Abstract group 115200.bv: $C_2^2\times S_5\wr C_2$

• Label: 115200.bv
• Order: \( 2^{9} \cdot 3^{2} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{3} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$C_2^2\times S_5\wr C_2$
Order:	\(115200\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5^{2} \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$A_5^2.D_4\times C_2^2:S_4$, of order \(2764800\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 5^{2} \)
Composition factors:	$C_2$ x 5, $A_5$ x 2
Derived length:	$2$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	30
Elements	1	3183	440	21840	624	23880	14400	18192	19200	960	5760	6720	115200
Conjugacy classes	1	27	2	24	2	38	4	18	12	1	4	7	140
Divisions	1	27	2	24	2	38	4	18	12	1	4	7	140
Autjugacy classes	1	10	2	8	2	13	1	6	4	1	1	2	51

Dimension	1	2	8	10	12	16	25	32	36	40	48	50	60
Irr. complex chars.	16	4	16	16	8	16	16	4	8	16	8	4	8	140
Irr. rational chars.	16	4	16	16	8	16	16	4	8	16	8	4	8	140

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$40$
Rank:	$4$
Inequivalent generating quadruples:	$19593826650720$

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 $14$ $\langle(1,3,8,2)(4,6)(5,10,9,7)(11,13)(12,14), (1,3,9,10)(2,6,4,5,7,8), (2,4)(3,7)(5,8,9)(11,12)(13,14), (1,2,5,10,9,4)(3,6,7,8)(11,13)(12,14)\rangle$
Transitive group:	40T45541	40T45558			more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(S_5\wr C_2)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(A_5.S_5)$ . $C_2^4$	$((C_2\times A_5).S_5)$ . $C_2^3$	$((C_2^2\times A_5).S_5)$ . $C_2^2$		more information
Aut. group:	$\Aut(A_5^2:D_4)$	$\Aut(C_4\times \PSOPlus(4,5))$	$\Aut(C_6\times \PSOPlus(4,5))$	$\Aut(S_5^2:C_4)$	all 10

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 6484530 subgroups in 17736 conjugacy classes, 83 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $A_5^2.D_4$
Commutator:	$G' \simeq$ $A_5.S_5$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2\times S_5\wr C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2$	$G/\operatorname{Fit} \simeq$ $A_5^2.D_4$
Radical:	$R \simeq$ $C_2^2$	$G/R \simeq$ $A_5^2.D_4$
Socle:	$\operatorname{soc} \simeq$ $(C_2^2\times A_5).A_5$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:C_2^3$
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

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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

Order 115200: $C_2^2\times S_5\wr C_2$

Order 57600: $C_2\times A_5^2.D_4$ x 12, $C_2^2\times A_5.S_5.C_2$, $C_2^2\times A_5^2.C_2^2$, $C_2^2.A_5^2.C_4$

Order 28800: $A_5^2.D_4$ x 16, $C_2\times A_5.S_5.C_2$ x 9, $C_2\times A_5^2.C_2^2$ x 9, $C_2.A_5^2.C_4$ x 6, $(C_2^2\times A_5).S_5$, $C_2^2.A_5^2.C_2$, $(C_2^2\times A_5).S_5$

Order 14400: $A_5.S_5.C_2$ x 10, $A_5^2.C_2^2$ x 10, $C_2.A_5^2.C_2$ x 6, $(C_2\times A_5).S_5$ x 6, $(C_2\times A_5).S_5$ x 6, $A_5^2.C_4$ x 4, $(C_2^2\times A_5).A_5$

Order 11520: $(C_2^2\times S_4).S_5$

Order 9600: $(C_2^2\times F_5).S_5$

Order 7200: $A_5.S_5$ x 4, $A_5.S_5$ x 4, $A_5^2.C_2$ x 4, $(C_2\times A_5).A_5$ x 3

Order 5760: $(C_2\times S_4).S_5$ x 12, $(C_2^3\times S_3).S_5$, $(C_2^2\times A_4).S_5$, $(C_2^2\times S_4).A_5$, $(C_2^2\times A_4).S_5$

Order 4800: $(C_2\times F_5).S_5$ x 12, $D_{10}.(C_2\times S_5)$, $(C_2^2\times F_5).A_5$, $(C_2^2\times D_5).S_5$

Order 4608: $A_4^2.C_2^4.C_2$

Order 3840: $(C_2^2\times D_4).S_5$

Order 3600: $A_5^2$

Order 3200: $D_5:F_5.C_2^4$

Order 2880: $S_4.S_5$ x 16, $C_2^2\times S_3.S_5$ x 13, $(C_2^2\times S_3).S_5$ x 13, $(C_2\times A_4).S_5$ x 6, $(C_2\times A_4).S_5$ x 6, $(C_2\times S_4).A_5$ x 6, $D_6.(C_2\times S_5)$ x 2, $(C_2^3\times S_3).A_5$, $(C_2^2\times C_6).S_5$, $(C_2^2\times A_4).A_5$, $(C_2^2\times C_6).S_5$

Order 2400: $F_5.S_5$ x 16, $D_{10}.S_5$ x 12, $(C_2\times F_5).A_5$ x 6, $D_{10}.S_5$ x 6, $(C_2\times C_{10}).S_5$, $(C_2^2\times D_5).A_5$, $(C_2\times C_{10}).S_5$

Order 2304: $A_4^2.C_2^3.C_2$ x 12, $A_4^2.C_2^4$, $A_4^2.C_2^3.C_2$, $A_4^2.C_2^4$

Order 1920: $C_2\times D_4\times S_5$ x 24, $C_2^4\times S_5$ x 2, $C_2^4:S_5$ x 2, $C_2^2\times D_4\times A_5$, $C_2^2\times C_4:S_5$, $C_4\times C_2^2\times S_5$, $C_2^2\times F_5\times S_4$

