Abstract group 176400.a: $C_{420}.D_{210}$

• Label: 176400.a
• Order: \( 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{Z(G)}$: 420
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{3} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3^{2} \)
• Perm deg.: $38$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_{420}.D_{210}$
Order:	\(176400\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(3870720\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_5$ x 2, $C_7$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	21	28	30	35	42	60	70	84	105	140	210	420
Elements	1	423	8	424	24	864	48	1752	872	2664	192	1776	384	2712	3936	1152	6192	4128	13536	6576	9216	14688	47808	57024	176400
Conjugacy classes	1	4	5	5	14	17	27	46	22	87	100	60	198	114	308	588	606	408	1788	804	4632	2376	13944	18576	44730
Divisions	1	4	3	4	4	9	5	12	8	15	14	10	18	12	40	26	52	30	76	38	100	54	294	202	1031

Minimal presentations

Permutation degree:	$38$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{210}=c^{420}=[a,c]=[b,c]=1, b^{a}=b^{209}c^{270} \rangle$
Permutation group:	Degree $38$ $\langle(1,2,4,5,3)(6,7,8)(19,20,22,24,25,23,21)(31,32,34,35)(33,36,37,38), (2,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 4 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 211 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{421})$
Direct product:	$C_3$ $\, \times\, $ $C_5$ $\, \times\, $ $C_7$ $\, \times\, $ $(D_{420}:C_2)$
Semidirect product:	$C_{105}^2$ $\,\rtimes\,$ $(D_4:C_2)$	$C_{35}^2$ $\,\rtimes\,$ $(D_{12}:C_6)$	$C_7^2$ $\,\rtimes\,$ $(D_{60}:C_{30})$	$C_5^2$ $\,\rtimes\,$ $(D_{84}:C_{42})$	all 7
Trans. wreath product:	not computed
Possibly split product:	$D_{420}$ . $C_{210}$	$C_{420}$ . $D_{210}$ (2)	$C_{210}^2$ . $C_2^2$	$(C_{70}\times C_{420})$ . $S_3$	all 355

Elements of the group are displayed as matrices in $\GL_{2}(\F_{421})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{210} \simeq C_{2}^{3} \times C_{3} \times C_{5} \times C_{7}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 472 normal subgroups (408 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_{420}$
Commutator:	a subgroup isomorphic to $C_{210}$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

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 252 (order at least 700) are shown, as well as all normal subgroups of any index.

Order 176400: $C_{420}.D_{210}$

Order 88200: $C_{420}.C_{210}$ x 2, $C_{105}^2:D_4$ x 2, $C_{105}^2:Q_8$, $C_{105}\times D_{420}$, $C_{210}\times C_{420}$

Order 58800: $C_{420}.D_{70}$, $C_{140}.D_{210}$

Order 44100: $C_{105}\times C_{420}$ x 2, $C_{105}:C_{420}$ x 2, $C_{210}.C_{210}$ x 2, $C_{210}^2$

Order 35280: $C_{84}.D_{210}$, $D_{84}.C_{210}$

Order 29400: $C_{70}\times C_{420}$ x 3, $C_{140}.C_{210}$ x 2, $C_{420}.C_{70}$ x 2, $D_{70}.C_{210}$ x 2, $C_{70}.D_{210}$ x 2, $C_{140}.C_{210}$, $C_{140}.C_{210}$, $C_{420}.C_{70}$, $C_{35}^2:D_{12}$

Order 25200: $C_{60}.D_{210}$, $D_{60}.C_{210}$

Order 22050: $C_{105}\times D_{105}$ x 2, $C_{105}\times C_{210}$ x 2

Order 19600: $C_{140}.D_{70}$

Order 17640: $C_{42}\times C_{420}$ x 4, $D_{21}\times C_{420}$ x 2, $C_{84}\times D_{105}$ x 2, $D_{42}.C_{210}$ x 2, $C_{42}.D_{210}$ x 2, $C_{84}.C_{210}$, $C_{420}.C_{42}$, $C_{21}\times D_{420}$, $C_{105}\times D_{84}$

Order 14700: $C_{35}\times C_{420}$ x 6, $C_{70}\times C_{210}$ x 3, $C_{105}:C_{140}$ x 2, $C_{105}\times D_{70}$ x 2, $C_{105}:C_{140}$ x 2, $C_{35}\times D_{210}$ x 2

Order 12600: $C_{30}\times C_{420}$ x 5, $D_{15}\times C_{420}$ x 2, $C_{60}\times D_{105}$ x 2, $D_{30}.C_{210}$ x 2, $C_{30}.D_{210}$ x 2, $C_{60}.C_{210}$, $C_{420}.C_{30}$, $C_{15}\times D_{420}$, $C_{105}\times D_{60}$

Order 11760: $C_{84}.D_{70}$, $C_{28}.D_{210}$, $D_{28}.C_{210}$, $C_{140}.D_{42}$

Order 11025: $C_{105}^2$

Order 9800: $D_{35}\times C_{140}$ x 2, $C_{70}\wr C_2$ x 2, $C_{35}^2:Q_8$, $C_{70}\times C_{140}$, $C_{35}\times D_{140}$

Order 8820: $C_{21}\times C_{420}$ x 8, $C_{42}\times C_{210}$ x 4, $C_{21}:C_{420}$ x 2, $D_{21}\times C_{210}$ x 2, $C_{105}:C_{84}$ x 2, $C_{21}\times D_{210}$ x 2

Order 8400: $D_{140}:C_{30}$, $D_{420}:C_{10}$, $D_{20}:C_{210}$, $D_{60}:C_{70}$

Order 7350: $C_{35}\times C_{210}$ x 7, $D_{35}\times C_{105}$ x 2, $C_{35}\times D_{105}$ x 2

