Abstract group 2688.eh: $C_8:C_2^3\times F_7$

• Label: 2688.eh
• Order: \( 2^{7} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $17$
• Trans deg.: $112$
• Rank: $5$

Group information

Description:	$C_8:C_2^3\times F_7$
Order:	\(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_3^4:(D_4\times D_6)$, of order \(172032\)\(\medspace = 2^{13} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 7, $C_3$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	7	8	12	14	24	28	56
Elements	1	255	14	128	882	6	128	448	186	448	96	96	2688
Conjugacy classes	1	23	2	12	46	1	8	24	11	16	6	4	154
Divisions	1	23	1	12	23	1	8	12	11	8	6	4	110
Autjugacy classes	1	9	2	7	18	1	2	14	5	4	4	1	68

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	96	24	12	16	0	4	2	154
Irr. rational chars.	32	40	12	16	4	4	2	110

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$112$
Rank:	$5$
Inequivalent generating 5-tuples:	$241984512000$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=d^{2}=e^{56}=[a,b]=[a,c]=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(1,2)(3,5)(6,8)(9,10)(12,13)(14,17)(15,16), (11,12,14,15,16,17,13), (1,3) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(D_8:C_2)$ $\, \times\, $ $F_7$
Semidirect product:	$(D_8\times D_{14})$ $\,\rtimes\,$ $C_6$	$(D_8:D_{14})$ $\,\rtimes\,$ $C_6$	$(D_8:D_{14})$ $\,\rtimes\,$ $C_6$	$D_{14}$ $\,\rtimes\,$ $(D_8:C_6)$	all 161
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_{14}:C_{12})$ . $D_4$ (3)	$(C_2^3\times F_7)$ . $D_4$	$(C_2^3.F_7)$ . $D_4$	$(D_4\times F_7)$ . $C_2^3$	all 60
Aut. group:	$\Aut(C_{56}.C_4)$	$\Aut(D_{56}:C_2)$	$\Aut(D_7\times D_8)$	$\Aut(D_8:D_7)$	all 22

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

Homology

Abelianization:	$C_{2}^{4} \times C_{6} \simeq C_{2}^{5} \times C_{3}$
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 23690 subgroups in 3440 conjugacy classes, 953 normal (99 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times D_4\times F_7$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^4\times C_6$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^4\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_{56}:C_2^3$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_8:C_2^3\times F_7$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_{12}:C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8:C_2^4$
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: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 2688: $C_8:C_2^3\times F_7$

Order 1344: $D_8:C_2\times F_7$ x 16, $C_2\times D_{56}:C_6$ x 2, $C_2\times \SD_{16}\times F_7$ x 2, $C_2\times D_8:F_7$ x 2, $C_2\times D_8\times F_7$ x 2, $(C_2^2\times D_{28}):C_6$, $D_{28}:(C_2^2\times C_6)$, $C_2\times D_{56}:C_6$, $C_2\times \OD_{16}\times F_7$, $D_4:C_2^2\times F_7$, $C_2^2\times D_4\times F_7$, $(C_{14}\times D_8):C_6$

Order 896: $C_2\times C_8:C_2^2\times D_7$

Order 672: $C_2\times D_4\times F_7$ x 20, $D_{56}:C_6$ x 16, $\SD_{16}\times F_7$ x 16, $D_8:F_7$ x 16, $D_8\times F_7$ x 16, $D_4:C_2\times F_7$ x 12, $C_{56}:(C_2\times C_6)$ x 8, $(C_2\times D_{28}):C_6$ x 8, $(D_4\times C_{14}):C_6$ x 8, $D_{56}:C_6$ x 8, $\OD_{16}\times F_7$ x 8, $C_2\times D_{28}:C_6$ x 4, $C_2\times C_{56}:C_6$ x 2, $C_2\times C_{56}:C_6$ x 2, $C_2\times Q_8:F_7$ x 2, $C_2\times D_4.F_7$ x 2, $D_{56}:C_6$ x 2, $C_2\times C_{56}:C_6$ x 2, $C_2\times C_{56}:C_6$ x 2, $D_{14}:C_{24}$ x 2, $C_2\times D_4:F_7$ x 2, $C_2\times D_{28}:C_6$ x 2, $C_4\times C_2^2\times F_7$ x 2, $C_2\times C_{56}:C_6$, $C_2\times C_{28}.C_{12}$, $C_2^4\times F_7$, $(D_4\times C_{14}):C_6$, $C_{14}:C_6\times D_4$, $C_2^4:F_7$, $C_2\times D_{28}:C_6$, $C_2\times Q_8\times F_7$, $C_2\times D_{28}:C_6$

Order 448: $D_8:D_{14}$ x 16, $C_{56}:C_2^3$ x 2, $\SD_{16}\times D_{14}$ x 2, $D_8:D_{14}$ x 2, $D_8\times D_{14}$ x 2, $D_{28}:C_2^3$, $C_{28}:C_2^4$, $C_{56}:C_2^3$, $C_{28}.C_2^4$, $D_{28}:C_2^3$, $C_{56}:C_2^3$, $\OD_{16}\times D_{14}$

Order 384: $C_{24}:C_2^4$

Order 336: $D_4\times F_7$ x 54, $C_2^3\times F_7$ x 23, $D_{14}:C_{12}$ x 23, $D_{28}:C_6$ x 16, $C_2^3:F_7$ x 15, $D_4:F_7$ x 14, $D_{28}:C_6$ x 14, $C_2\times C_{28}:C_6$ x 11, $D_{28}:C_6$ x 11, $C_{56}:C_6$ x 8, $C_{56}:C_6$ x 8, $C_8\times F_7$ x 8, $C_{56}:C_6$ x 8, $C_{56}:C_6$ x 8, $Q_8:F_7$ x 8, $D_4.F_7$ x 8, $C_{56}:C_6$ x 8, $(C_2\times C_{28}):C_6$ x 6, $D_{28}:C_6$ x 6, $Q_8\times F_7$ x 6, $C_{56}:C_6$ x 4, $C_{28}.C_{12}$ x 4, $C_{14}:C_{24}$ x 2, $C_2\times C_{14}:C_{12}$ x 2, $C_2^3.F_7$ x 2, $C_2\times C_{28}.C_6$ x 2, $C_{14}:C_{24}$ x 2, $C_2^2\times C_{14}:C_6$, $C_7:C_6\times Q_8$

