Abstract group 1344.8544: $C_{168}.C_2^3$

• Label: 1344.8544
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $4$

Group information

Description:	$C_{168}.C_2^3$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{21}.(C_6\times D_4).C_2^5$, of order \(32256\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7 \)
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	8	12	14	21	24	28	42	56	84	168
Elements	1	127	2	352	62	6	32	176	90	12	16	96	36	192	48	96	1344
Conjugacy classes	1	7	1	10	3	3	4	5	15	3	2	18	6	24	9	12	123
Divisions	1	7	1	10	3	1	4	5	5	1	2	4	2	4	2	2	54
Autjugacy classes	1	6	1	8	3	1	2	4	4	1	1	4	2	2	2	1	43

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	16	60	40	0	7	0	0	0	123
Irr. rational chars.	16	12	4	8	1	8	4	1	54

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$4$
Inequivalent generating quadruples:	$29877120$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	96
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{2}=d^{168}=[a,b]=[a,c]=[b,c]=1, a^{2}=d^{84}, d^{a}=d^{85}, d^{b}=d^{113}, d^{c}=d^{83} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(11,12)(13,17)(14,21)(15,24)(16,19)(18,22)(20,26)(23,25) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $(C_4.D_{28})$
Semidirect product:	$(C_{14}:Q_8)$ $\,\rtimes\,$ $D_6$	$(C_7:Q_{16})$ $\,\rtimes\,$ $D_6$ (2)	$C_7$ $\,\rtimes\,$ $(Q_{16}:D_6)$	$(C_{56}:D_6)$ $\,\rtimes\,$ $C_2$	all 36
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{84}$ . $C_2^3$	$C_{84}$ . $C_2^4$	$C_{168}$ . $C_2^3$ (2)	$(S_3\times C_{28})$ . $D_4$ (2)	all 56

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 4468 subgroups in 516 conjugacy classes, 140 normal (78 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6\times D_{28}$
Commutator:	$G' \simeq$ $C_{84}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $D_6\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $\OD_{16}\times C_{21}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{168}.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_2^2$
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: $C_{168}.C_2^3$

Order 672: $C_{24}.D_{14}$ x 2, $C_{24}.D_{14}$ x 2, $C_{24}.D_{14}$ x 2, $C_{24}:D_{14}$ x 2, $C_4.D_{84}$, $C_{56}:D_6$, $C_{12}.D_{28}$, $C_{12}.D_{28}$, $C_{12}.D_{28}$, $D_{28}:D_6$, $C_{84}.C_2^3$

Order 448: $\OD_{16}:D_{14}$

Order 336: $C_{12}.D_{14}$ x 7, $C_{21}:Q_{16}$ x 4, $C_{21}:Q_{16}$ x 2, $C_{168}:C_2$ x 2, $C_{168}:C_2$ x 2, $S_3\times C_{56}$ x 2, $C_{21}:Q_{16}$ x 2, $C_{168}:C_2$ x 2, $C_{21}:\SD_{16}$ x 2, $C_{21}:\SD_{16}$ x 2, $C_{12}.C_{28}$, $D_{84}:C_2$, $C_{42}:Q_8$, $D_6\times C_{28}$, $D_{28}:C_6$, $C_{42}:Q_8$, $D_6:D_{14}$, $C_{42}:Q_8$, $D_6.D_{14}$, $D_{14}.D_6$, $S_3\times D_{28}$, $C_{12}:D_{14}$, $C_{12}.D_{14}$, $D_{84}:C_2$, $D_{28}:S_3$, $\OD_{16}\times C_{21}$

Order 224: $C_4.D_{28}$ x 8, $C_8:D_{14}$ x 2, $C_{14}:Q_{16}$ x 2, $D_{28}:C_2^2$, $C_{14}.C_2^4$, $C_{14}\times \OD_{16}$