Order 7056: $D_{84}:C_{42}$

Order 6300: $C_{15}\times C_{420}$ x 10, $C_{30}\times C_{210}$ x 5, $D_{15}\times C_{210}$ x 2, $C_{15}:C_{420}$ x 2, $C_{105}:C_{60}$ x 2, $C_{15}\times D_{210}$ x 2

Order 5880: $C_{14}\times C_{420}$ x 14, $D_{35}\times C_{84}$ x 2, $D_7\times C_{420}$ x 2, $D_{21}\times C_{140}$ x 2, $C_{28}\times D_{105}$ x 2, $D_{14}:C_{210}$ x 2, $D_{42}:C_{70}$ x 2, $D_{70}:C_{42}$ x 2, $D_{210}:C_{14}$ x 2, $C_{28}.C_{210}$, $C_{140}.C_{42}$, $C_{21}\times D_{140}$, $C_{420}.C_{14}$, $C_{35}\times D_{84}$, $C_{420}.C_{14}$, $C_7\times D_{420}$, $D_{28}\times C_{105}$

Order 5040: $D_{60}:C_{42}$, $D_{420}:C_6$, $D_{12}:C_{210}$, $D_{84}:C_{30}$

Order 4900: $C_{35}\times C_{140}$ x 2, $C_{35}\times D_{70}$ x 2, $C_{35}:C_{140}$ x 2, $C_{70}^2$

Order 4410: $C_{21}\times C_{210}$ x 10, $D_{21}\times C_{105}$ x 2, $C_{21}\times D_{105}$ x 2

Order 4200: $C_{10}\times C_{420}$ x 18, $D_{15}\times C_{140}$ x 2, $D_5\times C_{420}$ x 2, $C_{20}\times D_{105}$ x 2, $C_{60}\times D_{35}$ x 2, $D_{10}:C_{210}$ x 2, $D_{30}:C_{70}$ x 2, $D_{70}:C_{30}$ x 2, $D_{210}:C_{10}$ x 2, $C_{20}.C_{210}$, $C_{60}.C_{70}$, $C_{35}\times D_{60}$, $C_{140}.C_{30}$, $C_{15}\times D_{140}$, $C_{420}.C_{10}$, $C_5\times D_{420}$, $D_{20}\times C_{105}$

Order 3920: $D_{28}:C_{70}$, $D_{140}:C_{14}$

Order 3675: $C_{35}\times C_{105}$ x 3

Order 3600: $D_{60}:C_{30}$

Order 3528: $D_{21}\times C_{84}$ x 2, $C_{42}\wr C_2$ x 2, $C_{42}\times C_{84}$, $C_{21}^2:Q_8$, $C_{21}\times D_{84}$

Order 3150: $C_{15}\times C_{210}$ x 13, $D_{15}\times C_{105}$ x 2, $C_{15}\times D_{105}$ x 2

Order 2940: $C_7\times C_{420}$ x 28, $C_{14}\times C_{210}$ x 14, $C_{35}:C_{84}$ x 2, $C_{21}\times D_{70}$ x 2, $D_7\times C_{210}$ x 2, $C_7:C_{420}$ x 2, $C_{105}:C_{28}$ x 2, $C_{21}:C_{140}$ x 2, $C_{35}\times D_{42}$ x 2, $C_7\times D_{210}$ x 2

Order 2800: $D_{140}:C_{10}$, $D_{20}:C_{70}$

Order 2520: $C_6\times C_{420}$ x 26, $S_3\times C_{420}$ x 2, $D_{21}\times C_{60}$ x 2, $C_{12}\times D_{105}$ x 2, $D_6:C_{210}$ x 2, $D_{30}:C_{42}$ x 2, $D_{42}:C_{30}$ x 2, $D_{210}:C_6$ x 2, $D_{15}\times C_{84}$ x 2, $C_{21}\times D_{60}$, $C_{420}.C_6$, $C_3\times D_{420}$, $D_{12}\times C_{105}$, $C_{15}\times D_{84}$, $C_{12}.C_{210}$, $C_{420}.C_6$, $C_{420}.C_6$

Order 2450: $C_{35}\times D_{35}$ x 2, $C_{35}\times C_{70}$ x 2

Order 2352: $D_{84}:C_{14}$, $D_{28}:C_{42}$

Order 2205: $C_{21}\times C_{105}$ x 4

Order 2100: $C_5\times C_{420}$ x 36, $C_{10}\times C_{210}$ x 18, $C_{35}\times D_{30}$ x 2, $C_5:C_{420}$ x 2, $D_5\times C_{210}$ x 2, $C_{15}\times D_{70}$ x 2, $C_{35}:C_{60}$ x 2, $C_{105}:C_{20}$ x 2, $C_{15}:C_{140}$ x 2, $C_5\times D_{210}$ x 2

Order 1960: $C_{14}\times C_{140}$ x 4, $D_{14}:C_{70}$ x 2, $D_7\times C_{140}$ x 2, $D_{70}:C_{14}$ x 2, $C_{28}\times D_{35}$ x 2, $C_{140}.C_{14}$, $C_{35}\times D_{28}$, $C_{28}.C_{70}$, $C_7\times D_{140}$

Order 1800: $C_{30}\wr C_2$ x 2, $D_{15}\times C_{60}$ x 2, $C_{30}\times C_{60}$, $C_{15}\times D_{60}$, $C_{15}^2:Q_8$

Order 1764: $C_{21}\times D_{42}$ x 2, $C_{21}\times C_{84}$ x 2, $C_{21}:C_{84}$ x 2, $C_{42}^2$

Order 1680: $D_4:C_{210}$, $D_{420}:C_2$, $D_{60}:C_{14}$, $D_{84}:C_{10}$, $D_{12}:C_{70}$, $D_{140}:C_6$, $D_{20}:C_{42}$, $D_{28}:C_{30}$