Order 224: $D_4\times D_{14}$ x 20, $C_8:D_{14}$ x 16, $D_7\times \SD_{16}$ x 16, $C_8:D_{14}$ x 16, $D_7\times D_8$ x 16, $D_4:D_{14}$ x 12, $D_8:C_{14}$ x 8, $D_4:D_{14}$ x 8, $D_{28}:C_2^2$ x 8, $C_8:D_{14}$ x 8, $D_7\times \OD_{16}$ x 8, $C_{14}:D_8$ x 4, $C_2\times D_{56}$ x 2, $C_8:D_{14}$ x 2, $C_8:D_{14}$ x 2, $C_8\times D_{14}$ x 2, $D_4:D_{14}$ x 2, $D_{28}:C_2^2$ x 2, $C_{28}:C_2^3$ x 2, $C_{14}\times \SD_{16}$ x 2, $C_{14}\times D_8$ x 2, $Q_8:D_{14}$ x 2, $C_{14}:\SD_{16}$ x 2, $C_2^3\times D_{14}$, $C_{14}.C_2^4$, $C_{28}:C_2^3$, $C_2^3:D_{14}$, $Q_8:D_{14}$, $Q_8\times D_{14}$, $C_2^2\times D_{28}$, $C_{14}\times \OD_{16}$, $C_{14}:\OD_{16}$

Order 192: $C_{24}:C_2^3$ x 24, $C_{24}:C_2^3$ x 2, $C_{24}:C_2^3$ x 2, $C_{12}.C_2^4$, $C_{12}:C_2^4$, $C_{24}:C_2^3$

Order 168: $C_2^2\times F_7$ x 93, $C_4\times F_7$ x 28, $D_{14}:C_6$ x 28, $C_{28}:C_6$ x 17, $C_{28}:C_6$ x 17, $C_2\times C_{14}:C_6$ x 12, $C_{14}:C_{12}$ x 11, $C_{14}:C_{12}$ x 11, $C_{28}.C_6$ x 7, $C_7:C_{24}$ x 4, $C_7:C_{24}$ x 4, $C_{28}.C_6$ x 3

Order 128: $C_8:C_2^4$

Order 112: $D_4\times D_7$ x 54, $C_2^2\times D_{14}$ x 23, $C_4\times D_{14}$ x 23, $C_7:D_8$ x 16, $C_{14}:D_4$ x 15, $D_4:D_7$ x 14, $D_{28}:C_2$ x 14, $D_4\times C_{14}$ x 11, $C_2\times D_{28}$ x 11, $D_{56}$ x 8, $C_{56}:C_2$ x 8, $C_{56}:C_2$ x 8, $C_8\times D_7$ x 8, $C_7\times \SD_{16}$ x 8, $C_7\times D_8$ x 8, $Q_8:D_7$ x 8, $C_7:\SD_{16}$ x 8, $D_4:C_{14}$ x 6, $D_{28}:C_2$ x 6, $Q_8\times D_7$ x 6, $C_7:\OD_{16}$ x 4, $C_7\times \OD_{16}$ x 4, $C_{14}:C_8$ x 2, $C_2^2\times C_{28}$ x 2, $C_2^2.D_{14}$ x 2, $C_{14}:Q_8$ x 2, $C_2\times C_{56}$ x 2, $C_2^3\times C_{14}$, $Q_8\times C_{14}$

Order 96: $D_8:C_6$ x 64, $C_6\times \SD_{16}$ x 24, $C_6\times D_8$ x 24, $C_{12}:C_2^3$ x 23, $C_6.C_2^4$ x 18, $C_6\times \OD_{16}$ x 12, $C_2^3\times C_{12}$ x 2, $C_2^2\times C_{24}$ x 2, $C_2^4\times C_6$, $C_6.C_2^4$

Order 84: $C_2\times F_7$ x 78, $C_{14}:C_6$ x 25, $C_7:C_{12}$ x 6, $C_7:C_{12}$ x 6

Order 64: $C_8:C_2^3$ x 24, $C_2^2\times \SD_{16}$ x 2, $C_2^2\times D_8$ x 2, $D_4:C_2^3$, $D_4\times C_2^3$, $C_2^2\times \OD_{16}$

Order 56: $C_2\times D_{14}$ x 93, $C_7:D_4$ x 28, $C_4\times D_7$ x 28, $C_7\times D_4$ x 17, $D_{28}$ x 17, $C_2^2\times C_{14}$ x 12, $C_2\times C_{28}$ x 11, $C_{14}:C_4$ x 11, $C_7:Q_8$ x 7, $C_{56}$ x 4, $C_7:C_8$ x 4, $C_7\times Q_8$ x 3

Order 48: $C_6\times D_4$ x 91, $D_4:C_6$ x 40, $C_3\times \SD_{16}$ x 32, $C_3\times D_8$ x 32, $C_2^2\times C_{12}$ x 27, $C_2^3\times C_6$ x 24, $C_3\times \OD_{16}$ x 16, $C_2\times C_{24}$ x 12, $C_6\times Q_8$ x 9

Order 42: $F_7$ x 12, $C_7:C_6$ x 11

Order 32: $D_8:C_2$ x 64, $C_2\times \SD_{16}$ x 24, $C_2\times D_8$ x 24, $C_2^2\times D_4$ x 23, $D_4:C_2^2$ x 18, $C_2\times \OD_{16}$ x 12, $C_2^3\times C_4$ x 2, $C_2^2\times C_8$ x 2, $C_2^5$, $C_2^2\times Q_8$

Order 28: $D_{14}$ x 78, $C_2\times C_{14}$ x 25, $C_{28}$ x 6, $C_7:C_4$ x 6

Order 24: $C_2^2\times C_6$ x 105, $C_3\times D_4$ x 62, $C_2\times C_{12}$ x 50, $C_3\times Q_8$ x 10, $C_{24}$ x 8

Order 21: $C_7:C_3$

Order 16: $C_2\times D_4$ x 91, $D_4:C_2$ x 40, $\SD_{16}$ x 32, $D_8$ x 32, $C_2^2\times C_4$ x 27, $C_2^4$ x 24, $\OD_{16}$ x 16, $C_2\times C_8$ x 12, $C_2\times Q_8$ x 9

