Abstract group 46080.b: $\GL(2,5)\times \GL(2,\mathbb{Z}/4)$

• Label: 46080.b
• Order: \( 2^{10} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $32$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$\GL(2,5)\times \GL(2,\mathbb{Z}/4)$
Order:	\(46080\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^5\times S_5\times C_2^2\times S_4$, of order \(368640\)\(\medspace = 2^{13} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 8, $C_3$, $A_5$
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	60
Elements	1	895	188	10880	24	3236	2560	1320	10144	192	4800	8960	1344	1536	46080
Conjugacy classes	1	20	3	79	1	31	20	13	62	1	26	64	7	8	336
Divisions	1	20	3	49	1	26	10	13	31	1	16	16	5	3	195
Autjugacy classes	1	17	3	34	1	21	8	10	24	1	13	12	4	3	152

Dimension	1	2	3	4	5	6	8	10	12	15	16	18	20	24	30	32	36	40	48	60	64	72	96
Irr. complex chars.	16	20	16	40	16	28	50	20	70	16	0	24	0	10	4	0	6	0	0	0	0	0	0	336
Irr. rational chars.	8	10	8	13	8	14	15	10	23	8	12	8	5	18	6	5	10	1	8	1	1	2	1	195

Minimal presentations

Permutation degree:	$32$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $32$ $\langle(1,4)(2,10)(3,13)(5,17)(6,14)(7,21)(8,23)(11,15)(16,20)(19,24), (25,27,26,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 5 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 12 \\ 4 & 3 \end{array}\right), \left(\begin{array}{rr} 8 & 11 \\ 17 & 11 \end{array}\right), \left(\begin{array}{rr} 7 & 6 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$\GL(2,\mathbb{Z}/4)$ $\, \times\, $ $\GL(2,5)$
Semidirect product:	$(A_4\times \GL(2,5))$ $\,\rtimes\,$ $D_4$	$A_4$ $\,\rtimes\,$ $(D_4\times \GL(2,5))$	$(C_2^4\times \GL(2,5))$ $\,\rtimes\,$ $S_3$	$(C_2^2\times \GL(2,5))$ $\,\rtimes\,$ $S_4$	all 33
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $(D_6.S_5)$	$(C_4\times A_4)$ . $(D_4:S_5)$	$(C_4\times A_4)$ . $(D_4\times S_5)$	$C_2^3$ . $(A_5:C_4\times S_4)$	all 58
Aut. group:	$\Aut(C_{20}^2)$	$\Aut(C_2^2.D_6^2)$

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{20}\Z)$.

Homology

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

Subgroups

There are 1341656 subgroups in 20933 conjugacy classes, 116 normal (96 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_2\times S_4\times S_5$
Commutator:	$G' \simeq$ $C_2^4.\GL(2,4)$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times S_4\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_4$	$G/\operatorname{Fit} \simeq$ $S_3\times S_5$
Radical:	$R \simeq$ $C_4\times \GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $A_5:C_4\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

All subgroups of index up to 72 (order at least 640) are shown, as well as all normal subgroups of any index.

Order 46080: $\GL(2,5)\times \GL(2,\mathbb{Z}/4)$

Order 23040: $C_2\times S_4\times \GL(2,5)$, $A_4:C_4\times \GL(2,5)$, $(C_2^2\times A_4):\GL(2,5)$, $\GL(2,\mathbb{Z}/4)\times \SL(2,5):C_2$, $C_2^2\times A_4\times \GL(2,5)$, $C_2.(S_4\times \GL(2,5))$, $(C_2\times S_4):\GL(2,5)$

Order 15360: $C_2^2\wr C_2\times \GL(2,5)$

Order 11520: $C_2\times A_4\times \GL(2,5)$ x 4, $S_4\times \GL(2,5)$ x 2, $C_2\times A_4:\GL(2,5)$ x 2, $C_3:D_4\times \GL(2,5)$, $\SL(2,5):\GL(2,\mathbb{Z}/4)$, $\GL(2,\mathbb{Z}/4)\times \SL(2,5)$, $A_4\times \SL(2,5):C_2^3$, $C_2^4.(D_6\times A_5)$, $S_4\times \SL(2,5):C_2^2$, $\SL(2,5):\GL(2,\mathbb{Z}/4)$, $\SL(2,5):\GL(2,\mathbb{Z}/4)$

Order 9216: $\GL(2,\mathbb{Z}/4)\times \Unitary(2,3)$

Order 7680: $(C_2\times D_4):\GL(2,5)$ x 3, $C_2^2:C_4\times \GL(2,5)$ x 3, $C_2\times D_4\times \GL(2,5)$ x 3, $C_2.(D_4\times \GL(2,5))$ x 3, $C_4\times F_5\times \GL(2,\mathbb{Z}/4)$, $C_2^4:\GL(2,5)$, $C_2^4\times \GL(2,5)$, $(C_2^3\times \SL(2,5)):C_2^3$

Order 5760: $A_4\times \SL(2,5):C_2^2$ x 4, $A_4\times \GL(2,5)$ x 3, $A_4:\GL(2,5)$ x 2, $S_4\times \SL(2,5):C_2$ x 2, $A_4:C_4\times \SL(2,5)$ x 2, $(C_2\times C_6):\GL(2,5)$, $C_6.(C_2\times \GL(2,5))$, $C_{12}.(D_4\times A_5)$, $D_6:\GL(2,5)$, $C_2\times C_6\times \GL(2,5)$, $C_3:C_4\times \GL(2,5)$, $C_2^5.\GL(2,4)$, $C_2\times S_4\times \SL(2,5)$, $C_2\times \SL(2,5):S_4$, $D_6\times \GL(2,5)$

Order 4608: $C_{24}.(C_2\times \GL(2,\mathbb{Z}/4))$, $(C_4\times A_4).\GL(2,\mathbb{Z}/4)$, $(C_2^4\times \SL(2,3)):C_{12}$, $(C_2\times C_4).S_4^2$, $(A_4\times \SL(2,3)):C_4^2$, $\GL(2,\mathbb{Z}/4)\times \SL(2,3):C_2$, $(C_2\times C_4).S_4^2$, $(C_2\times C_4).S_4^2$

Order 3840: $D_4\times \GL(2,5)$ x 12, $C_2^3\times \GL(2,5)$ x 11, $C_2^3:\GL(2,5)$ x 9, $(C_2\times C_4):\GL(2,5)$ x 6, $D_4\times \SL(2,5):C_2^2$ x 3, $C_2\times C_4\times \GL(2,5)$ x 3, $C_2^3.(C_2^3\times A_5)$ x 3, $C_2\times C_4:\GL(2,5)$ x 3, $C_2^3.(D_4\times A_5)$ x 3, $C_2^3.(D_4\times A_5)$ x 3, $F_5\times C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times F_5\times \GL(2,\mathbb{Z}/4)$ x 2, $(C_{10}\times A_4).(C_4\times D_4)$ x 2, $F_5\times C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.(D_4\times A_5)$, $\SL(2,5):C_2^5$, $C_2^3.(D_4\times A_5)$, $C_4\times D_5.\GL(2,\mathbb{Z}/4)$, $C_{10}:C_4^2\times S_4$, $C_4\times D_5\times \GL(2,\mathbb{Z}/4)$, $A_4:C_4^2\times F_5$, $A_4\times D_{10}:C_4^2$, $(C_{10}\times S_4):C_4^2$, $C_4\times D_5.\GL(2,\mathbb{Z}/4)$

Order 3072: $C_4.(D_4\times \GL(2,\mathbb{Z}/4))$, $(C_2^3\times Q_8).C_6.C_4.C_2$

