Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
278 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 24 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×#5931 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C5×#395⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×#1604 | — | 3 | 1 | no | maximal, Hall |
| 4 | C5×C22⋊SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 5 | C5×C4⋊S4 | — | 4 | 1 | no | maximal |
| 6 | #73003 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 7 | C5×#537 | — | 3 | 1 | no | — |
| 8 | C5×#448 | — | 3 | 2 | no | — |
| 9 | C5×#395 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C5×#327 | — | 3 | 2 | no | — |
| 11 | C5×#324 | — | 3 | 1 | no | — |
| 12 | C2×C5×S4 | — | 4 | 2 | no | — |
| 13 | C4×C5×A4 | — | 4 | 1 | no | — |
| 14 | C5×GL(2,𝔽3) | — | 8 | 2 | no | — |
| 15 | #5931 | C2×C5 | 1 | 2 | yes | complemented |
| 16 | #395⋊C3 | C2×C5 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×C5×#41 | — | 6 | 1 | no | — |
| 18 | C22×C5×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 19 | C2×C5×SD16 | — | 6 | 2 | no | — |
| 20 | C5×C22⋊Q8 | — | 6 | 2 | no | — |
| 21 | C5×#121 | — | 3 | 2 | no | — |
| 22 | C5×#121 | — | 6 | 2 | no | — |
| 23 | C5×#121 | — | 6 | 1 | no | — |
| 24 | C2×C5×C4⋊C4 | — | 3 | 1 | no | — |
| 25 | C5×#103 | — | 6 | 2 | no | — |
| 26 | C5×C22⋊C8 | — | 3 | 2 | no | — |
| 27 | C5×#95 | — | 3 | 1 | no | — |
| 28 | #1604 | — | 3 | 1 | no | Sylow |
| 29 | C2×C5×A4 | — | 4 | 1 | no | — |
| 30 | C5×S4 | — | 8 | 2 | no | — |
| 31 | C5×D12 | — | 16 | 1 | no | — |
| 32 | C5×SL(2,𝔽3) | — | 8 | 2 | no | — |
| 33 | C22⋊SL(2,𝔽3) | C22×C5 | 1 | 1 | yes | characteristic |
| 34 | C4⋊S4 | — | 4 | 1 | no | — |
| 35 | C5×#41 | — | 12 | 2 | no | — |
| 36 | C5×#41 | — | 12 | 2 | no | — |
| 37 | C2×C5×Q8 | — | 6 | 2 | no | — |
| 38 | C2×C5×Q8 | — | 6 | 1 | no | — |
| 39 | C2×C5×D4 | — | 3 | 2 | no | — |
| 40 | C2×C5×D4 | — | 6 | 1 | no | — |
| 41 | C2×C5×D4 | — | 6 | 2 | no | — |
| 42 | C2×C5×D4 | — | 6 | 1 | no | — |
| 43 | C22×C4×C5 | — | 3 | 1 | no | — |
| 44 | C22×C4×C5 | — | 3 | 1 | no | — |
| 45 | C22×C4×C5 | — | 6 | 2 | no | — |
| 46 | C22×C4×C5 | S4 | 1 | 1 | yes | characteristic |
| 47 | C5×SD16 | — | 12 | 2 | no | — |
| 48 | C5×SD16 | — | 12 | 2 | no | — |
| 49 | C2×C5×C8 | — | 6 | 2 | no | — |
| 50 | C5×C4⋊C4 | — | 3 | 2 | no | — |
| 51 | C5×C4⋊C4 | — | 6 | 2 | no | — |
| 52 | C5×C4⋊C4 | — | 6 | 1 | no | — |
| 53 | C5×C22⋊C4 | — | 6 | 2 | no | — |
| 54 | C5×C22⋊C4 | — | 6 | 2 | no | — |
| 55 | #537 | — | 3 | 1 | no | — |
| 56 | #448 | — | 3 | 2 | no | — |
| 57 | #395 | C5×D3 | 1 | 1 | yes | characteristic, complemented |
