Abstract group 17640.i: $C_{420}.C_{42}$

• Label: 17640.i
• Order: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \cdot 7 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3^{2} \)
• Perm deg.: $33$
• Trans deg.: $840$
• Rank: $2$

Group information

Description:	$C_{420}.C_{42}$
Order:	\(17640\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_{35}.C_6^3.C_2^6$, of order \(483840\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_5$, $C_7$ x 2
Derived length:	$2$

This group is nonabelian and metacyclic (hence solvable, supersolvable, 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	1	8	422	4	8	48	4	856	48	32	8	384	2616	32	192	384	64	192	5808	1536	384	1536	3072	17640
Conjugacy classes	1	1	5	3	2	5	27	2	12	27	16	4	198	60	16	96	198	32	96	408	768	192	768	1536	4473
Divisions	1	1	3	3	1	3	5	1	5	5	3	1	18	7	3	5	18	3	5	20	18	5	18	18	170
Autjugacy classes	1	1	3	2	1	3	3	1	4	3	3	1	9	4	3	3	9	3	3	10	9	3	9	9	100

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$840$
Rank:	$2$
Inequivalent generating pairs:	$96$

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

Constructions

Presentation:	$\langle a, b \mid b^{420}=1, a^{42}=b^{210}, b^{a}=b^{419} \rangle$
Permutation group:	Degree $33$ $\langle(1,2,3,5,4)(6,7,8)(16,17,19,22)(18,21,23,20)(27,28,30,33,29,32,31), (2,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 2 & 0 \\ 0 & 211 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 286 \end{array}\right), \left(\begin{array}{rr} 0 & 182 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{421})$
Direct product:	$C_3$ $\, \times\, $ $C_7$ $\, \times\, $ $(C_{105}:Q_8)$
Semidirect product:	$C_{35}$ $\,\rtimes\,$ $(C_{84}.C_6)$	$C_7$ $\,\rtimes\,$ $(C_{420}.C_6)$	$(C_{21}\times C_{105})$ $\,\rtimes\,$ $Q_8$	$C_{21}^2$ $\,\rtimes\,$ $(C_5:Q_8)$	all 28
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{420}$ . $D_{21}$	$C_{210}$ . $D_{42}$	$C_{84}$ . $D_{105}$	$C_{42}$ . $D_{210}$	all 68

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

Homology

Abelianization:	$C_{2} \times C_{42} \simeq C_{2}^{2} \times C_{3} \times C_{7}$
Schur multiplier:	$C_1$
Commutator length:	not computed

Subgroups

There are 2664 subgroups in 324 conjugacy classes, 108 normal (100 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{42}$	$G/Z \simeq$ $D_{210}$
Commutator:	$G' \simeq$ $C_{210}$	$G/G' \simeq$ $C_2\times C_{42}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{21}\times D_{210}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{21}\times C_{420}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{420}.C_{42}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{21}\times C_{210}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

