Abstract group 324000.bm: $C_{15}^3.(C_4\times S_4)$

• Label: 324000.bm
• Order: \( 2^{5} \cdot 3^{4} \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $24$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_{15}^3.(C_4\times S_4)$
Order:	\(324000\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5^{3} \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$D_{15}\wr S_3.C_4$, of order \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \)
Composition factors:	$C_2$ x 5, $C_3$ 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	18	20	30	45	60
Elements	1	1475	1826	49500	124	18190	3600	9900	104400	10424	18000	27000	43560	14400	21600	324000
Conjugacy classes	1	5	4	10	5	12	1	9	10	52	1	3	37	2	4	156
Divisions	1	5	4	6	5	12	1	9	5	32	1	2	21	1	1	106
Autjugacy classes	1	5	4	10	5	12	1	9	10	32	1	3	21	1	2	117

Dimension	1	2	3	4	6	8	12	16	24	32	48	64	96	192
Irr. complex chars.	8	4	8	0	8	4	14	8	29	3	50	0	20	0	156
Irr. rational chars.	4	4	4	1	6	4	12	4	11	3	18	1	27	7	106

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$29016$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{12}=c^{2}=d^{30}=e^{15}=f^{15}=[c,e]=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,2,3,5,8,11,12,14,6,7)(9,13,10,15)(16,17)(18,19,21)(20,22,23,24), (2,4,7,10)(3,6)(5,9,14,13)(8,12)(11,15)(17,18,20,19)(21,23)(22,24)\rangle$
Transitive group:	45T1425				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{15}^3.S_4)$ $\,\rtimes\,$ $C_4$	$(C_{15}^3.S_4)$ $\,\rtimes\,$ $C_4$	$C_3^3$ $\,\rtimes\,$ $(D_5^3.D_6)$	$((C_{15}^3.A_4):C_4)$ $\,\rtimes\,$ $C_2$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{15}^3.C_2^3)$ . $D_6$	$(C_3^3:D_5\wr S_3)$ . $C_2$	$(C_5^3:C_2)$ . $(S_3\wr S_3)$	$(C_3^3:D_5\wr C_3)$ . $C_2^2$	all 5

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

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 1413136 subgroups in 2508 conjugacy classes, 18 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{15}^3.(C_4\times S_4)$
Commutator:	$G' \simeq$ $C_{15}^3.A_4$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{15}^3.(C_4\times S_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_{15}^3$	$G/\operatorname{Fit} \simeq$ $C_4\times S_4$
Radical:	$R \simeq$ $C_{15}^3.(C_4\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{15}^3$	$G/\operatorname{soc} \simeq$ $C_4\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times D_4$
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 324000: $C_{15}^3.(C_4\times S_4)$

Order 162000: $(C_{15}^3.A_4):C_4$, $C_3^3:D_5\wr S_3$, $(C_{15}^3.A_4):C_4$

Order 108000: $C_{15}^3.C_4^2.C_2$

Order 81000: $C_3^3:D_5\wr C_3$, $C_{15}\wr S_3:C_4$, $C_{15}^3.S_4$, $C_{15}^3.S_4$

Order 54000: $C_{15}^3.C_4^2$, $C_{15}^3.C_2.C_2^3$, $C_{15}^3.C_2.D_4$, $C_5^3.(C_6\times S_3^2).C_2$, $C_{15}^3.C_4.C_2^2$, $C_{15}^3.C_2.D_4$, $C_{15}^3.C_4.C_2^2$

Order 40500: $C_{15}\wr S_3:C_2$, $C_{15}^3.A_4$, $C_{15}^3.C_{12}$, $C_{15}\wr C_3:C_4$

Order 36000: $C_5^3.(C_4\times S_3\wr C_2)$

Order 27000: $C_{15}^3.C_2^2.C_2$, $C_5^3.C_3:S_3.C_6.C_2$, $C_{15}^2.C_{30}.C_2^2$, $C_{15}^3.C_4.C_2$, $C_{15}^3.C_4.C_2$, $C_{15}^3.D_4$, $C_{15}^3.C_4.C_2$, $C_{15}^3.D_4$, $C_{15}^3.C_2^3$, $C_{15}^3.D_4$, $C_3\times C_5^3.(C_6\times S_3).C_2$, $C_{15}^3.C_2^3$, $(C_5\times C_{15}^2).C_{12}.C_2$, $C_{15}^2:(S_3\times F_5)$

Order 21600: $(C_3\times C_{15}^2).C_4^2.C_2$

Order 20250: $C_{15}\wr S_3$, $(C_3\times C_{15}^2):D_{15}$, $C_{15}\wr C_3:C_2$

Order 18000: $C_5^3.(C_4\times S_3^2)$ x 2, $C_5^3.S_3^2:C_4$ x 2, $C_5^3.(C_6\times S_3).C_2^2$, $(C_5\times C_{15}^2).C_4^2$, $C_5^3.C_3:S_3.C_2^3$, $C_5^3.(C_3\times C_6).D_4$

Order 16200: $C_{15}^2:C_6.D_6$

Order 13500: $C_5\times C_5^2:(C_3\times C_3:S_3.C_2)$, $C_{15}^3.C_4$, $C_3\times C_5^3:(C_6\times S_3)$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_3\times C_5^3:(C_3\times C_3:C_4)$, $(C_3\times C_{15}^2).C_{20}$, $C_{15}^3.C_2^2$, $C_3\times C_5^3:(C_3^2:C_4)$, $C_{15}^3.C_2^2$, $C_{15}^3.C_4$, $(C_5\times C_{15}^2):D_6$, $(C_5\times C_{15}^2):C_{12}$, $C_5^2:C_{45}:C_{12}$, $C_{15}^2:C_{15}:C_4$

Order 12000: $C_5^3.C_6.C_4.C_2^2$, $D_5^3.D_6$

Order 10800: $(C_3\times C_{15}^2).C_4^2$, $C_{15}^2.C_6.C_2^3$, $C_{15}^2.C_6.D_4$, $C_{15}^2.C_6.D_4$, $C_{15}^2.C_6.C_2^3$, $C_{15}^2.C_6.C_2^3$, $C_{15}^2.C_2^2.C_6.C_2$, $(C_3\times C_{15}^2).C_4^2$, $C_{15}^2.C_6.C_2^3$

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

Order 9000: $C_5^3.(C_6\times S_3).C_2$ x 3, $C_5^3.S_3\wr C_2$ x 2, $C_5^3.(C_2.S_3^2)$ x 2, $C_5^3.(C_2\times S_3^2)$ x 2, $C_5^3.C_3^2:C_4.C_2$ x 2, $C_5^3.(S_3\times C_3:C_4)$, $C_5^3.(C_6\times S_3).C_2$, $C_5^3.(C_6\times S_3).C_2$, $C_5^3.C_3^2:C_4.C_2$, $C_5^3.S_3\wr C_2$, $C_5^3.C_3:S_3.C_2^2$, $C_5^3.C_6^2.C_2$, $C_5^3.C_6.C_6.C_2$, $C_3\times C_5^3:(C_4\times S_3)$, $C_5^3.C_3:S_3.C_2^2$, $C_{15}^2.(C_5\times D_4)$, $C_3\times C_5^3:(C_2\times C_3:C_4)$, $C_5^3:C_6.D_6$

Order 8100: $(C_3\times C_{15}^2):C_{12}$, $(C_3\times C_{15}^2):D_6$, $C_{15}^2:C_6.S_3$

Order 7200: $C_{15}^2:(C_4\times D_4)$

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

Order 6000: $C_5^3.C_6.D_4$, $C_5^3.C_6.D_4$, $C_3\times C_5^3:(C_2^2:C_4)$, $C_5^3.C_6.C_2^3$, $C_5^3.C_6.C_4.C_2$, $C_5^3.D_6.C_2^2$, $C_5^3.C_6.C_2^3$, $C_5^3.C_6.C_2^3$, $D_5\wr S_3$, $D_5^3.C_6$, $D_5^3.S_3$