Order 14: $D_7$ x 12, $C_{14}$ x 11

Order 12: $C_2\times C_6$ x 103, $C_{12}$ x 12

Order 8: $C_2^3$ x 105, $D_4$ x 62, $C_2\times C_4$ x 50, $Q_8$ x 10, $C_8$ x 8

Order 7: $C_7$

Order 6: $C_6$ x 23

Order 4: $C_2^2$ x 103, $C_4$ x 12

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2688: $C_8:C_2^3\times F_7$

Order 1344: $(C_2^2\times D_{28}):C_6$, $D_{28}:(C_2^2\times C_6)$, $D_8:C_2\times F_7$, $C_2\times D_{56}:C_6$, $C_2\times \SD_{16}\times F_7$, $C_2\times D_8:F_7$, $C_2\times D_8\times F_7$, $C_2\times D_{56}:C_6$, $C_2\times \OD_{16}\times F_7$, $D_4:C_2^2\times F_7$, $C_2^2\times D_4\times F_7$, $(C_{14}\times D_8):C_6$

Order 896: $C_2\times C_8:C_2^2\times D_7$

Order 672: $C_2\times D_4\times F_7$ x 8, $C_2\times D_{28}:C_6$ x 2, $D_4:C_2\times F_7$ x 2, $C_2\times D_4:F_7$ x 2, $C_2\times D_{28}:C_6$ x 2, $C_4\times C_2^2\times F_7$ x 2, $C_{56}:(C_2\times C_6)$, $C_2\times C_{56}:C_6$, $C_2\times C_{56}:C_6$, $C_2\times C_{56}:C_6$, $(C_2\times D_{28}):C_6$, $C_2\times Q_8:F_7$, $C_2\times D_4.F_7$, $(D_4\times C_{14}):C_6$, $C_2\times C_{28}.C_{12}$, $D_{56}:C_6$, $\SD_{16}\times F_7$, $D_8:F_7$, $D_8\times F_7$, $D_{56}:C_6$, $\OD_{16}\times F_7$, $D_{56}:C_6$, $C_2\times C_{56}:C_6$, $C_2\times C_{56}:C_6$, $D_{14}:C_{24}$, $C_2^4\times F_7$, $(D_4\times C_{14}):C_6$, $C_{14}:C_6\times D_4$, $C_2^4:F_7$, $C_2\times D_{28}:C_6$, $C_2\times Q_8\times F_7$, $C_2\times D_{28}:C_6$

Order 448: $D_{28}:C_2^3$, $C_{28}:C_2^4$, $C_{56}:C_2^3$, $C_{28}.C_2^4$, $D_{28}:C_2^3$, $D_8:D_{14}$, $C_{56}:C_2^3$, $\SD_{16}\times D_{14}$, $D_8:D_{14}$, $D_8\times D_{14}$, $C_{56}:C_2^3$, $\OD_{16}\times D_{14}$

Order 384: $C_{24}:C_2^4$

Order 336: $D_{14}:C_{12}$ x 10, $D_4\times F_7$ x 9, $C_2^3\times F_7$ x 7, $C_2\times C_{28}:C_6$ x 6, $C_2^3:F_7$ x 6, $D_{28}:C_6$ x 6, $D_4:F_7$ x 3, $D_{28}:C_6$ x 3, $D_{28}:C_6$ x 2, $(C_2\times C_{28}):C_6$ x 2, $C_2\times C_{14}:C_{12}$ x 2, $C_2^3.F_7$ x 2, $D_{28}:C_6$ x 2, $Q_8\times F_7$ x 2, $C_2\times C_{28}.C_6$ x 2, $C_{56}:C_6$, $C_{56}:C_6$, $C_8\times F_7$, $C_{56}:C_6$, $C_{56}:C_6$, $C_{56}:C_6$, $C_{14}:C_{24}$, $C_2^2\times C_{14}:C_6$, $Q_8:F_7$, $D_4.F_7$, $C_7:C_6\times Q_8$, $C_{28}.C_{12}$, $C_{14}:C_{24}$, $C_{56}:C_6$

Order 224: $D_4\times D_{14}$ x 8, $D_4:D_{14}$ x 2, $D_4:D_{14}$ x 2, $D_{28}:C_2^2$ x 2, $C_{28}:C_2^3$ x 2, $C_{14}:D_8$ x 2, $C_2\times D_{56}$, $C_8:D_{14}$, $C_8:D_{14}$, $C_8\times D_{14}$, $C_2^3\times D_{14}$, $C_{14}.C_2^4$, $C_{28}:C_2^3$, $C_2^3:D_{14}$, $Q_8:D_{14}$, $Q_8\times D_{14}$, $C_2^2\times D_{28}$, $D_8:C_{14}$, $C_{14}\times \SD_{16}$, $C_{14}\times D_8$, $C_{14}\times \OD_{16}$, $D_4:D_{14}$, $Q_8:D_{14}$, $C_{14}:\SD_{16}$, $D_{28}:C_2^2$, $C_{14}:\OD_{16}$, $C_8:D_{14}$, $D_7\times \SD_{16}$, $C_8:D_{14}$, $D_7\times D_8$, $C_8:D_{14}$, $D_7\times \OD_{16}$

Order 192: $C_{24}:C_2^3$ x 7, $C_{12}.C_2^4$, $C_{12}:C_2^4$, $C_{24}:C_2^3$, $C_{24}:C_2^3$, $C_{24}:C_2^3$

Order 168: $C_2^2\times F_7$ x 15, $C_4\times F_7$ x 9, $C_{28}:C_6$ x 6, $C_{28}:C_6$ x 6, $C_{14}:C_{12}$ x 6, $C_2\times C_{14}:C_6$ x 5, $D_{14}:C_6$ x 5, $C_{14}:C_{12}$ x 5, $C_{28}.C_6$ x 3, $C_{28}.C_6$ x 2, $C_7:C_{24}$, $C_7:C_{24}$

Order 128: $C_8:C_2^4$

Order 112: $C_4\times D_{14}$ x 10, $D_4\times D_7$ x 9, $C_2^2\times D_{14}$ x 7, $D_4\times C_{14}$ x 6, $C_{14}:D_4$ x 6, $C_2\times D_{28}$ x 6, $D_4:D_7$ x 3, $D_{28}:C_2$ x 3, $D_4:C_{14}$ x 2, $C_2^2\times C_{28}$ x 2, $C_2^2.D_{14}$ x 2, $D_{28}:C_2$ x 2, $Q_8\times D_7$ x 2, $C_{14}:Q_8$ x 2, $C_7:D_8$ x 2, $C_7:\OD_{16}$, $C_{14}:C_8$, $D_{56}$, $C_{56}:C_2$, $C_2^3\times C_{14}$, $C_{56}:C_2$, $Q_8\times C_{14}$, $C_8\times D_7$, $C_7\times \SD_{16}$, $C_7\times D_8$, $C_7\times \OD_{16}$, $C_2\times C_{56}$, $Q_8:D_7$, $C_7:\SD_{16}$