| 58 | #327 | — | 3 | 2 | no | — |
| 59 | #324 | — | 3 | 1 | no | — |
| 60 | C2×C5×D3 | — | 16 | 2 | no | — |
| 61 | C5×A4 | — | 4 | 1 | no | — |
| 62 | C3×C4×C5 | — | 16 | 1 | no | — |
| 63 | C2×S4 | — | 4 | 2 | no | — |
| 64 | C4×A4 | — | 4 | 1 | no | — |
| 65 | GL(2,𝔽3) | — | 8 | 2 | no | — |
| 66 | C23×C5 | — | 6 | 2 | no | — |
| 67 | C23×C5 | C2×S4 | 1 | 1 | yes | characteristic |
| 68 | C5×Q8 | — | 2 | 2 | no | — |
| 69 | C5×Q8 | — | 6 | 2 | no | — |
| 70 | C5×Q8 | — | 12 | 2 | no | — |
| 71 | C5×D4 | — | 12 | 2 | no | — |
| 72 | C5×D4 | — | 12 | 2 | no | — |
| 73 | C5×D4 | — | 12 | 2 | no | — |
| 74 | C5×D4 | — | 12 | 2 | no | — |
| 75 | C5×D4 | — | 24 | 1 | no | — |
| 76 | C2×C4×C5 | — | 3 | 2 | no | — |
| 77 | C2×C4×C5 | — | 3 | 2 | no | — |
| 78 | C2×C4×C5 | — | 6 | 1 | no | — |
| 79 | C2×C4×C5 | — | 6 | 1 | no | — |
| 80 | C2×C4×C5 | — | 6 | 1 | no | — |
| 81 | C2×C4×C5 | — | 6 | 2 | no | — |
| 82 | C2×C4×C5 | — | 12 | 1 | no | — |
| 83 | C2×C4×C5 | — | 12 | 4 | no | — |
| 84 | C5×C8 | — | 12 | 2 | no | — |
| 85 | C2×#41 | — | 6 | 1 | no | — |
| 86 | C22×Q8 | C2×C5×D3 | 1 | 1 | yes | characteristic |
| 87 | C2×SD16 | — | 6 | 2 | no | — |
| 88 | C22⋊Q8 | — | 6 | 2 | no | — |
| 89 | #121 | — | 3 | 2 | no | — |
| 90 | #121 | — | 6 | 2 | no | — |
| 91 | #121 | — | 6 | 1 | no | — |
| 92 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 93 | #103 | — | 6 | 2 | no | — |
| 94 | C22⋊C8 | — | 3 | 2 | no | — |
| 95 | #95 | — | 3 | 1 | no | — |
| 96 | C2×C3×C5 | — | 16 | 1 | no | — |
| 97 | C5×D3 | — | 32 | 2 | no | — |
| 98 | C2×A4 | — | 4 | 1 | no | — |
| 99 | S4 | — | 8 | 2 | no | — |
| 100 | D12 | — | 16 | 1 | no | — |
| 101 | SL(2,𝔽3) | — | 8 | 2 | no | — |
| 102 | C22×C5 | — | 3 | 1 | no | — |
| 103 | C22×C5 | — | 3 | 1 | no | — |
| 104 | C22×C5 | — | 12 | 2 | no | — |
| 105 | C22×C5 | — | 12 | 2 | no | — |
| 106 | C22×C5 | — | 12 | 2 | no | — |
| 107 | C22×C5 | #986 | 1 | 1 | yes | characteristic |
| 108 | C4×C5 | — | 4 | 1 | no | — |
| 109 | C4×C5 | — | 6 | 2 | no | — |
| 110 | C4×C5 | — | 6 | 2 | no | — |
| 111 | C4×C5 | — | 12 | 2 | no | — |
| 112 | #41 | — | 12 | 2 | no | — |
| 113 | #41 | — | 12 | 2 | no | — |
| 114 | C2×Q8 | — | 6 | 2 | no | — |
| 115 | C2×Q8 | — | 6 | 1 | no | — |
| 116 | C2×D4 | — | 3 | 2 | no | — |
| 117 | C2×D4 | — | 6 | 1 | no | — |
| 118 | C2×D4 | — | 6 | 2 | no | — |
| 119 | C2×D4 | — | 6 | 1 | no | — |
| 120 | C22×C4 | — | 3 | 1 | no | — |
| 121 | C22×C4 | — | 3 | 1 | no | — |
| 122 | C22×C4 | — | 6 | 2 | no | — |
| 123 | C22×C4 | C5×S4 | 1 | 1 | yes | characteristic |
| 124 | SD16 | — | 12 | 2 | no | — |
