Abstract group 68518981440.a: $\PSp(4,13)$

• Label: 68518981440.a
• Order: \( 2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13^{4} \cdot 17 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13^{4} \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $2380$
• Trans deg.: $2380$
• Rank: $2$

Group information

Description:	$\PSp(4,13)$
Order:	\(68518981440\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13^{4} \cdot 17 \)
Exponent:	\(92820\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \)
Automorphism group:	$\SO(5,13)$, of order \(137037962880\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13^{4} \cdot 17 \)
Composition factors:	$\PSp(4,13)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	5	6	7	12	13	14	17	21	26	28	39	42	52	78	84	85	91	156	182
Elements	1	2628795	10457720	413079940	806105664	1913762760	2124411120	3681117440	815730720	4383278640	3224422656	4894212960	849488640	4894212960	1756896960	4894212960	878448480	1756896960	9788425920	12897690624	4517735040	1756896960	2258867520	68518981440
Conjugacy classes	1	2	2	2	1	5	9	7	6	12	4	6	6	6	3	6	2	3	12	16	9	4	6	130
Divisions	1	2	2	2	1	5	3	4	4	4	1	2	4	2	2	2	1	2	2	1	2	1	1	51
Autjugacy classes	1	2	2	2	1	5	9	7	4	12	4	6	4	6	2	6	1	2	12	16	6	2	3	115

Dimension	1	85	170	936	1105	1190	1274	2040	2210	2380	4760	6120	12240	14280	14365	15470	16660	24480	26520	28224	28560	28561	28730	30940	33320	57120	61880	66640	73440	79560	85680	112896	171360	451584
Irr. complex chars.	1	2	0	1	2	2	1	9	1	7	0	0	6	12	2	2	4	6	9	21	30	1	0	7	4	0	0	0	0	0	0	0	0	0	130
Irr. rational chars.	1	0	1	1	2	0	1	0	1	6	1	3	0	0	0	0	0	0	0	1	1	1	1	6	4	1	1	1	3	3	7	1	2	1	51

Minimal presentations

Permutation degree:	$2380$
Transitive degree:	$2380$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	85	85	170
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSp(4,13)$, $\Omega(5,13)$, $\POmega(5,13)$, $\PSigmaSp(4,13)$
Permutation group:	Degree $2380$ $\langle(4,6,10,21,50,116,46,114,109,43,17,8)(7,13,30,73,177,421,383,307,225,92,37,15) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\Sp(4,13)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 2 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $\PSp(4,13)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}^3.C_{13}$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 9788425920 (order at least 7) are shown, as well as all normal subgroups of any index.

Order 68518981440: $\PSp(4,13)$

Order 28789488: $\He_{13}.C_{12}.\PSL(2,13)$, $C_{13}^3.C_6.\PSL(2,13).C_2$

Order 14394744: $\He_{13}.C_6.\PSL(2,13)$, $C_{13}^3.C_6.\PSL(2,13)$

Order 9596496: $\He_{13}.C_4.\PSL(2,13)$, $C_{13}^3.C_2.\PSL(2,13).C_2$

Order 7197372: $C_{13}^3.C_3.\PSL(2,13)$

Order 4826640: $\POMinus(4,13)$

Order 4798248: $\He_{13}.C_2.\PSL(2,13)$, $C_{13}^3.C_2.\PSL(2,13)$

Order 4769856: $C_2.\PSL(2,13)^2.C_2$

Order 2413320: $\PSL(2,169)$

Order 2399124: $C_{13}^3.\PSL(2,13)$

Order 2384928: $\OmegaPlus(4,13)$

Order 2056392: $\He_{13}.C_{13}.C_6^2.C_2$

Order 1028196: $\He_{13}.C_{13}.C_6^2$, $\He_{13}.C_{13}.C_3.C_{12}$, $C_{13}^3.C_{39}.C_{12}$

Order 685464: $\He_{13}.C_{13}.C_6.C_2^2$ x 3, $\He_{13}.C_{78}.C_2^2$

Order 514098: $(C_{13}^3.C_{13}):(C_3\times C_6)$ x 3

Order 369096: $\He_{13}.C_{42}.C_2^2$, $C_{13}^3.C_{42}.C_2^2$

Order 342732: $(C_{13}^3.C_{13}):C_{12}$ x 8, $(C_{13}^3.C_{13}):(C_2\times C_6)$ x 4

Order 316368: $\He_{13}.Q_8.C_3.C_6$, $\He_{13}.C_6.C_6.C_2^2$, $C_{13}^3.C_6^2.C_2^2$, $D_{13}\wr C_3.C_6$

