Abstract group 5376.cn: $C_2\wr D_4\times F_7$

• Label: 5376.cn
• Order: \( 2^{8} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $15$
• Trans deg.: $56$
• Rank: $4$

Group information

Description:	$C_2\wr D_4\times F_7$
Order:	\(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$(C_{14}\times D_4).C_6.C_2^6$, of order \(43008\)\(\medspace = 2^{11} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 8, $C_3$, $C_7$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	7	8	12	14	24	28	56
Elements	1	351	14	544	1218	6	128	1904	258	448	408	96	5376
Conjugacy classes	1	21	2	16	42	1	2	32	10	4	8	1	140
Divisions	1	21	1	16	21	1	2	16	10	2	8	1	100
Autjugacy classes	1	12	2	10	24	1	2	20	6	4	5	1	88

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	48	36	36	8	0	6	6	140
Irr. rational chars.	16	28	24	8	12	6	6	100

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$56$
Rank:	$4$
Inequivalent generating quadruples:	$5883494400$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{12}=c^{4}=d^{2}=e^{28}=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(8,15)(10,12), (12,15), (9,14)(11,13), (2,7)(3,4)(5,6)(8,15)(10,12), (9,13) \!\cdots\! \rangle$
Direct product:	$F_7$ $\, \times\, $ $(C_2\wr D_4)$
Semidirect product:	$(D_7\times D_4^2)$ $\,\rtimes\,$ $C_6$	$(D_4^2:D_7)$ $\,\rtimes\,$ $C_6$	$D_7$ $\,\rtimes\,$ $(D_4^2:C_6)$	$(C_2^4\times F_7)$ $\,\rtimes\,$ $D_4$	all 113
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4:D_4\times F_7)$ . $C_2^2$	$(C_2\times D_4\times F_7)$ . $C_2^3$	$C_2^3$ . $(C_2\times D_4\times F_7)$	$(D_{28}:C_6)$ . $(C_2\times D_4)$	all 45
Aut. group:	$\Aut(C_{28}.D_4)$	$\Aut(C_{28}.D_4)$	$\Aut(D_4:D_{14})$	$\Aut((D_4\times C_{14}).C_6)$	all 8

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

Homology

Abelianization:	$C_{2}^{3} \times C_{6} \simeq C_{2}^{4} \times C_{3}$
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 51824 subgroups in 3912 conjugacy classes, 250 normal (104 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\wr C_2^2\times F_7$
Commutator:	$G' \simeq$ $D_4\times C_{14}$	$G/G' \simeq$ $C_2^3\times C_6$
Frattini:	$\Phi \simeq$ $C_2\times D_4$	$G/\Phi \simeq$ $C_2^3\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $D_4^2:C_{14}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_2\wr D_4\times F_7$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_2\wr C_2^2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:C_2^2$
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: 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 5376: $C_2\wr D_4\times F_7$

Order 2688: $C_2\wr C_2^2\times F_7$ x 2, $(C_{14}\times D_4).C_6.C_2^2$ x 2, $(C_{14}\times D_4).C_6.C_2^2$ x 2, $C_2\wr C_4\times F_7$ x 2, $D_4^2\times F_7$, $C_4^2:C_4\times F_7$, $(C_{14}\times D_4).C_6.C_2^2$, $(C_{14}\times D_4).C_6.C_2^2$, $D_4:D_4\times F_7$, $C_4^2:C_4.C_2\times C_7:C_3$, $(C_{14}\times D_4).C_6.C_2^2$

Order 1792: $D_4^2:D_{14}$

Order 1344: $C_2^2\wr C_2\times F_7$ x 5, $C_2^3:C_4\times F_7$ x 4, $(C_2^2\times F_7):D_4$ x 4, $C_2^4:(C_2\times F_7)$ x 4, $D_{14}:(C_6\times D_4)$ x 3, $(C_2^3\times C_{14}):C_{12}$ x 2, $D_4\times D_{14}:C_6$ x 2, $C_{28}:(C_6\times D_4)$ x 2, $C_4:D_4\times F_7$ x 2, $C_{14}:C_{12}:D_4$ x 2, $(C_2^3\times C_{14}):C_{12}$ x 2, $(C_2^2\times D_{14}):C_{12}$ x 2, $(C_2\times D_{28}):C_{12}$ x 2, $(C_2^2\times D_{14}):C_{12}$ x 2, $(C_2^3\times F_7):C_2^2$ x 2, $C_2^2\times D_4\times F_7$ x 2, $C_{28}:C_6\times D_4$, $D_{28}:(C_3\times D_4)$, $\OD_{16}:C_2\times F_7$, $D_8:C_2\times F_7$, $C_{28}:(C_6\times D_4)$, $C_4:D_4\times F_7$, $C_{28}:C_6\times D_4$, $C_4\times D_4\times F_7$, $C_{28}.(C_6\times D_4)$, $(C_4\times C_{28}):C_{12}$, $C_{28}.(C_6\times D_4)$, $D_{28}:(C_3\times D_4)$, $D_{28}:(C_3\times D_4)$, $D_4:C_2^2\times F_7$, $(C_4\times C_{28}):C_{12}$, $D_{28}:(C_3\times D_4)$, $C_4\wr C_2\times F_7$

Order 896: $(C_2\times D_{28}):D_4$ x 2, $C_2\wr C_4\times D_7$ x 2, $C_2^4:D_{28}$ x 2, $C_2\wr C_2^2\times D_7$ x 2, $D_4^2:C_{14}$, $D_4^2:D_7$, $(C_4\times C_{28}):D_4$, $D_7\times D_4^2$, $C_4^2:C_4\times D_7$, $C_4^2:D_{28}$, $D_4:D_4\times D_7$

