Abstract group 46080.bx: $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$

• Label: 46080.bx
• Order: \( 2^{10} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $15$
• Trans deg.: not computed
• Rank: $5$

Group information

Description:	$(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$
Order:	\(46080\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_{11}^2:C_{44}$, of order \(2949120\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 8, $C_3$, $A_5$
Derived length:	$3$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	3119	188	11216	24	10948	2856	11392	192	3264	2112	768	46080
Conjugacy classes	1	71	3	88	1	85	23	52	1	16	7	2	350
Divisions	1	71	3	88	1	85	23	52	1	16	7	2	350
Autjugacy classes	1	25	3	30	1	29	9	20	1	6	3	1	129

Dimension	1	2	3	4	5	6	8	10	12	15	16	18	20	24	30	36
Irr. complex chars.	32	24	32	36	32	24	24	24	44	32	4	16	4	10	8	4	350
Irr. rational chars.	32	24	32	36	32	24	24	24	44	32	4	16	4	10	8	4	350

Minimal presentations

Permutation degree:	$15$
Transitive degree:	not computed
Rank:	$5$
Inequivalent generating 5-tuples:	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 $15$ $\langle(8,9)(10,11)(12,15,13,14), (1,2), (14,15), (1,4,5)(2,3)(6,8,11,7,9,10), (6,7) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $D_4$ $\, \times\, $ $S_4$ $\, \times\, $ $S_5$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^5\times S_5)$ . $D_6$	$(C_2^5:S_5)$ . $D_6$	$C_2^5$ . $(D_6\times S_5)$	$(D_4\times S_4\times S_5)$ . $C_2$	all 175
Aut. group:	$\Aut(C_2\times A_4\times \GL(2,5))$	$\Aut(C_2\times A_4:\GL(2,5))$	$\Aut(D_4\times A_4:S_5)$	$\Aut(D_4\times A_4\times S_5)$	all 8

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{12}$
Commutator length:	$1$

Subgroups

There are 697 normal subgroups (69 characteristic).

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

Special subgroups

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

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 36 (order at least 1280) are shown, as well as all normal subgroups of any index.

Order 46080: $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$

Order 23040: $D_4\times S_4\times S_5$ x 16, $C_2^3\times S_5\times S_4$ x 2, $(C_2\times \GL(2,\mathbb{Z}/4)).S_5$ x 2, $(C_2\times S_5).\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.(S_4\times S_5)$ x 2, $(C_2^4\times S_5):D_6$, $C_2^5.(C_6\times S_5)$, $(A_5\times \GL(2,\mathbb{Z}/4)).C_2^2$, $(C_2^5\times A_5):D_6$, $C_2^3:D_{12}\times S_5$, $C_2^4.D_6\times S_5$, $(C_2^4\times S_5):D_6$

Order 15360: $C_2\times S_5\times D_4^2$

Order 11520: $C_2^2\times S_4\times S_5$ x 39, $S_5\times \GL(2,\mathbb{Z}/4)$ x 16, $S_4\times C_2^2:S_5$ x 16, $\GL(2,\mathbb{Z}/4):S_5$ x 16, $C_4:(S_4\times S_5)$ x 8, $S_4\times C_4:S_5$ x 8, $C_4:S_4\times S_5$ x 8, $C_4\times S_4\times S_5$ x 8, $D_4\times S_4\times A_5$ x 8, $D_4\times A_4\times S_5$ x 8, $D_4\times A_4:S_5$ x 8, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times S_4):S_5$ x 2, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times A_4).S_5$ x 2, $(C_2^3\times A_4).S_5$ x 2, $A_5.(C_2^3\times S_4)$ x 2, $C_2^3\times A_4:S_5$ x 2, $C_2\times A_5\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^2.S_4\times S_5$, $C_2^3:D_{12}\times A_5$, $C_2^4.D_6\times A_5$, $D_4\times D_6\times S_5$, $C_2^2.(S_4\times S_5)$, $(C_2^4\times A_5).D_6$, $C_2^2.S_5\times S_4$, $C_2^5:C_6\times A_5$, $(C_2^3\times A_5):D_{12}$, $C_2^3:C_{12}\times S_5$, $(C_2^4\times S_5):C_6$

Order 9216: $C_2^6.D_6^2$

Order 7680: $D_4^2.S_5$ x 32, $C_2^3:D_4\times S_5$ x 4, $C_2\times A_5:D_4^2$ x 4, $A_5:(C_2\times D_4^2)$ x 4, $C_2^3\times S_5\times D_4$ x 4, $A_5.(C_2\times D_4^2)$ x 4, $C_2^3.(D_4\times S_5)$ x 4, $C_4^2:C_2^2\times S_5$ x 2, $C_2\times A_5:D_4^2$ x 2, $C_2\times A_5:D_4^2$, $D_{10}.(C_2^4\times S_4)$, $C_2\times A_5\times D_4^2$, $C_4^2.(C_2^2\times S_5)$

Order 5760: $C_2\times S_4\times S_5$ x 124, $C_2^2\times A_4:S_5$ x 19, $C_2^2\times A_4\times S_5$ x 19, $C_2^2\times S_4\times A_5$ x 19, $S_3\times D_4\times S_5$ x 16, $A_5:\GL(2,\mathbb{Z}/4)$ x 8, $A_4\times C_2^2:S_5$ x 8, $(C_2\times S_4):S_5$ x 8, $A_5:\GL(2,\mathbb{Z}/4)$ x 8, $A_5:\GL(2,\mathbb{Z}/4)$ x 8, $A_5\times \GL(2,\mathbb{Z}/4)$ x 8, $D_4\times A_4\times A_5$ x 4, $A_5\times C_4:S_4$ x 4, $C_4\times S_4\times A_5$ x 4, $(C_4\times A_4):S_5$ x 4, $A_4\times C_4:S_5$ x 4, $C_4\times A_4\times S_5$ x 4, $C_4\times A_4:S_5$ x 4, $A_4:(C_4\times S_5)$ x 4, $A_4:C_4\times S_5$ x 4, $A_5:C_4\times S_4$ x 4, $C_2^5:\GL(2,4)$ x 2, $S_3\times C_2^3:S_5$ x 2, $C_6:(D_4\times S_5)$ x 2, $C_6:D_4\times S_5$ x 2, $\GL(2,4):C_2^5$ x 2, $C_2^2.S_4\times A_5$, $(C_2^2\times A_4).S_5$, $A_4\times C_2^2.S_5$, $C_2^3:C_{12}\times A_5$, $C_4:S_5\times D_6$, $C_6\times D_4\times S_5$, $C_2\times D_{12}:S_5$, $C_6:S_5\times D_4$, $C_4\times D_6\times S_5$, $C_2\times D_{12}\times S_5$, $D_4\times D_6\times A_5$

Order 4608: $D_4\times S_4^2$ x 16, $C_2^3:S_4^2$ x 4, $C_2^3:S_4^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^3\times S_4^2$ x 2, $A_4\times \GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $A_4^2:C_2^2\times D_4$

