Abstract group 3849120.a: $C_3^5:(C_2\times M_{11})$

• Label: 3849120.a
• Order: \( 2^{5} \cdot 3^{7} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{7} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $33$
• Trans deg.: $33$
• Rank: $2$

Group information

Description:	$C_3^5:(C_2\times M_{11})$
Order:	\(3849120\)\(\medspace = 2^{5} \cdot 3^{7} \cdot 5 \cdot 11 \)
Exponent:	\(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Automorphism group:	$C_3^5:(C_2\times M_{11})$, of order \(3849120\)\(\medspace = 2^{5} \cdot 3^{7} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$, $C_3$ x 5, $M_{11}$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	11	12	15	18	22	24
Elements	1	6183	35882	106920	128304	608850	641520	71280	384912	349920	374220	256608	213840	349920	320760	3849120
Conjugacy classes	1	3	4	2	1	8	4	2	1	2	3	1	1	2	2	37
Divisions	1	3	4	2	1	8	2	2	1	1	3	1	1	1	1	32
Autjugacy classes	1	3	4	2	1	8	4	2	1	2	3	1	1	2	2	37

Dimension	1	10	11	16	20	22	32	44	45	55	198	220	352	440	880
Irr. complex chars.	2	6	2	4	0	2	0	2	2	2	2	7	1	1	4	37
Irr. rational chars.	2	2	2	0	2	2	2	2	2	2	2	5	1	2	4	32

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$33$
Rank:	$2$
Inequivalent generating pairs:	$2351514$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	22	22	22
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $33$ $\langle(1,22,10,15,26,3,23,12,13,27)(2,24,11,14,25)(4,32,29,19,9)(5,33,30,20,8,6,31,28,21,7) \!\cdots\! \rangle$
Transitive group:	33T83				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^4:S_3)$ $\,\rtimes\,$ $M_{11}$	$(C_3^5:M_{11})$ $\,\rtimes\,$ $C_2$	$C_3^5$ $\,\rtimes\,$ $(C_2\times M_{11})$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 33.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 14512729 subgroups in 1136 conjugacy classes, 5 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^5:(C_2\times M_{11})$
Commutator:	$G' \simeq$ $C_3^5:M_{11}$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^5:(C_2\times M_{11})$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^5$	$G/\operatorname{Fit} \simeq$ $C_2\times M_{11}$
Radical:	$R \simeq$ $C_3^4:S_3$	$G/R \simeq$ $M_{11}$
Socle:	$\operatorname{soc} \simeq$ $C_3^5$	$G/\operatorname{soc} \simeq$ $C_2\times M_{11}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times \SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5:C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

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 3849120: $C_3^5:(C_2\times M_{11})$

Order 1924560: $C_3^5:M_{11}$

Order 349920: $C_3^4:A_6.D_6$

Order 320760: $C_3^5:(C_2\times \PSL(2,11))$

Order 174960: $C_3^5:(C_2\times A_6)$, $C_3^4:A_6.S_3$, $C_3^4:A_6.C_6$

Order 160380: $C_3^5:\PSL(2,11)$

Order 116640: $C_3^4:A_6.C_2^2$

Order 87480: $C_3^5:A_6$

Order 69984: $C_3^5:F_9:C_2^2$

Order 58320: $C_3^4:A_6.C_2$ x 2, $C_3^4:(C_2\times A_6)$, $C_3^5:(C_2\times S_5)$

Order 34992: $(C_9^2\times C_{54}).D_4$, $(C_9^2\times C_{54}).C_2^3$, $(C_9^2\times C_{54}).D_4$, $C_3^5:F_9:C_2$, $(C_9^2\times C_{54}).C_2^3$, $(C_9^2\times C_{54}).D_4$, $C_3^5:(C_2\times F_9)$

Order 29160: $C_3^5:(C_2\times A_5)$, $C_3^4:A_6$, $C_3^5:S_5$, $C_3^5:(C_2\times A_5)$, $C_3^5:S_5$

Order 26730: $C_3^5:C_{11}:C_{10}$

Order 23328: $C_3^5:(C_2\times \GL(2,3))$

Order 17496: $C_3^5:\SOPlus(4,2)$, $(C_9^2\times C_{54}).C_2^2$, $C_3^5:\SOPlus(4,2)$, $C_3^5:F_9$, $C_3^5:F_9$, $(C_9^2\times C_{54}).C_2^2$, $C_3^5:\SOPlus(4,2)$, $C_3^5:\PSU(3,2)$, $C_3^5:\PSU(3,2)$, $C_3^5:\PSU(3,2)$, $C_3^4.S_3^3$

Order 15840: $C_2\times M_{11}$

Order 14580: $C_3^5:A_5$, $C_3^5:A_5$

Order 13365: $C_3^5:C_{11}:C_5$

Order 11664: $C_3^4:(C_2\times \PSU(3,2))$, $C_3^3:S_3^2:D_6$, $C_3^5:(C_2\times \SL(2,3))$, $C_3^5:\GL(2,3)$, $C_3^5:\GL(2,3)$, $C_3^5:(C_2\times S_4)$