Order 768: $C_2\wr D_4\times C_6$

Order 672: $C_2\times D_4\times F_7$ x 38, $(D_4\times C_{14}):C_6$ x 11, $(C_2\times D_{28}):C_6$ x 11, $C_2^2:C_4\times F_7$ x 9, $(C_2\times D_{14}):C_{12}$ x 6, $(D_4\times C_{14}):C_6$ x 5, $C_2^4:F_7$ x 5, $(C_2\times C_{28}):C_{12}$ x 4, $(C_2\times D_{28}):C_6$ x 4, $(D_4\times C_{14}):C_6$ x 4, $(C_2\times D_{28}):C_6$ x 4, $(C_2\times C_{28}):C_{12}$ x 4, $C_2^4:F_7$ x 4, $(D_4\times C_{14}):C_6$ x 4, $C_2^4\times F_7$ x 3, $(D_4\times C_{14}):C_6$ x 2, $(C_4\times D_{14}):C_6$ x 2, $(C_2\times D_{28}):C_6$ x 2, $C_4\times D_{14}:C_6$ x 2, $(C_2\times D_{28}):C_6$ x 2, $C_7:(D_4\times C_{12})$ x 2, $C_{14}:C_6\times D_4$ x 2, $(C_2\times D_{28}):C_6$ x 2, $D_4:C_2\times F_7$ x 2, $C_2\times D_{28}:C_6$ x 2, $C_4\times C_2^2\times F_7$ x 2, $C_{56}:(C_2\times C_6)$, $(C_4\times C_{28}):C_6$, $C_4\times C_{28}:C_6$, $(C_7\times D_4):C_{12}$, $(D_4\times C_{14}).C_6$, $(C_2\times D_{28}):C_6$, $D_4\times C_7:C_{12}$, $(D_4\times C_{14}):C_6$, $D_{28}:C_{12}$, $D_{56}:C_6$, $\SD_{16}\times F_7$, $\OD_{16}:F_7$, $D_8:F_7$, $D_8\times F_7$, $D_{56}:C_6$, $\OD_{16}\times F_7$, $D_{28}:C_{12}$, $C_4:C_4\times F_7$, $C_4^2:F_7$, $D_{28}:C_{12}$, $C_4^2\times F_7$, $D_{28}:C_{12}$, $(C_4\times C_{28}):C_6$, $(C_7\times \OD_{16}):C_6$, $(D_4\times C_{14}):C_6$

Order 448: $C_2^4:D_{14}$ x 5, $C_2^4:D_{14}$ x 4, $C_2^3:C_4\times D_7$ x 4, $C_2^3:D_{28}$ x 4, $C_2^4:D_{14}$ x 3, $(C_2^3\times C_{14}):C_4$ x 2, $C_2\wr C_2^2\times C_7$ x 2, $(C_2\times C_{28}):D_4$ x 2, $C_2^3.D_{28}$ x 2, $C_2^3.D_{28}$ x 2, $(C_2^2\times D_{14}):C_4$ x 2, $C_2^4:C_{28}$ x 2, $C_{28}:C_2^4$ x 2, $C_2^4:D_{14}$ x 2, $D_{28}:D_4$ x 2, $C_4:D_4\times D_7$ x 2, $C_4^2:D_{14}$, $C_7\times D_4:D_4$, $(C_7\times D_4):D_4$, $D_{28}:D_4$, $C_4^2:D_{14}$, $D_4:D_{28}$, $C_4^2:D_{14}$, $D_{28}:D_4$, $\OD_{16}:D_{14}$, $C_4^2:C_{28}$, $D_{28}:C_2^3$, $C_7\times D_4^2$, $D_8:D_{14}$, $C_4^2:D_{14}$, $C_4^2:D_{14}$, $(C_4\times C_{28}):C_4$, $D_4\times D_{28}$

Order 384: $D_4^2:C_6$ x 8, $C_2^5:C_{12}$ x 2, $C_2\wr C_2^2\times C_6$ x 2, $C_2\times C_4^2:C_{12}$, $C_6\times D_4^2$, $C_6\times D_4:D_4$

Order 336: $D_4\times F_7$ x 88, $C_2^3:F_7$ x 43, $C_2^3\times F_7$ x 36, $C_2\times C_{28}:C_6$ x 22, $D_{28}:C_6$ x 22, $D_{14}:C_{12}$ x 21, $D_{14}:C_{12}$ x 14, $(C_2\times C_{14}):C_{12}$ x 9, $C_2^3.F_7$ x 9, $D_4:F_7$ x 5, $D_{28}:C_6$ x 5, $C_2^2\times C_{14}:C_6$ 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, $C_{14}.(C_2\times C_{12})$ x 2, $C_{28}:C_{12}$ x 2, $C_2^3.F_7$ x 2, $D_{28}:C_6$ x 2, $C_{56}:C_6$, $C_{56}:C_6$, $C_8\times F_7$, $C_{56}:C_6$, $C_{56}:C_6$, $C_{56}:C_6$, $C_{28}:C_{12}$, $C_{28}:C_{12}$, $Q_8:F_7$, $D_4.F_7$, $C_{28}:C_{12}$, $C_{28}.C_{12}$, $Q_8\times F_7$, $C_{56}:C_6$

Order 256: $D_4^2:C_2^2$

Order 224: $D_4\times D_{14}$ x 38, $C_2^2:D_{28}$ x 11, $D_{14}:D_4$ x 11, $D_{14}.D_4$ x 9, $C_2^2.D_{28}$ x 6, $C_2^4:C_{14}$ x 5, $C_2^4:D_7$ x 5, $D_{14}:D_4$ x 4, $C_2^3:C_{28}$ x 4, $C_2^3.D_{14}$ x 4, $C_2^3:D_{14}$ x 4, $D_4:D_{14}$ x 4, $C_{28}:D_4$ x 4, $C_2^3.D_{14}$ x 4, $C_2^3\times D_{14}$ x 3, $C_4:D_{28}$ x 2, $C_2^3.D_{14}$ x 2, $C_{28}:C_2^3$ x 2, $D_4:D_{14}$ x 2, $D_4:D_{14}$ x 2, $C_2^2\times D_{28}$ x 2, $C_{28}:C_2^3$ x 2, $C_{28}:D_4$ x 2, $C_{28}:D_4$ x 2, $C_{28}:D_4$ x 2, $C_{28}:D_4$ x 2, $D_{28}:C_4$, $D_{14}.D_4$, $C_4:D_{28}$, $C_4\times D_{28}$, $D_7\times C_4^2$, $D_4:C_{28}$, $\OD_{16}:C_{14}$, $C_{28}.D_4$, $C_{28}.D_4$, $D_{28}:C_4$, $C_4.D_{28}$, $C_{14}.C_2^4$, $D_8:C_{14}$, $C_{28}:D_4$, $D_4\times C_{28}$, $D_4:D_{14}$, $C_2^3.D_{14}$, $D_{28}:C_2^2$, $D_{28}:C_4$, $C_8:D_{14}$, $D_7\times \SD_{16}$, $C_8:D_{14}$, $D_7\times D_8$, $C_8:D_{14}$, $D_7\times \OD_{16}$

Order 192: $C_3\times C_2\wr C_2^2$ x 12, $C_3\times D_4^2$ x 10, $C_2^4:C_{12}$ x 8, $C_3\times D_4:D_4$ x 6, $C_2^5:C_6$ x 5, $C_2^4:C_{12}$ x 4, $C_4^2:C_{12}$ x 4, $C_{12}:C_2^4$ x 2, $C_2\times C_{12}:D_4$ x 2, $C_4\wr C_2\times C_6$, $C_2^4.C_{12}$, $C_{12}.C_2^4$, $C_{24}:C_2^3$, $C_2\times C_{12}:D_4$, $C_2\times D_4\times C_{12}$