Order 2880: $C_6\times \GL(2,5)$ x 4, $A_4\times \SL(2,5):C_2$ x 3, $C_2^4.\GL(2,4)$ x 2, $C_6:\GL(2,5)$ x 2, $S_3\times \GL(2,5)$ x 2, $C_2^2\times \SL(2,5):C_6$, $C_{12}.(C_4\times A_5)$, $(C_3\times \SL(2,5)):D_4$, $C_3:D_4\times \SL(2,5)$, $(C_3\times \SL(2,5)):D_4$, $(C_3\times \SL(2,5)):D_4$, $S_4\times \SL(2,5)$, $\SL(2,5):S_4$, $C_2\times \SL(2,5):D_6$

Order 2560: $C_2^{16}.\PSL(2,7)$

Order 2304: $(C_2^3\times \SL(2,3)):C_{12}$ x 4, $C_2^3:C_{12}.S_4$ x 2, $C_4.S_4^2$ x 2, $A_4\times C_{24}:C_2^3$, $C_8:D_6\times S_4$, $C_{24}:\GL(2,\mathbb{Z}/4)$, $C_{24}:\GL(2,\mathbb{Z}/4)$, $C_{24}\times \GL(2,\mathbb{Z}/4)$, $C_6.(\OD_{16}\times S_4)$, $C_{24}:\GL(2,\mathbb{Z}/4)$, $\SL(2,3):\GL(2,\mathbb{Z}/4)$, $\SL(2,3):\GL(2,\mathbb{Z}/4)$, $A_4\times \SL(2,3):C_2^3$, $C_2^4.(A_4\times D_6)$, $S_4\times \SL(2,3):C_2^2$, $\GL(2,\mathbb{Z}/4)\times \SL(2,3)$, $C_{12}.(D_4\times S_4)$, $\SL(2,3):\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_2\times \GL(2,\mathbb{Z}/4))$, $(C_2^3\times C_{24}):D_6$, $C_2^4.D_6^2$, $C_{12}.(C_2\times \GL(2,\mathbb{Z}/4))$, $C_6.(C_4\times \GL(2,\mathbb{Z}/4))$, $(C_6\times S_4):\OD_{16}$, $C_3:C_8\times \GL(2,\mathbb{Z}/4)$, $C_6.(C_4\times \GL(2,\mathbb{Z}/4))$

Order 1920: $C_2^2\times \GL(2,5)$ x 43, $C_2^2:\GL(2,5)$ x 18, $F_5\times \GL(2,\mathbb{Z}/4)$ x 16, $D_4\times \SL(2,5):C_2$ x 12, $\SL(2,5):C_2^4$ x 11, $A_4\times C_{10}:C_4^2$ x 8, $C_2\times \SL(2,5):D_4$ x 6, $C_4:\GL(2,5)$ x 6, $C_2^2.(D_4\times A_5)$ x 6, $C_2^2:C_4\times \SL(2,5)$ x 6, $C_4\times \GL(2,5)$ x 6, $(C_{10}\times A_4):C_4^2$ x 6, $C_4\times F_5\times S_4$ x 4, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 4, $C_2\times \SL(2,5):D_4$ x 3, $C_2\times D_4\times \SL(2,5)$ x 3, $C_4\times \SL(2,5):C_2^2$ x 3, $C_2^3\times A_4\times F_5$ x 2, $C_2^2\times F_5\times S_4$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $C_4\times D_{10}.S_4$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $C_{12}:D_4\times F_5$, $C_2^4\times \SL(2,5)$, $C_2^3:C_{12}\times D_{10}$, $D_{10}\times \GL(2,\mathbb{Z}/4)$, $C_4\times D_{10}\times S_4$, $C_{20}:\GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $C_5:(C_4\times \GL(2,\mathbb{Z}/4))$, $C_5:C_4\times \GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $D_{10}:(C_4\times S_4)$, $C_{20}:\GL(2,\mathbb{Z}/4)$, $D_{10}:(C_4\times S_4)$, $C_{20}:\GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $D_5\times A_4:C_4^2$, $C_{20}\times \GL(2,\mathbb{Z}/4)$

Order 1536: $C_4^2.(C_2^2\times S_4)$ x 3, $(C_2^2\times C_4).\GL(2,\mathbb{Z}/4)$ x 3, $C_2^2:C_4^2.S_4$ x 3, $(C_2^2\times \SL(2,3)):C_4^2$ x 3, $D_4:C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^4\times \Unitary(2,3)$, $C_2^4:\Unitary(2,3)$, $\OD_{16}\times \GL(2,\mathbb{Z}/4)$, $C_4^2\times \GL(2,\mathbb{Z}/4)$, $C_4^2:C_2^2\times S_4$, $C_2^4.\GL(2,\mathbb{Z}/4)$, $C_2^4:\Unitary(2,3)$, $\OD_{16}:\GL(2,\mathbb{Z}/4)$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $\OD_{16}:\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4^2):D_6$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $\OD_{16}:\GL(2,\mathbb{Z}/4)$, $C_2^3.(D_4\times S_4)$, $(D_4\times A_4):C_4^2$, $A_4\times C_4^2:C_2^3$, $(C_2^3\times \SL(2,3)):C_2^3$, $(C_2^3\times C_{24}):C_2^3$

Order 1440: $C_2\times \SL(2,5):C_6$ x 4, $C_3\times \GL(2,5)$ x 3, $\SL(2,5):D_6$ x 2, $C_3:C_4\times \SL(2,5)$ x 2, $C_3:\GL(2,5)$ x 2, $C_2^3.\GL(2,4)$, $\SL(2,5):D_6$, $D_6\times \SL(2,5)$, $A_4\times \SL(2,5)$

Order 1280: $C_2^3.D_4\times F_5$ x 6, $(C_2\times D_{20}):C_4^2$ x 6, $D_4\times C_{10}:C_4^2$ x 3, $C_{10}.(D_4\times C_4^2)$ x 3, $(C_2\times D_{20}):C_4^2$ x 3, $D_{10}:C_4^3$ x 3, $C_2^3:D_4\times F_5$ x 2, $C_2^4:C_4\times F_5$ x 2, $C_4\times C_2^4:D_{10}$, $D_{10}.C_2^6$, $C_4\times C_2^4:F_5$

Order 1152: $C_8:D_6\times A_4$ x 8, $A_4\times \SL(2,3):C_2^2$ x 4, $C_{24}:C_2\times S_4$ x 4, $A_4\times \Unitary(2,3)$ x 3, $S_4\times \SL(2,3):C_2$ x 2, $(C_6\times A_4):\OD_{16}$ x 2, $A_4:\Unitary(2,3)$ x 2, $A_4:C_4\times \SL(2,3)$ x 2, $C_{12}.\GL(2,\mathbb{Z}/4)$ x 2, $C_{24}:(S_3\times D_4)$, $C_2^3:C_{12}\times D_6$, $C_2^3.A_4^2$, $D_6\times \GL(2,\mathbb{Z}/4)$, $C_2^3.D_6^2$, $C_2\times S_4\times \SL(2,3)$, $C_2\times \SL(2,3):S_4$, $C_{12}.(D_4\times A_4)$, $(C_2^3\times C_6):C_{24}$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $C_2\times C_{24}:S_4$, $C_2^2\times \SL(2,3):C_{12}$, $C_{12}\times \GL(2,\mathbb{Z}/4)$, $C_2\times C_{24}\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $D_6\times \Unitary(2,3)$, $D_6.\GL(2,\mathbb{Z}/4)$, $D_6:(C_4\times S_4)$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $D_6:(C_4\times S_4)$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $D_6.\GL(2,\mathbb{Z}/4)$, $S_3\times A_4:C_4^2$, $C_{12}.(C_4\times S_4)$, $C_6:C_8\times S_4$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $(A_4\times C_{24}):C_4$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_8\times A_4:C_{12}$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.C_4:S_4$, $C_{12}.(C_4\times S_4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_4\times S_4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_6.(C_8\times S_4)$, $C_2^2\times A_4\times C_{24}$