Order 1600: $F_5^2:C_2^2$ x 12, $D_{10}^2.C_2^2$, $C_{10}^2:\OD_{16}$, $C_2^2\times F_5^2$

Order 1440: $D_6\times S_5$ x 112, $\GL(2,4):C_2^3$ x 14, $\GL(2,4):C_2^3$ x 14, $\GL(2,4):C_2^3$ x 14, $S_4\times A_5$ x 4, $A_4:S_5$ x 4, $A_4\times S_5$ x 4, $C_2^3:\GL(2,4)$ x 3, $C_2^3\times \GL(2,4)$

Order 1200: $D_5\times S_5$ x 16, $D_{10}\times A_5$ x 6, $C_{10}\times S_5$ x 6, $C_{10}:S_5$ x 6, $F_5\times A_5$ x 4, $A_5:F_5$ x 4, $C_2\times C_{10}\times A_5$

Order 1152: $S_4\wr C_2$ x 16, $A_4^2:C_2^3$ x 9, $C_2\times S_4^2$ x 9, $C_2\times A_4^2:C_4$ x 6, $C_2^3:D_6^2$, $A_4^2:C_2^3$, $A_4^2:C_2^3$, $A_4^2:C_2^3$, $D_6^2:D_4$

Order 960: $D_4\times S_5$ x 64, $C_2^3\times S_5$ x 36, $C_2^3:S_5$ x 24, $C_2\times F_5\times S_4$ x 12, $C_2\times D_4\times A_5$ x 12, $C_2\times C_4:S_5$ x 12, $C_2\times C_4\times S_5$ x 12, $C_2^4\times A_5$ x 2, $C_2^2\times D_6\times F_5$, $C_2\times D_{10}\times S_4$, $C_2^2\times A_4\times F_5$, $C_2^2\times A_4:F_5$, $C_4\times C_2^2\times A_5$, $C_2^3.S_5$

Order 800: $D_5^2.C_2^3$ x 18, $F_5\wr C_2$ x 16, $C_2\times C_5^2:\OD_{16}$ x 9, $C_2\times F_5^2$ x 9, $C_{10}^2:D_4$ x 2, $D_{10}^2.C_2$ x 2, $C_{10}^2:Q_8$, $D_{10}^2.C_2$, $C_{10}^2:C_8$

Order 768: $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2^5:S_4$

Order 720: $S_3\times S_5$ x 64, $D_6\times A_5$ x 28, $C_6:S_5$ x 28, $C_6\times S_5$ x 28, $C_2^2\times \GL(2,4)$ x 7, $A_4\times A_5$

Order 640: $D_{10}.C_2^5$

Order 600: $D_5\times A_5$ x 4, $C_5:S_5$ x 4, $C_5\times S_5$ x 4, $C_{10}\times A_5$ x 3

Order 576: $C_2^2:D_6^2$ x 28, $D_6^2:C_2^2$ x 24, $\POPlus(4,3)$ x 10, $S_4^2$ x 10, $A_4^2:C_2^2$ x 6, $A_4^2:C_2^2$ x 6, $A_4^2:C_2^2$ x 6, $A_4^2:C_4$ x 4, $D_6^2:C_2^2$ x 2, $D_6^2:C_4$ x 2, $C_2^2\times D_6^2$, $C_2^2\times A_4\times D_6$, $C_2^2\times C_6:S_4$, $C_2^2\times C_6\times S_4$, $C_2^2\times A_4^2$, $C_2^2:D_6^2$, $C_2^2\times C_6^2:C_4$

Order 512: $D_4^2:C_2^3$

Order 480: $C_2^2\times S_5$ x 131, $C_2^2:S_5$ x 32, $C_2\times D_6\times F_5$ x 28, $C_2^3\times A_5$ x 18, $D_4\times A_5$ x 16, $C_4:S_5$ x 16, $C_4\times S_5$ x 16, $F_5\times S_4$ x 16, $D_{10}\times S_4$ x 12, $C_2\times C_4\times A_5$ x 6, $C_2^2.S_5$ x 6, $C_2\times A_4\times F_5$ x 6, $D_{10}.S_4$ x 6, $C_{15}:C_2^5$, $C_2^2\times C_{30}:C_4$, $C_2^2\times C_6\times F_5$, $C_2\times A_4\times D_{10}$, $C_2^3:D_{30}$, $C_2\times C_{10}\times S_4$

Order 400: $D_5^2.C_2^2$ x 40, $D_{10}:F_5$ x 24, $D_5^2:C_2^2$ x 21, $D_{10}\times F_5$ x 12, $C_5^2:\OD_{16}$ x 10, $F_5^2$ x 10, $C_2\times C_5^2:Q_8$ x 9, $C_2\times C_5^2:C_8$ x 6, $D_{10}^2$ x 2, $C_{10}^2:C_4$ x 2, $C_{10}^2:C_4$, $C_{10}^2:C_4$, $D_{10}:C_{20}$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 24, $C_2^4:S_4$ x 12, $C_2^2\wr S_3$ x 2, $C_2^4\times S_4$ x 2, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_{12}:C_2^5$, $C_2^6:C_6$, $C_2^4:D_{12}$, $C_2^5.D_6$, $C_2^5:A_4$

Order 360: $S_3\times A_5$ x 8, $C_3:S_5$ x 8, $C_3\times S_5$ x 8, $C_6\times A_5$ x 7

Order 320: $D_{10}.C_2^4$ x 24, $C_2^4\times F_5$ x 2, $C_2^4:F_5$ x 2, $C_{20}:C_2^4$, $D_{10}.C_2^4$, $D_{10}:C_4^2$