Order 257049: $(C_{13}^3.C_{13}):C_3^2$

Order 228488: $(C_{13}^3.C_{13}):(C_2\times C_4)$

Order 184548: $\He_{13}:C_{84}$, $\He_{13}:(C_6\times D_7)$, $\He_{13}:(C_3\times C_7:C_4)$, $C_{13}^3:C_{84}$, $C_{13}^3:(C_6\times D_7)$, $C_{13}^3:(C_3\times C_7:C_4)$

Order 171366: $(C_{13}^3.C_{13}):C_6$ x 12

Order 170352: $C_{13}:C_{12}.\PSL(2,13)$

Order 158184: $\He_{13}.C_6.C_6.C_2$ x 3, $\He_{13}.(C_{12}\times S_3)$ x 2, $C_{13}^3.C_6^2.C_2$ x 2, $\He_{13}.Q_8.C_3^2$, $\He_{13}.C_6^2.C_2$, $\He_{13}.(C_6\times C_{12})$, $C_{13}^3.C_3^2.C_2^3$, $C_{13}^3:(C_6\times A_4)$

Order 123032: $\He_{13}:(C_4\times D_7)$, $C_{13}^3:(C_4\times D_7)$

Order 114244: $(C_{13}^3.C_{13}):C_4$ x 2, $(C_{13}^3.C_{13}):C_2^2$

Order 105456: $\He_{13}.Q_8.C_6$ x 3, $\He_{13}.D_{12}:C_2$, $\He_{13}.C_6.C_2^3$, $C_{13}^3.D_6:C_4$, $C_{13}^3.C_6.C_2^2.C_2$, $D_{13}\wr C_3.C_2$

Order 92274: $\He_{13}:(C_3\times D_7)$ x 2, $C_{13}^3:(C_3\times D_7)$ x 2, $\He_{13}:C_{42}$, $C_{13}^3:C_{42}$

Order 85683: $(C_{13}^3.C_{13}):C_3$ x 4

Order 85176: $(C_2\times C_{13}:C_3).\PSL(2,13)$

Order 79092: $C_{13}^3:(C_6\times S_3)$ x 3, $\He_{13}:(C_6\times S_3)$ x 2, $\He_{13}:(C_3\times C_{12})$ x 2, $\He_{13}:(C_3\times C_3:C_4)$ x 2, $\He_{13}:C_6^2$, $C_{13}^3:C_6^2$, $C_{13}^3:(C_3\times C_{12})$, $C_{13}\wr C_3:C_{12}$, $C_{13}^3:(C_6\times S_3)$, $C_{13}^3:(C_3\times A_4)$

Order 61516: $\He_{13}:D_{14}$, $\He_{13}:C_{28}$, $\He_{13}:(C_7:C_4)$, $C_{13}^3:D_{14}$, $C_{13}^3:C_{28}$, $C_{13}^3:(C_7:C_4)$

Order 57122: $(C_{13}^3.C_{13}):C_2$ x 3

Order 56784: $C_{13}:C_4.\PSL(2,13)$

Order 52728: $C_{13}^3:(C_2\times C_{12})$ x 4, $\He_{13}:\SL(2,3)$ x 3, $\He_{13}:(C_2\times C_{12})$ x 3, $\He_{13}:(C_4\times S_3)$ x 2, $\He_{13}:(C_3:D_4)$ x 2, $D_{13}\wr C_3$ x 2, $\He_{13}:D_{12}$, $\He_{13}:(C_3\times Q_8)$, $\He_{13}:(C_3\times D_4)$, $\He_{13}:(C_3:Q_8)$, $C_{13}^3:(C_2^2\times S_3)$, $C_{13}^3:(C_2^2\times C_6)$, $C_{13}^3:(C_2\times C_3:C_4)$

Order 46137: $\He_{13}:C_{21}$, $C_{13}^3:C_{21}$

Order 39546: $\He_{13}:(C_3\times S_3)$ x 2, $\He_{13}:(C_3\times C_6)$ x 2, $C_{13}^3:(C_3\times C_6)$ x 2, $C_{13}^3:(C_3\times S_3)$, $C_{13}^3:(C_3\times S_3)$, $C_{13}^3:(C_3\times C_6)$

Order 35152: $\He_{13}:(D_4:C_2)$, $C_{13}^3:(C_2^2:C_4)$

Order 30758: $\He_{13}:D_7$ x 2, $C_{13}^3:D_7$ x 2, $\He_{13}:C_{14}$, $C_{13}^3:C_{14}$