Order 1024: $C_2^4.D_4^2$

Order 960: $C_2\times \GL(2,5)$ x 44, $\SL(2,5):C_2^3$ x 40, $C_2^2\times A_4\times F_5$ x 20, $C_2\times F_5\times S_4$ x 16, $A_4:C_4\times F_5$ x 16, $\SL(2,5):D_4$ x 12, $C_4\times A_4\times F_5$ x 10, $C_4\times A_4\times D_{10}$ x 8, $D_5\times \GL(2,\mathbb{Z}/4)$ x 8, $D_5.\GL(2,\mathbb{Z}/4)$ x 8, $D_5.\GL(2,\mathbb{Z}/4)$ x 8, $D_5.\GL(2,\mathbb{Z}/4)$ x 8, $D_4\times \SL(2,5)$ x 6, $C_2\times C_4\times \SL(2,5)$ x 6, $\SL(2,5):D_4$ x 6, $C_4^2.A_5$ x 6, $C_2^3\times \SL(2,5)$ x 6, $C_2^2\times A_4:F_5$ x 4, $C_4\times D_5\times S_4$ x 4, $C_4\times A_4:F_5$ x 4, $A_4:C_4\times D_{10}$ x 3, $(C_2\times C_{30}):C_4^2$ x 2, $D_{30}:C_4^2$ x 2, $C_6.D_4\times F_5$ x 2, $C_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $(C_5\times A_4):C_4^2$ x 2, $C_6:D_4\times F_5$ x 2, $C_4\times D_6:D_{10}$, $(C_{12}\times D_{10}):C_4$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{30}$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_4\times A_4:C_{20}$, $C_4\times D_{10}.D_6$, $D_{30}:C_4^2$, $C_{15}:C_4^3$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_2^2\times A_4\times D_{10}$, $C_2\times D_{10}\times S_4$, $C_2^4:C_{60}$, $C_2^2\times C_{12}\times F_5$, $(C_2^3\times C_{10}):C_{12}$, $C_{10}:\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{30}$, $C_{10}\times \GL(2,\mathbb{Z}/4)$, $C_2\times C_{20}\times S_4$, $D_{30}:C_4^2$, $C_{10}:\GL(2,\mathbb{Z}/4)$, $C_{10}:\GL(2,\mathbb{Z}/4)$, $C_{10}:(C_4\times S_4)$, $C_{10}:C_4\times S_4$

Order 768: $D_4\times \Unitary(2,3)$ x 12, $C_2^3\times \Unitary(2,3)$ x 11, $C_2\times C_4.\GL(2,\mathbb{Z}/4)$ x 9, $A_4\times C_4^2:C_2^2$ x 8, $(C_2\times C_4).\GL(2,\mathbb{Z}/4)$ x 6, $C_4\wr C_2\times S_4$ x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 4, $(C_2^2\times C_8):D_{12}$ x 3, $C_2^4.(C_4\times D_6)$ x 3, $(C_6\times D_4):\OD_{16}$ x 3, $(C_2^2\times C_{24}):D_4$ x 3, $(C_2^3\times C_8):D_6$ x 3, $C_2^4.(C_4\times D_6)$ x 3, $C_3\times C_2^4.C_4^2$ x 3, $C_2\times C_4^2.S_4$ x 3, $C_2\times \SL(2,3):C_4^2$ x 3, $D_4\times \SL(2,3):C_2^2$ x 3, $(C_2^2\times \SL(2,3)):D_4$ x 3, $(C_2^2\times \SL(2,3)):D_4$ x 3, $C_2^3.(C_2^3\times A_4)$ x 3, $D_4\times C_8:D_6$ x 3, $C_2^3:\Unitary(2,3)$ x 2, $C_2^6.D_6$ x 2, $C_8:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_{12}$ x 2, $C_2^6.D_6$ x 2, $C_4\times C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_4\times C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_{12}$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_4\times C_2^4:D_6$, $C_{12}.D_4^2$, $(C_2^2\times C_{24}):D_4$, $(C_2^3\times C_6):\OD_{16}$, $(C_2^2\times C_{12}).C_2^4$, $(C_2^2\times C_{24}):C_2^3$, $C_2^2\wr C_2\times C_{24}$, $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2\times \OD_{16}\times S_4$, $C_2^6.D_6$, $Q_8:\GL(2,\mathbb{Z}/4)$, $Q_8\times \GL(2,\mathbb{Z}/4)$, $Q_8:\GL(2,\mathbb{Z}/4)$, $Q_8:\GL(2,\mathbb{Z}/4)$, $D_4:\GL(2,\mathbb{Z}/4)$, $D_4:\GL(2,\mathbb{Z}/4)$, $D_4:\GL(2,\mathbb{Z}/4)$, $D_4\times \GL(2,\mathbb{Z}/4)$, $(C_2\times C_4):\GL(2,\mathbb{Z}/4)$, $(C_2\times C_4):\GL(2,\mathbb{Z}/4)$, $C_2^2.(C_2^3\times S_4)$, $C_2\times C_4^2\times S_4$, $C_8:\GL(2,\mathbb{Z}/4)$, $C_8:\GL(2,\mathbb{Z}/4)$, $C_8\times \GL(2,\mathbb{Z}/4)$, $(C_2\times S_4):\OD_{16}$, $(C_2\times C_4).\GL(2,\mathbb{Z}/4)$, $(C_2\times C_4).\GL(2,\mathbb{Z}/4)$, $A_4:C_4\times \OD_{16}$, $A_4:C_4^3$, $Q_8.(C_2^3\times A_4)$, $\SL(2,3):C_2^5$, $D_4:C_2^3\times A_4$, $C_2^6.C_{12}$, $C_2^2\times A_4\times C_4^2$, $(C_2^2\times \SL(2,3)):D_4$, $C_2^2\wr C_2\times \SL(2,3)$, $C_{24}:C_2^5$

Order 720: $\SL(2,5):C_6$ x 3, $C_6\times \SL(2,5)$ x 2, $\SL(2,5):S_3$, $S_3\times \SL(2,5)$

Order 640: $C_2^2\wr C_2\times F_5$ x 32, $D_{20}:C_4^2$ x 24, $C_2\times D_{10}:C_4^2$ x 24, $D_{10}.C_2^5$ x 21, $(C_2^3\times F_5):C_4$ x 12, $(C_2\times C_{20}):C_4^2$ x 12, $C_{10}.C_4^3$ x 12, $C_2\times D_{10}:C_4^2$ x 9, $(C_2^3\times C_{20}):C_4$ x 6, $(C_2\times C_{20}):C_4^2$ x 6, $(C_2\times D_{10}):C_4^2$ x 6, $C_{10}.C_4^3$ x 6, $D_{10}.C_2^5$ x 6, $(D_4\times D_{10}):C_4$ x 6, $C_2\times C_{20}:C_4^2$ x 6, $D_{10}.(C_4\times D_4)$ x 6, $(C_4\times D_{10}):D_4$ x 3, $D_{10}:(C_4\times D_4)$ x 3, $C_2^3.D_4\times D_5$ x 3, $D_{10}:(C_4\times D_4)$ x 3, $C_4\times C_2^2:D_{20}$ x 3, $C_2^2:C_4^2\times D_5$ x 3, $C_4\times D_4\times D_{10}$ x 3, $C_2\times C_{20}:C_4^2$ x 3, $C_{10}:C_4^3$ x 3, $C_4\times D_{10}:D_4$ x 3, $F_5\times C_2^5$ x 2, $C_2^5:F_5$ x 2, $C_2^5.D_{10}$, $C_2^5.D_{10}$, $C_{20}:C_2^5$, $C_2^5:D_{10}$, $C_2^2\wr C_2\times C_{20}$, $C_2^5.D_{10}$, $C_2^5.D_{10}$

