Abstract group 9539712.a: $\PGL(2,13)\wr C_2$

• Label: 9539712.a
• Order: \( 2^{7} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $28$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$\PGL(2,13)\wr C_2$
Order:	\(9539712\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \)
Exponent:	\(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
Automorphism group:	$\PGL(2,13)\wr C_2$, of order \(9539712\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \)
Composition factors:	$C_2$ x 3, $\PSL(2,13)$ x 2
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	12	13	14	21	24	26	28	39	42	52	78	84	91	156	182
Elements	1	31083	33488	464100	620256	219960	397488	1183728	28560	1996488	170352	794976	423696	1362816	61152	511056	61152	61152	681408	157248	122304	157248	9539712
Conjugacy classes	1	6	2	6	8	9	1	18	2	33	3	2	3	9	1	9	1	1	12	3	2	3	135
Divisions	1	6	2	6	8	3	1	11	2	11	1	1	3	3	1	3	1	1	2	1	1	1	70
Autjugacy classes	1	6	2	6	8	9	1	18	2	33	3	2	3	9	1	9	1	1	12	3	2	3	135

Dimension	1	2	24	26	28	56	72	144	169	196	288	312	336	338	364	392	432	728	784	864	936	1008	2016
Irr. complex chars.	4	1	12	4	10	0	0	12	4	10	15	12	30	1	10	10	0	0	0	0	0	0	0	135
Irr. rational chars.	4	1	0	4	6	2	4	0	4	6	0	0	0	1	6	6	4	2	3	5	4	6	2	70

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	26
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $28$ $\langle(1,28,13,18,2,26,3,20,9,16,7,25,14,19)(4,21,11,22,6,27,10,23,5,24,12,17,8,15), (1,10,6,13)(2,9,12,14)(5,7,8,11)(15,28,25,27,18,16)(17,26,19,20,23,24)\rangle$
Transitive group:	28T1474				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\PSL(2,13)^2$ $\,\rtimes\,$ $D_4$	$\POPlus(4,13)$ $\,\rtimes\,$ $C_2$	$(\PSL(2,13)^2.C_4)$ $\,\rtimes\,$ $C_2$		more information
Trans. wreath product:	$\PGL(2,13)$$\ \wr\ $$C_2$
Non-split product:	$\PSOPlus(4,13)$ . $C_2^2$				more information
Aut. group:	$\Aut(\PSL(2,13)^2)$	$\Aut(\OmegaPlus(4,13))$	$\Aut(\PSOPlus(4,13))$	$\Aut(\SpinPlus(4,13))$	all 5

Elements of the group are displayed as permutations of degree 28.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 53847924 subgroups in 1476 conjugacy classes, 7 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PGL(2,13)\wr C_2$
Commutator:	$G' \simeq$ $\PSOPlus(4,13)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PGL(2,13)\wr C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PGL(2,13)\wr C_2$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PGL(2,13)\wr C_2$
Socle:	$\operatorname{soc} \simeq$ $\PSL(2,13)^2$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 9539712: $\PGL(2,13)\wr C_2$

Order 4769856: $\PSL(2,13).\PSL(2,13).C_2^2$, $\PSL(2,13)^2.C_4$, $\POPlus(4,13)$

Order 2384928: $\PSL(2,13).\PSL(2,13).C_2$, $\PSL(2,13)\wr C_2$, $\PSOPlus(4,13)$

Order 1192464: $\PSL(2,13)^2$

Order 340704: $F_{13}.\PSL(2,13).C_2$

Order 170352: $D_{13}:C_3.\PSL(2,13).C_2$, $F_{13}.\PSL(2,13)$, $D_{13}:C_3.\PSL(2,13).C_2$

Order 113568: $D_{13}.C_2.\PSL(2,13).C_2$

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

Order 61152: $D_{14}.\PSL(2,13).C_2$

Order 56784: $D_{13}.\PSL(2,13).C_2$, $D_{13}.C_2.\PSL(2,13)$, $D_{13}.\PSL(2,13).C_2$

Order 52416: $S_4.\PSL(2,13).C_2$, $D_{12}.\PSL(2,13).C_2$

Order 48672: $F_{13}\wr C_2$

Order 42588: $C_{13}:C_3.\PSL(2,13)$

Order 30576: $D_7.\PSL(2,13).C_2$ x 4, $D_{14}.\PSL(2,13)$, $C_{14}.\PSL(2,13).C_2$, $C_{14}.\PSL(2,13).C_2$

Order 28392: $D_{13}.\PSL(2,13)$, $C_{13}.\PSL(2,13).C_2$, $C_{13}.\PSL(2,13).C_2$

Order 26208: $D_6.\PSL(2,13).C_2$ x 2, $D_6.\PSL(2,13).C_2$ x 2, $D_{12}.\PSL(2,13)$, $A_4.\PSL(2,13).C_2$, $A_4.\PSL(2,13).C_2$, $C_{12}.\PSL(2,13).C_2$, $S_4.\PSL(2,13)$, $C_{12}\times \PGL(2,13)$

Order 24336: $C_{13}^2.C_{12}^2$, $C_{13}:F_{13}.D_6$, $D_{13}^2.(C_6\times S_3)$

Order 17472: $D_4.\PSL(2,13).C_2$

Order 16224: $D_{13}\wr C_2:C_{12}$, $C_{13}^2:D_{12}:C_4$

Order 15288: $C_7.\PSL(2,13).C_2$ x 2, $D_7.\PSL(2,13)$ x 2, $C_7.\PSL(2,13).C_2$ x 2, $C_{14}.\PSL(2,13)$

Order 14196: $C_{13}\times \PSL(2,13)$

Order 13104: $S_3\times \PGL(2,13)$ x 4, $C_6.\PSL(2,13).C_2$ x 2, $D_6.\PSL(2,13)$ x 2, $A_4.\PSL(2,13)$, $C_6.\PSL(2,13).C_2$, $C_{12}.\PSL(2,13)$, $C_6.\PSL(2,13).C_2$