Order 30576: $D_7.\PSL(2,13).C_2$, $C_7:C_4.\PSL(2,13)$

Order 28561: $C_{13}^3.C_{13}$

Order 28392: $C_{26}.\PSL(2,13)$, $C_{13}^2:(C_{12}\times D_7)$

Order 26364: $\He_{13}:C_{12}$ x 6, $C_{13}^3:C_{12}$ x 5, $C_{13}^3:(C_2\times C_6)$ x 5, $\He_{13}:(C_2\times C_6)$ x 3, $C_{13}^3:D_6$ x 3, $\He_{13}:D_6$ x 2, $\He_{13}:(C_3:C_4)$ x 2, $C_{13}^3:A_4$ x 2, $C_{13}^3:D_6$, $C_{13}\wr C_3:C_4$

Order 26208: $\SL(2,3).\PSL(2,13)$, $D_6.\PSL(2,13).C_2$, $C_3:Q_8.\PSL(2,13)$

Order 24336: $C_{13}^2.C_6^2.C_2^2$

Order 19773: $\He_{13}:C_3^2$, $C_{13}^3:C_3^2$

Order 17576: $C_{13}^3:(C_2\times C_4)$ x 2, $\He_{13}:Q_8$, $\He_{13}:D_4$, $\He_{13}:(C_2\times C_4)$, $C_{13}^3:C_2^3$

Order 15379: $\He_{13}:C_7$, $C_{13}\times C_{13}^2:C_7$

Order 15288: $C_7.\PSL(2,13).C_2$ x 2, $D_7.\PSL(2,13)$, $C_7\times \SL(2,13)$

Order 14196: $C_{13}^2:(C_6\times D_7)$, $C_{13}^2:(C_3\times C_7:C_4)$, $C_{13}^2:C_{84}$

Order 13182: $\He_{13}:C_6$ x 8, $C_{13}^3:C_6$ x 7, $\He_{13}:S_3$ x 2, $C_{13}^3:C_6$ x 2, $C_{13}^2:D_{39}$, $C_{13}\wr S_3$

Order 13104: $C_3:C_4.\PSL(2,13)$ x 2, $D_6.\PSL(2,13)$, $C_6.\PSL(2,13).C_2$, $C_2\times C_3.\PSL(2,13).C_2$, $\SL(2,13):C_6$

Order 12168: $C_{13}^2:(C_6\times C_{12})$ x 2, $C_{13}^2:(C_6\times C_3:C_4)$, $C_{13}^2:(C_2\times C_6\times S_3)$

Order 9464: $C_{13}^2:(C_4\times D_7)$

Order 8788: $C_{13}^3:C_2^2$ x 3, $\He_{13}:C_4$ x 2, $C_{13}^3:C_4$ x 2, $\He_{13}:C_2^2$

Order 8736: $Q_8.\PSL(2,13)$, $C_2^2.\PSL(2,13).C_2$

Order 8112: $C_{13}^2:(D_6:C_4)$, $C_{13}^2:(C_3\times C_2^2:C_4)$

Order 7644: $C_7\times \PSL(2,13)$

Order 7098: $C_{13}^2:(C_3\times D_7)$ x 2, $C_{13}^2:C_{42}$

Order 6591: $\He_{13}:C_3$ x 2, $C_{13}\wr C_3$ x 2, $\He_{13}:C_3$, $\He_{13}:C_3$, $C_{13}^3:C_3$

Order 6552: $C_6.\PSL(2,13)$, $C_3:\PGL(2,13)$, $S_3\times \PSL(2,13)$, $C_3\times \SL(2,13)$

Order 6084: $C_{13}^2:(C_3\times C_{12})$ x 3, $C_{13}^2:(C_6\times S_3)$ x 3, $C_{13}^2:C_6^2$ x 2, $C_{13}^2:(C_6\times S_3)$, $C_{13}^2:C_3:C_{12}$

Order 4732: $C_{13}^2:D_{14}$, $C_{13}^2:(C_7:C_4)$, $C_{13}^2:C_{28}$

Order 4394: $\He_{13}:C_2$ x 2, $\He_{13}:C_2$, $C_{13}^3:C_2$, $C_{13}\times C_{13}:D_{13}$, $C_{13}^2\times D_{13}$

Order 4368: $C_2\times \PGL(2,13)$ x 2, $C_2^2.\PSL(2,13)$, $\PSL(2,13):C_4$, $\SL(2,13):C_2$

