Abstract group 162000.be: $C_3^3:D_5\wr S_3$

• Label: 162000.be
• Order: \( 2^{4} \cdot 3^{4} \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{5} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_3^3:D_5\wr S_3$
Order:	\(162000\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5^{3} \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_5^3.C_6^2.(C_{12}\times S_3^2)$, of order \(1944000\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 5^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4, $C_5$ x 3
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	5	6	9	10	12	15	20	30	45	60
Elements	1	4575	1826	7200	124	33090	3600	18300	17100	10424	1800	35160	14400	14400	162000
Conjugacy classes	1	5	6	2	9	15	1	17	5	91	2	40	4	16	214
Divisions	1	5	6	2	5	15	1	9	5	42	1	17	1	4	114
Autjugacy classes	1	5	4	2	5	9	1	9	3	24	1	11	1	2	78

Dimension	1	2	3	4	6	8	12	16	24	32	48	64	96	192
Irr. complex chars.	4	4	4	1	26	4	54	8	58	0	51	0	0	0	214
Irr. rational chars.	4	4	4	1	18	0	8	2	17	2	30	1	19	4	114

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$1674$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{6}=c^{6}=d^{30}=e^{5}=f^{15}=[a,c]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,3,7,2,5,11)(4,9,14,12,15,10)(6,8,13)(16,17,19,22,24,23)(18,21,20), (1,2,4,8,12)(3,6)(5,10)(7,9)(11,13)(14,15)(17,18,20,23,24,19)(21,22)\rangle$
Transitive group:	45T1161				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $(D_5\wr S_3)$	$C_5^3$ $\,\rtimes\,$ $(C_6^2.S_3^2)$	$(C_3^3:D_5\wr C_3)$ $\,\rtimes\,$ $C_2$	$((C_5\times C_{15}^2):S_4)$ $\,\rtimes\,$ $S_3$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3$ . $(C_3^2:D_5\wr S_3)$	$(C_5\times C_{15}^2)$ . $(S_3\times S_4)$	$C_3^2$ . $(C_5^3:(S_3\times S_4))$	$(C_{15}:D_5^2)$ . $(C_3^2:D_6)$	all 6

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 1145664 subgroups in 2672 conjugacy classes, 21 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^3:D_5\wr S_3$
Commutator:	$G' \simeq$ $C_3^3:(C_5^3:A_4)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_5^3:(S_3\times S_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_{15}^3$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $C_3^3:D_5\wr S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^2\times C_{15}$	$G/\operatorname{soc} \simeq$ $C_6^2:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$

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 162000: $C_3^3:D_5\wr S_3$

Order 81000: $C_3^3:(C_5^3:S_4)$, $C_3^3:D_5\wr C_3$, $C_3^3:(C_5^3:S_4)$

Order 54000: $C_{15}^3.C_2^3.C_2$, $C_{15}:D_5^2.S_3^2$

Order 40500: $C_{15}^3.D_6$, $C_3^3:(C_5^3:A_4)$

Order 27000: $C_{15}^3.C_2^3$, $C_3^2\times C_5^3:(C_3:D_4)$, $C_{15}^3.D_4$, $C_{15}^2.C_{30}.C_2^2$, $C_5^3.(C_3\times C_6^2:C_2)$, $C_{15}^3.C_2^3$, $C_{15}^3.C_4.C_2$, $(C_5\times C_{15}^2).S_4$, $(C_5\times C_{15}^2):S_4$, $C_5^3:(C_6^2:C_6)$, $(C_5\times C_{15}^2):S_4$

Order 20250: $C_{15}\wr C_3:C_2$, $(C_3\times C_{15}^2):D_{15}$, $C_{15}^3.C_6$

Order 18000: $C_5^3.D_6:D_6$ x 4, $C_5^3.(D_4\times C_3:S_3)$, $C_5^3:(S_3\times S_4)$

Order 13500: $C_{15}^3.C_2^2$, $C_3^3\times C_5^3:C_2^2$, $(C_5\times C_{15}^2).C_{12}$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_3^2\times C_5^3:(C_3:C_4)$, $C_3^2\times C_5^3:D_6$, $(C_5\times C_{15}^2):D_6$, $(C_5\times C_{15}^2).A_4$, $(C_5\times C_{15}^2):A_4$

Order 10800: $C_{15}^2.D_6.C_2^2$

Order 10125: $C_{15}\wr C_3$

Order 9000: $C_5^3.C_3:S_3.C_2^2$ x 6, $C_5^3.S_3^2.C_2$ x 4, $C_3\times C_5^3:(C_3:D_4)$ x 4, $C_{15}^2.(C_5\times D_4)$ x 4, $C_5^3.C_3^2:D_4$ x 4, $C_5^3.(C_2.S_3^2)$ x 3, $C_3\times C_5^3:(C_3:D_4)$ x 3, $C_5^3.C_6^2:C_2$ x 2, $C_5^2.(C_6.D_{30})$, $C_5^3.C_6.C_6.C_2$, $C_{15}^2.(C_5\times D_4)$, $C_3^2\times C_5^3:D_4$, $C_5^3.C_3^2:C_4.C_2$, $(C_5^2\times C_{15}).S_4$, $(C_5^2\times C_{15}):S_4$, $C_5^3:(S_3\times A_4)$, $C_3\times C_5^3:S_4$

Order 8100: $C_{15}^2.S_3^2$

Order 6750: $C_3^2\times C_5^3:S_3$, $C_5^3:(C_3^3:C_2)$, $C_5\times C_5^2:(C_3^3:C_2)$, $C_3^3\times C_5\times C_5:D_5$, $C_5^3:(C_3\times C_3:S_3)$, $C_5\times C_{15}^2.S_3$, $C_5\times C_{15}^2:S_3$, $C_{15}^2:D_{15}$, $(C_5\times C_{15}^2):C_6$

Order 6000: $C_5^2.C_{20}:D_6$ x 4, $C_5^2.D_{15}:D_4$, $D_5\wr S_3$

Order 5400: $C_{15}^2.D_6.C_2$, $C_{15}^2.C_6.C_2^2$, $C_3^2\times C_5^2:(C_3:D_4)$, $C_{15}^2.D_6.C_2$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_3:D_4$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.D_{12}$, $C_{15}^2.D_6.C_2$