Order 480: $\GL(2,5)$ x 10, $C_2^2\times \SL(2,5)$ x 2, $\SL(2,5):C_2^2$ x 2

Order 384: $C_4\times \GL(2,\mathbb{Z}/4)$ x 9

Order 240: $\SL(2,5):C_2$ x 2, $C_2\times \SL(2,5)$

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2.\GL(2,\mathbb{Z}/4)$, $A_4:C_4^2$, $C_2^4:C_{12}$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^4.D_6$

Order 120: $\SL(2,5)$

Order 96: $\GL(2,\mathbb{Z}/4)$ x 13, $C_4\times S_4$ x 8, $C_{12}:D_4$ x 8, $C_2^2.S_4$ x 3, $C_2^3:C_{12}$ x 2, $C_2^3\times A_4$, $C_2^2\times S_4$

Order 64: $C_2^4\times C_4$

Order 48: $C_2^2\times A_4$ x 3, $C_2\times S_4$ x 2, $C_4\times A_4$ x 2, $A_4:C_4$ x 2

Order 32: $C_4\times D_4$ x 64, $C_2^3\times C_4$ x 2, $C_2^5$

Order 24: $C_3:D_4$ x 13, $C_4\times S_3$ x 8, $S_4$ x 4, $C_2\times A_4$ x 3

Order 16: $C_2^4$ x 3, $C_2^2\times C_4$ x 3

Order 12: $A_4$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 66, $D_4$ x 44, $C_2^3$ x 4

Order 6: $S_3$ x 4

Order 5: $C_5$

Order 4: $C_4$ x 22, $C_2^2$ x 4

Order 2: $C_2$ x 12

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 72 (order at least 640) are shown, as well as all normal subgroups of any index.

Order 46080: $\GL(2,5)\times \GL(2,\mathbb{Z}/4)$

Order 23040: $C_2\times S_4\times \GL(2,5)$, $A_4:C_4\times \GL(2,5)$, $(C_2^2\times A_4):\GL(2,5)$, $\GL(2,\mathbb{Z}/4)\times \SL(2,5):C_2$, $C_2^2\times A_4\times \GL(2,5)$, $C_2.(S_4\times \GL(2,5))$, $(C_2\times S_4):\GL(2,5)$

Order 15360: $C_2^2\wr C_2\times \GL(2,5)$

Order 11520: $C_2\times A_4\times \GL(2,5)$ x 3, $C_2\times A_4:\GL(2,5)$ x 2, $S_4\times \GL(2,5)$, $C_3:D_4\times \GL(2,5)$, $\SL(2,5):\GL(2,\mathbb{Z}/4)$, $\GL(2,\mathbb{Z}/4)\times \SL(2,5)$, $A_4\times \SL(2,5):C_2^3$, $C_2^4.(D_6\times A_5)$, $S_4\times \SL(2,5):C_2^2$, $\SL(2,5):\GL(2,\mathbb{Z}/4)$, $\SL(2,5):\GL(2,\mathbb{Z}/4)$

Order 9216: $\GL(2,\mathbb{Z}/4)\times \Unitary(2,3)$

Order 7680: $(C_2\times D_4):\GL(2,5)$ x 3, $C_2^2:C_4\times \GL(2,5)$ x 3, $C_2\times D_4\times \GL(2,5)$ x 3, $C_2.(D_4\times \GL(2,5))$ x 3, $C_4\times F_5\times \GL(2,\mathbb{Z}/4)$, $C_2^4:\GL(2,5)$, $C_2^4\times \GL(2,5)$, $(C_2^3\times \SL(2,5)):C_2^3$

Order 5760: $A_4\times \SL(2,5):C_2^2$ x 4, $A_4\times \GL(2,5)$ x 2, $A_4:\GL(2,5)$ x 2, $S_4\times \SL(2,5):C_2$ x 2, $A_4:C_4\times \SL(2,5)$ x 2, $(C_2\times C_6):\GL(2,5)$, $C_6.(C_2\times \GL(2,5))$, $C_{12}.(D_4\times A_5)$, $D_6:\GL(2,5)$, $C_2\times C_6\times \GL(2,5)$, $C_3:C_4\times \GL(2,5)$, $C_2^5.\GL(2,4)$, $C_2\times S_4\times \SL(2,5)$, $C_2\times \SL(2,5):S_4$, $D_6\times \GL(2,5)$

Order 4608: $C_{24}.(C_2\times \GL(2,\mathbb{Z}/4))$, $(C_4\times A_4).\GL(2,\mathbb{Z}/4)$, $(C_2^4\times \SL(2,3)):C_{12}$, $(C_2\times C_4).S_4^2$, $(A_4\times \SL(2,3)):C_4^2$, $\GL(2,\mathbb{Z}/4)\times \SL(2,3):C_2$, $(C_2\times C_4).S_4^2$, $(C_2\times C_4).S_4^2$

Order 3840: $C_2^3\times \GL(2,5)$ x 9, $C_2^3:\GL(2,5)$ x 9, $D_4\times \GL(2,5)$ x 7, $(C_2\times C_4):\GL(2,5)$ x 6, $D_4\times \SL(2,5):C_2^2$ x 3, $C_2\times C_4\times \GL(2,5)$ x 3, $C_2^3.(C_2^3\times A_5)$ x 3, $C_2\times C_4:\GL(2,5)$ x 3, $C_2^3.(D_4\times A_5)$ x 3, $C_2^3.(D_4\times A_5)$ x 3, $C_2^3.(D_4\times A_5)$, $\SL(2,5):C_2^5$, $C_2^3.(D_4\times A_5)$, $C_4\times D_5.\GL(2,\mathbb{Z}/4)$, $C_{10}:C_4^2\times S_4$, $C_4\times D_5\times \GL(2,\mathbb{Z}/4)$, $A_4:C_4^2\times F_5$, $F_5\times C_2.\GL(2,\mathbb{Z}/4)$, $C_2\times F_5\times \GL(2,\mathbb{Z}/4)$, $A_4\times D_{10}:C_4^2$, $(C_{10}\times S_4):C_4^2$, $C_4\times D_5.\GL(2,\mathbb{Z}/4)$, $(C_{10}\times A_4).(C_4\times D_4)$, $F_5\times C_2.\GL(2,\mathbb{Z}/4)$

Order 3072: $C_4.(D_4\times \GL(2,\mathbb{Z}/4))$, $(C_2^3\times Q_8).C_6.C_4.C_2$

Order 2880: $A_4\times \SL(2,5):C_2$ x 3, $C_6\times \GL(2,5)$ x 3, $C_2^4.\GL(2,4)$ x 2, $C_6:\GL(2,5)$ x 2, $C_2^2\times \SL(2,5):C_6$, $C_{12}.(C_4\times A_5)$, $(C_3\times \SL(2,5)):D_4$, $C_3:D_4\times \SL(2,5)$, $(C_3\times \SL(2,5)):D_4$, $(C_3\times \SL(2,5)):D_4$, $S_4\times \SL(2,5)$, $\SL(2,5):S_4$, $C_2\times \SL(2,5):D_6$, $S_3\times \GL(2,5)$

