Abstract group 88200.b: $C_{420}.C_{210}$

• Label: 88200.b
• Order: \( 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \)
• 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
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3^{2} \)
• Perm deg.: $34$
• Trans deg.: $840$
• Rank: $2$

Group information

Description:	$C_{420}.C_{210}$
Order:	\(88200\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(967680\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_5$ x 2, $C_7$ x 2
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

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	211	8	212	24	428	48	864	436	1308	192	888	384	1356	1872	1152	2904	2064	6192	3288	9216	7344	19296	28512	88200
Conjugacy classes	1	3	5	4	14	9	27	22	14	39	100	36	198	66	116	588	222	216	636	420	4632	1224	4728	9360	22680
Divisions	1	3	3	2	4	5	5	6	4	7	14	5	18	6	16	26	20	15	28	19	100	27	102	101	537

Minimal presentations

Permutation degree:	$34$
Transitive degree:	$840$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Presentation:	$\langle a, b \mid a^{210}=b^{420}=1, b^{a}=b^{209} \rangle$
Permutation group:	Degree $34$ $\langle(20,21)(22,23)(26,27)(29,30)(31,32)(33,34), (16,17)(18,19)(20,22,24,23,21) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 4 & 0 \\ 0 & 316 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 2 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{421})$
Direct product:	$C_4$ $\, \times\, $ $C_3$ $\, \times\, $ $C_5$ $\, \times\, $ $C_7$ $\, \times\, $ $D_{105}$
Semidirect product:	$C_{35}^2$ $\,\rtimes\,$ $(S_3\times C_{12})$	$C_{21}^2$ $\,\rtimes\,$ $(D_5\times C_{20})$	$C_{15}^2$ $\,\rtimes\,$ $(D_7\times C_{28})$	$C_7^2$ $\,\rtimes\,$ $(D_{15}\times C_{60})$	all 7
Trans. wreath product:	not computed
Possibly split product:	$D_{210}$ . $C_{210}$	$C_{420}$ . $D_{105}$	$D_{105}$ . $C_{420}$ (2)	$C_{210}$ . $D_{210}$	all 215

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

Homology

Abelianization:	$C_{2} \times C_{420} \simeq C_{2} \times C_{4} \times C_{3} \times C_{5} \times C_{7}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 9120 subgroups in 1136 conjugacy classes, 232 normal (216 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 $C_{105}$
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 2520 (order at least 35) are shown, as well as all normal subgroups of any index.

Order 88200: $C_{420}.C_{210}$

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

Order 29400: $C_{140}.C_{210}$, $C_{420}.C_{70}$

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

Order 17640: $D_{21}\times C_{420}$, $C_{84}\times D_{105}$

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

Order 12600: $D_{15}\times C_{420}$, $C_{60}\times D_{105}$

Order 11025: $C_{105}^2$

Order 9800: $D_{35}\times C_{140}$

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

Order 7350: $C_{35}\times C_{210}$ x 3, $D_{35}\times C_{105}$ x 2, $C_{35}\times D_{105}$ x 2

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

Order 5880: $D_{35}\times C_{84}$, $D_7\times C_{420}$, $D_{21}\times C_{140}$, $C_{28}\times D_{105}$

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

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

Order 4200: $D_{15}\times C_{140}$, $D_5\times C_{420}$, $C_{20}\times D_{105}$, $C_{60}\times D_{35}$

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

Order 3528: $D_{21}\times C_{84}$

Order 3150: $C_{15}\times C_{210}$ x 3, $D_{15}\times C_{105}$ x 2, $C_{15}\times D_{105}$ x 2, $C_{15}\times C_{210}$ x 2

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

Order 2520: $S_3\times C_{420}$, $D_{21}\times C_{60}$, $C_{12}\times D_{105}$, $D_{15}\times C_{84}$

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

Order 2205: $C_{21}\times C_{105}$ x 2, $C_{21}\times C_{105}$ x 2

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

Order 1960: $D_7\times C_{140}$, $C_{28}\times D_{35}$

Order 1800: $D_{15}\times C_{60}$

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

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

Order 1470: $C_7\times C_{210}$ x 14, $C_7\times D_{105}$ 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: $C_{20}\times D_{35}$, $D_5\times C_{140}$

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

Order 1225: $C_{35}^2$

Order 1176: $C_{28}\times D_{21}$, $D_7\times C_{84}$

Order 1050: $C_5\times C_{210}$ x 18, $C_5\times D_{105}$ x 2, $D_{15}\times C_{35}$ x 2, $C_{15}\times D_{35}$ x 2, $D_5\times C_{105}$ x 2