Order 96: $C_{12}:C_2^3$ x 11, $C_6.C_2^4$ x 8, $D_8:C_6$ x 6, $C_6\times \SD_{16}$ x 5, $C_6\times D_8$ x 5, $C_6\times \OD_{16}$ x 4, $C_2^3\times C_{12}$ x 2, $C_2^4\times C_6$, $C_6.C_2^4$, $C_2^2\times C_{24}$

Order 84: $C_2\times F_7$ x 13, $C_{14}:C_6$ x 8, $C_7:C_{12}$ x 4, $C_7:C_{12}$ x 3

Order 64: $C_8:C_2^3$ x 7, $D_4:C_2^3$, $D_4\times C_2^3$, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_2^2\times \OD_{16}$

Order 56: $C_2\times D_{14}$ x 15, $C_4\times D_7$ x 9, $C_7\times D_4$ x 6, $C_2\times C_{28}$ x 6, $D_{28}$ x 6, $C_7:D_4$ x 5, $C_{14}:C_4$ x 5, $C_2^2\times C_{14}$ x 5, $C_7:Q_8$ x 3, $C_7\times Q_8$ x 2, $C_{56}$, $C_7:C_8$

Order 48: $C_6\times D_4$ x 27, $C_2^2\times C_{12}$ x 14, $D_4:C_6$ x 10, $C_2^3\times C_6$ x 8, $C_6\times Q_8$ x 5, $C_3\times \SD_{16}$ x 4, $C_3\times D_8$ x 4, $C_3\times \OD_{16}$ x 3, $C_2\times C_{24}$ x 3

Order 42: $C_7:C_6$ x 5, $F_7$ x 4

Order 32: $C_2^2\times D_4$ x 11, $D_4:C_2^2$ x 8, $D_8:C_2$ x 6, $C_2\times \SD_{16}$ x 5, $C_2\times D_8$ x 5, $C_2\times \OD_{16}$ x 4, $C_2^3\times C_4$ x 2, $C_2^5$, $C_2^2\times Q_8$, $C_2^2\times C_8$

Order 28: $D_{14}$ x 13, $C_2\times C_{14}$ x 8, $C_{28}$ x 4, $C_7:C_4$ x 3

Order 24: $C_2\times C_{12}$ x 20, $C_2^2\times C_6$ x 20, $C_3\times D_4$ x 17, $C_3\times Q_8$ x 5, $C_{24}$ x 2

Order 21: $C_7:C_3$

Order 16: $C_2\times D_4$ x 27, $C_2^2\times C_4$ x 14, $D_4:C_2$ x 10, $C_2^4$ x 8, $C_2\times Q_8$ x 5, $\SD_{16}$ x 4, $D_8$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3

Order 14: $C_{14}$ x 5, $D_7$ x 4

Order 12: $C_2\times C_6$ x 21, $C_{12}$ x 7

Order 8: $C_2^3$ x 20, $C_2\times C_4$ x 20, $D_4$ x 17, $Q_8$ x 5, $C_8$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 9

Order 4: $C_2^2$ x 21, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2688: $C_8:C_2^3\times F_7$ ($C_1$)

Order 1344: $D_8:C_2\times F_7$ ($C_2$) x 16, $C_2\times D_{56}:C_6$ ($C_2$) x 2, $C_2\times \SD_{16}\times F_7$ ($C_2$) x 2, $C_2\times D_8:F_7$ ($C_2$) x 2, $C_2\times D_8\times F_7$ ($C_2$) x 2, $(C_2^2\times D_{28}):C_6$ ($C_2$), $D_{28}:(C_2^2\times C_6)$ ($C_2$), $C_2\times D_{56}:C_6$ ($C_2$), $C_2\times \OD_{16}\times F_7$ ($C_2$), $D_4:C_2^2\times F_7$ ($C_2$), $C_2^2\times D_4\times F_7$ ($C_2$), $(C_{14}\times D_8):C_6$ ($C_2$)

Order 896: $C_2\times C_8:C_2^2\times D_7$ ($C_3$)

Order 672: $D_{56}:C_6$ ($C_2^2$) x 16, $\SD_{16}\times F_7$ ($C_2^2$) x 16, $D_8:F_7$ ($C_2^2$) x 16, $D_8\times F_7$ ($C_2^2$) x 16, $C_2\times D_4\times F_7$ ($C_2^2$) x 13, $C_{56}:(C_2\times C_6)$ ($C_2^2$) x 8, $(C_2\times D_{28}):C_6$ ($C_2^2$) x 8, $(D_4\times C_{14}):C_6$ ($C_2^2$) x 8, $D_{56}:C_6$ ($C_2^2$) x 8, $\OD_{16}\times F_7$ ($C_2^2$) x 8, $D_4:C_2\times F_7$ ($C_2^2$) x 8, $C_2\times D_{28}:C_6$ ($C_2^2$) x 4, $C_2\times C_{56}:C_6$ ($C_2^2$) x 2, $C_2\times C_{56}:C_6$ ($C_2^2$) x 2, $C_2\times Q_8:F_7$ ($C_2^2$) x 2, $C_2\times D_4.F_7$ ($C_2^2$) x 2, $D_{56}:C_6$ ($C_2^2$) x 2, $C_2\times C_{56}:C_6$ ($C_2^2$) x 2, $C_2\times C_{56}:C_6$ ($C_2^2$) x 2, $D_{14}:C_{24}$ ($C_2^2$) x 2, $C_2\times C_{56}:C_6$ ($C_2^2$), $C_2\times C_{28}.C_{12}$ ($C_2^2$), $(D_4\times C_{14}):C_6$ ($C_2^2$), $C_{14}:C_6\times D_4$ ($C_2^2$), $C_2\times D_{28}:C_6$ ($C_2^2$), $C_2\times Q_8\times F_7$ ($C_2^2$), $C_2\times D_4:F_7$ ($C_2^2$), $C_2\times D_{28}:C_6$ ($C_2^2$), $C_2\times D_{28}:C_6$ ($C_2^2$), $C_4\times C_2^2\times F_7$ ($C_2^2$)