Order 192: $Q_{16}:D_6$

Order 168: $C_{21}:Q_8$ x 5, $C_6.D_{14}$ x 5, $C_{21}:Q_8$ x 4, $S_3\times C_{28}$ x 4, $C_{21}:Q_8$ x 4, $C_{168}$ x 2, $C_3:C_{56}$ x 2, $D_6\times C_{14}$, $S_3\times D_{14}$, $C_2\times C_{84}$, $C_{21}:D_4$, $C_{42}:C_4$, $D_{84}$, $C_4\times D_{21}$, $C_6:C_{28}$, $C_{21}:D_4$, $C_{14}:C_{12}$, $C_3\times D_{28}$, $C_{12}\times D_7$, $C_7:D_{12}$, $C_3:D_{28}$, $C_{21}:D_4$, $D_{21}:C_4$, $C_6.D_{14}$

Order 112: $C_{14}:Q_8$ x 10, $C_7:Q_{16}$ x 8, $C_{56}:C_2$ x 8, $D_{28}:C_2$ x 6, $C_7\times \OD_{16}$ x 4, $C_2\times C_{56}$ x 2, $C_2^2\times C_{28}$, $C_{14}:D_4$, $C_2^2.D_{14}$, $C_2\times D_{28}$, $C_4\times D_{14}$

Order 96: $Q_{16}:S_3$ x 2, $S_3\times Q_{16}$ x 2, $C_8.D_6$ x 2, $S_3\times \SD_{16}$ x 2, $D_4:D_6$, $Q_8\times D_6$, $Q_{16}:C_6$, $C_{12}.D_4$, $C_{12}.D_4$, $C_4.D_{12}$, $S_3\times \OD_{16}$

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

Order 64: $Q_{16}:C_2^2$

Order 56: $C_7:Q_8$ x 13, $C_{14}:C_4$ x 7, $C_2\times C_{28}$ x 6, $C_7:D_4$ x 4, $C_4\times D_7$ x 4, $C_{56}$ x 4, $D_{28}$ x 3, $C_2^2\times C_{14}$, $C_2\times D_{14}$

Order 48: $S_3\times Q_8$ x 7, $C_3:Q_{16}$ x 4, $C_4\times D_6$ x 3, $C_3:Q_{16}$ x 2, $C_{24}:C_2$ x 2, $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $D_4:S_3$ x 2, $S_3\times D_4$ x 2, $D_{12}:C_2$ x 2, $C_6:Q_8$ x 2, $C_3\times Q_{16}$ x 2, $C_3\times \SD_{16}$ x 2, $Q_8:S_3$ x 2, $C_3:\SD_{16}$ x 2, $D_4:C_6$, $C_6\times Q_8$, $D_{12}:C_2$, $C_3\times \OD_{16}$, $C_3:\OD_{16}$

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

Order 32: $Q_{16}:C_2$ x 8, $C_2\times Q_{16}$ x 2, $C_2\times \SD_{16}$ x 2, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_2\times \OD_{16}$

Order 28: $C_7:C_4$ x 6, $C_2\times C_{14}$ x 5, $D_{14}$ x 4, $C_{28}$ x 4

Order 24: $C_4\times S_3$ x 11, $C_3:Q_8$ x 9, $C_3\times Q_8$ x 4, $C_2\times C_{12}$ x 3, $C_3:D_4$ x 3, $C_6:C_4$ x 3, $D_{12}$ x 2, $C_{24}$ x 2, $C_2\times D_6$ x 2, $C_3\times D_4$ x 2, $C_3:C_8$ x 2

Order 21: $C_{21}$

Order 16: $C_2\times Q_8$ x 10, $Q_{16}$ x 8, $\SD_{16}$ x 8, $D_4:C_2$ x 6, $\OD_{16}$ x 4, $C_2^2\times C_4$ x 3, $C_2\times C_8$ x 2, $C_2\times D_4$ x 2

Order 14: $C_{14}$ x 5, $D_7$ x 2

Order 12: $D_6$ x 7, $C_{12}$ x 5, $C_3:C_4$ x 5, $C_2\times C_6$ x 2

Order 8: $C_2\times C_4$ x 17, $Q_8$ x 13, $D_4$ x 7, $C_8$ x 4, $C_2^3$ x 2

Order 7: $C_7$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_4$ x 10, $C_2^2$ x 9