Order 168: $C_2^2\times F_7$ x 112, $D_{14}:C_6$ x 50, $C_4\times F_7$ x 27, $C_{28}:C_6$ x 26, $C_{28}:C_6$ x 26, $C_2\times C_{14}:C_6$ x 18, $C_{14}:C_{12}$ x 13, $C_{14}:C_{12}$ x 13, $C_{28}.C_6$ x 2, $C_{28}.C_6$, $C_7:C_{24}$, $C_7:C_{24}$

Order 128: $C_2\wr D_4$ x 8, $C_2^5:C_4$ x 2, $C_2^4:D_4$ x 2, $C_2^4.D_4$, $C_2\times D_4^2$, $C_4^2:C_2^3$

Order 112: $D_4\times D_7$ x 88, $C_{14}:D_4$ x 43, $C_2^2\times D_{14}$ x 36, $D_4\times C_{14}$ x 22, $C_2\times D_{28}$ x 22, $C_4\times D_{14}$ x 21, $D_{14}:C_4$ x 14, $C_2^2:C_{28}$ x 9, $C_{14}.D_4$ x 9, $D_4:D_7$ x 5, $D_{28}:C_2$ x 5, $C_2^3\times C_{14}$ 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, $C_7:D_8$ x 2, $C_{14}.D_4$ x 2, $C_{28}:C_4$ x 2, $C_7:\OD_{16}$, $D_{56}$, $C_{56}:C_2$, $C_{56}:C_2$, $Q_8\times D_7$, $C_8\times D_7$, $C_7\times \SD_{16}$, $C_7\times D_8$, $C_7\times \OD_{16}$, $C_4:C_{28}$, $C_4\times C_{28}$, $Q_8:D_7$, $C_7:\SD_{16}$, $C_{28}:C_4$

Order 96: $C_{12}:C_2^3$ x 46, $C_2^4:C_6$ x 32, $C_{12}:D_4$ x 16, $C_2^3:C_{12}$ x 14, $C_2^3:C_{12}$ x 9, $D_4\times C_{12}$ x 8, $C_6.C_2^4$ x 7, $D_8:C_6$ x 6, $C_{12}:D_4$ x 6, $D_4:C_{12}$ x 4, $\OD_{16}:C_6$ x 3, $C_2^4\times C_6$ x 3, $C_6.C_2^4$ x 2, $C_2^3\times C_{12}$ x 2, $C_6\times \SD_{16}$, $C_6\times D_8$, $C_6\times \OD_{16}$, $C_2\times C_4:C_{12}$, $C_2\times C_4\times C_{12}$

Order 84: $C_2\times F_7$ x 78, $C_{14}:C_6$ x 29, $C_7:C_{12}$ x 8, $C_7:C_{12}$ x 8

Order 64: $C_2\wr C_2^2$ x 12, $D_4^2$ x 10, $C_2\wr C_4$ x 8, $D_4:D_4$ x 6, $C_2^3:D_4$ x 5, $C_2^4:C_4$ x 4, $C_4^2:C_4$ x 4, $D_4\times C_2^3$ x 2, $C_2^3:D_4$ x 2, $\OD_{16}:C_2^2$, $D_4:C_2^3$, $C_8:C_2^3$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$

Order 56: $C_2\times D_{14}$ x 112, $C_7:D_4$ x 50, $C_4\times D_7$ x 27, $C_7\times D_4$ x 26, $D_{28}$ x 26, $C_2^2\times C_{14}$ x 18, $C_2\times C_{28}$ x 13, $C_{14}:C_4$ x 13, $C_7:Q_8$ x 2, $C_{56}$, $C_7\times Q_8$, $C_7:C_8$

Order 48: $C_6\times D_4$ x 175, $C_2^3\times C_6$ x 39, $C_2^2:C_{12}$ x 32, $C_2^2\times C_{12}$ x 25, $D_4:C_6$ x 14, $C_3\times \SD_{16}$ x 4, $C_3\times D_8$ x 4, $C_4:C_{12}$ x 4, $C_3\times \OD_{16}$ x 3, $C_4\times C_{12}$ x 3, $C_6\times Q_8$, $C_2\times C_{24}$

Order 42: $F_7$ x 11, $C_7:C_6$ x 10

Order 32: $C_2^2\times D_4$ x 46, $C_2^2\wr C_2$ x 32, $C_4:D_4$ x 16, $C_2^3:C_4$ x 14, $C_2^3:C_4$ x 9, $C_4\times D_4$ x 8, $D_4:C_2^2$ x 7, $D_8:C_2$ x 6, $C_4:D_4$ x 6, $C_4\wr C_2$ x 4, $\OD_{16}:C_2$ x 3, $C_2^5$ x 3, $D_4:C_2^2$ x 2, $C_2^3\times C_4$ x 2, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2\times \OD_{16}$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 28: $D_{14}$ x 78, $C_2\times C_{14}$ x 29, $C_{28}$ x 8, $C_7:C_4$ x 8

Order 24: $C_2^2\times C_6$ x 130, $C_3\times D_4$ x 102, $C_2\times C_{12}$ x 53, $C_3\times Q_8$ x 3, $C_{24}$ x 2

Order 21: $C_7:C_3$

Order 16: $C_2\times D_4$ x 175, $C_2^4$ x 39, $C_2^2:C_4$ x 32, $C_2^2\times C_4$ x 25, $D_4:C_2$ x 14, $\SD_{16}$ x 4, $D_8$ x 4, $C_4:C_4$ x 4, $\OD_{16}$ x 3, $C_4^2$ x 3, $C_2\times C_8$, $C_2\times Q_8$

Order 14: $D_7$ x 11, $C_{14}$ x 10

Order 12: $C_2\times C_6$ x 107, $C_{12}$ x 16

Order 8: $C_2^3$ x 130, $D_4$ x 102, $C_2\times C_4$ x 53, $Q_8$ x 3, $C_8$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 21

Order 4: $C_2^2$ x 107, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 21

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5376: $C_2\wr D_4\times F_7$

Order 2688: $C_2\wr C_2^2\times F_7$, $D_4^2\times F_7$, $C_4^2:C_4\times F_7$, $(C_{14}\times D_4).C_6.C_2^2$, $(C_{14}\times D_4).C_6.C_2^2$, $(C_{14}\times D_4).C_6.C_2^2$, $D_4:D_4\times F_7$, $(C_{14}\times D_4).C_6.C_2^2$, $C_4^2:C_4.C_2\times C_7:C_3$, $(C_{14}\times D_4).C_6.C_2^2$, $C_2\wr C_4\times F_7$

