Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
238 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 25 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×Q8⋊SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10602732 | — | 3 | 1 | no | maximal, Hall |
| 3 | C2×C5⋊GL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 4 | C2×C5⋊GL(2,𝔽3) | — | 4 | 1 | no | maximal |
| 5 | #85555 | — | 5 | 1 | no | maximal, Hall |
| 6 | C5×Q82 | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | #65374 | — | 3 | 1 | no | — |
| 8 | #65316 | — | 3 | 2 | no | — |
| 9 | #64766 | — | 3 | 1 | no | — |
| 10 | Q8⋊#168 | — | 3 | 2 | no | — |
| 11 | C2×C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 12 | C2×C5×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 13 | C2×C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 14 | C5⋊GL(2,𝔽3) | — | 4 | 2 | no | — |
| 15 | C5⋊GL(2,𝔽3) | — | 4 | 4 | no | — |
| 16 | Q8⋊SL(2,𝔽3) | D5 | 1 | 1 | yes | characteristic, complemented |
| 17 | C5×C4⋊Q8 | — | 3 | 1 | no | — |
| 18 | C5×C4⋊Q8 | — | 6 | 1 | no | — |
| 19 | C4×C5×Q8 | — | 3 | 2 | no | — |
| 20 | C2×#688 | — | 6 | 1 | no | — |
| 21 | C2×#688 | — | 6 | 2 | no | — |
| 22 | C2×#688 | — | 6 | 1 | no | — |
| 23 | #4020 | — | 3 | 1 | no | — |
| 24 | #3941 | — | 6 | 2 | no | — |
| 25 | C4⋊#168 | — | 3 | 2 | no | — |
| 26 | C4×#168 | — | 3 | 1 | no | — |
| 27 | #1623 | — | 15 | 1 | no | Sylow |
| 28 | C22×D15 | — | 16 | 1 | no | — |
| 29 | C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 30 | C5×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 31 | C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 32 | C2×GL(2,𝔽3) | — | 20 | 2 | no | — |
| 33 | C2×GL(2,𝔽3) | — | 20 | 1 | no | — |
| 34 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 35 | C2×C5×Q8 | — | 3 | 1 | no | — |
| 36 | C2×C5×Q8 | S4 | 1 | 2 | yes | — |
| 37 | C2×C5×Q8 | S4 | 1 | 1 | yes | characteristic |
| 38 | C2×D20 | — | 6 | 2 | no | — |
| 39 | C2×D20 | — | 6 | 1 | no | — |
| 40 | C5×C4⋊C4 | — | 3 | 2 | no | — |
| 41 | C5×C4⋊C4 | — | 6 | 2 | no | — |
| 42 | C42×C5 | — | 3 | 1 | no | — |
| 43 | C42×C5 | — | 6 | 1 | no | — |
| 44 | #688 | — | 12 | 2 | no | — |
| 45 | #688 | — | 12 | 2 | no | — |
| 46 | #688 | — | 12 | 2 | no | — |
| 47 | #688 | — | 12 | 2 | no | — |
| 48 | C2×#168 | — | 6 | 1 | no | — |
| 49 | C2×#168 | — | 6 | 2 | no | — |
| 50 | Q82 | D15 | 1 | 1 | yes | characteristic, complemented |
| 51 | #492 | — | 15 | 1 | no | — |
| 52 | #460 | — | 15 | 2 | no | — |
| 53 | #333 | — | 15 | 1 | no | — |
| 54 | Q8⋊C8 | — | 15 | 2 | no | — |
| 55 | C22×C3×C5 | — | 16 | 1 | no | — |
| 56 | C2×D15 | — | 16 | 4 | no | — |
| 57 | C2×D15 | — | 16 | 2 | no | — |
| 58 | C2×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 59 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 60 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 61 | GL(2,𝔽3) | — | 20 | 2 | no | — |
| 62 | GL(2,𝔽3) | — | 20 | 4 | no | — |
| 63 | C22×D5 | — | 12 | 1 | no | — |
| 64 | C5×Q8 | — | 3 | 2 | no | — |
| 65 | C5×Q8 | — | 4 | 1 | no | — |
| 66 | C5×Q8 | — | 8 | 1 | no | — |
| 67 | C5×Q8 | — | 12 | 1 | no | — |
| 68 | C5×Q8 | GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 69 | C2×C4×C5 | — | 3 | 2 | no | — |
| 70 | C2×C4×C5 | — | 3 | 1 | no | — |
| 71 | C2×C4×C5 | — | 6 | 1 | no | — |
| 72 | D20 | — | 12 | 2 | no | — |
| 73 | D20 | — | 12 | 2 | no | — |
| 74 | D20 | — | 12 | 1 | no | — |
| 75 | D20 | — | 12 | 1 | no | — |
| 76 | #168 | — | 6 | 2 | no | — |
| 77 | #168 | — | 12 | 2 | no | — |
| 78 | C2×SD16 | — | 30 | 1 | no | — |