Order 300: $C_5\times A_5$

Order 288: $D_6\times S_4$ x 112, $D_6\wr C_2$ x 64, $C_6^2:D_4$ x 40, $C_2\times D_6^2$ x 38, $C_6^2.D_4$ x 24, $D_6^2:C_2$ x 18, $C_2\times A_4\times D_6$ x 14, $C_2\times C_6:S_4$ x 14, $C_2\times C_6\times S_4$ x 14, $C_2\times C_6^2:C_4$ x 12, $\PSOPlus(4,3)$ x 4, $A_4\wr C_2$ x 4, $A_4\times S_4$ x 4, $C_2\times A_4^2$ x 3, $C_2\times C_6^2:C_4$ x 2, $C_6^2:D_4$, $C_2.D_6^2$, $C_6^2:C_2^3$, $C_6^2:C_2^3$, $C_2^3:C_6^2$, $C_6^2:D_4$

Order 256: $D_4^2:C_2^2$ x 24, $C_2^5:D_4$ x 2, $C_2^6:C_4$ x 2, $C_2^2\times D_4^2$, $C_4^2:C_2^4$, $C_2^5.D_4$

Order 240: $C_2\times S_5$ x 114, $D_6\times F_5$ x 112, $C_2^2\times A_5$ x 35, $D_6\times D_{10}$ x 28, $D_5\times S_4$ x 16, $D_{10}.D_6$ x 14, $D_{10}:C_{12}$ x 14, $A_4\times D_{10}$ x 6, $C_{10}:S_4$ x 6, $C_{10}\times S_4$ x 6, $C_4\times A_5$ x 4, $A_5:C_4$ x 4, $A_4\times F_5$ x 4, $A_4:F_5$ x 4, $C_2^2\times D_{30}$, $C_{15}:C_2^4$, $C_{15}:C_2^4$, $C_2^3:C_{30}$

Order 200: $D_5:F_5$ x 32, $D_5\wr C_2$ x 26, $D_5\times D_{10}$ x 18, $D_5\times F_5$ x 16, $C_{10}:F_5$ x 12, $C_5^2:Q_8$ x 10, $C_{10}:F_5$ x 6, $C_{10}:F_5$ x 6, $C_{10}\times F_5$ x 6, $C_5^2:C_8$ x 4, $C_{10}\times D_{10}$ x 2, $C_{10}:D_{10}$

Order 192: $D_4\times S_4$ x 64, $C_{12}:C_2^4$ x 57, $C_2^3\times S_4$ x 37, $C_2\times \GL(2,\mathbb{Z}/4)$ x 24, $C_2^3:S_4$ x 16, $C_2^3:S_4$ x 12, $C_2^5:C_6$ x 12, $C_2^3:D_{12}$ x 12, $C_2^4.D_6$ x 12, $C_2^4:A_4$ x 6, $D_6\times C_2^4$ x 2, $A_4\times C_2^4$ x 2, $D_6:C_2^4$ x 2, $C_2^2\wr C_3$, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $C_2^4:C_{12}$, $C_2^3.S_4$

Order 180: $\GL(2,4)$

Order 160: $D_4\times F_5$ x 64, $C_2^3\times F_5$ x 37, $D_4\times D_{10}$ x 24, $C_2^3:F_5$ x 24, $D_{10}.D_4$ x 12, $C_{10}:C_4^2$ x 12, $C_2^3\times D_{10}$ x 2, $C_2^3:D_{10}$ x 2, $C_{20}:C_2^3$, $C_2^2\times D_{20}$, $C_{20}:C_2^3$

Order 144: $D_6^2$ x 319, $S_3^2:C_2^2$ x 128, $S_3\times S_4$ x 64, $D_6:D_6$ x 40, $S_3^2:C_4$ x 32, $C_6^2:C_2^2$ x 31, $A_4\times D_6$ x 28, $C_6:S_4$ x 28, $C_6\times S_4$ x 28, $C_6^2:C_2^2$ x 22, $C_6^2:C_4$ x 20, $C_6^2:C_4$ x 16, $C_6^2:C_2^2$ x 12, $C_6:D_{12}$ x 12, $C_2^2.S_3^2$ x 9, $C_2^2:C_6^2$ x 7, $C_2^2\times C_6^2$, $A_4^2$, $C_6^2:C_4$

Order 128: $C_2\wr D_4$ x 64, $C_2^4:D_4$ x 36, $C_2\times D_4^2$ x 30, $C_2^5:C_4$ x 24, $C_4^2:C_2^3$ x 18, $C_2^4.D_4$ x 12, $C_2^4:D_4$ x 5, $C_2^5:C_4$ x 4, $D_4\times C_2^4$ x 2, $C_2^4:D_4$ x 2, $D_4:C_2^4$, $C_8:C_2^4$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $\OD_{16}:C_2^3$

Order 120: $S_3\times D_{10}$ x 112, $S_3\times F_5$ x 64, $C_{30}:C_4$ x 28, $C_6\times F_5$ x 28, $S_5$ x 20, $C_2\times A_5$ x 18, $C_2\times D_{30}$ x 14, $C_{10}\times D_6$ x 14, $C_6\times D_{10}$ x 14, $D_5\times A_4$ x 4, $C_5:S_4$ x 4, $C_5\times S_4$ x 4, $C_{10}\times A_4$ x 3, $C_2^2\times C_{30}$

Order 100: $D_5^2$ x 20, $C_5\times D_{10}$ x 12, $C_5:F_5$ x 8, $C_5:D_{10}$ x 6, $C_5\times F_5$ x 4, $C_5:F_5$ x 4, $C_5:F_5$ x 4, $C_{10}^2$

Order 96: $D_4\times D_6$ x 472, $C_2^2\times S_4$ x 145, $C_2^3\times D_6$ x 77, $C_2^3:D_6$ x 58, $\GL(2,\mathbb{Z}/4)$ x 32, $C_{12}:C_2^3$ x 29, $C_2^2\times D_{12}$ x 29, $C_{12}:C_2^3$ x 29, $C_2^3\times A_4$ x 19, $D_4\times A_4$ x 16, $C_4:S_4$ x 16, $C_4\times S_4$ x 16, $C_2^2:S_4$ x 8, $C_2^3:C_{12}$ x 6, $C_2^2.S_4$ x 6, $C_2^3:A_4$ x 4, $C_2^3:A_4$ x 3, $C_2^4\times C_6$ x 2, $C_2^3\times C_{12}$, $C_6.C_2^4$