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $C_{168}.C_2^3$

Order 672: $C_4.D_{84}$, $C_{56}:D_6$, $C_{12}.D_{28}$, $C_{12}.D_{28}$, $C_{12}.D_{28}$, $C_{24}.D_{14}$, $C_{24}.D_{14}$, $C_{24}.D_{14}$, $C_{24}:D_{14}$, $D_{28}:D_6$, $C_{84}.C_2^3$

Order 448: $\OD_{16}:D_{14}$

Order 336: $C_{12}.D_{14}$ x 5, $C_{21}:Q_{16}$ x 2, $C_{21}:Q_{16}$, $C_{168}:C_2$, $C_{12}.C_{28}$, $C_{168}:C_2$, $S_3\times C_{56}$, $C_{21}:Q_{16}$, $C_{168}:C_2$, $C_{21}:\SD_{16}$, $C_{21}:\SD_{16}$, $D_{84}:C_2$, $C_{42}:Q_8$, $D_6\times C_{28}$, $D_{28}:C_6$, $C_{42}:Q_8$, $D_6:D_{14}$, $C_{42}:Q_8$, $D_6.D_{14}$, $D_{14}.D_6$, $S_3\times D_{28}$, $C_{12}:D_{14}$, $C_{12}.D_{14}$, $D_{84}:C_2$, $D_{28}:S_3$, $\OD_{16}\times C_{21}$

Order 224: $C_4.D_{28}$ x 6, $C_8:D_{14}$, $D_{28}:C_2^2$, $C_{14}.C_2^4$, $C_{14}\times \OD_{16}$, $C_{14}:Q_{16}$

Order 192: $Q_{16}:D_6$

Order 168: $S_3\times C_{28}$ x 4, $C_{21}:Q_8$ x 3, $C_{21}:Q_8$ x 3, $C_{21}:Q_8$ x 3, $C_6.D_{14}$ x 3, $C_{168}$, $D_6\times C_{14}$, $S_3\times D_{14}$, $C_2\times C_{84}$, $C_{21}:D_4$, $C_{42}:C_4$, $D_{84}$, $C_4\times D_{21}$, $C_6:C_{28}$, $C_3:C_{56}$, $C_{21}:D_4$, $C_{14}:C_{12}$, $C_3\times D_{28}$, $C_{12}\times D_7$, $C_7:D_{12}$, $C_3:D_{28}$, $C_{21}:D_4$, $D_{21}:C_4$, $C_6.D_{14}$

Order 112: $C_{14}:Q_8$ x 8, $D_{28}:C_2$ x 6, $C_7:Q_{16}$ x 4, $C_{56}:C_2$ x 4, $C_7\times \OD_{16}$ x 3, $C_2^2\times C_{28}$, $C_{14}:D_4$, $C_2^2.D_{14}$, $C_2\times D_{28}$, $C_4\times D_{14}$, $C_2\times C_{56}$

Order 96: $D_4:D_6$, $Q_8\times D_6$, $Q_{16}:C_6$, $C_{12}.D_4$, $C_{12}.D_4$, $Q_{16}:S_3$, $S_3\times Q_{16}$, $C_8.D_6$, $S_3\times \SD_{16}$, $C_4.D_{12}$, $S_3\times \OD_{16}$

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

Order 64: $Q_{16}:C_2^2$

Order 56: $C_7:Q_8$ x 9, $C_2\times C_{28}$ x 6, $C_{14}:C_4$ x 5, $C_7:D_4$ x 4, $C_4\times D_7$ x 4, $D_{28}$ x 3, $C_{56}$ x 2, $C_2^2\times C_{14}$, $C_2\times D_{14}$