Order 2560: $C_2^{16}.\PSL(2,7)$

Order 2304: $(C_2^3\times \SL(2,3)):C_{12}$ x 3, $C_2^3:C_{12}.S_4$ x 2, $A_4\times C_{24}:C_2^3$, $C_8:D_6\times S_4$, $C_{24}:\GL(2,\mathbb{Z}/4)$, $C_{24}:\GL(2,\mathbb{Z}/4)$, $C_{24}\times \GL(2,\mathbb{Z}/4)$, $C_6.(\OD_{16}\times S_4)$, $C_{24}:\GL(2,\mathbb{Z}/4)$, $\SL(2,3):\GL(2,\mathbb{Z}/4)$, $\SL(2,3):\GL(2,\mathbb{Z}/4)$, $A_4\times \SL(2,3):C_2^3$, $C_2^4.(A_4\times D_6)$, $S_4\times \SL(2,3):C_2^2$, $\GL(2,\mathbb{Z}/4)\times \SL(2,3)$, $C_4.S_4^2$, $C_{12}.(D_4\times S_4)$, $\SL(2,3):\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_2\times \GL(2,\mathbb{Z}/4))$, $(C_2^3\times C_{24}):D_6$, $C_2^4.D_6^2$, $C_{12}.(C_2\times \GL(2,\mathbb{Z}/4))$, $C_6.(C_4\times \GL(2,\mathbb{Z}/4))$, $(C_6\times S_4):\OD_{16}$, $C_3:C_8\times \GL(2,\mathbb{Z}/4)$, $C_6.(C_4\times \GL(2,\mathbb{Z}/4))$

Order 1920: $C_2^2\times \GL(2,5)$ x 29, $C_2^2:\GL(2,5)$ x 14, $\SL(2,5):C_2^4$ x 11, $D_4\times \SL(2,5):C_2$ x 10, $C_2\times \SL(2,5):D_4$ x 6, $C_2^2.(D_4\times A_5)$ x 6, $C_2^2:C_4\times \SL(2,5)$ x 6, $A_4\times C_{10}:C_4^2$ x 5, $C_4:\GL(2,5)$ x 4, $C_4\times \GL(2,5)$ x 4, $C_2\times \SL(2,5):D_4$ x 3, $C_2\times D_4\times \SL(2,5)$ x 3, $C_4\times \SL(2,5):C_2^2$ x 3, $(C_{10}\times A_4):C_4^2$ x 3, $C_4\times D_{10}.S_4$ x 2, $C_4\times F_5\times S_4$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $C_{12}:D_4\times F_5$, $C_2^4\times \SL(2,5)$, $C_2^3\times A_4\times F_5$, $C_2^2\times F_5\times S_4$, $C_2^3:C_{12}\times D_{10}$, $D_{10}\times \GL(2,\mathbb{Z}/4)$, $C_4\times D_{10}\times S_4$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $F_5\times \GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $C_{20}:\GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $C_5:(C_4\times \GL(2,\mathbb{Z}/4))$, $C_5:C_4\times \GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $D_{10}:(C_4\times S_4)$, $C_{20}:\GL(2,\mathbb{Z}/4)$, $D_{10}:(C_4\times S_4)$, $C_{20}:\GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $D_5\times A_4:C_4^2$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $C_{20}\times \GL(2,\mathbb{Z}/4)$

Order 1536: $C_4^2.(C_2^2\times S_4)$ x 3, $(C_2^2\times C_4).\GL(2,\mathbb{Z}/4)$ x 3, $C_2^2:C_4^2.S_4$ x 3, $(C_2^2\times \SL(2,3)):C_4^2$ x 3, $D_4:C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^4\times \Unitary(2,3)$, $C_2^4:\Unitary(2,3)$, $\OD_{16}\times \GL(2,\mathbb{Z}/4)$, $C_4^2\times \GL(2,\mathbb{Z}/4)$, $C_4^2:C_2^2\times S_4$, $C_2^4.\GL(2,\mathbb{Z}/4)$, $C_2^4:\Unitary(2,3)$, $\OD_{16}:\GL(2,\mathbb{Z}/4)$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $\OD_{16}:\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4^2):D_6$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $\OD_{16}:\GL(2,\mathbb{Z}/4)$, $C_2^3.(D_4\times S_4)$, $(D_4\times A_4):C_4^2$, $A_4\times C_4^2:C_2^3$, $(C_2^3\times \SL(2,3)):C_2^3$, $(C_2^3\times C_{24}):C_2^3$

Order 1440: $C_2\times \SL(2,5):C_6$ x 4, $\SL(2,5):D_6$ x 2, $C_3:C_4\times \SL(2,5)$ x 2, $C_3:\GL(2,5)$ x 2, $C_3\times \GL(2,5)$ x 2, $C_2^3.\GL(2,4)$, $\SL(2,5):D_6$, $D_6\times \SL(2,5)$, $A_4\times \SL(2,5)$

Order 1280: $D_4\times C_{10}:C_4^2$ x 3, $C_2^3.D_4\times F_5$ x 3, $C_{10}.(D_4\times C_4^2)$ x 3, $(C_2\times D_{20}):C_4^2$ x 3, $(C_2\times D_{20}):C_4^2$ x 3, $D_{10}:C_4^3$ x 3, $C_4\times C_2^4:D_{10}$, $D_{10}.C_2^6$, $C_2^3:D_4\times F_5$, $C_4\times C_2^4:F_5$, $C_2^4:C_4\times F_5$

Order 1152: $C_8:D_6\times A_4$ x 6, $A_4\times \SL(2,3):C_2^2$ x 4, $S_4\times \SL(2,3):C_2$ x 2, $(C_6\times A_4):\OD_{16}$ x 2, $C_{24}:C_2\times S_4$ x 2, $A_4:\Unitary(2,3)$ x 2, $A_4\times \Unitary(2,3)$ x 2, $A_4:C_4\times \SL(2,3)$ x 2, $C_{12}.\GL(2,\mathbb{Z}/4)$ x 2, $C_{24}:(S_3\times D_4)$, $C_2^3:C_{12}\times D_6$, $C_2^3.A_4^2$, $D_6\times \GL(2,\mathbb{Z}/4)$, $C_2^3.D_6^2$, $C_2\times S_4\times \SL(2,3)$, $C_2\times \SL(2,3):S_4$, $C_{12}.(D_4\times A_4)$, $(C_2^3\times C_6):C_{24}$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $C_2\times C_{24}:S_4$, $C_2^2\times \SL(2,3):C_{12}$, $C_{12}\times \GL(2,\mathbb{Z}/4)$, $C_2\times C_{24}\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $D_6\times \Unitary(2,3)$, $D_6.\GL(2,\mathbb{Z}/4)$, $D_6:(C_4\times S_4)$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $D_6:(C_4\times S_4)$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $D_6.\GL(2,\mathbb{Z}/4)$, $S_3\times A_4:C_4^2$, $C_{12}.(C_4\times S_4)$, $C_6:C_8\times S_4$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $(A_4\times C_{24}):C_4$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_8\times A_4:C_{12}$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.C_4:S_4$, $C_{12}.(C_4\times S_4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_4\times S_4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_6.(C_8\times S_4)$, $C_2^2\times A_4\times C_{24}$