Order 5400: $C_{15}^2.C_6.C_2^2$ x 4, $C_{15}^2.C_6.C_2^2$ x 3, $C_{15}^2.C_3:D_4$ x 2, $C_5^2.S_3^2:S_3$ x 2, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_{12}.C_2$, $C_3\times C_5^2:(S_3\times C_3:C_4)$, $C_{15}^2.D_6.C_2$, $C_3\times S_3\times C_3:(C_5:F_5)$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_6.C_2^2$, $C_3\times C_5^2:(S_3\times C_3:C_4)$, $(C_3\times C_{15}^2).C_4.C_2$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_{12}.C_2$, $C_{15}^2.C_{12}.C_2$, $C_{15}^2.C_{12}.C_2$, $(C_3\times C_{15}^2).C_4.C_2$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_{12}.C_2$, $C_{15}^2:(C_4\times S_3)$

Order 4500: $C_3\times C_5^3:(C_3:C_4)$ x 4, $C_5^3:(C_3^2:C_4)$ x 3, $C_5^3:(C_6\times S_3)$ x 3, $C_5^3:S_3^2$ x 2, $C_5^3:S_3^2$ x 2, $C_3\times C_5^3:D_6$ x 2, $C_5^3:S_3^2$ x 2, $C_5^3:(C_3^2:C_4)$, $C_3^2\times D_5\times C_5:D_5$, $C_5^3:(C_6\times S_3)$, $C_5\times D_5\times C_3\times D_{15}$, $C_5^3:(C_3\times C_3:C_4)$, $C_5^3:(C_3:S_3.C_2)$, $C_5^3:(C_6\times S_3)$, $C_5\times C_5^2:(C_3:S_3.C_2)$, $C_5^3:(C_6\times S_3)$, $C_5^3:(C_2\times C_3:S_3)$, $C_5^3:(C_3:S_3.C_2)$, $C_3^2\times C_5:(C_5:F_5)$, $C_3\times C_5^3:D_6$, $(C_5^2\times C_{15}):C_{12}$, $C_5^2:C_{45}:C_4$, $C_5^3:C_3^2:C_4$

Order 4320: $S_3^2:(S_3\times F_5)$

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

Order 4000: $D_5^3.C_2^2$

Order 3600: $C_{15}^2.C_2^2.C_2^2$ x 2, $D_{15}^2:C_4$ x 2, $D_{15}^2:C_4$ x 2, $C_{15}^2.C_4^2$, $C_{15}^2.C_2^2.C_2^2$, $C_{15}^2.C_4.C_2^2$, $C_5^2.(C_4\times S_3^2)$, $C_{15}^2:C_4^2$, $D_{15}^2:C_2^2$, $C_{15}^2:C_4:C_4$

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

Order 3240: $C_3^3:(S_3\times F_5)$

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

Order 2700: $C_5^2:(C_3^3:C_4)$ x 5, $C_3\times C_5^2:(C_3^2:C_4)$ x 4, $C_3\times C_5^2:(C_6\times S_3)$ x 4, $C_3\times C_5^2:(C_3\times C_3:C_4)$ x 3, $C_5^2:(C_6\times C_3:S_3)$ x 3, $C_5^2:(C_3^3:C_2^2)$ x 2, $C_5^2:(C_3\times S_3^2)$ x 2, $C_5^2:(C_3^2:(C_3:C_4))$ x 2, $C_5^2:(C_3\times S_3^2)$ x 2, $C_5^2:(C_3\times S_3^2)$, $C_5\times C_5:(C_3\times S_3^2)$, $C_3\times C_5^2:(C_3\times C_3:C_4)$, $C_5\times C_5:(C_3\times C_3:S_3.C_2)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5^2:(C_6\times C_3:S_3)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5^2:(C_3\times C_3:S_3.C_2)$, $C_5\times C_5:(C_3\times C_3:S_3.C_2)$, $C_5^2:(C_3^3:C_2^2)$, $C_3\times C_5^2:(C_3^2:C_4)$, $C_5^2:(C_3\times S_3^2)$, $C_3\times C_3\times S_3\times C_5:D_5$, $C_{15}^2:C_{12}$, $C_{15}^2:D_6$, $C_5^2:C_9:C_{12}$, $C_{15}^2:C_3:C_4$

Order 2592: $C_3^3:(C_4\times S_4)$

Order 2400: $(C_6\times C_5:D_5).C_2^3$

Order 2250: $C_5^3:(C_3\times S_3)$ x 3, $C_3^2\times C_5^2:D_5$ x 3, $C_5^3:(C_3\times S_3)$ x 2, $C_3\times C_5^3:S_3$ x 2, $C_3^2\times C_5\times C_5:D_5$, $C_5\times C_5\times C_3:D_{15}$, $C_3^2\times C_5^2\times D_5$, $C_5^3:(C_3:S_3)$, $C_{15}\times C_5\times D_{15}$, $C_5\times C_{15}\times D_{15}$, $C_3\times C_5^3:S_3$, $C_3\times C_5\wr S_3$, $C_3\times C_5^2:D_{15}$, $C_5^3:C_{18}$, $C_3\times C_5^3:C_6$

Order 2160: $D_5.S_3^3$ x 2, $(C_3^2\times C_{15}):C_4^2$, $(C_3^2\times C_{15}):C_4^2$, $C_3\times S_3^2:F_5$, $(C_3^2\times D_5).D_{12}$, $D_5.S_3^3$, $D_5\times S_3^2:S_3$, $(D_5\times C_3^3).D_4$

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

Order 2000: $D_5^2:F_5$ x 2, $D_5^2:F_5$ x 2, $D_5^3:C_2$, $(D_5\times C_5^2).Q_8$, $C_5:F_5^2$

Order 1800: $(C_{15}\times D_{15}):C_4$ x 24, $S_3\times C_{15}:F_5$ x 5, $D_{15}\wr C_2$ x 4, $C_{15}^2:(C_2\times C_4)$ x 4, $C_{15}:(S_3\times F_5)$ x 3, $D_{15}^2:C_2$ x 2, $C_{15}^2:C_2^3$ x 2, $C_{15}^2:C_2^3$ x 2, $C_{15}^2:(C_2\times C_4)$ x 2, $C_{15}:(S_3\times F_5)$ x 2, $(C_{15}\times D_{15}):C_4$ x 2, $C_{15}:(S_3\times F_5)$ x 2, $S_3\times C_{15}:F_5$ x 2, $C_3\times D_{15}:F_5$ x 2, $C_3\times D_{15}:F_5$ x 2, $(D_5\times C_{15}):C_{12}$ x 2, $S_3\times C_{15}:F_5$ x 2, $C_{15}^2:C_2^3$, $C_2\times C_{15}^2:C_4$, $C_6\times C_{15}:F_5$, $C_{15}^2:(C_2\times C_4)$, $C_{15}^2:(C_2\times C_4)$, $C_{15}:(S_3\times F_5)$, $C_{15}:F_5:S_3$

Order 1620: $C_3\wr C_3:F_5$, $(C_3^2\times C_{15}):C_{12}$, $C_3\wr S_3\times D_5$

Order 1500: $C_{15}:D_5^2$ x 4, $(C_5\times C_{15}):F_5$ x 4, $C_{15}:D_5^2$ x 3, $C_5^2:D_{30}$ x 3, $(C_5\times C_{15}):F_5$ x 3, $C_5^3:C_{12}$ x 3, $C_5^2:D_{30}$ x 2, $C_{15}:D_5^2$ x 2, $C_{15}\times D_5^2$ x 2, $C_{15}:D_5^2$, $(C_5\times C_{15}):F_5$, $C_5\times C_{15}:F_5$, $C_5^2:C_{60}$, $C_5^3:D_6$, $C_5^3:A_4$, $C_5^3:C_{12}$, $C_5\wr C_3:C_4$