Order 48: $S_3\times Q_8$ x 5, $C_4\times D_6$ x 3, $D_4:S_3$ x 2, $S_3\times D_4$ x 2, $D_{12}:C_2$ x 2, $C_6:Q_8$ x 2, $C_3:Q_{16}$ x 2, $C_3:Q_{16}$, $C_{24}:C_2$, $C_{24}:C_2$, $D_4:C_6$, $C_6\times Q_8$, $D_{12}:C_2$, $S_3\times C_8$, $C_3\times Q_{16}$, $C_3\times \SD_{16}$, $C_3\times \OD_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$, $C_3:\OD_{16}$

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

Order 32: $Q_{16}:C_2$ x 6, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_2\times Q_{16}$, $C_2\times \SD_{16}$, $C_2\times \OD_{16}$

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

Order 24: $C_4\times S_3$ x 9, $C_3:Q_8$ x 6, $C_2\times C_{12}$ x 3, $C_3:D_4$ x 3, $C_6:C_4$ x 3, $C_3\times Q_8$ x 3, $D_{12}$ x 2, $C_2\times D_6$ x 2, $C_3\times D_4$ x 2, $C_{24}$, $C_3:C_8$

Order 21: $C_{21}$

Order 16: $C_2\times Q_8$ x 8, $D_4:C_2$ x 6, $Q_{16}$ x 4, $\SD_{16}$ x 4, $\OD_{16}$ x 3, $C_2^2\times C_4$ x 3, $C_2\times D_4$ x 2, $C_2\times C_8$

Order 14: $C_{14}$ x 4, $D_7$ x 2

Order 12: $D_6$ x 5, $C_{12}$ x 4, $C_3:C_4$ x 4, $C_2\times C_6$ x 2

Order 8: $C_2\times C_4$ x 15, $Q_8$ x 9, $D_4$ x 7, $C_2^3$ x 2, $C_8$ x 2

Order 7: $C_7$

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

Order 4: $C_4$ x 8, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $C_{168}.C_2^3$ ($C_1$)

Order 672: $C_{24}.D_{14}$ ($C_2$) x 2, $C_{24}.D_{14}$ ($C_2$) x 2, $C_{24}.D_{14}$ ($C_2$) x 2, $C_{24}:D_{14}$ ($C_2$) x 2, $C_4.D_{84}$ ($C_2$), $C_{56}:D_6$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $D_{28}:D_6$ ($C_2$), $C_{84}.C_2^3$ ($C_2$)

Order 336: $C_{12}.D_{14}$ ($C_2^2$) x 5, $C_{21}:Q_{16}$ ($C_2^2$) x 4, $C_{21}:Q_{16}$ ($C_2^2$) x 2, $C_{168}:C_2$ ($C_2^2$) x 2, $C_{168}:C_2$ ($C_2^2$) x 2, $S_3\times C_{56}$ ($C_2^2$) x 2, $C_{21}:Q_{16}$ ($C_2^2$) x 2, $C_{168}:C_2$ ($C_2^2$) x 2, $C_{21}:\SD_{16}$ ($C_2^2$) x 2, $C_{21}:\SD_{16}$ ($C_2^2$) x 2, $C_{12}.C_{28}$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $D_6\times C_{28}$ ($C_2^2$), $D_{28}:C_6$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $S_3\times D_{28}$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $D_{28}:S_3$ ($C_2^2$), $\OD_{16}\times C_{21}$ ($C_2^2$)

Order 224: $C_4.D_{28}$ ($S_3$)

Order 168: $C_{21}:Q_8$ ($C_2^3$) x 3, $C_{21}:Q_8$ ($C_2^3$) x 3, $C_{168}$ ($C_2^3$) x 2, $S_3\times C_{28}$ ($C_2^3$) x 2, $S_3\times C_{28}$ ($D_4$) x 2, $C_3:C_{56}$ ($C_2^3$) x 2, $D_6\times C_{14}$ ($D_4$), $C_2\times C_{84}$ ($C_2^3$), $D_{84}$ ($C_2^3$), $C_6:C_{28}$ ($D_4$), $C_3\times D_{28}$ ($C_2^3$)

Order 112: $C_7:Q_{16}$ ($D_6$) x 2, $C_{56}:C_2$ ($D_6$) x 2, $D_{28}:C_2$ ($D_6$), $C_{14}:Q_8$ ($D_6$), $C_7\times \OD_{16}$ ($D_6$)