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 17640: $C_{420}.C_{42}$

Order 8820: $C_{105}:C_{84}$ x 2, $C_{21}\times C_{420}$

Order 5880: $C_{140}.C_{42}$, $C_{420}.C_{14}$

Order 4410: $C_{21}\times C_{210}$

Order 3528: $C_{21}^2:Q_8$

Order 2940: $C_7\times C_{420}$ x 3, $C_{35}:C_{84}$ x 2, $C_{105}:C_{28}$ x 2

Order 2520: $C_{420}.C_6$, $C_{420}.C_6$

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

Order 1960: $C_{140}.C_{14}$

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

Order 1470: $C_7\times C_{210}$ x 3

Order 1260: $C_3\times C_{420}$ x 5, $C_{105}:C_{12}$ x 2, $C_{15}:C_{84}$ x 2

Order 1176: $C_{84}.C_{14}$, $C_{28}.C_{42}$

Order 980: $C_{35}:C_{28}$ x 2, $C_7\times C_{140}$

Order 882: $C_{21}\times C_{42}$

Order 840: $C_{105}:Q_8$, $C_{105}:Q_8$, $C_{105}:Q_8$, $C_{105}:Q_8$

Order 735: $C_7\times C_{105}$ x 3

Order 630: $C_3\times C_{210}$ x 5

Order 588: $C_7\times C_{84}$ x 3, $C_{21}:C_{28}$ x 2, $C_7:C_{84}$ x 2

Order 504: $C_{84}.C_6$, $C_{12}.C_{42}$

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

Order 441: $C_{21}^2$

Order 420: $C_{420}$ x 18, $C_{105}:C_4$ x 2, $C_5:C_{84}$ x 2, $C_{105}:C_4$ x 2, $C_{15}:C_{28}$ x 2