Order 448: $D_8:D_{14}$ ($C_6$) x 16, $C_{56}:C_2^3$ ($C_6$) x 2, $\SD_{16}\times D_{14}$ ($C_6$) x 2, $D_8:D_{14}$ ($C_6$) x 2, $D_8\times D_{14}$ ($C_6$) x 2, $D_{28}:C_2^3$ ($C_6$), $C_{28}:C_2^4$ ($C_6$), $C_{56}:C_2^3$ ($C_6$), $C_{28}.C_2^4$ ($C_6$), $D_{28}:C_2^3$ ($C_6$), $C_{56}:C_2^3$ ($C_6$), $\OD_{16}\times D_{14}$ ($C_6$)

Order 336: $D_4\times F_7$ ($C_2^3$) x 20, $D_{28}:C_6$ ($C_2^3$) x 16, $C_{56}:C_6$ ($C_2^3$) x 8, $C_{56}:C_6$ ($C_2^3$) x 8, $C_8\times F_7$ ($C_2^3$) x 8, $C_{56}:C_6$ ($C_2^3$) x 8, $C_{56}:C_6$ ($C_2^3$) x 8, $Q_8:F_7$ ($C_2^3$) x 8, $D_4.F_7$ ($C_2^3$) x 8, $C_{56}:C_6$ ($C_2^3$) x 8, $C_2\times C_{28}:C_6$ ($C_2^3$) x 7, $D_{28}:C_6$ ($C_2^3$) x 7, $D_{14}:C_{12}$ ($C_2^3$) x 6, $D_{14}:C_{12}$ ($D_4$) x 6, $C_{56}:C_6$ ($C_2^3$) x 4, $(C_2\times C_{28}):C_6$ ($C_2^3$) x 4, $C_{28}.C_{12}$ ($C_2^3$) x 4, $D_{28}:C_6$ ($C_2^3$) x 4, $Q_8\times F_7$ ($C_2^3$) x 4, $D_4:F_7$ ($C_2^3$) x 4, $D_{28}:C_6$ ($C_2^3$) x 4, $C_{14}:C_{24}$ ($C_2^3$) x 2, $C_{14}:C_{24}$ ($C_2^3$) x 2, $C_2^3\times F_7$ ($D_4$), $C_7:C_6\times Q_8$ ($C_2^3$), $C_2\times C_{14}:C_{12}$ ($C_2^3$), $C_2^3.F_7$ ($D_4$), $C_2\times C_{28}.C_6$ ($C_2^3$)

Order 224: $C_8:D_{14}$ ($C_2\times C_6$) x 16, $D_7\times \SD_{16}$ ($C_2\times C_6$) x 16, $C_8:D_{14}$ ($C_2\times C_6$) x 16, $D_7\times D_8$ ($C_2\times C_6$) x 16, $D_4\times D_{14}$ ($C_2\times C_6$) x 13, $D_4:D_{14}$ ($C_2\times C_6$) x 8, $D_8:C_{14}$ ($C_2\times C_6$) x 8, $D_4:D_{14}$ ($C_2\times C_6$) x 8, $D_{28}:C_2^2$ ($C_2\times C_6$) x 8, $C_8:D_{14}$ ($C_2\times C_6$) x 8, $D_7\times \OD_{16}$ ($C_2\times C_6$) x 8, $C_{14}:D_8$ ($C_2\times C_6$) x 4, $C_2\times D_{56}$ ($C_2\times C_6$) x 2, $C_8:D_{14}$ ($C_2\times C_6$) x 2, $C_8:D_{14}$ ($C_2\times C_6$) x 2, $C_8\times D_{14}$ ($C_2\times C_6$) x 2, $C_{14}\times \SD_{16}$ ($C_2\times C_6$) x 2, $C_{14}\times D_8$ ($C_2\times C_6$) x 2, $Q_8:D_{14}$ ($C_2\times C_6$) x 2, $C_{14}:\SD_{16}$ ($C_2\times C_6$) x 2, $C_{14}.C_2^4$ ($C_2\times C_6$), $C_{28}:C_2^3$ ($C_2\times C_6$), $Q_8:D_{14}$ ($C_2\times C_6$), $Q_8\times D_{14}$ ($C_2\times C_6$), $D_4:D_{14}$ ($C_2\times C_6$), $D_{28}:C_2^2$ ($C_2\times C_6$), $C_2^2\times D_{28}$ ($C_2\times C_6$), $C_{28}:C_2^3$ ($C_2\times C_6$), $C_{14}\times \OD_{16}$ ($C_2\times C_6$), $C_{14}:\OD_{16}$ ($C_2\times C_6$)

Order 168: $C_4\times F_7$ ($C_2\times D_4$) x 12, $C_{28}:C_6$ ($C_2^4$) x 6, $C_2^2\times F_7$ ($C_2\times D_4$) x 6, $C_{28}:C_6$ ($C_2^4$) x 6, $C_{14}:C_{12}$ ($C_2\times D_4$) x 6, $C_4\times F_7$ ($C_2^4$) x 4, $C_7:C_{24}$ ($C_2^4$) x 4, $C_7:C_{24}$ ($C_2^4$) x 4, $C_{14}:C_{12}$ ($C_2^4$) x 3, $C_{14}:C_{12}$ ($C_2\times D_4$) x 3, $C_{28}.C_6$ ($C_2^4$) x 2, $C_{28}.C_6$ ($C_2^4$) x 2, $C_2\times C_{14}:C_6$ ($C_2\times D_4$)

Order 112: $D_4\times D_7$ ($C_2^2\times C_6$) x 20, $C_7:D_8$ ($C_2^2\times C_6$) x 16, $D_{56}$ ($C_2^2\times C_6$) x 8, $C_{56}:C_2$ ($C_2^2\times C_6$) x 8, $C_{56}:C_2$ ($C_2^2\times C_6$) x 8, $C_8\times D_7$ ($C_2^2\times C_6$) x 8, $C_7\times \SD_{16}$ ($C_2^2\times C_6$) x 8, $C_7\times D_8$ ($C_2^2\times C_6$) x 8, $Q_8:D_7$ ($C_2^2\times C_6$) x 8, $C_7:\SD_{16}$ ($C_2^2\times C_6$) x 8, $D_4\times C_{14}$ ($C_2^2\times C_6$) x 7, $C_2\times D_{28}$ ($C_2^2\times C_6$) x 7, $C_4\times D_{14}$ ($C_2^2\times C_6$) x 6, $C_4\times D_{14}$ ($C_3\times D_4$) x 6, $C_7:\OD_{16}$ ($C_2^2\times C_6$) x 4, $D_4:C_{14}$ ($C_2^2\times C_6$) x 4, $D_{28}:C_2$ ($C_2^2\times C_6$) x 4, $Q_8\times D_7$ ($C_2^2\times C_6$) x 4, $D_4:D_7$ ($C_2^2\times C_6$) x 4, $D_{28}:C_2$ ($C_2^2\times C_6$) x 4, $C_7\times \OD_{16}$ ($C_2^2\times C_6$) x 4, $C_{14}:C_8$ ($C_2^2\times C_6$) x 2, $C_2\times C_{56}$ ($C_2^2\times C_6$) x 2, $C_2^2\times D_{14}$ ($C_3\times D_4$), $Q_8\times C_{14}$ ($C_2^2\times C_6$), $C_2^2\times C_{28}$ ($C_2^2\times C_6$), $C_2^2.D_{14}$ ($C_3\times D_4$), $C_{14}:Q_8$ ($C_2^2\times C_6$)

