Abstract group 1344.9365: $C_{21}:D_4^2$

• Label: 1344.9365
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $336$
• Rank: $4$

Group information

Description:	$C_{21}:D_4^2$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^8\times S_3\times F_7$, of order \(64512\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42	84
Elements	1	275	2	236	190	6	64	138	12	240	132	48	1344
Conjugacy classes	1	15	1	9	15	3	4	21	3	18	21	6	117
Divisions	1	15	1	9	11	1	3	7	1	5	6	1	61
Autjugacy classes	1	11	1	7	8	1	2	6	1	4	5	1	48

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	16	48	47	0	6	0	0	117
Irr. rational chars.	16	16	7	8	1	8	5	61

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$336$
Rank:	$4$
Inequivalent generating quadruples:	$14938560$

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

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{28}=d^{2}=e^{6}=[a,b]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(2,3)(4,5)(6,7)(15,16)(17,18), (9,10)(15,17,18,16), (12,14)(15,18)(16,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 27 & 14 \\ 70 & 69 \end{array}\right), \left(\begin{array}{rr} 22 & 63 \\ 63 & 1 \end{array}\right), \left(\begin{array}{rr} 69 & 14 \\ 70 & 27 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 41 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 43 & 42 \\ 21 & 1 \end{array}\right), \left(\begin{array}{rr} 36 & 35 \\ 77 & 36 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/84\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{21}$ $\,\rtimes\,$ $D_4^2$	$(D_4\times D_{14})$ $\,\rtimes\,$ $S_3$	$D_{14}$ $\,\rtimes\,$ $(S_3\times D_4)$ (2)	$(D_{42}:D_4)$ $\,\rtimes\,$ $C_2$	all 70
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{14}:D_4)$ . $D_6$	$C_{14}$ . $(D_4\times D_6)$	$C_{14}$ . $(D_4:D_6)$	$C_6$ . $(D_4\times D_{14})$ (2)	all 35

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{84}\Z)$.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 8564 subgroups in 856 conjugacy classes, 158 normal (120 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_6\times D_{14}$
Commutator:	$G' \simeq$ $C_2\times C_{42}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $D_6\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{21}:D_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1344: $C_{21}:D_4^2$

Order 672: $(C_2\times C_{42}):D_4$, $C_{14}.(S_3\times D_4)$, $(C_2\times C_6):D_{28}$, $(C_2\times C_{42}):D_4$, $C_{21}:(C_4\times D_4)$, $D_{42}:D_4$, $D_6:D_{28}$, $D_{14}:D_{12}$, $C_{21}:(C_4\times D_4)$, $(C_6\times D_{28}):C_2$, $C_{84}:C_2^3$, $D_{42}:D_4$, $D_{42}:C_2^3$, $D_{42}:C_2^3$, $D_{42}:D_4$

Order 448: $D_{28}:D_4$

Order 336: $C_6:D_{28}$ x 9, $C_{12}:D_{14}$ x 4, $D_6:D_{14}$ x 4, $D_6:D_{14}$ x 4, $C_6.D_{28}$ x 3, $C_{42}:C_2^3$ x 2, $C_{42}:D_4$ x 2, $C_{42}:D_4$ x 2, $C_{42}:D_4$ x 2, $C_{42}:D_4$ x 2, $D_{42}:C_4$ x 2, $C_{42}.D_4$, $D_6:C_{28}$, $C_{42}.D_4$, $C_{14}.D_{12}$, $C_{21}:C_4^2$, $C_2^2\times D_{42}$, $D_6\times D_{14}$, $D_4\times C_{42}$, $C_{42}.C_2^3$, $C_6\times D_{28}$, $C_{12}\times D_{14}$, $C_{14}:D_{12}$, $D_{42}:C_4$, $D_{14}.D_6$, $C_{42}.D_4$

Order 224: $D_4\times D_{14}$ x 3, $D_{14}:D_4$ x 2, $C_2^2:D_{28}$ x 2, $D_{14}:D_4$ x 2, $C_4:D_{28}$, $D_{28}:C_4$, $C_2^2\times D_{28}$, $C_{28}:D_4$, $C_{28}:D_4$, $C_{28}:D_4$

