Abstract group 28800.bb: $(S_3\times F_5)\wr C_2$

• Label: 28800.bb
• Order: \( 2^{7} \cdot 3^{2} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $16$
• Trans deg.: $30$
• Rank: $3$

Group information

Description:	$(S_3\times F_5)\wr C_2$
Order:	\(28800\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5^{2} \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_{15}^2.C_4.C_2^3.C_2^3$, of order \(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \)
Composition factors:	$C_2$ x 7, $C_3$ x 2, $C_5$ x 2
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	60
Elements	1	695	8	6280	24	952	4800	1480	5600	192	2720	2400	2048	1600	28800
Conjugacy classes	1	10	2	29	2	10	4	10	32	5	9	2	9	10	135
Divisions	1	10	2	17	2	10	2	10	17	5	5	1	9	5	96
Autjugacy classes	1	9	2	24	2	8	4	8	22	4	7	2	6	6	105

Dimension	1	2	4	8	16	32	64
Irr. complex chars.	16	28	40	22	20	5	4	135
Irr. rational chars.	8	12	22	24	18	7	5	96

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$30$
Rank:	$3$
Inequivalent generating triples:	$134184960$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{4}=c^{4}=d^{60}=e^{15}=[a,c]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(11,12)(13,15)(14,16), (2,4,7,9)(11,13)(12,15), (1,2,3,5)(4,6)(7,8,9,10)(12,14,15)(13,16)\rangle$
Transitive group:	30T1012				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_{15}^2:Q_8)$ $\,\rtimes\,$ $C_4$	$(D_{15}^2:D_4)$ $\,\rtimes\,$ $C_4$	$S_3^2$ $\,\rtimes\,$ $(F_5\wr C_2)$	$(F_5^2:S_3^2)$ $\,\rtimes\,$ $C_2$	all 21
Trans. wreath product:	$S_3 imes F_5$ $\ \wr\ $$C_2$
Non-split product:	$(D_{15}^2:C_4)$ . $D_4$	$(S_3^2\times D_5^2)$ . $D_4$ (3)	$D_5^2$ . $(D_6\wr C_2)$	$(C_5:F_5)$ . $(S_3^2:D_4)$	all 37
Aut. group:	$\Aut(D_{15}^2)$

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

Homology

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

Subgroups

There are 166320 subgroups in 2750 conjugacy classes, 71 normal (67 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(S_3\times F_5)\wr C_2$
Commutator:	$G' \simeq$ $C_{15}:(S_3\times F_5)$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $(S_3\times F_5)\wr C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{15}^2$	$G/\operatorname{Fit} \simeq$ $C_4^2:D_4$
Radical:	$R \simeq$ $(S_3\times F_5)\wr C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{15}^2$	$G/\operatorname{soc} \simeq$ $C_4^2:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 28800: $(S_3\times F_5)\wr C_2$

Order 14400: $C_5^2.C_6^2.C_2^4$, $C_{15}^2:\OD_{16}:C_4$, $F_5^2:S_3^2$, $C_{15}^2:\OD_{16}:C_4$, $D_{15}^2:\OD_{16}$, $C_{15}^2:\OD_{16}:C_2^2$, $D_{15}^2:C_4:C_2^2$

Order 7200: $C_5^2.C_6^2.C_2^3$ x 2, $D_5^2.\SOPlus(4,2)$ x 2, $C_{15}^2.C_4^2.C_2$, $C_{15}^2.C_4^2.C_2$, $C_{15}^2.C_2^3.C_2^2$, $C_5^2.C_6^2.C_2^3$, $C_5^2.C_6^2.C_2^3$, $C_{15}^2.C_2^2.C_2^3$, $C_5^2.C_6^2.C_2^3$, $(C_5^2\times S_3^2):D_4$, $C_{15}^2:(C_2\times \OD_{16})$, $D_{15}^2:D_4$, $(C_3\times F_5)\wr C_2$, $D_5^2.\SOPlus(4,2)$, $D_{15}^2:Q_8$, $(C_5^2\times S_3^2):Q_8$, $D_{15}^2:C_8$, $C_{15}^2:C_4\wr C_2$, $C_{15}^2:(C_2\times \OD_{16})$, $D_5^2.(S_3\times D_6)$, $(C_5^2\times S_3^2):C_8$, $D_{15}^2:D_4$, $C_{15}^2:(C_4\times D_4)$, $C_{15}^2:(C_4\times D_4)$, $C_{15}^2:C_4^2:C_2$

Order 4800: $D_6\times F_5^2$, $F_5^2:D_6$

Order 3600: $C_5^2.(C_4\times S_3^2)$ x 2, $S_3\times C_5^2:(C_2\times C_3:C_4)$ x 2, $C_5^2.C_6^2.C_2^2$ x 2, $C_{15}^2.C_4^2$ x 2, $C_5^2.(C_4\times S_3^2)$ x 2, $C_{15}^2.C_4^2$ x 2, $C_5.(C_{60}\times S_3).C_2$ x 2, $D_{15}^2:C_4$ x 2, $D_{15}^2:C_4$ x 2, $C_5^2.(C_4\times S_3^2)$, $C_5.(C_{60}\times S_3).C_2$, $C_3^2\times F_5^2$, $C_{15}^2.C_2^2.C_2^2$, $C_{15}^2.C_4^2$, $C_{15}^2.C_4.C_2^2$, $C_5^2.C_6^2.C_2^2$, $C_3\times S_3\times D_5:F_5$, $C_5^2.(C_4\times S_3^2)$, $C_{15}^2.C_2^3.C_2$, $C_{15}^2.C_2^3.C_2$, $S_3^2\times D_5^2$, $C_{15}^2.C_2^3.C_2$, $C_{15}^2.C_4.C_2^2$, $C_5^2.(C_4\times S_3^2)$, $C_{15}^2.C_2^2.C_2^2$, $D_5\times F_5\times C_3:S_3$, $C_{15}^2.C_4^2$, $C_{15}^2.C_2^3.C_2$, $C_{15}^2.C_2^3.C_2$, $C_{15}^2:\OD_{16}$, $D_5^2:S_3^2$, $C_{15}^2:C_4^2$, $C_{15}:D_6:F_5$, $C_{15}^2:\OD_{16}$, $C_{15}^2:\OD_{16}$, $C_{15}^2:\OD_{16}$, $C_{15}^2:(C_2\times C_8)$, $C_{15}^2:C_4:C_4$, $C_{15}^2:C_4:C_4$, $(C_5\times S_3^2):F_5$, $C_3\times D_5^2.D_6$, $D_{15}^2:C_2^2$, $C_{15}^2:C_4:C_4$, $(C_3^2\times D_5^2):C_4$, $D_5^2.S_3^2$, $C_{15}^2:C_4:C_4$, $C_{15}^2:(C_2\times D_4)$, $C_{15}^2:D_4:C_2$, $C_{15}^2:D_4:C_2$, $C_{15}^2:(C_2\times Q_8)$, $C_{15}^2:(C_2\times C_8)$, $S_3^2:D_5^2$

Order 3200: $F_5^2:D_4$

Order 2880: $D_{10}.D_6^2$

Order 2400: $S_3\times F_5^2$ x 9, $S_3\times D_{10}\times F_5$ x 2, $C_6:F_5^2$ x 2, $C_6\times F_5^2$, $C_6:F_5^2$, $S_3\times D_{10}:F_5$, $S_3\times C_5^2:\OD_{16}$, $F_5^2:C_6$, $F_5^2:S_3$, $C_3:F_5\wr C_2$, $D_5^2.D_{12}$, $D_5^2.C_2^2\times S_3$