Order 4056: $C_{13}^2:(C_2\times C_{12})$ x 8, $C_{13}^2:(C_2^2\times S_3)$, $C_{13}^2:(C_2^2\times C_6)$, $C_{13}^2:(C_2\times C_3:C_4)$, $D_{13}:F_{13}$

Order 3549: $C_{13}^2:C_{21}$

Order 3276: $C_3\times \PSL(2,13)$

Order 3042: $C_{13}^2:(C_3\times C_6)$ x 3, $C_{13}^2:(C_3\times C_6)$ x 3, $C_{13}^2:(C_3\times S_3)$ x 2

Order 2704: $C_{13}^2:(C_2^2:C_4)$

Order 2366: $C_{13}^2:D_7$ x 2, $C_{13}^2:C_{14}$

Order 2197: $\He_{13}$ x 2, $C_{13}^3$

Order 2184: $C_2\times \PSL(2,13)$ x 3, $\PGL(2,13)$ x 3, $C_{13}:(C_{12}\times D_7)$ x 2, $\SL(2,13)$

Order 2028: $C_{13}^2:C_{12}$ x 12, $C_{13}^2:(C_2\times C_6)$ x 6, $D_{13}^2:C_3$ x 5, $C_{13}^2:D_6$ x 3, $C_{13}:F_{13}$ x 2, $C_{13}^2:D_6$, $C_{13}^2:C_3:C_4$, $C_{13}:F_{13}$, $C_2\times C_{13}^2:C_6$

Order 1872: $D_{156}:C_6$, $(C_{13}\times \SL(2,3)):C_6$, $D_{78}:C_{12}$

Order 1521: $C_{13}^2:C_3^2$ x 2

Order 1352: $D_{13}\times D_{26}$, $C_2\times C_{13}^2:C_4$, $D_{13}.D_{26}$, $D_{13}^2.C_2$, $C_{26}.D_{26}$

Order 1183: $C_{13}^2:C_7$

Order 1092: $\PSL(2,13)$ x 3, $C_{182}:C_6$, $D_{91}:C_6$, $C_{91}:C_{12}$, $C_7\times F_{13}$, $C_{91}:C_{12}$, $C_{13}:C_{84}$

Order 1014: $C_{13}^2:C_6$ x 6, $C_{13}^2:C_6$ x 5, $C_{13}^2:C_6$ x 4, $C_{13}^2:C_6$ x 3, $C_{13}:C_{78}$ x 2, $C_{13}^2:S_3$ x 2, $C_{13}^2:C_6$ x 2, $C_{13}^2:C_6$ x 2, $C_{13}^2:C_6$, $C_{13}^2:C_6$, $C_{13}:C_{78}$

Order 960: $C_2^4:A_5$

Order 936: $D_{78}:C_6$ x 2, $D_{39}:C_{12}$ x 2, $C_{156}:C_6$, $C_{156}.C_6$, $C_{156}:C_6$, $C_{13}:C_3\times \SL(2,3)$, $D_{78}:C_6$, $C_{78}:C_{12}$, $C_6\times F_{13}$

Order 784: $D_7^2:C_4$

Order 728: $D_{91}:C_4$, $D_{91}:C_4$

Order 720: $S_6$ x 2

Order 676: $D_{13}^2$ x 5, $C_{13}^2:C_4$ x 2, $C_{13}:C_{52}$, $C_{13}:C_{52}$, $C_{13}:D_{26}$, $C_{13}\times D_{26}$, $C_{13}^2:C_4$, $C_{13}^2:C_4$

Order 624: $C_{26}.(C_2\times A_4)$ x 2, $D_{156}:C_2$, $\SL(2,3):D_{13}$, $D_{52}:C_6$, $D_{78}:C_4$, $D_{26}:C_{12}$

Order 576: $\OmegaPlus(4,3):C_2$, $D_6^2:C_2^2$

Order 546: $C_{91}:C_6$ x 3, $C_{91}:C_6$, $C_{13}:C_{42}$, $C_{13}:C_{42}$

Order 507: $C_{13}:C_{39}$ x 4, $C_{13}^2:C_3$ x 3, $C_{13}^2:C_3$ x 2

Order 468: $C_{78}:C_6$ x 3, $C_{78}:C_6$ x 2, $D_{39}:C_6$ x 2, $C_{39}:C_{12}$ x 2, $C_{78}:C_6$, $C_{39}:C_{12}$, $C_3\times F_{13}$, $C_{156}:C_3$, $C_{39}:C_{12}$