Order 1440: $F_5\times \SOPlus(4,2)$

Order 1350: $C_{15}^2:C_6$ x 8, $C_{15}^2:C_6$ x 6, $C_{15}^2:C_6$ x 5, $C_{15}^2:S_3$ x 2, $C_{15}^2:C_6$, $C_{15}^2.C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$, $S_3\times C_{15}^2$

Order 1296: $C_2\times C_3^3:S_4$, $C_3^3:(C_4\times A_4)$, $(C_3^2\times C_6).S_4$

Order 1200: $D_5^2.D_6$ x 3, $D_{30}:F_5$ x 2, $D_5^2.D_6$ x 2, $C_3:F_5^2$, $(C_5\times C_{30}).Q_8$, $(C_5\times C_{30}).D_4$, $D_5^2:C_{12}$, $C_{15}:(C_4\times F_5)$, $D_5^2:D_6$

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

Order 1080: $S_3\times C_3^2:F_5$ x 5, $C_3^2:(S_3\times F_5)$ x 4, $S_3\times C_{15}:C_{12}$ x 3, $C_3\times D_5\times S_3^2$ x 2, $C_3^2:(S_3\times F_5)$ x 2, $C_{15}:(S_3\times C_{12})$ x 2, $D_5\times C_3:S_3^2$, $C_3^3:D_{20}$, $C_{15}:\SOPlus(4,2)$, $S_3^2:D_{15}$, $C_5\times S_3^2:S_3$, $C_3^3:C_4\times D_5$, $(D_5\times C_3^3):C_4$, $C_3^2:C_{12}\times D_5$, $(C_3^2\times D_5):C_{12}$, $C_3^2:(S_3\times F_5)$

Order 1000: $C_5^3:(C_2\times C_4)$ x 3, $D_5^3$ x 2, $C_5^3:D_4$ x 2, $C_5^3:(C_2\times C_4)$ x 2, $C_5^2:D_{20}$, $D_5^2:C_{10}$, $D_5.D_5^2$, $D_5\times C_5:F_5$

Order 900: $C_{15}^2:C_4$ x 32, $C_{15}^2:C_2^2$ x 24, $C_{15}^2:C_4$ x 22, $C_{15}^2:C_2^2$ x 5, $C_{15}^2:C_4$ x 5, $C_{15}:D_{30}$ x 4, $D_{15}^2$ x 4, $C_{15}:D_{30}$ x 4, $C_{15}:D_{30}$ x 4, $D_{15}:C_{30}$ x 4, $C_3^2\times D_5^2$ x 4, $C_{15}^2:C_4$ x 4, $C_{15}^2:C_2^2$ x 3, $C_{15}^2:C_4$ x 3, $C_{15}\times D_{30}$ x 2, $C_5^2:S_3^2$ x 2, $C_{15}^2:C_4$ x 2, $C_{15}^2:C_4$ x 2, $C_{15}^2:C_4$ x 2, $C_3\times C_5^2:D_6$, $(C_5\times C_{15}):C_{12}$, $C_5^2:C_3^2:C_4$, $C_5^2:C_9:C_4$, $C_5^2:C_6^2$, $C_{15}^2:C_2^2$, $C_{15}^2:C_4$, $C_{15}^2:C_4$

Order 864: $S_3^2:(C_4\times S_3)$

Order 810: $C_3^3:D_{15}$, $\He_3:C_{30}$, $C_3\wr C_3\times D_5$

Order 800: $C_5^2:(C_4\times D_4)$

Order 750: $C_5^2:C_{30}$ x 4, $C_5^2:C_{30}$ x 4, $C_5^2:D_{15}$ x 3, $C_5^2\times D_{15}$ x 3, $C_5^3:C_6$ x 3, $C_5^3:C_6$, $C_5^2:D_{15}$, $C_5\wr S_3$

Order 720: $F_5\times S_3^2$ x 4, $D_{10}.S_3^2$ x 3, $S_3^2:F_5$ x 2, $C_3^2:C_4\times F_5$ x 2, $S_3^2:D_{10}$, $D_5.\SOPlus(4,2)$

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

Order 648: $C_3^3:S_4$ x 2, $C_2\times C_3^3:A_4$, $C_3^3:(C_4\times S_3)$

Order 600: $D_5^2.S_3$ x 20, $D_{15}:F_5$ x 15, $C_{30}:F_5$ x 4, $D_5^2:S_3$ x 4, $C_{30}:F_5$ x 3, $S_3\times D_5^2$ x 3, $D_{15}:F_5$ x 3, $S_3\times C_5:F_5$ x 3, $S_3\times C_5:F_5$ x 3, $D_5^2.C_6$ x 3, $C_{30}:D_{10}$ x 2, $C_6\times D_5^2$ x 2, $C_{30}:F_5$ x 2, $C_{30}.D_{10}$, $D_5^2.S_3$, $D_5^2.C_6$, $C_5^2:(C_4\times S_3)$

Order 540: $C_{15}:C_6^2$ x 5, $C_3^3:F_5$ x 5, $C_3^3:F_5$ x 5, $C_3^3:F_5$ x 4, $C_3^3:D_{10}$ x 4, $C_{15}:S_3^2$ x 3, $C_3^3:F_5$ x 2, $\He_3:F_5$ x 2, $C_{15}:S_3^2$ x 2, $C_{15}:S_3^2$ x 2, $C_{15}\times S_3^2$ x 2, $(C_3^2\times C_{15}):C_4$, $C_3^3:C_{20}$, $(C_3\times C_{15}):C_{12}$, $C_3^3:F_5$, $C_3^2:C_{60}$, $\He_3:D_{10}$, $C_{45}:C_{12}$, $C_{15}:S_3^2$

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

Order 480: $C_{15}:(C_4\times D_4)$

Order 450: $C_{15}\times D_{15}$ x 44, $C_{15}^2:C_2$ x 32, $C_{15}:C_{30}$ x 8, $C_{15}:D_{15}$ x 6, $C_{15}:D_{15}$ x 5, $C_{15}:C_{30}$ x 5, $C_3\times C_5^2:S_3$ x 2, $C_{15}\times C_{30}$, $C_{15}:D_{15}$, $C_{15}^2:C_2$, $C_3\times C_5^2:C_6$, $C_5^2:C_{18}$

Order 432: $S_3^2:D_6$, $C_2.S_3^3$, $C_2.S_3^3$, $C_6.\SOPlus(4,2)$, $C_6.\SOPlus(4,2)$, $S_3^2:C_{12}$, $C_3^3:C_4^2$

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

Order 400: $D_{10}:F_5$ x 3, $D_5^2:C_4$ x 2, $D_5^2:C_2^2$, $F_5^2$, $(C_5\times C_{10}).Q_8$, $C_5^2:C_4^2$

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

Order 360: $D_5.S_3^2$ x 34, $D_{15}:C_{12}$ x 6, $C_2\times C_3^2:F_5$ x 5, $D_5\times S_3^2$ x 4, $D_5.S_3^2$ x 4, $C_3:S_3\times F_5$ x 4, $C_{30}:D_6$ x 3, $C_{30}:C_{12}$ x 3, $S_3^2:D_5$ x 2, $(C_3^2\times D_5):C_4$ x 2, $C_3^2:C_4\times D_5$ x 2, $C_3^2:D_{20}$, $S_3^2:C_{10}$