Order 3840: $C_2^2\times D_4\times S_5$ x 144, $C_4:D_4\times S_5$ x 64, $A_5.D_4^2$ x 64, $A_5:D_4^2$ x 64, $A_5:D_4^2$ x 64, $S_5\times C_2^2\wr C_2$ x 64, $C_4\times D_4\times S_5$ x 32, $A_5:D_4^2$ x 32, $C_4:D_4\times S_5$ x 16, $A_5:D_4^2$ x 16, $D_4\times F_5\times S_4$ x 16, $A_5\times D_4^2$ x 16, $C_2^2.(D_4\times S_5)$ x 12, $(C_2^2\times D_4).S_5$ x 12, $(C_2^2\times S_5):D_4$ x 8, $(C_2^2\times D_4):S_5$ x 8, $C_2^5:S_5$ x 8, $C_2^3:C_4\times S_5$ x 8, $(C_2^2\times D_4):S_5$ x 6, $(C_2^3\times C_4):S_5$ x 4, $(C_2^2\times D_4):S_5$ x 4, $D_4\times C_2^3\times A_5$ x 4, $C_4\times C_2^3\times S_5$ x 4, $(C_2^2\times S_5):D_4$ x 4, $C_2^3:D_4\times A_5$ x 4, $C_2^3:(C_4\times S_5)$ x 4, $C_2^3\times C_4:S_5$ x 4, $C_4\times C_2^3:S_5$ x 4, $C_2^5\times S_5$ x 4, $C_2^5.S_5$ x 4, $C_2^3.(D_4\times A_5)$ x 4, $(C_2\times C_4^2):S_5$ x 2, $C_2^2.(D_4\times S_5)$ x 2, $C_2^2.D_4\times S_5$ x 2, $C_4^2:C_2^2\times A_5$ x 2, $C_2^3\times F_5\times S_4$ x 2, $C_2^2.S_5\times D_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4):F_5$ x 2, $C_2^3:F_5\times S_4$ x 2, $C_2\times F_5\times \GL(2,\mathbb{Z}/4)$ x 2, $D_{10}.(D_4\times S_4)$, $C_2^5:C_6\times F_5$, $C_2\times C_4^2\times S_5$, $D_4\times D_{10}.S_4$, $C_4^2:C_2^2\times A_5$, $D_{10}.D_4\times S_4$, $D_4\times D_{10}\times S_4$, $C_2^3:D_{12}\times F_5$, $(C_2\times C_4^2).S_5$, $C_{10}:C_4^2\times S_4$

Order 3072: $A_4.C_2^5.C_2^3$ x 2

Order 2880: $S_4\times S_5$ x 64, $\GL(2,4):C_2^4$ x 39, $C_2\times S_4\times A_5$ x 30, $C_2\times A_4\times S_5$ x 30, $C_2\times A_4:S_5$ x 30, $C_3:(D_4\times S_5)$ x 16, $S_3\times C_2^2:S_5$ x 16, $C_3:D_4\times S_5$ x 16, $C_2^4:\GL(2,4)$ x 9, $D_{12}:S_5$ x 8, $S_3\times C_4:S_5$ x 8, $D_{12}\times S_5$ x 8, $S_3\times D_4\times A_5$ x 8, $C_4\times S_3\times S_5$ x 8, $D_4\times C_3:S_5$ x 8, $C_3\times D_4\times S_5$ x 8, $\GL(2,4):C_2^4$ x 2, $C_4\times A_4\times A_5$ x 2, $C_6:D_4\times A_5$ x 2, $A_4:C_4\times A_5$ x 2, $\GL(2,4):C_2^4$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times \GL(2,4):D_4$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times \GL(2,4):D_4$ x 2, $C_2^2:S_5\times C_6$ x 2, $\GL(2,4):C_2^4$ x 2, $(A_4\times A_5):C_4$ x 2, $A_4\times A_5:C_4$ x 2, $C_2\times D_{12}\times A_5$, $C_4\times D_6\times A_5$, $A_5:C_4\times D_6$, $C_6\times D_4\times A_5$, $C_2\times C_{12}:S_5$, $C_6:(C_4\times S_5)$, $C_6:C_4\times S_5$, $C_2\times C_{12}:S_5$, $C_2\times C_{12}\times S_5$, $C_2\times C_{12}:S_5$

Order 2560: $C_2\times F_5\times D_4^2$

Order 2304: $C_2^4:D_6^2$ x 49, $C_2^2\times S_4^2$ x 39, $S_4\times \GL(2,\mathbb{Z}/4)$ x 32, $C_4:S_4^2$ x 16, $\GL(2,\mathbb{Z}/4):S_4$ x 16, $D_4\times A_4\times S_4$ x 16, $C_4:S_4^2$ x 8, $C_4\times S_4^2$ x 8, $D_4\times \PSOPlus(4,3)$ x 8, $C_2^2.S_4^2$ x 4, $C_2\times A_4\times \GL(2,\mathbb{Z}/4)$ x 4, $A_4^2:C_2^4$ x 4, $C_2^2.S_4^2$ x 2, $A_4^2:C_2^4$ x 2, $C_2^4.D_6^2$ x 2, $C_2^4:D_6^2$ x 2, $C_2^4:D_6^2$ x 2, $C_2^4:D_6^2$ x 2, $A_4\times C_2^3:D_{12}$ x 2, $C_2^2.S_4^2$ x 2, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times C_4\times S_4\times A_4$ x 2, $C_2\times A_4^2:D_4$, $C_4\times A_4^2:C_2^2$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_{12}:C_2^3\times S_4$, $C_{12}:(C_2^3\times S_4)$, $C_{12}.(A_4\times C_2^4)$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^2.S_4^2$, $C_2^6:C_6^2$

Order 1920: $C_2\times D_4\times S_5$ x 883, $C_2^4:S_5$ x 144, $C_2^4\times S_5$ x 108, $C_2.(D_4\times S_5)$ x 96, $(C_2\times D_4).S_5$ x 96, $C_2^2\times D_4\times A_5$ x 72, $C_2^2\times C_4:S_5$ x 72, $C_4\times C_2^2\times S_5$ x 71, $(C_2\times S_5):D_4$ x 64, $(C_2\times D_4):S_5$ x 64, $C_2^2:C_4\times S_5$ x 64, $(C_2\times D_4):S_5$ x 48, $C_2^2\times F_5\times S_4$ x 40, $C_4:D_4\times A_5$ x 32, $(C_2^2\times C_4):S_5$ x 32, $(C_2\times S_5):D_4$ x 32, $(C_2\times D_4):S_5$ x 32, $A_5\times C_2^2\wr C_2$ x 32, $C_2^4.S_5$ x 32, $C_2^2:(C_4\times S_5)$ x 32, $C_4\times C_2^2:S_5$ x 32, $C_4\times D_4\times A_5$ x 16, $C_4:(C_4\times S_5)$ x 16, $A_5:C_4\times D_4$ x 16, $C_4^2:S_5$ x 16, $C_4:C_4\times S_5$ x 16, $(C_2^2\times C_4):S_5$ x 16, $\GL(2,\mathbb{Z}/4):D_{10}$ x 16, $\GL(2,\mathbb{Z}/4):F_5$ x 16, $D_{10}:C_4\times S_4$ x 16, $F_5\times \GL(2,\mathbb{Z}/4)$ x 16, $C_4:D_4\times A_5$ x 8, $C_4^2:S_5$ x 8, $C_2^3:C_4\times A_5$ x 8, $C_2^4.S_5$ x 8, $C_4^2\times S_5$ x 8, $D_4\times A_4\times F_5$ x 8, $D_4\times A_4:F_5$ x 8, $C_{20}:(C_4\times S_4)$ x 8, $C_{20}:C_4\times S_4$ x 8, $F_5\times C_4:S_4$ x 8, $C_4\times F_5\times S_4$ x 8, $C_2^5\times A_5$ x 4, $C_4\times C_2^3\times A_5$ x 4, $C_2^4.S_5$ x 4, $C_2^2.(C_4\times S_5)$ x 4, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 4, $C_2\times A_5:C_4^2$ x 3, $C_2^2.D_4\times A_5$ x 2, $(C_2^2\times C_4).S_5$ x 2, $C_2^2\times D_{10}\times S_4$ x 2, $C_2^3\times A_4\times F_5$ x 2, $C_2^3\times A_4:F_5$ x 2, $\GL(2,\mathbb{Z}/4):D_{10}$ x 2, $C_{10}:D_4\times S_4$ x 2, $D_{10}\times \GL(2,\mathbb{Z}/4)$ x 2, $A_4\times C_2^3:F_5$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $(C_{10}\times A_4):C_4^2$ x 2, $C_2\times C_4^2\times A_5$, $D_4\times A_4\times D_{10}$, $C_2^5:D_{30}$, $C_2\times D_{20}:S_4$, $C_2\times S_4\times D_{20}$, $C_4:S_4\times D_{10}$, $C_4\times D_{10}\times S_4$, $(C_2^3\times C_{20}):C_{12}$, $A_4\times C_{10}:C_4^2$, $(C_4\times D_{10}).S_4$, $C_4\times D_{10}.S_4$, $D_4\times C_{10}\times S_4$, $D_4\times D_6\times F_5$