Order 96: $S_3\times \OD_{16}$ ($D_7$)

Order 84: $C_3:C_{28}$ ($C_2\times D_4$) x 2, $S_3\times C_{14}$ ($C_2\times D_4$) x 2, $C_{84}$ ($C_2^4$), $C_{84}$ ($C_2\times D_4$), $C_2\times C_{42}$ ($C_2\times D_4$)

Order 56: $C_7:Q_8$ ($C_2\times D_6$) x 3, $C_{56}$ ($C_2\times D_6$) x 2, $C_2\times C_{28}$ ($C_2\times D_6$), $D_{28}$ ($C_2\times D_6$)

Order 48: $C_{24}:C_2$ ($D_{14}$) x 2, $S_3\times C_8$ ($D_{14}$) x 2, $C_4\times D_6$ ($D_{14}$), $C_3\times \OD_{16}$ ($D_{14}$), $C_3:\OD_{16}$ ($D_{14}$)

Order 42: $S_3\times C_7$ ($Q_{16}:C_2$) x 2, $C_{42}$ ($C_2^2\times D_4$)

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

Order 24: $C_4\times S_3$ ($D_{28}$) x 2, $C_4\times S_3$ ($C_2\times D_{14}$) x 2, $C_{24}$ ($C_2\times D_{14}$) x 2, $C_3:C_8$ ($C_2\times D_{14}$) x 2, $C_2\times C_{12}$ ($C_2\times D_{14}$), $C_6:C_4$ ($D_{28}$), $C_2\times D_6$ ($D_{28}$)

Order 21: $C_{21}$ ($Q_{16}:C_2^2$)

Order 16: $\OD_{16}$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($D_4\times D_6$)

Order 12: $D_6$ ($C_2\times D_{28}$) x 2, $C_3:C_4$ ($C_2\times D_{28}$) x 2, $C_2\times C_6$ ($C_2\times D_{28}$), $C_{12}$ ($C_2^2\times D_{14}$), $C_{12}$ ($C_2\times D_{28}$)

Order 8: $C_8$ ($S_3\times D_{14}$) x 2, $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($Q_{16}:D_6$)

Order 6: $S_3$ ($C_4.D_{28}$) x 2, $C_6$ ($C_2^2\times D_{28}$)

Order 4: $C_2^2$ ($S_3\times D_{28}$), $C_4$ ($D_6\times D_{14}$), $C_4$ ($S_3\times D_{28}$)

Order 3: $C_3$ ($\OD_{16}:D_{14}$)

Order 2: $C_2$ ($D_6\times D_{28}$)

Order 1: $C_1$ ($C_{168}.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $C_{168}.C_2^3$ ($C_1$)

Order 672: $C_4.D_{84}$ ($C_2$), $C_{56}:D_6$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{24}.D_{14}$ ($C_2$), $C_{24}.D_{14}$ ($C_2$), $C_{24}.D_{14}$ ($C_2$), $C_{24}:D_{14}$ ($C_2$), $D_{28}:D_6$ ($C_2$), $C_{84}.C_2^3$ ($C_2$)

Order 336: $C_{12}.D_{14}$ ($C_2^2$) x 3, $C_{21}:Q_{16}$ ($C_2^2$) x 2, $C_{21}:Q_{16}$ ($C_2^2$), $C_{168}:C_2$ ($C_2^2$), $C_{12}.C_{28}$ ($C_2^2$), $C_{168}:C_2$ ($C_2^2$), $S_3\times C_{56}$ ($C_2^2$), $C_{21}:Q_{16}$ ($C_2^2$), $C_{168}:C_2$ ($C_2^2$), $C_{21}:\SD_{16}$ ($C_2^2$), $C_{21}:\SD_{16}$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $D_6\times C_{28}$ ($C_2^2$), $D_{28}:C_6$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $S_3\times D_{28}$ ($C_2^2$), $D_{84}:C_2$ ($C_2^2$), $D_{28}:S_3$ ($C_2^2$), $\OD_{16}\times C_{21}$ ($C_2^2$)