Order 9720: $C_3^4:(S_3\times F_5)$, $C_3^4:(C_2\times A_5)$

Order 8748: $C_3^5:S_3^2$, $C_3^5:C_6:S_3$, $C_3^5:C_3^2:C_4$, $C_3^5:S_3^2$, $C_3^5:(C_6\times S_3)$, $(C_9^2\times C_{54}).C_2$, $(C_9^2\times C_{54}).C_2$, $C_3^5:S_3^2$, $C_3^3.C_3\wr C_4$

Order 7920: $M_{11}$ x 2

Order 7776: $C_3^3:F_9:C_2^2$, $C_3^4:(S_3\times \SD_{16})$

Order 5832: $C_3^4:(C_3\times C_6):C_4$ x 2, $C_3^4:\PSU(3,2)$ x 2, $C_3^3.S_3^3$, $C_3^3.S_3^3$, $C_3^4:\SOPlus(4,2)$, $C_3^4:(C_3\times C_6):C_4$, $C_3^4:\SOPlus(4,2)$, $C_3^4:\SOPlus(4,2)$, $C_3^4:(C_3\times C_6):C_4$, $C_3^4:\SOPlus(4,2)$, $C_3^4:\PSU(3,2)$, $C_3^4:\PSU(3,2)$, $C_3^5:\SL(2,3)$, $C_3^5:S_4$, $C_3\wr A_4:S_3$, $C_3^5:S_4$, $C_3^3.S_3^3$

Order 5346: $C_3^5:C_{22}$

Order 4860: $C_3\wr F_5$, $C_3^5:D_{10}$, $C_3^5:F_5$, $C_3^4:A_5$

Order 4374: $C_3^5:(C_3\times S_3)$, $C_3^3.C_3^2\wr C_2$, $C_3^5:C_3:S_3$, $C_3^5:(C_3\times C_6)$, $C_3^5:(C_3\times S_3)$

Order 3888: $C_3^3:S_3^2.C_2^2$, $C_3^3:S_3^2:C_2^2$, $C_3^3:F_9:C_2$, $C_3^3:F_9:C_2$, $C_3^3:(C_2\times F_9)$, $C_3^3:F_9:C_2$, $C_3^3:F_9:C_2$, $C_3^2:(S_3\times F_9)$, $C_3^4:(S_3\times Q_8)$, $C_3\wr C_4:D_6$, $C_3^5:\SD_{16}$, $C_3^5:\SD_{16}$, $C_3^5:\SD_{16}$, $C_3^5:\SD_{16}$, $C_3^3:(S_3\times S_4)$

Order 3240: $C_3^4:(C_2\times F_5)$

Order 2916: $C_3^3:C_3^2:C_{12}$ x 2, $C_3^4:C_3^2:C_4$ x 2, $C_3^5:D_6$, $C_3^4:(C_6\times S_3)$, $(C_3\times \He_3).S_3^2$, $(C_3\times \He_3).S_3^2$, $C_3^4.S_3^2$, $C_3^4:S_3^2$, $C_3^4:C_3^2:C_4$, $C_3^4.S_3^2$, $C_3^3:C_3^2:C_{12}$, $C_3^4:(C_6\times S_3)$, $C_3^4:C_3^2:C_4$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^3:C_3^2:C_{12}$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:A_4$, $C_3^5:D_6$, $C_3^5:(C_2\times C_6)$, $C_3^5:D_6$

Order 2673: $C_3^5:C_{11}$

Order 2592: $C_3^4:(C_2\times \SD_{16})$, $C_3^3:(C_2\times \GL(2,3))$

Order 2430: $C_3\wr D_5$, $C_3^4:D_{15}$, $C_3^5:C_{10}$

Order 2187: $C_3^5:C_3^2$

Order 1944: $C_3^4:S_4$ x 2, $C_3^3:\PSU(3,2)$ x 2, $C_3^3:S_3^2.C_2$ x 2, $C_3^2:S_3^3$, $C_3^5:Q_8$, $C_3^5:Q_8$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3\wr C_4:S_3$, $C_3^3:F_9$, $C_3^4:C_{24}$, $C_3^3:(S_3\times A_4)$, $C_3^4:S_4$, $C_3^2.S_3^3$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$, $C_3^3:F_9$, $C_3^2.S_3^3$, $C_3^2.S_3^3$

Order 1620: $C_3^4:F_5$ x 2, $C_3^4:D_{10}$

Order 1458: $C_3^4:C_3:S_3$, $C_3^4:(C_3\times C_6)$, $C_3^4:C_3:S_3$, $C_3^4:C_3:S_3$, $C_3^4.(C_3\times S_3)$, $C_3^4:C_3:S_3$, $C_3^4.(C_3\times S_3)$, $C_3^4:C_3:S_3$, $C_3^4.(C_3\times S_3)$, $C_3^4.(C_3\times S_3)$, $C_3^4:(C_3\times S_3)$, $C_3^4:(C_3\times S_3)$, $C_3^4:(C_3\times C_6)$, $C_3^4:C_3:S_3$, $C_3^4:(C_3\times S_3)$, $C_3^4:C_3:S_3$, $C_3^4:(C_3\times S_3)$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1440: $A_6.C_2^2$