Order 192: $C_2^4:D_6$

Order 168: $C_6\times D_{14}$ x 13, $C_3:D_{28}$ x 12, $C_2\times D_{42}$ x 7, $C_{21}:D_4$ x 6, $S_3\times D_{14}$ x 5, $C_6:C_{28}$ x 5, $D_6:C_{14}$ x 4, $C_{21}:D_4$ x 3, $C_{21}:D_4$ x 3, $C_2^2\times C_{42}$ x 2, $D_4\times C_{21}$ x 2, $C_{42}:C_4$ x 2, $C_3\times D_{28}$ x 2, $C_{12}\times D_7$ x 2, $C_7:D_{12}$ x 2, $D_{21}:C_4$ x 2, $C_6.D_{14}$ x 2, $D_6\times C_{14}$, $C_2\times C_{84}$, $C_{14}:C_{12}$

Order 112: $D_4\times D_7$ x 12, $C_2\times D_{28}$ x 10, $C_{14}:D_4$ x 7, $D_{14}:C_4$ x 5, $C_2^2\times D_{14}$ x 4, $D_4\times C_{14}$ x 3, $C_4\times D_{14}$ x 3, $C_2^2:C_{28}$ x 2, $C_2^2\times C_{28}$, $C_4:C_{28}$, $C_{14}.D_4$, $C_{14}.D_4$, $C_{28}:C_4$

Order 96: $C_2^3:D_6$ x 2, $C_2^4:S_3$ x 2, $C_2^3.D_6$ x 2, $D_6:D_4$ x 2, $C_{12}:C_2^3$, $D_4\times D_6$, $C_{12}:D_4$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_{12}:D_4$

Order 84: $C_3\times D_{14}$ x 19, $D_{42}$ x 11, $C_2\times C_{42}$ x 8, $S_3\times D_7$ x 4, $C_3:C_{28}$ x 4, $S_3\times C_{14}$ x 3, $C_{21}:C_4$ x 2, $C_7:C_{12}$ x 2, $C_{84}$

Order 64: $D_4^2$

Order 56: $C_2\times D_{14}$ x 25, $C_7:D_4$ x 14, $D_{28}$ x 14, $C_7\times D_4$ x 6, $C_2\times C_{28}$ x 6, $C_4\times D_7$ x 6, $C_{14}:C_4$ x 3, $C_2^2\times C_{14}$ x 3

Order 48: $C_6:D_4$ x 19, $C_6\times D_4$ x 8, $C_6.D_4$ x 5, $S_3\times D_4$ x 4, $D_6:C_4$ x 3, $C_2^3\times C_6$ x 2, $C_2^2\times D_6$ x 2, $C_6.C_2^3$ x 2, $C_2^2\times C_{12}$, $C_2\times D_{12}$, $C_4\times D_6$, $C_{12}:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 42: $C_{42}$ x 6, $C_3\times D_7$ x 5, $D_{21}$ x 3, $S_3\times C_7$

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

Order 28: $D_{14}$ x 34, $C_2\times C_{14}$ x 11, $C_{28}$ x 5, $C_7:C_4$ x 4

Order 24: $C_3:D_4$ x 22, $C_2^2\times C_6$ x 15, $C_2\times D_6$ x 13, $C_3\times D_4$ x 10, $C_6:C_4$ x 9, $C_2\times C_{12}$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2

Order 21: $C_{21}$

Order 16: $C_2\times D_4$ x 32, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 14: $D_7$ x 8, $C_{14}$ x 7

Order 12: $C_2\times C_6$ x 27, $D_6$ x 18, $C_3:C_4$ x 6, $C_{12}$ x 3

Order 8: $D_4$ x 34, $C_2^3$ x 28, $C_2\times C_4$ x 15

Order 7: $C_7$

Order 6: $C_6$ x 11, $S_3$ x 4

Order 4: $C_2^2$ x 45, $C_4$ x 9

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $C_{21}:D_4^2$

Order 672: $(C_2\times C_{42}):D_4$, $C_{14}.(S_3\times D_4)$, $(C_2\times C_6):D_{28}$, $(C_2\times C_{42}):D_4$, $C_{21}:(C_4\times D_4)$, $D_{42}:D_4$, $D_6:D_{28}$, $D_{14}:D_{12}$, $C_{21}:(C_4\times D_4)$, $(C_6\times D_{28}):C_2$, $C_{84}:C_2^3$, $D_{42}:D_4$, $D_{42}:C_2^3$, $D_{42}:C_2^3$, $D_{42}:D_4$