Order 4500: $C_5^3:(C_2\times C_3:S_3)$ x 10, $C_5^3:(C_2\times C_3:S_3)$ x 7, $C_3^2\times C_5^3:C_2^2$ x 6, $C_3\times C_5^3:(C_3:C_4)$ x 4, $C_5\times C_5^2:(C_2\times C_3:S_3)$ x 4, $C_5^3:S_3^2$ x 4, $C_5^3:S_3^2$ x 4, $C_3\times C_5^3:D_6$ x 4, $C_3\times C_5^3:(C_3:C_4)$ x 3, $C_3\times C_5^3:D_6$ x 3, $C_5^3:(C_3^2:C_4)$, $C_3^2\times C_5\times C_5:F_5$, $C_5^3:(C_3\times C_3:C_4)$, $C_5^3:(C_6\times S_3)$, $C_5^3:S_3^2$, $(C_5^2\times C_{15}).A_4$, $C_3\times C_5^3:A_4$

Order 4050: $(C_3\times C_{15}^2):S_3$, $(C_3\times C_{15}^2):S_3$, $(C_3\times C_{15}^2):C_6$

Order 3600: $D_5^2:S_3^2$ x 4, $C_{15}^2.C_2^3.C_2$

Order 3375: $C_{15}^3$, $C_{15}^2.C_{15}$, $C_{15}^2:C_{15}$

Order 3000: $C_5^3:(C_2^2\times S_3)$ x 9, $C_5^3:(C_3:D_4)$ x 8, $C_5^3:(C_4\times S_3)$ x 4, $C_5^3:D_{12}$ x 4, $C_3\times C_5^3:D_4$ x 4, $C_3\times C_5^3:D_4$, $C_5^3:(C_4\times S_3)$, $C_5^3:D_{12}$, $C_5^3:(C_3:D_4)$, $C_5^3:(C_2^2\times S_3)$, $C_5^3:(C_3:D_4)$, $C_5^3:S_4$, $D_5\wr C_3$, $C_5^3:S_4$

Order 2700: $C_5^2:(S_3\times C_3:S_3)$ x 4, $C_5^2:(C_2\times C_3^3:C_2)$ x 3, $C_5^2:(C_3\times C_3^2:C_4)$, $C_3^3\times D_5^2$, $C_5^2:(C_2\times C_3^3:C_2)$, $C_5^2:(S_3\times C_3:S_3)$, $C_5^2:(S_3\times C_3:S_3)$, $C_5^2:(C_3\times C_3^2:C_4)$, $C_5^2:(S_3\times C_3:S_3)$, $C_3^2\times C_3:(C_5:F_5)$, $C_3^2\times D_{15}:D_5$, $C_5\times C_3\times C_3:S_3\times D_5$, $C_5^2:(C_6\times C_3:S_3)$, $C_5^2:(S_3\times C_3:S_3)$, $C_5^2:(C_2\times C_3^3:C_2)$, $C_5\times C_5:(C_2\times C_3^3:C_2)$, $C_3\times C_{15}\times C_3:F_5$, $C_3^2\times D_5\times D_{15}$, $C_{15}^2:D_6$

Order 2250: $C_3^2\times C_5\times C_5:D_5$ x 6, $C_5\times C_5\times C_3:D_{15}$ x 5, $C_5^3:(C_3:S_3)$ x 5, $C_3\times C_{15}\times C_5:D_5$ x 4, $C_5^2\times C_3:D_{15}$ x 4, $C_3\times C_5^3:S_3$ x 4, $C_3\times C_5^3:S_3$ x 3, $C_5^3:(C_3:S_3)$, $C_5^3:(C_3\times S_3)$, $C_5^3:D_9$, $(C_5^2\times C_{15}):S_3$, $C_3\times C_5^2:D_{15}$, $C_5^2:(C_3\times D_{15})$

Order 2160: $C_{15}.(D_4\times C_3:S_3)$

Order 2025: $C_{15}^2.C_3^2$

Order 2000: $D_5^3:C_2$

Order 1800: $C_{15}^2:C_2^3$ x 11, $C_{15}^2:D_4$ x 8, $C_{15}^2:D_4$ x 8, $D_{15}^2:C_2$ x 4, $D_{15}:D_{30}$ x 4, $C_{15}:(C_4\times D_{15})$ x 4, $C_{15}:(S_3\times F_5)$ x 4, $C_{15}^2:D_4$ x 2, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $F_5\times C_3:D_{15}$, $C_{15}:(S_3\times F_5)$

Order 1620: $(C_3^2\times C_{15}):D_6$

Order 1500: $C_{15}:D_5^2$ x 17, $C_5^2:D_{30}$ x 13, $C_{15}:D_5^2$ x 9, $(C_5\times C_{15}):F_5$ x 4, $C_5^2:C_{60}$ x 4, $C_5^2:D_{30}$, $C_5^2:D_{30}$, $C_5^2:D_{30}$, $C_{15}\times D_5^2$, $C_5\times C_{15}:F_5$, $C_5^3:C_{12}$, $C_5^3:D_6$, $C_5^3:A_4$

Order 1350: $C_{15}^2:S_3$ x 5, $C_{15}^2:C_6$ x 4, $C_{15}^2:C_6$ x 4, $C_{15}^2:S_3$ x 3, $C_{15}^2:C_6$ x 3, $C_{15}^2:C_6$, $C_{15}^2:S_3$, $C_{15}^2:S_3$, $C_{15}^2.S_3$, $C_{15}^2:S_3$, $C_{15}^2:C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$

Order 1296: $C_6^2.S_3^2$

Order 1200: $D_5^2:D_6$ x 5

Order 1125: $C_5\times C_{15}^2$ x 5, $C_5^3:C_3^2$, $C_5^2:C_{45}$

Order 1080: $C_{30}:S_3^2$, $C_3^2:S_3\times D_{10}$, $S_3\times C_{15}:D_6$, $C_3^2:(S_3\times F_5)$, $C_{30}.S_3^2$, $C_{30}.S_3^2$, $C_{30}.S_3^2$, $C_3\times C_{30}.D_6$, $C_3^3:D_{20}$

Order 1000: $D_5^3$ x 2, $C_5^3:D_4$ x 2, $C_5^2:D_{20}$, $D_5^2:C_{10}$, $D_5\times C_5:F_5$