Order 392: $C_7^2:Q_8$

Order 360: $C_{60}.C_6$

Order 315: $C_3\times C_{105}$ x 5

Order 294: $C_7\times C_{42}$ x 3

Order 280: $C_{35}:Q_8$, $C_{35}:Q_8$

Order 252: $C_3\times C_{84}$ x 5, $C_{21}:C_{12}$ x 2, $C_3:C_{84}$ x 2

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

Order 210: $C_{210}$ x 18

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

Order 180: $C_{15}:C_{12}$ x 2, $C_3\times C_{60}$

Order 168: $Q_8\times C_{21}$, $C_{21}:Q_8$, $C_{21}:Q_8$, $C_{21}:Q_8$

Order 147: $C_7\times C_{21}$ x 3

Order 140: $C_{140}$ x 5, $C_{35}:C_4$ x 2, $C_5:C_{28}$ x 2

Order 126: $C_3\times C_{42}$ x 5

Order 120: $C_{15}:Q_8$, $C_{15}:Q_8$

Order 105: $C_{105}$ x 18

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

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

Order 84: $C_{84}$ x 20, $C_{21}:C_4$ x 2, $C_7:C_{12}$ x 2, $C_3:C_{28}$ x 2

Order 72: $C_3^2:Q_8$

Order 70: $C_{70}$ x 5

Order 63: $C_3\times C_{21}$ x 5

Order 60: $C_{60}$ x 3, $C_{15}:C_4$ x 2, $C_5:C_{12}$ x 2

Order 56: $C_7:Q_8$, $C_7\times Q_8$

Order 49: $C_7^2$

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

Order 42: $C_{42}$ x 18

Order 40: $C_5:Q_8$

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

Order 35: $C_{35}$ x 5

Order 30: $C_{30}$ x 3

Order 28: $C_{28}$ x 7, $C_7:C_4$ x 2

Order 24: $C_3:Q_8$, $C_3\times Q_8$

Order 21: $C_{21}$ x 18

Order 20: $C_5:C_4$ x 2, $C_{20}$

Order 18: $C_3\times C_6$

Order 15: $C_{15}$ x 3

Order 14: $C_{14}$ x 5

Order 12: $C_{12}$ x 5, $C_3:C_4$ x 2

Order 10: $C_{10}$

Order 9: $C_3^2$

Order 8: $Q_8$

Order 7: $C_7$ x 5

Order 6: $C_6$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 3

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 17640: $C_{420}.C_{42}$

Order 8820: $C_{21}\times C_{420}$, $C_{105}:C_{84}$

Order 5880: $C_{140}.C_{42}$, $C_{420}.C_{14}$

Order 4410: $C_{21}\times C_{210}$

Order 3528: $C_{21}^2:Q_8$

Order 2940: $C_7\times C_{420}$ x 3, $C_{35}:C_{84}$, $C_{105}:C_{28}$

Order 2520: $C_{420}.C_6$, $C_{420}.C_6$

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

Order 1960: $C_{140}.C_{14}$

Order 1764: $C_{21}\times C_{84}$, $C_{21}:C_{84}$

Order 1470: $C_7\times C_{210}$ x 3

Order 1260: $C_3\times C_{420}$ x 3, $C_{105}:C_{12}$, $C_{15}:C_{84}$

Order 1176: $C_{84}.C_{14}$, $C_{28}.C_{42}$

Order 980: $C_7\times C_{140}$, $C_{35}:C_{28}$

Order 882: $C_{21}\times C_{42}$

Order 840: $C_{105}:Q_8$, $C_{105}:Q_8$, $C_{105}:Q_8$, $C_{105}:Q_8$

Order 735: $C_7\times C_{105}$ x 3

Order 630: $C_3\times C_{210}$ x 3

Order 588: $C_7\times C_{84}$ x 3, $C_{21}:C_{28}$, $C_7:C_{84}$

Order 504: $C_{84}.C_6$, $C_{12}.C_{42}$

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

Order 441: $C_{21}^2$

Order 420: $C_{420}$ x 9, $C_{105}:C_4$, $C_5:C_{84}$, $C_{105}:C_4$, $C_{15}:C_{28}$

Order 392: $C_7^2:Q_8$

Order 360: $C_{60}.C_6$

Order 315: $C_3\times C_{105}$ x 3

Order 294: $C_7\times C_{42}$ x 3

Order 280: $C_{35}:Q_8$, $C_{35}:Q_8$

Order 252: $C_3\times C_{84}$ x 3, $C_{21}:C_{12}$, $C_3:C_{84}$

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

Order 210: $C_{210}$ x 9

Order 196: $C_7\times C_{28}$, $C_7:C_{28}$

Order 180: $C_3\times C_{60}$, $C_{15}:C_{12}$

Order 168: $Q_8\times C_{21}$, $C_{21}:Q_8$, $C_{21}:Q_8$, $C_{21}:Q_8$

Order 147: $C_7\times C_{21}$ x 3

Order 140: $C_{140}$ x 3, $C_{35}:C_4$, $C_5:C_{28}$

Order 126: $C_3\times C_{42}$ x 3

Order 120: $C_{15}:Q_8$, $C_{15}:Q_8$

Order 105: $C_{105}$ x 9

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

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

Order 84: $C_{84}$ x 10, $C_{21}:C_4$, $C_7:C_{12}$, $C_3:C_{28}$

Order 72: $C_3^2:Q_8$

Order 70: $C_{70}$ x 3

Order 63: $C_3\times C_{21}$ x 3

Order 60: $C_{60}$ x 3, $C_{15}:C_4$, $C_5:C_{12}$

Order 56: $C_7:Q_8$, $C_7\times Q_8$

Order 49: $C_7^2$

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

Order 42: $C_{42}$ x 9

Order 40: $C_5:Q_8$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$

Order 35: $C_{35}$ x 3

Order 30: $C_{30}$ x 3

Order 28: $C_{28}$ x 4, $C_7:C_4$

Order 24: $C_3:Q_8$, $C_3\times Q_8$

Order 21: $C_{21}$ x 9

Order 20: $C_{20}$, $C_5:C_4$

Order 18: $C_3\times C_6$

Order 15: $C_{15}$ x 3

Order 14: $C_{14}$ x 3

Order 12: $C_{12}$ x 4, $C_3:C_4$

Order 10: $C_{10}$

Order 9: $C_3^2$

Order 8: $Q_8$

Order 7: $C_7$ x 3

Order 6: $C_6$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 17640: $C_{420}.C_{42}$ ($C_1$)

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

Order 5880: $C_{420}.C_{14}$ ($C_3$)

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

Order 2940: $C_{105}:C_{28}$ ($C_6$) x 2, $C_7\times C_{420}$ ($C_6$), $C_7\times C_{420}$ ($S_3$)

Order 2520: $C_{420}.C_6$ ($C_7$)

Order 2205: $C_{21}\times C_{105}$ ($Q_8$)

Order 1764: $C_{21}\times C_{84}$ ($D_5$)

Order 1470: $C_7\times C_{210}$ ($C_2\times C_6$), $C_7\times C_{210}$ ($D_6$)

Order 1260: $C_{105}:C_{12}$ ($C_{14}$) x 2, $C_3\times C_{420}$ ($C_{14}$), $C_3\times C_{420}$ ($D_7$)