Order 80: $C_2^2\times F_5$ x 141, $D_4\times D_5$ x 64, $C_2^2\times D_{10}$ x 36, $D_{10}:C_4$ x 32, $C_{10}:D_4$ x 24, $C_{20}:C_4$ x 16, $C_4\times F_5$ x 16, $D_4\times C_{10}$ x 12, $C_2\times D_{20}$ x 12, $C_4\times D_{10}$ x 12, $C_2^3\times C_{10}$ x 2, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 72: $S_3\times D_6$ x 554, $C_6\times D_6$ x 150, $C_6:D_6$ x 88, $\SOPlus(4,2)$ x 64, $C_2\times C_3^2:C_4$ x 32, $C_6\wr C_2$ x 16, $C_3:D_{12}$ x 16, $C_2\times C_6^2$ x 11, $C_6.D_6$ x 10, $S_3\times A_4$ x 8, $C_3:S_4$ x 8, $C_3\times S_4$ x 8, $C_6\times A_4$ x 7, $C_6:C_{12}$ x 6

Order 64: $D_4^2$ x 136, $C_2^3:D_4$ x 90, $C_2\wr C_2^2$ x 80, $D_4\times C_2^3$ x 77, $C_2^3:D_4$ x 48, $C_2^4:C_4$ x 42, $D_4:D_4$ x 40, $C_2\wr C_4$ x 32, $C_4^2:C_2^2$ x 24, $D_4:C_2^3$ x 21, $C_8:C_2^3$ x 18, $C_4^2:C_2^2$ x 18, $C_4^2:C_4$ x 16, $C_4^2:C_2^2$ x 12, $\OD_{16}:C_2^2$ x 9, $C_2^4:C_4$ x 8, $C_2^6$ x 3, $D_4:C_2^3$ x 2, $C_2^4\times C_4$ x 2, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_2^2\times \OD_{16}$, $C_2^3.D_4$, $C_2^2\times C_4^2$

Order 60: $S_3\times D_5$ x 64, $D_{30}$ x 28, $S_3\times C_{10}$ x 28, $C_3\times D_{10}$ x 28, $C_{15}:C_4$ x 8, $C_3\times F_5$ x 8, $C_2\times C_{30}$ x 7, $A_5$ x 2, $C_5\times A_4$

Order 50: $C_5\times D_5$ x 8, $C_5:D_5$ x 4, $C_5\times C_{10}$ x 3

Order 48: $C_2^2\times D_6$ x 592, $S_3\times D_4$ x 576, $C_6:D_4$ x 248, $C_2\times S_4$ x 142, $C_6\times D_4$ x 124, $C_2\times D_{12}$ x 124, $C_4\times D_6$ x 124, $C_2^2\times A_4$ x 42, $C_2^3\times C_6$ x 39, $C_2^2\times C_{12}$ x 15, $C_6.C_2^3$ x 15, $C_4\times A_4$ x 4, $A_4:C_4$ x 4, $C_2^2:A_4$

Order 40: $C_2\times D_{10}$ x 131, $C_2\times F_5$ x 130, $C_5:D_4$ x 32, $C_2^2\times C_{10}$ x 18, $D_{20}$ x 16, $C_4\times D_5$ x 16, $C_5\times D_4$ x 16, $C_2\times C_{20}$ x 6, $C_{10}:C_4$ x 6

Order 36: $S_3^2$ x 156, $C_6\times S_3$ x 136, $C_6:S_3$ x 76, $C_6^2$ x 23, $C_3^2:C_4$ x 8, $C_3:C_{12}$ x 4, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 592, $C_2^2\wr C_2$ x 204, $C_4:D_4$ x 128, $C_2^3:C_4$ x 84, $D_4:C_2^2$ x 76, $C_4\times D_4$ x 64, $C_2^5$ x 57, $C_2^3:C_4$ x 52, $D_4:C_2^2$ x 42, $C_2^3\times C_4$ x 40, $D_8:C_2$ x 40, $C_4:D_4$ x 40, $C_4\wr C_2$ x 16, $C_2\times \SD_{16}$ x 12, $C_2\times D_8$ x 12, $C_2^2.D_4$ x 12, $\OD_{16}:C_2$ x 10, $C_2\times \OD_{16}$ x 9, $C_2\times C_4^2$ x 9, $C_2^2\times Q_8$, $C_2^2\times C_8$

Order 30: $D_{15}$ x 8, $C_3\times D_5$ x 8, $C_5\times S_3$ x 8, $C_{30}$ x 7

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 1088, $C_3:D_4$ x 160, $C_2^2\times C_6$ x 155, $D_{12}$ x 80, $C_4\times S_3$ x 80, $C_3\times D_4$ x 80, $C_2\times C_{12}$ x 34, $C_6:C_4$ x 34, $S_4$ x 28, $C_2\times A_4$ x 25

Order 20: $D_{10}$ x 114, $C_2\times C_{10}$ x 35, $F_5$ x 24, $C_{20}$ x 4, $C_5:C_4$ x 4

Order 18: $C_3\times S_3$ x 20, $C_3:S_3$ x 12, $C_3\times C_6$ x 11

Order 16: $C_2\times D_4$ x 1013, $C_2^4$ x 311, $C_2^2\times C_4$ x 168, $C_2^2:C_4$ x 104, $D_4:C_2$ x 104, $\SD_{16}$ x 16, $D_8$ x 16, $C_4:C_4$ x 16, $\OD_{16}$ x 10, $C_4^2$ x 10, $C_2\times Q_8$ x 9, $C_2\times C_8$ x 6

Order 15: $C_{15}$

Order 12: $D_6$ x 472, $C_2\times C_6$ x 153, $C_{12}$ x 12, $C_3:C_4$ x 12, $A_4$ x 3