Order 1536: $S_4\times D_4^2$ x 64, $\GL(2,\mathbb{Z}/4):C_2^4$ x 8, $C_2^7:D_6$ x 8, $C_2\times A_4:D_4^2$ x 8, $C_2^3:D_4\times S_4$ x 8, $C_2^7:D_6$ x 8, $C_2^7:D_6$ x 8, $C_2\times A_4:D_4^2$ x 4, $C_4\times \GL(2,\mathbb{Z}/4):C_2^2$ x 4, $C_2\times A_4:D_4^2$ x 2, $C_4^2:C_2^2\times S_4$ x 2, $C_2\times A_4\times D_4^2$ x 2, $C_2^7:D_6$, $C_2\times D_6\times D_4^2$

Order 1440: $D_6\times S_5$ x 124, $\GL(2,4):C_2^3$ x 19, $\GL(2,4):C_2^3$ x 19, $\GL(2,4):C_2^3$ x 19, $S_4\times A_5$ x 8, $A_4:S_5$ x 8, $A_4\times S_5$ x 8, $\GL(2,4):D_4$ x 8, $\GL(2,4):D_4$ x 8, $\GL(2,4):D_4$ x 8, $D_6:S_5$ x 8, $\GL(2,4):D_4$ x 8, $D_6:S_5$ x 8, $C_2^3:\GL(2,4)$ x 7, $D_4\times \GL(2,4)$ x 4, $D_{12}\times A_5$ x 4, $C_4\times S_3\times A_5$ x 4, $C_{12}:S_5$ x 4, $C_{12}:S_5$ x 4, $C_{12}:S_5$ x 4, $C_{12}\times S_5$ x 4, $C_3:C_4\times S_5$ x 4, $D_6.S_5$ x 4, $C_3:(C_4\times S_5)$ x 4, $C_2^3\times \GL(2,4)$ x 2, $C_2\times C_{12}\times A_5$, $C_6:C_4\times A_5$, $C_2\times \GL(2,4):C_4$, $A_5:C_4\times C_6$

Order 1280: $F_5\times D_4^2$ x 32, $D_{10}.C_2^6$ x 4, $D_{10}.D_4^2$ x 4, $D_{10}.D_4^2$ x 4, $D_{10}.D_4^2$ x 4, $C_2^3:D_4\times F_5$ x 4, $C_2^3:D_4\times F_5$ x 4, $D_{10}.D_4^2$ x 2, $D_4\times C_{10}:C_4^2$ x 2, $D_{10}.D_4^2$, $C_4^2:C_2^2\times F_5$, $D_{10}\times D_4^2$

Order 1024: $C_2^5.C_2^5$

Order 960: $C_2^3\times S_5$ x 12, $D_4\times S_5$ x 8, $C_2^4\times A_5$ x 5, $C_4\times C_2^2\times A_5$ x 2, $C_2^3:S_5$ x 2, $C_2^3.S_5$ x 2, $C_2\times D_4\times A_5$, $C_2\times C_4:S_5$, $C_2\times C_4\times S_5$

Order 720: $A_4\times A_5$

Order 480: $C_2^2\times S_5$ x 15, $C_2^2:S_5$ x 8, $C_2^3\times A_5$ x 5, $D_4\times A_5$ x 4, $C_4:S_5$ x 4, $C_4\times S_5$ x 4, $C_2\times C_4\times A_5$, $C_2^2.S_5$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$

Order 240: $C_2\times S_5$ x 10, $C_2^2\times A_5$ x 6, $C_4\times A_5$ x 2, $A_5:C_4$ x 2

Order 192: $D_4\times S_4$ x 8, $C_2^3\times S_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:C_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 120: $S_5$ x 4, $C_2\times A_5$ x 3

Order 96: $C_2^2\times S_4$ x 11, $\GL(2,\mathbb{Z}/4)$ x 8, $D_4\times A_4$ x 4, $C_4:S_4$ x 4, $C_4\times S_4$ x 4, $C_2^3\times A_4$ x 2, $C_2^3:C_{12}$, $C_2^2.S_4$

Order 64: $D_4\times C_2^3$

Order 60: $A_5$

Order 48: $C_2\times S_4$ x 10, $C_2^2\times A_4$ x 5, $C_4\times A_4$ x 2, $A_4:C_4$ x 2

Order 32: $C_2^2\times D_4$ x 4, $C_2^5$ x 2, $C_2^3\times C_4$

Order 24: $S_4$ x 4, $C_2\times A_4$ x 3

Order 16: $C_2^4$ x 5, $C_2^2\times C_4$ x 2, $C_2\times D_4$

Order 12: $A_4$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 5, $D_4$ x 4, $C_2\times C_4$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 36 (order at least 1280) are shown, as well as all normal subgroups of any index.

Order 46080: $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$

Order 23040: $C_2^3\times S_5\times S_4$, $(C_2^4\times S_5):D_6$, $(C_2\times \GL(2,\mathbb{Z}/4)).S_5$, $(C_2\times S_5).\GL(2,\mathbb{Z}/4)$, $C_2^5.(C_6\times S_5)$, $(A_5\times \GL(2,\mathbb{Z}/4)).C_2^2$, $(C_2^5\times A_5):D_6$, $C_2^3:D_{12}\times S_5$, $C_2^4.D_6\times S_5$, $(C_2^4\times S_5):D_6$, $D_4\times S_4\times S_5$, $C_2^3.(S_4\times S_5)$

Order 15360: $C_2\times S_5\times D_4^2$

Order 11520: $C_2^2\times S_4\times S_5$ x 6, $C_2^2.S_4\times S_5$, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}\times A_5$, $C_2^4.D_6\times A_5$, $D_4\times D_6\times S_5$, $C_2^2.(S_4\times S_5)$, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$, $C_4:(S_4\times S_5)$, $S_4\times C_4:S_5$, $(C_2^4\times A_5).D_6$, $C_2^2.S_5\times S_4$, $(C_2^2\times S_4):S_5$, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$, $C_4:S_4\times S_5$, $C_4\times S_4\times S_5$, $(C_2^3\times A_4).S_5$, $(C_2^3\times A_4).S_5$, $A_5.(C_2^3\times S_4)$, $C_2^5:C_6\times A_5$, $C_2^3\times A_4:S_5$, $S_5\times \GL(2,\mathbb{Z}/4)$, $S_4\times C_2^2:S_5$, $D_4\times S_4\times A_5$, $(C_2^3\times A_5):D_{12}$, $C_2^3:C_{12}\times S_5$, $(C_2^4\times S_5):C_6$, $D_4\times A_4\times S_5$, $D_4\times A_4:S_5$, $\GL(2,\mathbb{Z}/4):S_5$, $C_2\times A_5\times \GL(2,\mathbb{Z}/4)$