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

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

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

Order 840: $D_7\times C_{60}$, $C_2\times C_{420}$, $C_4\times D_{105}$, $C_{28}\times D_{15}$, $C_{20}\times D_{21}$, $S_3\times C_{140}$, $C_{12}\times D_{35}$, $D_5\times C_{84}$

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

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

Order 630: $C_3\times C_{210}$ x 26, $C_3\times D_{105}$ x 2, $C_{21}\times D_{15}$ x 2, $C_{15}\times D_{21}$ x 2, $S_3\times C_{105}$ x 2

Order 600: $C_{20}\times D_{15}$, $D_5\times C_{60}$

Order 588: $C_7\times C_{84}$ x 3, $C_7\times D_{42}$, $C_{21}\times D_{14}$, $C_{21}:C_{28}$, $C_7:C_{84}$

Order 525: $C_5\times C_{105}$ x 18

Order 504: $C_{12}\times D_{21}$, $S_3\times C_{84}$

Order 490: $C_7\times C_{70}$ x 4, $C_7\times D_{35}$ x 2, $D_7\times C_{35}$ x 2

Order 450: $C_{15}\times D_{15}$ x 2, $C_{15}\times C_{30}$

Order 441: $C_{21}^2$

Order 420: $C_{420}$ x 101, $C_{105}:C_4$, $C_3:C_{140}$, $C_{105}:C_4$, $C_5:C_{84}$, $C_7:C_{60}$, $C_2\times C_{210}$, $D_{210}$, $C_7\times D_{30}$, $C_5\times D_{42}$, $S_3\times C_{70}$, $C_3\times D_{70}$, $D_5\times C_{42}$, $C_{15}\times D_{14}$, $C_{105}:C_4$, $C_{15}:C_{28}$

Order 392: $D_7\times C_{28}$

Order 360: $S_3\times C_{60}$, $C_{12}\times D_{15}$

Order 350: $C_5\times C_{70}$ x 5, $C_5\times D_{35}$ x 2, $D_5\times C_{35}$ x 2

Order 315: $C_3\times C_{105}$ x 26

Order 300: $C_5\times C_{60}$ x 3, $C_5\times D_{30}$, $C_{15}\times D_{10}$, $C_{15}:C_{20}$, $C_5:C_{60}$

Order 294: $C_7\times C_{42}$ x 3, $C_7\times D_{21}$ x 2, $D_7\times C_{21}$ x 2

Order 280: $C_2\times C_{140}$, $C_4\times D_{35}$, $D_5\times C_{28}$, $D_7\times C_{20}$

Order 252: $C_3\times C_{84}$ x 5, $C_3\times D_{42}$, $S_3\times C_{42}$, $C_{21}:C_{12}$, $C_3:C_{84}$

Order 245: $C_7\times C_{35}$ x 4

Order 225: $C_{15}^2$

Order 210: $C_{210}$ x 102, $C_5\times D_{21}$ x 2, $S_3\times C_{35}$ x 2, $C_3\times D_{35}$ x 2, $D_5\times C_{21}$ x 2, $D_7\times C_{15}$ x 2, $D_{105}$ x 2, $C_7\times D_{15}$ x 2

Order 200: $D_5\times C_{20}$

Order 196: $C_7\times C_{28}$, $C_7:C_{28}$, $C_7\times D_{14}$

Order 180: $C_3\times C_{60}$ x 4, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_{15}:C_{12}$, $C_3:C_{60}$

Order 175: $C_5\times C_{35}$ x 5

Order 168: $C_2\times C_{84}$, $C_4\times D_{21}$, $S_3\times C_{28}$, $C_{12}\times D_7$

Order 150: $C_5\times C_{30}$ x 3, $D_5\times C_{15}$ x 2, $C_5\times D_{15}$ x 2

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

Order 140: $C_{140}$ x 27, $C_7\times D_{10}$, $C_5\times D_{14}$, $C_{35}:C_4$, $C_7:C_{20}$, $C_2\times C_{70}$, $D_{70}$, $C_5:C_{28}$

Order 126: $C_3\times C_{42}$ x 5, $C_3\times D_{21}$ x 2, $S_3\times C_{21}$ x 2

Order 120: $C_2\times C_{60}$, $C_4\times D_{15}$, $S_3\times C_{20}$, $D_5\times C_{12}$