Order 1575: $C_{15}\times C_{105}$ x 5

Order 1470: $C_7\times C_{210}$ x 38, $C_7\times D_{105}$ x 2, $C_{35}\times D_{21}$ x 2, $C_{21}\times D_{35}$ x 2, $D_7\times C_{105}$ x 2

Order 1400: $C_{10}\times C_{140}$ x 5, $C_{20}\times D_{35}$ x 2, $D_{10}:C_{70}$ x 2, $D_5\times C_{140}$ x 2, $D_{70}:C_{10}$ x 2, $C_5\times D_{140}$, $C_{140}.C_{10}$, $C_{35}\times D_{20}$, $C_{20}.C_{70}$

Order 1260: $C_3\times C_{420}$ x 52, $C_6\times C_{210}$ x 26, $C_{105}:C_{12}$ x 2, $C_{15}:C_{84}$ x 2, $C_{21}:C_{60}$ x 2, $C_3:C_{420}$ x 2, $C_3\times D_{210}$ x 2, $C_{21}\times D_{30}$ x 2, $C_{15}\times D_{42}$ x 2, $S_3\times C_{210}$ x 2

Order 1225: $C_{35}^2$

Order 1200: $D_{60}:C_{10}$, $D_{20}:C_{30}$

Order 1176: $C_{14}\times C_{84}$ x 3, $D_{42}:C_{14}$ x 2, $C_{28}\times D_{21}$ x 2, $D_{14}:C_{42}$ x 2, $D_7\times C_{84}$ x 2, $C_7\times D_{84}$, $C_{84}.C_{14}$, $C_{21}\times D_{28}$, $C_{28}.C_{42}$

Order 1050: $C_5\times C_{210}$ x 50, $C_5\times D_{105}$ x 2, $D_{15}\times C_{35}$ x 2, $C_{15}\times D_{35}$ x 2, $D_5\times C_{105}$ x 2

Order 1008: $D_{84}:C_6$, $D_{12}:C_{42}$

Order 980: $C_7\times C_{140}$ x 8, $C_{14}\times C_{70}$ x 4, $C_7\times D_{70}$ x 2, $D_7\times C_{70}$ x 2, $C_{35}:C_{28}$ x 2, $C_7:C_{140}$ x 2

Order 900: $C_{15}\times C_{60}$ x 2, $C_{15}:C_{60}$ x 2, $C_{15}\times D_{30}$ x 2, $C_{30}^2$

Order 882: $C_{21}\times C_{42}$ x 2, $C_{21}\times D_{21}$ x 2

Order 840: $C_2\times C_{420}$ x 102, $D_4\times C_{105}$ x 3, $D_7\times C_{60}$ x 2, $C_{105}:D_4$ x 2, $C_4\times D_{105}$ x 2, $C_{105}:D_4$ x 2, $C_{28}\times D_{15}$ x 2, $C_{105}:D_4$ x 2, $C_{20}\times D_{21}$ x 2, $D_6:C_{70}$ x 2, $S_3\times C_{140}$ x 2, $C_{105}:D_4$ x 2, $C_{12}\times D_{35}$ x 2, $D_{10}:C_{42}$ x 2, $D_5\times C_{84}$ x 2, $D_{14}:C_{30}$ x 2, $C_{15}\times D_{28}$, $C_{105}:Q_8$, $Q_8\times C_{105}$, $D_{420}$, $C_{105}:Q_8$, $C_7\times D_{60}$, $C_{105}:Q_8$, $C_5\times D_{84}$, $C_{105}:Q_8$, $D_{12}\times C_{35}$, $C_{105}:Q_8$, $C_3\times D_{140}$, $C_{105}:Q_8$, $C_{21}\times D_{20}$, $C_{105}:Q_8$

Order 784: $D_{28}:C_{14}$

Order 735: $C_7\times C_{105}$ x 14

Order 720: $D_{60}:C_6$, $D_{12}:C_{30}$

Order 700: $C_5\times C_{140}$ x 10, $C_{10}\times C_{70}$ x 5, $C_5\times D_{70}$ x 2, $D_5\times C_{70}$ x 2, $C_{35}:C_{20}$ x 2, $C_5:C_{140}$ x 2

Order 630: $C_3\times C_{210}$ x 4

Order 600: $C_{10}\times C_{60}$ x 2

Order 588: $C_7\times C_{84}$ x 4, $C_{14}\times C_{42}$ x 2

Order 525: $C_5\times C_{105}$ x 4

Order 504: $C_6\times C_{84}$ x 2

Order 490: $C_7\times C_{70}$ x 2

Order 450: $C_{15}\times C_{30}$

Order 441: $C_{21}^2$

Order 420: $C_{420}$ x 16, $C_2\times C_{210}$ x 8, $D_{210}$ x 2, $C_{105}:C_4$ x 2