Order 84: $C_2\times F_7$ ($C_2^2\times D_4$) x 4, $C_2\times F_7$ ($D_8:C_2$) x 4, $C_7:C_{12}$ ($C_2^2\times D_4$) x 4, $C_{14}:C_6$ ($C_2^2\times D_4$) x 3, $C_7:C_{12}$ ($C_2^2\times D_4$) x 3, $C_7:C_{12}$ ($C_2^5$)

Order 64: $C_8:C_2^3$ ($F_7$)

Order 56: $C_4\times D_7$ ($C_6\times D_4$) x 12, $C_7\times D_4$ ($C_2^3\times C_6$) x 6, $C_{14}:C_4$ ($C_6\times D_4$) x 6, $D_{28}$ ($C_2^3\times C_6$) x 6, $C_2\times D_{14}$ ($C_6\times D_4$) x 6, $C_4\times D_7$ ($C_2^3\times C_6$) x 4, $C_{56}$ ($C_2^3\times C_6$) x 4, $C_7:C_8$ ($C_2^3\times C_6$) x 4, $C_2\times C_{28}$ ($C_2^3\times C_6$) x 3, $C_2\times C_{28}$ ($C_6\times D_4$) x 3, $C_7:Q_8$ ($C_2^3\times C_6$) x 2, $C_7\times Q_8$ ($C_2^3\times C_6$) x 2, $C_2^2\times C_{14}$ ($C_6\times D_4$)

Order 42: $F_7$ ($C_8:C_2^3$) x 4, $C_7:C_6$ ($C_8:C_2^3$) x 2, $C_7:C_6$ ($D_4\times C_2^3$)

Order 32: $D_8:C_2$ ($C_2\times F_7$) x 8, $C_2\times \SD_{16}$ ($C_2\times F_7$) x 2, $C_2\times D_8$ ($C_2\times F_7$) x 2, $D_4:C_2^2$ ($C_2\times F_7$), $C_2^2\times D_4$ ($C_2\times F_7$), $C_2\times \OD_{16}$ ($C_2\times F_7$)

Order 28: $D_{14}$ ($C_{12}:C_2^3$) x 4, $D_{14}$ ($D_8:C_6$) x 4, $C_7:C_4$ ($C_{12}:C_2^3$) x 4, $C_2\times C_{14}$ ($C_{12}:C_2^3$) x 3, $C_{28}$ ($C_{12}:C_2^3$) x 3, $C_{28}$ ($C_2^4\times C_6$)

Order 21: $C_7:C_3$ ($C_8:C_2^4$)

Order 16: $\SD_{16}$ ($C_2^2\times F_7$) x 8, $D_8$ ($C_2^2\times F_7$) x 8, $C_2\times D_4$ ($C_2^2\times F_7$) x 7, $\OD_{16}$ ($C_2^2\times F_7$) x 4, $D_4:C_2$ ($C_2^2\times F_7$) x 4, $C_2\times C_8$ ($C_2^2\times F_7$) x 2, $C_2\times Q_8$ ($C_2^2\times F_7$), $C_2^2\times C_4$ ($C_2^2\times F_7$)

Order 14: $D_7$ ($C_{24}:C_2^3$) x 4, $C_{14}$ ($C_{24}:C_2^3$) x 2, $C_{14}$ ($C_{12}:C_2^4$)

Order 8: $D_4$ ($C_2^3\times F_7$) x 6, $C_8$ ($C_2^3\times F_7$) x 4, $C_2\times C_4$ ($C_2^3\times F_7$) x 3, $C_2\times C_4$ ($D_4\times F_7$) x 3, $Q_8$ ($C_2^3\times F_7$) x 2, $C_2^3$ ($D_4\times F_7$)

Order 7: $C_7$ ($C_{24}:C_2^4$)

Order 4: $C_2^2$ ($C_2\times D_4\times F_7$) x 3, $C_4$ ($C_2\times D_4\times F_7$) x 3, $C_4$ ($C_2^4\times F_7$)

Order 2: $C_2$ ($D_8:C_2\times F_7$) x 2, $C_2$ ($C_2^2\times D_4\times F_7$)

Order 1: $C_1$ ($C_8:C_2^3\times F_7$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2688: $C_8:C_2^3\times F_7$ ($C_1$)

Order 1344: $(C_2^2\times D_{28}):C_6$ ($C_2$), $D_{28}:(C_2^2\times C_6)$ ($C_2$), $D_8:C_2\times F_7$ ($C_2$), $C_2\times D_{56}:C_6$ ($C_2$), $C_2\times \SD_{16}\times F_7$ ($C_2$), $C_2\times D_8:F_7$ ($C_2$), $C_2\times D_8\times F_7$ ($C_2$), $C_2\times D_{56}:C_6$ ($C_2$), $C_2\times \OD_{16}\times F_7$ ($C_2$), $D_4:C_2^2\times F_7$ ($C_2$), $C_2^2\times D_4\times F_7$ ($C_2$), $(C_{14}\times D_8):C_6$ ($C_2$)

Order 896: $C_2\times C_8:C_2^2\times D_7$ ($C_3$)