Order 9216: $C_2^6.D_6^2$

Order 7680: $C_2^3:D_4\times S_5$ x 3, $C_2\times A_5:D_4^2$ x 3, $C_2^3\times S_5\times D_4$ x 3, $A_5.(C_2\times D_4^2)$ x 3, $D_4^2.S_5$ x 2, $A_5:(C_2\times D_4^2)$ x 2, $C_4^2:C_2^2\times S_5$ x 2, $C_2\times A_5:D_4^2$ x 2, $C_2^3.(D_4\times S_5)$ x 2, $C_2\times A_5:D_4^2$, $D_{10}.(C_2^4\times S_4)$, $C_2\times A_5\times D_4^2$, $C_4^2.(C_2^2\times S_5)$

Order 5760: $C_2\times S_4\times S_5$ x 9, $C_2^2\times A_4:S_5$ x 4, $C_2^2\times A_4\times S_5$ x 4, $C_2^2\times S_4\times A_5$ x 4, $C_2^2.S_4\times A_5$, $(C_2^2\times A_4).S_5$, $A_4\times C_2^2.S_5$, $C_2^3:C_{12}\times A_5$, $C_2^5:\GL(2,4)$, $C_4:S_5\times D_6$, $C_6\times D_4\times S_5$, $C_2\times D_{12}:S_5$, $S_3\times C_2^3:S_5$, $D_4\times A_4\times A_5$, $A_5\times C_4:S_4$, $C_4\times S_4\times A_5$, $C_6:(D_4\times S_5)$, $C_6:S_5\times D_4$, $(C_4\times A_4):S_5$, $A_4\times C_4:S_5$, $C_4\times A_4\times S_5$, $C_4\times A_4:S_5$, $C_6:D_4\times S_5$, $C_4\times D_6\times S_5$, $C_2\times D_{12}\times S_5$, $\GL(2,4):C_2^5$, $A_5:\GL(2,\mathbb{Z}/4)$, $A_4\times C_2^2:S_5$, $A_4:(C_4\times S_5)$, $(C_2\times S_4):S_5$, $A_4:C_4\times S_5$, $A_5:\GL(2,\mathbb{Z}/4)$, $A_5:\GL(2,\mathbb{Z}/4)$, $D_4\times D_6\times A_5$, $S_3\times D_4\times S_5$, $A_5\times \GL(2,\mathbb{Z}/4)$, $A_5:C_4\times S_4$

Order 4608: $C_2^3:S_4^2$ x 2, $C_2^5.D_6^2$ x 2, $A_4\times \GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^3:S_4^2$, $C_2^3\times S_4^2$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $A_4^2:C_2^2\times D_4$, $D_4\times S_4^2$

Order 3840: $C_2^2\times D_4\times S_5$ x 34, $C_2^2.(D_4\times S_5)$ x 6, $(C_2^2\times D_4).S_5$ x 6, $A_5:D_4^2$ x 5, $C_2^3:C_4\times S_5$ x 5, $(C_2^2\times S_5):D_4$ x 4, $(C_2^2\times D_4):S_5$ x 4, $C_4:D_4\times S_5$ x 4, $(C_2^2\times D_4):S_5$ x 4, $C_2^5:S_5$ x 4, $A_5:D_4^2$ x 4, $(C_2^3\times C_4):S_5$ x 3, $(C_2^2\times D_4):S_5$ x 3, $D_4\times C_2^3\times A_5$ x 3, $C_4\times C_2^3\times S_5$ x 3, $(C_2^2\times S_5):D_4$ x 3, $C_2^3:D_4\times A_5$ x 3, $C_2^3:(C_4\times S_5)$ x 3, $C_2^3\times C_4:S_5$ x 3, $C_4\times C_2^3:S_5$ x 3, $C_4\times D_4\times S_5$ x 3, $A_5.D_4^2$ x 3, $A_5:D_4^2$ x 3, $S_5\times C_2^2\wr C_2$ x 3, $(C_2\times C_4^2):S_5$ x 2, $C_2^2.(D_4\times S_5)$ x 2, $C_2^2.D_4\times S_5$ x 2, $C_4^2:C_2^2\times A_5$ x 2, $C_2^2.S_5\times D_4$ x 2, $C_2^5\times S_5$ x 2, $C_2^5.S_5$ x 2, $C_2^3.(D_4\times A_5)$ x 2, $A_5\times D_4^2$ x 2, $D_{10}.(D_4\times S_4)$, $C_2^5:C_6\times F_5$, $C_2\times C_4^2\times S_5$, $D_4\times D_{10}.S_4$, $C_4^2:C_2^2\times A_5$, $D_{10}.D_4\times S_4$, $D_4\times D_{10}\times S_4$, $C_2^3:D_{12}\times F_5$, $C_2^3\times F_5\times S_4$, $(C_2\times C_4^2).S_5$, $C_4:D_4\times S_5$, $A_5:D_4^2$, $C_2\times \GL(2,\mathbb{Z}/4):F_5$, $D_4\times F_5\times S_4$, $C_2^3:F_5\times S_4$, $C_{10}:C_4^2\times S_4$, $C_2\times F_5\times \GL(2,\mathbb{Z}/4)$

Order 3072: $A_4.C_2^5.C_2^3$ x 2

Order 2880: $\GL(2,4):C_2^4$ x 6, $C_2\times S_4\times A_5$ x 5, $C_2\times A_4\times S_5$ x 5, $C_2\times A_4:S_5$ x 5, $S_4\times S_5$ x 4, $C_2^4:\GL(2,4)$ x 3, $C_2\times D_{12}\times A_5$, $C_4\times D_6\times A_5$, $\GL(2,4):C_2^4$, $C_4\times A_4\times A_5$, $D_{12}:S_5$, $A_5:C_4\times D_6$, $S_3\times C_4:S_5$, $C_6:D_4\times A_5$, $A_4:C_4\times A_5$, $D_{12}\times S_5$, $C_6\times D_4\times A_5$, $S_3\times D_4\times A_5$, $C_2\times C_{12}:S_5$, $\GL(2,4):C_2^4$, $C_2\times D_6:S_5$, $C_6:(C_4\times S_5)$, $C_2\times \GL(2,4):D_4$, $C_2\times D_6:S_5$, $C_6:C_4\times S_5$, $C_2\times \GL(2,4):D_4$, $C_2\times C_{12}:S_5$, $C_2^2:S_5\times C_6$, $\GL(2,4):C_2^4$, $C_2\times C_{12}\times S_5$, $C_2\times C_{12}:S_5$, $C_3:(D_4\times S_5)$, $S_3\times C_2^2:S_5$, $C_3:D_4\times S_5$, $C_4\times S_3\times S_5$, $D_4\times C_3:S_5$, $C_3\times D_4\times S_5$, $(A_4\times A_5):C_4$, $A_4\times A_5:C_4$

Order 2560: $C_2\times F_5\times D_4^2$

Order 2304: $C_2^4:D_6^2$ x 15, $C_2^2\times S_4^2$ x 6, $C_2^2.S_4^2$ x 2, $C_2^2.S_4^2$ x 2, $C_2\times A_4\times \GL(2,\mathbb{Z}/4)$ x 2, $A_4\times C_2^3:D_{12}$ x 2, $A_4^2:C_2^4$ x 2, $C_2\times C_4\times S_4\times A_4$ x 2, $C_4:S_4^2$ x 2, $S_4\times \GL(2,\mathbb{Z}/4)$ x 2, $D_4\times A_4\times S_4$ x 2, $C_2\times A_4^2:D_4$, $C_4\times A_4^2:C_2^2$, $A_4^2:C_2^4$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^4:D_6^2$, $C_{12}:C_2^3\times S_4$, $C_{12}:(C_2^3\times S_4)$, $C_{12}.(A_4\times C_2^4)$, $C_2^4:D_6^2$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^4:D_6^2$, $C_2^2.S_4^2$, $C_2^2.S_4^2$, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$, $C_2^6:C_6^2$, $C_4:S_4^2$, $C_4\times S_4^2$, $D_4\times \PSOPlus(4,3)$, $\GL(2,\mathbb{Z}/4):S_4$