Order 1800: $D_{15}^2:C_2$ x 2, $D_{15}:D_{30}$ x 2, $D_{15}\wr C_2$ x 2, $C_{15}:(C_4\times D_{15})$ x 2, $(C_{15}\times D_{15}):C_4$ x 2, $C_{15}:C_4\times D_{15}$ x 2, $S_3\times C_{15}:F_5$ x 2, $(C_{15}\times D_{15}):C_4$ x 2, $C_{15}:(S_3\times F_5)$ x 2, $S_3\times C_{15}:C_{20}$ x 2, $C_3\times D_{15}:F_5$ x 2, $C_3\times F_5\times D_{15}$ x 2, $D_5\times C_{15}:C_{12}$ x 2, $S_3\times C_{15}:F_5$ x 2, $C_3\times D_{15}:F_5$ x 2, $D_{15}:C_{60}$ x 2, $C_{15}^2:C_2^3$, $C_{15}^2:C_2^3$, $C_{15}^2:C_2^3$, $C_{15}^2:C_2^3$, $C_{15}^2:C_2^3$, $C_{15}^2:C_2^3$, $C_{15}^2:Q_8$, $C_{15}^2:Q_8$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:(C_2\times C_4)$, $C_{15}^2:(C_2\times C_4)$, $C_{15}:(S_3\times F_5)$, $C_{15}:(S_3\times F_5)$, $(C_{15}\times D_{15}):C_4$, $S_3\times C_{15}:F_5$, $C_{15}:(S_3\times F_5)$, $S_3\times C_{15}:F_5$, $C_{15}:(S_3\times C_{20})$, $C_{15}:(S_3\times F_5)$, $F_5\times C_3:D_{15}$, $D_5\times C_3^2:F_5$, $C_{15}:(S_3\times F_5)$, $C_{15}:(S_3\times F_5)$, $C_{15}:(S_3\times F_5)$, $C_{15}^2:(C_2\times C_4)$, $C_{15}:(S_3\times F_5)$, $C_{15}:(S_3\times F_5)$, $C_{15}:S_3\times F_5$, $C_3\times D_{15}:F_5$, $(D_5\times C_{15}):C_{12}$, $S_3\times C_{15}:F_5$, $S_3\times C_{15}:F_5$, $C_3^2\times D_5\times F_5$, $C_3^2\times D_5:F_5$, $C_{15}^2:C_8$, $C_{15}^2:C_8$

Order 1600: $F_5^2:C_4$ x 2, $F_5^2:C_2^2$ x 2, $D_{10}^2.C_2^2$, $C_{10}^2:\OD_{16}$, $C_2^2\times F_5^2$

Order 1440: $S_3\times C_{15}:C_4^2$ x 4, $D_5.D_6^2$ x 2, $C_{60}:C_4\times S_3$ x 2, $S_3\times C_{12}\times F_5$ x 2, $C_{15}:(S_3\times C_4^2)$ x 2, $F_5\times \SOPlus(4,2)$, $C_{10}.D_6^2$, $C_{60}:(C_4\times S_3)$, $C_{12}:S_3\times F_5$

Order 1200: $D_5^2.D_6$ x 17, $D_5^2.D_6$ x 10, $C_3:F_5^2$ x 9, $C_3:F_5^2$ x 5, $C_3\times F_5^2$ x 5, $F_5\times D_{30}$ x 2, $D_5^2.D_6$ x 2, $S_3\times C_{10}:F_5$ x 2, $D_{30}:C_{20}$ x 2, $D_5^2:C_{12}$ x 2, $D_5^2.D_6$ x 2, $D_5^2.D_6$ x 2, $D_5^2:D_6$ x 2, $D_{30}:F_5$ x 2, $D_{30}:F_5$, $D_{30}:F_5$, $D_5^2.D_6$, $S_3\times C_{10}:F_5$, $S_3\times C_{10}:F_5$, $D_5^2:C_{12}$, $C_5^2:D_{12}:C_2$, $C_3\times D_5^2.C_2^2$, $S_3\times C_5^2:Q_8$, $(C_5\times C_{15}):\OD_{16}$, $(C_5\times C_{15}):\OD_{16}$, $D_5^2.C_{12}$, $S_3\times C_5^2:C_8$, $D_6\times D_5^2$

Order 1152: $C_{12}^2:D_4$

Order 960: $D_{30}:C_4^2$ x 2

Order 900: $D_{15}^2$ x 3, $C_{15}:D_{30}$ x 2, $C_5^2:S_3^2$ x 2, $C_{15}:D_{30}$ x 2, $C_{15}^2:C_2^2$ x 2, $C_{15}:D_{30}$ x 2, $D_{15}:C_{30}$ x 2, $C_{15}^2:C_4$ x 2, $C_{15}^2:C_4$ x 2, $C_{15}:C_{60}$ x 2, $C_5^2:S_3^2$, $C_{15}:D_{30}$, $C_5^2\times S_3^2$, $C_{15}^2:C_2^2$, $C_{15}:D_{30}$, $C_{15}^2:C_2^2$, $C_{15}^2:C_2^2$, $C_{15}^2:C_2^2$, $C_3^2\times D_5^2$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}:C_{60}$

Order 800: $C_2\times F_5^2$ x 7, $F_5\wr C_2$ x 4, $C_2\times C_5^2:\OD_{16}$ x 2, $C_{10}^2:C_8$, $D_5^2.D_4$, $D_5^2:Q_8$, $D_5^2:D_4$, $C_5^2:(C_4\times D_4)$, $D_5^2.(C_2\times C_4)$, $D_5^2:C_8$, $D_{10}^2.C_2$, $D_{10}^2.C_2$, $D_5^2.C_2^3$, $C_{10}^2:Q_8$, $C_{10}^2:D_4$, $D_{10}\wr C_2$, $C_{10}^2:(C_2\times C_4)$

Order 720: $F_5\times S_3^2$ x 18, $D_{10}.S_3^2$ x 4, $D_{30}:C_{12}$ x 4, $D_{10}.S_3^2$ x 4, $C_3:C_{12}\times F_5$ x 4, $D_{10}.S_3^2$ x 2, $C_6:S_3\times F_5$ x 2, $S_3^2:F_5$ x 2, $C_{12}:D_{30}$ x 2, $C_{60}:D_6$ x 2, $C_{60}:C_{12}$ x 2, $D_{30}.D_6$ x 2, $D_{30}.D_6$ x 2, $D_{10}.S_3^2$ x 2, $C_3^2:C_4\times F_5$ x 2, $D_{10}\times S_3^2$, $S_3^2:D_{10}$, $D_5.\SOPlus(4,2)$, $C_3^2:C_4\times F_5$, $C_{60}:D_6$, $C_{20}\times S_3^2$, $C_{60}:D_6$, $C_4\times C_3^2:F_5$, $C_{60}:C_{12}$, $D_{30}.D_6$, $C_5:C_4\times S_3^2$, $D_{10}.S_3^2$

Order 600: $S_3\times D_5^2$ x 10, $S_3\times C_5:F_5$ x 9, $D_{15}:F_5$ x 9, $D_5^2.S_3$ x 9, $F_5\times D_{15}$ x 9, $D_{15}:C_{20}$ x 9, $D_5^2.C_6$ x 9, $D_{15}:F_5$ x 6, $D_{15}:F_5$ x 6, $D_5^2.S_3$ x 6, $S_3\times C_5:F_5$ x 6, $D_5^2.C_6$ x 6, $S_3\times C_5:F_5$ x 5, $D_5^2:S_3$ x 3, $D_5\times D_{30}$ x 2, $C_{30}:D_{10}$ x 2, $C_{30}:F_5$ x 2, $C_{30}:C_{20}$ x 2, $C_{30}:F_5$ x 2, $F_5\times C_{30}$ x 2, $(C_5\times C_{15}):Q_8$ x 2, $C_5^2:D_{12}$ x 2, $D_5^2:C_6$ x 2, $C_{30}:D_{10}$, $C_{30}:D_{10}$, $C_6\times D_5^2$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{30}:F_5$, $C_3\times C_5^2:Q_8$, $(C_5\times C_{15}):C_8$, $C_5^2:C_{24}$

Order 576: $D_6^2.C_2^2$, $C_4^2\times S_3^2$, $(C_6\times C_{12}).D_4$, $S_3^2:\OD_{16}$, $C_4^2:S_3^2$, $C_{12}^2:C_4$, $(C_6\times C_{12}).D_4$

Order 480: $D_{15}:C_4^2$ x 17, $C_{30}:C_4^2$ x 4, $C_2\times D_6\times F_5$ x 4, $C_{60}:C_2^3$ x 2, $C_{30}:C_4^2$ x 2, $C_{30}:C_4^2$ x 2

Order 450: $C_{15}\times D_{15}$ x 3, $C_{15}:D_{15}$, $C_{15}:D_{15}$, $C_{15}^2:C_2$, $C_{15}:D_{15}$, $C_{15}:C_{30}$, $C_{15}^2:C_2$, $C_{15}:C_{30}$