Order 392: $C_{14}\times C_{28}$

Order 360: $C_6\times C_{60}$ x 2

Order 350: $C_5\times C_{70}$ x 2

Order 315: $C_3\times C_{105}$ x 4

Order 300: $C_5\times C_{60}$ x 4, $C_{10}\times C_{30}$ x 2

Order 294: $C_7\times C_{42}$ x 2

Order 280: $C_2\times C_{140}$ x 4

Order 252: $C_3\times C_{84}$ x 4, $C_6\times C_{42}$ x 2

Order 245: $C_7\times C_{35}$ x 2

Order 225: $C_{15}^2$

Order 210: $C_{210}$ x 8

Order 200: $C_{10}\times C_{20}$

Order 196: $C_7\times C_{28}$ x 2, $C_{14}^2$

Order 180: $C_3\times C_{60}$ x 4, $C_6\times C_{30}$ x 2

Order 175: $C_5\times C_{35}$ x 2

Order 168: $C_2\times C_{84}$ x 4

Order 150: $C_5\times C_{30}$ x 2

Order 147: $C_7\times C_{21}$ x 2

Order 140: $C_{140}$ x 8, $C_2\times C_{70}$ x 4

Order 126: $C_3\times C_{42}$ x 2

Order 120: $C_2\times C_{60}$ x 4

Order 105: $C_{105}$ x 8

Order 100: $C_5\times C_{20}$ x 2, $C_{10}^2$

Order 98: $C_7\times C_{14}$

Order 90: $C_3\times C_{30}$ x 2

Order 84: $C_{84}$ x 8, $C_2\times C_{42}$ x 4

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

Order 72: $C_6\times C_{12}$

Order 70: $C_{70}$ x 4

Order 63: $C_3\times C_{21}$ x 2

Order 60: $C_{60}$ x 8, $C_2\times C_{30}$ x 4

Order 56: $C_2\times C_{28}$ x 2

Order 50: $C_5\times C_{10}$

Order 49: $C_7^2$

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

Order 42: $C_{42}$ x 4

Order 40: $C_2\times C_{20}$ x 2

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

Order 35: $C_{35}$ x 4

Order 30: $C_{30}$ x 4

Order 28: $C_{28}$ x 4, $C_2\times C_{14}$ x 2

Order 25: $C_5^2$

Order 24: $C_2\times C_{12}$ x 2

Order 21: $C_{21}$ x 4

Order 20: $C_{20}$ x 4, $C_2\times C_{10}$ x 2

Order 18: $C_3\times C_6$

Order 15: $C_{15}$ x 4

Order 14: $C_{14}$ x 2

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 2

Order 10: $C_{10}$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$

Order 7: $C_7$ x 2

Order 6: $C_6$ x 2

Order 5: $C_5$ x 2

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

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 176400: $C_{420}.D_{210}$ ($C_1$)

Order 88200: $C_{420}.C_{210}$ ($C_2$) x 2, $C_{105}^2:D_4$ ($C_2$) x 2, $C_{105}^2:Q_8$ ($C_2$), $C_{105}\times D_{420}$ ($C_2$), $C_{210}\times C_{420}$ ($C_2$)

Order 58800: $C_{140}.D_{210}$ ($C_3$)

Order 44100: $C_{105}\times C_{420}$ ($C_2^2$) x 2, $C_{105}:C_{420}$ ($C_2^2$) x 2, $C_{210}.C_{210}$ ($C_2^2$) x 2, $C_{210}^2$ ($C_2^2$)

Order 35280: $C_{84}.D_{210}$ ($C_5$)

Order 29400: $C_{420}.C_{70}$ ($C_6$) x 2, $C_{70}.D_{210}$ ($C_6$) x 2, $C_{420}.C_{70}$ ($C_6$), $C_{35}^2:D_{12}$ ($C_6$), $C_{70}\times C_{420}$ ($C_6$), $C_{70}\times C_{420}$ ($S_3$)

Order 25200: $C_{60}.D_{210}$ ($C_7$)

Order 22050: $C_{105}\times C_{210}$ ($C_2^3$)

Order 17640: $C_{84}\times D_{105}$ ($C_{10}$) x 2, $C_{42}.D_{210}$ ($C_{10}$) x 2, $C_{420}.C_{42}$ ($C_{10}$), $C_{21}\times D_{420}$ ($C_{10}$), $C_{42}\times C_{420}$ ($C_{10}$), $C_{42}\times C_{420}$ ($D_5$)

Order 14700: $C_{35}\times C_{420}$ ($C_2\times C_6$) x 2, $C_{35}\times C_{420}$ ($D_6$) x 2, $C_{105}:C_{140}$ ($C_2\times C_6$) x 2, $C_{35}\times D_{210}$ ($C_2\times C_6$) x 2, $C_{70}\times C_{210}$ ($C_2\times C_6$), $C_{70}\times C_{210}$ ($D_6$)

Order 12600: $C_{60}\times D_{105}$ ($C_{14}$) x 2, $C_{30}.D_{210}$ ($C_{14}$) x 2, $C_{420}.C_{30}$ ($C_{14}$), $C_{15}\times D_{420}$ ($C_{14}$), $C_{30}\times C_{420}$ ($C_{14}$), $C_{30}\times C_{420}$ ($D_7$)

Order 11760: $C_{28}.D_{210}$ ($C_{15}$)

Order 11025: $C_{105}^2$ ($D_4:C_2$)

Order 9800: $C_{70}\times C_{140}$ ($C_3\times S_3$)

Order 8820: $C_{21}\times C_{420}$ ($C_2\times C_{10}$) x 2, $C_{21}\times C_{420}$ ($D_{10}$) x 2, $C_{105}:C_{84}$ ($C_2\times C_{10}$) x 2, $C_{21}\times D_{210}$ ($C_2\times C_{10}$) x 2, $C_{42}\times C_{210}$ ($C_2\times C_{10}$), $C_{42}\times C_{210}$ ($D_{10}$)

Order 8400: $D_{420}:C_{10}$ ($C_{21}$)

Order 7350: $C_{35}\times C_{210}$ ($C_2^2\times C_6$), $C_{35}\times C_{210}$ ($C_2\times D_6$)

Order 6300: $C_{15}\times C_{420}$ ($C_2\times C_{14}$) x 2, $C_{15}\times C_{420}$ ($D_{14}$) x 2, $C_{105}:C_{60}$ ($C_2\times C_{14}$) x 2, $C_{15}\times D_{210}$ ($C_2\times C_{14}$) x 2, $C_{30}\times C_{210}$ ($C_2\times C_{14}$), $C_{30}\times C_{210}$ ($D_{14}$)