| 79 | C2×SD16 | — | 30 | 2 | no | — |
| 80 | C2×SD16 | — | 30 | 1 | no | — |
| 81 | C4⋊Q8 | — | 3 | 1 | no | — |
| 82 | C4⋊Q8 | — | 6 | 1 | no | — |
| 83 | #127 | — | 15 | 1 | no | — |
| 84 | C4×Q8 | — | 3 | 2 | no | — |
| 85 | C4⋊C8 | — | 15 | 2 | no | — |
| 86 | #102 | — | 30 | 2 | no | — |
| 87 | C4×C8 | — | 15 | 1 | no | — |
| 88 | C2×C3×C5 | — | 16 | 2 | no | — |
| 89 | C2×C3×C5 | — | 16 | 1 | no | — |
| 90 | D15 | — | 16 | 4 | no | — |
| 91 | C22×D3 | — | 80 | 1 | no | — |
| 92 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 93 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 94 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 95 | C22×C5 | #1020 | 1 | 1 | yes | characteristic |
| 96 | C2×D5 | — | 24 | 2 | no | — |
| 97 | C2×D5 | — | 24 | 1 | no | — |
| 98 | C4×C5 | — | 3 | 2 | no | — |
| 99 | C4×C5 | — | 3 | 2 | no | — |
| 100 | C4×C5 | — | 6 | 1 | no | — |
| 101 | C4×C5 | — | 12 | 1 | no | — |
| 102 | C2×Q8 | — | 2 | 1 | no | — |
| 103 | C2×Q8 | — | 3 | 1 | no | — |
| 104 | C2×Q8 | C5⋊S4 | 1 | 2 | yes | — |
| 105 | C2×Q8 | C5⋊S4 | 1 | 1 | yes | characteristic |
| 106 | C2×D4 | — | 30 | 2 | no | — |
| 107 | C2×D4 | — | 30 | 1 | no | — |
| 108 | SD16 | — | 60 | 2 | no | — |
| 109 | SD16 | — | 60 | 2 | no | — |
| 110 | SD16 | — | 60 | 2 | no | — |
| 111 | SD16 | — | 60 | 2 | no | — |
| 112 | C2×C8 | — | 30 | 1 | no | — |
| 113 | C2×C8 | — | 30 | 2 | no | — |
| 114 | C4⋊C4 | — | 3 | 2 | no | — |
| 115 | C4⋊C4 | — | 6 | 2 | no | — |
| 116 | C42 | — | 3 | 1 | no | — |
| 117 | C42 | — | 6 | 1 | no | — |
| 118 | C3×C5 | — | 16 | 1 | no | Hall |
| 119 | C22×C3 | — | 16 | 1 | no | — |
| 120 | C2×D3 | — | 80 | 4 | no | — |
| 121 | C2×D3 | — | 80 | 2 | no | — |
| 122 | C2×C5 | #5931 | 1 | 2 | yes | — |
| 123 | C2×C5 | #5935 | 1 | 1 | yes | characteristic |
| 124 | D5 | — | 48 | 1 | no | — |
| 125 | C23 | — | 60 | 1 | no | — |
| 126 | Q8 | — | 3 | 2 | no | — |
| 127 | Q8 | — | 4 | 1 | no | — |
| 128 | Q8 | — | 8 | 1 | no | — |
| 129 | Q8 | — | 12 | 1 | no | — |
| 130 | Q8 | C5⋊GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 131 | D4 | — | 60 | 2 | no | — |
| 132 | D4 | — | 60 | 2 | no | — |
| 133 | D4 | — | 60 | 1 | no | — |
| 134 | D4 | — | 60 | 1 | no | — |
| 135 | C2×C4 | — | 3 | 2 | no | — |
| 136 | C2×C4 | — | 3 | 1 | no | — |
| 137 | C2×C4 | — | 6 | 1 | no | — |
| 138 | C8 | — | 30 | 2 | no | — |
| 139 | C8 | — | 60 | 2 | no | — |
| 140 | C2×C3 | — | 16 | 2 | no | — |
| 141 | C2×C3 | — | 16 | 1 | no | — |
| 142 | D3 | — | 80 | 4 | no | — |
| 143 | C5 | #85555 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 144 | C22 | — | 120 | 2 | no | — |
| 145 | C22 | — | 120 | 1 | no | — |
| 146 | C22 | C5⋊#1020 | 1 | 1 | yes | characteristic |
| 147 | C4 | — | 3 | 2 | no | — |
| 148 | C4 | — | 3 | 2 | no | — |
| 149 | C4 | — | 6 | 1 | no | — |
| 150 | C4 | — | 12 | 1 | no | — |
| 151 | C3 | — | 16 | 1 | no | Sylow |
| 152 | C2 | — | 240 | 1 | no | — |
| 153 | C2 | C5⋊#5931 | 1 | 2 | yes | — |
| 154 | C2 | C5⋊#5935 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C5⋊#1020 |
| commutator, cosocle, hypocenter | C5×Q8⋊SL(2,𝔽3) | C2 |
| Fitting | C5×Q82 | D3 |
| socle | C22×C5 | #1020 |
| LCS | G ▷ C5×Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×Q8⋊SL(2,𝔽3) | C2 | semidirect |
| 6. C5×Q82 | D3 | semidirect |
| 16. Q8⋊SL(2,𝔽3) | D5 | semidirect |
| 50. Q82 | D15 | semidirect |
| 68. C5×Q8 | GL(2,𝔽3) | semidirect |
| 130. Q8 | C5⋊GL(2,𝔽3) | semidirect |
| 143. C5 | #85555 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×Q8⋊SL(2,𝔽3) | 2 |
| #10602732 | 3 |
| C2×C5⋊GL(2,𝔽3) | 4 |
| #85555 | 5 |