Order 105: $C_{105}$ x 100

Order 100: $C_5\times C_{20}$, $C_5:C_{20}$, $C_5\times D_{10}$

Order 98: $C_7\times D_7$ x 2, $C_7\times C_{14}$

Order 90: $C_3\times C_{30}$ x 4, $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2

Order 84: $C_{84}$ x 19, $C_{21}:C_4$, $C_7:C_{12}$, $C_3:C_{28}$, $C_2\times C_{42}$, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 75: $C_5\times C_{15}$ x 3

Order 72: $S_3\times C_{12}$

Order 70: $C_{70}$ x 28, $D_{35}$ x 2, $C_5\times D_7$ x 2, $C_7\times D_5$ x 2

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

Order 60: $C_{60}$ x 15, $C_{15}:C_4$, $C_5:C_{12}$, $C_2\times C_{30}$, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$, $C_3:C_{20}$

Order 56: $C_2\times C_{28}$, $C_4\times D_7$

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

Order 49: $C_7^2$

Order 45: $C_3\times C_{15}$ x 4

Order 42: $C_{42}$ x 20, $D_{21}$ x 2, $C_3\times D_7$ x 2, $S_3\times C_7$ x 2

Order 40: $C_2\times C_{20}$, $C_4\times D_5$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$

Order 35: $C_{35}$ x 26

Order 30: $C_{30}$ x 4

Order 28: $C_{28}$ x 2

Order 25: $C_5^2$

Order 21: $C_{21}$ x 4

Order 20: $C_{20}$ 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 2

Order 10: $C_{10}$ x 2

Order 9: $C_3^2$

Order 7: $C_7$ x 2

Order 6: $C_6$ x 2

Order 5: $C_5$ x 2

Order 4: $C_4$

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 88200: $C_{420}.C_{210}$ ($C_1$)

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

Order 29400: $C_{420}.C_{70}$ ($C_3$)

Order 22050: $C_{105}\times D_{105}$ ($C_4$) x 2, $C_{105}\times C_{210}$ ($C_2^2$)

Order 17640: $C_{84}\times D_{105}$ ($C_5$)

Order 14700: $C_{35}\times C_{420}$ ($C_6$), $C_{35}\times C_{420}$ ($S_3$), $C_{105}:C_{140}$ ($C_6$), $C_{35}\times D_{210}$ ($C_6$)

Order 12600: $C_{60}\times D_{105}$ ($C_7$)

Order 11025: $C_{105}^2$ ($C_2\times C_4$)

Order 8820: $C_{21}\times C_{420}$ ($C_{10}$), $C_{21}\times C_{420}$ ($D_5$), $C_{105}:C_{84}$ ($C_{10}$), $C_{21}\times D_{210}$ ($C_{10}$)

Order 7350: $C_{35}\times D_{105}$ ($C_{12}$) x 2, $C_{35}\times C_{210}$ ($C_2\times C_6$), $C_{35}\times C_{210}$ ($D_6$)

Order 6300: $C_{15}\times C_{420}$ ($C_{14}$), $C_{15}\times C_{420}$ ($D_7$), $C_{105}:C_{60}$ ($C_{14}$), $C_{15}\times D_{210}$ ($C_{14}$)

Order 5880: $C_{28}\times D_{105}$ ($C_{15}$)

Order 4900: $C_{35}\times C_{140}$ ($C_3\times S_3$)

Order 4410: $C_{21}\times D_{105}$ ($C_{20}$) x 2, $C_{21}\times C_{210}$ ($C_2\times C_{10}$), $C_{21}\times C_{210}$ ($D_{10}$)

Order 4200: $C_{20}\times D_{105}$ ($C_{21}$)

Order 3675: $C_{35}\times C_{105}$ ($C_2\times C_{12}$), $C_{35}\times C_{105}$ ($C_4\times S_3$)

Order 3150: $C_{15}\times D_{105}$ ($C_{28}$) x 2, $C_{15}\times C_{210}$ ($C_2\times C_{14}$), $C_{15}\times C_{210}$ ($D_{14}$)

Order 2940: $C_7\times C_{420}$ ($C_{30}$), $C_7\times C_{420}$ ($D_{15}$), $C_7\times C_{420}$ ($C_3\times D_5$), $C_7\times C_{420}$ ($C_5\times S_3$), $C_{105}:C_{28}$ ($C_{30}$), $C_7\times D_{210}$ ($C_{30}$)

Order 2520: $C_{12}\times D_{105}$ ($C_{35}$)

Order 2450: $C_{35}\times C_{70}$ ($C_6\times S_3$)