Order 10: $D_5$ x 20, $C_{10}$ x 18

Order 9: $C_3^2$

Order 8: $C_2^3$ x 498, $D_4$ x 292, $C_2\times C_4$ x 160, $Q_8$ x 10, $C_8$ x 4

Order 6: $S_3$ x 40, $C_6$ x 38

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 215, $C_4$ x 24

Order 3: $C_3$ x 2

Order 2: $C_2$ x 27

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 115200: $C_2^2\times S_5\wr C_2$

Order 57600: $C_2^2\times A_5.S_5.C_2$, $C_2^2\times A_5^2.C_2^2$, $C_2\times A_5^2.D_4$, $C_2^2.A_5^2.C_4$

Order 28800: $C_2\times A_5.S_5.C_2$ x 2, $C_2\times A_5^2.C_2^2$ x 2, $(C_2^2\times A_5).S_5$, $C_2.A_5^2.C_4$, $A_5^2.D_4$, $C_2^2.A_5^2.C_2$, $(C_2^2\times A_5).S_5$

Order 14400: $A_5.S_5.C_2$ x 2, $(C_2\times A_5).S_5$ x 2, $A_5^2.C_2^2$ x 2, $C_2.A_5^2.C_2$, $A_5^2.C_4$, $(C_2^2\times A_5).A_5$, $(C_2\times A_5).S_5$

Order 11520: $(C_2^2\times S_4).S_5$

Order 9600: $(C_2^2\times F_5).S_5$

Order 7200: $A_5.S_5$ x 2, $A_5.S_5$, $(C_2\times A_5).A_5$, $A_5^2.C_2$

Order 5760: $(C_2\times S_4).S_5$ x 2, $(C_2^3\times S_3).S_5$, $(C_2^2\times A_4).S_5$, $(C_2^2\times S_4).A_5$, $(C_2^2\times A_4).S_5$

Order 4800: $(C_2\times F_5).S_5$ x 2, $D_{10}.(C_2\times S_5)$, $(C_2^2\times F_5).A_5$, $(C_2^2\times D_5).S_5$

Order 4608: $A_4^2.C_2^4.C_2$

Order 3840: $(C_2^2\times D_4).S_5$

Order 3600: $A_5^2$

Order 3200: $D_5:F_5.C_2^4$

Order 2880: $C_2^2\times S_3.S_5$ x 3, $(C_2^2\times S_3).S_5$ x 3, $(C_2\times A_4).S_5$ x 2, $D_6.(C_2\times S_5)$ x 2, $S_4.S_5$ x 2, $(C_2^3\times S_3).A_5$, $(C_2\times A_4).S_5$, $(C_2^2\times C_6).S_5$, $(C_2^2\times A_4).A_5$, $(C_2\times S_4).A_5$, $(C_2^2\times C_6).S_5$

Order 2400: $D_{10}.S_5$ x 2, $F_5.S_5$ x 2, $D_{10}.S_5$ x 2, $(C_2\times F_5).A_5$, $(C_2\times C_{10}).S_5$, $(C_2^2\times D_5).A_5$, $(C_2\times C_{10}).S_5$

Order 2304: $A_4^2.C_2^4$, $A_4^2.C_2^3.C_2$, $A_4^2.C_2^3.C_2$, $A_4^2.C_2^4$

Order 1920: $C_2\times D_4\times S_5$ x 4, $C_2^4\times S_5$ x 2, $C_2^4:S_5$ x 2, $C_2^2\times D_4\times A_5$, $C_2^2\times C_4:S_5$, $C_4\times C_2^2\times S_5$, $C_2^2\times F_5\times S_4$

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

Order 1440: $D_6\times S_5$ x 15, $\GL(2,4):C_2^3$ x 4, $\GL(2,4):C_2^3$ x 4, $\GL(2,4):C_2^3$ x 4, $A_4:S_5$ x 2, $C_2^3\times \GL(2,4)$, $C_2^3:\GL(2,4)$, $S_4\times A_5$, $A_4\times S_5$

Order 1200: $D_{10}\times A_5$ x 2, $D_5\times S_5$ x 2, $A_5:F_5$ x 2, $C_2\times C_{10}\times A_5$, $C_{10}\times S_5$, $C_{10}:S_5$, $F_5\times A_5$

Order 1152: $A_4^2:C_2^3$ x 2, $C_2\times S_4^2$ x 2, $C_2^3:D_6^2$, $A_4^2:C_2^3$, $A_4^2:C_2^3$, $A_4^2:C_2^3$, $C_2\times A_4^2:C_4$, $S_4\wr C_2$, $D_6^2:D_4$

Order 960: $C_2^3\times S_5$ x 11, $C_2^3:S_5$ x 6, $D_4\times S_5$ x 5, $C_2\times F_5\times S_4$ x 2, $C_2^4\times A_5$ x 2, $C_2\times D_4\times A_5$ x 2, $C_2\times C_4:S_5$ x 2, $C_2\times C_4\times S_5$ x 2, $C_2^2\times D_6\times F_5$, $C_2\times D_{10}\times S_4$, $C_2^2\times A_4\times F_5$, $C_2^2\times A_4:F_5$, $C_4\times C_2^2\times A_5$, $C_2^3.S_5$

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

Order 768: $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2^5:S_4$

Order 720: $S_3\times S_5$ x 11, $D_6\times A_5$ x 5, $C_6:S_5$ x 5, $C_6\times S_5$ x 5, $C_2^2\times \GL(2,4)$ x 2, $A_4\times A_5$