Order 12168: $C_{13}^2:(C_6\times C_{12})$, $C_{13}^2:(C_6\times C_{12})$, $C_{13}^2:C_3:C_{24}$, $C_{13}^2:(C_3\times D_{12})$, $D_{13}^2:(C_3\times S_3)$, $C_{13}^2:(S_3\times C_{12})$, $C_{13}^2:C_3^2:Q_8$

Order 8736: $C_2^2.\PSL(2,13).C_2$ x 2, $C_4.\PSL(2,13).C_2$, $C_4.\PSL(2,13).C_2$, $C_2^2\times \PSL(2,13).C_2$, $C_2^2.\PSL(2,13).C_2$, $D_4.\PSL(2,13)$

Order 8112: $C_{13}^2:(C_4\times C_{12})$ x 2, $C_{13}^2:(C_4\times C_{12})$, $D_{13}^2.D_6$, $D_{13}^2.C_{12}$, $C_{13}^2:C_3:\OD_{16}$, $D_{13}\wr C_2:C_6$

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

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

Order 6084: $C_{13}^2:(C_3\times C_{12})$ x 2, $C_{13}^2:(C_6\times S_3)$, $C_{13}^2:C_6^2$, $C_{13}^2:(C_3\times C_{12})$, $C_{13}^2:(C_3\times C_3:C_4)$, $C_{13}^2:(C_3\times C_{12})$

Order 5408: $D_{13}^2.D_4$

Order 4368: $C_2\times \PGL(2,13)$ x 6, $C_{13}:(C_2\times C_{12}\times D_7)$, $C_4.\PSL(2,13)$, $C_2^2\times \PSL(2,13)$, $C_2^2.\PSL(2,13)$, $\PSL(2,13):C_4$

Order 4056: $C_{13}^2:(C_2\times C_{12})$ x 2, $D_{13}:F_{13}$ x 2, $D_{13}^2:C_6$ x 2, $C_{13}^2:(C_2\times C_{12})$, $C_{13}^2:(C_2\times C_{12})$, $C_{13}^2:(C_2\times C_{12})$, $C_{13}^2:(C_2\times C_{12})$, $C_{13}^2:C_{24}$, $C_{13}^2:D_{12}$, $C_{13}^2:C_3:Q_8$, $C_{13}^2:(C_3\times Q_8)$, $C_{13}^2:C_3:C_8$, $C_{13}^2:(C_4\times S_3)$, $D_{13}^2:S_3$

Order 3744: $C_{39}.(C_{12}\times D_4)$, $S_4\times F_{13}$

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

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

Order 2704: $C_{13}^2:C_4^2$, $C_{13}^2:\OD_{16}$, $D_{13}^2.C_2^2$

Order 2184: $\PGL(2,13)$ x 5, $C_{13}:(C_{12}\times D_7)$ x 4, $C_2\times \PSL(2,13)$ x 3, $C_{13}:(C_6\times C_7:C_4)$, $C_{13}:(C_2\times C_6\times D_7)$, $C_{14}\times F_{13}$

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

Order 1872: $D_{78}:C_{12}$ x 2, $D_6\times F_{13}$ x 2, $D_{156}:C_6$, $C_{156}:C_{12}$, $C_{12}\times F_{13}$, $C_{13}:C_6\times S_4$, $A_4\times F_{13}$, $A_4:F_{13}$

Order 1568: $D_{14}\wr C_2$

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

Order 1456: $D_{182}:C_4$

Order 1352: $D_{13}\wr C_2$ x 2, $D_{13}^2.C_2$ x 2, $C_{13}^2:Q_8$, $D_{13}.D_{26}$, $C_{13}^2:C_8$

Order 1248: $C_{13}:C_4\times S_4$, $D_{156}:C_4$, $D_4\times F_{13}$

Order 1152: $D_{12}\wr C_2$, $S_4\wr C_2$

Order 1092: $D_{91}:C_6$ x 4, $C_{91}:C_{12}$ x 2, $C_7\times F_{13}$ x 2, $\PSL(2,13)$ x 2, $C_{182}:C_6$, $C_{182}:C_6$, $C_{26}:C_{42}$

Order 1014: $C_{13}^2:C_6$ x 2, $C_{13}^2:C_6$, $C_{13}:C_{78}$, $C_{13}^2:S_3$, $C_{13}^2:C_6$, $C_{13}^2:C_6$, $C_{13}^2:C_6$, $C_{13}^2:C_6$

Order 936: $S_3\times F_{13}$ x 4, $D_{78}:C_6$ x 2, $D_{78}:C_6$ x 2, $C_{78}:C_{12}$ x 2, $C_6\times F_{13}$ x 2, $C_{156}:C_6$, $C_{156}:C_6$, $C_{156}:C_6$, $A_4\times C_{13}:C_6$, $C_{13}:(C_3\times S_4)$, $C_{13}:C_3\times S_4$

Order 784: $D_7^2:C_2^2$ x 2, $D_7^2:C_4$ x 2, $D_{14}^2$, $D_{14}:D_{14}$, $C_{14}^2:C_4$

Order 728: $D_{91}:C_4$ x 4, $D_7\times D_{26}$, $C_{182}:C_4$, $C_{26}:C_{28}$

Order 676: $D_{13}^2$ x 2, $C_{13}^2:C_4$ x 2, $C_{13}:C_{52}$, $C_{13}^2:C_4$, $C_{13}^2:C_4$

Order 672: $S_4\times D_{14}$, $D_{12}\times D_{14}$

Order 624: $D_{78}:C_4$ x 2, $C_2^2\times F_{13}$ x 2, $D_{78}:C_4$ x 2, $D_{26}:C_{12}$ x 2, $C_4\times F_{13}$ x 2, $S_4\times D_{13}$, $C_2^2:F_{13}$, $A_4\times C_{13}:C_4$, $D_{13}.S_4$, $D_{12}\times D_{13}$, $C_{156}:C_4$, $C_{156}:C_4$, $D_{52}:C_6$, $C_{52}:C_{12}$