Order 1792: $D_4^2:D_{14}$

Order 1344: $C_2^2\wr C_2\times F_7$ x 3, $C_2^3:C_4\times F_7$ x 2, $(C_2^2\times F_7):D_4$ x 2, $D_{14}:(C_6\times D_4)$ x 2, $C_2^4:(C_2\times F_7)$ x 2, $C_{28}:C_6\times D_4$, $(C_2^3\times C_{14}):C_{12}$, $D_{28}:(C_3\times D_4)$, $\OD_{16}:C_2\times F_7$, $D_4\times D_{14}:C_6$, $D_8:C_2\times F_7$, $C_{28}:(C_6\times D_4)$, $C_4:D_4\times F_7$, $C_{28}:(C_6\times D_4)$, $C_4:D_4\times F_7$, $C_{28}:C_6\times D_4$, $C_4\times D_4\times F_7$, $C_{14}:C_{12}:D_4$, $C_{28}.(C_6\times D_4)$, $(C_4\times C_{28}):C_{12}$, $(C_2^3\times C_{14}):C_{12}$, $(C_2^2\times D_{14}):C_{12}$, $(C_2\times D_{28}):C_{12}$, $(C_2^2\times D_{14}):C_{12}$, $(C_2^3\times F_7):C_2^2$, $C_{28}.(C_6\times D_4)$, $D_{28}:(C_3\times D_4)$, $D_{28}:(C_3\times D_4)$, $D_4:C_2^2\times F_7$, $C_2^2\times D_4\times F_7$, $(C_4\times C_{28}):C_{12}$, $D_{28}:(C_3\times D_4)$, $C_4\wr C_2\times F_7$

Order 896: $D_4^2:C_{14}$, $D_4^2:D_7$, $(C_2\times D_{28}):D_4$, $(C_4\times C_{28}):D_4$, $D_7\times D_4^2$, $C_4^2:C_4\times D_7$, $C_4^2:D_{28}$, $C_2\wr C_4\times D_7$, $C_2^4:D_{28}$, $C_2\wr C_2^2\times D_7$, $D_4:D_4\times D_7$

Order 768: $C_2\wr D_4\times C_6$

Order 672: $C_2\times D_4\times F_7$ x 14, $(D_4\times C_{14}):C_6$ x 6, $(C_2\times D_{28}):C_6$ x 6, $C_2^2:C_4\times F_7$ x 5, $(D_4\times C_{14}):C_6$ x 3, $C_2^4:F_7$ x 3, $(C_2\times D_{28}):C_6$ x 3, $(C_2\times D_{14}):C_{12}$ x 3, $(D_4\times C_{14}):C_6$ x 3, $(C_2\times C_{28}):C_{12}$ x 2, $(D_4\times C_{14}):C_6$ x 2, $(C_2\times D_{28}):C_6$ x 2, $C_2^4\times F_7$ x 2, $(C_2\times C_{28}):C_{12}$ x 2, $C_2^4:F_7$ x 2, $(C_2\times D_{28}):C_6$ x 2, $D_4:C_2\times F_7$ x 2, $C_{56}:(C_2\times C_6)$, $(C_4\times C_{28}):C_6$, $(D_4\times C_{14}):C_6$, $C_4\times C_{28}:C_6$, $(C_7\times D_4):C_{12}$, $(D_4\times C_{14}).C_6$, $(C_2\times D_{28}):C_6$, $(C_4\times D_{14}):C_6$, $D_4\times C_7:C_{12}$, $(D_4\times C_{14}):C_6$, $D_{28}:C_{12}$, $(C_2\times D_{28}):C_6$, $C_4\times D_{14}:C_6$, $D_{56}:C_6$, $\SD_{16}\times F_7$, $\OD_{16}:F_7$, $D_8:F_7$, $D_8\times F_7$, $D_{56}:C_6$, $\OD_{16}\times F_7$, $(C_2\times D_{28}):C_6$, $D_{28}:C_{12}$, $C_4:C_4\times F_7$, $C_7:(D_4\times C_{12})$, $C_4^2:F_7$, $D_{28}:C_{12}$, $C_4^2\times F_7$, $D_{28}:C_{12}$, $(C_4\times C_{28}):C_6$, $(C_7\times \OD_{16}):C_6$, $(D_4\times C_{14}):C_6$, $C_{14}:C_6\times D_4$, $C_2\times D_{28}:C_6$, $C_4\times C_2^2\times F_7$

Order 448: $C_2^4:D_{14}$ x 3, $C_2^4:D_{14}$ x 2, $C_2^3:C_4\times D_7$ x 2, $C_2^3:D_{28}$ x 2, $C_2^4:D_{14}$ x 2, $C_4^2:D_{14}$, $(C_2^3\times C_{14}):C_4$, $C_2\wr C_2^2\times C_7$, $C_7\times D_4:D_4$, $(C_2\times C_{28}):D_4$, $(C_7\times D_4):D_4$, $D_{28}:D_4$, $C_4^2:D_{14}$, $D_4:D_{28}$, $C_4^2:D_{14}$, $C_2^3.D_{28}$, $C_2^3.D_{28}$, $D_{28}:D_4$, $(C_2^2\times D_{14}):C_4$, $\OD_{16}:D_{14}$, $C_4^2:C_{28}$, $C_2^4:C_{28}$, $D_{28}:C_2^3$, $C_{28}:C_2^4$, $C_7\times D_4^2$, $C_2^4:D_{14}$, $D_8:D_{14}$, $C_4^2:D_{14}$, $C_4^2:D_{14}$, $D_{28}:D_4$, $C_4:D_4\times D_7$, $(C_4\times C_{28}):C_4$, $D_4\times D_{28}$

Order 384: $D_4^2:C_6$ x 6, $C_2\times C_4^2:C_{12}$, $C_2^5:C_{12}$, $C_6\times D_4^2$, $C_2\wr C_2^2\times C_6$, $C_6\times D_4:D_4$