Order 2205: $C_{21}\times C_{105}$ ($C_2\times C_{20}$), $C_{21}\times C_{105}$ ($C_4\times D_5$)

Order 2100: $C_5\times C_{420}$ ($C_{42}$), $C_5\times C_{420}$ ($D_{21}$), $C_5\times C_{420}$ ($C_3\times D_7$), $C_5\times C_{420}$ ($S_3\times C_7$), $C_{105}:C_{20}$ ($C_{42}$), $C_5\times D_{210}$ ($C_{42}$)

Order 1764: $C_{21}\times C_{84}$ ($C_5\times D_5$)

Order 1575: $C_{15}\times C_{105}$ ($C_2\times C_{28}$), $C_{15}\times C_{105}$ ($C_4\times D_7$)

Order 1470: $C_7\times D_{105}$ ($C_{60}$) x 2, $C_7\times C_{210}$ ($C_2\times C_{30}$), $C_7\times C_{210}$ ($D_{30}$), $C_7\times C_{210}$ ($S_3\times C_{10}$), $C_7\times C_{210}$ ($C_3\times D_{10}$)

Order 1260: $C_3\times C_{420}$ ($C_{70}$), $C_3\times C_{420}$ ($D_{35}$), $C_3\times C_{420}$ ($C_5\times D_7$), $C_3\times C_{420}$ ($C_7\times D_5$), $C_{105}:C_{12}$ ($C_{70}$), $C_3\times D_{210}$ ($C_{70}$)

Order 1225: $C_{35}^2$ ($S_3\times C_{12}$)

Order 1050: $C_5\times D_{105}$ ($C_{84}$) x 2, $C_5\times C_{210}$ ($C_2\times C_{42}$), $C_5\times C_{210}$ ($D_{42}$), $C_5\times C_{210}$ ($S_3\times C_{14}$), $C_5\times C_{210}$ ($C_3\times D_{14}$)

Order 980: $C_7\times C_{140}$ ($C_3\times D_{15}$), $C_7\times C_{140}$ ($S_3\times C_{15}$)

Order 900: $C_{15}\times C_{60}$ ($C_7\times D_7$)

Order 882: $C_{21}\times C_{42}$ ($C_5\times D_{10}$)

Order 840: $C_4\times D_{105}$ ($C_{105}$)

Order 735: $C_7\times C_{105}$ ($C_2\times C_{60}$), $C_7\times C_{105}$ ($C_4\times D_{15}$), $C_7\times C_{105}$ ($S_3\times C_{20}$), $C_7\times C_{105}$ ($D_5\times C_{12}$)

Order 700: $C_5\times C_{140}$ ($C_3\times D_{21}$), $C_5\times C_{140}$ ($S_3\times C_{21}$)

Order 630: $C_3\times D_{105}$ ($C_{140}$) x 2, $C_3\times C_{210}$ ($C_7\times D_{10}$), $C_3\times C_{210}$ ($C_5\times D_{14}$), $C_3\times C_{210}$ ($C_2\times C_{70}$), $C_3\times C_{210}$ ($D_{70}$)

Order 588: $C_7\times C_{84}$ ($D_5\times C_{15}$), $C_7\times C_{84}$ ($C_5\times D_{15}$)

Order 525: $C_5\times C_{105}$ ($C_2\times C_{84}$), $C_5\times C_{105}$ ($C_4\times D_{21}$), $C_5\times C_{105}$ ($S_3\times C_{28}$), $C_5\times C_{105}$ ($C_{12}\times D_7$)

Order 490: $C_7\times C_{70}$ ($C_3\times D_{30}$), $C_7\times C_{70}$ ($S_3\times C_{30}$)

Order 450: $C_{15}\times C_{30}$ ($C_7\times D_{14}$)

Order 441: $C_{21}^2$ ($D_5\times C_{20}$)

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

Order 350: $C_5\times C_{70}$ ($C_3\times D_{42}$), $C_5\times C_{70}$ ($S_3\times C_{42}$)

Order 315: $C_3\times C_{105}$ ($C_2\times C_{140}$), $C_3\times C_{105}$ ($C_4\times D_{35}$), $C_3\times C_{105}$ ($D_5\times C_{28}$), $C_3\times C_{105}$ ($D_7\times C_{20}$)

Order 300: $C_5\times C_{60}$ ($C_7\times D_{21}$), $C_5\times C_{60}$ ($D_7\times C_{21}$)

Order 294: $C_7\times C_{42}$ ($C_5\times D_{30}$), $C_7\times C_{42}$ ($C_{15}\times D_{10}$)