Order 576: $D_6^2:C_2^2$ x 2, $D_6^2:C_4$ x 2, $\POPlus(4,3)$, $S_4^2$, $A_4^2:C_4$, $S_4\times D_{12}$, $D_{12}^2$, $D_{12}:D_{12}$, $C_{12}^2:C_4$

Order 546: $C_{91}:C_6$ x 2, $C_{91}:C_6$ x 2, $C_{13}:C_{42}$ x 2, $C_{13}:C_{42}$

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

Order 468: $D_{39}:C_6$ x 4, $C_3\times F_{13}$ x 3, $C_{78}:C_6$ x 2, $C_{78}:C_6$ x 2, $C_{39}:C_{12}$ x 2, $C_{78}:C_6$, $A_4\times C_{13}:C_3$, $C_{156}:C_3$, $C_{39}:C_{12}$

Order 416: $D_{52}:C_4$

Order 392: $D_7\times D_{14}$ x 8, $D_7\wr C_2$ x 4, $C_2\times C_7^2:C_4$ x 2, $C_{14}:D_{14}$, $C_{14}\times D_{14}$, $C_{14}\wr C_2$, $C_7:D_{28}$, $C_{14}.D_{14}$

Order 364: $D_7\times D_{13}$ x 4, $C_{91}:C_4$ x 2, $C_{13}:C_{28}$ x 2, $C_{13}\times D_{14}$, $C_7\times D_{26}$, $D_{182}$

Order 338: $C_{13}\times D_{13}$ x 2, $C_{13}:D_{13}$

Order 336: $D_7\times D_{12}$ x 8, $D_7\times S_4$ x 4, $D_6\times D_{14}$ x 2, $C_{14}:D_{12}$ x 2, $A_4\times D_{14}$, $C_{14}:S_4$, $C_{14}\times S_4$, $C_2\times D_{84}$, $C_{14}\times D_{12}$, $C_{12}\times D_{14}$

Order 312: $C_2\times F_{13}$ x 9, $D_{39}:C_4$ x 4, $C_{52}:C_6$ x 2, $S_3\times D_{26}$ x 2, $C_{78}:C_4$ x 2, $C_{26}:C_{12}$ x 2, $D_{26}:C_6$ x 2, $C_{13}:D_{12}$ x 2, $D_{26}:C_6$ x 2, $D_{13}:A_4$, $A_4\times D_{13}$, $C_{13}:S_4$, $C_{13}\times S_4$, $D_{156}$, $C_{13}\times D_{12}$, $C_{12}\times D_{13}$, $C_{52}:C_6$, $C_{52}:C_6$

Order 288: $D_6:D_{12}$ x 3, $D_6\times D_{12}$ x 2, $D_6\wr C_2$ x 2, $D_6:S_4$ x 2, $C_{12}:D_{12}$ x 2, $(C_6\times C_{12}):C_4$ x 2, $C_6^2.D_4$ x 2, $D_6\times S_4$ x 2, $D_{12}:D_6$, $A_4\times D_{12}$, $C_{12}:S_4$, $C_{12}\times S_4$, $C_{12}:D_{12}$, $C_{12}\times D_{12}$, $D_{12}:D_6$, $C_{12}\wr C_2$, $C_{12}.D_{12}$, $\PSOPlus(4,3)$, $A_4\wr C_2$, $A_4\times S_4$

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

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

Order 224: $D_4\times D_{14}$

Order 208: $D_{26}:C_4$ x 2, $D_{26}:C_4$ x 2, $D_4\times D_{13}$, $C_{52}:C_4$, $C_{52}:C_4$

Order 196: $D_7^2$ x 12, $C_7\times D_{14}$ x 7, $C_7:D_{14}$ x 4, $C_7^2:C_4$ x 2, $C_7:C_{28}$, $C_{14}^2$

Order 192: $C_2^3:S_4$, $D_4:D_{12}$, $D_4\times S_4$, $D_4\times D_{12}$

Order 182: $D_{91}$ x 2, $C_7\times D_{13}$ x 2, $C_{13}\times D_7$ x 2, $C_{182}$

Order 169: $C_{13}^2$

Order 168: $S_3\times D_{14}$ x 16, $C_7:D_{12}$ x 8, $D_{84}$ x 4, $C_7\times D_{12}$ x 4, $C_{12}\times D_7$ x 4, $C_2\times D_{42}$ x 2, $D_6\times C_{14}$ x 2, $A_4\times D_7$ x 2, $C_7:S_4$ x 2, $C_7\times S_4$ x 2, $C_6\times D_{14}$, $A_4\times C_{14}$, $C_2\times C_{84}$, $C_{14}:C_{12}$

Order 156: $F_{13}$ x 10, $C_{26}:C_6$ x 7, $S_3\times D_{13}$ x 4, $C_{13}:C_{12}$ x 3, $C_{13}:C_{12}$ x 2, $D_{78}$ x 2, $S_3\times C_{26}$ x 2, $C_{26}:C_6$ x 2, $C_{39}:C_4$ x 2, $C_{13}:C_{12}$ x 2, $C_{156}$, $C_{13}:C_{12}$, $C_3\times D_{26}$, $C_{13}:A_4$, $A_4\times C_{13}$

Order 144: $S_3\times D_{12}$ x 9, $S_3\times S_4$ x 4, $C_6:D_{12}$ x 4, $C_6:D_{12}$ x 4, $C_6.D_{12}$ x 3, $D_6^2$ x 3, $D_6:C_{12}$ x 2, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $S_3^2:C_2^2$ x 2, $C_{12}\times D_6$ x 2, $D_6:D_6$ x 2, $C_6^2:C_4$ x 2, $S_3^2:C_4$ x 2, $C_{12}:C_{12}$, $C_{12}.C_{12}$, $C_6.D_{12}$, $C_3:D_{24}$, $C_6\times S_4$, $A_4^2$, $D_{12}:C_6$, $C_6\times D_{12}$, $C_{12}\times A_4$, $C_{12}.D_6$, $A_4:C_{12}$, $C_{12}^2$