Order 672: $C_2\times D_4\times F_7$ ($C_2^2$) x 4, $C_2\times D_{28}:C_6$ ($C_2^2$) x 2, $C_{56}:(C_2\times C_6)$ ($C_2^2$), $C_2\times C_{56}:C_6$ ($C_2^2$), $C_2\times C_{56}:C_6$ ($C_2^2$), $C_2\times C_{56}:C_6$ ($C_2^2$), $(C_2\times D_{28}):C_6$ ($C_2^2$), $C_2\times Q_8:F_7$ ($C_2^2$), $C_2\times D_4.F_7$ ($C_2^2$), $(D_4\times C_{14}):C_6$ ($C_2^2$), $C_2\times C_{28}.C_{12}$ ($C_2^2$), $D_{56}:C_6$ ($C_2^2$), $\SD_{16}\times F_7$ ($C_2^2$), $D_8:F_7$ ($C_2^2$), $D_8\times F_7$ ($C_2^2$), $D_{56}:C_6$ ($C_2^2$), $\OD_{16}\times F_7$ ($C_2^2$), $D_{56}:C_6$ ($C_2^2$), $C_2\times C_{56}:C_6$ ($C_2^2$), $C_2\times C_{56}:C_6$ ($C_2^2$), $D_{14}:C_{24}$ ($C_2^2$), $(D_4\times C_{14}):C_6$ ($C_2^2$), $C_{14}:C_6\times D_4$ ($C_2^2$), $D_4:C_2\times F_7$ ($C_2^2$), $C_2\times D_{28}:C_6$ ($C_2^2$), $C_2\times Q_8\times F_7$ ($C_2^2$), $C_2\times D_4:F_7$ ($C_2^2$), $C_2\times D_{28}:C_6$ ($C_2^2$), $C_2\times D_{28}:C_6$ ($C_2^2$), $C_4\times C_2^2\times F_7$ ($C_2^2$)

Order 448: $D_{28}:C_2^3$ ($C_6$), $C_{28}:C_2^4$ ($C_6$), $C_{56}:C_2^3$ ($C_6$), $C_{28}.C_2^4$ ($C_6$), $D_{28}:C_2^3$ ($C_6$), $D_8:D_{14}$ ($C_6$), $C_{56}:C_2^3$ ($C_6$), $\SD_{16}\times D_{14}$ ($C_6$), $D_8:D_{14}$ ($C_6$), $D_8\times D_{14}$ ($C_6$), $C_{56}:C_2^3$ ($C_6$), $\OD_{16}\times D_{14}$ ($C_6$)

Order 336: $C_2\times C_{28}:C_6$ ($C_2^3$) x 3, $D_4\times F_7$ ($C_2^3$) x 3, $D_{28}:C_6$ ($C_2^3$) x 3, $D_{14}:C_{12}$ ($C_2^3$) x 3, $D_{14}:C_{12}$ ($D_4$) x 3, $D_{28}:C_6$ ($C_2^3$) x 2, $C_{56}:C_6$ ($C_2^3$), $C_{56}:C_6$ ($C_2^3$), $C_8\times F_7$ ($C_2^3$), $C_{56}:C_6$ ($C_2^3$), $C_{56}:C_6$ ($C_2^3$), $C_{56}:C_6$ ($C_2^3$), $C_{14}:C_{24}$ ($C_2^3$), $C_2^3\times F_7$ ($D_4$), $Q_8:F_7$ ($C_2^3$), $D_4.F_7$ ($C_2^3$), $(C_2\times C_{28}):C_6$ ($C_2^3$), $C_7:C_6\times Q_8$ ($C_2^3$), $C_2\times C_{14}:C_{12}$ ($C_2^3$), $C_{28}.C_{12}$ ($C_2^3$), $C_2^3.F_7$ ($D_4$), $D_{28}:C_6$ ($C_2^3$), $Q_8\times F_7$ ($C_2^3$), $D_4:F_7$ ($C_2^3$), $D_{28}:C_6$ ($C_2^3$), $C_2\times C_{28}.C_6$ ($C_2^3$), $C_{14}:C_{24}$ ($C_2^3$), $C_{56}:C_6$ ($C_2^3$)

Order 224: $D_4\times D_{14}$ ($C_2\times C_6$) x 4, $C_{14}:D_8$ ($C_2\times C_6$) x 2, $C_2\times D_{56}$ ($C_2\times C_6$), $C_8:D_{14}$ ($C_2\times C_6$), $C_8:D_{14}$ ($C_2\times C_6$), $C_8\times D_{14}$ ($C_2\times C_6$), $C_{14}.C_2^4$ ($C_2\times C_6$), $C_{28}:C_2^3$ ($C_2\times C_6$), $D_4:D_{14}$ ($C_2\times C_6$), $Q_8:D_{14}$ ($C_2\times C_6$), $Q_8\times D_{14}$ ($C_2\times C_6$), $D_4:D_{14}$ ($C_2\times C_6$), $D_{28}:C_2^2$ ($C_2\times C_6$), $C_2^2\times D_{28}$ ($C_2\times C_6$), $C_{28}:C_2^3$ ($C_2\times C_6$), $D_8:C_{14}$ ($C_2\times C_6$), $C_{14}\times \SD_{16}$ ($C_2\times C_6$), $C_{14}\times D_8$ ($C_2\times C_6$), $C_{14}\times \OD_{16}$ ($C_2\times C_6$), $D_4:D_{14}$ ($C_2\times C_6$), $Q_8:D_{14}$ ($C_2\times C_6$), $C_{14}:\SD_{16}$ ($C_2\times C_6$), $D_{28}:C_2^2$ ($C_2\times C_6$), $C_{14}:\OD_{16}$ ($C_2\times C_6$), $C_8:D_{14}$ ($C_2\times C_6$), $D_7\times \SD_{16}$ ($C_2\times C_6$), $C_8:D_{14}$ ($C_2\times C_6$), $D_7\times D_8$ ($C_2\times C_6$), $C_8:D_{14}$ ($C_2\times C_6$), $D_7\times \OD_{16}$ ($C_2\times C_6$)

Order 168: $C_4\times F_7$ ($C_2\times D_4$) x 4, $C_2^2\times F_7$ ($C_2\times D_4$) x 3, $C_{14}:C_{12}$ ($C_2\times D_4$) x 3, $C_{28}:C_6$ ($C_2^4$) x 2, $C_4\times F_7$ ($C_2^4$) x 2, $C_{28}:C_6$ ($C_2^4$) x 2, $C_{14}:C_{12}$ ($C_2^4$) x 2, $C_{14}:C_{12}$ ($C_2\times D_4$) x 2, $C_{28}.C_6$ ($C_2^4$), $C_2\times C_{14}:C_6$ ($C_2\times D_4$), $C_{28}.C_6$ ($C_2^4$), $C_7:C_{24}$ ($C_2^4$), $C_7:C_{24}$ ($C_2^4$)

