Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
1648 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1648 conjugacy classes of subgroups — too many to draw. Only its 42 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 | C5×C22⋊(C22⋊A4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×C5×#120 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C2×C5×C22⋊A4 | — | 4 | 1 | no | maximal |
| 4 | C5×#120⋊C3 | — | 4 | 4 | no | maximal |
| 5 | #85645 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 6 | C26×C5 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C23×C5×D4 | — | 3 | 1 | no | — |
| 8 | C2×C5×#120 | — | 3 | 4 | no | — |
| 9 | C2×C5×#120 | — | 3 | 4 | no | — |
| 10 | C22×C5×C22⋊C4 | — | 3 | 1 | no | — |
| 11 | C5×C22⋊A4 | — | 4 | 1 | no | — |
| 12 | C5×C22⋊A4 | — | 4 | 4 | no | — |
| 13 | C5×C22⋊A4 | — | 8 | 8 | no | — |
| 14 | C22⋊(C22⋊A4) | C2×C5 | 1 | 1 | yes | characteristic, complemented |
| 15 | C25×C5 | — | 3 | 1 | no | — |
| 16 | C25×C5 | — | 3 | 4 | no | — |
| 17 | C25×C5 | — | 6 | 8 | no | — |
| 18 | C25×C5 | A4 | 1 | 1 | yes | characteristic, complemented |
| 19 | C22×C5×D4 | — | 3 | 6 | no | — |
| 20 | C22×C5×D4 | — | 3 | 6 | no | — |
| 21 | C22×C5×D4 | — | 6 | 8 | no | — |
| 22 | C23×C4×C5 | — | 3 | 1 | no | — |
| 23 | C5×#120 | — | 3 | 4 | no | — |
| 24 | C5×#120 | — | 3 | 12 | no | — |
| 25 | C5×#120 | — | 3 | 4 | no | — |
| 26 | C5×#120 | A4 | 1 | 4 | yes | complemented |
| 27 | C2×C5×C22⋊C4 | — | 3 | 12 | no | — |
| 28 | C22×#120 | C3×C5 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 29 | C2×C5×A4 | — | 16 | 4 | no | — |
| 30 | C2×C5×A4 | — | 16 | 1 | no | — |
| 31 | C2×C22⋊A4 | — | 4 | 1 | no | — |
| 32 | #120⋊C3 | — | 4 | 4 | no | — |
| 33 | C24×C5 | — | 2 | 8 | no | — |
| 34 | C24×C5 | — | 3 | 6 | no | — |
| 35 | C24×C5 | — | 3 | 8 | no | — |
| 36 | C24×C5 | — | 3 | 4 | no | — |
| 37 | C24×C5 | — | 3 | 1 | no | — |
| 38 | C24×C5 | — | 3 | 1 | no | — |
| 39 | C24×C5 | — | 6 | 4 | no | — |
| 40 | C24×C5 | — | 6 | 8 | no | — |
| 41 | C24×C5 | — | 6 | 48 | no | — |
| 42 | C24×C5 | — | 6 | 8 | no | — |
| 43 | C24×C5 | — | 6 | 32 | no | — |
| 44 | C24×C5 | C2×A4 | 1 | 1 | yes | characteristic |
| 45 | C24×C5 | C2×A4 | 1 | 4 | yes | complemented |
| 46 | C2×C5×D4 | — | 3 | 12 | no | — |
| 47 | C2×C5×D4 | — | 3 | 4 | no | — |
| 48 | C2×C5×D4 | — | 6 | 24 | no | — |
| 49 | C2×C5×D4 | — | 6 | 24 | no | — |
| 50 | C22×C4×C5 | — | 3 | 6 | no | — |
| 51 | C22×C4×C5 | — | 6 | 4 | no | — |
| 52 | C5×C22⋊C4 | — | 3 | 12 | no | — |
| 53 | C5×C22⋊C4 | — | 3 | 4 | no | — |
| 54 | C26 | C2×C3×C5 | 1 | 1 | yes | characteristic, complemented |
| 55 | C23×D4 | — | 3 | 1 | no | — |
| 56 | C2×#120 | — | 3 | 4 | no | — |
| 57 | C2×#120 | — | 3 | 4 | no | — |
| 58 | C22×C22⋊C4 | — | 3 | 1 | no | — |
| 59 | C5×A4 | — | 16 | 4 | no | — |
| 60 | C5×A4 | — | 16 | 1 | no | — |
| 61 | C5×A4 | — | 32 | 8 | no | — |
| 62 | C22⋊A4 | — | 4 | 1 | no | — |
| 63 | C22⋊A4 | — | 4 | 4 | no | — |