Order 128: $C_2\wr D_4$

Order 117: $C_{39}:C_3$

Order 112: $D_4\times D_7$ x 9, $C_2^2\times D_{14}$ x 2, $C_{14}:D_4$ x 2, $D_4\times C_{14}$, $C_2\times D_{28}$, $C_4\times D_{14}$

Order 104: $C_{26}:C_4$ x 7, $C_{13}:D_4$ x 2, $C_2\times D_{26}$ x 2, $D_{52}$, $C_4\times D_{13}$, $D_4\times C_{13}$

Order 98: $C_7\times D_7$ x 5, $C_7:D_7$ x 3, $C_7\times C_{14}$ x 2

Order 96: $C_2^2:D_{12}$ x 4, $C_4:D_{12}$ x 2, $C_2^2:S_4$ x 2, $C_2^2\times S_4$ x 2, $D_4\times D_6$ x 2, $C_2^2\times D_{12}$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_4:S_4$ x 2, $C_{12}:D_4$ x 2, $C_4:D_{12}$ x 2, $C_4\times D_{12}$, $C_2^3:A_4$, $D_4:C_{12}$, $C_4.D_{12}$, $D_4:D_6$, $D_4\times A_4$, $C_4\times S_4$, $D_4\times C_{12}$, $D_4:D_6$, $D_{12}:C_4$, $C_8:D_6$

Order 91: $C_{91}$

Order 84: $S_3\times D_7$ x 16, $D_{42}$ x 8, $S_3\times C_{14}$ x 8, $C_3\times D_{14}$ x 6, $C_{84}$ x 2, $C_7:C_{12}$ x 2, $C_2\times C_{42}$, $C_7\times A_4$

Order 78: $C_{13}:C_6$ x 7, $C_{13}:C_6$ x 4, $D_{39}$ x 2, $C_3\times D_{13}$ x 2, $S_3\times C_{13}$ x 2, $C_{78}$

Order 72: $S_3\times D_6$ x 13, $C_3:D_{12}$ x 10, $C_3:D_{12}$ x 6, $C_3\times D_{12}$ x 5, $S_3\times C_{12}$ x 5, $\SOPlus(4,2)$ x 4, $C_6:D_6$ x 3, $C_6\times D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $C_3\times S_4$ x 2, $C_6\times C_{12}$ x 2, $C_6:C_{12}$ x 2, $C_6.D_6$ x 2, $C_6\times A_4$, $C_6\wr C_2$, $C_3^2:Q_8$, $C_3:C_{24}$

Order 64: $C_2\wr C_4$ x 2, $C_2\wr C_2^2$ x 2, $C_4^2:C_4$, $D_4^2$, $D_4:D_4$

Order 56: $C_2\times D_{14}$ x 18, $C_7:D_4$ x 10, $C_7\times D_4$ x 5, $D_{28}$ x 5, $C_4\times D_7$ x 5, $C_2^2\times C_{14}$ x 2, $C_2\times C_{28}$, $C_{14}:C_4$

Order 52: $D_{26}$ x 6, $C_{13}:C_4$ x 6, $C_2\times C_{26}$ x 2, $C_{52}$, $C_{13}:C_4$

Order 49: $C_7^2$

Order 48: $C_2\times D_{12}$ x 20, $S_3\times D_4$ x 11, $C_2\times S_4$ x 8, $D_6:C_4$ x 6, $C_2^2\times D_6$ x 4, $C_6:D_4$ x 4, $C_2^2\times A_4$ x 2, $C_2^2\times C_{12}$ x 2, $D_{12}:C_2$ x 2, $C_4\times D_6$ x 2, $C_4\times A_4$ x 2, $C_2^2:C_{12}$ x 2, $C_4\times C_{12}$ x 2, $D_{24}$, $C_{24}:C_2$, $C_2^2:A_4$, $D_4:C_6$, $C_6\times D_4$, $D_{12}:C_2$, $A_4:C_4$, $C_3\times \OD_{16}$, $C_4:C_{12}$, $Q_8:S_3$, $C_3:D_8$, $C_{12}:C_4$, $C_3:\OD_{16}$

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

Order 39: $C_{13}:C_3$ x 3, $C_{39}$

Order 36: $S_3^2$ x 12, $C_6\times S_3$ x 9, $C_6:S_3$ x 7, $C_3\times C_{12}$ x 3, $C_3:C_{12}$ x 3, $C_3^2:C_4$ x 2, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\wr C_2$ x 5, $C_2^3:C_4$ x 4, $C_2^2\times D_4$ x 2, $C_4:D_4$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $D_8:C_2$, $C_4:D_4$, $C_4\times D_4$, $C_4\wr C_2$

Order 28: $D_{14}$ x 34, $C_2\times C_{14}$ x 10, $C_{28}$ x 3, $C_7:C_4$ x 3

Order 26: $D_{13}$ x 5, $C_{26}$ x 3

Order 24: $D_{12}$ x 28, $C_2\times D_6$ x 25, $C_3:D_4$ x 11, $C_2\times C_{12}$ x 10, $C_4\times S_3$ x 7, $S_4$ x 7, $C_3\times D_4$ x 6, $C_2\times A_4$ x 4, $C_6:C_4$ x 2, $C_2^2\times C_6$ x 2, $C_3:Q_8$, $C_{24}$, $C_3\times Q_8$, $C_3:C_8$

Order 21: $C_{21}$

Order 18: $C_3\times S_3$ x 5, $C_3:S_3$ x 3, $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$ x 17, $C_2^2:C_4$ x 8, $C_2^4$ x 3, $D_4:C_2$ x 2, $C_2^2\times C_4$ x 2, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_4:C_4$, $C_4^2$