| 125 | SD16 | — | 12 | 2 | no | — |
| 126 | C2×C8 | — | 6 | 2 | no | — |
| 127 | C4⋊C4 | — | 3 | 2 | no | — |
| 128 | C4⋊C4 | — | 6 | 2 | no | — |
| 129 | C4⋊C4 | — | 6 | 1 | no | — |
| 130 | C22⋊C4 | — | 6 | 2 | no | — |
| 131 | C22⋊C4 | — | 6 | 2 | no | — |
| 132 | C3×C5 | — | 16 | 1 | no | Hall |
| 133 | C2×D3 | — | 16 | 2 | no | — |
| 134 | A4 | — | 4 | 1 | no | — |
| 135 | C3×C4 | — | 16 | 1 | no | — |
| 136 | C2×C5 | — | 3 | 1 | no | — |
| 137 | C2×C5 | — | 3 | 1 | no | — |
| 138 | C2×C5 | — | 24 | 2 | no | — |
| 139 | C2×C5 | #1020⋊C2 | 1 | 1 | yes | characteristic |
| 140 | C23 | — | 6 | 2 | no | — |
| 141 | C23 | C2×C5×S4 | 1 | 1 | yes | characteristic |
| 142 | Q8 | — | 2 | 2 | no | — |
| 143 | Q8 | — | 6 | 2 | no | — |
| 144 | Q8 | — | 12 | 2 | no | — |
| 145 | D4 | — | 12 | 2 | no | — |
| 146 | D4 | — | 12 | 2 | no | — |
| 147 | D4 | — | 12 | 2 | no | — |
| 148 | D4 | — | 12 | 2 | no | — |
| 149 | D4 | — | 24 | 1 | no | — |
| 150 | C2×C4 | — | 3 | 2 | no | — |
| 151 | C2×C4 | — | 3 | 2 | no | — |
| 152 | C2×C4 | — | 6 | 1 | no | — |
| 153 | C2×C4 | — | 6 | 1 | no | — |
| 154 | C2×C4 | — | 6 | 1 | no | — |
| 155 | C2×C4 | — | 6 | 2 | no | — |
| 156 | C2×C4 | — | 12 | 1 | no | — |
| 157 | C2×C4 | — | 12 | 4 | no | — |
| 158 | C8 | — | 12 | 2 | no | — |
| 159 | C2×C3 | — | 16 | 1 | no | — |
| 160 | D3 | — | 32 | 2 | no | — |
| 161 | C5 | #73003 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 162 | C22 | — | 3 | 1 | no | — |
| 163 | C22 | — | 3 | 1 | no | — |
| 164 | C22 | — | 12 | 2 | no | — |
| 165 | C22 | — | 12 | 2 | no | — |
| 166 | C22 | — | 12 | 2 | no | — |
| 167 | C22 | C5×#986 | 1 | 1 | yes | characteristic |
| 168 | C4 | — | 4 | 1 | no | — |
| 169 | C4 | — | 6 | 2 | no | — |
| 170 | C4 | — | 6 | 2 | no | — |
| 171 | C4 | — | 12 | 2 | no | — |
| 172 | C3 | — | 16 | 1 | no | Sylow |
| 173 | C2 | — | 3 | 1 | no | — |
| 174 | C2 | — | 3 | 1 | no | — |
| 175 | C2 | — | 24 | 2 | no | — |
| 176 | C2 | C5×#1020⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | #1020⋊C2 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C22×C5 |
| Frattini | C23 | C2×C5×S4 |
| Fitting | C5×#395 | D3 |
| socle | C23×C5 | C2×S4 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#5931 | C2 | semidirect |
| 2. C5×#395⋊C3 | C2 | semidirect |
| 6. #73003 | C5 | direct (N × H) |
| 9. C5×#395 | D3 | semidirect |
| 15. #5931 | C2×C5 | semidirect |
| 16. #395⋊C3 | C2×C5 | semidirect |
| 57. #395 | C5×D3 | semidirect |
| 161. C5 | #73003 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#5931 | 2 |
| C5×#395⋊C3 | 2 |
| C5×#1604 | 3 |
| C5×C4⋊S4 | 4 |
| #73003 | 5 |