Order 448: $D_{28}:D_4$

Order 336: $C_6:D_{28}$ x 6, $C_6.D_{28}$ x 3, $C_{42}:C_2^3$ x 2, $C_{42}:D_4$ x 2, $C_{42}:D_4$ x 2, $C_{42}:D_4$ x 2, $C_{42}:D_4$ x 2, $D_{42}:C_4$ x 2, $C_{42}.D_4$, $D_6:C_{28}$, $C_{42}.D_4$, $C_{14}.D_{12}$, $C_{21}:C_4^2$, $C_2^2\times D_{42}$, $D_6\times D_{14}$, $D_4\times C_{42}$, $C_{42}.C_2^3$, $C_{12}:D_{14}$, $C_6\times D_{28}$, $C_{12}\times D_{14}$, $D_6:D_{14}$, $D_6:D_{14}$, $C_{14}:D_{12}$, $D_{42}:C_4$, $D_{14}.D_6$, $C_{42}.D_4$

Order 224: $D_4\times D_{14}$ x 3, $D_{14}:D_4$ x 2, $C_2^2:D_{28}$ x 2, $D_{14}:D_4$ x 2, $C_4:D_{28}$, $D_{28}:C_4$, $C_2^2\times D_{28}$, $C_{28}:D_4$, $C_{28}:D_4$, $C_{28}:D_4$

Order 192: $C_2^4:D_6$

Order 168: $C_6\times D_{14}$ x 9, $C_2\times D_{42}$ x 5, $C_3:D_{28}$ x 5, $C_6:C_{28}$ x 4, $S_3\times D_{14}$ x 3, $C_2^2\times C_{42}$ x 2, $C_{21}:D_4$ x 2, $C_{42}:C_4$ x 2, $D_6:C_{14}$ x 2, $C_{21}:D_4$ x 2, $C_{21}:D_4$ x 2, $D_6\times C_{14}$, $D_4\times C_{21}$, $C_2\times C_{84}$, $C_{14}:C_{12}$, $C_3\times D_{28}$, $C_{12}\times D_7$, $C_7:D_{12}$, $D_{21}:C_4$, $C_6.D_{14}$

Order 112: $C_{14}:D_4$ x 7, $C_2\times D_{28}$ x 7, $D_{14}:C_4$ x 5, $C_2^2\times D_{14}$ x 4, $D_4\times C_{14}$ x 3, $D_4\times D_7$ x 3, $C_4\times D_{14}$ x 3, $C_2^2:C_{28}$ x 2, $C_2^2\times C_{28}$, $C_4:C_{28}$, $C_{14}.D_4$, $C_{14}.D_4$, $C_{28}:C_4$

Order 96: $C_2^3:D_6$ x 2, $C_2^4:S_3$ x 2, $C_2^3.D_6$ x 2, $D_6:D_4$ x 2, $C_{12}:C_2^3$, $D_4\times D_6$, $C_{12}:D_4$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_{12}:D_4$

Order 84: $C_3\times D_{14}$ x 11, $C_2\times C_{42}$ x 7, $D_{42}$ x 7, $C_3:C_{28}$ x 3, $S_3\times C_{14}$ x 3, $C_{21}:C_4$ x 2, $S_3\times D_7$, $C_{84}$, $C_7:C_{12}$

Order 64: $D_4^2$

Order 56: $C_2\times D_{14}$ x 17, $C_7:D_4$ x 7, $D_{28}$ x 6, $C_2\times C_{28}$ x 5, $C_7\times D_4$ x 3, $C_{14}:C_4$ x 3, $C_4\times D_7$ x 3, $C_2^2\times C_{14}$ x 3

Order 48: $C_6:D_4$ x 13, $C_6\times D_4$ x 5, $C_6.D_4$ x 5, $D_6:C_4$ x 3, $C_2^3\times C_6$ x 2, $C_2^2\times D_6$ x 2, $C_6.C_2^3$ x 2, $C_2^2\times C_{12}$, $S_3\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_{12}:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 42: $C_{42}$ x 5, $C_3\times D_7$ x 3, $D_{21}$ x 2, $S_3\times C_7$

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

Order 28: $D_{14}$ x 19, $C_2\times C_{14}$ x 10, $C_{28}$ x 4, $C_7:C_4$ x 3

Order 24: $C_3:D_4$ x 11, $C_2^2\times C_6$ x 11, $C_2\times D_6$ x 9, $C_6:C_4$ x 7, $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 3, $D_{12}$, $C_4\times S_3$