Order 14: $D_7$ x 14, $C_{14}$ x 11

Order 13: $C_{13}$ x 2

Order 12: $D_6$ x 36, $C_{12}$ x 11, $C_2\times C_6$ x 9, $A_4$ x 3, $C_3:C_4$ x 3

Order 9: $C_3^2$

Order 8: $D_4$ x 19, $C_2^3$ x 12, $C_2\times C_4$ x 10, $Q_8$, $C_8$

Order 7: $C_7$ x 3

Order 6: $S_3$ x 10, $C_6$ x 8

Order 4: $C_2^2$ x 17, $C_4$ x 6

Order 3: $C_3$ x 2

Order 2: $C_2$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 9539712: $\PGL(2,13)\wr C_2$

Order 4769856: $\PSL(2,13).\PSL(2,13).C_2^2$, $\PSL(2,13)^2.C_4$, $\POPlus(4,13)$

Order 2384928: $\PSL(2,13).\PSL(2,13).C_2$, $\PSL(2,13)\wr C_2$, $\PSOPlus(4,13)$

Order 1192464: $\PSL(2,13)^2$

Order 340704: $F_{13}.\PSL(2,13).C_2$

Order 170352: $D_{13}:C_3.\PSL(2,13).C_2$, $F_{13}.\PSL(2,13)$, $D_{13}:C_3.\PSL(2,13).C_2$

Order 113568: $D_{13}.C_2.\PSL(2,13).C_2$

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

Order 61152: $D_{14}.\PSL(2,13).C_2$

Order 56784: $D_{13}.\PSL(2,13).C_2$, $D_{13}.C_2.\PSL(2,13)$, $D_{13}.\PSL(2,13).C_2$

Order 52416: $S_4.\PSL(2,13).C_2$, $D_{12}.\PSL(2,13).C_2$

Order 48672: $F_{13}\wr C_2$

Order 42588: $C_{13}:C_3.\PSL(2,13)$

Order 30576: $D_7.\PSL(2,13).C_2$ x 4, $D_{14}.\PSL(2,13)$, $C_{14}.\PSL(2,13).C_2$, $C_{14}.\PSL(2,13).C_2$

Order 28392: $D_{13}.\PSL(2,13)$, $C_{13}.\PSL(2,13).C_2$, $C_{13}.\PSL(2,13).C_2$

Order 26208: $D_6.\PSL(2,13).C_2$ x 2, $D_6.\PSL(2,13).C_2$ x 2, $D_{12}.\PSL(2,13)$, $A_4.\PSL(2,13).C_2$, $A_4.\PSL(2,13).C_2$, $C_{12}.\PSL(2,13).C_2$, $S_4.\PSL(2,13)$, $C_{12}\times \PGL(2,13)$

Order 24336: $C_{13}^2.C_{12}^2$, $C_{13}:F_{13}.D_6$, $D_{13}^2.(C_6\times S_3)$

Order 17472: $D_4.\PSL(2,13).C_2$

Order 16224: $D_{13}\wr C_2:C_{12}$, $C_{13}^2:D_{12}:C_4$

Order 15288: $C_7.\PSL(2,13).C_2$ x 2, $D_7.\PSL(2,13)$ x 2, $C_7.\PSL(2,13).C_2$ x 2, $C_{14}.\PSL(2,13)$

Order 14196: $C_{13}\times \PSL(2,13)$

Order 13104: $S_3\times \PGL(2,13)$ x 4, $C_6.\PSL(2,13).C_2$ x 2, $D_6.\PSL(2,13)$ x 2, $A_4.\PSL(2,13)$, $C_6.\PSL(2,13).C_2$, $C_{12}.\PSL(2,13)$, $C_6.\PSL(2,13).C_2$

Order 12168: $C_{13}^2:(C_6\times C_{12})$, $C_{13}^2:(C_6\times C_{12})$, $C_{13}^2:C_3:C_{24}$, $C_{13}^2:(C_3\times D_{12})$, $D_{13}^2:(C_3\times S_3)$, $C_{13}^2:(S_3\times C_{12})$, $C_{13}^2:C_3^2:Q_8$

Order 8736: $C_2^2.\PSL(2,13).C_2$ x 2, $C_4.\PSL(2,13).C_2$, $C_4.\PSL(2,13).C_2$, $C_2^2\times \PSL(2,13).C_2$, $C_2^2.\PSL(2,13).C_2$, $D_4.\PSL(2,13)$

Order 8112: $C_{13}^2:(C_4\times C_{12})$ x 2, $C_{13}^2:(C_4\times C_{12})$, $D_{13}^2.D_6$, $D_{13}^2.C_{12}$, $C_{13}^2:C_3:\OD_{16}$, $D_{13}\wr C_2:C_6$

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

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

Order 6084: $C_{13}^2:(C_3\times C_{12})$ x 2, $C_{13}^2:(C_6\times S_3)$, $C_{13}^2:C_6^2$, $C_{13}^2:(C_3\times C_{12})$, $C_{13}^2:(C_3\times C_3:C_4)$, $C_{13}^2:(C_3\times C_{12})$

Order 5408: $D_{13}^2.D_4$

Order 4368: $C_2\times \PGL(2,13)$ x 6, $C_{13}:(C_2\times C_{12}\times D_7)$, $C_4.\PSL(2,13)$, $C_2^2\times \PSL(2,13)$, $C_2^2.\PSL(2,13)$, $\PSL(2,13):C_4$

Order 4056: $C_{13}^2:(C_2\times C_{12})$ x 2, $D_{13}:F_{13}$ x 2, $D_{13}^2:C_6$ x 2, $C_{13}^2:(C_2\times C_{12})$, $C_{13}^2:(C_2\times C_{12})$, $C_{13}^2:(C_2\times C_{12})$, $C_{13}^2:(C_2\times C_{12})$, $C_{13}^2:C_{24}$, $C_{13}^2:D_{12}$, $C_{13}^2:C_3:Q_8$, $C_{13}^2:(C_3\times Q_8)$, $C_{13}^2:C_3:C_8$, $C_{13}^2:(C_4\times S_3)$, $D_{13}^2:S_3$