Order 400: $D_{10}\times F_5$ x 12, $F_5^2$ x 10, $D_{10}:F_5$ x 9, $D_5^2.C_2^2$ x 4, $C_5^2:\OD_{16}$ x 4, $D_5^2:C_2^2$ x 3, $D_5^2:C_4$ x 3, $C_2\times C_5^2:C_8$ x 2, $(C_5\times C_{10}).Q_8$ x 2, $D_{10}^2$, $C_{10}^2:C_4$, $C_{10}^2:C_4$, $C_{10}^2:C_4$, $D_{10}:C_{20}$, $C_2\times C_5^2:Q_8$, $D_{10}:D_{10}$, $C_{10}^2:C_4$, $(C_5\times C_{10}).D_4$, $C_{10}.(C_2\times F_5)$, $D_{10}:F_5$, $C_5^2:C_4^2$

Order 360: $D_5.S_3^2$ x 18, $D_{15}:C_{12}$ x 18, $D_5\times S_3^2$ x 10, $D_5.S_3^2$ x 10, $C_3:S_3\times F_5$ x 9, $C_{30}:C_{12}$ x 4, $S_3\times C_{60}$ x 2, $C_{30}.D_6$ x 2, $C_6.D_{30}$ x 2, $C_6.D_{30}$ x 2, $C_6.D_{30}$ x 2, $C_{30}.D_6$ x 2, $D_{15}:C_{12}$ x 2, $C_{30}.D_6$ x 2, $C_{30}.D_6$ x 2, $S_3\times D_{30}$ x 2, $C_{30}:D_6$ x 2, $C_2\times C_3^2:F_5$ x 2, $C_{30}:C_{12}$ x 2, $S_3^2:D_5$ x 2, $C_{12}\times D_{15}$ x 2, $C_{60}:C_6$, $C_{30}.D_6$, $C_{30}.D_6$, $C_{30}.D_6$, $C_{30}.D_6$, $C_{30}.D_6$, $C_{30}:D_6$, $C_{10}\times S_3^2$, $C_{30}:D_6$, $C_3^2:D_{20}$, $S_3^2:C_{10}$, $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $C_{12}:D_{15}$, $C_{60}:S_3$

Order 320: $D_{10}:C_4^2$

Order 300: $D_5\times D_{15}$ x 10, $C_{15}:D_{10}$ x 10, $C_{15}:D_{10}$ x 6, $C_3\times D_5^2$ x 6, $C_{15}:D_{10}$ x 5, $C_{15}:F_5$ x 5, $C_{15}:C_{20}$ x 5, $C_{15}:F_5$ x 5, $C_{15}\times F_5$ x 5, $C_{15}:F_5$ x 4, $C_{15}:F_5$ x 4, $C_{15}:F_5$ x 3, $C_{15}:F_5$ x 3, $C_5\times D_{30}$ x 2, $C_{15}\times D_{10}$ x 2, $C_5:D_{30}$, $D_6\times C_5^2$, $C_{15}:D_{10}$

Order 288: $C_6^2.C_2^3$ x 3, $C_2.D_6^2$ x 2, $S_3^2:D_4$ x 2, $C_4\times \SOPlus(4,2)$ x 2, $S_3^2:Q_8$ x 2, $C_6^2.C_2^3$ x 2, $C_6^2.D_4$ x 2, $S_3^2:C_8$ x 2, $D_{12}:D_6$, $(C_6\times C_{12}):C_4$, $(C_6\times C_{12}).C_4$, $D_6\wr C_2$, $C_6^2.D_4$, $C_{12}^2:C_2$, $C_{12}^2:C_2$, $C_3:(S_3\times \OD_{16})$, $C_{12}\wr C_2$, $C_6^2.D_4$

Order 240: $D_6\times F_5$ x 52, $C_{15}:C_4^2$ x 18, $C_{12}:D_{10}$ x 17, $C_{60}:C_4$ x 9, $C_{12}\times F_5$ x 9, $D_{10}.D_6$ x 4, $D_{10}:C_{12}$ x 4, $D_6\times D_{10}$ x 2, $C_4\times D_{30}$ x 2, $D_6\times C_{20}$ x 2, $C_{12}\times D_{10}$ x 2, $D_{30}:C_4$ x 2, $D_6.D_{10}$ x 2, $D_{10}.D_6$ x 2

Order 225: $C_{15}^2$

Order 200: $D_5\times F_5$ x 16, $D_5:F_5$ x 12, $D_5\times D_{10}$ x 8, $C_{10}:F_5$ x 6, $C_{10}:F_5$ x 6, $C_{10}\times F_5$ x 6, $D_5\wr C_2$ x 6, $C_{10}:F_5$ x 4, $C_5^2:Q_8$ x 2, $C_5^2:C_8$ x 2, $C_{10}:D_{10}$, $C_{10}\times D_{10}$, $C_{10}\wr C_2$, $C_5:D_{20}$, $C_{10}.D_{10}$

Order 192: $C_4^2:D_6$, $D_6\times C_4^2$

Order 180: $S_3\times D_{15}$ x 10, $C_{15}:D_6$ x 10, $C_{15}:C_{12}$ x 10, $C_{15}:D_6$ x 6, $C_5\times S_3^2$ x 6, $C_{15}:D_6$ x 5, $C_3^2:F_5$ x 5, $C_3^2\times F_5$ x 5, $C_3\times D_{30}$ x 2, $S_3\times C_{30}$ x 2, $C_{15}:C_{12}$ x 2, $C_3:C_{60}$ x 2, $C_3:D_{30}$, $C_{30}:S_3$, $C_3^2\times D_{10}$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$, $C_3\times C_{60}$, $C_6.D_{15}$, $C_3^2:C_{20}$, $C_{15}:C_{12}$

Order 160: $C_{10}:C_4^2$ x 12, $C_2^3\times F_5$ x 2, $C_{20}:C_2^3$, $D_4\times F_5$

Order 150: $D_5\times C_{15}$ x 6, $C_5\times D_{15}$ x 6, $C_{15}:D_5$ x 3, $C_5:D_{15}$ x 3, $S_3\times C_5^2$ x 3, $C_5\times C_{30}$

Order 144: $C_4\times S_3^2$ x 15, $C_3^2:C_4^2$ x 4, $S_3^2:C_4$ x 4, $C_{12}:C_{12}$ x 3, $C_{12}\times D_6$ x 3, $D_6.D_6$ x 3, $C_3^2:C_4^2$ x 2, $S_3^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $C_2^2.S_3^2$ x 2, $(C_3\times C_{12}):C_4$ x 2, $C_2.\SOPlus(4,2)$ x 2, $C_{12}.C_{12}$, $C_{12}.D_6$, $C_{12}.D_6$, $D_6^2$, $D_{12}:C_6$, $D_6:D_6$, $D_6.D_6$, $C_{12}:D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $D_{12}:S_3$, $C_6^2:C_4$, $C_3^2:\OD_{16}$, $C_3^2:C_4^2$, $C_3^2:\OD_{16}$, $C_3^2:(C_2\times C_8)$, $C_{12}^2$

Order 128: $C_4^2:D_4$

Order 120: $S_3\times F_5$ x 80, $C_{30}:C_4$ x 28, $C_6\times F_5$ x 28, $S_3\times D_{10}$ x 27, $C_6.D_{10}$ x 9, $C_6.D_{10}$ x 9, $C_4\times D_{15}$ x 9, $S_3\times C_{20}$ x 9, $D_5\times C_{12}$ x 9, $D_{15}:C_4$ x 9, $C_2\times D_{30}$ x 2, $C_{10}\times D_6$ x 2, $C_6\times D_{10}$ x 2, $C_2\times C_{60}$ x 2, $C_{30}:C_4$ x 2, $C_6:C_{20}$ x 2, $C_{10}:C_{12}$ x 2

Order 100: $D_5^2$ x 12, $C_5\times D_{10}$ x 7, $C_5:F_5$ x 5, $C_5\times F_5$ x 4, $C_5:D_{10}$ x 4, $C_5:F_5$ x 4, $C_5:F_5$ x 3, $C_5:C_{20}$, $C_{10}^2$

Order 96: $S_3\times C_4^2$ x 9, $C_{12}:C_2^3$ x 3, $C_6:C_4^2$ x 3, $D_4:C_{12}$, $C_{12}.D_4$, $D_{12}:C_4$, $D_4:D_6$, $C_2\times C_4\times C_{12}$, $D_{12}:C_4$, $S_3\times \OD_{16}$