Order 392: $D_7\times D_{14}$, $C_2\times C_7^2:C_4$, $C_{14}.D_{14}$

Order 364: $D_7\times D_{13}$, $C_{91}:C_4$, $C_{13}:C_{28}$, $C_{13}:C_{28}$, $D_{182}$, $C_7:C_{52}$

Order 360: $A_6$ x 2

Order 340: $C_{85}:C_4$

Order 338: $C_{13}\times D_{13}$ x 6, $C_{13}:D_{13}$ x 5, $C_{13}\times C_{26}$

Order 336: $D_7\times S_4$, $D_7\times D_{12}$, $D_{84}:C_2$, $\SL(2,3):D_7$

Order 312: $C_2\times F_{13}$ x 4, $C_{52}:C_6$ x 2, $C_{13}:\SL(2,3)$ x 2, $C_3:D_{52}$ x 2, $D_{39}:C_4$ x 2, $S_3\times D_{26}$, $C_{78}:C_4$, $C_{26}:C_{12}$, $D_{26}:C_6$, $D_{156}$, $C_{39}:Q_8$, $C_{12}\times D_{13}$, $C_{13}\times \SL(2,3)$, $C_{52}.C_6$, $C_{52}:C_6$

Order 288: $D_6\wr C_2$ x 2, $C_6^2.D_4$ x 2, $D_{12}:D_6$, $\SL(2,3):D_6$, $\OmegaPlus(4,3)$, $D_6:S_4$, $D_6:D_{12}$, $(C_6\times C_{12}):C_4$

Order 273: $C_{13}:C_{21}$ x 2

Order 234: $C_{39}:C_6$ x 3, $C_{39}:C_6$ x 3, $C_{39}:C_6$ x 2, $C_{39}:C_6$

Order 208: $D_{52}:C_2$, $D_{26}:C_4$

Order 196: $D_7^2$ x 3, $C_7^2:C_4$, $C_7:C_{28}$, $C_7:D_{14}$, $C_7\times D_{14}$

Order 192: $C_2\wr A_4$, $C_2^3:S_4$

Order 182: $D_{91}$ x 3, $C_{182}$, $C_7\times D_{13}$, $C_{13}\times D_7$

Order 170: $D_{85}$

Order 169: $C_{13}^2$ x 4

Order 168: $S_3\times D_{14}$ x 2, $D_{84}$ x 2, $C_{12}\times D_7$ x 2, $C_7:D_{12}$ x 2, $C_3:D_{28}$ x 2, $D_{21}:C_4$ x 2, $A_4\times D_7$, $C_7:S_4$, $C_7\times S_4$, $C_7\times D_{12}$, $C_{21}:Q_8$, $C_7\times \SL(2,3)$

Order 160: $C_2^4:D_5$

Order 156: $C_{26}:C_6$ x 8, $F_{13}$ x 6, $D_{78}$ x 3, $C_3:C_{52}$ x 2, $C_{13}:C_{12}$ x 2, $C_3\times D_{26}$ x 2, $S_3\times D_{13}$ x 2, $C_{13}:C_{12}$ x 2, $C_{13}:C_{12}$, $C_{156}$, $C_{13}:C_{12}$, $S_3\times C_{26}$, $C_{26}:C_6$, $C_{39}:C_4$

Order 144: $D_6:D_6$ x 3, $S_3^2:C_2^2$ x 2, $S_3\times D_{12}$ x 2, $C_{12}.D_6$ x 2, $C_6^2:C_4$ x 2, $\SL(2,3):S_3$ x 2, $S_3^2:C_4$ x 2, $D_6:C_{12}$, $C_6.D_{12}$, $D_6^2$, $A_4\times D_6$, $C_6:S_4$, $C_6:D_{12}$, $D_{12}:C_6$, $\SL(2,3):C_6$, $C_6:D_{12}$, $A_4:C_{12}$

Order 120: $S_5$ x 4

Order 117: $C_{39}:C_3$ x 2

Order 112: $D_{28}:C_2$, $D_4\times D_7$

Order 104: $C_{26}:C_4$ x 3, $D_{52}$, $C_4\times D_{13}$, $C_2\times D_{26}$, $Q_8\times C_{13}$

Order 98: $C_7:D_7$ x 2, $C_7\times D_7$ x 2, $C_7\times C_{14}$

Order 96: $Q_8:A_4$ x 2, $C_2^2:D_{12}$, $D_4:D_6$, $\SL(2,3):C_2^2$, $\GL(2,\mathbb{Z}/4)$