Order 3744: $C_{39}.(C_{12}\times D_4)$, $S_4\times F_{13}$

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

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

Order 2704: $C_{13}^2:C_4^2$, $C_{13}^2:\OD_{16}$, $D_{13}^2.C_2^2$

Order 2184: $\PGL(2,13)$ x 5, $C_{13}:(C_{12}\times D_7)$ x 4, $C_2\times \PSL(2,13)$ x 3, $C_{13}:(C_6\times C_7:C_4)$, $C_{13}:(C_2\times C_6\times D_7)$, $C_{14}\times F_{13}$

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

Order 1872: $D_{78}:C_{12}$ x 2, $D_6\times F_{13}$ x 2, $D_{156}:C_6$, $C_{156}:C_{12}$, $C_{12}\times F_{13}$, $C_{13}:C_6\times S_4$, $A_4\times F_{13}$, $A_4:F_{13}$

Order 1568: $D_{14}\wr C_2$

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

Order 1456: $D_{182}:C_4$

Order 1352: $D_{13}\wr C_2$ x 2, $D_{13}^2.C_2$ x 2, $C_{13}^2:Q_8$, $D_{13}.D_{26}$, $C_{13}^2:C_8$

Order 1248: $C_{13}:C_4\times S_4$, $D_{156}:C_4$, $D_4\times F_{13}$

Order 1152: $D_{12}\wr C_2$, $S_4\wr C_2$

Order 1092: $D_{91}:C_6$ x 4, $C_{91}:C_{12}$ x 2, $C_7\times F_{13}$ x 2, $\PSL(2,13)$ x 2, $C_{182}:C_6$, $C_{182}:C_6$, $C_{26}:C_{42}$

Order 1014: $C_{13}^2:C_6$ x 2, $C_{13}^2:C_6$, $C_{13}:C_{78}$, $C_{13}^2:S_3$, $C_{13}^2:C_6$, $C_{13}^2:C_6$, $C_{13}^2:C_6$, $C_{13}^2:C_6$

Order 936: $S_3\times F_{13}$ x 4, $D_{78}:C_6$ x 2, $D_{78}:C_6$ x 2, $C_{78}:C_{12}$ x 2, $C_6\times F_{13}$ x 2, $C_{156}:C_6$, $C_{156}:C_6$, $C_{156}:C_6$, $A_4\times C_{13}:C_6$, $C_{13}:(C_3\times S_4)$, $C_{13}:C_3\times S_4$

Order 784: $D_7^2:C_2^2$ x 2, $D_7^2:C_4$ x 2, $D_{14}^2$, $D_{14}:D_{14}$, $C_{14}^2:C_4$

Order 728: $D_{91}:C_4$ x 4, $D_7\times D_{26}$, $C_{182}:C_4$, $C_{26}:C_{28}$

Order 676: $D_{13}^2$ x 2, $C_{13}^2:C_4$ x 2, $C_{13}:C_{52}$, $C_{13}^2:C_4$, $C_{13}^2:C_4$

Order 672: $S_4\times D_{14}$, $D_{12}\times D_{14}$

Order 624: $D_{78}:C_4$ x 2, $C_2^2\times F_{13}$ x 2, $D_{78}:C_4$ x 2, $D_{26}:C_{12}$ x 2, $C_4\times F_{13}$ x 2, $S_4\times D_{13}$, $C_2^2:F_{13}$, $A_4\times C_{13}:C_4$, $D_{13}.S_4$, $D_{12}\times D_{13}$, $C_{156}:C_4$, $C_{156}:C_4$, $D_{52}:C_6$, $C_{52}:C_{12}$

Order 576: $D_6^2:C_2^2$ x 2, $D_6^2:C_4$ x 2, $\POPlus(4,3)$, $S_4^2$, $A_4^2:C_4$, $S_4\times D_{12}$, $D_{12}^2$, $D_{12}:D_{12}$, $C_{12}^2:C_4$

Order 546: $C_{91}:C_6$ x 2, $C_{91}:C_6$ x 2, $C_{13}:C_{42}$ x 2, $C_{13}:C_{42}$

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

Order 468: $D_{39}:C_6$ x 4, $C_3\times F_{13}$ x 3, $C_{78}:C_6$ x 2, $C_{78}:C_6$ x 2, $C_{39}:C_{12}$ x 2, $C_{78}:C_6$, $A_4\times C_{13}:C_3$, $C_{156}:C_3$, $C_{39}:C_{12}$

Order 416: $D_{52}:C_4$

Order 392: $D_7\times D_{14}$ x 8, $D_7\wr C_2$ x 4, $C_2\times C_7^2:C_4$ x 2, $C_{14}:D_{14}$, $C_{14}\times D_{14}$, $C_{14}\wr C_2$, $C_7:D_{28}$, $C_{14}.D_{14}$

Order 364: $D_7\times D_{13}$ x 4, $C_{91}:C_4$ x 2, $C_{13}:C_{28}$ x 2, $C_{13}\times D_{14}$, $C_7\times D_{26}$, $D_{182}$

Order 338: $C_{13}\times D_{13}$ x 2, $C_{13}:D_{13}$

Order 336: $D_7\times D_{12}$ x 8, $D_7\times S_4$ x 4, $D_6\times D_{14}$ x 2, $C_{14}:D_{12}$ x 2, $A_4\times D_{14}$, $C_{14}:S_4$, $C_{14}\times S_4$, $C_2\times D_{84}$, $C_{14}\times D_{12}$, $C_{12}\times D_{14}$