Order 90: $C_3\times D_{15}$ x 6, $S_3\times C_{15}$ x 6, $C_3:D_{15}$ x 3, $C_{15}:S_3$ x 3, $C_3^2\times D_5$ x 3, $C_3\times C_{30}$

Order 80: $C_2^2\times F_5$ x 30, $C_4\times F_5$ x 17, $C_4\times D_{10}$ x 12, $D_{10}:C_4$ x 2, $C_2^2\times D_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$, $D_4\times D_5$, $C_{20}:C_4$

Order 75: $C_5\times C_{15}$ x 2

Order 72: $S_3\times C_{12}$ x 13, $C_6.D_6$ x 13, $S_3\times D_6$ x 8, $C_{12}:S_3$ x 8, $C_6.D_6$ x 8, $\SOPlus(4,2)$ x 4, $C_6:C_{12}$ x 3, $C_2\times C_3^2:C_4$ x 2, $C_6\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_6:D_6$, $C_6\times D_6$, $C_6\wr C_2$, $C_3\times D_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3:D_{12}$, $D_6:S_3$, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $C_4^2:C_4$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2:\OD_{16}$, $C_4^2:C_2^2$, $C_2^2\times C_4^2$

Order 60: $S_3\times D_5$ x 44, $C_{15}:C_4$ x 24, $C_3\times F_5$ x 24, $D_{30}$ x 15, $S_3\times C_{10}$ x 15, $C_3\times D_{10}$ x 15, $C_{60}$ x 5, $C_{15}:C_4$ x 5, $C_5:C_{12}$ x 5, $C_3:C_{20}$ x 5, $C_2\times C_{30}$ x 2

Order 50: $C_5\times D_5$ x 5, $C_5:D_5$ x 3, $C_5\times C_{10}$ x 2

Order 48: $C_4\times D_6$ x 39, $C_{12}:C_4$ x 14, $C_4\times C_{12}$ x 5, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3, $D_4:S_3$ x 2, $S_3\times D_4$ x 2, $D_{12}:C_2$ x 2, $C_2^2\times D_6$, $C_{24}:C_2$, $D_4:C_6$, $D_{12}:C_2$, $S_3\times Q_8$, $S_3\times C_8$, $C_3\times \OD_{16}$, $C_3:\OD_{16}$

Order 45: $C_3\times C_{15}$ x 2

Order 40: $C_2\times F_5$ x 64, $C_4\times D_5$ x 17, $C_2\times D_{10}$ x 16, $C_2\times C_{20}$ x 6, $C_{10}:C_4$ x 6, $C_5:D_4$ x 2, $D_{20}$, $C_2^2\times C_{10}$, $C_5\times D_4$

Order 36: $S_3^2$ x 12, $C_3:C_{12}$ x 7, $C_6\times S_3$ x 7, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4, $C_6:S_3$ x 4, $C_3^2:C_4$ x 2, $C_6^2$

Order 32: $C_2\times C_4^2$ x 7, $C_4\wr C_2$ x 4, $C_2^2:C_8$ x 2, $C_2^3\times C_4$ x 2, $C_2\times \OD_{16}$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_4\times D_4$ x 2, $D_4:C_2^2$, $C_2^2.D_4$, $C_2^2\wr C_2$, $C_4^2:C_2$

Order 30: $D_{15}$ x 14, $C_3\times D_5$ x 14, $C_5\times S_3$ x 14, $C_{30}$ x 9

Order 25: $C_5^2$

Order 24: $C_4\times S_3$ x 59, $C_2\times C_{12}$ x 21, $C_6:C_4$ x 21, $C_2\times D_6$ x 16, $C_3:D_4$ x 3, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_3\times D_4$ x 2, $C_{24}$, $C_2^2\times C_6$, $C_3\times Q_8$, $C_3:C_8$

Order 20: $D_{10}$ x 35, $F_5$ x 21, $C_2\times C_{10}$ x 9, $C_{20}$ x 5, $C_5:C_4$ x 5

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

Order 16: $C_2^2\times C_4$ x 25, $C_4^2$ x 11, $C_2^2:C_4$ x 5, $\OD_{16}$ x 4, $C_4:C_4$ x 4, $D_4:C_2$ x 4, $C_2\times D_4$ x 4, $C_2\times C_8$ x 2, $C_2^4$, $C_2\times Q_8$

Order 15: $C_{15}$ x 5

Order 12: $D_6$ x 35, $C_{12}$ x 17, $C_3:C_4$ x 17, $C_2\times C_6$ x 9

Order 10: $D_5$ x 12, $C_{10}$ x 10

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 51, $C_2^3$ x 10, $D_4$ x 8, $Q_8$ x 2, $C_8$ x 2

Order 6: $S_3$ x 12, $C_6$ x 10

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 24, $C_4$ x 17

Order 3: $C_3$ x 2

Order 2: $C_2$ x 10

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 28800: $(S_3\times F_5)\wr C_2$

Order 14400: $C_5^2.C_6^2.C_2^4$, $C_{15}^2:\OD_{16}:C_4$, $F_5^2:S_3^2$, $C_{15}^2:\OD_{16}:C_4$, $D_{15}^2:\OD_{16}$, $C_{15}^2:\OD_{16}:C_2^2$, $D_{15}^2:C_4:C_2^2$

Order 7200: $C_{15}^2.C_4^2.C_2$, $C_{15}^2.C_4^2.C_2$, $C_{15}^2.C_2^3.C_2^2$, $C_5^2.C_6^2.C_2^3$, $C_5^2.C_6^2.C_2^3$, $C_5^2.C_6^2.C_2^3$, $C_{15}^2.C_2^2.C_2^3$, $C_5^2.C_6^2.C_2^3$, $(C_5^2\times S_3^2):D_4$, $C_{15}^2:(C_2\times \OD_{16})$, $D_5^2.\SOPlus(4,2)$, $D_{15}^2:D_4$, $(C_3\times F_5)\wr C_2$, $D_5^2.\SOPlus(4,2)$, $D_{15}^2:Q_8$, $(C_5^2\times S_3^2):Q_8$, $D_{15}^2:C_8$, $C_{15}^2:C_4\wr C_2$, $C_{15}^2:(C_2\times \OD_{16})$, $D_5^2.(S_3\times D_6)$, $(C_5^2\times S_3^2):C_8$, $D_{15}^2:D_4$, $C_{15}^2:(C_4\times D_4)$, $C_{15}^2:(C_4\times D_4)$, $C_{15}^2:C_4^2:C_2$

Order 4800: $D_6\times F_5^2$, $F_5^2:D_6$

Order 3600: $D_{15}^2:C_4$ x 2, $D_{15}^2:C_4$ x 2, $C_5^2.(C_4\times S_3^2)$, $C_5.(C_{60}\times S_3).C_2$, $C_3^2\times F_5^2$, $C_{15}^2.C_2^2.C_2^2$, $C_{15}^2.C_4^2$, $C_{15}^2.C_4.C_2^2$, $C_5^2.C_6^2.C_2^2$, $C_3\times S_3\times D_5:F_5$, $C_5^2.(C_4\times S_3^2)$, $C_{15}^2.C_2^3.C_2$, $C_{15}^2.C_2^3.C_2$, $S_3^2\times D_5^2$, $C_{15}^2.C_2^3.C_2$, $C_{15}^2.C_4.C_2^2$, $C_5^2.(C_4\times S_3^2)$, $C_5^2.(C_4\times S_3^2)$, $S_3\times C_5^2:(C_2\times C_3:C_4)$, $C_5^2.C_6^2.C_2^2$, $C_{15}^2.C_4^2$, $C_{15}^2.C_2^2.C_2^2$, $C_5^2.(C_4\times S_3^2)$, $D_5\times F_5\times C_3:S_3$, $C_{15}^2.C_4^2$, $C_{15}^2.C_4^2$, $C_5.(C_{60}\times S_3).C_2$, $C_{15}^2.C_2^3.C_2$, $C_{15}^2.C_2^3.C_2$, $C_{15}^2:\OD_{16}$, $D_5^2:S_3^2$, $C_{15}^2:C_4^2$, $C_{15}:D_6:F_5$, $C_{15}^2:\OD_{16}$, $C_{15}^2:\OD_{16}$, $C_{15}^2:\OD_{16}$, $C_{15}^2:(C_2\times C_8)$, $C_{15}^2:C_4:C_4$, $C_{15}^2:C_4:C_4$, $(C_5\times S_3^2):F_5$, $C_3\times D_5^2.D_6$, $D_{15}^2:C_2^2$, $C_{15}^2:C_4:C_4$, $(C_3^2\times D_5^2):C_4$, $D_5^2.S_3^2$, $C_{15}^2:C_4:C_4$, $C_{15}^2:(C_2\times D_4)$, $C_{15}^2:D_4:C_2$, $C_{15}^2:D_4:C_2$, $C_{15}^2:(C_2\times Q_8)$, $C_{15}^2:(C_2\times C_8)$, $S_3^2:D_5^2$