Order 1320: $C_2\times \PSL(2,11)$

Order 1296: $C_3^4:\SD_{16}$ x 4, $C_3^3:(S_3\times Q_8)$, $C_3^3:(S_3\times D_4)$, $S_3^2:S_3^2$, $C_3^2:(C_2\times F_9)$, $S_3\wr S_3$, $C_3^3:(C_2\times \SL(2,3))$, $C_3^3:\GL(2,3)$, $C_3^3:\GL(2,3)$

Order 1215: $C_3\wr C_5$

Order 972: $C_3\wr A_4$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:C_3^2:C_4$ x 2, $C_3^3:C_3^2:C_4$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^4:D_6$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^3:S_3^2$

Order 864: $F_9:D_6$ x 2, $C_2\times C_3^2:\GL(2,3)$, $C_3^2:(S_3\times \SD_{16})$

Order 810: $C_3^4:D_5$ x 2, $C_3^4:C_{10}$

Order 729: $C_3^3.C_3^3$, $C_3^4:C_3^2$, $C_3^3.C_3^3$, $C_3^3.C_3^3$, $C_3^4:C_3^2$, $C_3^5:C_3$

Order 720: $A_6.C_2$ x 3, $S_3\times S_5$, $C_2\times A_6$

Order 660: $\PSL(2,11)$

Order 648: $C_3.S_3^3$ x 3, $C_3:S_3^3$ x 2, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3\wr D_4$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^2:F_9$ x 2, $C_3^3:S_4$ x 2, $C_3:S_3^3$, $C_3^4:Q_8$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $S_3\wr C_3$, $C_3^3:S_4$, $C_3^3:\SL(2,3)$, $C_2\times C_3^3:D_6$

Order 486: $C_3^4:S_3$ x 4, $C_3^3:(C_3\times C_6)$ x 3, $C_9.(S_3\times C_3^2)$ x 2, $(C_3\times \He_3).S_3$ x 2, $C_3\wr C_3:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$, $(C_3\times \He_3):S_3$, $C_3^3.(C_3\times S_3)$, $C_3^3:(C_3\times S_3)$, $C_3^4:S_3$, $C_3^4:S_3$, $C_9:C_3^2.S_3$, $(C_3\times \He_3):S_3$, $C_3^4.S_3$, $C_3^4:C_6$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^3:(C_3\times C_6)$, $C_3\wr C_3:C_6$, $C_3^4:C_6$, $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4:S_3$

Order 432: $S_3^3:C_2$ x 3, $C_3^3:\SD_{16}$ x 3, $S_3\times \PSU(3,2)$ x 2, $F_9:S_3$ x 2, $C_3^3:\SD_{16}$ x 2, $F_9:C_6$ x 2, $S_3\times F_9$ x 2, $C_3^2:\GL(2,3)$ x 2, $C_2\times \PU(3,2)$, $C_2.S_3^3$, $D_6:S_3^2$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $C_3^2:(S_3\times C_8)$, $C_6^2:D_6$, $\He_3:\SD_{16}$

Order 405: $C_3^4:C_5$

Order 360: $C_3\times S_5$ x 2, $A_6$ x 2, $S_3\times A_5$, $C_3:S_5$

Order 324: $C_3^3:D_6$ x 7, $C_3^2:S_3^2$ x 6, $C_3^2:S_3^2$ x 6, $C_3\wr C_2^2$ x 5, $C_3\wr C_4$ x 4, $C_3^3:D_6$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^3:A_4$ x 2, $C_3^2.S_3^2$ x 2, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^2.S_3^2$, $C_3^2.S_3^2$, $C_3^2:D_{18}$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2\times S_3^2$, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_9.S_3^2$

Order 288: $F_9:C_2^2$ x 2, $C_4.\SOPlus(4,2)$

Order 243: $C_3^4:C_3$ x 4, $C_3^2.C_3^3$ x 2, $C_3^3:C_3^2$ x 2, $C_3^5$, $C_9.C_3^3$, $C_3^3:C_3^2$, $C_3^2.C_3^3$, $\He_3:C_3^2$, $C_3^4.C_3$, $C_3^4:C_3$

Order 240: $C_2\times S_5$

Order 216: $S_3^3$ x 6, $C_3^3:Q_8$ x 5, $C_3^2:C_4\times S_3$ x 5, $S_3^2:S_3$ x 4, $S_3^2:C_6$ x 4, $C_3^2:D_{12}$ x 3, $C_3:F_9$ x 3, $\He_3:C_8$ x 2, $C_3^3:Q_8$ x 2, $C_3\times F_9$ x 2, $C_6.S_3^2$ x 2, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $\SU(3,2)$, $\He_3:D_4$, $C_6:S_3^2$, $\PU(3,2)$, $C_3^3:Q_8$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_3^3:Q_8$, $C_3^2:D_{12}$, $C_3^3:D_4$, $C_3^3:C_8$, $C_3^2:C_{24}$