Order 336: $D_4\times F_7$ x 21, $C_2^3:F_7$ x 16, $C_2^3\times F_7$ x 14, $C_2\times C_{28}:C_6$ x 10, $D_{28}:C_6$ x 10, $D_{14}:C_{12}$ x 9, $D_{14}:C_{12}$ x 7, $(C_2\times C_{14}):C_{12}$ x 5, $C_2^3.F_7$ x 5, $D_4:F_7$ x 4, $D_{28}:C_6$ x 4, $C_2^2\times C_{14}:C_6$ x 2, $D_{28}:C_6$ x 2, $(C_2\times C_{28}):C_6$ x 2, $C_{28}:C_{12}$ x 2, $D_{28}:C_6$ x 2, $C_{56}:C_6$, $C_{56}:C_6$, $C_8\times F_7$, $C_{56}:C_6$, $C_{56}:C_6$, $C_{56}:C_6$, $C_{28}:C_{12}$, $C_{28}:C_{12}$, $Q_8:F_7$, $D_4.F_7$, $C_2\times C_{14}:C_{12}$, $C_{28}:C_{12}$, $C_{14}.(C_2\times C_{12})$, $C_{28}.C_{12}$, $C_2^3.F_7$, $Q_8\times F_7$, $C_{56}:C_6$

Order 256: $D_4^2:C_2^2$

Order 224: $D_4\times D_{14}$ x 14, $C_2^2:D_{28}$ x 6, $D_{14}:D_4$ x 6, $D_{14}.D_4$ x 5, $D_4:D_{14}$ x 3, $C_2^4:C_{14}$ x 3, $C_2^4:D_7$ x 3, $C_{28}:D_4$ x 3, $C_2^2.D_{28}$ x 3, $D_{14}:D_4$ x 2, $C_2^3:C_{28}$ x 2, $C_2^3.D_{14}$ x 2, $C_2^3\times D_{14}$ x 2, $C_2^3:D_{14}$ x 2, $D_4:D_{14}$ x 2, $D_4:D_{14}$ x 2, $C_2^3.D_{14}$ x 2, $C_4:D_{28}$, $D_{28}:C_4$, $D_{14}.D_4$, $C_2^3.D_{14}$, $C_4:D_{28}$, $C_4\times D_{28}$, $D_7\times C_4^2$, $D_4:C_{28}$, $\OD_{16}:C_{14}$, $C_{28}.D_4$, $C_{28}.D_4$, $D_{28}:C_4$, $C_4.D_{28}$, $C_{14}.C_2^4$, $C_{28}:C_2^3$, $C_2^2\times D_{28}$, $C_{28}:C_2^3$, $D_8:C_{14}$, $C_{28}:D_4$, $C_{28}:D_4$, $D_4\times C_{28}$, $D_4:D_{14}$, $C_{28}:D_4$, $C_2^3.D_{14}$, $D_{28}:C_2^2$, $C_{28}:D_4$, $C_{28}:D_4$, $D_{28}:C_4$, $C_8:D_{14}$, $D_7\times \SD_{16}$, $C_8:D_{14}$, $D_7\times D_8$, $C_8:D_{14}$, $D_7\times \OD_{16}$

Order 192: $C_3\times D_4^2$ x 7, $C_3\times C_2\wr C_2^2$ x 6, $C_3\times D_4:D_4$ x 6, $C_2^4:C_{12}$ x 4, $C_4^2:C_{12}$ x 3, $C_2^5:C_6$ x 3, $C_2^4:C_{12}$ x 2, $C_4\wr C_2\times C_6$, $C_2^4.C_{12}$, $C_{12}.C_2^4$, $C_{12}:C_2^4$, $C_{24}:C_2^3$, $C_2\times C_{12}:D_4$, $C_2\times C_{12}:D_4$, $C_2\times D_4\times C_{12}$

Order 168: $C_2^2\times F_7$ x 34, $D_{14}:C_6$ x 16, $C_4\times F_7$ x 11, $C_{28}:C_6$ x 10, $C_{28}:C_6$ x 10, $C_2\times C_{14}:C_6$ x 8, $C_{14}:C_{12}$ x 7, $C_{14}:C_{12}$ x 7, $C_{28}.C_6$ x 2, $C_{28}.C_6$, $C_7:C_{24}$, $C_7:C_{24}$

Order 128: $C_2\wr D_4$ x 6, $C_2^4.D_4$, $C_2^5:C_4$, $C_2\times D_4^2$, $C_2^4:D_4$, $C_4^2:C_2^3$

Order 112: $D_4\times D_7$ x 21, $C_{14}:D_4$ x 16, $C_2^2\times D_{14}$ x 14, $D_4\times C_{14}$ x 10, $C_2\times D_{28}$ x 10, $C_4\times D_{14}$ x 9, $D_{14}:C_4$ x 7, $C_2^2:C_{28}$ x 5, $C_{14}.D_4$ x 5, $D_4:D_7$ x 4, $D_{28}:C_2$ x 4, $C_2^3\times C_{14}$ x 2, $D_4:C_{14}$ x 2, $D_{28}:C_2$ x 2, $C_7:D_8$ x 2, $C_{28}:C_4$ x 2, $C_7:\OD_{16}$, $D_{56}$, $C_{56}:C_2$, $C_{56}:C_2$, $C_2^2\times C_{28}$, $C_2^2.D_{14}$, $Q_8\times D_7$, $C_8\times D_7$, $C_7\times \SD_{16}$, $C_7\times D_8$, $C_7\times \OD_{16}$, $C_4:C_{28}$, $C_4\times C_{28}$, $Q_8:D_7$, $C_7:\SD_{16}$, $C_{28}:C_4$, $C_{14}.D_4$

Order 96: $C_{12}:C_2^3$ x 18, $C_2^4:C_6$ x 18, $C_{12}:D_4$ x 8, $C_2^3:C_{12}$ x 7, $C_6.C_2^4$ x 6, $D_8:C_6$ x 6, $D_4\times C_{12}$ x 6, $C_{12}:D_4$ x 5, $C_2^3:C_{12}$ x 5, $D_4:C_{12}$ x 4, $\OD_{16}:C_6$ x 3, $C_2^4\times C_6$ x 2, $C_6.C_2^4$ x 2, $C_2^3\times C_{12}$, $C_6\times \SD_{16}$, $C_6\times D_8$, $C_6\times \OD_{16}$, $C_2\times C_4:C_{12}$, $C_2\times C_4\times C_{12}$

Order 84: $C_2\times F_7$ x 25, $C_{14}:C_6$ x 13, $C_7:C_{12}$ x 5, $C_7:C_{12}$ x 5

Order 64: $D_4^2$ x 7, $C_2\wr C_2^2$ x 6, $D_4:D_4$ x 6, $C_2\wr C_4$ x 4, $C_4^2:C_4$ x 3, $C_2^3:D_4$ x 3, $C_2^4:C_4$ x 2, $\OD_{16}:C_2^2$, $D_4:C_2^3$, $D_4\times C_2^3$, $C_8:C_2^3$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_4^2:C_2^2$