Order 324: $C_3^3:D_6$, $C_3\wr C_3:C_4$, $C_3^3:A_4$, $C_3^3:C_{12}$

Order 300: $C_{15}:F_5$ x 35, $C_3\times D_5^2$ x 21, $C_{15}:F_5$ x 19, $C_{15}:D_{10}$ x 16, $C_{15}:F_5$ x 12, $D_5\times D_{15}$ x 6, $C_{15}:D_{10}$ x 6, $C_{15}:D_{10}$ x 4, $C_5\times D_{30}$ x 3, $C_{15}\times D_{10}$ x 3, $C_{15}:D_{10}$ x 3, $C_{15}:F_5$ x 3, $C_{15}:F_5$, $C_{15}:C_{20}$, $C_{15}:F_5$, $C_{15}\times F_5$, $C_5^2:D_6$, $C_5^2:C_{12}$, $C_5^2:C_3:C_4$, $C_5:C_{60}$

Order 288: $C_4\times \SOPlus(4,2)$

Order 270: $C_3^2\times D_{15}$ x 5, $C_3^2:C_{30}$ x 5, $D_5\times C_3^3$ x 5, $C_3^2:D_{15}$ x 4, $C_3^2:C_{30}$ x 4, $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

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

Order 240: $D_6\times F_5$ x 5, $D_{10}.D_6$ x 2, $D_{10}.D_6$, $D_{30}:C_4$, $C_{15}:C_4^2$, $D_6:D_{10}$, $C_{60}:C_4$, $D_{10}:C_{12}$

Order 225: $C_{15}^2$ x 31, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $S_3^2:S_3$ x 4, $C_6.S_3^2$, $C_6:S_3^2$, $C_6\times S_3^2$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$

Order 200: $D_5:F_5$ x 11, $C_{10}:F_5$ x 4, $D_5\wr C_2$ x 4, $D_5\times D_{10}$ x 3, $C_{10}:F_5$ x 2, $D_5\times F_5$ x 2, $C_{10}.D_{10}$

Order 180: $C_3^2:F_5$ x 37, $C_{15}:D_6$ x 34, $C_{15}:C_{12}$ x 28, $S_3\times D_{15}$ x 6, $C_3^2\times D_{10}$ x 5, $C_{15}:D_6$ x 4, $C_5\times S_3^2$ x 4, $C_{15}:D_6$ x 4, $C_3^2\times F_5$ x 4, $C_3\times D_{30}$ x 3, $S_3\times C_{30}$ x 3, $C_3^2:F_5$ x 3, $(C_3\times C_{15}):C_4$ x 2, $C_3^2:C_{20}$ x 2, $C_{45}:C_4$

Order 162: $C_3\wr S_3$ x 2, $C_3^3:C_6$

Order 160: $D_4\times F_5$

Order 150: $C_{15}:D_5$ x 35, $D_5\times C_{15}$ x 34, $C_5\times D_{15}$ x 24, $C_5\times C_{30}$ x 4, $C_5:D_{15}$ x 3, $S_3\times C_5^2$ x 3, $C_5^2:S_3$ x 2, $C_5^2:C_6$

Order 144: $C_4\times S_3^2$ x 2, $S_3^2:C_4$ x 2, $S_3^2:C_2^2$, $D_6.D_6$, $C_3^2:C_4^2$, $C_2.\SOPlus(4,2)$

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: $C_{30}:C_4$ x 27, $S_3\times F_5$ x 25, $C_6\times F_5$ x 6, $S_3\times D_{10}$ x 5, $C_6\times D_{10}$ x 2, $C_6.D_{10}$, $C_{15}:D_4$, $C_{15}:D_4$, $D_5\times C_{12}$, $C_3:D_{20}$, $C_{15}:D_4$

Order 108: $C_3\times S_3^2$ x 3, $C_3^2:C_{12}$ x 2, $C_3:S_3^2$ x 2, $C_3^3:C_4$ x 2, $C_9:C_{12}$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:C_{12}$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:C_{12}$, $C_3^2:D_6$

Order 100: $D_5^2$ x 13, $C_5:F_5$ x 9, $C_5:F_5$ x 7, $C_5\times D_{10}$ x 4, $C_5\times F_5$ x 2, $C_5:D_{10}$ x 2, $C_5:F_5$ x 2, $C_5:C_{20}$

Order 96: $C_2^3.D_6$, $C_4\times S_4$

Order 90: $C_3^2\times D_5$ x 37, $C_3\times D_{15}$ x 28, $S_3\times C_{15}$ x 28, $C_3\times C_{30}$ x 5, $C_3:D_{15}$ x 4, $C_{15}:S_3$ x 4, $C_9\times D_5$

Order 81: $C_3\wr C_3$

Order 80: $C_2^2\times F_5$ x 3, $D_{10}:C_4$ x 2, $C_4\times F_5$ x 2, $D_4\times D_5$, $C_{20}:C_4$

Order 75: $C_5\times C_{15}$ x 31, $C_5^2:C_3$

Order 72: $C_6.D_6$ x 8, $\SOPlus(4,2)$ x 4, $S_3\times D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $S_3\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_6\times D_6$, $C_6.D_6$, $C_{12}:S_3$, $C_6:C_{12}$

Order 60: $C_{15}:C_4$ x 55, $C_3\times D_{10}$ x 26, $S_3\times D_5$ x 25, $C_3\times F_5$ x 17, $D_{30}$ x 5, $S_3\times C_{10}$ x 5, $C_2\times C_{30}$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_3:C_{20}$

Order 54: $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

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

Order 48: $C_4\times D_6$ x 2, $C_2\times S_4$, $C_6:D_4$, $C_6.C_2^3$, $C_4\times A_4$, $A_4:C_4$, $C_2^2:C_{12}$, $D_6:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 45: $C_3\times C_{15}$ x 32, $C_{45}$

Order 40: $C_2\times F_5$ x 18, $C_2\times D_{10}$ x 3, $C_5:D_4$ x 2, $C_4\times D_5$ x 2, $D_{20}$, $C_5\times D_4$

Order 36: $C_6\times S_3$ x 10, $C_3:C_{12}$ x 6, $S_3^2$ x 6, $C_3^2:C_4$ x 5, $C_3^2:C_4$ x 3, $C_3\times C_{12}$, $C_6^2$, $C_6:S_3$, $C_9:C_4$

Order 32: $C_4\times D_4$

Order 30: $C_3\times D_5$ x 53, $C_{30}$ x 21, $D_{15}$ x 16, $C_5\times S_3$ x 16

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

Order 25: $C_5^2$ x 5

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

Order 20: $F_5$ x 17, $D_{10}$ x 16, $C_2\times C_{10}$ x 3, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 18: $C_3\times S_3$ x 12, $C_3\times C_6$ x 6, $C_3:S_3$ x 2, $C_{18}$

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

Order 15: $C_{15}$ x 32

Order 12: $D_6$ x 11, $C_3:C_4$ x 10, $C_2\times C_6$ x 7, $C_{12}$ x 5, $A_4$

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

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

Order 8: $C_2\times C_4$ x 7, $D_4$ x 4, $C_2^3$ x 2

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

Order 5: $C_5$ x 5

Order 4: $C_2^2$ x 7, $C_4$ x 6

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 324000: $C_{15}^3.(C_4\times S_4)$

Order 162000: $(C_{15}^3.A_4):C_4$, $C_3^3:D_5\wr S_3$, $(C_{15}^3.A_4):C_4$

Order 108000: $C_{15}^3.C_4^2.C_2$

Order 81000: $C_3^3:D_5\wr C_3$, $C_{15}\wr S_3:C_4$, $C_{15}^3.S_4$, $C_{15}^3.S_4$

Order 54000: $C_{15}^3.C_4^2$, $C_{15}^3.C_2.C_2^3$, $C_{15}^3.C_2.D_4$, $C_5^3.(C_6\times S_3^2).C_2$, $C_{15}^3.C_4.C_2^2$, $C_{15}^3.C_2.D_4$, $C_{15}^3.C_4.C_2^2$