Order 180: $\GL(2,4)$ x 2

Order 162: $C_3^3:C_6$ x 8, $C_3^3:S_3$ x 7, $C_3^2\wr C_2$ x 6, $C_3^3:C_6$ x 6, $C_3^3:C_6$ x 5, $C_3\wr S_3$ x 5, $C_3^3:S_3$ x 4, $C_3^3:S_3$ x 3, $\He_3.S_3$ x 3, $D_9:C_3^2$ x 3, $S_3\times \He_3$ x 3, $C_3^3:S_3$ x 2, $C_3^3.S_3$ x 2, $C_3^3.C_6$ x 2, $\He_3.S_3$ x 2, $C_3^3:C_6$, $S_3\times C_3^3$, $C_3^2:D_9$, $\He_3.C_6$, $C_3^2:C_{18}$, $C_3^3:C_6$, $(C_3\times C_9).S_3$, $\He_3:S_3$, $C_3^2:D_9$, $\He_3.S_3$, $\He_3.C_6$, $\He_3.S_3$, $\He_3.C_6$

Order 144: $F_9:C_2$ x 9, $C_3^2:\SD_{16}$ x 4, $C_2\times \PSU(3,2)$ x 2, $S_3^2:C_2^2$ x 2, $C_2\times F_9$ x 2, $S_3\times S_4$, $D_6:D_6$, $C_{12}:D_6$, $C_{12}.D_6$, $C_3^2:(C_2\times C_8)$

Order 120: $S_5$ x 3, $C_2\times A_5$ x 2, $S_3\times F_5$

Order 110: $C_{11}:C_{10}$

Order 108: $C_3\times S_3^2$ x 15, $C_3:S_3^2$ x 14, $C_3^2:D_6$ x 12, $C_3:S_3^2$ x 6, $C_3^3:C_4$ x 6, $C_3^2:C_{12}$ x 6, $C_3^2:D_6$ x 4, $\He_3:C_4$ x 3, $C_3^2:D_6$ x 2, $S_3\times D_9$ x 2, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:C_{12}$, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_3^2:A_4$

Order 96: $C_2\times \GL(2,3)$, $S_3\times \SD_{16}$

Order 81: $C_3\wr C_3$ x 6, $C_3\times \He_3$ x 4, $C_9:C_3^2$ x 3, $C_3.\He_3$ x 2, $C_3^4$ x 2, $C_9.C_3^2$ x 2, $\He_3:C_3$, $C_3^2:C_9$, $C_3.\He_3$

Order 72: $S_3\times D_6$ x 7, $\SOPlus(4,2)$ x 7, $\PSU(3,2)$ x 6, $F_9$ x 5, $C_2\times C_3^2:C_4$ x 3, $C_3\times S_4$ x 2, $C_6\wr C_2$ x 2, $C_3^2:Q_8$ x 2, $C_3:D_{12}$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$ x 2, $C_6:D_6$, $S_3\times A_4$, $C_3:S_4$, $C_{12}:S_3$, $C_3\times D_{12}$, $C_3^2:Q_8$

Order 60: $A_5$ x 3, $S_3\times D_5$, $C_{15}:C_4$, $C_3\times F_5$

Order 55: $C_{11}:C_5$

Order 54: $C_3^2:C_6$ x 16, $S_3\times C_3^2$ x 16, $C_3^2:C_6$ x 15, $C_3^2:S_3$ x 9, $C_9:C_6$ x 7, $C_3^2:S_3$ x 7, $S_3\times C_9$ x 3, $C_3\times D_9$ x 3, $C_3:D_9$ x 2, $C_2\times \He_3$ x 2, $C_3^2\times C_6$, $C_9:C_6$

Order 48: $S_3\times D_4$ x 2, $\GL(2,3)$ x 2, $C_{24}:C_2$, $C_2\times S_4$, $S_3\times Q_8$, $S_3\times C_8$, $C_2\times \SL(2,3)$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 40: $C_2\times F_5$

Order 36: $S_3^2$ x 24, $C_6\times S_3$ x 13, $C_3^2:C_4$ x 8, $C_6:S_3$ x 8, $C_3:C_{12}$ x 4, $C_3\times A_4$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $D_{18}$, $C_6^2$

Order 32: $C_2\times \SD_{16}$

Order 30: $D_{15}$, $C_3\times D_5$, $C_5\times S_3$

Order 27: $C_3^3$ x 8, $\He_3$ x 8, $C_9:C_3$ x 7, $C_3\times C_9$ x 4

Order 24: $C_2\times D_6$ x 4, $C_3:D_4$ x 3, $C_4\times S_3$ x 3, $S_4$ x 3, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_3\times D_4$ x 2, $\SL(2,3)$, $C_{24}$, $C_2\times A_4$, $C_3\times Q_8$, $C_3:C_8$

Order 22: $C_{22}$

Order 20: $F_5$ x 2, $D_{10}$

Order 18: $C_3\times S_3$ x 25, $C_3:S_3$ x 16, $C_3\times C_6$ x 8, $D_9$ x 3, $C_{18}$