Order 91: $C_{91}$ x 2

Order 85: $C_{85}$

Order 84: $S_3\times D_7$ x 4, $D_{42}$ x 4, $C_{84}$ x 2, $C_7:C_{12}$ x 2, $C_3:C_{28}$ x 2, $S_3\times C_{14}$ x 2, $C_3\times D_{14}$ x 2, $C_7\times A_4$

Order 80: $C_2^4:C_5$

Order 78: $C_{13}:C_6$ x 11, $C_{13}:C_6$ x 6, $D_{39}$ x 3, $C_3\times D_{13}$ x 3, $C_{78}$ x 2, $S_3\times C_{13}$

Order 72: $S_3\times D_6$ x 7, $C_3:D_{12}$ x 6, $\SOPlus(4,2)$ x 4, $C_6.D_6$ x 3, $C_6:D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_3:D_{12}$ x 2, $C_6\wr C_2$ x 2, $S_3\times C_{12}$ x 2, $C_6\times D_6$, $C_6\times A_4$, $S_3\times A_4$, $C_3:S_4$, $C_6\times C_{12}$, $C_6:C_{12}$, $C_3\times D_{12}$, $C_3^2:Q_8$, $C_3\times \SL(2,3)$

Order 68: $C_{17}:C_4$

Order 64: $C_2\wr C_2^2$

Order 60: $A_5$ x 4

Order 56: $C_7:D_4$ x 2, $D_{28}$ x 2, $C_4\times D_7$ x 2, $C_2\times D_{14}$ x 2, $C_7\times D_4$, $C_7\times Q_8$

Order 52: $D_{26}$ x 6, $C_{13}:C_4$ x 4, $C_2\times C_{26}$, $C_{52}$, $C_{13}:C_4$

Order 49: $C_7^2$

Order 48: $C_2\times S_4$ x 3, $C_2\times D_{12}$ x 3, $D_{12}:C_2$ x 2, $S_3\times D_4$ x 2, $\SL(2,3):C_2$ x 2, $D_6:C_4$ x 2, $C_2^2\times D_6$, $C_2^2\times A_4$, $D_4:C_6$, $C_6:D_4$, $D_{12}:C_2$, $A_4:C_4$, $C_2^2:C_{12}$

Order 42: $D_{21}$ x 4, $C_3\times D_7$ x 3, $C_{42}$ x 2, $S_3\times C_7$ x 2

Order 39: $C_{13}:C_3$ x 6, $C_{39}$ x 2

Order 36: $S_3^2$ x 9, $C_6\times S_3$ x 6, $C_6:S_3$ x 5, $C_3:C_{12}$ x 3, $C_3^2:C_4$ x 2, $C_3\times C_{12}$, $C_6^2$, $C_3\times A_4$

Order 34: $D_{17}$

Order 32: $C_2^3:C_4$, $D_4:C_2^2$, $C_2^2\wr C_2$

Order 28: $D_{14}$ x 7, $C_2\times C_{14}$ x 2, $C_{28}$ x 2, $C_7:C_4$ x 2

Order 26: $D_{13}$ x 7, $C_{26}$ x 4

Order 24: $D_{12}$ x 7, $C_2\times D_6$ x 7, $S_4$ x 5, $C_3:D_4$ x 4, $C_2\times A_4$ x 4, $C_2\times C_{12}$ x 3, $C_4\times S_3$ x 2, $\SL(2,3)$ x 2, $C_6:C_4$, $C_3:Q_8$, $C_2^2\times C_6$, $C_3\times Q_8$, $C_3\times D_4$

Order 21: $C_{21}$ x 2

Order 20: $F_5$

Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 3, $C_3\times C_6$ x 2

Order 17: $C_{17}$

Order 16: $C_2^2:C_4$ x 2, $C_2\times D_4$ x 2, $C_2^4$, $D_4:C_2$

Order 14: $D_7$ x 6, $C_{14}$ x 4

Order 13: $C_{13}$ x 4

Order 12: $D_6$ x 14, $C_2\times C_6$ x 4, $C_{12}$ x 4, $A_4$ x 3, $C_3:C_4$ x 3

Order 10: $D_5$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 3, $D_4$ x 3, $C_2\times C_4$ x 2, $Q_8$

Order 7: $C_7$ x 3

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 68518981440: $\PSp(4,13)$ ($C_1$)

Order 1: $C_1$ ($\PSp(4,13)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $130 \times 130$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $51 \times 51$ rational character table.