Order 312: $C_2\times F_{13}$ x 9, $D_{39}:C_4$ x 4, $C_{52}:C_6$ x 2, $S_3\times D_{26}$ x 2, $C_{78}:C_4$ x 2, $C_{26}:C_{12}$ x 2, $D_{26}:C_6$ x 2, $C_{13}:D_{12}$ x 2, $D_{26}:C_6$ x 2, $D_{13}:A_4$, $A_4\times D_{13}$, $C_{13}:S_4$, $C_{13}\times S_4$, $D_{156}$, $C_{13}\times D_{12}$, $C_{12}\times D_{13}$, $C_{52}:C_6$, $C_{52}:C_6$

Order 288: $D_6:D_{12}$ x 3, $D_6\times D_{12}$ x 2, $D_6\wr C_2$ x 2, $D_6:S_4$ x 2, $C_{12}:D_{12}$ x 2, $(C_6\times C_{12}):C_4$ x 2, $C_6^2.D_4$ x 2, $D_6\times S_4$ x 2, $D_{12}:D_6$, $A_4\times D_{12}$, $C_{12}:S_4$, $C_{12}\times S_4$, $C_{12}:D_{12}$, $C_{12}\times D_{12}$, $D_{12}:D_6$, $C_{12}\wr C_2$, $C_{12}.D_{12}$, $\PSOPlus(4,3)$, $A_4\wr C_2$, $A_4\times S_4$

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

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

Order 224: $D_4\times D_{14}$

Order 208: $D_{26}:C_4$ x 2, $D_{26}:C_4$ x 2, $D_4\times D_{13}$, $C_{52}:C_4$, $C_{52}:C_4$

Order 196: $D_7^2$ x 12, $C_7\times D_{14}$ x 7, $C_7:D_{14}$ x 4, $C_7^2:C_4$ x 2, $C_7:C_{28}$, $C_{14}^2$

Order 192: $C_2^3:S_4$, $D_4:D_{12}$, $D_4\times S_4$, $D_4\times D_{12}$

Order 182: $D_{91}$ x 2, $C_7\times D_{13}$ x 2, $C_{13}\times D_7$ x 2, $C_{182}$

Order 169: $C_{13}^2$

Order 168: $S_3\times D_{14}$ x 16, $C_7:D_{12}$ x 8, $D_{84}$ x 4, $C_7\times D_{12}$ x 4, $C_{12}\times D_7$ x 4, $C_2\times D_{42}$ x 2, $D_6\times C_{14}$ x 2, $A_4\times D_7$ x 2, $C_7:S_4$ x 2, $C_7\times S_4$ x 2, $C_6\times D_{14}$, $A_4\times C_{14}$, $C_2\times C_{84}$, $C_{14}:C_{12}$

Order 156: $F_{13}$ x 10, $C_{26}:C_6$ x 7, $S_3\times D_{13}$ x 4, $C_{13}:C_{12}$ x 3, $C_{13}:C_{12}$ x 2, $D_{78}$ x 2, $S_3\times C_{26}$ x 2, $C_{26}:C_6$ x 2, $C_{39}:C_4$ x 2, $C_{13}:C_{12}$ x 2, $C_{156}$, $C_{13}:C_{12}$, $C_3\times D_{26}$, $C_{13}:A_4$, $A_4\times C_{13}$

Order 144: $S_3\times D_{12}$ x 9, $S_3\times S_4$ x 4, $C_6:D_{12}$ x 4, $C_6:D_{12}$ x 4, $C_6.D_{12}$ x 3, $D_6^2$ x 3, $D_6:C_{12}$ x 2, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $S_3^2:C_2^2$ x 2, $C_{12}\times D_6$ x 2, $D_6:D_6$ x 2, $C_6^2:C_4$ x 2, $S_3^2:C_4$ x 2, $C_{12}:C_{12}$, $C_{12}.C_{12}$, $C_6.D_{12}$, $C_3:D_{24}$, $C_6\times S_4$, $A_4^2$, $D_{12}:C_6$, $C_6\times D_{12}$, $C_{12}\times A_4$, $C_{12}.D_6$, $A_4:C_{12}$, $C_{12}^2$

Order 128: $C_2\wr D_4$

Order 117: $C_{39}:C_3$

Order 112: $D_4\times D_7$ x 9, $C_2^2\times D_{14}$ x 2, $C_{14}:D_4$ x 2, $D_4\times C_{14}$, $C_2\times D_{28}$, $C_4\times D_{14}$

Order 104: $C_{26}:C_4$ x 7, $C_{13}:D_4$ x 2, $C_2\times D_{26}$ x 2, $D_{52}$, $C_4\times D_{13}$, $D_4\times C_{13}$

Order 98: $C_7\times D_7$ x 5, $C_7:D_7$ x 3, $C_7\times C_{14}$ x 2

Order 96: $C_2^2:D_{12}$ x 4, $C_4:D_{12}$ x 2, $C_2^2:S_4$ x 2, $C_2^2\times S_4$ x 2, $D_4\times D_6$ x 2, $C_2^2\times D_{12}$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_4:S_4$ x 2, $C_{12}:D_4$ x 2, $C_4:D_{12}$ x 2, $C_4\times D_{12}$, $C_2^3:A_4$, $D_4:C_{12}$, $C_4.D_{12}$, $D_4:D_6$, $D_4\times A_4$, $C_4\times S_4$, $D_4\times C_{12}$, $D_4:D_6$, $D_{12}:C_4$, $C_8:D_6$

Order 91: $C_{91}$

Order 84: $S_3\times D_7$ x 16, $D_{42}$ x 8, $S_3\times C_{14}$ x 8, $C_3\times D_{14}$ x 6, $C_{84}$ x 2, $C_7:C_{12}$ x 2, $C_2\times C_{42}$, $C_7\times A_4$

Order 78: $C_{13}:C_6$ x 7, $C_{13}:C_6$ x 4, $D_{39}$ x 2, $C_3\times D_{13}$ x 2, $S_3\times C_{13}$ x 2, $C_{78}$