Order 16: $\SD_{16}$ x 4, $C_2\times C_8$, $C_2\times Q_8$, $C_2\times D_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 16, $C_2\times C_6$ x 4, $C_{12}$ x 3, $C_3:C_4$ x 3, $A_4$ x 2

Order 11: $C_{11}$

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 10, $C_9$ x 2

Order 8: $D_4$ x 3, $Q_8$ x 2, $C_8$ x 2, $C_2^3$, $C_2\times C_4$

Order 6: $S_3$ x 12, $C_6$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3849120: $C_3^5:(C_2\times M_{11})$

Order 1924560: $C_3^5:M_{11}$

Order 349920: $C_3^4:A_6.D_6$

Order 320760: $C_3^5:(C_2\times \PSL(2,11))$

Order 174960: $C_3^5:(C_2\times A_6)$, $C_3^4:A_6.S_3$, $C_3^4:A_6.C_6$

Order 160380: $C_3^5:\PSL(2,11)$

Order 116640: $C_3^4:A_6.C_2^2$

Order 87480: $C_3^5:A_6$

Order 69984: $C_3^5:F_9:C_2^2$

Order 58320: $C_3^4:A_6.C_2$ x 2, $C_3^4:(C_2\times A_6)$, $C_3^5:(C_2\times S_5)$

Order 34992: $(C_9^2\times C_{54}).D_4$, $(C_9^2\times C_{54}).C_2^3$, $(C_9^2\times C_{54}).D_4$, $C_3^5:F_9:C_2$, $(C_9^2\times C_{54}).C_2^3$, $(C_9^2\times C_{54}).D_4$, $C_3^5:(C_2\times F_9)$

Order 29160: $C_3^5:(C_2\times A_5)$, $C_3^4:A_6$, $C_3^5:S_5$, $C_3^5:(C_2\times A_5)$, $C_3^5:S_5$

Order 26730: $C_3^5:C_{11}:C_{10}$

Order 23328: $C_3^5:(C_2\times \GL(2,3))$

Order 17496: $C_3^5:\SOPlus(4,2)$, $(C_9^2\times C_{54}).C_2^2$, $C_3^5:\SOPlus(4,2)$, $C_3^5:F_9$, $C_3^5:F_9$, $(C_9^2\times C_{54}).C_2^2$, $C_3^5:\SOPlus(4,2)$, $C_3^5:\PSU(3,2)$, $C_3^5:\PSU(3,2)$, $C_3^5:\PSU(3,2)$, $C_3^4.S_3^3$

Order 15840: $C_2\times M_{11}$

Order 14580: $C_3^5:A_5$, $C_3^5:A_5$

Order 13365: $C_3^5:C_{11}:C_5$

Order 11664: $C_3^4:(C_2\times \PSU(3,2))$, $C_3^3:S_3^2:D_6$, $C_3^5:(C_2\times \SL(2,3))$, $C_3^5:\GL(2,3)$, $C_3^5:\GL(2,3)$, $C_3^5:(C_2\times S_4)$

Order 9720: $C_3^4:(S_3\times F_5)$, $C_3^4:(C_2\times A_5)$

Order 8748: $C_3^5:S_3^2$, $C_3^5:C_6:S_3$, $C_3^5:C_3^2:C_4$, $C_3^5:S_3^2$, $C_3^5:(C_6\times S_3)$, $(C_9^2\times C_{54}).C_2$, $(C_9^2\times C_{54}).C_2$, $C_3^5:S_3^2$, $C_3^3.C_3\wr C_4$

Order 7920: $M_{11}$ x 2

Order 7776: $C_3^3:F_9:C_2^2$, $C_3^4:(S_3\times \SD_{16})$

Order 5832: $C_3^4:(C_3\times C_6):C_4$ x 2, $C_3^4:\PSU(3,2)$ x 2, $C_3^3.S_3^3$, $C_3^3.S_3^3$, $C_3^4:\SOPlus(4,2)$, $C_3^4:(C_3\times C_6):C_4$, $C_3^4:\SOPlus(4,2)$, $C_3^4:\SOPlus(4,2)$, $C_3^4:(C_3\times C_6):C_4$, $C_3^4:\SOPlus(4,2)$, $C_3^4:\PSU(3,2)$, $C_3^4:\PSU(3,2)$, $C_3^5:\SL(2,3)$, $C_3^5:S_4$, $C_3\wr A_4:S_3$, $C_3^5:S_4$, $C_3^3.S_3^3$

Order 5346: $C_3^5:C_{22}$

Order 4860: $C_3\wr F_5$, $C_3^5:D_{10}$, $C_3^5:F_5$, $C_3^4:A_5$

Order 4374: $C_3^5:(C_3\times S_3)$, $C_3^3.C_3^2\wr C_2$, $C_3^5:C_3:S_3$, $C_3^5:(C_3\times C_6)$, $C_3^5:(C_3\times S_3)$