Order 5880: $C_{28}\times D_{105}$ ($C_{30}$) x 2, $D_{210}:C_{14}$ ($C_{30}$) x 2, $C_{420}.C_{14}$ ($C_{30}$), $C_7\times D_{420}$ ($C_{30}$), $C_{14}\times C_{420}$ ($C_{30}$), $C_{14}\times C_{420}$ ($D_{15}$), $C_{14}\times C_{420}$ ($C_3\times D_5$), $C_{14}\times C_{420}$ ($C_5\times S_3$)

Order 5040: $D_{420}:C_6$ ($C_{35}$)

Order 4900: $C_{35}\times C_{140}$ ($C_6\times S_3$) x 2, $C_{70}^2$ ($C_6\times S_3$)

Order 4410: $C_{21}\times C_{210}$ ($C_2^2\times C_{10}$), $C_{21}\times C_{210}$ ($C_2\times D_{10}$)

Order 4200: $C_{20}\times D_{105}$ ($C_{42}$) x 2, $D_{210}:C_{10}$ ($C_{42}$) x 2, $C_{420}.C_{10}$ ($C_{42}$), $C_5\times D_{420}$ ($C_{42}$), $C_{10}\times C_{420}$ ($C_{42}$), $C_{10}\times C_{420}$ ($D_{21}$), $C_{10}\times C_{420}$ ($C_3\times D_7$), $C_{10}\times C_{420}$ ($S_3\times C_7$)

Order 3675: $C_{35}\times C_{105}$ ($D_4:C_6$), $C_{35}\times C_{105}$ ($D_{12}:C_2$)

Order 3528: $C_{42}\times C_{84}$ ($C_5\times D_5$)

Order 3150: $C_{15}\times C_{210}$ ($C_2^2\times C_{14}$), $C_{15}\times C_{210}$ ($C_2\times D_{14}$)

Order 2940: $C_7\times C_{420}$ ($C_2\times C_{30}$) x 2, $C_7\times C_{420}$ ($D_{30}$) x 2, $C_7\times C_{420}$ ($S_3\times C_{10}$) x 2, $C_7\times C_{420}$ ($C_3\times D_{10}$) x 2, $C_{105}:C_{28}$ ($C_2\times C_{30}$) x 2, $C_7\times D_{210}$ ($C_2\times C_{30}$) x 2, $C_{14}\times C_{210}$ ($C_2\times C_{30}$), $C_{14}\times C_{210}$ ($D_{30}$), $C_{14}\times C_{210}$ ($S_3\times C_{10}$), $C_{14}\times C_{210}$ ($C_3\times D_{10}$)

Order 2520: $C_{12}\times D_{105}$ ($C_{70}$) x 2, $D_{210}:C_6$ ($C_{70}$) x 2, $C_{420}.C_6$ ($C_{70}$), $C_3\times D_{420}$ ($C_{70}$), $C_6\times C_{420}$ ($C_{70}$), $C_6\times C_{420}$ ($D_{35}$), $C_6\times C_{420}$ ($C_5\times D_7$), $C_6\times C_{420}$ ($C_7\times D_5$)

Order 2450: $C_{35}\times C_{70}$ ($C_6\times D_6$)

Order 2205: $C_{21}\times C_{105}$ ($D_4:C_{10}$), $C_{21}\times C_{105}$ ($D_{20}:C_2$)

Order 2100: $C_5\times C_{420}$ ($C_2\times C_{42}$) x 2, $C_5\times C_{420}$ ($D_{42}$) x 2, $C_5\times C_{420}$ ($S_3\times C_{14}$) x 2, $C_5\times C_{420}$ ($C_3\times D_{14}$) x 2, $C_{105}:C_{20}$ ($C_2\times C_{42}$) x 2, $C_5\times D_{210}$ ($C_2\times C_{42}$) x 2, $C_{10}\times C_{210}$ ($C_2\times C_{42}$), $C_{10}\times C_{210}$ ($D_{42}$), $C_{10}\times C_{210}$ ($S_3\times C_{14}$), $C_{10}\times C_{210}$ ($C_3\times D_{14}$)

Order 1960: $C_{14}\times C_{140}$ ($C_3\times D_{15}$), $C_{14}\times C_{140}$ ($S_3\times C_{15}$)

Order 1800: $C_{30}\times C_{60}$ ($C_7\times D_7$)

Order 1764: $C_{21}\times C_{84}$ ($C_5\times D_{10}$) x 2, $C_{42}^2$ ($C_5\times D_{10}$)

Order 1680: $D_{420}:C_2$ ($C_{105}$)

Order 1575: $C_{15}\times C_{105}$ ($D_4:C_{14}$), $C_{15}\times C_{105}$ ($D_{28}:C_2$)

Order 1470: $C_7\times C_{210}$ ($C_2^2\times C_{30}$), $C_7\times C_{210}$ ($C_2\times D_{30}$), $C_7\times C_{210}$ ($C_{10}\times D_6$), $C_7\times C_{210}$ ($C_6\times D_{10}$)

Order 1400: $C_{10}\times C_{140}$ ($C_3\times D_{21}$), $C_{10}\times C_{140}$ ($S_3\times C_{21}$)

Order 1260: $C_3\times C_{420}$ ($C_7\times D_{10}$) x 2, $C_3\times C_{420}$ ($C_5\times D_{14}$) x 2, $C_3\times C_{420}$ ($C_2\times C_{70}$) x 2, $C_3\times C_{420}$ ($D_{70}$) x 2, $C_{105}:C_{12}$ ($C_2\times C_{70}$) x 2, $C_3\times D_{210}$ ($C_2\times C_{70}$) x 2, $C_6\times C_{210}$ ($C_7\times D_{10}$), $C_6\times C_{210}$ ($C_5\times D_{14}$), $C_6\times C_{210}$ ($C_2\times C_{70}$), $C_6\times C_{210}$ ($D_{70}$)

