 | #41 | — | 75 | 1 | no | — |
| 117 | #41 | — | 75 | 1 | no | — |
| 118 | C2×Q8 | — | 5 | 1 | no | — |
| 119 | C2×Q8 | — | 25 | 1 | no | — |
| 120 | C2×Q8 | — | 75 | 1 | no | — |
| 121 | C3×C5 | — | 10 | 1 | no | — |
| 122 | C22×C3 | — | 10 | 1 | no | — |
| 123 | C22×C3 | — | 50 | 1 | no | — |
| 124 | C2×D3 | — | 50 | 1 | no | — |
| 125 | C3×C4 | — | 50 | 1 | no | — |
| 126 | Dic3 | — | 10 | 1 | no | — |
| 127 | Dic3 | — | 10 | 1 | no | — |
| 128 | Dic3 | — | 50 | 1 | no | — |
| 129 | C2×C5 | — | 2 | 1 | no | — |
| 130 | C2×C5 | — | 6 | 1 | no | — |
| 131 | C2×C5 | — | 12 | 1 | no | — |
| 132 | C2×C5 | — | 12 | 2 | no | — |
| 133 | C2×C5 | — | 24 | 2 | no | — |
| 134 | C2×C5 | — | 60 | 1 | no | — |
| 135 | C2×C5 | C22×A5 | 1 | 1 | yes | characteristic |
| 136 | D5 | — | 2 | 1 | no | — |
| 137 | D5 | — | 30 | 1 | no | — |
| 138 | D5 | — | 60 | 1 | no | — |
| 139 | D5 | — | 120 | 2 | no | — |
| 140 | Q8 | — | 5 | 1 | no | — |
| 141 | Q8 | — | 15 | 1 | no | — |
| 142 | Q8 | — | 75 | 1 | no | — |
| 143 | Q8 | — | 75 | 1 | no | — |
| 144 | D4 | — | 5 | 1 | no | — |
| 145 | D4 | — | 75 | 1 | no | — |
| 146 | D4 | — | 75 | 1 | no | — |
| 147 | D4 | — | 75 | 1 | no | — |
| 148 | C2×C4 | — | 15 | 1 | no | — |
| 149 | C2×C4 | — | 75 | 1 | no | — |
| 150 | C2×C4 | — | 75 | 1 | no | — |
| 151 | C2×C4 | — | 75 | 2 | no | — |
| 152 | C2×C3 | — | 10 | 1 | no | — |
| 153 | C2×C3 | — | 20 | 1 | no | — |
| 154 | C2×C3 | — | 100 | 1 | no | — |
| 155 | D3 | — | 100 | 1 | no | — |
| 156 | C5 | — | 6 | 1 | no | — |
| 157 | C5 | — | 12 | 2 | no | — |
| 158 | C5 | #91881 | 1 | 1 | yes | characteristic, complemented |
| 159 | C22 | — | 5 | 1 | no | — |
| 160 | C22 | — | 75 | 1 | no | — |
| 161 | C22 | D5×A5 | 1 | 1 | yes | characteristic |
| 162 | C4 | — | 5 | 1 | no | — |
| 163 | C4 | — | 15 | 1 | no | — |
| 164 | C4 | — | 15 | 1 | no | — |
| 165 | C4 | — | 75 | 1 | no | — |
| 166 | C3 | — | 10 | 1 | no | Sylow |
| 167 | C2 | — | 2 | 1 | no | — |
| 168 | C2 | — | 10 | 1 | no | — |
| 169 | C2 | — | 150 | 1 | no | — |
| 170 | C2 | C2×D5×A5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5×A5 |
| commutator, hypocenter | C5×SL(2,𝔽5) | C22 |
| Fitting | C22×C5 | C2×A5 |
| socle, cosocle | C2×C5 | C22×A5 |
| hypercenter | C22 | D5×A5 |
| LCS | G ▷ C5×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×SL(2,𝔽5) | C2 | semidirect |
| 2. C5⋊#6732 | C2 | semidirect |
| 3. D5×SL(2,𝔽5) | C2 | semidirect |
| 13. C2×SL(2,𝔽5) | D5 | semidirect |
| 158. C5 | #91881 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×SL(2,𝔽5) | 2 |
| C5⋊#6732 | 2 |
| D5×SL(2,𝔽5) | 2 |
| #4147⋊C3 | 5 |
| #91881 | 5 |
| #87879 | 6 |
| #6779 | 10 |