Order 252: $C_3\times C_{84}$ ($C_5\times D_{35}$), $C_3\times C_{84}$ ($D_5\times C_{35}$)

Order 245: $C_7\times C_{35}$ ($S_3\times C_{60}$), $C_7\times C_{35}$ ($C_{12}\times D_{15}$)

Order 225: $C_{15}^2$ ($D_7\times C_{28}$)

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

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

Order 180: $C_3\times C_{60}$ ($C_7\times D_{35}$), $C_3\times C_{60}$ ($D_7\times C_{35}$)

Order 175: $C_5\times C_{35}$ ($C_{12}\times D_{21}$), $C_5\times C_{35}$ ($S_3\times C_{84}$)

Order 150: $C_5\times C_{30}$ ($C_7\times D_{42}$), $C_5\times C_{30}$ ($C_{21}\times D_{14}$)

Order 147: $C_7\times C_{21}$ ($C_{20}\times D_{15}$), $C_7\times C_{21}$ ($D_5\times C_{60}$)

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

Order 126: $C_3\times C_{42}$ ($C_5\times D_{70}$), $C_3\times C_{42}$ ($D_5\times C_{70}$)

Order 105: $C_{105}$ ($D_7\times C_{60}$), $C_{105}$ ($C_2\times C_{420}$), $C_{105}$ ($C_4\times D_{105}$), $C_{105}$ ($C_{28}\times D_{15}$), $C_{105}$ ($C_{20}\times D_{21}$), $C_{105}$ ($S_3\times C_{140}$), $C_{105}$ ($C_{12}\times D_{35}$), $C_{105}$ ($D_5\times C_{84}$)

Order 100: $C_5\times C_{20}$ ($C_{21}\times D_{21}$)

Order 98: $C_7\times C_{14}$ ($C_{15}\times D_{30}$)

Order 90: $C_3\times C_{30}$ ($C_7\times D_{70}$), $C_3\times C_{30}$ ($D_7\times C_{70}$)

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

Order 75: $C_5\times C_{15}$ ($C_{28}\times D_{21}$), $C_5\times C_{15}$ ($D_7\times C_{84}$)

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

Order 63: $C_3\times C_{21}$ ($C_{20}\times D_{35}$), $C_3\times C_{21}$ ($D_5\times C_{140}$)

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

Order 50: $C_5\times C_{10}$ ($C_{21}\times D_{42}$)

Order 49: $C_7^2$ ($D_{15}\times C_{60}$)

Order 45: $C_3\times C_{15}$ ($D_7\times C_{140}$), $C_3\times C_{15}$ ($C_{28}\times D_{35}$)

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

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

Order 35: $C_{35}$ ($S_3\times C_{420}$), $C_{35}$ ($D_{21}\times C_{60}$), $C_{35}$ ($C_{12}\times D_{105}$), $C_{35}$ ($D_{15}\times C_{84}$)

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

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

Order 25: $C_5^2$ ($D_{21}\times C_{84}$)

Order 21: $C_{21}$ ($D_{15}\times C_{140}$), $C_{21}$ ($D_5\times C_{420}$), $C_{21}$ ($C_{20}\times D_{105}$), $C_{21}$ ($C_{60}\times D_{35}$)

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

Order 18: $C_3\times C_6$ ($C_{35}\times D_{70}$)

Order 15: $C_{15}$ ($D_{35}\times C_{84}$), $C_{15}$ ($D_7\times C_{420}$), $C_{15}$ ($D_{21}\times C_{140}$), $C_{15}$ ($C_{28}\times D_{105}$)

Order 14: $C_{14}$ ($D_{15}\times C_{210}$), $C_{14}$ ($C_{15}\times D_{210}$)

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

Order 10: $C_{10}$ ($D_{21}\times C_{210}$), $C_{10}$ ($C_{21}\times D_{210}$)

Order 9: $C_3^2$ ($D_{35}\times C_{140}$)

Order 7: $C_7$ ($D_{15}\times C_{420}$), $C_7$ ($C_{60}\times D_{105}$)

Order 6: $C_6$ ($C_{105}\times D_{70}$), $C_6$ ($C_{35}\times D_{210}$)

Order 5: $C_5$ ($D_{21}\times C_{420}$), $C_5$ ($C_{84}\times D_{105}$)

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

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

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

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

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 $22680 \times 22680$ character table is not available for this group.

Rational character table

The $537 \times 537$ rational character table is not available for this group.