Order 56: $C_2\times D_{14}$ x 34, $C_7:D_4$ x 16, $C_4\times D_7$ x 11, $C_7\times D_4$ x 10, $D_{28}$ x 10, $C_2^2\times C_{14}$ x 8, $C_2\times C_{28}$ x 7, $C_{14}:C_4$ x 7, $C_7:Q_8$ x 2, $C_{56}$, $C_7\times Q_8$, $C_7:C_8$

Order 48: $C_6\times D_4$ x 57, $C_2^2:C_{12}$ x 17, $C_2^3\times C_6$ x 16, $D_4:C_6$ x 12, $C_2^2\times C_{12}$ x 11, $C_3\times \SD_{16}$ x 4, $C_3\times D_8$ x 4, $C_3\times \OD_{16}$ x 3, $C_4:C_{12}$ x 3, $C_4\times C_{12}$ x 3, $C_6\times Q_8$, $C_2\times C_{24}$

Order 42: $C_7:C_6$ x 6, $F_7$ x 6

Order 32: $C_2^2\times D_4$ x 18, $C_2^2\wr C_2$ x 18, $C_4:D_4$ x 8, $C_2^3:C_4$ x 7, $D_4:C_2^2$ x 6, $D_8:C_2$ x 6, $C_4\times D_4$ x 6, $C_4:D_4$ x 5, $C_2^3:C_4$ x 5, $C_4\wr C_2$ x 4, $\OD_{16}:C_2$ x 3, $C_2^5$ x 2, $D_4:C_2^2$ x 2, $C_2^3\times C_4$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2\times \OD_{16}$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 28: $D_{14}$ x 25, $C_2\times C_{14}$ x 13, $C_{28}$ x 5, $C_7:C_4$ x 5

Order 24: $C_2^2\times C_6$ x 42, $C_3\times D_4$ x 36, $C_2\times C_{12}$ x 25, $C_3\times Q_8$ x 3, $C_{24}$ x 2

Order 21: $C_7:C_3$

Order 16: $C_2\times D_4$ x 57, $C_2^2:C_4$ x 17, $C_2^4$ x 16, $D_4:C_2$ x 12, $C_2^2\times C_4$ x 11, $\SD_{16}$ x 4, $D_8$ x 4, $\OD_{16}$ x 3, $C_4:C_4$ x 3, $C_4^2$ x 3, $C_2\times C_8$, $C_2\times Q_8$

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

Order 12: $C_2\times C_6$ x 38, $C_{12}$ x 10

Order 8: $C_2^3$ x 42, $D_4$ x 36, $C_2\times C_4$ x 25, $Q_8$ x 3, $C_8$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 12

Order 4: $C_2^2$ x 38, $C_4$ x 10

Order 3: $C_3$

Order 2: $C_2$ x 12

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5376: $C_2\wr D_4\times F_7$ ($C_1$)

Order 2688: $C_2\wr C_2^2\times F_7$ ($C_2$) x 2, $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$) x 2, $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$) x 2, $C_2\wr C_4\times F_7$ ($C_2$) x 2, $D_4^2\times F_7$ ($C_2$), $C_4^2:C_4\times F_7$ ($C_2$), $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$), $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$), $D_4:D_4\times F_7$ ($C_2$), $C_4^2:C_4.C_2\times C_7:C_3$ ($C_2$), $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$)

Order 1792: $D_4^2:D_{14}$ ($C_3$)

Order 1344: $(C_2^3\times C_{14}):C_{12}$ ($C_2^2$) x 2, $C_2^3:C_4\times F_7$ ($C_2^2$) x 2, $(C_2^2\times F_7):D_4$ ($C_2^2$) x 2, $D_{14}:(C_6\times D_4)$ ($C_2^2$) x 2, $C_2^2\wr C_2\times F_7$ ($C_2^2$) x 2, $C_{14}:C_{12}:D_4$ ($C_2^2$) x 2, $(C_2^3\times C_{14}):C_{12}$ ($C_2^2$) x 2, $(C_2^2\times D_{14}):C_{12}$ ($C_2^2$) x 2, $(C_2\times D_{28}):C_{12}$ ($C_2^2$) x 2, $(C_2^2\times D_{14}):C_{12}$ ($C_2^2$) x 2, $(C_2^3\times F_7):C_2^2$ ($C_2^2$) x 2, $C_2^4:(C_2\times F_7)$ ($C_2^2$) x 2, $C_{28}:C_6\times D_4$ ($C_2^2$), $D_{28}:(C_3\times D_4)$ ($C_2^2$), $\OD_{16}:C_2\times F_7$ ($C_2^2$), $C_{28}:(C_6\times D_4)$ ($C_2^2$), $C_4:D_4\times F_7$ ($C_2^2$), $C_{28}.(C_6\times D_4)$ ($C_2^2$), $(C_4\times C_{28}):C_{12}$ ($C_2^2$), $C_{28}.(C_6\times D_4)$ ($C_2^2$), $D_{28}:(C_3\times D_4)$ ($C_2^2$), $D_4:C_2^2\times F_7$ ($C_2^2$), $(C_4\times C_{28}):C_{12}$ ($C_2^2$)

Order 896: $(C_2\times D_{28}):D_4$ ($C_6$) x 2, $C_2\wr C_4\times D_7$ ($C_6$) x 2, $C_2^4:D_{28}$ ($C_6$) x 2, $C_2\wr C_2^2\times D_7$ ($C_6$) x 2, $D_4^2:C_{14}$ ($C_6$), $D_4^2:D_7$ ($C_6$), $(C_4\times C_{28}):D_4$ ($C_6$), $D_7\times D_4^2$ ($C_6$), $C_4^2:C_4\times D_7$ ($C_6$), $C_4^2:D_{28}$ ($C_6$), $D_4:D_4\times D_7$ ($C_6$)

Order 672: $(D_4\times C_{14}):C_6$ ($C_2^3$) x 2, $(C_2\times C_{28}):C_{12}$ ($C_2^3$) x 2, $C_2^4:F_7$ ($D_4$) x 2, $(D_4\times C_{14}):C_6$ ($C_2^3$) x 2, $(C_2\times D_{28}):C_6$ ($D_4$) x 2, $C_2^2:C_4\times F_7$ ($D_4$) x 2, $C_2^4\times F_7$ ($D_4$) x 2, $(C_2\times C_{28}):C_{12}$ ($C_2^3$) x 2, $(C_4\times C_{28}):C_6$ ($C_2^3$), $(D_4\times C_{14}).C_6$ ($C_2^3$), $(C_2\times D_{28}):C_6$ ($C_2^3$), $(C_2\times D_{28}):C_6$ ($D_4$), $C_4^2:F_7$ ($D_4$), $C_4^2\times F_7$ ($D_4$), $(C_7\times \OD_{16}):C_6$ ($C_2^3$), $(D_4\times C_{14}):C_6$ ($C_2^3$), $(D_4\times C_{14}):C_6$ ($C_2^3$), $C_2\times D_4\times F_7$ ($C_2^3$), $C_2\times D_4\times F_7$ ($D_4$)