Order 21: $C_{21}$

Order 16: $C_2\times D_4$ x 20, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 14: $C_{14}$ x 6, $D_7$ x 5

Order 12: $C_2\times C_6$ x 18, $D_6$ x 11, $C_3:C_4$ x 5, $C_{12}$ x 2

Order 8: $C_2^3$ x 20, $D_4$ x 16, $C_2\times C_4$ x 11

Order 7: $C_7$

Order 6: $C_6$ x 8, $S_3$ x 3

Order 4: $C_2^2$ x 29, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $C_{21}:D_4^2$ ($C_1$)

Order 672: $(C_2\times C_{42}):D_4$ ($C_2$), $C_{14}.(S_3\times D_4)$ ($C_2$), $(C_2\times C_6):D_{28}$ ($C_2$), $(C_2\times C_{42}):D_4$ ($C_2$), $C_{21}:(C_4\times D_4)$ ($C_2$), $D_{42}:D_4$ ($C_2$), $D_6:D_{28}$ ($C_2$), $D_{14}:D_{12}$ ($C_2$), $C_{21}:(C_4\times D_4)$ ($C_2$), $(C_6\times D_{28}):C_2$ ($C_2$), $C_{84}:C_2^3$ ($C_2$), $D_{42}:D_4$ ($C_2$), $D_{42}:C_2^3$ ($C_2$), $D_{42}:C_2^3$ ($C_2$), $D_{42}:D_4$ ($C_2$)

Order 336: $C_6:D_{28}$ ($C_2^2$) x 5, $C_6.D_{28}$ ($C_2^2$) x 3, $C_{42}:C_2^3$ ($C_2^2$) x 2, $C_{42}:D_4$ ($C_2^2$) x 2, $C_{42}:D_4$ ($C_2^2$) x 2, $C_{42}:D_4$ ($C_2^2$) x 2, $C_{42}:D_4$ ($C_2^2$) x 2, $D_{42}:C_4$ ($C_2^2$) x 2, $C_{42}.D_4$ ($C_2^2$), $D_6:C_{28}$ ($C_2^2$), $C_{42}.D_4$ ($C_2^2$), $C_{14}.D_{12}$ ($C_2^2$), $C_{21}:C_4^2$ ($C_2^2$), $C_2^2\times D_{42}$ ($C_2^2$), $D_6\times D_{14}$ ($C_2^2$), $D_4\times C_{42}$ ($C_2^2$), $C_{42}.C_2^3$ ($C_2^2$), $C_6\times D_{28}$ ($C_2^2$), $C_{12}\times D_{14}$ ($C_2^2$), $C_{14}:D_{12}$ ($C_2^2$), $D_{42}:C_4$ ($C_2^2$), $D_{14}.D_6$ ($C_2^2$), $C_{42}.D_4$ ($C_2^2$)

Order 224: $D_4\times D_{14}$ ($S_3$)

Order 168: $C_{21}:D_4$ ($D_4$) x 4, $C_3:D_{28}$ ($D_4$) x 4, $C_6\times D_{14}$ ($C_2^3$) x 3, $C_6:C_{28}$ ($C_2^3$) x 3, $C_2^2\times C_{42}$ ($C_2^3$) x 2, $C_2\times D_{42}$ ($C_2^3$) x 2, $C_{42}:C_4$ ($C_2^3$) x 2, $D_6\times C_{14}$ ($C_2^3$), $C_2\times C_{84}$ ($C_2^3$), $C_{14}:C_{12}$ ($C_2^3$)

Order 112: $C_2^2\times D_{14}$ ($D_6$) x 2, $C_{14}:D_4$ ($D_6$) x 2, $D_4\times C_{14}$ ($D_6$), $C_2\times D_{28}$ ($D_6$), $C_4\times D_{14}$ ($D_6$)

Order 96: $C_2^3.D_6$ ($D_7$)

Order 84: $C_3\times D_{14}$ ($C_2\times D_4$) x 4, $C_7:C_{12}$ ($C_2\times D_4$) x 2, $C_3:C_{28}$ ($C_2\times D_4$) x 2, $C_2\times C_{42}$ ($C_2\times D_4$) x 2, $D_{42}$ ($C_2\times D_4$) x 2, $C_2\times C_{42}$ ($C_2^4$)