Order 900: $C_{15}:D_{30}$ x 35, $D_{15}^2$ x 16, $C_{15}^2:C_2^2$ x 16, $C_{15}^2:C_2^2$ x 11, $C_3^2\times D_5^2$ x 11, $C_{15}^2:C_2^2$ x 10, $C_{15}^2:C_2^2$ x 8, $C_{15}^2:C_4$ x 8, $C_{15}:D_{30}$ x 4, $C_{15}:D_{30}$ x 4, $C_5^2:S_3^2$ x 4, $C_{15}:D_{30}$ x 4, $C_{15}:D_{30}$ x 4, $D_{15}:C_{30}$ x 4, $C_{15}^2:C_4$ x 4, $C_{15}:C_{60}$ x 4, $C_5^2:S_3^2$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}:C_{60}$

Order 810: $C_5\times C_3^3:S_3$, $\He_3:D_{15}$, $C_3^3:D_{15}$

Order 750: $C_5^2\times D_{15}$ x 13, $C_5^2:C_{30}$ x 13, $C_5^2:D_{15}$ x 5, $C_5^2:D_{15}$, $C_5^2:C_{30}$, $C_5^3:C_6$, $C_5^2:D_{15}$, $C_5\wr S_3$

Order 720: $D_{30}:D_6$ x 4, $C_{60}:D_6$

Order 675: $C_3\times C_{15}^2$ x 5, $C_{15}^2:C_3$, $C_{15}^2.C_3$

Order 648: $(C_3\times C_6^2):C_6$, $C_3^3:S_4$, $C_3^3:S_4$

Order 600: $S_3\times D_5^2$ x 18, $D_5^2:S_3$ x 10, $C_5^2:D_{12}$ x 5, $D_5^2:C_6$ x 5, $S_3\times C_5:F_5$ x 5, $F_5\times D_{15}$ x 4, $D_5\times D_{30}$, $D_{15}:F_5$

Order 540: $C_{15}:S_3^2$ x 5, $C_3^3:D_{10}$ x 4, $C_{15}:C_6^2$, $C_3^3:F_5$, $C_3^3:F_5$, $(C_3\times C_{15}):C_{12}$, $C_3^2:C_{60}$, $C_3^2:D_{30}$, $C_3^2:D_{30}$, $C_3^2:S_3\times C_{10}$, $C_3:S_3\times C_{30}$, $C_3^2\times D_{30}$, $D_{10}\times C_3^3$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_3^3:D_{10}$

Order 500: $C_5\times D_5^2$ x 3, $C_5:D_5^2$ x 2, $C_5:D_5^2$ x 2, $C_5^3:C_4$, $C_5^2:C_{20}$

Order 450: $C_{15}:D_{15}$ x 41, $C_{15}:D_{15}$ x 35, $C_{15}:C_{30}$ x 35, $C_{15}\times D_{15}$ x 32, $C_{15}^2:C_2$ x 10, $C_{15}^2:C_2$ x 10, $C_{15}:D_{15}$ x 4, $C_{15}:C_{30}$ x 4, $(C_5\times C_{15}):C_6$, $(C_5\times C_{15}):S_3$, $C_3\times C_5^2:S_3$, $C_5^2:D_9$

Order 432: $C_6^2:D_6$, $C_6^2:D_6$

Order 405: $C_3^3:C_{15}$

Order 400: $D_5^2:C_2^2$

Order 375: $C_5^2\times C_{15}$ x 5, $C_5\wr C_3$

Order 360: $C_{30}:D_6$ x 11, $C_{15}:D_{12}$ x 4, $C_6.D_{30}$ x 4, $C_{15}:D_{12}$ x 4, $D_{30}:C_6$ x 4, $C_{30}.D_6$ x 4, $S_3\times D_{30}$ x 4, $D_5\times S_3^2$ x 4, $D_5.S_3^2$ x 4, $C_{30}.D_6$ x 2, $C_3^2\times D_{20}$, $C_3:S_3\times F_5$, $C_{12}:D_{15}$, $C_{15}:D_{12}$

Order 324: $C_3^3:A_4$, $C_3^3:D_6$

Order 300: $D_5\times D_{15}$ x 53, $C_{15}:D_{10}$ x 26, $C_{15}:D_{10}$ x 18, $C_3\times D_5^2$ x 18, $C_{15}:D_{10}$ x 13, $C_{15}:F_5$ x 5, $C_{15}:F_5$ x 5, $C_{15}:F_5$ x 4, $C_{15}\times F_5$ x 4, $C_5:D_{30}$, $C_5\times D_{30}$, $C_{15}\times D_{10}$, $C_{15}:C_{20}$, $C_{15}:F_5$, $C_5^2:D_6$

Order 270: $C_3^2:D_{15}$ x 6, $C_3^2:C_{30}$ x 4, $C_3^2\times D_{15}$ x 4, $C_3^3:C_{10}$ x 3, $D_5\times C_3^3$ x 3, $C_3^2\times C_{30}$, $C_3^2:D_{15}$, $C_3^2:C_{30}$, $C_3^2:D_{15}$, $C_3^2:D_{15}$, $C_9:C_{30}$, $C_3^2:C_{30}$

Order 250: $C_5^2:C_{10}$ x 2, $D_5\times C_5^2$ x 2, $C_5^3:C_2$

Order 240: $C_{12}:D_{10}$ x 4, $D_6:D_{10}$

Order 225: $C_{15}^2$ x 41, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $C_3^2.S_4$, $C_6^2:S_3$, $C_6:S_3^2$, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_6.S_3^2$

Order 200: $D_5\wr C_2$ x 4, $D_5\times D_{10}$ x 3, $C_{10}:F_5$, $D_5\times F_5$

Order 180: $C_{15}:D_6$ x 51, $S_3\times D_{15}$ x 24, $C_{30}:S_3$ x 11, $C_3^2\times D_{10}$ x 11, $C_3:D_{30}$ x 10, $C_{15}:D_6$ x 8, $C_{15}:C_{12}$ x 8, $C_{15}:D_6$ x 5, $C_3\times D_{30}$ x 4, $S_3\times C_{30}$ x 4, $C_5\times S_3^2$ x 4, $C_{15}:C_{12}$ x 4, $C_3:C_{60}$ x 4, $C_3^2:F_5$, $C_3^2\times F_5$, $C_3\times C_{60}$, $C_6.D_{15}$