Order 3888: $C_3^3:S_3^2.C_2^2$, $C_3^3:S_3^2:C_2^2$, $C_3^3:F_9:C_2$, $C_3^3:F_9:C_2$, $C_3^3:(C_2\times F_9)$, $C_3^3:F_9:C_2$, $C_3^3:F_9:C_2$, $C_3^2:(S_3\times F_9)$, $C_3^4:(S_3\times Q_8)$, $C_3\wr C_4:D_6$, $C_3^5:\SD_{16}$, $C_3^5:\SD_{16}$, $C_3^5:\SD_{16}$, $C_3^5:\SD_{16}$, $C_3^3:(S_3\times S_4)$

Order 3240: $C_3^4:(C_2\times F_5)$

Order 2916: $C_3^3:C_3^2:C_{12}$ x 2, $C_3^4:C_3^2:C_4$ x 2, $C_3^5:D_6$, $C_3^4:(C_6\times S_3)$, $(C_3\times \He_3).S_3^2$, $(C_3\times \He_3).S_3^2$, $C_3^4.S_3^2$, $C_3^4:S_3^2$, $C_3^4:C_3^2:C_4$, $C_3^4.S_3^2$, $C_3^3:C_3^2:C_{12}$, $C_3^4:(C_6\times S_3)$, $C_3^4:C_3^2:C_4$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^3:C_3^2:C_{12}$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:A_4$, $C_3^5:D_6$, $C_3^5:(C_2\times C_6)$, $C_3^5:D_6$

Order 2673: $C_3^5:C_{11}$

Order 2592: $C_3^4:(C_2\times \SD_{16})$, $C_3^3:(C_2\times \GL(2,3))$

Order 2430: $C_3\wr D_5$, $C_3^4:D_{15}$, $C_3^5:C_{10}$

Order 2187: $C_3^5:C_3^2$

Order 1944: $C_3^4:S_4$ x 2, $C_3^3:\PSU(3,2)$ x 2, $C_3^3:S_3^2.C_2$ x 2, $C_3^2:S_3^3$, $C_3^5:Q_8$, $C_3^5:Q_8$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3\wr C_4:S_3$, $C_3^3:F_9$, $C_3^4:C_{24}$, $C_3^3:(S_3\times A_4)$, $C_3^4:S_4$, $C_3^2.S_3^3$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$, $C_3^3:F_9$, $C_3^2.S_3^3$, $C_3^2.S_3^3$

Order 1620: $C_3^4:F_5$ x 2, $C_3^4:D_{10}$

Order 1458: $C_3^4:C_3:S_3$, $C_3^4:(C_3\times C_6)$, $C_3^4:C_3:S_3$, $C_3^4:C_3:S_3$, $C_3^4.(C_3\times S_3)$, $C_3^4:C_3:S_3$, $C_3^4.(C_3\times S_3)$, $C_3^4:C_3:S_3$, $C_3^4.(C_3\times S_3)$, $C_3^4.(C_3\times S_3)$, $C_3^4:(C_3\times S_3)$, $C_3^4:(C_3\times S_3)$, $C_3^4:(C_3\times C_6)$, $C_3^4:C_3:S_3$, $C_3^4:(C_3\times S_3)$, $C_3^4:C_3:S_3$, $C_3^4:(C_3\times S_3)$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1440: $A_6.C_2^2$

Order 1320: $C_2\times \PSL(2,11)$

Order 1296: $C_3^4:\SD_{16}$ x 4, $C_3^3:(S_3\times Q_8)$, $C_3^3:(S_3\times D_4)$, $S_3^2:S_3^2$, $C_3^2:(C_2\times F_9)$, $S_3\wr S_3$, $C_3^3:(C_2\times \SL(2,3))$, $C_3^3:\GL(2,3)$, $C_3^3:\GL(2,3)$

Order 1215: $C_3\wr C_5$

Order 972: $C_3\wr A_4$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:C_3^2:C_4$ x 2, $C_3^3:C_3^2:C_4$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^4:D_6$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^3:S_3^2$

Order 864: $F_9:D_6$ x 2, $C_2\times C_3^2:\GL(2,3)$, $C_3^2:(S_3\times \SD_{16})$

Order 810: $C_3^4:D_5$ x 2, $C_3^4:C_{10}$

Order 729: $C_3^3.C_3^3$, $C_3^4:C_3^2$, $C_3^3.C_3^3$, $C_3^3.C_3^3$, $C_3^4:C_3^2$, $C_3^5:C_3$

Order 720: $A_6.C_2$ x 3, $S_3\times S_5$, $C_2\times A_6$

Order 660: $\PSL(2,11)$

Order 648: $C_3.S_3^3$ x 3, $C_3:S_3^3$ x 2, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3\wr D_4$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^2:F_9$ x 2, $C_3^3:S_4$ x 2, $C_3:S_3^3$, $C_3^4:Q_8$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $S_3\wr C_3$, $C_3^3:S_4$, $C_3^3:\SL(2,3)$, $C_2\times C_3^3:D_6$