Order 640: $D_{10}.C_2^5$

Order 600: $D_5\times A_5$ x 2, $C_{10}\times A_5$, $C_5:S_5$, $C_5\times S_5$

Order 576: $C_2^2:D_6^2$ x 8, $D_6^2:C_2^2$ x 2, $A_4^2:C_2^2$ x 2, $\POPlus(4,3)$ x 2, $S_4^2$ x 2, $D_6^2:C_2^2$ x 2, $D_6^2:C_4$ x 2, $C_2^2\times D_6^2$, $C_2^2\times A_4\times D_6$, $C_2^2\times C_6:S_4$, $C_2^2\times C_6\times S_4$, $C_2^2\times A_4^2$, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $A_4^2:C_4$, $C_2^2:D_6^2$, $C_2^2\times C_6^2:C_4$

Order 512: $D_4^2:C_2^3$

Order 480: $C_2^2\times S_5$ x 25, $C_2\times D_6\times F_5$ x 8, $C_2^2:S_5$ x 7, $C_2^3\times A_5$ x 7, $D_4\times A_5$ x 2, $C_2^2.S_5$ x 2, $C_4:S_5$ x 2, $C_4\times S_5$ x 2, $D_{10}\times S_4$ x 2, $D_{10}.S_4$ x 2, $F_5\times S_4$ x 2, $C_2\times C_4\times A_5$, $C_{15}:C_2^5$, $C_2^2\times C_{30}:C_4$, $C_2^2\times C_6\times F_5$, $C_2\times A_4\times D_{10}$, $C_2^3:D_{30}$, $C_2\times C_{10}\times S_4$, $C_2\times A_4\times F_5$

Order 400: $D_{10}:F_5$ x 6, $D_5^2.C_2^2$ x 5, $D_5^2:C_2^2$ x 4, $D_{10}^2$ x 2, $C_{10}^2:C_4$ x 2, $C_2\times C_5^2:Q_8$ x 2, $D_{10}\times F_5$ x 2, $C_5^2:\OD_{16}$ x 2, $F_5^2$ x 2, $C_{10}^2:C_4$, $C_{10}^2:C_4$, $D_{10}:C_{20}$, $C_2\times C_5^2:C_8$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 4, $C_2^2\wr S_3$ x 2, $C_2^4\times S_4$ x 2, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:S_4$ x 2, $C_{12}:C_2^5$, $C_2^6:C_6$, $C_2^4:D_{12}$, $C_2^5.D_6$, $C_2^5:A_4$

Order 360: $S_3\times A_5$ x 3, $C_3:S_5$ x 3, $C_3\times S_5$ x 3, $C_6\times A_5$ x 2

Order 320: $D_{10}.C_2^4$ x 4, $C_2^4\times F_5$ x 2, $C_2^4:F_5$ x 2, $C_{20}:C_2^4$, $D_{10}.C_2^4$, $D_{10}:C_4^2$

Order 300: $C_5\times A_5$

Order 288: $D_6\times S_4$ x 15, $C_2\times D_6^2$ x 14, $C_6^2:D_4$ x 10, $C_2\times A_4\times D_6$ x 4, $C_2\times C_6:S_4$ x 4, $C_2\times C_6\times S_4$ x 4, $C_6^2.D_4$ x 3, $D_6^2:C_2$ x 2, $D_6\wr C_2$ x 2, $C_2\times C_6^2:C_4$ x 2, $\PSOPlus(4,3)$ x 2, $C_6^2:D_4$, $C_2.D_6^2$, $C_2\times C_6^2:C_4$, $C_6^2:C_2^3$, $C_6^2:C_2^3$, $C_2^3:C_6^2$, $C_2\times A_4^2$, $A_4\wr C_2$, $A_4\times S_4$, $C_6^2:D_4$

Order 256: $C_2^5:D_4$ x 2, $D_4^2:C_2^2$ x 2, $C_2^6:C_4$ x 2, $C_2^2\times D_4^2$, $C_4^2:C_2^4$, $C_2^5.D_4$

Order 240: $C_2\times S_5$ x 21, $D_6\times F_5$ x 15, $C_2^2\times A_5$ x 11, $D_6\times D_{10}$ x 8, $D_{10}.D_6$ x 4, $D_{10}:C_{12}$ x 4, $A_5:C_4$ x 2, $A_4\times D_{10}$ x 2, $D_5\times S_4$ x 2, $A_4:F_5$ x 2, $C_4\times A_5$, $C_2^2\times D_{30}$, $C_{15}:C_2^4$, $C_{15}:C_2^4$, $C_2^3:C_{30}$, $C_{10}:S_4$, $C_{10}\times S_4$, $A_4\times F_5$

Order 200: $D_5:F_5$ x 7, $D_5\times D_{10}$ x 5, $D_5\wr C_2$ x 4, $C_{10}:F_5$ x 3, $C_{10}\times D_{10}$ x 2, $C_{10}:F_5$ x 2, $C_5^2:Q_8$ x 2, $D_5\times F_5$ x 2, $C_{10}:D_{10}$, $C_{10}:F_5$, $C_{10}\times F_5$, $C_5^2:C_8$

Order 192: $C_{12}:C_2^4$ x 17, $C_2^3\times S_4$ x 12, $C_2\times \GL(2,\mathbb{Z}/4)$ x 6, $D_4\times S_4$ x 5, $C_2^3:S_4$ x 3, $C_2^3:S_4$ x 2, $D_6\times C_2^4$ x 2, $A_4\times C_2^4$ x 2, $D_6:C_2^4$ x 2, $C_2^5:C_6$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^2\wr C_3$, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $C_2^4:C_{12}$, $C_2^3.S_4$, $C_2^4:A_4$

Order 180: $\GL(2,4)$

Order 160: $C_2^3\times F_5$ x 12, $C_2^3:F_5$ x 6, $D_4\times F_5$ x 5, $D_4\times D_{10}$ x 4, $C_2^3\times D_{10}$ x 2, $C_2^3:D_{10}$ x 2, $D_{10}.D_4$ x 2, $C_{10}:C_4^2$ x 2, $C_{20}:C_2^3$, $C_2^2\times D_{20}$, $C_{20}:C_2^3$