Order 162: $C_3^3:S_3$, $C_3^3:C_6$, $C_3\wr S_3$

Order 150: $D_5\times C_{15}$ x 51, $C_5\times D_{15}$ x 51, $C_5:D_{15}$ x 42, $C_{15}:D_5$ x 13, $S_3\times C_5^2$ x 13, $C_5^2:S_3$ x 2, $C_5^2:C_6$, $C_5\times C_{30}$

Order 144: $D_6:D_6$ x 4, $S_3\times S_4$, $C_{12}:D_6$

Order 135: $C_3^2\times C_{15}$ x 5, $C_9:C_{15}$, $C_5\times \He_3$

Order 125: $C_5^3$

Order 120: $S_3\times D_{10}$ x 18, $C_{15}:D_4$ x 8, $S_3\times F_5$ x 5, $C_4\times D_{15}$ x 4, $C_5\times D_{12}$ x 4, $C_3\times D_{20}$ x 4, $C_2\times D_{30}$, $C_{15}:D_4$, $C_{15}:D_4$, $C_5:D_{12}$, $C_3:D_{20}$, $D_{15}:C_4$

Order 108: $C_3^2:D_6$ x 2, $C_3:S_3^2$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:C_{12}$, $C_3^2:C_{12}$, $C_3^2:A_4$, $C_3^2.A_4$, $C_3^2:D_6$

Order 100: $D_5^2$ x 13, $C_5\times D_{10}$ x 4, $C_5:D_{10}$ x 2, $C_5:F_5$ x 2, $C_5\times F_5$, $C_5:F_5$

Order 90: $C_3:D_{15}$ x 51, $C_{15}:S_3$ x 36, $C_3^2\times D_5$ x 35, $C_3\times D_{15}$ x 22, $S_3\times C_{15}$ x 20, $C_3\times C_{30}$ x 10, $C_5\times D_9$

Order 81: $C_3\wr C_3$

Order 80: $D_4\times D_5$

Order 75: $C_5\times C_{15}$ x 41, $C_5^2:C_3$

Order 72: $C_6\wr C_2$ x 8, $C_3:D_{12}$ x 8, $C_6:D_6$ x 6, $S_3\times D_6$ x 4, $C_6.D_6$ x 4, $C_6^2:C_2$ x 2, $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$, $D_4\times C_3^2$, $C_3:D_{12}$, $C_{12}:S_3$, $C_2^2:D_9$

Order 60: $S_3\times D_5$ x 76, $D_{30}$ x 19, $S_3\times C_{10}$ x 18, $C_3\times D_{10}$ x 18, $C_{15}:C_4$ x 5, $C_3\times F_5$ x 5, $C_{60}$ x 4, $C_{15}:C_4$ x 4, $C_5:C_{12}$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 54: $C_3^2:C_6$ x 2, $C_3^2:S_3$ x 2, $C_3^2:S_3$, $C_9:C_6$, $C_3^2\times C_6$, $C_3^2:C_6$, $S_3\times C_3^2$

Order 50: $C_5\times D_5$ x 14, $C_5:D_5$ x 7, $C_5\times C_{10}$ x 2

Order 48: $S_3\times D_4$ x 5, $C_2\times S_4$

Order 45: $C_3\times C_{15}$ x 42, $C_{45}$

Order 40: $C_2\times D_{10}$ x 3, $C_5:D_4$ x 2, $D_{20}$, $C_4\times D_5$, $C_2\times F_5$, $C_5\times D_4$

Order 36: $C_6:S_3$ x 21, $S_3^2$ x 9, $C_3:C_{12}$ x 8, $C_6\times S_3$ x 8, $C_6^2$ x 6, $C_3\times C_{12}$, $C_3^2:C_4$, $C_2^2:C_9$, $C_3\times A_4$

Order 30: $D_{15}$ x 59, $C_3\times D_5$ x 49, $C_5\times S_3$ x 49, $C_{30}$ x 17

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 25: $C_5^2$ x 5

Order 24: $C_3:D_4$ x 10, $C_2\times D_6$ x 10, $D_{12}$ x 5, $C_4\times S_3$ x 5, $C_3\times D_4$ x 5, $S_4$ x 2, $C_2\times A_4$

Order 20: $D_{10}$ x 16, $C_2\times C_{10}$ x 3, $F_5$ x 2, $C_{20}$, $C_5:C_4$

Order 18: $C_3:S_3$ x 16, $C_3\times C_6$ x 10, $C_3\times S_3$ x 10, $D_9$

Order 16: $C_2\times D_4$

Order 15: $C_{15}$ x 42

Order 12: $D_6$ x 34, $C_2\times C_6$ x 10, $C_{12}$ x 5, $C_3:C_4$ x 5, $A_4$

Order 10: $D_5$ x 14, $C_{10}$ x 9

Order 9: $C_3^2$ x 6, $C_9$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $S_3$ x 21, $C_6$ x 15

Order 5: $C_5$ x 5

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 6

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 162000: $C_3^3:D_5\wr S_3$

Order 81000: $C_3^3:(C_5^3:S_4)$, $C_3^3:D_5\wr C_3$, $C_3^3:(C_5^3:S_4)$

Order 54000: $C_{15}^3.C_2^3.C_2$, $C_{15}:D_5^2.S_3^2$

Order 40500: $C_{15}^3.D_6$, $C_3^3:(C_5^3:A_4)$

Order 27000: $C_{15}^3.C_2^3$, $C_3^2\times C_5^3:(C_3:D_4)$, $C_{15}^3.D_4$, $C_{15}^2.C_{30}.C_2^2$, $C_5^3.(C_3\times C_6^2:C_2)$, $C_{15}^3.C_2^3$, $C_{15}^3.C_4.C_2$, $(C_5\times C_{15}^2).S_4$, $(C_5\times C_{15}^2):S_4$, $C_5^3:(C_6^2:C_6)$, $(C_5\times C_{15}^2):S_4$

Order 20250: $C_{15}\wr C_3:C_2$, $(C_3\times C_{15}^2):D_{15}$, $C_{15}^3.C_6$

Order 18000: $C_5^3.D_6:D_6$ x 2, $C_5^3.(D_4\times C_3:S_3)$, $C_5^3:(S_3\times S_4)$