Order 980: $C_7\times C_{140}$ ($C_3\times S_3$)

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

Order 840: $C_{105}:Q_8$ ($C_{21}$)

Order 735: $C_7\times C_{105}$ ($C_3:Q_8$), $C_7\times C_{105}$ ($C_3\times Q_8$)

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

Order 588: $C_7\times C_{84}$ ($D_{15}$), $C_7\times C_{84}$ ($C_3\times D_5$)

Order 490: $C_7\times C_{70}$ ($C_6\times S_3$)

Order 441: $C_{21}^2$ ($C_5:Q_8$)

Order 420: $C_{105}:C_4$ ($C_{42}$) x 2, $C_{420}$ ($C_{42}$), $C_{420}$ ($D_{21}$), $C_{420}$ ($C_3\times D_7$), $C_{420}$ ($S_3\times C_7$)

Order 315: $C_3\times C_{105}$ ($C_7:Q_8$), $C_3\times C_{105}$ ($C_7\times Q_8$)

Order 294: $C_7\times C_{42}$ ($D_{30}$), $C_7\times C_{42}$ ($C_3\times D_{10}$)

Order 252: $C_3\times C_{84}$ ($D_{35}$), $C_3\times C_{84}$ ($C_7\times D_5$)

Order 245: $C_7\times C_{35}$ ($C_3^2:Q_8$)

Order 210: $C_{210}$ ($C_2\times C_{42}$), $C_{210}$ ($D_{42}$), $C_{210}$ ($S_3\times C_{14}$), $C_{210}$ ($C_3\times D_{14}$)

Order 196: $C_7\times C_{28}$ ($C_3\times D_{15}$)

Order 180: $C_3\times C_{60}$ ($C_7\times D_7$)

Order 147: $C_7\times C_{21}$ ($C_{15}:Q_8$), $C_7\times C_{21}$ ($C_{15}:Q_8$)

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

Order 126: $C_3\times C_{42}$ ($C_7\times D_{10}$), $C_3\times C_{42}$ ($D_{70}$)

Order 105: $C_{105}$ ($Q_8\times C_{21}$), $C_{105}$ ($C_{21}:Q_8$), $C_{105}$ ($C_{21}:Q_8$), $C_{105}$ ($C_{21}:Q_8$)

Order 98: $C_7\times C_{14}$ ($C_3\times D_{30}$)

Order 90: $C_3\times C_{30}$ ($C_7\times D_{14}$)

Order 84: $C_{84}$ ($C_3\times D_{35}$), $C_{84}$ ($D_5\times C_{21}$), $C_{84}$ ($D_{105}$), $C_{84}$ ($C_7\times D_{15}$)

Order 70: $C_{70}$ ($C_3\times D_{42}$), $C_{70}$ ($S_3\times C_{42}$)

Order 63: $C_3\times C_{21}$ ($C_{35}:Q_8$), $C_3\times C_{21}$ ($C_{35}:Q_8$)

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

Order 49: $C_7^2$ ($C_{60}.C_6$)

Order 45: $C_3\times C_{15}$ ($C_7^2:Q_8$)

Order 42: $C_{42}$ ($D_{210}$), $C_{42}$ ($C_7\times D_{30}$), $C_{42}$ ($C_3\times D_{70}$), $C_{42}$ ($D_5\times C_{42}$)

Order 36: $C_3\times C_{12}$ ($C_7\times D_{35}$)

Order 35: $C_{35}$ ($C_{84}.C_6$), $C_{35}$ ($C_{12}.C_{42}$)

Order 30: $C_{30}$ ($C_7\times D_{42}$), $C_{30}$ ($C_{21}\times D_{14}$)

Order 28: $C_{28}$ ($C_3\times D_{105}$), $C_{28}$ ($C_{21}\times D_{15}$)

Order 21: $C_{21}$ ($C_{105}:Q_8$), $C_{21}$ ($C_{105}:Q_8$), $C_{21}$ ($C_{105}:Q_8$), $C_{21}$ ($C_{105}:Q_8$)