Order 40500: $C_{15}\wr S_3:C_2$, $C_{15}^3.A_4$, $C_{15}^3.C_{12}$, $C_{15}\wr C_3:C_4$

Order 36000: $C_5^3.(C_4\times S_3\wr C_2)$

Order 27000: $C_{15}^3.C_2^2.C_2$, $C_5^3.C_3:S_3.C_6.C_2$, $C_{15}^2.C_{30}.C_2^2$, $C_{15}^3.C_4.C_2$, $C_{15}^3.C_4.C_2$, $C_{15}^3.D_4$, $C_{15}^3.C_4.C_2$, $C_{15}^3.D_4$, $C_{15}^3.C_2^3$, $C_{15}^3.D_4$, $C_3\times C_5^3.(C_6\times S_3).C_2$, $C_{15}^3.C_2^3$, $(C_5\times C_{15}^2).C_{12}.C_2$, $C_{15}^2:(S_3\times F_5)$

Order 21600: $(C_3\times C_{15}^2).C_4^2.C_2$

Order 20250: $C_{15}\wr S_3$, $(C_3\times C_{15}^2):D_{15}$, $C_{15}\wr C_3:C_2$

Order 18000: $C_5^3.(C_4\times S_3^2)$ x 2, $C_5^3.S_3^2:C_4$ x 2, $C_5^3.(C_6\times S_3).C_2^2$, $(C_5\times C_{15}^2).C_4^2$, $C_5^3.C_3:S_3.C_2^3$, $C_5^3.(C_3\times C_6).D_4$

Order 16200: $C_{15}^2:C_6.D_6$

Order 13500: $C_5\times C_5^2:(C_3\times C_3:S_3.C_2)$, $C_{15}^3.C_4$, $C_3\times C_5^3:(C_6\times S_3)$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_2^2$, $C_3\times C_5^3:(C_3\times C_3:C_4)$, $(C_3\times C_{15}^2).C_{20}$, $C_{15}^3.C_2^2$, $C_3\times C_5^3:(C_3^2:C_4)$, $C_{15}^3.C_2^2$, $C_{15}^3.C_4$, $(C_5\times C_{15}^2):D_6$, $(C_5\times C_{15}^2):C_{12}$, $C_5^2:C_{45}:C_{12}$, $C_{15}^2:C_{15}:C_4$

Order 12000: $C_5^3.C_6.C_4.C_2^2$, $D_5^3.D_6$

Order 10800: $(C_3\times C_{15}^2).C_4^2$, $C_{15}^2.C_6.C_2^3$, $C_{15}^2.C_6.D_4$, $C_{15}^2.C_6.D_4$, $C_{15}^2.C_6.C_2^3$, $C_{15}^2.C_6.C_2^3$, $C_{15}^2.C_2^2.C_6.C_2$, $(C_3\times C_{15}^2).C_4^2$, $C_{15}^2.C_6.C_2^3$

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

Order 9000: $C_5^3.(C_6\times S_3).C_2$ x 3, $C_5^3.S_3\wr C_2$ x 2, $C_5^3.(C_2.S_3^2)$ x 2, $C_5^3.(C_2\times S_3^2)$ x 2, $C_5^3.C_3^2:C_4.C_2$ x 2, $C_5^3.(S_3\times C_3:C_4)$, $C_5^3.(C_6\times S_3).C_2$, $C_5^3.(C_6\times S_3).C_2$, $C_5^3.C_3^2:C_4.C_2$, $C_5^3.S_3\wr C_2$, $C_5^3.C_3:S_3.C_2^2$, $C_5^3.C_6^2.C_2$, $C_5^3.C_6.C_6.C_2$, $C_3\times C_5^3:(C_4\times S_3)$, $C_5^3.C_3:S_3.C_2^2$, $C_{15}^2.(C_5\times D_4)$, $C_3\times C_5^3:(C_2\times C_3:C_4)$, $C_5^3:C_6.D_6$

Order 8100: $(C_3\times C_{15}^2):C_{12}$, $(C_3\times C_{15}^2):D_6$, $C_{15}^2:C_6.S_3$

Order 7200: $C_{15}^2:(C_4\times D_4)$

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

Order 6000: $C_5^3.C_6.D_4$, $C_5^3.C_6.D_4$, $C_3\times C_5^3:(C_2^2:C_4)$, $C_5^3.C_6.C_2^3$, $C_5^3.C_6.C_4.C_2$, $C_5^3.D_6.C_2^2$, $C_5^3.C_6.C_2^3$, $C_5^3.C_6.C_2^3$, $D_5\wr S_3$, $D_5^3.C_6$, $D_5^3.S_3$

Order 5400: $C_{15}^2.C_6.C_2^2$ x 4, $C_{15}^2.C_6.C_2^2$ x 3, $C_{15}^2.C_3:D_4$ x 2, $C_5^2.S_3^2:S_3$ x 2, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_{12}.C_2$, $C_3\times C_5^2:(S_3\times C_3:C_4)$, $C_{15}^2.D_6.C_2$, $C_3\times S_3\times C_3:(C_5:F_5)$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_6.C_2^2$, $C_3\times C_5^2:(S_3\times C_3:C_4)$, $(C_3\times C_{15}^2).C_4.C_2$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_{12}.C_2$, $C_{15}^2.C_{12}.C_2$, $C_{15}^2.C_{12}.C_2$, $(C_3\times C_{15}^2).C_4.C_2$, $C_{15}^2.C_6.C_2^2$, $C_{15}^2.C_{12}.C_2$, $C_{15}^2:(C_4\times S_3)$

Order 4500: $C_3\times C_5^3:(C_3:C_4)$ x 4, $C_5^3:(C_3^2:C_4)$ x 3, $C_5^3:(C_6\times S_3)$ x 3, $C_5^3:S_3^2$ x 2, $C_5^3:S_3^2$ x 2, $C_3\times C_5^3:D_6$ x 2, $C_5^3:S_3^2$ x 2, $C_5^3:(C_3^2:C_4)$, $C_3^2\times D_5\times C_5:D_5$, $C_5^3:(C_6\times S_3)$, $C_5\times D_5\times C_3\times D_{15}$, $C_5^3:(C_3\times C_3:C_4)$, $C_5^3:(C_3:S_3.C_2)$, $C_5^3:(C_6\times S_3)$, $C_5\times C_5^2:(C_3:S_3.C_2)$, $C_5^3:(C_6\times S_3)$, $C_5^3:(C_2\times C_3:S_3)$, $C_5^3:(C_3:S_3.C_2)$, $C_3^2\times C_5:(C_5:F_5)$, $C_3\times C_5^3:D_6$, $(C_5^2\times C_{15}):C_{12}$, $C_5^2:C_{45}:C_4$, $C_5^3:C_3^2:C_4$

Order 4320: $S_3^2:(S_3\times F_5)$

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

Order 4000: $D_5^3.C_2^2$

Order 3600: $C_{15}^2.C_2^2.C_2^2$ x 2, $D_{15}^2:C_4$ x 2, $D_{15}^2:C_4$ x 2, $C_{15}^2.C_4^2$, $C_{15}^2.C_2^2.C_2^2$, $C_{15}^2.C_4.C_2^2$, $C_5^2.(C_4\times S_3^2)$, $C_{15}^2:C_4^2$, $D_{15}^2:C_2^2$, $C_{15}^2:C_4:C_4$