Order 1920: $C_2\times D_4\times S_5$ x 86, $C_2^4:S_5$ x 31, $C_2^4\times S_5$ x 23, $C_4\times C_2^2\times S_5$ x 22, $C_2^2\times D_4\times A_5$ x 21, $C_2^2\times C_4:S_5$ x 19, $(C_2^2\times C_4):S_5$ x 10, $C_2.(D_4\times S_5)$ x 7, $(C_2\times D_4).S_5$ x 7, $C_2^2\times F_5\times S_4$ x 7, $C_2^2:C_4\times S_5$ x 6, $(C_2\times D_4):S_5$ x 5, $C_2^3:C_4\times A_5$ x 5, $C_2^4.S_5$ x 5, $(C_2\times D_4):S_5$ x 4, $C_4:D_4\times A_5$ x 4, $(C_2^2\times C_4):S_5$ x 4, $(C_2\times S_5):D_4$ x 4, $C_2^2:(C_4\times S_5)$ x 4, $C_4\times C_2^2:S_5$ x 4, $C_4\times C_2^3\times A_5$ x 3, $C_2^4.S_5$ x 3, $(C_2\times S_5):D_4$ x 3, $(C_2\times D_4):S_5$ x 3, $A_5\times C_2^2\wr C_2$ x 3, $C_2^4.S_5$ x 3, $C_4\times D_4\times A_5$ x 3, $A_5:C_4\times D_4$ x 3, $C_2^2.(C_4\times S_5)$ x 3, $C_2\times A_5:C_4^2$ x 3, $C_2^5\times A_5$ x 2, $C_4:(C_4\times S_5)$ x 2, $C_4^2:S_5$ x 2, $C_2^2.D_4\times A_5$ x 2, $C_4:C_4\times S_5$ x 2, $(C_2^2\times C_4).S_5$ x 2, $\GL(2,\mathbb{Z}/4):D_{10}$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $(C_{10}\times A_4):C_4^2$ x 2, $C_4:D_4\times A_5$, $C_4^2:S_5$, $C_2\times C_4^2\times A_5$, $C_4^2\times S_5$, $C_2^2\times D_{10}\times S_4$, $C_2^3\times A_4\times F_5$, $C_2^3\times A_4:F_5$, $D_4\times A_4\times D_{10}$, $C_2^5:D_{30}$, $\GL(2,\mathbb{Z}/4):D_{10}$, $C_{10}:D_4\times S_4$, $D_{10}\times \GL(2,\mathbb{Z}/4)$, $C_2\times D_{20}:S_4$, $C_2\times S_4\times D_{20}$, $C_4:S_4\times D_{10}$, $C_4\times D_{10}\times S_4$, $A_4\times C_2^3:F_5$, $D_4\times A_4\times F_5$, $(C_2^3\times C_{20}):C_{12}$, $A_4\times C_{10}:C_4^2$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $D_4\times A_4:F_5$, $(C_4\times D_{10}).S_4$, $C_4\times D_{10}.S_4$, $\GL(2,\mathbb{Z}/4):F_5$, $D_{10}:C_4\times S_4$, $F_5\times \GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $C_{20}:(C_4\times S_4)$, $C_{20}:C_4\times S_4$, $F_5\times C_4:S_4$, $C_4\times F_5\times S_4$, $D_4\times C_{10}\times S_4$, $D_4\times D_6\times F_5$

Order 1536: $\GL(2,\mathbb{Z}/4):C_2^4$ x 6, $C_2^7:D_6$ x 6, $C_2\times A_4:D_4^2$ x 6, $C_2^3:D_4\times S_4$ x 6, $S_4\times D_4^2$ x 4, $C_2^7:D_6$ x 4, $C_2^7:D_6$ x 4, $C_2\times A_4:D_4^2$ x 4, $C_4\times \GL(2,\mathbb{Z}/4):C_2^2$ x 4, $C_2\times A_4:D_4^2$ x 2, $C_4^2:C_2^2\times S_4$ x 2, $C_2\times A_4\times D_4^2$ x 2, $C_2^7:D_6$, $C_2\times D_6\times D_4^2$

Order 1440: $D_6\times S_5$ x 9, $\GL(2,4):C_2^3$ x 4, $\GL(2,4):C_2^3$ x 4, $\GL(2,4):C_2^3$ x 4, $C_2^3:\GL(2,4)$ x 3, $S_4\times A_5$ x 2, $A_4:S_5$ x 2, $A_4\times S_5$ x 2, $C_2^3\times \GL(2,4)$, $D_4\times \GL(2,4)$, $C_2\times C_{12}\times A_5$, $\GL(2,4):D_4$, $C_6:C_4\times A_5$, $D_{12}\times A_5$, $C_4\times S_3\times A_5$, $\GL(2,4):D_4$, $C_2\times \GL(2,4):C_4$, $C_{12}:S_5$, $C_{12}:S_5$, $\GL(2,4):D_4$, $A_5:C_4\times C_6$, $C_{12}:S_5$, $C_{12}\times S_5$, $D_6:S_5$, $\GL(2,4):D_4$, $C_3:C_4\times S_5$, $D_6:S_5$, $D_6.S_5$, $C_3:(C_4\times S_5)$

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

Order 1024: $C_2^5.C_2^5$

Order 960: $C_2^3\times S_5$ x 4, $C_2^4\times A_5$ x 2, $C_2\times D_4\times A_5$, $C_4\times C_2^2\times A_5$, $C_2^3:S_5$, $C_2^3.S_5$, $D_4\times S_5$, $C_2\times C_4:S_5$, $C_2\times C_4\times S_5$

Order 720: $A_4\times A_5$

Order 480: $C_2^2\times S_5$ x 4, $C_2^3\times A_5$ x 3, $D_4\times A_5$, $C_2\times C_4\times A_5$, $C_2^2.S_5$, $C_2^2:S_5$, $C_4:S_5$, $C_4\times S_5$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$

Order 240: $C_2^2\times A_5$ x 3, $C_2\times S_5$ x 3, $C_4\times A_5$, $A_5:C_4$

Order 192: $C_2^3\times S_4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $D_4\times S_4$, $C_2^3:D_{12}$, $C_2^4.D_6$

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

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

Order 64: $D_4\times C_2^3$

Order 60: $A_5$

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

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

Order 24: $C_2\times A_4$ x 2, $S_4$

Order 16: $C_2^4$ x 2, $C_2\times D_4$, $C_2^2\times C_4$

Order 12: $A_4$

Order 9: $C_3^2$

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

Order 5: $C_5$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 46080: $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ ($C_1$)

Order 23040: $D_4\times S_4\times S_5$ ($C_2$) x 16, $C_2^3\times S_5\times S_4$ ($C_2$) x 2, $(C_2\times \GL(2,\mathbb{Z}/4)).S_5$ ($C_2$) x 2, $(C_2\times S_5).\GL(2,\mathbb{Z}/4)$ ($C_2$) x 2, $C_2^3.(S_4\times S_5)$ ($C_2$) x 2, $(C_2^4\times S_5):D_6$ ($C_2$), $C_2^5.(C_6\times S_5)$ ($C_2$), $(A_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ ($C_2$), $(C_2^5\times A_5):D_6$ ($C_2$), $C_2^3:D_{12}\times S_5$ ($C_2$), $C_2^4.D_6\times S_5$ ($C_2$), $(C_2^4\times S_5):D_6$ ($C_2$)