Order 1024: $C_2^4.D_4^2$

Order 960: $\SL(2,5):C_2^3$ x 36, $C_2\times \GL(2,5)$ x 31, $\SL(2,5):D_4$ x 10, $C_4\times A_4\times D_{10}$ x 8, $C_2\times C_4\times \SL(2,5)$ x 6, $C_4^2.A_5$ x 6, $C_2^3\times \SL(2,5)$ x 6, $D_4\times \SL(2,5)$ x 5, $\SL(2,5):D_4$ x 5, $C_2^2\times A_4\times F_5$ x 5, $C_4\times A_4\times F_5$ x 5, $C_4\times D_5\times S_4$ x 4, $C_4\times A_4:F_5$ x 4, $C_2\times F_5\times S_4$ x 3, $A_4:C_4\times D_{10}$ x 3, $A_4:C_4\times F_5$ x 3, $C_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $(C_5\times A_4):C_4^2$ x 2, $C_2^2\times A_4:F_5$ x 2, $C_4\times D_6:D_{10}$, $(C_{12}\times D_{10}):C_4$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{30}$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_4\times A_4:C_{20}$, $(C_2\times C_{30}):C_4^2$, $D_{30}:C_4^2$, $C_4\times D_{10}.D_6$, $D_{30}:C_4^2$, $C_6.D_4\times F_5$, $C_{15}:C_4^3$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_{10}.\GL(2,\mathbb{Z}/4)$, $C_2^2\times A_4\times D_{10}$, $C_2\times D_{10}\times S_4$, $C_2^4:C_{60}$, $C_2^2\times C_{12}\times F_5$, $(C_2^3\times C_{10}):C_{12}$, $C_{10}:\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{30}$, $C_{10}\times \GL(2,\mathbb{Z}/4)$, $C_2\times C_{20}\times S_4$, $C_6:D_4\times F_5$, $D_{30}:C_4^2$, $D_5\times \GL(2,\mathbb{Z}/4)$, $C_{10}:\GL(2,\mathbb{Z}/4)$, $C_{10}:\GL(2,\mathbb{Z}/4)$, $C_{10}:(C_4\times S_4)$, $C_{10}:C_4\times S_4$, $D_5.\GL(2,\mathbb{Z}/4)$, $D_5.\GL(2,\mathbb{Z}/4)$, $D_5.\GL(2,\mathbb{Z}/4)$

Order 768: $C_2^3\times \Unitary(2,3)$ x 9, $C_2\times C_4.\GL(2,\mathbb{Z}/4)$ x 9, $D_4\times \Unitary(2,3)$ x 7, $(C_2\times C_4).\GL(2,\mathbb{Z}/4)$ x 6, $A_4\times C_4^2:C_2^2$ x 6, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 4, $(C_2^2\times C_8):D_{12}$ x 3, $C_2^4.(C_4\times D_6)$ x 3, $(C_6\times D_4):\OD_{16}$ x 3, $(C_2^2\times C_{24}):D_4$ x 3, $(C_2^3\times C_8):D_6$ x 3, $C_2^4.(C_4\times D_6)$ x 3, $C_3\times C_2^4.C_4^2$ x 3, $C_2\times C_4^2.S_4$ x 3, $C_2\times \SL(2,3):C_4^2$ x 3, $D_4\times \SL(2,3):C_2^2$ x 3, $(C_2^2\times \SL(2,3)):D_4$ x 3, $(C_2^2\times \SL(2,3)):D_4$ x 3, $C_2^3.(C_2^3\times A_4)$ x 3, $D_4\times C_8:D_6$ x 3, $C_2^3:\Unitary(2,3)$ x 2, $C_2^6.D_6$ x 2, $C_4\wr C_2\times S_4$ x 2, $C_8:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_{12}$ x 2, $C_2^6.D_6$ x 2, $C_4\times C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_4\times C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_{12}$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_4\times C_2^4:D_6$, $C_{12}.D_4^2$, $(C_2^2\times C_{24}):D_4$, $(C_2^3\times C_6):\OD_{16}$, $(C_2^2\times C_{12}).C_2^4$, $(C_2^2\times C_{24}):C_2^3$, $C_2^2\wr C_2\times C_{24}$, $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2\times \OD_{16}\times S_4$, $C_2^6.D_6$, $Q_8:\GL(2,\mathbb{Z}/4)$, $Q_8\times \GL(2,\mathbb{Z}/4)$, $Q_8:\GL(2,\mathbb{Z}/4)$, $Q_8:\GL(2,\mathbb{Z}/4)$, $D_4:\GL(2,\mathbb{Z}/4)$, $D_4:\GL(2,\mathbb{Z}/4)$, $D_4:\GL(2,\mathbb{Z}/4)$, $D_4\times \GL(2,\mathbb{Z}/4)$, $(C_2\times C_4):\GL(2,\mathbb{Z}/4)$, $(C_2\times C_4):\GL(2,\mathbb{Z}/4)$, $C_2^2.(C_2^3\times S_4)$, $C_2\times C_4^2\times S_4$, $C_8:\GL(2,\mathbb{Z}/4)$, $C_8:\GL(2,\mathbb{Z}/4)$, $C_8\times \GL(2,\mathbb{Z}/4)$, $(C_2\times S_4):\OD_{16}$, $(C_2\times C_4).\GL(2,\mathbb{Z}/4)$, $(C_2\times C_4).\GL(2,\mathbb{Z}/4)$, $A_4:C_4\times \OD_{16}$, $A_4:C_4^3$, $Q_8.(C_2^3\times A_4)$, $\SL(2,3):C_2^5$, $D_4:C_2^3\times A_4$, $C_2^6.C_{12}$, $C_2^2\times A_4\times C_4^2$, $(C_2^2\times \SL(2,3)):D_4$, $C_2^2\wr C_2\times \SL(2,3)$, $C_{24}:C_2^5$

Order 720: $\SL(2,5):C_6$ x 3, $C_6\times \SL(2,5)$ x 2, $\SL(2,5):S_3$, $S_3\times \SL(2,5)$

Order 640: $D_{10}.C_2^5$ x 14, $D_{20}:C_4^2$ x 12, $C_2\times D_{10}:C_4^2$ x 12, $C_2\times D_{10}:C_4^2$ x 9, $(C_2^3\times F_5):C_4$ x 6, $C_{10}.C_4^3$ x 6, $(C_2\times C_{20}):C_4^2$ x 6, $C_{10}.C_4^3$ x 6, $(C_4\times D_{10}):D_4$ x 3, $D_{10}:(C_4\times D_4)$ x 3, $C_2^3.D_4\times D_5$ x 3, $D_{10}:(C_4\times D_4)$ x 3, $C_4\times C_2^2:D_{20}$ x 3, $C_2^2:C_4^2\times D_5$ x 3, $(C_2^3\times C_{20}):C_4$ x 3, $(C_2\times C_{20}):C_4^2$ x 3, $(C_2\times D_{10}):C_4^2$ x 3, $D_{10}.C_2^5$ x 3, $C_4\times D_4\times D_{10}$ x 3, $(D_4\times D_{10}):C_4$ x 3, $C_2\times C_{20}:C_4^2$ x 3, $D_{10}.(C_4\times D_4)$ x 3, $C_2\times C_{20}:C_4^2$ x 3, $C_{10}:C_4^3$ x 3, $C_4\times D_{10}:D_4$ x 3, $C_2^2\wr C_2\times F_5$ x 2, $C_2^5.D_{10}$, $C_2^5.D_{10}$, $F_5\times C_2^5$, $C_{20}:C_2^5$, $C_2^5:D_{10}$, $C_2^5:F_5$, $C_2^2\wr C_2\times C_{20}$, $C_2^5.D_{10}$, $C_2^5.D_{10}$

Order 480: $\GL(2,5)$ x 8, $C_2^2\times \SL(2,5)$ x 2, $\SL(2,5):C_2^2$ x 2

Order 384: $C_4\times \GL(2,\mathbb{Z}/4)$ x 2

Order 240: $\SL(2,5):C_2$ x 2, $C_2\times \SL(2,5)$

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2.\GL(2,\mathbb{Z}/4)$, $A_4:C_4^2$, $C_2^4:C_{12}$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^4.D_6$

Order 120: $\SL(2,5)$

Order 96: $\GL(2,\mathbb{Z}/4)$ x 6, $C_2^2.S_4$ x 3, $C_2^3:C_{12}$ x 2, $C_2^3\times A_4$, $C_2^2\times S_4$, $C_4\times S_4$, $C_{12}:D_4$

Order 64: $C_2^4\times C_4$

Order 48: $C_2^2\times A_4$ x 2, $C_4\times A_4$ x 2, $C_2\times S_4$, $A_4:C_4$

Order 32: $C_4\times D_4$ x 5, $C_2^3\times C_4$ x 2, $C_2^5$

Order 24: $C_3:D_4$ x 6, $C_2\times A_4$ x 3, $S_4$ x 3, $C_4\times S_3$

Order 16: $C_2^2\times C_4$ x 3, $C_2^4$ x 2

Order 12: $A_4$

Order 9: $C_3^2$

Order 8: $D_4$ x 15, $C_2\times C_4$ x 9, $C_2^3$ x 4

Order 6: $S_3$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 10, $C_2^2$ x 3