Order 13500: $C_{15}^3.C_2^2$, $C_3^3\times C_5^3:C_2^2$, $(C_5\times C_{15}^2).C_{12}$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_3^2\times C_5^3:(C_3:C_4)$, $C_3^2\times C_5^3:D_6$, $(C_5\times C_{15}^2):D_6$, $(C_5\times C_{15}^2).A_4$, $(C_5\times C_{15}^2):A_4$

Order 10800: $C_{15}^2.D_6.C_2^2$

Order 10125: $C_{15}\wr C_3$

Order 9000: $C_5^3.C_3:S_3.C_2^2$ x 4, $C_5^3.S_3^2.C_2$ x 2, $C_3\times C_5^3:(C_3:D_4)$ x 2, $C_{15}^2.(C_5\times D_4)$ x 2, $C_5^3.C_3^2:D_4$ x 2, $C_5^3.C_6^2:C_2$ x 2, $C_5^3.(C_2.S_3^2)$, $C_5^2.(C_6.D_{30})$, $C_5^3.C_6.C_6.C_2$, $C_3\times C_5^3:(C_3:D_4)$, $C_{15}^2.(C_5\times D_4)$, $C_3^2\times C_5^3:D_4$, $C_5^3.C_3^2:C_4.C_2$, $(C_5^2\times C_{15}).S_4$, $(C_5^2\times C_{15}):S_4$, $C_5^3:(S_3\times A_4)$, $C_3\times C_5^3:S_4$

Order 8100: $C_{15}^2.S_3^2$

Order 6750: $C_3^2\times C_5^3:S_3$, $C_5^3:(C_3^3:C_2)$, $C_5\times C_5^2:(C_3^3:C_2)$, $C_3^3\times C_5\times C_5:D_5$, $C_5^3:(C_3\times C_3:S_3)$, $C_5\times C_{15}^2.S_3$, $C_5\times C_{15}^2:S_3$, $C_{15}^2:D_{15}$, $(C_5\times C_{15}^2):C_6$

Order 6000: $C_5^2.C_{20}:D_6$ x 2, $C_5^2.D_{15}:D_4$, $D_5\wr S_3$

Order 5400: $C_{15}^2.D_6.C_2$, $C_{15}^2.C_6.C_2^2$, $C_3^2\times C_5^2:(C_3:D_4)$, $C_{15}^2.D_6.C_2$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_3:D_4$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.D_{12}$, $C_{15}^2.D_6.C_2$

Order 4500: $C_5^3:(C_2\times C_3:S_3)$ x 6, $C_5^3:(C_2\times C_3:S_3)$ x 5, $C_3^2\times C_5^3:C_2^2$ x 4, $C_3\times C_5^3:(C_3:C_4)$ x 2, $C_5\times C_5^2:(C_2\times C_3:S_3)$ x 2, $C_5^3:S_3^2$ x 2, $C_5^3:S_3^2$ x 2, $C_3\times C_5^3:D_6$ x 2, $C_5^3:(C_3^2:C_4)$, $C_3^2\times C_5\times C_5:F_5$, $C_5^3:(C_3\times C_3:C_4)$, $C_3\times C_5^3:(C_3:C_4)$, $C_5^3:(C_6\times S_3)$, $C_3\times C_5^3:D_6$, $C_5^3:S_3^2$, $(C_5^2\times C_{15}).A_4$, $C_3\times C_5^3:A_4$

Order 4050: $(C_3\times C_{15}^2):S_3$, $(C_3\times C_{15}^2):S_3$, $(C_3\times C_{15}^2):C_6$

Order 3600: $D_5^2:S_3^2$ x 2, $C_{15}^2.C_2^3.C_2$

Order 3375: $C_{15}^3$, $C_{15}^2.C_{15}$, $C_{15}^2:C_{15}$

Order 3000: $C_5^3:(C_2^2\times S_3)$ x 5, $C_5^3:(C_3:D_4)$ x 4, $C_5^3:(C_4\times S_3)$ x 2, $C_5^3:D_{12}$ x 2, $C_3\times C_5^3:D_4$ x 2, $C_3\times C_5^3:D_4$, $C_5^3:(C_4\times S_3)$, $C_5^3:D_{12}$, $C_5^3:(C_3:D_4)$, $C_5^3:(C_2^2\times S_3)$, $C_5^3:(C_3:D_4)$, $C_5^3:S_4$, $D_5\wr C_3$, $C_5^3:S_4$

Order 2700: $C_5^2:(S_3\times C_3:S_3)$ x 4, $C_5^2:(C_2\times C_3^3:C_2)$ x 3, $C_5^2:(C_3\times C_3^2:C_4)$, $C_3^3\times D_5^2$, $C_5^2:(C_2\times C_3^3:C_2)$, $C_5^2:(S_3\times C_3:S_3)$, $C_5^2:(S_3\times C_3:S_3)$, $C_5^2:(C_3\times C_3^2:C_4)$, $C_5^2:(S_3\times C_3:S_3)$, $C_3^2\times C_3:(C_5:F_5)$, $C_3^2\times D_{15}:D_5$, $C_5\times C_3\times C_3:S_3\times D_5$, $C_5^2:(C_6\times C_3:S_3)$, $C_5^2:(S_3\times C_3:S_3)$, $C_5^2:(C_2\times C_3^3:C_2)$, $C_5\times C_5:(C_2\times C_3^3:C_2)$, $C_3\times C_{15}\times C_3:F_5$, $C_3^2\times D_5\times D_{15}$, $C_{15}^2:D_6$

Order 2250: $C_3^2\times C_5\times C_5:D_5$ x 4, $C_5\times C_5\times C_3:D_{15}$ x 3, $C_5^3:(C_3:S_3)$ x 3, $C_3\times C_{15}\times C_5:D_5$ x 2, $C_5^2\times C_3:D_{15}$ x 2, $C_3\times C_5^3:S_3$ x 2, $C_5^3:(C_3:S_3)$, $C_5^3:(C_3\times S_3)$, $C_3\times C_5^3:S_3$, $C_5^3:D_9$, $(C_5^2\times C_{15}):S_3$, $C_3\times C_5^2:D_{15}$, $C_5^2:(C_3\times D_{15})$