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

Order 18: $C_3\times C_6$ ($C_7\times D_{70}$)

Order 15: $C_{15}$ ($C_{84}.C_{14}$), $C_{15}$ ($C_{28}.C_{42}$)

Order 14: $C_{14}$ ($C_3\times D_{210}$), $C_{14}$ ($C_{21}\times D_{30}$)

Order 12: $C_{12}$ ($C_7\times D_{105}$), $C_{12}$ ($C_{21}\times D_{35}$)

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

Order 9: $C_3^2$ ($C_{140}.C_{14}$)

Order 7: $C_7$ ($C_{420}.C_6$), $C_7$ ($C_{420}.C_6$)

Order 6: $C_6$ ($C_{21}\times D_{70}$), $C_6$ ($C_7\times D_{210}$)

Order 5: $C_5$ ($C_{21}^2:Q_8$)

Order 4: $C_4$ ($C_{21}\times D_{105}$)

Order 3: $C_3$ ($C_{140}.C_{42}$), $C_3$ ($C_{420}.C_{14}$)

Order 2: $C_2$ ($C_{21}\times D_{210}$)

Order 1: $C_1$ ($C_{420}.C_{42}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 17640: $C_{420}.C_{42}$ ($C_1$)

Order 8820: $C_{21}\times C_{420}$ ($C_2$), $C_{105}:C_{84}$ ($C_2$)

Order 5880: $C_{420}.C_{14}$ ($C_3$)

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

Order 2940: $C_7\times C_{420}$ ($C_6$), $C_7\times C_{420}$ ($S_3$), $C_{105}:C_{28}$ ($C_6$)

Order 2520: $C_{420}.C_6$ ($C_7$)

Order 2205: $C_{21}\times C_{105}$ ($Q_8$)

Order 1764: $C_{21}\times C_{84}$ ($D_5$)

Order 1470: $C_7\times C_{210}$ ($C_2\times C_6$), $C_7\times C_{210}$ ($D_6$)

Order 1260: $C_3\times C_{420}$ ($C_{14}$), $C_3\times C_{420}$ ($D_7$), $C_{105}:C_{12}$ ($C_{14}$)

Order 980: $C_7\times C_{140}$ ($C_3\times S_3$)

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

Order 840: $C_{105}:Q_8$ ($C_{21}$)

Order 735: $C_7\times C_{105}$ ($C_3:Q_8$), $C_7\times C_{105}$ ($C_3\times Q_8$)

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

Order 588: $C_7\times C_{84}$ ($D_{15}$), $C_7\times C_{84}$ ($C_3\times D_5$)

Order 490: $C_7\times C_{70}$ ($C_6\times S_3$)

Order 441: $C_{21}^2$ ($C_5:Q_8$)

Order 420: $C_{420}$ ($C_{42}$), $C_{420}$ ($D_{21}$), $C_{420}$ ($C_3\times D_7$), $C_{420}$ ($S_3\times C_7$), $C_{105}:C_4$ ($C_{42}$)

Order 315: $C_3\times C_{105}$ ($C_7:Q_8$), $C_3\times C_{105}$ ($C_7\times Q_8$)

Order 294: $C_7\times C_{42}$ ($D_{30}$), $C_7\times C_{42}$ ($C_3\times D_{10}$)

Order 252: $C_3\times C_{84}$ ($D_{35}$), $C_3\times C_{84}$ ($C_7\times D_5$)

Order 245: $C_7\times C_{35}$ ($C_3^2:Q_8$)

Order 210: $C_{210}$ ($C_2\times C_{42}$), $C_{210}$ ($D_{42}$), $C_{210}$ ($S_3\times C_{14}$), $C_{210}$ ($C_3\times D_{14}$)

Order 196: $C_7\times C_{28}$ ($C_3\times D_{15}$)

Order 180: $C_3\times C_{60}$ ($C_7\times D_7$)