Order 112: $D_4\times C_{14}$ ($C_2^2\times C_6$) x 3, $D_4\times D_7$ ($C_2^2\times C_6$) x 3, $C_2\times D_{28}$ ($C_2^2\times C_6$) x 3, $C_4\times D_{14}$ ($C_2^2\times C_6$) x 3, $C_4\times D_{14}$ ($C_3\times D_4$) x 3, $C_7:D_8$ ($C_2^2\times C_6$) x 2, $C_7:\OD_{16}$ ($C_2^2\times C_6$), $C_{14}:C_8$ ($C_2^2\times C_6$), $D_{56}$ ($C_2^2\times C_6$), $C_{56}:C_2$ ($C_2^2\times C_6$), $C_2^2\times D_{14}$ ($C_3\times D_4$), $D_4:C_{14}$ ($C_2^2\times C_6$), $C_{56}:C_2$ ($C_2^2\times C_6$), $Q_8\times C_{14}$ ($C_2^2\times C_6$), $C_2^2\times C_{28}$ ($C_2^2\times C_6$), $C_2^2.D_{14}$ ($C_3\times D_4$), $D_{28}:C_2$ ($C_2^2\times C_6$), $Q_8\times D_7$ ($C_2^2\times C_6$), $D_4:D_7$ ($C_2^2\times C_6$), $D_{28}:C_2$ ($C_2^2\times C_6$), $C_8\times D_7$ ($C_2^2\times C_6$), $C_{14}:Q_8$ ($C_2^2\times C_6$), $C_7\times \SD_{16}$ ($C_2^2\times C_6$), $C_7\times D_8$ ($C_2^2\times C_6$), $C_7\times \OD_{16}$ ($C_2^2\times C_6$), $C_2\times C_{56}$ ($C_2^2\times C_6$), $Q_8:D_7$ ($C_2^2\times C_6$), $C_7:\SD_{16}$ ($C_2^2\times C_6$)

Order 84: $C_{14}:C_6$ ($C_2^2\times D_4$) x 2, $C_2\times F_7$ ($C_2^2\times D_4$) x 2, $C_7:C_{12}$ ($C_2^2\times D_4$) x 2, $C_7:C_{12}$ ($C_2^2\times D_4$) x 2, $C_2\times F_7$ ($D_8:C_2$), $C_7:C_{12}$ ($C_2^5$)

Order 64: $C_8:C_2^3$ ($F_7$)

Order 56: $C_4\times D_7$ ($C_6\times D_4$) x 4, $C_{14}:C_4$ ($C_6\times D_4$) x 3, $C_2\times D_{14}$ ($C_6\times D_4$) x 3, $C_7\times D_4$ ($C_2^3\times C_6$) x 2, $C_2\times C_{28}$ ($C_2^3\times C_6$) x 2, $C_2\times C_{28}$ ($C_6\times D_4$) x 2, $D_{28}$ ($C_2^3\times C_6$) x 2, $C_4\times D_7$ ($C_2^3\times C_6$) x 2, $C_7:Q_8$ ($C_2^3\times C_6$), $C_{56}$ ($C_2^3\times C_6$), $C_2^2\times C_{14}$ ($C_6\times D_4$), $C_7\times Q_8$ ($C_2^3\times C_6$), $C_7:C_8$ ($C_2^3\times C_6$)

Order 42: $C_7:C_6$ ($D_4\times C_2^3$), $C_7:C_6$ ($C_8:C_2^3$), $F_7$ ($C_8:C_2^3$)

Order 32: $D_4:C_2^2$ ($C_2\times F_7$), $C_2^2\times D_4$ ($C_2\times F_7$), $D_8:C_2$ ($C_2\times F_7$), $C_2\times \SD_{16}$ ($C_2\times F_7$), $C_2\times D_8$ ($C_2\times F_7$), $C_2\times \OD_{16}$ ($C_2\times F_7$)

Order 28: $C_2\times C_{14}$ ($C_{12}:C_2^3$) x 2, $D_{14}$ ($C_{12}:C_2^3$) x 2, $C_{28}$ ($C_{12}:C_2^3$) x 2, $C_7:C_4$ ($C_{12}:C_2^3$) x 2, $D_{14}$ ($D_8:C_6$), $C_{28}$ ($C_2^4\times C_6$)

Order 21: $C_7:C_3$ ($C_8:C_2^4$)

Order 16: $C_2\times D_4$ ($C_2^2\times F_7$) x 3, $\SD_{16}$ ($C_2^2\times F_7$), $D_8$ ($C_2^2\times F_7$), $\OD_{16}$ ($C_2^2\times F_7$), $C_2\times C_8$ ($C_2^2\times F_7$), $D_4:C_2$ ($C_2^2\times F_7$), $C_2\times Q_8$ ($C_2^2\times F_7$), $C_2^2\times C_4$ ($C_2^2\times F_7$)

Order 14: $C_{14}$ ($C_{12}:C_2^4$), $C_{14}$ ($C_{24}:C_2^3$), $D_7$ ($C_{24}:C_2^3$)

Order 8: $D_4$ ($C_2^3\times F_7$) x 2, $C_2\times C_4$ ($C_2^3\times F_7$) x 2, $C_2\times C_4$ ($D_4\times F_7$) x 2, $C_2^3$ ($D_4\times F_7$), $Q_8$ ($C_2^3\times F_7$), $C_8$ ($C_2^3\times F_7$)

Order 7: $C_7$ ($C_{24}:C_2^4$)

Order 4: $C_2^2$ ($C_2\times D_4\times F_7$) x 2, $C_4$ ($C_2\times D_4\times F_7$) x 2, $C_4$ ($C_2^4\times F_7$)

Order 2: $C_2$ ($D_8:C_2\times F_7$), $C_2$ ($C_2^2\times D_4\times F_7$)

Order 1: $C_1$ ($C_8:C_2^3\times F_7$)

Series

Derived series

$C_8:C_2^3\times F_7$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$C_8:C_2^3\times F_7$

$\rhd$

$C_2\times \OD_{16}\times F_7$

$\rhd$

$D_{14}:C_{24}$

$\rhd$

$C_{14}:C_{24}$

$\rhd$

$C_2\times C_{56}$

$\rhd$

$C_{56}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_8:C_2^3\times F_7$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_8:C_2^3$

Supergroups

Character theory

Complex character table

See the $154 \times 154$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $110 \times 110$ rational character table.