Order 1225: $C_{35}^2$ ($D_{12}:C_6$)

Order 1176: $C_{14}\times C_{84}$ ($D_5\times C_{15}$), $C_{14}\times C_{84}$ ($C_5\times D_{15}$)

Order 1050: $C_5\times C_{210}$ ($C_2^2\times C_{42}$), $C_5\times C_{210}$ ($C_2\times D_{42}$), $C_5\times C_{210}$ ($D_6\times C_{14}$), $C_5\times C_{210}$ ($C_6\times D_{14}$)

Order 980: $C_7\times C_{140}$ ($C_3\times D_{30}$) x 2, $C_7\times C_{140}$ ($S_3\times C_{30}$) x 2, $C_{14}\times C_{70}$ ($C_3\times D_{30}$), $C_{14}\times C_{70}$ ($S_3\times C_{30}$)

Order 900: $C_{15}\times C_{60}$ ($C_7\times D_{14}$) x 2, $C_{30}^2$ ($C_7\times D_{14}$)

Order 882: $C_{21}\times C_{42}$ ($C_{10}\times D_{10}$)

Order 840: $C_{105}:D_4$ ($C_{210}$) x 2, $C_4\times D_{105}$ ($C_{210}$) x 2, $C_2\times C_{420}$ ($C_5\times D_{21}$), $C_2\times C_{420}$ ($S_3\times C_{35}$), $C_2\times C_{420}$ ($C_3\times D_{35}$), $C_2\times C_{420}$ ($D_5\times C_{21}$), $C_2\times C_{420}$ ($D_7\times C_{15}$), $C_2\times C_{420}$ ($C_{210}$), $C_2\times C_{420}$ ($D_{105}$), $C_2\times C_{420}$ ($C_7\times D_{15}$), $D_{420}$ ($C_{210}$), $C_{105}:Q_8$ ($C_{210}$)

Order 735: $C_7\times C_{105}$ ($D_4:C_{30}$), $C_7\times C_{105}$ ($D_{60}:C_2$), $C_7\times C_{105}$ ($D_{12}:C_{10}$), $C_7\times C_{105}$ ($D_{20}:C_6$)

Order 700: $C_5\times C_{140}$ ($C_3\times D_{42}$) x 2, $C_5\times C_{140}$ ($S_3\times C_{42}$) x 2, $C_{10}\times C_{70}$ ($C_3\times D_{42}$), $C_{10}\times C_{70}$ ($S_3\times C_{42}$)

Order 630: $C_3\times C_{210}$ ($C_2^2\times C_{70}$), $C_3\times C_{210}$ ($C_2\times D_{70}$), $C_3\times C_{210}$ ($C_{14}\times D_{10}$), $C_3\times C_{210}$ ($C_{10}\times D_{14}$)

Order 600: $C_{10}\times C_{60}$ ($C_7\times D_{21}$), $C_{10}\times C_{60}$ ($D_7\times C_{21}$)

Order 588: $C_7\times C_{84}$ ($C_5\times D_{30}$) x 2, $C_7\times C_{84}$ ($C_{15}\times D_{10}$) x 2, $C_{14}\times C_{42}$ ($C_5\times D_{30}$), $C_{14}\times C_{42}$ ($C_{15}\times D_{10}$)

Order 525: $C_5\times C_{105}$ ($D_4:C_{42}$), $C_5\times C_{105}$ ($D_{84}:C_2$), $C_5\times C_{105}$ ($D_{12}:C_{14}$), $C_5\times C_{105}$ ($D_{28}:C_6$)

Order 504: $C_6\times C_{84}$ ($C_5\times D_{35}$), $C_6\times C_{84}$ ($D_5\times C_{35}$)

Order 490: $C_7\times C_{70}$ ($C_6\times D_{30}$), $C_7\times C_{70}$ ($D_6\times C_{30}$)

Order 450: $C_{15}\times C_{30}$ ($C_{14}\times D_{14}$)

Order 441: $C_{21}^2$ ($D_{20}:C_{10}$)

Order 420: $D_{210}$ ($C_2\times C_{210}$) x 2, $C_{420}$ ($C_2\times C_{210}$) x 2, $C_{420}$ ($D_{210}$) x 2, $C_{420}$ ($C_7\times D_{30}$) x 2, $C_{420}$ ($C_5\times D_{42}$) x 2, $C_{420}$ ($S_3\times C_{70}$) x 2, $C_{420}$ ($C_3\times D_{70}$) x 2, $C_{420}$ ($D_5\times C_{42}$) x 2, $C_{420}$ ($C_{15}\times D_{14}$) x 2, $C_{105}:C_4$ ($C_2\times C_{210}$) x 2, $C_2\times C_{210}$ ($C_2\times C_{210}$), $C_2\times C_{210}$ ($D_{210}$), $C_2\times C_{210}$ ($C_7\times D_{30}$), $C_2\times C_{210}$ ($C_5\times D_{42}$), $C_2\times C_{210}$ ($S_3\times C_{70}$), $C_2\times C_{210}$ ($C_3\times D_{70}$), $C_2\times C_{210}$ ($D_5\times C_{42}$), $C_2\times C_{210}$ ($C_{15}\times D_{14}$)

Order 392: $C_{14}\times C_{28}$ ($C_{15}\times D_{15}$)

Order 360: $C_6\times C_{60}$ ($C_7\times D_{35}$), $C_6\times C_{60}$ ($D_7\times C_{35}$)

Order 350: $C_5\times C_{70}$ ($C_6\times D_{42}$), $C_5\times C_{70}$ ($D_6\times C_{42}$)