Order 2160: $C_{15}.(D_4\times C_3:S_3)$

Order 2025: $C_{15}^2.C_3^2$

Order 2000: $D_5^3:C_2$

Order 1800: $C_{15}^2:C_2^3$ x 7, $C_{15}^2:D_4$ x 4, $C_{15}^2:D_4$ x 4, $D_{15}^2:C_2$ x 2, $D_{15}:D_{30}$ x 2, $C_{15}^2:D_4$ x 2, $C_{15}:(C_4\times D_{15})$ x 2, $C_{15}:(S_3\times F_5)$ x 2, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $F_5\times C_3:D_{15}$, $C_{15}:(S_3\times F_5)$

Order 1620: $(C_3^2\times C_{15}):D_6$

Order 1500: $C_{15}:D_5^2$ x 9, $C_5^2:D_{30}$ x 7, $C_{15}:D_5^2$ x 5, $(C_5\times C_{15}):F_5$ x 2, $C_5^2:C_{60}$ x 2, $C_5^2:D_{30}$, $C_5^2:D_{30}$, $C_5^2:D_{30}$, $C_{15}\times D_5^2$, $C_5\times C_{15}:F_5$, $C_5^3:C_{12}$, $C_5^3:D_6$, $C_5^3:A_4$

Order 1350: $C_{15}^2:S_3$ x 5, $C_{15}^2:C_6$ x 4, $C_{15}^2:C_6$ x 4, $C_{15}^2:S_3$ x 3, $C_{15}^2:C_6$ x 3, $C_{15}^2:C_6$, $C_{15}^2:S_3$, $C_{15}^2:S_3$, $C_{15}^2.S_3$, $C_{15}^2:S_3$, $C_{15}^2:C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$

Order 1296: $C_6^2.S_3^2$

Order 1200: $D_5^2:D_6$ x 3

Order 1125: $C_5\times C_{15}^2$ x 3, $C_5^3:C_3^2$, $C_5^2:C_{45}$

Order 1080: $C_{30}:S_3^2$, $C_3^2:S_3\times D_{10}$, $S_3\times C_{15}:D_6$, $C_3^2:(S_3\times F_5)$, $C_{30}.S_3^2$, $C_{30}.S_3^2$, $C_{30}.S_3^2$, $C_3\times C_{30}.D_6$, $C_3^3:D_{20}$

Order 1000: $D_5^3$ x 2, $C_5^3:D_4$ x 2, $C_5^2:D_{20}$, $D_5^2:C_{10}$, $D_5\times C_5:F_5$

Order 900: $C_{15}:D_{30}$ x 21, $C_{15}^2:C_2^2$ x 10, $D_{15}^2$ x 8, $C_{15}^2:C_2^2$ x 7, $C_3^2\times D_5^2$ x 7, $C_{15}^2:C_2^2$ x 6, $C_{15}^2:C_2^2$ x 4, $C_{15}^2:C_4$ x 4, $C_{15}:D_{30}$ x 2, $C_{15}:D_{30}$ x 2, $C_5^2:S_3^2$ x 2, $C_{15}:D_{30}$ x 2, $C_{15}:D_{30}$ x 2, $D_{15}:C_{30}$ x 2, $C_{15}^2:C_4$ x 2, $C_{15}:C_{60}$ x 2, $C_5^2:S_3^2$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}:C_{60}$

Order 810: $C_5\times C_3^3:S_3$, $\He_3:D_{15}$, $C_3^3:D_{15}$

Order 750: $C_5^2\times D_{15}$ x 7, $C_5^2:C_{30}$ x 7, $C_5^2:D_{15}$ x 3, $C_5^2:D_{15}$, $C_5^2:C_{30}$, $C_5^3:C_6$, $C_5^2:D_{15}$, $C_5\wr S_3$

Order 720: $D_{30}:D_6$ x 2, $C_{60}:D_6$

Order 675: $C_3\times C_{15}^2$ x 5, $C_{15}^2:C_3$, $C_{15}^2.C_3$

Order 648: $(C_3\times C_6^2):C_6$, $C_3^3:S_4$, $C_3^3:S_4$

Order 600: $S_3\times D_5^2$ x 10, $D_5^2:S_3$ x 6, $C_5^2:D_{12}$ x 3, $D_5^2:C_6$ x 3, $S_3\times C_5:F_5$ x 3, $F_5\times D_{15}$ x 2, $D_5\times D_{30}$, $D_{15}:F_5$

Order 540: $C_{15}:S_3^2$ x 5, $C_3^3:D_{10}$ x 4, $C_{15}:C_6^2$, $C_3^3:F_5$, $C_3^3:F_5$, $(C_3\times C_{15}):C_{12}$, $C_3^2:C_{60}$, $C_3^2:D_{30}$, $C_3^2:D_{30}$, $C_3^2:S_3\times C_{10}$, $C_3:S_3\times C_{30}$, $C_3^2\times D_{30}$, $D_{10}\times C_3^3$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_3^3:D_{10}$

Order 500: $C_5\times D_5^2$ x 3, $C_5:D_5^2$ x 2, $C_5:D_5^2$ x 2, $C_5^3:C_4$, $C_5^2:C_{20}$

Order 450: $C_{15}:D_{15}$ x 23, $C_{15}:D_{15}$ x 21, $C_{15}:C_{30}$ x 21, $C_{15}\times D_{15}$ x 16, $C_{15}^2:C_2$ x 6, $C_{15}^2:C_2$ x 6, $C_{15}:D_{15}$ x 2, $C_{15}:C_{30}$ x 2, $(C_5\times C_{15}):C_6$, $(C_5\times C_{15}):S_3$, $C_3\times C_5^2:S_3$, $C_5^2:D_9$

Order 432: $C_6^2:D_6$, $C_6^2:D_6$

Order 405: $C_3^3:C_{15}$

Order 400: $D_5^2:C_2^2$

Order 375: $C_5^2\times C_{15}$ x 3, $C_5\wr C_3$

Order 360: $C_{30}:D_6$ x 7, $C_{15}:D_{12}$ x 2, $C_6.D_{30}$ x 2, $C_{15}:D_{12}$ x 2, $C_{30}.D_6$ x 2, $D_{30}:C_6$ x 2, $C_{30}.D_6$ x 2, $S_3\times D_{30}$ x 2, $D_5\times S_3^2$ x 2, $D_5.S_3^2$ x 2, $C_3^2\times D_{20}$, $C_3:S_3\times F_5$, $C_{12}:D_{15}$, $C_{15}:D_{12}$