Order 448: $(C_2^3\times C_{14}):C_4$ ($C_2\times C_6$) x 2, $C_2\wr C_2^2\times C_7$ ($C_2\times C_6$) x 2, $(C_2\times C_{28}):D_4$ ($C_2\times C_6$) x 2, $C_2^4:D_{14}$ ($C_2\times C_6$) x 2, $C_2^3.D_{28}$ ($C_2\times C_6$) x 2, $C_2^3.D_{28}$ ($C_2\times C_6$) x 2, $(C_2^2\times D_{14}):C_4$ ($C_2\times C_6$) x 2, $C_2^3:C_4\times D_7$ ($C_2\times C_6$) x 2, $C_2^3:D_{28}$ ($C_2\times C_6$) x 2, $C_2^4:C_{28}$ ($C_2\times C_6$) x 2, $C_2^4:D_{14}$ ($C_2\times C_6$) x 2, $C_2^4:D_{14}$ ($C_2\times C_6$) x 2, $C_7\times D_4:D_4$ ($C_2\times C_6$), $(C_7\times D_4):D_4$ ($C_2\times C_6$), $C_4^2:D_{14}$ ($C_2\times C_6$), $D_{28}:D_4$ ($C_2\times C_6$), $\OD_{16}:D_{14}$ ($C_2\times C_6$), $C_4^2:C_{28}$ ($C_2\times C_6$), $D_{28}:C_2^3$ ($C_2\times C_6$), $C_7\times D_4^2$ ($C_2\times C_6$), $C_4^2:D_{14}$ ($C_2\times C_6$), $C_4^2:D_{14}$ ($C_2\times C_6$), $(C_4\times C_{28}):C_4$ ($C_2\times C_6$)

Order 336: $C_2^3\times F_7$ ($C_2\times D_4$) x 4, $(C_2\times C_{14}):C_{12}$ ($C_2\times D_4$) x 2, $C_2^2\times C_{14}:C_6$ ($C_2\times D_4$) x 2, $C_2^3.F_7$ ($C_2\times D_4$) x 2, $C_2^3:F_7$ ($C_2\times D_4$) x 2, $D_{28}:C_6$ ($C_2\times D_4$) x 2, $C_{28}:C_{12}$ ($C_2\times D_4$), $C_2\times C_{28}:C_6$ ($C_2^4$), $C_2\times C_{28}:C_6$ ($C_2\times D_4$), $C_{28}:C_{12}$ ($C_2\times D_4$), $D_{14}:C_{12}$ ($C_2\times D_4$)

Order 224: $C_2^2:D_{28}$ ($C_3\times D_4$) x 2, $D_{14}.D_4$ ($C_3\times D_4$) x 2, $C_2^3:C_{28}$ ($C_2^2\times C_6$) x 2, $C_2^3.D_{14}$ ($C_2^2\times C_6$) x 2, $C_2^3\times D_{14}$ ($C_3\times D_4$) x 2, $C_2^4:C_{14}$ ($C_2^2\times C_6$) x 2, $C_2^4:D_7$ ($C_3\times D_4$) x 2, $D_{14}:D_4$ ($C_2^2\times C_6$) x 2, $C_4:D_{28}$ ($C_3\times D_4$), $D_7\times C_4^2$ ($C_3\times D_4$), $\OD_{16}:C_{14}$ ($C_2^2\times C_6$), $C_{28}.D_4$ ($C_2^2\times C_6$), $C_{14}.C_2^4$ ($C_2^2\times C_6$), $D_4:D_{14}$ ($C_2^2\times C_6$), $D_4\times D_{14}$ ($C_2^2\times C_6$), $D_4\times D_{14}$ ($C_3\times D_4$), $C_{28}:D_4$ ($C_2^2\times C_6$), $C_{28}:D_4$ ($C_2^2\times C_6$), $C_{28}:D_4$ ($C_3\times D_4$)

Order 168: $C_2^2\times F_7$ ($C_2^2\wr C_2$) x 3, $C_2\times C_{14}:C_6$ ($C_2^2\times D_4$) x 2, $C_{14}:C_{12}$ ($C_2^2\times D_4$), $C_{14}:C_{12}$ ($C_2^2\wr C_2$)

Order 128: $C_2\wr D_4$ ($F_7$)

Order 112: $C_2^2\times D_{14}$ ($C_6\times D_4$) x 4, $C_2^3\times C_{14}$ ($C_6\times D_4$) x 2, $C_{14}:D_4$ ($C_6\times D_4$) x 2, $C_2\times D_{28}$ ($C_6\times D_4$) x 2, $C_2^2:C_{28}$ ($C_6\times D_4$) x 2, $C_{14}.D_4$ ($C_6\times D_4$) x 2, $D_4\times C_{14}$ ($C_2^3\times C_6$), $D_4\times C_{14}$ ($C_6\times D_4$), $C_4\times D_{14}$ ($C_6\times D_4$), $C_4\times C_{28}$ ($C_6\times D_4$), $C_{28}:C_4$ ($C_6\times D_4$)

Order 84: $C_2\times F_7$ ($C_2\wr C_2^2$) x 2, $C_{14}:C_6$ ($C_2^3:D_4$)

Order 64: $C_2\wr C_4$ ($C_2\times F_7$) x 2, $C_2\wr C_2^2$ ($C_2\times F_7$) x 2, $C_4^2:C_4$ ($C_2\times F_7$), $D_4^2$ ($C_2\times F_7$), $D_4:D_4$ ($C_2\times F_7$)

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

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

Order 32: $C_2^3:C_4$ ($C_2^2\times F_7$) x 2, $C_2^2\wr C_2$ ($C_2^2\times F_7$) x 2, $\OD_{16}:C_2$ ($C_2^2\times F_7$), $D_4:C_2^2$ ($C_2^2\times F_7$), $C_4:D_4$ ($C_2^2\times F_7$)

Order 28: $D_{14}$ ($C_3\times C_2\wr C_2^2$) x 2, $C_2\times C_{14}$ ($C_2^5:C_6$)