Order 315: $C_3\times C_{105}$ ($D_4:C_{70}$), $C_3\times C_{105}$ ($D_{140}:C_2$), $C_3\times C_{105}$ ($D_{20}:C_{14}$), $C_3\times C_{105}$ ($D_{28}:C_{10}$)

Order 300: $C_5\times C_{60}$ ($C_7\times D_{42}$) x 2, $C_5\times C_{60}$ ($C_{21}\times D_{14}$) x 2, $C_{10}\times C_{30}$ ($C_7\times D_{42}$), $C_{10}\times C_{30}$ ($C_{21}\times D_{14}$)

Order 294: $C_7\times C_{42}$ ($C_{10}\times D_{30}$), $C_7\times C_{42}$ ($D_{10}\times C_{30}$)

Order 280: $C_2\times C_{140}$ ($C_3\times D_{105}$), $C_2\times C_{140}$ ($C_{21}\times D_{15}$), $C_2\times C_{140}$ ($C_{15}\times D_{21}$), $C_2\times C_{140}$ ($S_3\times C_{105}$)

Order 252: $C_3\times C_{84}$ ($C_5\times D_{70}$) x 2, $C_3\times C_{84}$ ($D_5\times C_{70}$) x 2, $C_6\times C_{42}$ ($C_5\times D_{70}$), $C_6\times C_{42}$ ($D_5\times C_{70}$)

Order 245: $C_7\times C_{35}$ ($D_{60}:C_6$), $C_7\times C_{35}$ ($D_{12}:C_{30}$)

Order 225: $C_{15}^2$ ($D_{28}:C_{14}$)

Order 210: $C_{210}$ ($C_2^2\times C_{210}$), $C_{210}$ ($C_2\times D_{210}$), $C_{210}$ ($C_{14}\times D_{30}$), $C_{210}$ ($C_{10}\times D_{42}$), $C_{210}$ ($D_6\times C_{70}$), $C_{210}$ ($C_6\times D_{70}$), $C_{210}$ ($D_{10}\times C_{42}$), $C_{210}$ ($D_{14}\times C_{30}$)

Order 200: $C_{10}\times C_{20}$ ($C_{21}\times D_{21}$)

Order 196: $C_7\times C_{28}$ ($C_{15}\times D_{30}$) x 2, $C_{14}^2$ ($C_{15}\times D_{30}$)

Order 180: $C_3\times C_{60}$ ($C_7\times D_{70}$) x 2, $C_3\times C_{60}$ ($D_7\times C_{70}$) x 2, $C_6\times C_{30}$ ($C_7\times D_{70}$), $C_6\times C_{30}$ ($D_7\times C_{70}$)

Order 175: $C_5\times C_{35}$ ($D_{84}:C_6$), $C_5\times C_{35}$ ($D_{12}:C_{42}$)

Order 168: $C_2\times C_{84}$ ($C_5\times D_{105}$), $C_2\times C_{84}$ ($D_{15}\times C_{35}$), $C_2\times C_{84}$ ($C_{15}\times D_{35}$), $C_2\times C_{84}$ ($D_5\times C_{105}$)

Order 150: $C_5\times C_{30}$ ($C_{14}\times D_{42}$), $C_5\times C_{30}$ ($D_{14}\times C_{42}$)

Order 147: $C_7\times C_{21}$ ($D_{60}:C_{10}$), $C_7\times C_{21}$ ($D_{20}:C_{30}$)

Order 140: $C_{140}$ ($C_3\times D_{210}$) x 2, $C_{140}$ ($C_{21}\times D_{30}$) x 2, $C_{140}$ ($C_{15}\times D_{42}$) x 2, $C_{140}$ ($S_3\times C_{210}$) x 2, $C_2\times C_{70}$ ($C_3\times D_{210}$), $C_2\times C_{70}$ ($C_{21}\times D_{30}$), $C_2\times C_{70}$ ($C_{15}\times D_{42}$), $C_2\times C_{70}$ ($S_3\times C_{210}$)

Order 126: $C_3\times C_{42}$ ($C_{10}\times D_{70}$), $C_3\times C_{42}$ ($D_{10}\times C_{70}$)

Order 120: $C_2\times C_{60}$ ($C_7\times D_{105}$), $C_2\times C_{60}$ ($C_{35}\times D_{21}$), $C_2\times C_{60}$ ($C_{21}\times D_{35}$), $C_2\times C_{60}$ ($D_7\times C_{105}$)

Order 105: $C_{105}$ ($D_4:C_{210}$), $C_{105}$ ($D_{420}:C_2$), $C_{105}$ ($D_{60}:C_{14}$), $C_{105}$ ($D_{84}:C_{10}$), $C_{105}$ ($D_{12}:C_{70}$), $C_{105}$ ($D_{140}:C_6$), $C_{105}$ ($D_{20}:C_{42}$), $C_{105}$ ($D_{28}:C_{30}$)

Order 100: $C_5\times C_{20}$ ($C_{21}\times D_{42}$) x 2, $C_{10}^2$ ($C_{21}\times D_{42}$)

Order 98: $C_7\times C_{14}$ ($C_{30}\times D_{30}$)

Order 90: $C_3\times C_{30}$ ($C_{14}\times D_{70}$), $C_3\times C_{30}$ ($D_{14}\times C_{70}$)

Order 84: $C_{84}$ ($C_{35}\times D_{30}$) x 2, $C_{84}$ ($D_5\times C_{210}$) x 2, $C_{84}$ ($C_{15}\times D_{70}$) x 2, $C_{84}$ ($C_5\times D_{210}$) x 2, $C_2\times C_{42}$ ($C_{35}\times D_{30}$), $C_2\times C_{42}$ ($D_5\times C_{210}$), $C_2\times C_{42}$ ($C_{15}\times D_{70}$), $C_2\times C_{42}$ ($C_5\times D_{210}$)