Order 324: $C_3^3:A_4$, $C_3^3:D_6$

Order 300: $D_5\times D_{15}$ x 31, $C_{15}:D_{10}$ x 14, $C_{15}:D_{10}$ x 10, $C_3\times D_5^2$ x 10, $C_{15}:D_{10}$ x 7, $C_{15}:F_5$ x 3, $C_{15}:F_5$ x 3, $C_{15}:F_5$ x 2, $C_{15}\times F_5$ x 2, $C_5:D_{30}$, $C_5\times D_{30}$, $C_{15}\times D_{10}$, $C_{15}:C_{20}$, $C_{15}:F_5$, $C_5^2:D_6$

Order 270: $C_3^2:D_{15}$ x 6, $C_3^2:C_{30}$ x 4, $C_3^2\times D_{15}$ x 4, $C_3^3:C_{10}$ x 3, $D_5\times C_3^3$ x 3, $C_3^2\times C_{30}$, $C_3^2:D_{15}$, $C_3^2:C_{30}$, $C_3^2:D_{15}$, $C_3^2:D_{15}$, $C_9:C_{30}$, $C_3^2:C_{30}$

Order 250: $C_5^2:C_{10}$ x 2, $D_5\times C_5^2$ x 2, $C_5^3:C_2$

Order 240: $C_{12}:D_{10}$ x 2, $D_6:D_{10}$

Order 225: $C_{15}^2$ x 23, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $C_3^2.S_4$, $C_6^2:S_3$, $C_6:S_3^2$, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_6.S_3^2$

Order 200: $D_5\wr C_2$ x 4, $D_5\times D_{10}$ x 3, $C_{10}:F_5$, $D_5\times F_5$

Order 180: $C_{15}:D_6$ x 31, $S_3\times D_{15}$ x 12, $C_{30}:S_3$ x 7, $C_3^2\times D_{10}$ x 7, $C_3:D_{30}$ x 6, $C_{15}:D_6$ x 4, $C_{15}:C_{12}$ x 4, $C_{15}:D_6$ x 3, $C_3\times D_{30}$ x 2, $S_3\times C_{30}$ x 2, $C_5\times S_3^2$ x 2, $C_{15}:C_{12}$ x 2, $C_3:C_{60}$ x 2, $C_3^2:F_5$, $C_3^2\times F_5$, $C_3\times C_{60}$, $C_6.D_{15}$

Order 162: $C_3^3:S_3$, $C_3^3:C_6$, $C_3\wr S_3$

Order 150: $D_5\times C_{15}$ x 29, $C_5\times D_{15}$ x 29, $C_5:D_{15}$ x 24, $C_{15}:D_5$ x 7, $S_3\times C_5^2$ x 7, $C_5^2:S_3$ x 2, $C_5^2:C_6$, $C_5\times C_{30}$

Order 144: $D_6:D_6$ x 2, $S_3\times S_4$, $C_{12}:D_6$

Order 135: $C_3^2\times C_{15}$ x 5, $C_9:C_{15}$, $C_5\times \He_3$

Order 125: $C_5^3$

Order 120: $S_3\times D_{10}$ x 10, $C_{15}:D_4$ x 4, $S_3\times F_5$ x 3, $C_4\times D_{15}$ x 2, $C_5\times D_{12}$ x 2, $C_3\times D_{20}$ x 2, $C_2\times D_{30}$, $C_{15}:D_4$, $C_{15}:D_4$, $C_5:D_{12}$, $C_3:D_{20}$, $D_{15}:C_4$

Order 108: $C_3^2:D_6$ x 2, $C_3:S_3^2$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:C_{12}$, $C_3^2:C_{12}$, $C_3^2:A_4$, $C_3^2.A_4$, $C_3^2:D_6$

Order 100: $D_5^2$ x 13, $C_5\times D_{10}$ x 4, $C_5:D_{10}$ x 2, $C_5:F_5$ x 2, $C_5\times F_5$, $C_5:F_5$

Order 90: $C_3:D_{15}$ x 29, $C_{15}:S_3$ x 22, $C_3^2\times D_5$ x 21, $C_3\times D_{15}$ x 12, $S_3\times C_{15}$ x 10, $C_3\times C_{30}$ x 6, $C_5\times D_9$

Order 81: $C_3\wr C_3$

Order 80: $D_4\times D_5$

Order 75: $C_5\times C_{15}$ x 23, $C_5^2:C_3$

Order 72: $C_6:D_6$ x 4, $C_6\wr C_2$ x 4, $C_3:D_{12}$ x 4, $S_3\times D_6$ x 2, $C_6^2:C_2$ x 2, $C_6.D_6$ x 2, $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$, $D_4\times C_3^2$, $C_3:D_{12}$, $C_{12}:S_3$, $C_2^2:D_9$

Order 60: $S_3\times D_5$ x 42, $D_{30}$ x 13, $S_3\times C_{10}$ x 10, $C_3\times D_{10}$ x 10, $C_{15}:C_4$ x 3, $C_3\times F_5$ x 3, $C_{60}$ x 2, $C_{15}:C_4$ x 2, $C_5:C_{12}$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 54: $C_3^2:C_6$ x 2, $C_3^2:S_3$ x 2, $C_3^2:S_3$, $C_9:C_6$, $C_3^2\times C_6$, $C_3^2:C_6$, $S_3\times C_3^2$

Order 50: $C_5\times D_5$ x 14, $C_5:D_5$ x 7, $C_5\times C_{10}$ x 2

Order 48: $S_3\times D_4$ x 3, $C_2\times S_4$

Order 45: $C_3\times C_{15}$ x 24, $C_{45}$

Order 40: $C_2\times D_{10}$ x 3, $C_5:D_4$ x 2, $D_{20}$, $C_4\times D_5$, $C_2\times F_5$, $C_5\times D_4$

Order 36: $C_6:S_3$ x 13, $S_3^2$ x 5, $C_3:C_{12}$ x 4, $C_6^2$ x 4, $C_6\times S_3$ x 4, $C_3\times C_{12}$, $C_3^2:C_4$, $C_2^2:C_9$, $C_3\times A_4$