Order 144: $D_6^2$ x 57, $S_3^2:C_2^2$ x 13, $C_6^2:C_2^2$ x 11, $S_3\times S_4$ x 11, $C_6^2:C_2^2$ x 8, $C_6^2:C_4$ x 6, $A_4\times D_6$ x 5, $C_6:S_4$ x 5, $C_6\times S_4$ x 5, $S_3^2:C_4$ x 3, $C_2^2:C_6^2$ x 2, $D_6:D_6$ x 2, $C_2^2.S_3^2$ x 2, $C_2^2\times C_6^2$, $A_4^2$, $C_6^2:C_2^2$, $C_6^2:C_4$, $C_6:D_{12}$, $C_6^2:C_4$

Order 128: $C_2\times D_4^2$ x 6, $C_2^4:D_4$ x 6, $C_2^4:D_4$ x 5, $C_2^5:C_4$ x 4, $C_2^5:C_4$ x 3, $C_2\wr D_4$ x 2, $D_4\times C_2^4$ x 2, $C_2^4:D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^4.D_4$, $D_4:C_2^4$, $C_8:C_2^4$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $\OD_{16}:C_2^3$

Order 120: $S_3\times D_{10}$ x 15, $S_3\times F_5$ x 11, $S_5$ x 7, $C_2\times A_5$ x 6, $C_{30}:C_4$ x 5, $C_6\times F_5$ x 5, $C_2\times D_{30}$ x 4, $C_{10}\times D_6$ x 4, $C_6\times D_{10}$ x 4, $D_5\times A_4$ x 2, $C_2^2\times C_{30}$, $C_{10}\times A_4$, $C_5:S_4$, $C_5\times S_4$

Order 100: $D_5^2$ x 6, $C_5\times D_{10}$ x 3, $C_5:F_5$ x 3, $C_5:D_{10}$ x 2, $C_5:F_5$ x 2, $C_5\times F_5$, $C_{10}^2$, $C_5:F_5$

Order 96: $D_4\times D_6$ x 53, $C_2^2\times S_4$ x 29, $C_2^3\times D_6$ x 27, $C_2^3:D_6$ x 18, $C_{12}:C_2^3$ x 9, $C_2^2\times D_{12}$ x 9, $C_{12}:C_2^3$ x 9, $C_2^3\times A_4$ x 8, $\GL(2,\mathbb{Z}/4)$ x 7, $C_2^2:S_4$ x 3, $C_2^4\times C_6$ x 2, $D_4\times A_4$ x 2, $C_2^2.S_4$ x 2, $C_4:S_4$ x 2, $C_4\times S_4$ x 2, $C_2^3:A_4$, $C_2^3:A_4$, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_2^3:C_{12}$

Order 80: $C_2^2\times F_5$ x 29, $C_2^2\times D_{10}$ x 14, $D_{10}:C_4$ x 7, $D_4\times D_5$ x 5, $C_{10}:D_4$ x 4, $C_2^3\times C_{10}$ x 2, $D_4\times C_{10}$ x 2, $C_2\times D_{20}$ x 2, $C_4\times D_{10}$ x 2, $C_{20}:C_4$ x 2, $C_4\times F_5$ x 2, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 72: $S_3\times D_6$ x 74, $C_6\times D_6$ x 26, $C_6:D_6$ x 18, $C_2\times C_3^2:C_4$ x 6, $\SOPlus(4,2)$ x 6, $C_2\times C_6^2$ x 4, $S_3\times A_4$ x 3, $C_3:S_4$ x 3, $C_3\times S_4$ x 3, $C_6\times A_4$ x 2, $C_6.D_6$ x 2, $C_6\wr C_2$, $C_6:C_{12}$, $C_3:D_{12}$

Order 64: $D_4\times C_2^3$ x 27, $C_2^3:D_4$ x 18, $D_4^2$ x 11, $C_2^3:D_4$ x 8, $C_2^4:C_4$ x 8, $C_2^4:C_4$ x 7, $C_2\wr C_2^2$ x 7, $D_4:C_2^3$ x 5, $C_4^2:C_2^2$ x 4, $C_4^2:C_2^2$ x 4, $C_2\wr C_4$ x 3, $C_2^6$ x 3, $\OD_{16}:C_2^2$ x 2, $D_4:C_2^3$ x 2, $C_2^4\times C_4$ x 2, $C_8:C_2^3$ x 2, $D_4:D_4$ x 2, $C_4^2:C_4$, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_2^2\times \OD_{16}$, $C_2^3.D_4$, $C_2^2\times C_4^2$, $C_4^2:C_2^2$

Order 60: $S_3\times D_5$ x 11, $D_{30}$ x 5, $S_3\times C_{10}$ x 5, $C_3\times D_{10}$ x 5, $C_{15}:C_4$ x 3, $C_3\times F_5$ x 3, $A_5$ x 2, $C_2\times C_{30}$ x 2, $C_5\times A_4$

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

Order 48: $C_2^2\times D_6$ x 109, $S_3\times D_4$ x 52, $C_6:D_4$ x 32, $C_2\times S_4$ x 26, $C_4\times D_6$ x 17, $C_6\times D_4$ x 16, $C_2\times D_{12}$ x 16, $C_2^3\times C_6$ x 14, $C_2^2\times A_4$ x 13, $C_2^2\times C_{12}$ x 5, $C_6.C_2^3$ x 5, $A_4:C_4$ x 2, $C_2^2:A_4$, $C_4\times A_4$

Order 40: $C_2\times D_{10}$ x 32, $C_2\times F_5$ x 27, $C_2^2\times C_{10}$ x 7, $C_5:D_4$ x 4, $D_{20}$ x 2, $C_4\times D_5$ x 2, $C_5\times D_4$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$