Order 224: $C_4.D_{28}$ ($S_3$)

Order 168: $C_{21}:Q_8$ ($C_2^3$) x 2, $S_3\times C_{28}$ ($C_2^3$) x 2, $S_3\times C_{28}$ ($D_4$) x 2, $C_{21}:Q_8$ ($C_2^3$) x 2, $C_{168}$ ($C_2^3$), $D_6\times C_{14}$ ($D_4$), $C_2\times C_{84}$ ($C_2^3$), $D_{84}$ ($C_2^3$), $C_6:C_{28}$ ($D_4$), $C_3:C_{56}$ ($C_2^3$), $C_3\times D_{28}$ ($C_2^3$)

Order 112: $C_7:Q_{16}$ ($D_6$), $C_{56}:C_2$ ($D_6$), $D_{28}:C_2$ ($D_6$), $C_{14}:Q_8$ ($D_6$), $C_7\times \OD_{16}$ ($D_6$)

Order 96: $S_3\times \OD_{16}$ ($D_7$)

Order 84: $C_3:C_{28}$ ($C_2\times D_4$) x 2, $S_3\times C_{14}$ ($C_2\times D_4$) x 2, $C_{84}$ ($C_2^4$), $C_{84}$ ($C_2\times D_4$), $C_2\times C_{42}$ ($C_2\times D_4$)

Order 56: $C_7:Q_8$ ($C_2\times D_6$) x 2, $C_2\times C_{28}$ ($C_2\times D_6$), $D_{28}$ ($C_2\times D_6$), $C_{56}$ ($C_2\times D_6$)

Order 48: $C_{24}:C_2$ ($D_{14}$), $S_3\times C_8$ ($D_{14}$), $C_4\times D_6$ ($D_{14}$), $C_3\times \OD_{16}$ ($D_{14}$), $C_3:\OD_{16}$ ($D_{14}$)

Order 42: $C_{42}$ ($C_2^2\times D_4$), $S_3\times C_7$ ($Q_{16}:C_2$)

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

Order 24: $C_4\times S_3$ ($D_{28}$) x 2, $C_4\times S_3$ ($C_2\times D_{14}$) x 2, $C_2\times C_{12}$ ($C_2\times D_{14}$), $C_6:C_4$ ($D_{28}$), $C_{24}$ ($C_2\times D_{14}$), $C_2\times D_6$ ($D_{28}$), $C_3:C_8$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($Q_{16}:C_2^2$)

Order 16: $\OD_{16}$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($D_4\times D_6$)

Order 12: $D_6$ ($C_2\times D_{28}$) x 2, $C_3:C_4$ ($C_2\times D_{28}$) x 2, $C_2\times C_6$ ($C_2\times D_{28}$), $C_{12}$ ($C_2^2\times D_{14}$), $C_{12}$ ($C_2\times D_{28}$)

Order 8: $C_2\times C_4$ ($S_3\times D_{14}$), $C_8$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($Q_{16}:D_6$)

Order 6: $C_6$ ($C_2^2\times D_{28}$), $S_3$ ($C_4.D_{28}$)

Order 4: $C_2^2$ ($S_3\times D_{28}$), $C_4$ ($D_6\times D_{14}$), $C_4$ ($S_3\times D_{28}$)

Order 3: $C_3$ ($\OD_{16}:D_{14}$)

Order 2: $C_2$ ($D_6\times D_{28}$)

Order 1: $C_1$ ($C_{168}.C_2^3$)

Series

Derived series

$C_{168}.C_2^3$

$\rhd$

$C_{84}$

$\rhd$

$C_1$

Chief series

$C_{168}.C_2^3$

$\rhd$

$D_{28}:D_6$

$\rhd$

$D_6\times C_{28}$

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{168}.C_2^3$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$\OD_{16}$

Character theory

Complex character table

See the $123 \times 123$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $54 \times 54$ rational character table.