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

Order 3240: $C_3^3:(S_3\times F_5)$

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

Order 2700: $C_5^2:(C_3^3:C_4)$ x 5, $C_3\times C_5^2:(C_3^2:C_4)$ x 4, $C_3\times C_5^2:(C_6\times S_3)$ x 4, $C_3\times C_5^2:(C_3\times C_3:C_4)$ x 3, $C_5^2:(C_6\times C_3:S_3)$ x 3, $C_5^2:(C_3^3:C_2^2)$ x 2, $C_5^2:(C_3\times S_3^2)$ x 2, $C_5^2:(C_3^2:(C_3:C_4))$ x 2, $C_5^2:(C_3\times S_3^2)$ x 2, $C_5^2:(C_3\times S_3^2)$, $C_5\times C_5:(C_3\times S_3^2)$, $C_3\times C_5^2:(C_3\times C_3:C_4)$, $C_5\times C_5:(C_3\times C_3:S_3.C_2)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5^2:(C_6\times C_3:S_3)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5^2:(C_3\times C_3:S_3.C_2)$, $C_5\times C_5:(C_3\times C_3:S_3.C_2)$, $C_5^2:(C_3^3:C_2^2)$, $C_3\times C_5^2:(C_3^2:C_4)$, $C_5^2:(C_3\times S_3^2)$, $C_3\times C_3\times S_3\times C_5:D_5$, $C_{15}^2:C_{12}$, $C_{15}^2:D_6$, $C_5^2:C_9:C_{12}$, $C_{15}^2:C_3:C_4$

Order 2592: $C_3^3:(C_4\times S_4)$

Order 2400: $(C_6\times C_5:D_5).C_2^3$

Order 2250: $C_5^3:(C_3\times S_3)$ x 3, $C_3^2\times C_5^2:D_5$ x 3, $C_5^3:(C_3\times S_3)$ x 2, $C_3\times C_5^3:S_3$ x 2, $C_3^2\times C_5\times C_5:D_5$, $C_5\times C_5\times C_3:D_{15}$, $C_3^2\times C_5^2\times D_5$, $C_5^3:(C_3:S_3)$, $C_{15}\times C_5\times D_{15}$, $C_5\times C_{15}\times D_{15}$, $C_3\times C_5^3:S_3$, $C_3\times C_5\wr S_3$, $C_3\times C_5^2:D_{15}$, $C_5^3:C_{18}$, $C_3\times C_5^3:C_6$

Order 2160: $D_5.S_3^3$ x 2, $(C_3^2\times C_{15}):C_4^2$, $(C_3^2\times C_{15}):C_4^2$, $C_3\times S_3^2:F_5$, $(C_3^2\times D_5).D_{12}$, $D_5.S_3^3$, $D_5\times S_3^2:S_3$, $(D_5\times C_3^3).D_4$

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

Order 2000: $D_5^2:F_5$ x 2, $D_5^2:F_5$ x 2, $D_5^3:C_2$, $(D_5\times C_5^2).Q_8$, $C_5:F_5^2$

Order 1800: $(C_{15}\times D_{15}):C_4$ x 24, $S_3\times C_{15}:F_5$ x 5, $D_{15}\wr C_2$ x 4, $C_{15}^2:(C_2\times C_4)$ x 4, $C_{15}:(S_3\times F_5)$ x 3, $D_{15}^2:C_2$ x 2, $C_{15}^2:C_2^3$ x 2, $C_{15}^2:C_2^3$ x 2, $C_{15}^2:(C_2\times C_4)$ x 2, $C_{15}:(S_3\times F_5)$ x 2, $(C_{15}\times D_{15}):C_4$ x 2, $C_{15}:(S_3\times F_5)$ x 2, $S_3\times C_{15}:F_5$ x 2, $C_3\times D_{15}:F_5$ x 2, $C_3\times D_{15}:F_5$ x 2, $(D_5\times C_{15}):C_{12}$ x 2, $S_3\times C_{15}:F_5$ x 2, $C_{15}^2:C_2^3$, $C_2\times C_{15}^2:C_4$, $C_6\times C_{15}:F_5$, $C_{15}^2:(C_2\times C_4)$, $C_{15}^2:(C_2\times C_4)$, $C_{15}:(S_3\times F_5)$, $C_{15}:F_5:S_3$

Order 1620: $C_3\wr C_3:F_5$, $(C_3^2\times C_{15}):C_{12}$, $C_3\wr S_3\times D_5$

Order 1500: $C_{15}:D_5^2$ x 4, $(C_5\times C_{15}):F_5$ x 4, $C_{15}:D_5^2$ x 3, $C_5^2:D_{30}$ x 3, $(C_5\times C_{15}):F_5$ x 3, $C_5^3:C_{12}$ x 3, $C_5^2:D_{30}$ x 2, $C_{15}:D_5^2$ x 2, $C_{15}\times D_5^2$ x 2, $C_{15}:D_5^2$, $(C_5\times C_{15}):F_5$, $C_5\times C_{15}:F_5$, $C_5^2:C_{60}$, $C_5^3:D_6$, $C_5^3:A_4$, $C_5^3:C_{12}$, $C_5\wr C_3:C_4$

Order 1440: $F_5\times \SOPlus(4,2)$

Order 1350: $C_{15}^2:C_6$ x 8, $C_{15}^2:C_6$ x 6, $C_{15}^2:C_6$ x 5, $C_{15}^2:S_3$ x 2, $C_{15}^2:C_6$, $C_{15}^2.C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$, $S_3\times C_{15}^2$

Order 1296: $C_2\times C_3^3:S_4$, $C_3^3:(C_4\times A_4)$, $(C_3^2\times C_6).S_4$

Order 1200: $D_5^2.D_6$ x 3, $D_{30}:F_5$ x 2, $D_5^2.D_6$ x 2, $C_3:F_5^2$, $(C_5\times C_{30}).Q_8$, $(C_5\times C_{30}).D_4$, $D_5^2:C_{12}$, $C_{15}:(C_4\times F_5)$, $D_5^2:D_6$

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

Order 1080: $S_3\times C_3^2:F_5$ x 5, $C_3^2:(S_3\times F_5)$ x 4, $S_3\times C_{15}:C_{12}$ x 3, $C_3\times D_5\times S_3^2$ x 2, $C_3^2:(S_3\times F_5)$ x 2, $C_{15}:(S_3\times C_{12})$ x 2, $D_5\times C_3:S_3^2$, $C_3^3:D_{20}$, $C_{15}:\SOPlus(4,2)$, $S_3^2:D_{15}$, $C_5\times S_3^2:S_3$, $C_3^3:C_4\times D_5$, $(D_5\times C_3^3):C_4$, $C_3^2:C_{12}\times D_5$, $(C_3^2\times D_5):C_{12}$, $C_3^2:(S_3\times F_5)$

Order 1000: $C_5^3:(C_2\times C_4)$ x 3, $D_5^3$ x 2, $C_5^3:D_4$ x 2, $C_5^3:(C_2\times C_4)$ x 2, $C_5^2:D_{20}$, $D_5^2:C_{10}$, $D_5.D_5^2$, $D_5\times C_5:F_5$

Order 900: $C_{15}^2:C_4$ x 32, $C_{15}^2:C_2^2$ x 24, $C_{15}^2:C_4$ x 22, $C_{15}^2:C_2^2$ x 5, $C_{15}^2:C_4$ x 5, $C_{15}:D_{30}$ x 4, $D_{15}^2$ x 4, $C_{15}:D_{30}$ x 4, $C_{15}:D_{30}$ x 4, $D_{15}:C_{30}$ x 4, $C_3^2\times D_5^2$ x 4, $C_{15}^2:C_4$ x 4, $C_{15}^2:C_2^2$ x 3, $C_{15}^2:C_4$ x 3, $C_{15}\times D_{30}$ x 2, $C_5^2:S_3^2$ x 2, $C_{15}^2:C_4$ x 2, $C_{15}^2:C_4$ x 2, $C_{15}^2:C_4$ x 2, $C_3\times C_5^2:D_6$, $(C_5\times C_{15}):C_{12}$, $C_5^2:C_3^2:C_4$, $C_5^2:C_9:C_4$, $C_5^2:C_6^2$, $C_{15}^2:C_2^2$, $C_{15}^2:C_4$, $C_{15}^2:C_4$