Order 11520: $C_2^2\times S_4\times S_5$ ($C_2^2$) x 23, $S_5\times \GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 16, $S_4\times C_2^2:S_5$ ($C_2^2$) x 16, $\GL(2,\mathbb{Z}/4):S_5$ ($C_2^2$) x 16, $C_4:(S_4\times S_5)$ ($C_2^2$) x 8, $S_4\times C_4:S_5$ ($C_2^2$) x 8, $C_4:S_4\times S_5$ ($C_2^2$) x 8, $C_4\times S_4\times S_5$ ($C_2^2$) x 8, $D_4\times S_4\times A_5$ ($C_2^2$) x 8, $D_4\times A_4\times S_5$ ($C_2^2$) x 8, $D_4\times A_4:S_5$ ($C_2^2$) x 8, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $(C_2^2\times S_4):S_5$ ($C_2^2$) x 2, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $(C_2^3\times A_4).S_5$ ($C_2^2$) x 2, $(C_2^3\times A_4).S_5$ ($C_2^2$) x 2, $A_5.(C_2^3\times S_4)$ ($C_2^2$) x 2, $C_2^3\times A_4:S_5$ ($C_2^2$) x 2, $C_2\times A_5\times \GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $C_2^2.S_4\times S_5$ ($C_2^2$), $C_2^3:D_{12}\times A_5$ ($C_2^2$), $C_2^4.D_6\times A_5$ ($C_2^2$), $C_2^2.(S_4\times S_5)$ ($C_2^2$), $(C_2^4\times A_5).D_6$ ($C_2^2$), $C_2^2.S_5\times S_4$ ($C_2^2$), $C_2^5:C_6\times A_5$ ($C_2^2$), $(C_2^3\times A_5):D_{12}$ ($C_2^2$), $C_2^3:C_{12}\times S_5$ ($C_2^2$), $(C_2^4\times S_5):C_6$ ($C_2^2$)

Order 7680: $C_2^3\times S_5\times D_4$ ($S_3$)

Order 5760: $C_2\times S_4\times S_5$ ($C_2^3$) x 28, $C_2^2\times A_4:S_5$ ($C_2^3$) x 11, $C_2^2\times A_4\times S_5$ ($C_2^3$) x 11, $C_2^2\times S_4\times A_5$ ($C_2^3$) x 11, $A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 8, $A_4\times C_2^2:S_5$ ($C_2^3$) x 8, $(C_2\times S_4):S_5$ ($C_2^3$) x 8, $A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 8, $A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 8, $A_5\times \GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 8, $C_2\times S_4\times S_5$ ($D_4$) x 8, $D_4\times A_4\times A_5$ ($C_2^3$) x 4, $A_5\times C_4:S_4$ ($C_2^3$) x 4, $C_4\times S_4\times A_5$ ($C_2^3$) x 4, $(C_4\times A_4):S_5$ ($C_2^3$) x 4, $A_4\times C_4:S_5$ ($C_2^3$) x 4, $C_4\times A_4\times S_5$ ($C_2^3$) x 4, $C_4\times A_4:S_5$ ($C_2^3$) x 4, $A_4:(C_4\times S_5)$ ($C_2^3$) x 4, $A_4:C_4\times S_5$ ($C_2^3$) x 4, $A_5:C_4\times S_4$ ($C_2^3$) x 4, $C_2^5:\GL(2,4)$ ($C_2^3$) x 2, $C_2^2.S_4\times A_5$ ($C_2^3$), $(C_2^2\times A_4).S_5$ ($C_2^3$), $A_4\times C_2^2.S_5$ ($C_2^3$), $C_2^3:C_{12}\times A_5$ ($C_2^3$)

Order 3840: $C_2^2\times D_4\times S_5$ ($D_6$) x 8, $C_2^5\times S_5$ ($D_6$) x 2, $C_2^5:S_5$ ($D_6$) x 2, $D_4\times C_2^3\times A_5$ ($D_6$), $C_4\times C_2^3\times S_5$ ($D_6$), $C_2^3\times C_4:S_5$ ($D_6$)

Order 2880: $S_4\times S_5$ ($C_2\times D_4$) x 16, $C_2\times S_4\times A_5$ ($C_2^4$) x 6, $C_2\times A_4\times S_5$ ($C_2^4$) x 6, $C_2\times A_4:S_5$ ($C_2^4$) x 6, $C_2^4:\GL(2,4)$ ($C_2^4$) x 5, $C_2\times S_4\times A_5$ ($C_2\times D_4$) x 4, $C_2\times A_4\times S_5$ ($C_2\times D_4$) x 4, $C_2\times A_4:S_5$ ($C_2\times D_4$) x 4, $C_4\times A_4\times A_5$ ($C_2^4$) x 2, $A_4:C_4\times A_5$ ($C_2^4$) x 2, $(A_4\times A_5):C_4$ ($C_2^4$) x 2, $A_4\times A_5:C_4$ ($C_2^4$) x 2

Order 1920: $C_2^4\times S_5$ ($C_2\times D_6$) x 11, $C_2^4:S_5$ ($C_2\times D_6$) x 8, $C_2^2\times D_4\times A_5$ ($C_2\times D_6$) x 4, $C_2^2\times C_4:S_5$ ($C_2\times D_6$) x 4, $C_4\times C_2^2\times S_5$ ($C_2\times D_6$) x 4, $C_2^5\times A_5$ ($C_2\times D_6$) x 2, $C_2\times D_4\times S_5$ ($S_4$), $C_4\times C_2^3\times A_5$ ($C_2\times D_6$), $C_2^4.S_5$ ($C_2\times D_6$)

Order 1440: $S_4\times A_5$ ($C_2^2\times D_4$) x 4, $A_4:S_5$ ($C_2^2\times D_4$) x 4, $A_4\times S_5$ ($C_2^2\times D_4$) x 4, $C_2^3:\GL(2,4)$ ($C_2^2\times D_4$) x 2, $C_2^3:\GL(2,4)$ ($C_2^5$)

Order 960: $D_4\times S_5$ ($C_2\times S_4$) x 8, $C_2^3\times S_5$ ($C_2^2\times D_6$) x 6, $C_2^4\times A_5$ ($C_2^2\times D_6$) x 5, $C_2^3\times S_5$ ($S_3\times D_4$) x 4, $C_2^3\times S_5$ ($C_2\times S_4$) x 2, $C_4\times C_2^2\times A_5$ ($C_2^2\times D_6$) x 2, $C_2^3:S_5$ ($C_2\times S_4$) x 2, $C_2^3.S_5$ ($C_2^2\times D_6$) x 2, $C_2\times D_4\times A_5$ ($C_2\times S_4$), $C_2\times C_4:S_5$ ($C_2\times S_4$), $C_2\times C_4\times S_5$ ($C_2\times S_4$)

Order 720: $A_4\times A_5$ ($D_4\times C_2^3$)

Order 480: $C_2^2\times S_5$ ($C_2^2\times S_4$) x 11, $C_2^2:S_5$ ($C_2^2\times S_4$) x 8, $D_4\times A_5$ ($C_2^2\times S_4$) x 4, $C_4:S_5$ ($C_2^2\times S_4$) x 4, $C_4\times S_5$ ($C_2^2\times S_4$) x 4, $C_2^2\times S_5$ ($D_4\times D_6$) x 4, $C_2^3\times A_5$ ($C_2^2\times S_4$) x 2, $C_2^3\times A_5$ ($D_4\times D_6$) x 2, $C_2\times C_4\times A_5$ ($C_2^2\times S_4$), $C_2^2.S_5$ ($C_2^2\times S_4$), $C_2^3\times A_5$ ($C_2^3\times D_6$)