Order 486: $C_3^4:S_3$ x 4, $C_3^3:(C_3\times C_6)$ x 3, $C_9.(S_3\times C_3^2)$ x 2, $(C_3\times \He_3).S_3$ x 2, $C_3\wr C_3:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$, $(C_3\times \He_3):S_3$, $C_3^3.(C_3\times S_3)$, $C_3^3:(C_3\times S_3)$, $C_3^4:S_3$, $C_3^4:S_3$, $C_9:C_3^2.S_3$, $(C_3\times \He_3):S_3$, $C_3^4.S_3$, $C_3^4:C_6$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^3:(C_3\times C_6)$, $C_3\wr C_3:C_6$, $C_3^4:C_6$, $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4:S_3$

Order 432: $S_3^3:C_2$ x 3, $C_3^3:\SD_{16}$ x 3, $S_3\times \PSU(3,2)$ x 2, $F_9:S_3$ x 2, $C_3^3:\SD_{16}$ x 2, $F_9:C_6$ x 2, $S_3\times F_9$ x 2, $C_3^2:\GL(2,3)$ x 2, $C_2\times \PU(3,2)$, $C_2.S_3^3$, $D_6:S_3^2$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $C_3^2:(S_3\times C_8)$, $C_6^2:D_6$, $\He_3:\SD_{16}$

Order 405: $C_3^4:C_5$

Order 360: $C_3\times S_5$ x 2, $A_6$ x 2, $S_3\times A_5$, $C_3:S_5$

Order 324: $C_3^3:D_6$ x 7, $C_3^2:S_3^2$ x 6, $C_3^2:S_3^2$ x 6, $C_3\wr C_2^2$ x 5, $C_3\wr C_4$ x 4, $C_3^3:D_6$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^3:A_4$ x 2, $C_3^2.S_3^2$ x 2, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^2.S_3^2$, $C_3^2.S_3^2$, $C_3^2:D_{18}$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2\times S_3^2$, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_9.S_3^2$

Order 288: $F_9:C_2^2$ x 2, $C_4.\SOPlus(4,2)$

Order 243: $C_3^4:C_3$ x 4, $C_3^2.C_3^3$ x 2, $C_3^3:C_3^2$ x 2, $C_3^5$, $C_9.C_3^3$, $C_3^3:C_3^2$, $C_3^2.C_3^3$, $\He_3:C_3^2$, $C_3^4.C_3$, $C_3^4:C_3$

Order 240: $C_2\times S_5$

Order 216: $S_3^3$ x 6, $C_3^3:Q_8$ x 5, $C_3^2:C_4\times S_3$ x 5, $S_3^2:S_3$ x 4, $S_3^2:C_6$ x 4, $C_3^2:D_{12}$ x 3, $C_3:F_9$ x 3, $\He_3:C_8$ x 2, $C_3^3:Q_8$ x 2, $C_3\times F_9$ x 2, $C_6.S_3^2$ x 2, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $\SU(3,2)$, $\He_3:D_4$, $C_6:S_3^2$, $\PU(3,2)$, $C_3^3:Q_8$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_3^3:Q_8$, $C_3^2:D_{12}$, $C_3^3:D_4$, $C_3^3:C_8$, $C_3^2:C_{24}$

Order 180: $\GL(2,4)$ x 2

Order 162: $C_3^3:C_6$ x 8, $C_3^3:S_3$ x 7, $C_3^2\wr C_2$ x 6, $C_3^3:C_6$ x 6, $C_3^3:C_6$ x 5, $C_3\wr S_3$ x 5, $C_3^3:S_3$ x 4, $C_3^3:S_3$ x 3, $\He_3.S_3$ x 3, $D_9:C_3^2$ x 3, $S_3\times \He_3$ x 3, $C_3^3:S_3$ x 2, $C_3^3.S_3$ x 2, $C_3^3.C_6$ x 2, $\He_3.S_3$ x 2, $C_3^3:C_6$, $S_3\times C_3^3$, $C_3^2:D_9$, $\He_3.C_6$, $C_3^2:C_{18}$, $C_3^3:C_6$, $(C_3\times C_9).S_3$, $\He_3:S_3$, $C_3^2:D_9$, $\He_3.S_3$, $\He_3.C_6$, $\He_3.S_3$, $\He_3.C_6$

Order 144: $F_9:C_2$ x 9, $C_3^2:\SD_{16}$ x 4, $C_2\times \PSU(3,2)$ x 2, $S_3^2:C_2^2$ x 2, $C_2\times F_9$ x 2, $S_3\times S_4$, $D_6:D_6$, $C_{12}:D_6$, $C_{12}.D_6$, $C_3^2:(C_2\times C_8)$

Order 120: $S_5$ x 3, $C_2\times A_5$ x 2, $S_3\times F_5$

Order 110: $C_{11}:C_{10}$

Order 108: $C_3\times S_3^2$ x 15, $C_3:S_3^2$ x 14, $C_3^2:D_6$ x 12, $C_3:S_3^2$ x 6, $C_3^3:C_4$ x 6, $C_3^2:C_{12}$ x 6, $C_3^2:D_6$ x 4, $\He_3:C_4$ x 3, $C_3^2:D_6$ x 2, $S_3\times D_9$ x 2, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:C_{12}$, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_3^2:A_4$