Order 36: $S_3^2$ x 31, $C_6\times S_3$ x 24, $C_6:S_3$ x 16, $C_6^2$ x 6, $C_3^2:C_4$ x 2, $C_3:C_{12}$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 101, $C_2^2\wr C_2$ x 26, $C_2^5$ x 23, $C_2^3:C_4$ x 18, $C_2^3\times C_4$ x 15, $C_4:D_4$ x 10, $D_4:C_2^2$ x 9, $D_4:C_2^2$ x 8, $C_2^3:C_4$ x 7, $C_4:D_4$ x 5, $C_4\times D_4$ x 5, $\OD_{16}:C_2$ x 2, $D_8:C_2$ x 2, $C_2\times \OD_{16}$ x 2, $C_2^2.D_4$ x 2, $C_2\times C_4^2$ x 2, $C_2^2\times Q_8$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2^2\times C_8$, $C_4\wr C_2$

Order 30: $D_{15}$ x 3, $C_3\times D_5$ x 3, $C_5\times S_3$ x 3, $C_{30}$ x 2

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 162, $C_2^2\times C_6$ x 34, $C_3:D_4$ x 24, $C_4\times S_3$ x 13, $D_{12}$ x 12, $C_3\times D_4$ x 12, $S_4$ x 10, $C_2\times A_4$ x 8, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6

Order 20: $D_{10}$ x 25, $C_2\times C_{10}$ x 11, $F_5$ x 9, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 7, $C_3:S_3$ x 5, $C_3\times C_6$ x 4

Order 16: $C_2\times D_4$ x 129, $C_2^4$ x 78, $C_2^2\times C_4$ x 35, $C_2^2:C_4$ x 20, $D_4:C_2$ x 10, $\OD_{16}$ x 2, $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2\times Q_8$ x 2, $\SD_{16}$, $D_8$, $C_2\times C_8$

Order 15: $C_{15}$

Order 12: $D_6$ x 87, $C_2\times C_6$ x 34, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$ x 3

Order 10: $D_5$ x 8, $C_{10}$ x 6

Order 9: $C_3^2$

Order 8: $C_2^3$ x 102, $D_4$ x 45, $C_2\times C_4$ x 32, $Q_8$ x 2, $C_8$

Order 6: $S_3$ x 15, $C_6$ x 13

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 54, $C_4$ x 8

Order 3: $C_3$ x 2

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 115200: $C_2^2\times S_5\wr C_2$ ($C_1$)

Order 57600: $C_2\times A_5^2.D_4$ ($C_2$) x 12, $C_2^2\times A_5.S_5.C_2$ ($C_2$), $C_2^2\times A_5^2.C_2^2$ ($C_2$), $C_2^2.A_5^2.C_4$ ($C_2$)

Order 28800: $A_5^2.D_4$ ($C_2^2$) x 16, $C_2\times A_5.S_5.C_2$ ($C_2^2$) x 6, $C_2.A_5^2.C_4$ ($C_2^2$) x 6, $C_2\times A_5^2.C_2^2$ ($C_2^2$) x 6, $(C_2^2\times A_5).S_5$ ($C_2^2$)

Order 14400: $A_5^2.C_4$ ($C_2^3$) x 4, $A_5.S_5.C_2$ ($C_2^3$) x 4, $A_5^2.C_2^2$ ($C_2^3$) x 4, $(C_2\times A_5).S_5$ ($C_2^3$) x 3, $(C_2\times A_5).S_5$ ($D_4$) x 3, $(C_2^2\times A_5).A_5$ ($D_4$)

Order 7200: $A_5.S_5$ ($C_2\times D_4$) x 3, $(C_2\times A_5).A_5$ ($C_2\times D_4$) x 3, $A_5.S_5$ ($C_2^4$)

Order 3600: $A_5^2$ ($C_2^2\times D_4$)

Order 4: $C_2^2$ ($A_5^2.D_4$)

Order 2: $C_2$ ($C_2.A_5^2.D_4$) x 3

Order 1: $C_1$ ($C_2^2\times S_5\wr C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 115200: $C_2^2\times S_5\wr C_2$ ($C_1$)

Order 57600: $C_2^2\times A_5.S_5.C_2$ ($C_2$), $C_2^2\times A_5^2.C_2^2$ ($C_2$), $C_2\times A_5^2.D_4$ ($C_2$), $C_2^2.A_5^2.C_4$ ($C_2$)

Order 28800: $(C_2^2\times A_5).S_5$ ($C_2^2$), $C_2\times A_5.S_5.C_2$ ($C_2^2$), $C_2.A_5^2.C_4$ ($C_2^2$), $A_5^2.D_4$ ($C_2^2$), $C_2\times A_5^2.C_2^2$ ($C_2^2$)

Order 14400: $A_5^2.C_4$ ($C_2^3$), $A_5.S_5.C_2$ ($C_2^3$), $(C_2\times A_5).S_5$ ($C_2^3$), $(C_2\times A_5).S_5$ ($D_4$), $(C_2^2\times A_5).A_5$ ($D_4$), $A_5^2.C_2^2$ ($C_2^3$)

Order 7200: $A_5.S_5$ ($C_2^4$), $A_5.S_5$ ($C_2\times D_4$), $(C_2\times A_5).A_5$ ($C_2\times D_4$)

Order 3600: $A_5^2$ ($C_2^2\times D_4$)

Order 4: $C_2^2$ ($A_5^2.D_4$)

Order 2: $C_2$ ($C_2.A_5^2.D_4$)

Order 1: $C_1$ ($C_2^2\times S_5\wr C_2$)

Series

Derived series

$C_2^2\times S_5\wr C_2$

$\rhd$

$A_5.S_5$

$\rhd$

$A_5^2$

Chief series

$C_2^2\times S_5\wr C_2$

$\rhd$

$C_2^2\times A_5.S_5.C_2$

$\rhd$

$(C_2^2\times A_5).S_5$

$\rhd$

$(C_2^2\times A_5).A_5$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^2\times S_5\wr C_2$

$\rhd$

$A_5.S_5$

$\rhd$

$A_5^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 10 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 $140 \times 140$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.