| 64 | C22⋊A4 | — | 8 | 8 | no | — |
| 65 | C23×C5 | — | 3 | 4 | no | — |
| 66 | C23×C5 | — | 3 | 1 | no | — |
| 67 | C23×C5 | — | 3 | 24 | no | — |
| 68 | C23×C5 | — | 3 | 4 | no | — |
| 69 | C23×C5 | — | 3 | 1 | no | — |
| 70 | C23×C5 | — | 4 | 4 | no | — |
| 71 | C23×C5 | — | 6 | 8 | no | — |
| 72 | C23×C5 | — | 6 | 32 | no | — |
| 73 | C23×C5 | — | 6 | 24 | no | — |
| 74 | C23×C5 | — | 6 | 24 | no | — |
| 75 | C23×C5 | — | 6 | 24 | no | — |
| 76 | C23×C5 | — | 6 | 6 | no | — |
| 77 | C23×C5 | — | 6 | 6 | no | — |
| 78 | C23×C5 | — | 6 | 96 | no | — |
| 79 | C23×C5 | — | 6 | 8 | no | — |
| 80 | C23×C5 | — | 12 | 4 | no | — |
| 81 | C23×C5 | C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 82 | C5×D4 | — | 6 | 8 | no | — |
| 83 | C5×D4 | — | 6 | 24 | no | — |
| 84 | C2×C4×C5 | — | 3 | 4 | no | — |
| 85 | C2×C4×C5 | — | 6 | 12 | no | — |
| 86 | C25 | — | 3 | 1 | no | — |
| 87 | C25 | — | 3 | 4 | no | — |
| 88 | C25 | — | 6 | 8 | no | — |
| 89 | C25 | C5×A4 | 1 | 1 | yes | characteristic, complemented |
| 90 | C22×D4 | — | 3 | 6 | no | — |
| 91 | C22×D4 | — | 3 | 6 | no | — |
| 92 | C22×D4 | — | 6 | 8 | no | — |
| 93 | C23×C4 | — | 3 | 1 | no | — |
| 94 | #120 | — | 3 | 4 | no | — |
| 95 | #120 | — | 3 | 12 | no | — |
| 96 | #120 | — | 3 | 4 | no | — |
| 97 | #120 | C5×A4 | 1 | 4 | yes | complemented |
| 98 | C2×C22⋊C4 | — | 3 | 12 | no | — |
| 99 | C2×C3×C5 | — | 64 | 1 | no | — |
| 100 | C2×A4 | — | 16 | 4 | no | — |
| 101 | C2×A4 | — | 16 | 1 | no | — |
| 102 | C22×C5 | — | 2 | 8 | no | — |
| 103 | C22×C5 | — | 3 | 6 | no | — |
| 104 | C22×C5 | — | 3 | 4 | no | — |
| 105 | C22×C5 | — | 3 | 8 | no | — |
| 106 | C22×C5 | — | 6 | 1 | no | — |
| 107 | C22×C5 | — | 6 | 8 | no | — |
| 108 | C22×C5 | — | 6 | 48 | no | — |
| 109 | C22×C5 | — | 6 | 3 | no | — |
| 110 | C22×C5 | — | 6 | 8 | no | — |
| 111 | C22×C5 | — | 6 | 32 | no | — |
| 112 | C22×C5 | — | 12 | 4 | no | — |
| 113 | C22×C5 | — | 12 | 4 | no | — |
| 114 | C22×C5 | #120⋊C3 | 1 | 4 | yes | complemented |
| 115 | C22×C5 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 116 | C4×C5 | — | 6 | 4 | no | — |
| 117 | C24 | — | 2 | 8 | no | — |
| 118 | C24 | — | 3 | 6 | no | — |
| 119 | C24 | — | 3 | 8 | no | — |
| 120 | C24 | — | 3 | 4 | no | — |
| 121 | C24 | — | 3 | 1 | no | — |
| 122 | C24 | — | 3 | 1 | no | — |
| 123 | C24 | — | 6 | 4 | no | — |
| 124 | C24 | — | 6 | 8 | no | — |
| 125 | C24 | — | 6 | 48 | no | — |
| 126 | C24 | — | 6 | 8 | no | — |
| 127 | C24 | — | 6 | 32 | no | — |
| 128 | C24 | C2×C5×A4 | 1 | 1 | yes | characteristic |
| 129 | C24 | C2×C5×A4 | 1 | 4 | yes | complemented |
| 130 | C2×D4 | — | 3 | 12 | no | — |
| 131 | C2×D4 | — | 3 | 4 | no | — |
| 132 | C2×D4 | — | 6 | 24 | no | — |
| 133 | C2×D4 | — | 6 | 24 | no | — |
| 134 | C22×C4 | — | 3 | 6 | no | — |
| 135 | C22×C4 | — | 6 | 4 | no | — |
| 136 | C22⋊C4 | — | 3 | 12 | no | — |
| 137 | C22⋊C4 | — | 3 | 4 | no | — |
| 138 | C3×C5 | — | 64 | 1 | no | Hall |
| 139 | A4 | — | 16 | 4 | no | — |