Order 72: $S_3\times D_6$ x 13, $C_3:D_{12}$ x 10, $C_3:D_{12}$ x 6, $C_3\times D_{12}$ x 5, $S_3\times C_{12}$ x 5, $\SOPlus(4,2)$ x 4, $C_6:D_6$ x 3, $C_6\times D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $C_3\times S_4$ x 2, $C_6\times C_{12}$ x 2, $C_6:C_{12}$ x 2, $C_6.D_6$ x 2, $C_6\times A_4$, $C_6\wr C_2$, $C_3^2:Q_8$, $C_3:C_{24}$

Order 64: $C_2\wr C_4$ x 2, $C_2\wr C_2^2$ x 2, $C_4^2:C_4$, $D_4^2$, $D_4:D_4$

Order 56: $C_2\times D_{14}$ x 18, $C_7:D_4$ x 10, $C_7\times D_4$ x 5, $D_{28}$ x 5, $C_4\times D_7$ x 5, $C_2^2\times C_{14}$ x 2, $C_2\times C_{28}$, $C_{14}:C_4$

Order 52: $D_{26}$ x 6, $C_{13}:C_4$ x 6, $C_2\times C_{26}$ x 2, $C_{52}$, $C_{13}:C_4$

Order 49: $C_7^2$

Order 48: $C_2\times D_{12}$ x 20, $S_3\times D_4$ x 11, $C_2\times S_4$ x 8, $D_6:C_4$ x 6, $C_2^2\times D_6$ x 4, $C_6:D_4$ x 4, $C_2^2\times A_4$ x 2, $C_2^2\times C_{12}$ x 2, $D_{12}:C_2$ x 2, $C_4\times D_6$ x 2, $C_4\times A_4$ x 2, $C_2^2:C_{12}$ x 2, $C_4\times C_{12}$ x 2, $D_{24}$, $C_{24}:C_2$, $C_2^2:A_4$, $D_4:C_6$, $C_6\times D_4$, $D_{12}:C_2$, $A_4:C_4$, $C_3\times \OD_{16}$, $C_4:C_{12}$, $Q_8:S_3$, $C_3:D_8$, $C_{12}:C_4$, $C_3:\OD_{16}$

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

Order 39: $C_{13}:C_3$ x 3, $C_{39}$

Order 36: $S_3^2$ x 12, $C_6\times S_3$ x 9, $C_6:S_3$ x 7, $C_3\times C_{12}$ x 3, $C_3:C_{12}$ x 3, $C_3^2:C_4$ x 2, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\wr C_2$ x 5, $C_2^3:C_4$ x 4, $C_2^2\times D_4$ x 2, $C_4:D_4$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $D_8:C_2$, $C_4:D_4$, $C_4\times D_4$, $C_4\wr C_2$

Order 28: $D_{14}$ x 34, $C_2\times C_{14}$ x 10, $C_{28}$ x 3, $C_7:C_4$ x 3

Order 26: $D_{13}$ x 5, $C_{26}$ x 3

Order 24: $D_{12}$ x 28, $C_2\times D_6$ x 25, $C_3:D_4$ x 11, $C_2\times C_{12}$ x 10, $C_4\times S_3$ x 7, $S_4$ x 7, $C_3\times D_4$ x 6, $C_2\times A_4$ x 4, $C_6:C_4$ x 2, $C_2^2\times C_6$ x 2, $C_3:Q_8$, $C_{24}$, $C_3\times Q_8$, $C_3:C_8$

Order 21: $C_{21}$

Order 18: $C_3\times S_3$ x 5, $C_3:S_3$ x 3, $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$ x 17, $C_2^2:C_4$ x 8, $C_2^4$ x 3, $D_4:C_2$ x 2, $C_2^2\times C_4$ x 2, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_4:C_4$, $C_4^2$

Order 14: $D_7$ x 14, $C_{14}$ x 11

Order 13: $C_{13}$ x 2

Order 12: $D_6$ x 36, $C_{12}$ x 11, $C_2\times C_6$ x 9, $A_4$ x 3, $C_3:C_4$ x 3

Order 9: $C_3^2$

Order 8: $D_4$ x 19, $C_2^3$ x 12, $C_2\times C_4$ x 10, $Q_8$, $C_8$

Order 7: $C_7$ x 3

Order 6: $S_3$ x 10, $C_6$ x 8

Order 4: $C_2^2$ x 17, $C_4$ x 6

Order 3: $C_3$ x 2

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 9539712: $\PGL(2,13)\wr C_2$ ($C_1$)

Order 4769856: $\PSL(2,13).\PSL(2,13).C_2^2$ ($C_2$), $\PSL(2,13)^2.C_4$ ($C_2$), $\POPlus(4,13)$ ($C_2$)

Order 2384928: $\PSOPlus(4,13)$ ($C_2^2$)

Order 1192464: $\PSL(2,13)^2$ ($D_4$)

Order 1: $C_1$ ($\PGL(2,13)\wr C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 9539712: $\PGL(2,13)\wr C_2$ ($C_1$)

Order 4769856: $\PSL(2,13).\PSL(2,13).C_2^2$ ($C_2$), $\PSL(2,13)^2.C_4$ ($C_2$), $\POPlus(4,13)$ ($C_2$)

Order 2384928: $\PSOPlus(4,13)$ ($C_2^2$)

Order 1192464: $\PSL(2,13)^2$ ($D_4$)

Order 1: $C_1$ ($\PGL(2,13)\wr C_2$)

Series

Derived series	$\PGL(2,13)\wr C_2$	$\rhd$	$\PSOPlus(4,13)$	$\rhd$	$\PSL(2,13)^2$
Chief series	$\PGL(2,13)\wr C_2$	$\rhd$	$\PSL(2,13).\PSL(2,13).C_2^2$	$\rhd$	$\PSOPlus(4,13)$	$\rhd$	$\PSL(2,13)^2$	$\rhd$	$C_1$
Lower central series	$\PGL(2,13)\wr C_2$	$\rhd$	$\PSOPlus(4,13)$	$\rhd$	$\PSL(2,13)^2$
Upper central series	$C_1$

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 $135 \times 135$ 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 $70 \times 70$ rational character table.