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 46080: $\GL(2,5)\times \GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 23040: $C_2\times S_4\times \GL(2,5)$ ($C_2$), $A_4:C_4\times \GL(2,5)$ ($C_2$), $(C_2^2\times A_4):\GL(2,5)$ ($C_2$), $\GL(2,\mathbb{Z}/4)\times \SL(2,5):C_2$ ($C_2$), $C_2^2\times A_4\times \GL(2,5)$ ($C_2$), $C_2.(S_4\times \GL(2,5))$ ($C_2$), $(C_2\times S_4):\GL(2,5)$ ($C_2$)

Order 11520: $C_2\times A_4:\GL(2,5)$ ($C_2^2$) x 2, $C_2\times A_4\times \GL(2,5)$ ($C_2^2$) x 2, $\SL(2,5):\GL(2,\mathbb{Z}/4)$ ($C_4$), $\GL(2,\mathbb{Z}/4)\times \SL(2,5)$ ($C_4$), $A_4\times \SL(2,5):C_2^3$ ($C_2^2$), $C_2^4.(D_6\times A_5)$ ($C_2^2$), $S_4\times \SL(2,5):C_2^2$ ($C_2^2$), $\SL(2,5):\GL(2,\mathbb{Z}/4)$ ($C_4$), $\SL(2,5):\GL(2,\mathbb{Z}/4)$ ($C_4$)

Order 7680: $C_2^4\times \GL(2,5)$ ($S_3$)

Order 5760: $A_4\times \GL(2,5)$ ($D_4$) x 2, $A_4:C_4\times \SL(2,5)$ ($C_2\times C_4$) x 2, $C_2^5.\GL(2,4)$ ($C_2\times C_4$), $C_2\times S_4\times \SL(2,5)$ ($C_2\times C_4$), $C_2\times \SL(2,5):S_4$ ($C_2\times C_4$), $A_4\times \SL(2,5):C_2^2$ ($C_2^3$), $A_4\times \SL(2,5):C_2^2$ ($C_2\times C_4$)

Order 3840: $C_2^3\times \GL(2,5)$ ($D_6$) x 2, $\SL(2,5):C_2^5$ ($D_6$)

Order 2880: $A_4\times \SL(2,5):C_2$ ($D_4:C_2$), $A_4\times \SL(2,5):C_2$ ($C_2\times D_4$), $C_2^4.\GL(2,4)$ ($C_2^2\times C_4$)

Order 1920: $C_2^2\times \GL(2,5)$ ($C_3:D_4$) x 2, $\SL(2,5):C_2^4$ ($C_4\times S_3$), $\SL(2,5):C_2^4$ ($C_2\times D_6$), $C_2^4\times \SL(2,5)$ ($C_4\times S_3$), $C_2^2\times \GL(2,5)$ ($S_4$)

Order 1440: $A_4\times \SL(2,5)$ ($C_4\times D_4$)

Order 960: $C_2\times \GL(2,5)$ ($C_2\times S_4$) x 2, $C_2^3\times \SL(2,5)$ ($C_4\times D_6$), $\SL(2,5):C_2^3$ ($C_2\times S_4$), $\SL(2,5):C_2^3$ ($C_6:D_4$), $\SL(2,5):C_2^3$ ($D_{12}:C_2$)

Order 480: $\GL(2,5)$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_2^2\times \SL(2,5)$ ($C_4\times S_4$), $C_2^2\times \SL(2,5)$ ($C_{12}:D_4$), $\SL(2,5):C_2^2$ ($C_2^2\times S_4$), $\SL(2,5):C_2^2$ ($C_4\times S_4$)

Order 384: $C_4\times \GL(2,\mathbb{Z}/4)$ ($S_5$)

Order 240: $C_2\times \SL(2,5)$ ($C_2^4.D_6$), $\SL(2,5):C_2$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $\SL(2,5):C_2$ ($\GL(2,\mathbb{Z}/4):C_2$)

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$ ($A_5:C_4$), $C_2.\GL(2,\mathbb{Z}/4)$ ($A_5:C_4$), $C_2.\GL(2,\mathbb{Z}/4)$ ($A_5:C_4$), $A_4:C_4^2$ ($C_2\times S_5$), $C_2^4:C_{12}$ ($C_2\times S_5$), $C_2\times \GL(2,\mathbb{Z}/4)$ ($A_5:C_4$), $C_2^4.D_6$ ($C_2\times S_5$)

Order 120: $\SL(2,5)$ ($C_4\times \GL(2,\mathbb{Z}/4)$)

Order 96: $\GL(2,\mathbb{Z}/4)$ ($\GL(2,5)$) x 4, $C_2^2.S_4$ ($C_2^2.S_5$) x 3, $C_2^3\times A_4$ ($C_2^2.S_5$), $C_2^2\times S_4$ ($C_2^2.S_5$), $C_2^3:C_{12}$ ($C_2^2.S_5$), $C_2^3:C_{12}$ ($C_2^2\times S_5$)

Order 64: $C_2^4\times C_4$ ($S_3\times S_5$)

Order 48: $C_2^2\times A_4$ ($C_2\times \GL(2,5)$) x 2, $C_2\times S_4$ ($C_2\times \GL(2,5)$) x 2, $A_4:C_4$ ($C_2\times \GL(2,5)$) x 2, $C_2^2\times A_4$ ($C_2^3.S_5$), $C_4\times A_4$ ($D_4:S_5$), $C_4\times A_4$ ($D_4\times S_5$)

Order 32: $C_2^5$ ($D_6.S_5$), $C_2^3\times C_4$ ($D_6\times S_5$), $C_2^3\times C_4$ ($D_6.S_5$)

Order 24: $C_2\times A_4$ ($\GL(2,5):C_2^2$), $C_2\times A_4$ ($C_2^2\times \GL(2,5)$), $C_2\times A_4$ ($A_5:C_4\times D_4$)

Order 16: $C_2^4$ ($S_3\times \GL(2,5)$) x 2, $C_2^4$ ($A_5:C_4\times D_6$), $C_2^2\times C_4$ ($C_3:D_4.S_5$), $C_2^2\times C_4$ ($S_4.S_5$), $C_2^2\times C_4$ ($C_3:D_4\times S_5$)

Order 12: $A_4$ ($D_4\times \GL(2,5)$)

Order 8: $C_2^3$ ($C_3:D_4.A_5:C_4$), $C_2^3$ ($D_{12}:C_2.S_5$), $C_2^3$ ($D_6\times \GL(2,5)$), $C_2^3$ ($A_5:C_4\times S_4$), $C_2\times C_4$ ($A_5:C_4\times S_4$), $C_2\times C_4$ ($C_2\times S_4\times S_5$)

Order 4: $C_2^2$ ($S_4\times \GL(2,5)$) x 2, $C_2^2$ ($C_3:D_4\times \GL(2,5)$), $C_2^2$ ($C_2^2.S_5\times S_4$), $C_4$ ($\GL(2,\mathbb{Z}/4).S_5$), $C_4$ ($S_5\times \GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($\GL(2,\mathbb{Z}/4):C_2.S_5$), $C_2$ ($\GL(2,\mathbb{Z}/4).A_5:C_4$), $C_2$ ($C_2\times S_4\times \GL(2,5)$)