Order 96: $C_2\times \GL(2,3)$, $S_3\times \SD_{16}$

Order 81: $C_3\wr C_3$ x 6, $C_3\times \He_3$ x 4, $C_9:C_3^2$ x 3, $C_3.\He_3$ x 2, $C_3^4$ x 2, $C_9.C_3^2$ x 2, $\He_3:C_3$, $C_3^2:C_9$, $C_3.\He_3$

Order 72: $S_3\times D_6$ x 7, $\SOPlus(4,2)$ x 7, $\PSU(3,2)$ x 6, $F_9$ x 5, $C_2\times C_3^2:C_4$ x 3, $C_3\times S_4$ x 2, $C_6\wr C_2$ x 2, $C_3^2:Q_8$ x 2, $C_3:D_{12}$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$ x 2, $C_6:D_6$, $S_3\times A_4$, $C_3:S_4$, $C_{12}:S_3$, $C_3\times D_{12}$, $C_3^2:Q_8$

Order 60: $A_5$ x 3, $S_3\times D_5$, $C_{15}:C_4$, $C_3\times F_5$

Order 55: $C_{11}:C_5$

Order 54: $C_3^2:C_6$ x 16, $S_3\times C_3^2$ x 16, $C_3^2:C_6$ x 15, $C_3^2:S_3$ x 9, $C_9:C_6$ x 7, $C_3^2:S_3$ x 7, $S_3\times C_9$ x 3, $C_3\times D_9$ x 3, $C_3:D_9$ x 2, $C_2\times \He_3$ x 2, $C_3^2\times C_6$, $C_9:C_6$

Order 48: $S_3\times D_4$ x 2, $\GL(2,3)$ x 2, $C_{24}:C_2$, $C_2\times S_4$, $S_3\times Q_8$, $S_3\times C_8$, $C_2\times \SL(2,3)$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 40: $C_2\times F_5$

Order 36: $S_3^2$ x 24, $C_6\times S_3$ x 13, $C_3^2:C_4$ x 8, $C_6:S_3$ x 8, $C_3:C_{12}$ x 4, $C_3\times A_4$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $D_{18}$, $C_6^2$

Order 32: $C_2\times \SD_{16}$

Order 30: $D_{15}$, $C_3\times D_5$, $C_5\times S_3$

Order 27: $C_3^3$ x 8, $\He_3$ x 8, $C_9:C_3$ x 7, $C_3\times C_9$ x 4

Order 24: $C_2\times D_6$ x 4, $C_3:D_4$ x 3, $C_4\times S_3$ x 3, $S_4$ x 3, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_3\times D_4$ x 2, $\SL(2,3)$, $C_{24}$, $C_2\times A_4$, $C_3\times Q_8$, $C_3:C_8$

Order 22: $C_{22}$

Order 20: $F_5$ x 2, $D_{10}$

Order 18: $C_3\times S_3$ x 25, $C_3:S_3$ x 16, $C_3\times C_6$ x 8, $D_9$ x 3, $C_{18}$

Order 16: $\SD_{16}$ x 4, $C_2\times C_8$, $C_2\times Q_8$, $C_2\times D_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 16, $C_2\times C_6$ x 4, $C_{12}$ x 3, $C_3:C_4$ x 3, $A_4$ x 2

Order 11: $C_{11}$

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 10, $C_9$ x 2

Order 8: $D_4$ x 3, $Q_8$ x 2, $C_8$ x 2, $C_2^3$, $C_2\times C_4$

Order 6: $S_3$ x 12, $C_6$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3849120: $C_3^5:(C_2\times M_{11})$ ($C_1$)

Order 1924560: $C_3^5:M_{11}$ ($C_2$)

Order 486: $C_3^4:S_3$ ($M_{11}$)

Order 243: $C_3^5$ ($C_2\times M_{11}$)

Order 1: $C_1$ ($C_3^5:(C_2\times M_{11})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3849120: $C_3^5:(C_2\times M_{11})$ ($C_1$)

Order 1924560: $C_3^5:M_{11}$ ($C_2$)

Order 486: $C_3^4:S_3$ ($M_{11}$)

Order 243: $C_3^5$ ($C_2\times M_{11}$)

Order 1: $C_1$ ($C_3^5:(C_2\times M_{11})$)

Series

Derived series

$C_3^5:(C_2\times M_{11})$

$\rhd$

$C_3^5:(C_2\times M_{11})$

$\rhd$

$C_3^5:M_{11}$

$\rhd$

$C_3^5:M_{11}$

Chief series

$C_3^5:(C_2\times M_{11})$

$\rhd$

$C_3^5:(C_2\times M_{11})$

$\rhd$

$C_3^4:S_3$

$\rhd$

$C_3^4:S_3$

$\rhd$

$C_3^5$

$\rhd$

$C_3^5$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^5:(C_2\times M_{11})$

$\rhd$

$C_3^5:(C_2\times M_{11})$

$\rhd$

$C_3^5:M_{11}$

$\rhd$

$C_3^5:M_{11}$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $37 \times 37$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.