Order 75: $C_5\times C_{15}$ ($D_{84}:C_{14}$), $C_5\times C_{15}$ ($D_{28}:C_{42}$)

Order 72: $C_6\times C_{12}$ ($C_{35}\times D_{35}$)

Order 70: $C_{70}$ ($C_7:(C_2\times C_{30}\times S_3)$) x 2, $C_{70}$ ($C_7:(C_2\times C_6\times D_{15})$) x 2

Order 63: $C_3\times C_{21}$ ($D_{140}:C_{10}$), $C_3\times C_{21}$ ($D_{20}:C_{70}$)

Order 60: $C_{60}$ ($C_{21}\times D_{70}$) x 2, $C_{60}$ ($D_7\times C_{210}$) x 2, $C_{60}$ ($C_{35}\times D_{42}$) x 2, $C_{60}$ ($C_7\times D_{210}$) x 2, $C_2\times C_{30}$ ($C_{21}\times D_{70}$), $C_2\times C_{30}$ ($D_7\times C_{210}$), $C_2\times C_{30}$ ($C_{35}\times D_{42}$), $C_2\times C_{30}$ ($C_7\times D_{210}$)

Order 56: $C_2\times C_{28}$ ($D_{15}\times C_{105}$), $C_2\times C_{28}$ ($C_{15}\times D_{105}$)

Order 50: $C_5\times C_{10}$ ($C_7^2:(C_2\times C_6\times S_3)$)

Order 49: $C_7^2$ ($D_{60}:C_{30}$)

Order 45: $C_3\times C_{15}$ ($D_{28}:C_{70}$), $C_3\times C_{15}$ ($D_{140}:C_{14}$)

Order 42: $C_{42}$ ($C_7:(C_2^2\times C_5\times D_{15})$) x 2, $C_{42}$ ($C_7:(C_2\times C_6\times C_5\times D_5)$) x 2

Order 40: $C_2\times C_{20}$ ($D_{21}\times C_{105}$), $C_2\times C_{20}$ ($C_{21}\times D_{105}$)

Order 36: $C_3\times C_{12}$ ($C_{35}\times D_{70}$) x 2, $C_6^2$ ($C_{35}\times D_{70}$)

Order 35: $C_{35}$ ($D_{60}:C_{42}$), $C_{35}$ ($D_{420}:C_6$), $C_{35}$ ($D_{12}:C_{210}$), $C_{35}$ ($D_{84}:C_{30}$)

Order 30: $C_{30}$ ($C_7^2:(C_2^2\times D_{15})$), $C_{30}$ ($C_7^2:(C_2^2\times C_{30})$), $C_{30}$ ($C_7^2:(C_2\times C_{10}\times S_3)$), $C_{30}$ ($C_7^2:(C_2\times C_6\times D_5)$)

Order 28: $C_{28}$ ($D_{15}\times C_{210}$) x 2, $C_{28}$ ($C_{15}\times D_{210}$) x 2, $C_2\times C_{14}$ ($D_{15}\times C_{210}$), $C_2\times C_{14}$ ($C_{15}\times D_{210}$)

Order 25: $C_5^2$ ($D_{84}:C_{42}$)

Order 24: $C_2\times C_{12}$ ($D_{35}\times C_{105}$), $C_2\times C_{12}$ ($C_{35}\times D_{105}$)

Order 21: $C_{21}$ ($D_{140}:C_{30}$), $C_{21}$ ($D_{420}:C_{10}$), $C_{21}$ ($D_{20}:C_{210}$), $C_{21}$ ($D_{60}:C_{70}$)

Order 20: $C_{20}$ ($D_{21}\times C_{210}$) x 2, $C_{20}$ ($C_{21}\times D_{210}$) x 2, $C_2\times C_{10}$ ($D_{21}\times C_{210}$), $C_2\times C_{10}$ ($C_{21}\times D_{210}$)

Order 18: $C_3\times C_6$ ($C_7^2:(C_2\times C_{10}\times D_5)$)

Order 15: $C_{15}$ ($C_{84}.D_{70}$), $C_{15}$ ($C_{28}.D_{210}$), $C_{15}$ ($D_{28}.C_{210}$), $C_{15}$ ($C_{140}.D_{42}$)

Order 14: $C_{14}$ ($C_{105}.C_{30}.C_2^2$) x 2

Order 12: $C_{12}$ ($C_{105}\times D_{70}$) x 2, $C_{12}$ ($C_{35}\times D_{210}$) x 2, $C_2\times C_6$ ($C_{105}\times D_{70}$), $C_2\times C_6$ ($C_{35}\times D_{210}$)

Order 10: $C_{10}$ ($(C_{42}\times C_{210}).C_2$) x 2

Order 9: $C_3^2$ ($C_{140}.D_{70}$)

Order 8: $C_2\times C_4$ ($C_{105}\times D_{105}$)

Order 7: $C_7$ ($C_{60}.D_{210}$), $C_7$ ($D_{60}.C_{210}$)

Order 6: $C_6$ ($(C_{70}\times C_{210}).C_2$) x 2

Order 5: $C_5$ ($C_{84}.D_{210}$), $C_5$ ($D_{84}.C_{210}$)

Order 4: $C_4$ ($C_{210}.C_{210}$) x 2, $C_2^2$ ($C_{210}.C_{210}$)

Order 3: $C_3$ ($C_{420}.D_{70}$), $C_3$ ($C_{140}.D_{210}$)

Order 2: $C_2$ ($C_{210}^2.C_2$)

Order 1: $C_1$ ($C_{420}.D_{210}$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

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

Character theory

Complex character table

The $44730 \times 44730$ character table is not available for this group.

Rational character table

The $1031 \times 1031$ rational character table is not available for this group.