Order 1: $C_1$ ($\GL(2,5)\times \GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 46080: $\GL(2,5)\times \GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 23040: $C_2\times S_4\times \GL(2,5)$ ($C_2$), $A_4:C_4\times \GL(2,5)$ ($C_2$), $(C_2^2\times A_4):\GL(2,5)$ ($C_2$), $\GL(2,\mathbb{Z}/4)\times \SL(2,5):C_2$ ($C_2$), $C_2^2\times A_4\times \GL(2,5)$ ($C_2$), $C_2.(S_4\times \GL(2,5))$ ($C_2$), $(C_2\times S_4):\GL(2,5)$ ($C_2$)

Order 11520: $C_2\times A_4:\GL(2,5)$ ($C_2^2$) x 2, $C_2\times A_4\times \GL(2,5)$ ($C_2^2$) x 2, $\SL(2,5):\GL(2,\mathbb{Z}/4)$ ($C_4$), $\GL(2,\mathbb{Z}/4)\times \SL(2,5)$ ($C_4$), $A_4\times \SL(2,5):C_2^3$ ($C_2^2$), $C_2^4.(D_6\times A_5)$ ($C_2^2$), $S_4\times \SL(2,5):C_2^2$ ($C_2^2$), $\SL(2,5):\GL(2,\mathbb{Z}/4)$ ($C_4$), $\SL(2,5):\GL(2,\mathbb{Z}/4)$ ($C_4$)

Order 7680: $C_2^4\times \GL(2,5)$ ($S_3$)

Order 5760: $A_4:C_4\times \SL(2,5)$ ($C_2\times C_4$) x 2, $A_4\times \GL(2,5)$ ($D_4$), $C_2^5.\GL(2,4)$ ($C_2\times C_4$), $C_2\times S_4\times \SL(2,5)$ ($C_2\times C_4$), $C_2\times \SL(2,5):S_4$ ($C_2\times C_4$), $A_4\times \SL(2,5):C_2^2$ ($C_2^3$), $A_4\times \SL(2,5):C_2^2$ ($C_2\times C_4$)

Order 3840: $C_2^3\times \GL(2,5)$ ($D_6$) x 2, $\SL(2,5):C_2^5$ ($D_6$)

Order 2880: $A_4\times \SL(2,5):C_2$ ($D_4:C_2$), $A_4\times \SL(2,5):C_2$ ($C_2\times D_4$), $C_2^4.\GL(2,4)$ ($C_2^2\times C_4$)

Order 1920: $\SL(2,5):C_2^4$ ($C_4\times S_3$), $\SL(2,5):C_2^4$ ($C_2\times D_6$), $C_2^4\times \SL(2,5)$ ($C_4\times S_3$), $C_2^2\times \GL(2,5)$ ($C_3:D_4$), $C_2^2\times \GL(2,5)$ ($S_4$)

Order 1440: $A_4\times \SL(2,5)$ ($C_4\times D_4$)

Order 960: $C_2\times \GL(2,5)$ ($C_2\times S_4$) x 2, $C_2^3\times \SL(2,5)$ ($C_4\times D_6$), $\SL(2,5):C_2^3$ ($C_2\times S_4$), $\SL(2,5):C_2^3$ ($C_6:D_4$), $\SL(2,5):C_2^3$ ($D_{12}:C_2$)

Order 480: $C_2^2\times \SL(2,5)$ ($C_4\times S_4$), $C_2^2\times \SL(2,5)$ ($C_{12}:D_4$), $\SL(2,5):C_2^2$ ($C_2^2\times S_4$), $\SL(2,5):C_2^2$ ($C_4\times S_4$), $\GL(2,5)$ ($\GL(2,\mathbb{Z}/4)$)

Order 384: $C_4\times \GL(2,\mathbb{Z}/4)$ ($S_5$)

Order 240: $C_2\times \SL(2,5)$ ($C_2^4.D_6$), $\SL(2,5):C_2$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $\SL(2,5):C_2$ ($\GL(2,\mathbb{Z}/4):C_2$)

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$ ($A_5:C_4$), $C_2.\GL(2,\mathbb{Z}/4)$ ($A_5:C_4$), $C_2.\GL(2,\mathbb{Z}/4)$ ($A_5:C_4$), $A_4:C_4^2$ ($C_2\times S_5$), $C_2^4:C_{12}$ ($C_2\times S_5$), $C_2\times \GL(2,\mathbb{Z}/4)$ ($A_5:C_4$), $C_2^4.D_6$ ($C_2\times S_5$)

Order 120: $\SL(2,5)$ ($C_4\times \GL(2,\mathbb{Z}/4)$)

Order 96: $C_2^2.S_4$ ($C_2^2.S_5$) x 3, $C_2^3\times A_4$ ($C_2^2.S_5$), $C_2^2\times S_4$ ($C_2^2.S_5$), $C_2^3:C_{12}$ ($C_2^2.S_5$), $C_2^3:C_{12}$ ($C_2^2\times S_5$), $\GL(2,\mathbb{Z}/4)$ ($\GL(2,5)$)

Order 64: $C_2^4\times C_4$ ($S_3\times S_5$)

Order 48: $C_2^2\times A_4$ ($C_2\times \GL(2,5)$), $C_2^2\times A_4$ ($C_2^3.S_5$), $C_2\times S_4$ ($C_2\times \GL(2,5)$), $C_4\times A_4$ ($D_4:S_5$), $C_4\times A_4$ ($D_4\times S_5$), $A_4:C_4$ ($C_2\times \GL(2,5)$)

Order 32: $C_2^5$ ($D_6.S_5$), $C_2^3\times C_4$ ($D_6\times S_5$), $C_2^3\times C_4$ ($D_6.S_5$)

Order 24: $C_2\times A_4$ ($\GL(2,5):C_2^2$), $C_2\times A_4$ ($C_2^2\times \GL(2,5)$), $C_2\times A_4$ ($A_5:C_4\times D_4$)

Order 16: $C_2^4$ ($A_5:C_4\times D_6$), $C_2^4$ ($S_3\times \GL(2,5)$), $C_2^2\times C_4$ ($C_3:D_4.S_5$), $C_2^2\times C_4$ ($S_4.S_5$), $C_2^2\times C_4$ ($C_3:D_4\times S_5$)

Order 12: $A_4$ ($D_4\times \GL(2,5)$)

Order 8: $C_2^3$ ($C_3:D_4.A_5:C_4$), $C_2^3$ ($D_{12}:C_2.S_5$), $C_2^3$ ($D_6\times \GL(2,5)$), $C_2^3$ ($A_5:C_4\times S_4$), $C_2\times C_4$ ($A_5:C_4\times S_4$), $C_2\times C_4$ ($C_2\times S_4\times S_5$)

Order 4: $C_2^2$ ($S_4\times \GL(2,5)$), $C_2^2$ ($C_3:D_4\times \GL(2,5)$), $C_2^2$ ($C_2^2.S_5\times S_4$), $C_4$ ($\GL(2,\mathbb{Z}/4).S_5$), $C_4$ ($S_5\times \GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($\GL(2,\mathbb{Z}/4):C_2.S_5$), $C_2$ ($\GL(2,\mathbb{Z}/4).A_5:C_4$), $C_2$ ($C_2\times S_4\times \GL(2,5)$)

Order 1: $C_1$ ($\GL(2,5)\times \GL(2,\mathbb{Z}/4)$)

Series

Derived series

$\GL(2,5)\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^4.\GL(2,4)$

$\rhd$

$C_2^2\times \SL(2,5)$

$\rhd$

$\SL(2,5)$

Chief series

$\GL(2,5)\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$\GL(2,\mathbb{Z}/4)\times \SL(2,5):C_2$

$\rhd$

$C_4\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^2\times S_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$\GL(2,5)\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^4.\GL(2,4)$

$\rhd$

$A_4\times \SL(2,5)$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2^2\times C_4$

Character theory

Complex character table

See the $336 \times 336$ 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 $195 \times 195$ rational character table (warning: may be slow to load).