Order 3200: $F_5^2:D_4$

Order 2880: $D_{10}.D_6^2$

Order 2400: $S_3\times F_5^2$ x 7, $S_3\times D_{10}\times F_5$, $C_6\times F_5^2$, $C_6:F_5^2$, $C_6:F_5^2$, $S_3\times D_{10}:F_5$, $S_3\times C_5^2:\OD_{16}$, $F_5^2:C_6$, $F_5^2:S_3$, $C_3:F_5\wr C_2$, $D_5^2.D_{12}$, $D_5^2.C_2^2\times S_3$

Order 1800: $D_{15}^2:C_2$ x 2, $(C_{15}\times D_{15}):C_4$ x 2, $C_{15}:(S_3\times F_5)$ x 2, $C_3\times D_{15}:F_5$ x 2, $C_{15}^2:C_2^3$, $C_{15}^2:C_2^3$, $D_{15}:D_{30}$, $C_{15}^2:C_2^3$, $C_{15}^2:C_2^3$, $C_{15}^2:C_2^3$, $C_{15}^2:C_2^3$, $C_{15}^2:Q_8$, $C_{15}^2:Q_8$, $D_{15}\wr C_2$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:D_4$, $C_{15}^2:(C_2\times C_4)$, $C_{15}^2:(C_2\times C_4)$, $C_{15}:(C_4\times D_{15})$, $(C_{15}\times D_{15}):C_4$, $C_{15}:C_4\times D_{15}$, $C_{15}:(S_3\times F_5)$, $S_3\times C_{15}:F_5$, $C_{15}:(S_3\times F_5)$, $(C_{15}\times D_{15}):C_4$, $S_3\times C_{15}:F_5$, $C_{15}:(S_3\times F_5)$, $S_3\times C_{15}:F_5$, $C_{15}:(S_3\times C_{20})$, $S_3\times C_{15}:C_{20}$, $C_{15}:(S_3\times F_5)$, $F_5\times C_3:D_{15}$, $D_5\times C_3^2:F_5$, $C_{15}:(S_3\times F_5)$, $C_{15}:(S_3\times F_5)$, $C_{15}:(S_3\times F_5)$, $C_{15}^2:(C_2\times C_4)$, $C_{15}:(S_3\times F_5)$, $C_{15}:(S_3\times F_5)$, $C_{15}:S_3\times F_5$, $C_3\times D_{15}:F_5$, $C_3\times F_5\times D_{15}$, $D_5\times C_{15}:C_{12}$, $S_3\times C_{15}:F_5$, $C_3\times D_{15}:F_5$, $(D_5\times C_{15}):C_{12}$, $S_3\times C_{15}:F_5$, $S_3\times C_{15}:F_5$, $D_{15}:C_{60}$, $C_3^2\times D_5\times F_5$, $C_3^2\times D_5:F_5$, $C_{15}^2:C_8$, $C_{15}^2:C_8$

Order 1600: $F_5^2:C_4$ x 2, $F_5^2:C_2^2$ x 2, $D_{10}^2.C_2^2$, $C_{10}^2:\OD_{16}$, $C_2^2\times F_5^2$

Order 1440: $D_5.D_6^2$ x 2, $C_{15}:(S_3\times C_4^2)$ x 2, $S_3\times C_{15}:C_4^2$ x 2, $F_5\times \SOPlus(4,2)$, $C_{10}.D_6^2$, $C_{60}:(C_4\times S_3)$, $C_{60}:C_4\times S_3$, $C_{12}:S_3\times F_5$, $S_3\times C_{12}\times F_5$

Order 1200: $D_5^2.D_6$ x 10, $D_5^2.D_6$ x 9, $C_3:F_5^2$ x 5, $C_3:F_5^2$ x 4, $C_3\times F_5^2$ x 4, $D_5^2.D_6$ x 2, $D_5^2.D_6$ x 2, $D_5^2:D_6$ x 2, $F_5\times D_{30}$, $D_5^2.D_6$, $S_3\times C_{10}:F_5$, $D_{30}:F_5$, $D_{30}:F_5$, $D_5^2.D_6$, $S_3\times C_{10}:F_5$, $S_3\times C_{10}:F_5$, $D_{30}:C_{20}$, $D_5^2:C_{12}$, $D_5^2:C_{12}$, $C_5^2:D_{12}:C_2$, $C_3\times D_5^2.C_2^2$, $S_3\times C_5^2:Q_8$, $(C_5\times C_{15}):\OD_{16}$, $(C_5\times C_{15}):\OD_{16}$, $D_5^2.C_{12}$, $S_3\times C_5^2:C_8$, $D_6\times D_5^2$, $D_{30}:F_5$

Order 1152: $C_{12}^2:D_4$

Order 960: $D_{30}:C_4^2$

Order 900: $D_{15}^2$ x 2, $C_5^2:S_3^2$ x 2, $C_{15}^2:C_2^2$ x 2, $C_{15}^2:C_4$ x 2, $C_{15}:D_{30}$, $C_5^2:S_3^2$, $C_{15}:D_{30}$, $C_{15}:D_{30}$, $C_5^2\times S_3^2$, $C_{15}^2:C_2^2$, $C_{15}:D_{30}$, $C_{15}^2:C_2^2$, $C_{15}^2:C_2^2$, $C_{15}:D_{30}$, $C_{15}^2:C_2^2$, $D_{15}:C_{30}$, $C_3^2\times D_5^2$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}:C_{60}$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}^2:C_4$, $C_{15}:C_{60}$

Order 800: $C_2\times F_5^2$ x 6, $F_5\wr C_2$ x 3, $C_2\times C_5^2:\OD_{16}$ x 2, $C_{10}^2:C_8$, $D_5^2.D_4$, $D_5^2:Q_8$, $D_5^2:D_4$, $C_5^2:(C_4\times D_4)$, $D_5^2.(C_2\times C_4)$, $D_5^2:C_8$, $D_{10}^2.C_2$, $D_{10}^2.C_2$, $D_5^2.C_2^3$, $C_{10}^2:Q_8$, $C_{10}^2:D_4$, $D_{10}\wr C_2$, $C_{10}^2:(C_2\times C_4)$

Order 720: $F_5\times S_3^2$ x 14, $D_{10}.S_3^2$ x 2, $D_{10}.S_3^2$ x 2, $C_6:S_3\times F_5$ x 2, $D_{30}:C_{12}$ x 2, $S_3^2:F_5$ x 2, $D_{10}.S_3^2$ x 2, $C_3^2:C_4\times F_5$ x 2, $C_3:C_{12}\times F_5$ x 2, $D_{10}\times S_3^2$, $S_3^2:D_{10}$, $D_5.\SOPlus(4,2)$, $C_3^2:C_4\times F_5$, $C_{60}:D_6$, $C_{12}:D_{30}$, $C_{20}\times S_3^2$, $C_{60}:D_6$, $C_{60}:D_6$, $C_4\times C_3^2:F_5$, $C_{60}:C_{12}$, $C_{60}:C_{12}$, $D_{30}.D_6$, $D_{30}.D_6$, $C_5:C_4\times S_3^2$, $D_{10}.S_3^2$, $D_{30}.D_6$, $D_{10}.S_3^2$

Order 600: $S_3\times D_5^2$ x 8, $D_{15}:F_5$ x 6, $D_{15}:F_5$ x 6, $D_5^2.S_3$ x 6, $S_3\times C_5:F_5$ x 6, $D_5^2.C_6$ x 6, $S_3\times C_5:F_5$ x 5, $D_{15}:F_5$ x 5, $D_5^2.S_3$ x 5, $F_5\times D_{15}$ x 5, $S_3\times C_5:F_5$ x 5, $D_{15}:C_{20}$ x 5, $D_5^2.C_6$ x 5, $D_5^2:S_3$ x 3, $(C_5\times C_{15}):Q_8$ x 2, $C_5^2:D_{12}$ x 2, $D_5^2:C_6$ x 2, $C_{30}:D_{10}$, $D_5\times D_{30}$, $C_{30}:D_{10}$, $C_{30}:D_{10}$, $C_6\times D_5^2$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{30}:C_{20}$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{30}:F_5$, $F_5\times C_{30}$, $C_3\times C_5^2:Q_8$, $(C_5\times C_{15}):C_8$, $C_5^2:C_{24}$