Order 56: $C_7:D_4$ ($C_3:D_4$) x 4, $C_2\times D_{14}$ ($C_2\times D_6$) x 3, $C_2^2\times C_{14}$ ($C_2\times D_6$) x 2, $C_2\times C_{28}$ ($C_2\times D_6$), $C_{14}:C_4$ ($C_2\times D_6$)

Order 48: $C_6:D_4$ ($D_{14}$) x 2, $C_6\times D_4$ ($D_{14}$), $C_6.C_2^3$ ($D_{14}$), $C_6.D_4$ ($D_{14}$), $D_6:C_4$ ($D_{14}$), $C_6.D_4$ ($D_{14}$)

Order 42: $C_{42}$ ($C_2^2\times D_4$) x 2, $C_{42}$ ($D_4:C_2^2$)

Order 28: $C_2\times C_{14}$ ($C_6:D_4$) x 2, $D_{14}$ ($C_6:D_4$) x 2, $D_{14}$ ($S_3\times D_4$) x 2, $C_7:C_4$ ($C_6:D_4$) x 2, $C_2\times C_{14}$ ($C_2^2\times D_6$)

Order 24: $C_6:C_4$ ($C_2\times D_{14}$) x 3, $C_2^2\times C_6$ ($C_2\times D_{14}$) x 2, $C_2\times C_{12}$ ($C_2\times D_{14}$), $C_2\times D_6$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($D_4^2$)

Order 16: $C_2\times D_4$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_2^3:D_6$), $C_{14}$ ($D_4:D_6$), $C_{14}$ ($D_4\times D_6$)

Order 12: $C_2\times C_6$ ($D_4\times D_7$) x 2, $C_3:C_4$ ($D_4\times D_7$) x 2, $C_2\times C_6$ ($C_2^2\times D_{14}$)

Order 8: $C_2^3$ ($S_3\times D_{14}$) x 2, $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_2^4:D_6$)

Order 6: $C_6$ ($D_4\times D_{14}$) x 2, $C_6$ ($D_4:D_{14}$)

Order 4: $C_2^2$ ($D_6:D_{14}$) x 2, $C_2^2$ ($D_6\times D_{14}$)

Order 3: $C_3$ ($D_{28}:D_4$)

Order 2: $C_2$ ($D_{42}:C_2^3$), $C_2$ ($D_{28}:D_6$), $C_2$ ($D_{12}:D_{14}$)

Order 1: $C_1$ ($C_{21}:D_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $C_{21}:D_4^2$ ($C_1$)

Order 672: $(C_2\times C_{42}):D_4$ ($C_2$), $C_{14}.(S_3\times D_4)$ ($C_2$), $(C_2\times C_6):D_{28}$ ($C_2$), $(C_2\times C_{42}):D_4$ ($C_2$), $C_{21}:(C_4\times D_4)$ ($C_2$), $D_{42}:D_4$ ($C_2$), $D_6:D_{28}$ ($C_2$), $D_{14}:D_{12}$ ($C_2$), $C_{21}:(C_4\times D_4)$ ($C_2$), $(C_6\times D_{28}):C_2$ ($C_2$), $C_{84}:C_2^3$ ($C_2$), $D_{42}:D_4$ ($C_2$), $D_{42}:C_2^3$ ($C_2$), $D_{42}:C_2^3$ ($C_2$), $D_{42}:D_4$ ($C_2$)

Order 336: $C_6:D_{28}$ ($C_2^2$) x 5, $C_6.D_{28}$ ($C_2^2$) x 3, $C_{42}:C_2^3$ ($C_2^2$) x 2, $C_{42}:D_4$ ($C_2^2$) x 2, $C_{42}:D_4$ ($C_2^2$) x 2, $C_{42}:D_4$ ($C_2^2$) x 2, $C_{42}:D_4$ ($C_2^2$) x 2, $D_{42}:C_4$ ($C_2^2$) x 2, $C_{42}.D_4$ ($C_2^2$), $D_6:C_{28}$ ($C_2^2$), $C_{42}.D_4$ ($C_2^2$), $C_{14}.D_{12}$ ($C_2^2$), $C_{21}:C_4^2$ ($C_2^2$), $C_2^2\times D_{42}$ ($C_2^2$), $D_6\times D_{14}$ ($C_2^2$), $D_4\times C_{42}$ ($C_2^2$), $C_{42}.C_2^3$ ($C_2^2$), $C_6\times D_{28}$ ($C_2^2$), $C_{12}\times D_{14}$ ($C_2^2$), $C_{14}:D_{12}$ ($C_2^2$), $D_{42}:C_4$ ($C_2^2$), $D_{14}.D_6$ ($C_2^2$), $C_{42}.D_4$ ($C_2^2$)