Order 864: $S_3^2:(C_4\times S_3)$

Order 810: $C_3^3:D_{15}$, $\He_3:C_{30}$, $C_3\wr C_3\times D_5$

Order 800: $C_5^2:(C_4\times D_4)$

Order 750: $C_5^2:C_{30}$ x 4, $C_5^2:C_{30}$ x 4, $C_5^2:D_{15}$ x 3, $C_5^2\times D_{15}$ x 3, $C_5^3:C_6$ x 3, $C_5^3:C_6$, $C_5^2:D_{15}$, $C_5\wr S_3$

Order 720: $F_5\times S_3^2$ x 4, $D_{10}.S_3^2$ x 3, $S_3^2:F_5$ x 2, $C_3^2:C_4\times F_5$ x 2, $S_3^2:D_{10}$, $D_5.\SOPlus(4,2)$

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

Order 648: $C_3^3:S_4$ x 2, $C_2\times C_3^3:A_4$, $C_3^3:(C_4\times S_3)$

Order 600: $D_5^2.S_3$ x 20, $D_{15}:F_5$ x 15, $C_{30}:F_5$ x 4, $D_5^2:S_3$ x 4, $C_{30}:F_5$ x 3, $S_3\times D_5^2$ x 3, $D_{15}:F_5$ x 3, $S_3\times C_5:F_5$ x 3, $S_3\times C_5:F_5$ x 3, $D_5^2.C_6$ x 3, $C_{30}:D_{10}$ x 2, $C_6\times D_5^2$ x 2, $C_{30}:F_5$ x 2, $C_{30}.D_{10}$, $D_5^2.S_3$, $D_5^2.C_6$, $C_5^2:(C_4\times S_3)$

Order 540: $C_{15}:C_6^2$ x 5, $C_3^3:F_5$ x 5, $C_3^3:F_5$ x 5, $C_3^3:F_5$ x 4, $C_3^3:D_{10}$ x 4, $C_{15}:S_3^2$ x 3, $C_3^3:F_5$ x 2, $\He_3:F_5$ x 2, $C_{15}:S_3^2$ x 2, $C_{15}:S_3^2$ x 2, $C_{15}\times S_3^2$ x 2, $(C_3^2\times C_{15}):C_4$, $C_3^3:C_{20}$, $(C_3\times C_{15}):C_{12}$, $C_3^3:F_5$, $C_3^2:C_{60}$, $\He_3:D_{10}$, $C_{45}:C_{12}$, $C_{15}:S_3^2$

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

Order 480: $C_{15}:(C_4\times D_4)$

Order 450: $C_{15}\times D_{15}$ x 44, $C_{15}^2:C_2$ x 32, $C_{15}:C_{30}$ x 8, $C_{15}:D_{15}$ x 6, $C_{15}:D_{15}$ x 5, $C_{15}:C_{30}$ x 5, $C_3\times C_5^2:S_3$ x 2, $C_{15}\times C_{30}$, $C_{15}:D_{15}$, $C_{15}^2:C_2$, $C_3\times C_5^2:C_6$, $C_5^2:C_{18}$

Order 432: $S_3^2:D_6$, $C_2.S_3^3$, $C_2.S_3^3$, $C_6.\SOPlus(4,2)$, $C_6.\SOPlus(4,2)$, $S_3^2:C_{12}$, $C_3^3:C_4^2$

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

Order 400: $D_{10}:F_5$ x 3, $D_5^2:C_4$ x 2, $D_5^2:C_2^2$, $F_5^2$, $(C_5\times C_{10}).Q_8$, $C_5^2:C_4^2$

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

Order 360: $D_5.S_3^2$ x 34, $D_{15}:C_{12}$ x 6, $C_2\times C_3^2:F_5$ x 5, $D_5\times S_3^2$ x 4, $D_5.S_3^2$ x 4, $C_3:S_3\times F_5$ x 4, $C_{30}:D_6$ x 3, $C_{30}:C_{12}$ x 3, $S_3^2:D_5$ x 2, $(C_3^2\times D_5):C_4$ x 2, $C_3^2:C_4\times D_5$ x 2, $C_3^2:D_{20}$, $S_3^2:C_{10}$

Order 324: $C_3^3:D_6$, $C_3\wr C_3:C_4$, $C_3^3:A_4$, $C_3^3:C_{12}$

Order 300: $C_{15}:F_5$ x 35, $C_3\times D_5^2$ x 21, $C_{15}:F_5$ x 19, $C_{15}:D_{10}$ x 16, $C_{15}:F_5$ x 12, $D_5\times D_{15}$ x 6, $C_{15}:D_{10}$ x 6, $C_{15}:D_{10}$ x 4, $C_5\times D_{30}$ x 3, $C_{15}\times D_{10}$ x 3, $C_{15}:D_{10}$ x 3, $C_{15}:F_5$ x 3, $C_{15}:F_5$, $C_{15}:C_{20}$, $C_{15}:F_5$, $C_{15}\times F_5$, $C_5^2:D_6$, $C_5^2:C_{12}$, $C_5^2:C_3:C_4$, $C_5:C_{60}$

Order 288: $C_4\times \SOPlus(4,2)$

Order 270: $C_3^2\times D_{15}$ x 5, $C_3^2:C_{30}$ x 5, $D_5\times C_3^3$ x 5, $C_3^2:D_{15}$ x 4, $C_3^2:C_{30}$ x 4, $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

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

Order 240: $D_6\times F_5$ x 5, $D_{10}.D_6$ x 2, $D_{10}.D_6$, $D_{30}:C_4$, $C_{15}:C_4^2$, $D_6:D_{10}$, $C_{60}:C_4$, $D_{10}:C_{12}$

Order 225: $C_{15}^2$ x 31, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $S_3^2:S_3$ x 4, $C_6.S_3^2$, $C_6:S_3^2$, $C_6\times S_3^2$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$

Order 200: $D_5:F_5$ x 11, $C_{10}:F_5$ x 4, $D_5\wr C_2$ x 4, $D_5\times D_{10}$ x 3, $C_{10}:F_5$ x 2, $D_5\times F_5$ x 2, $C_{10}.D_{10}$

Order 180: $C_3^2:F_5$ x 37, $C_{15}:D_6$ x 34, $C_{15}:C_{12}$ x 28, $S_3\times D_{15}$ x 6, $C_3^2\times D_{10}$ x 5, $C_{15}:D_6$ x 4, $C_5\times S_3^2$ x 4, $C_{15}:D_6$ x 4, $C_3^2\times F_5$ x 4, $C_3\times D_{30}$ x 3, $S_3\times C_{30}$ x 3, $C_3^2:F_5$ x 3, $(C_3\times C_{15}):C_4$ x 2, $C_3^2:C_{20}$ x 2, $C_{45}:C_4$

Order 162: $C_3\wr S_3$ x 2, $C_3^3:C_6$

Order 160: $D_4\times F_5$

Order 150: $C_{15}:D_5$ x 35, $D_5\times C_{15}$ x 34, $C_5\times D_{15}$ x 24, $C_5\times C_{30}$ x 4, $C_5:D_{15}$ x 3, $S_3\times C_5^2$ x 3, $C_5^2:S_3$ x 2, $C_5^2:C_6$

Order 144: $C_4\times S_3^2$ x 2, $S_3^2:C_4$ x 2, $S_3^2:C_2^2$, $D_6.D_6$, $C_3^2:C_4^2$, $C_2.\SOPlus(4,2)$

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: $C_{30}:C_4$ x 27, $S_3\times F_5$ x 25, $C_6\times F_5$ x 6, $S_3\times D_{10}$ x 5, $C_6\times D_{10}$ x 2, $C_6.D_{10}$, $C_{15}:D_4$, $C_{15}:D_4$, $D_5\times C_{12}$, $C_3:D_{20}$, $C_{15}:D_4$