| 140 | A4 | — | 16 | 1 | no | — |
| 141 | A4 | — | 32 | 8 | no | — |
| 142 | C2×C5 | — | 3 | 1 | no | — |
| 143 | C2×C5 | — | 3 | 4 | no | — |
| 144 | C2×C5 | — | 4 | 1 | no | — |
| 145 | C2×C5 | — | 6 | 8 | no | — |
| 146 | C2×C5 | — | 12 | 1 | no | — |
| 147 | C23 | — | 3 | 4 | no | — |
| 148 | C23 | — | 3 | 1 | no | — |
| 149 | C23 | — | 3 | 24 | no | — |
| 150 | C23 | — | 3 | 4 | no | — |
| 151 | C23 | — | 3 | 1 | no | — |
| 152 | C23 | — | 4 | 4 | no | — |
| 153 | C23 | — | 6 | 8 | no | — |
| 154 | C23 | — | 6 | 32 | no | — |
| 155 | C23 | — | 6 | 24 | no | — |
| 156 | C23 | — | 6 | 24 | no | — |
| 157 | C23 | — | 6 | 24 | no | — |
| 158 | C23 | — | 6 | 6 | no | — |
| 159 | C23 | — | 6 | 6 | no | — |
| 160 | C23 | — | 6 | 96 | no | — |
| 161 | C23 | — | 6 | 8 | no | — |
| 162 | C23 | — | 12 | 4 | no | — |
| 163 | C23 | C5×C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 164 | D4 | — | 6 | 8 | no | — |
| 165 | D4 | — | 6 | 24 | no | — |
| 166 | C2×C4 | — | 3 | 4 | no | — |
| 167 | C2×C4 | — | 6 | 12 | no | — |
| 168 | C2×C3 | — | 64 | 1 | no | — |
| 169 | C5 | #85645 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 170 | C22 | — | 2 | 8 | no | — |
| 171 | C22 | — | 3 | 6 | no | — |
| 172 | C22 | — | 3 | 4 | no | — |
| 173 | C22 | — | 3 | 8 | no | — |
| 174 | C22 | — | 6 | 1 | no | — |
| 175 | C22 | — | 6 | 8 | no | — |
| 176 | C22 | — | 6 | 48 | no | — |
| 177 | C22 | — | 6 | 3 | no | — |
| 178 | C22 | — | 6 | 8 | no | — |
| 179 | C22 | — | 6 | 32 | no | — |
| 180 | C22 | — | 12 | 4 | no | — |
| 181 | C22 | — | 12 | 4 | no | — |
| 182 | C22 | C5×#120⋊C3 | 1 | 4 | yes | complemented |
| 183 | C22 | C2×C5×C22⋊A4 | 1 | 1 | yes | characteristic |
| 184 | C4 | — | 6 | 4 | no | — |
| 185 | C3 | — | 64 | 1 | no | Sylow |
| 186 | C2 | — | 3 | 1 | no | — |
| 187 | C2 | — | 3 | 4 | no | — |
| 188 | C2 | — | 4 | 1 | no | — |
| 189 | C2 | — | 6 | 8 | no | — |
| 190 | C2 | — | 12 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #85645 |
| commutator, cosocle, hypocenter | C26 | C2×C3×C5 |
| Frattini | C22 | C2×C5×C22⋊A4 |
| Fitting | C22×C5×#120 | C3 |
| socle | C24×C5 | C2×A4 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C22⋊(C22⋊A4) | C2 | semidirect |
| 2. C22×C5×#120 | C3 | semidirect |
| 5. #85645 | C5 | direct (N × H) |
| 6. C26×C5 | C2×C3 | semidirect |
| 14. C22⋊(C22⋊A4) | C2×C5 | semidirect |
| 18. C25×C5 | A4 | semidirect |
| 26. C5×#120 | A4 | semidirect |
| 28. C22×#120 | C3×C5 | semidirect |
| 45. C24×C5 | C2×A4 | semidirect |
| 54. C26 | C2×C3×C5 | semidirect |
| 81. C23×C5 | C22⋊A4 | semidirect |
| 89. C25 | C5×A4 | semidirect |
| 97. #120 | C5×A4 | semidirect |
| 114. C22×C5 | #120⋊C3 | semidirect |
| 129. C24 | C2×C5×A4 | semidirect |
| 163. C23 | C5×C22⋊A4 | semidirect |
| 169. C5 | #85645 | direct (N × H) |
| 182. C22 | C5×#120⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C22⋊(C22⋊A4) | 2 |
| C22×C5×#120 | 3 |
| C2×C5×C22⋊A4 | 4 |
| C5×#120⋊C3 | 4 |
| #85645 | 5 |