Order 576: $D_6^2.C_2^2$, $C_4^2\times S_3^2$, $(C_6\times C_{12}).D_4$, $S_3^2:\OD_{16}$, $C_4^2:S_3^2$, $C_{12}^2:C_4$, $(C_6\times C_{12}).D_4$

Order 480: $D_{15}:C_4^2$ x 9, $C_{30}:C_4^2$ x 2, $C_2\times D_6\times F_5$ x 2, $C_{60}:C_2^3$, $C_{30}:C_4^2$, $C_{30}:C_4^2$

Order 450: $C_{15}\times D_{15}$ x 2, $C_{15}:D_{15}$, $C_{15}:D_{15}$, $C_{15}^2:C_2$, $C_{15}:D_{15}$, $C_{15}:C_{30}$, $C_{15}^2:C_2$, $C_{15}:C_{30}$

Order 400: $D_{10}:F_5$ x 9, $D_{10}\times F_5$ x 8, $F_5^2$ x 7, $D_5^2.C_2^2$ x 4, $C_5^2:\OD_{16}$ x 4, $D_5^2:C_2^2$ x 3, $D_5^2:C_4$ x 3, $C_2\times C_5^2:C_8$ x 2, $(C_5\times C_{10}).Q_8$ x 2, $D_{10}^2$, $C_{10}^2:C_4$, $C_{10}^2:C_4$, $C_{10}^2:C_4$, $D_{10}:C_{20}$, $C_2\times C_5^2:Q_8$, $D_{10}:D_{10}$, $C_{10}^2:C_4$, $(C_5\times C_{10}).D_4$, $C_{10}.(C_2\times F_5)$, $D_{10}:F_5$, $C_5^2:C_4^2$

Order 360: $D_5.S_3^2$ x 10, $D_{15}:C_{12}$ x 10, $C_3:S_3\times F_5$ x 9, $D_5\times S_3^2$ x 8, $D_5.S_3^2$ x 8, $C_2\times C_3^2:F_5$ x 2, $C_{30}:C_{12}$ x 2, $C_{30}:C_{12}$ x 2, $S_3^2:D_5$ x 2, $S_3\times C_{60}$, $C_{60}:C_6$, $C_{30}.D_6$, $C_{30}.D_6$, $C_6.D_{30}$, $C_6.D_{30}$, $C_6.D_{30}$, $C_{30}.D_6$, $C_{30}.D_6$, $C_{30}.D_6$, $C_{30}.D_6$, $C_{30}.D_6$, $D_{15}:C_{12}$, $C_{30}.D_6$, $C_{30}.D_6$, $C_{30}:D_6$, $S_3\times D_{30}$, $C_{10}\times S_3^2$, $C_{30}:D_6$, $C_{30}:D_6$, $C_3^2:D_{20}$, $S_3^2:C_{10}$, $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $C_{12}:D_{15}$, $C_{60}:S_3$, $C_{12}\times D_{15}$

Order 320: $D_{10}:C_4^2$

Order 300: $D_5\times D_{15}$ x 6, $C_{15}:D_{10}$ x 6, $C_{15}:D_{10}$ x 5, $C_{15}:D_{10}$ x 5, $C_3\times D_5^2$ x 5, $C_{15}:F_5$ x 4, $C_{15}:F_5$ x 4, $C_{15}:F_5$ x 3, $C_{15}:F_5$ x 3, $C_{15}:C_{20}$ x 3, $C_{15}:F_5$ x 3, $C_{15}:F_5$ x 3, $C_{15}\times F_5$ x 3, $C_5:D_{30}$, $C_5\times D_{30}$, $D_6\times C_5^2$, $C_{15}:D_{10}$, $C_{15}\times D_{10}$

Order 288: $C_2.D_6^2$ x 2, $S_3^2:D_4$ x 2, $C_4\times \SOPlus(4,2)$ x 2, $S_3^2:Q_8$ x 2, $C_6^2.C_2^3$ x 2, $C_6^2.C_2^3$ x 2, $S_3^2:C_8$ x 2, $D_{12}:D_6$, $(C_6\times C_{12}):C_4$, $(C_6\times C_{12}).C_4$, $D_6\wr C_2$, $C_6^2.D_4$, $C_{12}^2:C_2$, $C_{12}^2:C_2$, $C_3:(S_3\times \OD_{16})$, $C_6^2.D_4$, $C_{12}\wr C_2$, $C_6^2.D_4$

Order 240: $D_6\times F_5$ x 28, $C_{15}:C_4^2$ x 10, $C_{12}:D_{10}$ x 9, $C_{60}:C_4$ x 5, $C_{12}\times F_5$ x 5, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2, $D_6\times D_{10}$, $C_4\times D_{30}$, $D_6\times C_{20}$, $C_{12}\times D_{10}$, $D_{30}:C_4$, $D_6.D_{10}$, $D_{10}.D_6$

Order 225: $C_{15}^2$

Order 200: $D_5:F_5$ x 12, $D_5\times F_5$ x 10, $D_5\times D_{10}$ x 7, $C_{10}:F_5$ x 6, $D_5\wr C_2$ x 5, $C_{10}:F_5$ x 4, $C_{10}:F_5$ x 4, $C_{10}\times F_5$ x 4, $C_5^2:Q_8$ x 2, $C_5^2:C_8$ x 2, $C_{10}:D_{10}$, $C_{10}\times D_{10}$, $C_{10}\wr C_2$, $C_5:D_{20}$, $C_{10}.D_{10}$

Order 192: $C_4^2:D_6$, $D_6\times C_4^2$

Order 180: $S_3\times D_{15}$ x 6, $C_{15}:D_6$ x 6, $C_{15}:C_{12}$ x 6, $C_{15}:D_6$ x 5, $C_5\times S_3^2$ x 5, $C_{15}:D_6$ x 5, $C_3^2:F_5$ x 5, $C_3^2\times F_5$ x 5, $C_3:D_{30}$, $C_{30}:S_3$, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$, $C_3\times C_{60}$, $C_6.D_{15}$, $C_3^2:C_{20}$, $C_{15}:C_{12}$, $C_3:C_{60}$, $C_{15}:C_{12}$

Order 160: $C_{10}:C_4^2$ x 8, $C_2^3\times F_5$ x 2, $C_{20}:C_2^3$, $D_4\times F_5$

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

Order 144: $C_4\times S_3^2$ x 13, $S_3^2:C_4$ x 4, $C_3^2:C_4^2$ x 2, $C_{12}:C_{12}$ x 2, $C_3^2:C_4^2$ x 2, $S_3^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $C_{12}\times D_6$ x 2, $C_2^2.S_3^2$ x 2, $D_6.D_6$ x 2, $(C_3\times C_{12}):C_4$ x 2, $C_2.\SOPlus(4,2)$ x 2, $C_{12}.C_{12}$, $C_{12}.D_6$, $C_{12}.D_6$, $D_6^2$, $D_{12}:C_6$, $D_6:D_6$, $D_6.D_6$, $C_{12}:D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $D_{12}:S_3$, $C_6^2:C_4$, $C_3^2:\OD_{16}$, $C_3^2:C_4^2$, $C_3^2:\OD_{16}$, $C_3^2:(C_2\times C_8)$, $C_{12}^2$

Order 128: $C_4^2:D_4$

Order 120: $S_3\times F_5$ x 48, $C_{30}:C_4$ x 16, $C_6\times F_5$ x 16, $S_3\times D_{10}$ x 15, $C_6.D_{10}$ x 5, $C_6.D_{10}$ x 5, $C_4\times D_{15}$ x 5, $S_3\times C_{20}$ x 5, $D_5\times C_{12}$ x 5, $D_{15}:C_4$ x 5, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_2\times C_{60}$, $C_{30}:C_4$, $C_6:C_{20}$, $C_{10}:C_{12}$