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ ($S_5$)

Order 240: $C_2\times S_5$ ($C_2^3\times S_4$) x 6, $C_2^2\times A_5$ ($C_2^3\times S_4$) x 5, $C_2\times S_5$ ($D_4\times S_4$) x 4, $C_4\times A_5$ ($C_2^3\times S_4$) x 2, $A_5:C_4$ ($C_2^3\times S_4$) x 2, $C_2^2\times A_5$ ($C_{12}:C_2^4$)

Order 192: $D_4\times S_4$ ($C_2\times S_5$) x 8, $C_2^3\times S_4$ ($C_2\times S_5$) x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ ($C_2\times S_5$) x 2, $C_2^5:C_6$ ($C_2\times S_5$), $C_2^3:D_{12}$ ($C_2\times S_5$), $C_2^4.D_6$ ($C_2\times S_5$)

Order 120: $S_5$ ($\GL(2,\mathbb{Z}/4):C_2^2$) x 4, $C_2\times A_5$ ($\GL(2,\mathbb{Z}/4):C_2^2$) x 2, $C_2\times A_5$ ($C_2^4\times S_4$)

Order 96: $C_2^2\times S_4$ ($C_2^2\times S_5$) x 11, $\GL(2,\mathbb{Z}/4)$ ($C_2^2\times S_5$) x 8, $D_4\times A_4$ ($C_2^2\times S_5$) x 4, $C_4:S_4$ ($C_2^2\times S_5$) x 4, $C_4\times S_4$ ($C_2^2\times S_5$) x 4, $C_2^3\times A_4$ ($C_2^2\times S_5$) x 2, $C_2^3:C_{12}$ ($C_2^2\times S_5$), $C_2^2.S_4$ ($C_2^2\times S_5$)

Order 64: $D_4\times C_2^3$ ($S_3\times S_5$)

Order 60: $A_5$ ($\GL(2,\mathbb{Z}/4):C_2^3$)

Order 48: $C_2\times S_4$ ($C_2^3\times S_5$) x 6, $C_2^2\times A_4$ ($C_2^3\times S_5$) x 5, $C_2\times S_4$ ($D_4\times S_5$) x 4, $C_4\times A_4$ ($C_2^3\times S_5$) x 2, $A_4:C_4$ ($C_2^3\times S_5$) x 2

Order 32: $C_2^2\times D_4$ ($D_6\times S_5$) x 4, $C_2^5$ ($D_6\times S_5$) x 2, $C_2^3\times C_4$ ($D_6\times S_5$)

Order 24: $S_4$ ($C_2\times D_4\times S_5$) x 4, $C_2\times A_4$ ($C_2\times D_4\times S_5$) x 2, $C_2\times A_4$ ($C_2^4\times S_5$)

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

Order 12: $A_4$ ($C_2^2\times D_4\times S_5$)

Order 8: $D_4$ ($C_2\times S_4\times S_5$) x 4, $C_2^3$ ($S_3\times D_4\times S_5$) x 2, $C_2^3$ ($C_2\times S_4\times S_5$) x 2, $C_2^3$ ($\GL(2,4):C_2^5$), $C_2\times C_4$ ($C_2\times S_4\times S_5$)

Order 4: $C_2^2$ ($C_2^2\times S_4\times S_5$) x 5, $C_4$ ($C_2^2\times S_4\times S_5$) x 2, $C_2^2$ ($D_4\times D_6\times S_5$)

Order 2: $C_2$ ($D_4\times S_4\times S_5$) x 2, $C_2$ ($C_2^3\times S_5\times S_4$)

Order 1: $C_1$ ($(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 46080: $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ ($C_1$)

Order 23040: $C_2^3\times S_5\times S_4$ ($C_2$), $(C_2^4\times S_5):D_6$ ($C_2$), $(C_2\times \GL(2,\mathbb{Z}/4)).S_5$ ($C_2$), $(C_2\times S_5).\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^5.(C_6\times S_5)$ ($C_2$), $(A_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ ($C_2$), $(C_2^5\times A_5):D_6$ ($C_2$), $C_2^3:D_{12}\times S_5$ ($C_2$), $C_2^4.D_6\times S_5$ ($C_2$), $(C_2^4\times S_5):D_6$ ($C_2$), $D_4\times S_4\times S_5$ ($C_2$), $C_2^3.(S_4\times S_5)$ ($C_2$)

Order 11520: $C_2^2\times S_4\times S_5$ ($C_2^2$) x 5, $C_2^2.S_4\times S_5$ ($C_2^2$), $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_2^3:D_{12}\times A_5$ ($C_2^2$), $C_2^4.D_6\times A_5$ ($C_2^2$), $C_2^2.(S_4\times S_5)$ ($C_2^2$), $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_4:(S_4\times S_5)$ ($C_2^2$), $S_4\times C_4:S_5$ ($C_2^2$), $(C_2^4\times A_5).D_6$ ($C_2^2$), $C_2^2.S_5\times S_4$ ($C_2^2$), $(C_2^2\times S_4):S_5$ ($C_2^2$), $C_2\times A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_4:S_4\times S_5$ ($C_2^2$), $C_4\times S_4\times S_5$ ($C_2^2$), $(C_2^3\times A_4).S_5$ ($C_2^2$), $(C_2^3\times A_4).S_5$ ($C_2^2$), $A_5.(C_2^3\times S_4)$ ($C_2^2$), $C_2^5:C_6\times A_5$ ($C_2^2$), $C_2^3\times A_4:S_5$ ($C_2^2$), $S_5\times \GL(2,\mathbb{Z}/4)$ ($C_2^2$), $S_4\times C_2^2:S_5$ ($C_2^2$), $D_4\times S_4\times A_5$ ($C_2^2$), $(C_2^3\times A_5):D_{12}$ ($C_2^2$), $C_2^3:C_{12}\times S_5$ ($C_2^2$), $(C_2^4\times S_5):C_6$ ($C_2^2$), $D_4\times A_4\times S_5$ ($C_2^2$), $D_4\times A_4:S_5$ ($C_2^2$), $\GL(2,\mathbb{Z}/4):S_5$ ($C_2^2$), $C_2\times A_5\times \GL(2,\mathbb{Z}/4)$ ($C_2^2$)

Order 7680: $C_2^3\times S_5\times D_4$ ($S_3$)

Order 5760: $C_2\times S_4\times S_5$ ($C_2^3$) x 4, $C_2^2\times A_4:S_5$ ($C_2^3$) x 3, $C_2^2\times A_4\times S_5$ ($C_2^3$) x 3, $C_2^2\times S_4\times A_5$ ($C_2^3$) x 3, $C_2^2.S_4\times A_5$ ($C_2^3$), $(C_2^2\times A_4).S_5$ ($C_2^3$), $A_4\times C_2^2.S_5$ ($C_2^3$), $C_2^3:C_{12}\times A_5$ ($C_2^3$), $C_2^5:\GL(2,4)$ ($C_2^3$), $D_4\times A_4\times A_5$ ($C_2^3$), $A_5\times C_4:S_4$ ($C_2^3$), $C_4\times S_4\times A_5$ ($C_2^3$), $(C_4\times A_4):S_5$ ($C_2^3$), $A_4\times C_4:S_5$ ($C_2^3$), $C_4\times A_4\times S_5$ ($C_2^3$), $C_4\times A_4:S_5$ ($C_2^3$), $A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^3$), $A_4\times C_2^2:S_5$ ($C_2^3$), $A_4:(C_4\times S_5)$ ($C_2^3$), $(C_2\times S_4):S_5$ ($C_2^3$), $A_4:C_4\times S_5$ ($C_2^3$), $A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^3$), $A_5:\GL(2,\mathbb{Z}/4)$ ($C_2^3$), $A_5\times \GL(2,\mathbb{Z}/4)$ ($C_2^3$), $A_5:C_4\times S_4$ ($C_2^3$), $C_2\times S_4\times S_5$ ($D_4$)