Order 224: $D_4\times D_{14}$ ($S_3$)

Order 168: $C_6\times D_{14}$ ($C_2^3$) x 3, $C_6:C_{28}$ ($C_2^3$) x 3, $C_2^2\times C_{42}$ ($C_2^3$) x 2, $C_2\times D_{42}$ ($C_2^3$) x 2, $C_{42}:C_4$ ($C_2^3$) x 2, $D_6\times C_{14}$ ($C_2^3$), $C_2\times C_{84}$ ($C_2^3$), $C_{21}:D_4$ ($D_4$), $C_{14}:C_{12}$ ($C_2^3$), $C_3:D_{28}$ ($D_4$)

Order 112: $C_2^2\times D_{14}$ ($D_6$) x 2, $C_{14}:D_4$ ($D_6$) x 2, $D_4\times C_{14}$ ($D_6$), $C_2\times D_{28}$ ($D_6$), $C_4\times D_{14}$ ($D_6$)

Order 96: $C_2^3.D_6$ ($D_7$)

Order 84: $C_3\times D_{14}$ ($C_2\times D_4$) x 2, $C_7:C_{12}$ ($C_2\times D_4$), $C_3:C_{28}$ ($C_2\times D_4$), $C_2\times C_{42}$ ($C_2^4$), $C_2\times C_{42}$ ($C_2\times D_4$), $D_{42}$ ($C_2\times D_4$)

Order 56: $C_2\times D_{14}$ ($C_2\times D_6$) x 3, $C_2^2\times C_{14}$ ($C_2\times D_6$) x 2, $C_2\times C_{28}$ ($C_2\times D_6$), $C_7:D_4$ ($C_3:D_4$), $C_{14}:C_4$ ($C_2\times D_6$)

Order 48: $C_6:D_4$ ($D_{14}$) x 2, $C_6\times D_4$ ($D_{14}$), $C_6.C_2^3$ ($D_{14}$), $C_6.D_4$ ($D_{14}$), $D_6:C_4$ ($D_{14}$), $C_6.D_4$ ($D_{14}$)

Order 42: $C_{42}$ ($C_2^2\times D_4$) x 2, $C_{42}$ ($D_4:C_2^2$)

Order 28: $C_2\times C_{14}$ ($C_2^2\times D_6$), $C_2\times C_{14}$ ($C_6:D_4$), $D_{14}$ ($C_6:D_4$), $D_{14}$ ($S_3\times D_4$), $C_7:C_4$ ($C_6:D_4$)

Order 24: $C_6:C_4$ ($C_2\times D_{14}$) x 3, $C_2^2\times C_6$ ($C_2\times D_{14}$) x 2, $C_2\times C_{12}$ ($C_2\times D_{14}$), $C_2\times D_6$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($D_4^2$)

Order 16: $C_2\times D_4$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_2^3:D_6$), $C_{14}$ ($D_4:D_6$), $C_{14}$ ($D_4\times D_6$)

Order 12: $C_2\times C_6$ ($C_2^2\times D_{14}$), $C_2\times C_6$ ($D_4\times D_7$), $C_3:C_4$ ($D_4\times D_7$)

Order 8: $C_2^3$ ($S_3\times D_{14}$) x 2, $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_2^4:D_6$)

Order 6: $C_6$ ($D_4\times D_{14}$) x 2, $C_6$ ($D_4:D_{14}$)

Order 4: $C_2^2$ ($D_6\times D_{14}$), $C_2^2$ ($D_6:D_{14}$)

Order 3: $C_3$ ($D_{28}:D_4$)

Order 2: $C_2$ ($D_{42}:C_2^3$), $C_2$ ($D_{28}:D_6$), $C_2$ ($D_{12}:D_{14}$)

Order 1: $C_1$ ($C_{21}:D_4^2$)

Series

Derived series

$C_{21}:D_4^2$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_1$

Chief series

$C_{21}:D_4^2$

$\rhd$

$D_{42}:C_2^3$

$\rhd$

$C_{42}:C_2^3$

$\rhd$

$C_6\times D_{14}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{21}:D_4^2$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

Character theory

Complex character table

See the $117 \times 117$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $61 \times 61$ rational character table.