Order 100: $D_5^2$ x 9, $C_5\times D_{10}$ x 5, $C_5:F_5$ x 5, $C_5:D_{10}$ x 4, $C_5\times F_5$ x 3, $C_5:F_5$ x 3, $C_5:F_5$ x 3, $C_5:C_{20}$, $C_{10}^2$

Order 96: $S_3\times C_4^2$ x 7, $C_{12}:C_2^3$ x 2, $C_6:C_4^2$ x 2, $D_4:C_{12}$, $C_{12}.D_4$, $D_{12}:C_4$, $D_4:D_6$, $C_2\times C_4\times C_{12}$, $D_{12}:C_4$, $S_3\times \OD_{16}$

Order 90: $C_3\times D_{15}$ x 4, $S_3\times C_{15}$ x 4, $C_3:D_{15}$ x 3, $C_{15}:S_3$ x 3, $C_3^2\times D_5$ x 3, $C_3\times C_{30}$

Order 80: $C_2^2\times F_5$ x 22, $C_4\times F_5$ x 11, $C_4\times D_{10}$ x 8, $D_{10}:C_4$ x 2, $C_2^2\times D_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$, $D_4\times D_5$, $C_{20}:C_4$

Order 75: $C_5\times C_{15}$ x 2

Order 72: $S_3\times C_{12}$ x 9, $C_6.D_6$ x 9, $C_{12}:S_3$ x 8, $S_3\times D_6$ x 7, $C_6.D_6$ x 7, $\SOPlus(4,2)$ x 3, $C_2\times C_3^2:C_4$ x 2, $C_6\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_6:C_{12}$ x 2, $C_6:D_6$, $C_6\times D_6$, $C_6\wr C_2$, $C_3\times D_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3:D_{12}$, $D_6:S_3$, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $C_4^2:C_4$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2:\OD_{16}$, $C_4^2:C_2^2$, $C_2^2\times C_4^2$

Order 60: $S_3\times D_5$ x 28, $C_{15}:C_4$ x 16, $C_3\times F_5$ x 16, $D_{30}$ x 9, $S_3\times C_{10}$ x 9, $C_3\times D_{10}$ x 9, $C_{60}$ x 3, $C_{15}:C_4$ x 3, $C_5:C_{12}$ x 3, $C_3:C_{20}$ x 3, $C_2\times C_{30}$

Order 50: $C_5\times D_5$ x 4, $C_5:D_5$ x 3, $C_5\times C_{10}$ x 2

Order 48: $C_4\times D_6$ x 27, $C_{12}:C_4$ x 9, $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 2, $C_6.C_2^3$ x 2, $D_4:S_3$ x 2, $S_3\times D_4$ x 2, $D_{12}:C_2$ x 2, $C_2^2\times D_6$, $C_{24}:C_2$, $D_4:C_6$, $D_{12}:C_2$, $S_3\times Q_8$, $S_3\times C_8$, $C_3\times \OD_{16}$, $C_3:\OD_{16}$

Order 45: $C_3\times C_{15}$ x 2

Order 40: $C_2\times F_5$ x 44, $C_2\times D_{10}$ x 12, $C_4\times D_5$ x 11, $C_2\times C_{20}$ x 4, $C_{10}:C_4$ x 4, $C_5:D_4$ x 2, $D_{20}$, $C_2^2\times C_{10}$, $C_5\times D_4$

Order 36: $S_3^2$ x 9, $C_3:C_{12}$ x 5, $C_6\times S_3$ x 5, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4, $C_6:S_3$ x 4, $C_3^2:C_4$ x 2, $C_6^2$

Order 32: $C_2\times C_4^2$ x 6, $C_4\wr C_2$ x 3, $C_2^2:C_8$ x 2, $C_2^3\times C_4$ x 2, $C_2\times \OD_{16}$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_4\times D_4$ x 2, $D_4:C_2^2$, $C_2^2.D_4$, $C_2^2\wr C_2$, $C_4^2:C_2$

Order 30: $D_{15}$ x 10, $C_3\times D_5$ x 10, $C_5\times S_3$ x 10, $C_{30}$ x 6

Order 25: $C_5^2$

Order 24: $C_4\times S_3$ x 43, $C_2\times C_{12}$ x 15, $C_6:C_4$ x 15, $C_2\times D_6$ x 12, $C_3:D_4$ x 3, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_3\times D_4$ x 2, $C_{24}$, $C_2^2\times C_6$, $C_3\times Q_8$, $C_3:C_8$

Order 20: $D_{10}$ x 25, $F_5$ x 17, $C_2\times C_{10}$ x 7, $C_{20}$ x 4, $C_5:C_4$ x 4

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

Order 16: $C_2^2\times C_4$ x 21, $C_4^2$ x 8, $C_2^2:C_4$ x 5, $\OD_{16}$ x 4, $C_4:C_4$ x 4, $D_4:C_2$ x 4, $C_2\times D_4$ x 4, $C_2\times C_8$ x 2, $C_2^4$, $C_2\times Q_8$

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 25, $C_{12}$ x 13, $C_3:C_4$ x 13, $C_2\times C_6$ x 7

Order 10: $D_5$ x 10, $C_{10}$ x 8

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 41, $C_2^3$ x 9, $D_4$ x 7, $Q_8$ x 2, $C_8$ x 2

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

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 19, $C_4$ x 15

Order 3: $C_3$ x 2

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 28800: $(S_3\times F_5)\wr C_2$ ($C_1$)

Order 14400: $C_5^2.C_6^2.C_2^4$ ($C_2$), $C_{15}^2:\OD_{16}:C_4$ ($C_2$), $F_5^2:S_3^2$ ($C_2$), $C_{15}^2:\OD_{16}:C_4$ ($C_2$), $D_{15}^2:\OD_{16}$ ($C_2$), $C_{15}^2:\OD_{16}:C_2^2$ ($C_2$), $D_{15}^2:C_4:C_2^2$ ($C_2$)

Order 7200: $C_{15}^2.C_2^3.C_2^2$ ($C_2^2$), $C_5^2.C_6^2.C_2^3$ ($C_2^2$), $C_5^2.C_6^2.C_2^3$ ($C_2^2$), $(C_5^2\times S_3^2):D_4$ ($C_4$), $C_{15}^2:(C_2\times \OD_{16})$ ($C_2^2$), $D_{15}^2:D_4$ ($C_4$), $D_{15}^2:Q_8$ ($C_4$), $(C_5^2\times S_3^2):Q_8$ ($C_4$), $C_{15}^2:(C_2\times \OD_{16})$ ($C_2^2$), $D_5^2.(S_3\times D_6)$ ($C_2^2$), $C_{15}^2:C_4^2:C_2$ ($C_2^2$)

Order 3600: $C_{15}^2.C_4^2$ ($D_4$) x 2, $C_3^2\times F_5^2$ ($D_4$), $C_{15}^2.C_2^2.C_2^2$ ($C_2^3$), $C_{15}^2.C_4^2$ ($D_4$), $C_{15}^2.C_4.C_2^2$ ($D_4$), $C_{15}^2.C_2^3.C_2$ ($C_2\times C_4$), $S_3^2\times D_5^2$ ($D_4$), $C_{15}^2.C_2^3.C_2$ ($D_4$), $C_{15}^2.C_2^3.C_2$ ($C_2\times C_4$), $C_{15}^2:C_4:C_4$ ($C_2\times C_4$), $C_{15}^2:C_4:C_4$ ($C_2\times C_4$), $D_{15}^2:C_4$ ($D_4$), $C_{15}^2:(C_2\times D_4)$ ($C_2\times C_4$), $C_{15}^2:(C_2\times Q_8)$ ($C_2\times C_4$)

Order 1800: $D_{15}^2:C_2$ ($C_2^2:C_4$), $C_{15}^2:C_2^3$ ($C_2^2:C_4$), $C_{15}^2:C_2^3$ ($C_2\times D_4$), $C_{15}:(S_3\times F_5)$ ($C_2^2:C_4$), $C_{15}:(S_3\times F_5)$ ($C_2^2:C_4$), $C_{15}:(S_3\times F_5)$ ($D_4:C_2$), $C_{15}:(S_3\times F_5)$ ($D_4:C_2$), $C_{15}^2:(C_2\times C_4)$ ($C_2\times D_4$), $C_{15}:(S_3\times F_5)$ ($C_2^2\times C_4$), $C_{15}:(S_3\times F_5)$ ($C_2\times D_4$), $C_3^2\times D_5:F_5$ ($C_2\times D_4$)