Order 108: $C_3\times S_3^2$ x 3, $C_3^2:C_{12}$ x 2, $C_3:S_3^2$ x 2, $C_3^3:C_4$ x 2, $C_9:C_{12}$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:C_{12}$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:C_{12}$, $C_3^2:D_6$

Order 100: $D_5^2$ x 13, $C_5:F_5$ x 9, $C_5:F_5$ x 7, $C_5\times D_{10}$ x 4, $C_5\times F_5$ x 2, $C_5:D_{10}$ x 2, $C_5:F_5$ x 2, $C_5:C_{20}$

Order 96: $C_2^3.D_6$, $C_4\times S_4$

Order 90: $C_3^2\times D_5$ x 37, $C_3\times D_{15}$ x 28, $S_3\times C_{15}$ x 28, $C_3\times C_{30}$ x 5, $C_3:D_{15}$ x 4, $C_{15}:S_3$ x 4, $C_9\times D_5$

Order 81: $C_3\wr C_3$

Order 80: $C_2^2\times F_5$ x 3, $D_{10}:C_4$ x 2, $C_4\times F_5$ x 2, $D_4\times D_5$, $C_{20}:C_4$

Order 75: $C_5\times C_{15}$ x 31, $C_5^2:C_3$

Order 72: $C_6.D_6$ x 8, $\SOPlus(4,2)$ x 4, $S_3\times D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $S_3\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_6\times D_6$, $C_6.D_6$, $C_{12}:S_3$, $C_6:C_{12}$

Order 60: $C_{15}:C_4$ x 55, $C_3\times D_{10}$ x 26, $S_3\times D_5$ x 25, $C_3\times F_5$ x 17, $D_{30}$ x 5, $S_3\times C_{10}$ x 5, $C_2\times C_{30}$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_3:C_{20}$

Order 54: $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

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

Order 48: $C_4\times D_6$ x 2, $C_2\times S_4$, $C_6:D_4$, $C_6.C_2^3$, $C_4\times A_4$, $A_4:C_4$, $C_2^2:C_{12}$, $D_6:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 45: $C_3\times C_{15}$ x 32, $C_{45}$

Order 40: $C_2\times F_5$ x 18, $C_2\times D_{10}$ x 3, $C_5:D_4$ x 2, $C_4\times D_5$ x 2, $D_{20}$, $C_5\times D_4$

Order 36: $C_6\times S_3$ x 10, $C_3:C_{12}$ x 6, $S_3^2$ x 6, $C_3^2:C_4$ x 5, $C_3^2:C_4$ x 3, $C_3\times C_{12}$, $C_6^2$, $C_6:S_3$, $C_9:C_4$

Order 32: $C_4\times D_4$

Order 30: $C_3\times D_5$ x 53, $C_{30}$ x 21, $D_{15}$ x 16, $C_5\times S_3$ x 16

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

Order 25: $C_5^2$ x 5

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

Order 20: $F_5$ x 17, $D_{10}$ x 16, $C_2\times C_{10}$ x 3, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 18: $C_3\times S_3$ x 12, $C_3\times C_6$ x 6, $C_3:S_3$ x 2, $C_{18}$

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

Order 15: $C_{15}$ x 32

Order 12: $D_6$ x 11, $C_3:C_4$ x 10, $C_2\times C_6$ x 7, $C_{12}$ x 5, $A_4$

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

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

Order 8: $C_2\times C_4$ x 7, $D_4$ x 4, $C_2^3$ x 2

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

Order 5: $C_5$ x 5

Order 4: $C_2^2$ x 7, $C_4$ x 6

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 324000: $C_{15}^3.(C_4\times S_4)$ ($C_1$)

Order 162000: $(C_{15}^3.A_4):C_4$ ($C_2$), $C_3^3:D_5\wr S_3$ ($C_2$), $(C_{15}^3.A_4):C_4$ ($C_2$)

Order 81000: $C_3^3:D_5\wr C_3$ ($C_2^2$), $C_{15}^3.S_4$ ($C_4$), $C_{15}^3.S_4$ ($C_4$)

Order 54000: $C_{15}^3.C_4.C_2^2$ ($S_3$)

Order 40500: $C_{15}^3.A_4$ ($C_2\times C_4$)

Order 27000: $C_{15}^3.C_2^3$ ($D_6$)

Order 13500: $C_{15}^3.C_4$ ($S_4$), $C_{15}^3.C_2^2$ ($C_4\times S_3$)

Order 6750: $C_3^3\times C_5^2:D_5$ ($C_2\times S_4$)

Order 3375: $C_{15}^3$ ($C_4\times S_4$)

Order 250: $C_5^3:C_2$ ($S_3\wr S_3$)

Order 125: $C_5^3$ ($C_3^3:(C_4\times S_4)$)

Order 27: $C_3^3$ ($D_5^3.D_6$)

Order 1: $C_1$ ($C_{15}^3.(C_4\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 324000: $C_{15}^3.(C_4\times S_4)$ ($C_1$)

Order 162000: $(C_{15}^3.A_4):C_4$ ($C_2$), $C_3^3:D_5\wr S_3$ ($C_2$), $(C_{15}^3.A_4):C_4$ ($C_2$)

Order 81000: $C_3^3:D_5\wr C_3$ ($C_2^2$), $C_{15}^3.S_4$ ($C_4$), $C_{15}^3.S_4$ ($C_4$)

Order 54000: $C_{15}^3.C_4.C_2^2$ ($S_3$)

Order 40500: $C_{15}^3.A_4$ ($C_2\times C_4$)

Order 27000: $C_{15}^3.C_2^3$ ($D_6$)

Order 13500: $C_{15}^3.C_4$ ($S_4$), $C_{15}^3.C_2^2$ ($C_4\times S_3$)

Order 6750: $C_3^3\times C_5^2:D_5$ ($C_2\times S_4$)

Order 3375: $C_{15}^3$ ($C_4\times S_4$)

Order 250: $C_5^3:C_2$ ($S_3\wr S_3$)

Order 125: $C_5^3$ ($C_3^3:(C_4\times S_4)$)

Order 27: $C_3^3$ ($D_5^3.D_6$)

Order 1: $C_1$ ($C_{15}^3.(C_4\times S_4)$)

Series

Derived series

$C_{15}^3.(C_4\times S_4)$

$\rhd$

$C_{15}^3.(C_4\times S_4)$

$\rhd$

$C_{15}^3.A_4$

$\rhd$

$C_{15}^3.A_4$

$\rhd$

$C_{15}^3.C_2^2$

$\rhd$

$C_{15}^3.C_2^2$

$\rhd$

$C_{15}^3$

$\rhd$

$C_{15}^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{15}^3.(C_4\times S_4)$

$\rhd$

$C_{15}^3.(C_4\times S_4)$

$\rhd$

$C_3^3:D_5\wr S_3$

$\rhd$

$C_3^3:D_5\wr S_3$

$\rhd$

$C_3^3:D_5\wr C_3$

$\rhd$

$C_3^3:D_5\wr C_3$

$\rhd$

$C_{15}^3.A_4$

$\rhd$

$C_{15}^3.A_4$

$\rhd$

$C_{15}^3.C_2^2$

$\rhd$

$C_{15}^3.C_2^2$

$\rhd$

$C_{15}^3$

$\rhd$

$C_{15}^3$

$\rhd$

$C_5^3$

$\rhd$

$C_5^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{15}^3.(C_4\times S_4)$

$\rhd$

$C_{15}^3.(C_4\times S_4)$

$\rhd$

$C_{15}^3.A_4$

$\rhd$

$C_{15}^3.A_4$

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 $156 \times 156$ 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 $106 \times 106$ rational character table.