Order 147: $C_7\times C_{21}$ ($C_{15}:Q_8$), $C_7\times C_{21}$ ($C_{15}:Q_8$)

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

Order 126: $C_3\times C_{42}$ ($C_7\times D_{10}$), $C_3\times C_{42}$ ($D_{70}$)

Order 105: $C_{105}$ ($Q_8\times C_{21}$), $C_{105}$ ($C_{21}:Q_8$), $C_{105}$ ($C_{21}:Q_8$), $C_{105}$ ($C_{21}:Q_8$)

Order 98: $C_7\times C_{14}$ ($C_3\times D_{30}$)

Order 90: $C_3\times C_{30}$ ($C_7\times D_{14}$)

Order 84: $C_{84}$ ($C_3\times D_{35}$), $C_{84}$ ($D_5\times C_{21}$), $C_{84}$ ($D_{105}$), $C_{84}$ ($C_7\times D_{15}$)

Order 70: $C_{70}$ ($C_3\times D_{42}$), $C_{70}$ ($S_3\times C_{42}$)

Order 63: $C_3\times C_{21}$ ($C_{35}:Q_8$), $C_3\times C_{21}$ ($C_{35}:Q_8$)

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

Order 49: $C_7^2$ ($C_{60}.C_6$)

Order 45: $C_3\times C_{15}$ ($C_7^2:Q_8$)

Order 42: $C_{42}$ ($D_{210}$), $C_{42}$ ($C_7\times D_{30}$), $C_{42}$ ($C_3\times D_{70}$), $C_{42}$ ($D_5\times C_{42}$)

Order 36: $C_3\times C_{12}$ ($C_7\times D_{35}$)

Order 35: $C_{35}$ ($C_{84}.C_6$), $C_{35}$ ($C_{12}.C_{42}$)

Order 30: $C_{30}$ ($C_7\times D_{42}$), $C_{30}$ ($C_{21}\times D_{14}$)

Order 28: $C_{28}$ ($C_3\times D_{105}$), $C_{28}$ ($C_{21}\times D_{15}$)

Order 21: $C_{21}$ ($C_{105}:Q_8$), $C_{21}$ ($C_{105}:Q_8$), $C_{21}$ ($C_{105}:Q_8$), $C_{21}$ ($C_{105}:Q_8$)

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

Order 18: $C_3\times C_6$ ($C_7\times D_{70}$)

Order 15: $C_{15}$ ($C_{84}.C_{14}$), $C_{15}$ ($C_{28}.C_{42}$)

Order 14: $C_{14}$ ($C_3\times D_{210}$), $C_{14}$ ($C_{21}\times D_{30}$)

Order 12: $C_{12}$ ($C_7\times D_{105}$), $C_{12}$ ($C_{21}\times D_{35}$)

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

Order 9: $C_3^2$ ($C_{140}.C_{14}$)

Order 7: $C_7$ ($C_{420}.C_6$), $C_7$ ($C_{420}.C_6$)

Order 6: $C_6$ ($C_{21}\times D_{70}$), $C_6$ ($C_7\times D_{210}$)

Order 5: $C_5$ ($C_{21}^2:Q_8$)

Order 4: $C_4$ ($C_{21}\times D_{105}$)

Order 3: $C_3$ ($C_{140}.C_{42}$), $C_3$ ($C_{420}.C_{14}$)

Order 2: $C_2$ ($C_{21}\times D_{210}$)

Order 1: $C_1$ ($C_{420}.C_{42}$)

Series

Derived series

$C_{420}.C_{42}$

$\rhd$

$C_{210}$

$\rhd$

$C_1$

Chief series

$C_{420}.C_{42}$

$\rhd$

$C_{21}\times C_{420}$

$\rhd$

$C_7\times C_{420}$

$\rhd$

$C_{420}$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{420}.C_{42}$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

Upper central series

$C_1$

$\lhd$

$C_{42}$

$\lhd$

$C_{84}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $4473 \times 4473$ character table is not available for this group.

Rational character table

The $170 \times 170$ rational character table is not available for this group.