Order 900: $D_{15}^2$ ($C_4\wr C_2$) x 2, $C_5^2:S_3^2$ ($C_4\wr C_2$), $C_5^2\times S_3^2$ ($C_4\wr C_2$), $C_{15}^2:C_2^2$ ($C_2^2.D_4$), $C_{15}^2:C_2^2$ ($C_2^3:C_4$), $C_3^2\times D_5^2$ ($C_2^2\wr C_2$), $C_{15}^2:C_4$ ($C_4\times D_4$), $C_{15}^2:C_4$ ($C_4:D_4$), $C_{15}^2:C_4$ ($C_4\times D_4$), $C_{15}^2:C_4$ ($C_4:D_4$)

Order 450: $C_{15}:D_{15}$ ($C_4^2:C_2^2$), $C_{15}^2:C_2$ ($C_4^2:C_2^2$), $C_{15}^2:C_2$ ($C_2^3.D_4$)

Order 400: $F_5^2$ ($\SOPlus(4,2)$)

Order 225: $C_{15}^2$ ($C_4^2:D_4$)

Order 200: $D_5:F_5$ ($S_3^2:C_2^2$)

Order 100: $D_5^2$ ($D_6\wr C_2$), $C_5:F_5$ ($C_4\times \SOPlus(4,2)$), $C_5:F_5$ ($S_3^2:D_4$)

Order 50: $C_5:D_5$ ($D_6^2.C_2^2$)

Order 36: $S_3^2$ ($F_5\wr C_2$)

Order 25: $C_5^2$ ($C_{12}^2:D_4$)

Order 18: $C_3:S_3$ ($F_5^2:C_2^2$)

Order 9: $C_3^2$ ($F_5^2:D_4$)

Order 1: $C_1$ ($(S_3\times F_5)\wr C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 28800: $(S_3\times F_5)\wr C_2$ ($C_1$)

Order 14400: $C_5^2.C_6^2.C_2^4$ ($C_2$), $C_{15}^2:\OD_{16}:C_4$ ($C_2$), $F_5^2:S_3^2$ ($C_2$), $C_{15}^2:\OD_{16}:C_4$ ($C_2$), $D_{15}^2:\OD_{16}$ ($C_2$), $C_{15}^2:\OD_{16}:C_2^2$ ($C_2$), $D_{15}^2:C_4:C_2^2$ ($C_2$)

Order 7200: $C_{15}^2.C_2^3.C_2^2$ ($C_2^2$), $C_5^2.C_6^2.C_2^3$ ($C_2^2$), $C_5^2.C_6^2.C_2^3$ ($C_2^2$), $(C_5^2\times S_3^2):D_4$ ($C_4$), $C_{15}^2:(C_2\times \OD_{16})$ ($C_2^2$), $D_{15}^2:D_4$ ($C_4$), $D_{15}^2:Q_8$ ($C_4$), $(C_5^2\times S_3^2):Q_8$ ($C_4$), $C_{15}^2:(C_2\times \OD_{16})$ ($C_2^2$), $D_5^2.(S_3\times D_6)$ ($C_2^2$), $C_{15}^2:C_4^2:C_2$ ($C_2^2$)

Order 3600: $C_3^2\times F_5^2$ ($D_4$), $C_{15}^2.C_2^2.C_2^2$ ($C_2^3$), $C_{15}^2.C_4^2$ ($D_4$), $C_{15}^2.C_4.C_2^2$ ($D_4$), $C_{15}^2.C_2^3.C_2$ ($C_2\times C_4$), $S_3^2\times D_5^2$ ($D_4$), $C_{15}^2.C_2^3.C_2$ ($D_4$), $C_{15}^2.C_4^2$ ($D_4$), $C_{15}^2.C_2^3.C_2$ ($C_2\times C_4$), $C_{15}^2:C_4:C_4$ ($C_2\times C_4$), $C_{15}^2:C_4:C_4$ ($C_2\times C_4$), $D_{15}^2:C_4$ ($D_4$), $C_{15}^2:(C_2\times D_4)$ ($C_2\times C_4$), $C_{15}^2:(C_2\times Q_8)$ ($C_2\times C_4$)

Order 1800: $D_{15}^2:C_2$ ($C_2^2:C_4$), $C_{15}^2:C_2^3$ ($C_2^2:C_4$), $C_{15}^2:C_2^3$ ($C_2\times D_4$), $C_{15}:(S_3\times F_5)$ ($C_2^2:C_4$), $C_{15}:(S_3\times F_5)$ ($C_2^2:C_4$), $C_{15}:(S_3\times F_5)$ ($D_4:C_2$), $C_{15}:(S_3\times F_5)$ ($D_4:C_2$), $C_{15}^2:(C_2\times C_4)$ ($C_2\times D_4$), $C_{15}:(S_3\times F_5)$ ($C_2^2\times C_4$), $C_{15}:(S_3\times F_5)$ ($C_2\times D_4$), $C_3^2\times D_5:F_5$ ($C_2\times D_4$)

Order 900: $D_{15}^2$ ($C_4\wr C_2$), $C_5^2:S_3^2$ ($C_4\wr C_2$), $C_5^2\times S_3^2$ ($C_4\wr C_2$), $C_{15}^2:C_2^2$ ($C_2^2.D_4$), $C_{15}^2:C_2^2$ ($C_2^3:C_4$), $C_3^2\times D_5^2$ ($C_2^2\wr C_2$), $C_{15}^2:C_4$ ($C_4\times D_4$), $C_{15}^2:C_4$ ($C_4:D_4$), $C_{15}^2:C_4$ ($C_4\times D_4$), $C_{15}^2:C_4$ ($C_4:D_4$)

Order 450: $C_{15}:D_{15}$ ($C_4^2:C_2^2$), $C_{15}^2:C_2$ ($C_4^2:C_2^2$), $C_{15}^2:C_2$ ($C_2^3.D_4$)

Order 400: $F_5^2$ ($\SOPlus(4,2)$)

Order 225: $C_{15}^2$ ($C_4^2:D_4$)

Order 200: $D_5:F_5$ ($S_3^2:C_2^2$)

Order 100: $D_5^2$ ($D_6\wr C_2$), $C_5:F_5$ ($C_4\times \SOPlus(4,2)$), $C_5:F_5$ ($S_3^2:D_4$)

Order 50: $C_5:D_5$ ($D_6^2.C_2^2$)

Order 36: $S_3^2$ ($F_5\wr C_2$)

Order 25: $C_5^2$ ($C_{12}^2:D_4$)

Order 18: $C_3:S_3$ ($F_5^2:C_2^2$)

Order 9: $C_3^2$ ($F_5^2:D_4$)

Order 1: $C_1$ ($(S_3\times F_5)\wr C_2$)

Series

Derived series

$(S_3\times F_5)\wr C_2$

$\rhd$

$(S_3\times F_5)\wr C_2$

$\rhd$

$C_{15}:(S_3\times F_5)$

$\rhd$

$C_{15}:(S_3\times F_5)$

$\rhd$

$C_{15}^2$

$\rhd$

$C_{15}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$(S_3\times F_5)\wr C_2$

$\rhd$

$(S_3\times F_5)\wr C_2$

$\rhd$

$D_{15}^2:C_4:C_2^2$

$\rhd$

$D_{15}^2:C_4:C_2^2$

$\rhd$

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

$\rhd$

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

$\rhd$

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

$\rhd$

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

$\rhd$

$C_{15}:(S_3\times F_5)$

$\rhd$

$C_{15}:(S_3\times F_5)$

$\rhd$

$C_{15}^2:C_2^2$

$\rhd$

$C_{15}^2:C_2^2$

$\rhd$

$C_{15}^2:C_2$

$\rhd$

$C_{15}^2:C_2$

$\rhd$

$C_{15}^2$

$\rhd$

$C_{15}^2$

$\rhd$

$C_5^2$

$\rhd$

$C_5^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$(S_3\times F_5)\wr C_2$

$\rhd$

$(S_3\times F_5)\wr C_2$

$\rhd$

$C_{15}:(S_3\times F_5)$

$\rhd$

$C_{15}:(S_3\times F_5)$

$\rhd$

$C_{15}^2:C_2$

$\rhd$

$C_{15}^2:C_2$

$\rhd$

$C_{15}^2$

$\rhd$

$C_{15}^2$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $135 \times 135$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $96 \times 96$ rational character table.