Order 30: $D_{15}$ x 35, $C_3\times D_5$ x 27, $C_5\times S_3$ x 27, $C_{30}$ x 11

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 25: $C_5^2$ x 5

Order 24: $C_3:D_4$ x 6, $C_2\times D_6$ x 6, $D_{12}$ x 3, $C_4\times S_3$ x 3, $C_3\times D_4$ x 3, $S_4$ x 2, $C_2\times A_4$

Order 20: $D_{10}$ x 16, $C_2\times C_{10}$ x 3, $F_5$ x 2, $C_{20}$, $C_5:C_4$

Order 18: $C_3:S_3$ x 10, $C_3\times C_6$ x 6, $C_3\times S_3$ x 6, $D_9$

Order 16: $C_2\times D_4$

Order 15: $C_{15}$ x 24

Order 12: $D_6$ x 20, $C_2\times C_6$ x 6, $C_{12}$ x 3, $C_3:C_4$ x 3, $A_4$

Order 10: $D_5$ x 14, $C_{10}$ x 9

Order 9: $C_3^2$ x 4, $C_9$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $S_3$ x 13, $C_6$ x 9

Order 5: $C_5$ x 5

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 162000: $C_3^3:D_5\wr S_3$ ($C_1$)

Order 81000: $C_3^3:(C_5^3:S_4)$ ($C_2$), $C_3^3:D_5\wr C_3$ ($C_2$), $C_3^3:(C_5^3:S_4)$ ($C_2$)

Order 40500: $C_3^3:(C_5^3:A_4)$ ($C_2^2$)

Order 27000: $C_{15}^3.C_2^3$ ($S_3$), $(C_5\times C_{15}^2):S_4$ ($S_3$)

Order 13500: $C_3^3\times C_5^3:C_2^2$ ($D_6$), $(C_5\times C_{15}^2):A_4$ ($D_6$)

Order 6750: $C_5^3:(C_3^3:C_2)$ ($S_4$)

Order 4500: $C_3^2\times C_5^3:C_2^2$ ($S_3^2$)

Order 3375: $C_{15}^3$ ($C_2\times S_4$)

Order 1500: $C_{15}:D_5^2$ ($C_3^2:D_6$)

Order 1125: $C_5\times C_{15}^2$ ($S_3\times S_4$)

Order 500: $C_5:D_5^2$ ($C_3^3:D_6$)

Order 375: $C_5^2\times C_{15}$ ($C_6^2:D_6$)

Order 125: $C_5^3$ ($C_6^2.S_3^2$)

Order 27: $C_3^3$ ($D_5\wr S_3$)

Order 9: $C_3^2$ ($C_5^3:(S_3\times S_4)$)

Order 3: $C_3$ ($C_3^2:D_5\wr S_3$)

Order 1: $C_1$ ($C_3^3:D_5\wr S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 162000: $C_3^3:D_5\wr S_3$ ($C_1$)

Order 81000: $C_3^3:(C_5^3:S_4)$ ($C_2$), $C_3^3:D_5\wr C_3$ ($C_2$), $C_3^3:(C_5^3:S_4)$ ($C_2$)

Order 40500: $C_3^3:(C_5^3:A_4)$ ($C_2^2$)

Order 27000: $C_{15}^3.C_2^3$ ($S_3$), $(C_5\times C_{15}^2):S_4$ ($S_3$)

Order 13500: $C_3^3\times C_5^3:C_2^2$ ($D_6$), $(C_5\times C_{15}^2):A_4$ ($D_6$)

Order 6750: $C_5^3:(C_3^3:C_2)$ ($S_4$)

Order 4500: $C_3^2\times C_5^3:C_2^2$ ($S_3^2$)

Order 3375: $C_{15}^3$ ($C_2\times S_4$)

Order 1500: $C_{15}:D_5^2$ ($C_3^2:D_6$)

Order 1125: $C_5\times C_{15}^2$ ($S_3\times S_4$)

Order 500: $C_5:D_5^2$ ($C_3^3:D_6$)

Order 375: $C_5^2\times C_{15}$ ($C_6^2:D_6$)

Order 125: $C_5^3$ ($C_6^2.S_3^2$)

Order 27: $C_3^3$ ($D_5\wr S_3$)

Order 9: $C_3^2$ ($C_5^3:(S_3\times S_4)$)

Order 3: $C_3$ ($C_3^2:D_5\wr S_3$)

Order 1: $C_1$ ($C_3^3:D_5\wr S_3$)

Series

Derived series

$C_3^3:D_5\wr S_3$

$\rhd$

$C_3^3:D_5\wr S_3$

$\rhd$

$C_3^3:(C_5^3:A_4)$

$\rhd$

$C_3^3:(C_5^3:A_4)$

$\rhd$

$C_3^2\times C_5^3:C_2^2$

$\rhd$

$C_3^2\times C_5^3:C_2^2$

$\rhd$

$C_5^3$

$\rhd$

$C_5^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^3:D_5\wr S_3$

$\rhd$

$C_3^3:D_5\wr S_3$

$\rhd$

$C_3^3:(C_5^3:S_4)$

$\rhd$

$C_3^3:(C_5^3:S_4)$

$\rhd$

$C_3^3:(C_5^3:A_4)$

$\rhd$

$C_3^3:(C_5^3:A_4)$

$\rhd$

$C_3^3\times C_5^3:C_2^2$

$\rhd$

$C_3^3\times C_5^3:C_2^2$

$\rhd$

$(C_5\times C_{15}^2):A_4$

$\rhd$

$C_3^2\times C_5^3:C_2^2$

$\rhd$

$C_3^2\times C_5^3:C_2^2$

$\rhd$

$C_{15}:D_5^2$

$\rhd$

$C_{15}:D_5^2$

$\rhd$

$C_5:D_5^2$

$\rhd$

$C_5:D_5^2$

$\rhd$

$C_5^3$

$\rhd$

$C_5^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^3:D_5\wr S_3$

$\rhd$

$C_3^3:D_5\wr S_3$

$\rhd$

$C_3^3:(C_5^3:A_4)$

$\rhd$

$C_3^3:(C_5^3:A_4)$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $214 \times 214$ 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 $114 \times 114$ rational character table.