Order 3840: $D_4\times C_2^3\times A_5$ ($D_6$), $C_4\times C_2^3\times S_5$ ($D_6$), $C_2^3\times C_4:S_5$ ($D_6$), $C_2^5\times S_5$ ($D_6$), $C_2^5:S_5$ ($D_6$), $C_2^2\times D_4\times S_5$ ($D_6$)

Order 2880: $C_2^4:\GL(2,4)$ ($C_2^4$) x 2, $C_2\times S_4\times A_5$ ($C_2^4$) x 2, $C_2\times A_4\times S_5$ ($C_2^4$) x 2, $C_2\times A_4:S_5$ ($C_2^4$) x 2, $C_4\times A_4\times A_5$ ($C_2^4$), $A_4:C_4\times A_5$ ($C_2^4$), $(A_4\times A_5):C_4$ ($C_2^4$), $A_4\times A_5:C_4$ ($C_2^4$), $C_2\times S_4\times A_5$ ($C_2\times D_4$), $C_2\times A_4\times S_5$ ($C_2\times D_4$), $C_2\times A_4:S_5$ ($C_2\times D_4$), $S_4\times S_5$ ($C_2\times D_4$)

Order 1920: $C_2^4\times S_5$ ($C_2\times D_6$) x 3, $C_2^5\times A_5$ ($C_2\times D_6$), $C_2^2\times D_4\times A_5$ ($C_2\times D_6$), $C_2\times D_4\times S_5$ ($S_4$), $C_2^2\times C_4:S_5$ ($C_2\times D_6$), $C_2^4:S_5$ ($C_2\times D_6$), $C_4\times C_2^3\times A_5$ ($C_2\times D_6$), $C_4\times C_2^2\times S_5$ ($C_2\times D_6$), $C_2^4.S_5$ ($C_2\times D_6$)

Order 1440: $C_2^3:\GL(2,4)$ ($C_2^5$), $C_2^3:\GL(2,4)$ ($C_2^2\times D_4$), $S_4\times A_5$ ($C_2^2\times D_4$), $A_4:S_5$ ($C_2^2\times D_4$), $A_4\times S_5$ ($C_2^2\times D_4$)

Order 960: $C_2^4\times A_5$ ($C_2^2\times D_6$) x 2, $C_2^3\times S_5$ ($C_2^2\times D_6$) x 2, $C_2^3\times S_5$ ($C_2\times S_4$), $C_2^3\times S_5$ ($S_3\times D_4$), $C_2\times D_4\times A_5$ ($C_2\times S_4$), $C_4\times C_2^2\times A_5$ ($C_2^2\times D_6$), $C_2^3:S_5$ ($C_2\times S_4$), $C_2^3.S_5$ ($C_2^2\times D_6$), $D_4\times S_5$ ($C_2\times S_4$), $C_2\times C_4:S_5$ ($C_2\times S_4$), $C_2\times C_4\times S_5$ ($C_2\times S_4$)

Order 720: $A_4\times A_5$ ($D_4\times C_2^3$)

Order 480: $C_2^2\times S_5$ ($C_2^2\times S_4$) x 3, $D_4\times A_5$ ($C_2^2\times S_4$), $C_2\times C_4\times A_5$ ($C_2^2\times S_4$), $C_2^2.S_5$ ($C_2^2\times S_4$), $C_2^2:S_5$ ($C_2^2\times S_4$), $C_4:S_5$ ($C_2^2\times S_4$), $C_4\times S_5$ ($C_2^2\times S_4$), $C_2^3\times A_5$ ($C_2^3\times D_6$), $C_2^3\times A_5$ ($C_2^2\times S_4$), $C_2^3\times A_5$ ($D_4\times D_6$), $C_2^2\times S_5$ ($D_4\times D_6$)

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ ($S_5$)

Order 240: $C_2^2\times A_5$ ($C_2^3\times S_4$) x 2, $C_2\times S_5$ ($C_2^3\times S_4$) x 2, $C_4\times A_5$ ($C_2^3\times S_4$), $A_5:C_4$ ($C_2^3\times S_4$), $C_2^2\times A_5$ ($C_{12}:C_2^4$), $C_2\times S_5$ ($D_4\times S_4$)

Order 192: $C_2^3\times S_4$ ($C_2\times S_5$), $C_2^5:C_6$ ($C_2\times S_5$), $C_2\times \GL(2,\mathbb{Z}/4)$ ($C_2\times S_5$), $D_4\times S_4$ ($C_2\times S_5$), $C_2^3:D_{12}$ ($C_2\times S_5$), $C_2^4.D_6$ ($C_2\times S_5$)

Order 120: $C_2\times A_5$ ($C_2^4\times S_4$), $C_2\times A_5$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $S_5$ ($\GL(2,\mathbb{Z}/4):C_2^2$)

Order 96: $C_2^2\times S_4$ ($C_2^2\times S_5$) x 3, $C_2^3\times A_4$ ($C_2^2\times S_5$), $D_4\times A_4$ ($C_2^2\times S_5$), $C_2^3:C_{12}$ ($C_2^2\times S_5$), $\GL(2,\mathbb{Z}/4)$ ($C_2^2\times S_5$), $C_2^2.S_4$ ($C_2^2\times S_5$), $C_4:S_4$ ($C_2^2\times S_5$), $C_4\times S_4$ ($C_2^2\times S_5$)

Order 64: $D_4\times C_2^3$ ($S_3\times S_5$)

Order 60: $A_5$ ($\GL(2,\mathbb{Z}/4):C_2^3$)

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

Order 32: $C_2^5$ ($D_6\times S_5$), $C_2^2\times D_4$ ($D_6\times S_5$), $C_2^3\times C_4$ ($D_6\times S_5$)

Order 24: $C_2\times A_4$ ($C_2^4\times S_5$), $C_2\times A_4$ ($C_2\times D_4\times S_5$), $S_4$ ($C_2\times D_4\times S_5$)

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

Order 12: $A_4$ ($C_2^2\times D_4\times S_5$)

Order 8: $C_2^3$ ($\GL(2,4):C_2^5$), $C_2^3$ ($S_3\times D_4\times S_5$), $C_2^3$ ($C_2\times S_4\times S_5$), $D_4$ ($C_2\times S_4\times S_5$), $C_2\times C_4$ ($C_2\times S_4\times S_5$)

Order 4: $C_2^2$ ($C_2^2\times S_4\times S_5$) x 2, $C_2^2$ ($D_4\times D_6\times S_5$), $C_4$ ($C_2^2\times S_4\times S_5$)

Order 2: $C_2$ ($C_2^3\times S_5\times S_4$), $C_2$ ($D_4\times S_4\times S_5$)

Order 1: $C_1$ ($(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$)

Series

Derived series

$(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$

$\rhd$

$C_2^3:\GL(2,4)$

$\rhd$

$C_2^2\times A_5$

$\rhd$

$A_5$

Chief series

$(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$

$\rhd$

$(A_5\times \GL(2,\mathbb{Z}/4)).C_2^2$

$\rhd$

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

$\rhd$

$C_2^5:C_6$

$\rhd$

$D_4\times A_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$

$\rhd$

$C_2^3:\GL(2,4)$

$\rhd$

$A_4\times A_5$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $350 \times 350$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.