Order 21: $C_7:C_3$ ($D_4^2:C_2^2$)

Order 16: $C_2^2:C_4$ ($D_4\times F_7$) x 2, $C_2^4$ ($D_4\times F_7$) x 2, $C_4^2$ ($D_4\times F_7$), $C_2\times D_4$ ($C_2^3\times F_7$), $C_2\times D_4$ ($D_4\times F_7$)

Order 14: $D_7$ ($D_4^2:C_6$) x 2, $C_{14}$ ($C_2\wr C_2^2\times C_6$)

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

Order 7: $C_7$ ($C_2\wr D_4\times C_6$)

Order 4: $C_2^2$ ($C_2^2\wr C_2\times F_7$)

Order 2: $C_2$ ($C_2\wr C_2^2\times F_7$)

Order 1: $C_1$ ($C_2\wr D_4\times F_7$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5376: $C_2\wr D_4\times F_7$ ($C_1$)

Order 2688: $C_2\wr C_2^2\times F_7$ ($C_2$), $D_4^2\times F_7$ ($C_2$), $C_4^2:C_4\times F_7$ ($C_2$), $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$), $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$), $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$), $D_4:D_4\times F_7$ ($C_2$), $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$), $C_4^2:C_4.C_2\times C_7:C_3$ ($C_2$), $(C_{14}\times D_4).C_6.C_2^2$ ($C_2$), $C_2\wr C_4\times F_7$ ($C_2$)

Order 1792: $D_4^2:D_{14}$ ($C_3$)

Order 1344: $C_{28}:C_6\times D_4$ ($C_2^2$), $(C_2^3\times C_{14}):C_{12}$ ($C_2^2$), $D_{28}:(C_3\times D_4)$ ($C_2^2$), $\OD_{16}:C_2\times F_7$ ($C_2^2$), $C_2^3:C_4\times F_7$ ($C_2^2$), $(C_2^2\times F_7):D_4$ ($C_2^2$), $C_{28}:(C_6\times D_4)$ ($C_2^2$), $C_4:D_4\times F_7$ ($C_2^2$), $D_{14}:(C_6\times D_4)$ ($C_2^2$), $C_2^2\wr C_2\times F_7$ ($C_2^2$), $C_{14}:C_{12}:D_4$ ($C_2^2$), $C_{28}.(C_6\times D_4)$ ($C_2^2$), $(C_4\times C_{28}):C_{12}$ ($C_2^2$), $(C_2^3\times C_{14}):C_{12}$ ($C_2^2$), $(C_2^2\times D_{14}):C_{12}$ ($C_2^2$), $(C_2\times D_{28}):C_{12}$ ($C_2^2$), $(C_2^2\times D_{14}):C_{12}$ ($C_2^2$), $(C_2^3\times F_7):C_2^2$ ($C_2^2$), $C_{28}.(C_6\times D_4)$ ($C_2^2$), $D_{28}:(C_3\times D_4)$ ($C_2^2$), $C_2^4:(C_2\times F_7)$ ($C_2^2$), $D_4:C_2^2\times F_7$ ($C_2^2$), $(C_4\times C_{28}):C_{12}$ ($C_2^2$)

Order 896: $D_4^2:C_{14}$ ($C_6$), $D_4^2:D_7$ ($C_6$), $(C_2\times D_{28}):D_4$ ($C_6$), $(C_4\times C_{28}):D_4$ ($C_6$), $D_7\times D_4^2$ ($C_6$), $C_4^2:C_4\times D_7$ ($C_6$), $C_4^2:D_{28}$ ($C_6$), $C_2\wr C_4\times D_7$ ($C_6$), $C_2^4:D_{28}$ ($C_6$), $C_2\wr C_2^2\times D_7$ ($C_6$), $D_4:D_4\times D_7$ ($C_6$)

Order 672: $(C_4\times C_{28}):C_6$ ($C_2^3$), $(D_4\times C_{14}):C_6$ ($C_2^3$), $(C_2\times C_{28}):C_{12}$ ($C_2^3$), $(D_4\times C_{14}).C_6$ ($C_2^3$), $C_2^4:F_7$ ($D_4$), $(C_2\times D_{28}):C_6$ ($C_2^3$), $(C_2\times D_{28}):C_6$ ($D_4$), $(D_4\times C_{14}):C_6$ ($C_2^3$), $(C_2\times D_{28}):C_6$ ($D_4$), $C_2^2:C_4\times F_7$ ($D_4$), $C_4^2:F_7$ ($D_4$), $C_4^2\times F_7$ ($D_4$), $C_2^4\times F_7$ ($D_4$), $(C_7\times \OD_{16}):C_6$ ($C_2^3$), $(C_2\times C_{28}):C_{12}$ ($C_2^3$), $(D_4\times C_{14}):C_6$ ($C_2^3$), $(D_4\times C_{14}):C_6$ ($C_2^3$), $C_2\times D_4\times F_7$ ($C_2^3$), $C_2\times D_4\times F_7$ ($D_4$)

Order 448: $(C_2^3\times C_{14}):C_4$ ($C_2\times C_6$), $C_2\wr C_2^2\times C_7$ ($C_2\times C_6$), $C_7\times D_4:D_4$ ($C_2\times C_6$), $(C_2\times C_{28}):D_4$ ($C_2\times C_6$), $(C_7\times D_4):D_4$ ($C_2\times C_6$), $C_4^2:D_{14}$ ($C_2\times C_6$), $C_2^4:D_{14}$ ($C_2\times C_6$), $C_2^3.D_{28}$ ($C_2\times C_6$), $C_2^3.D_{28}$ ($C_2\times C_6$), $D_{28}:D_4$ ($C_2\times C_6$), $(C_2^2\times D_{14}):C_4$ ($C_2\times C_6$), $\OD_{16}:D_{14}$ ($C_2\times C_6$), $C_2^3:C_4\times D_7$ ($C_2\times C_6$), $C_2^3:D_{28}$ ($C_2\times C_6$), $C_4^2:C_{28}$ ($C_2\times C_6$), $C_2^4:C_{28}$ ($C_2\times C_6$), $D_{28}:C_2^3$ ($C_2\times C_6$), $C_7\times D_4^2$ ($C_2\times C_6$), $C_4^2:D_{14}$ ($C_2\times C_6$), $C_4^2:D_{14}$ ($C_2\times C_6$), $C_2^4:D_{14}$ ($C_2\times C_6$), $C_2^4:D_{14}$ ($C_2\times C_6$), $(C_4\times C_{28}):C_4$ ($C_2\times C_6$)

Order 336: $C_2^3\times F_7$ ($C_2\times D_4$) x 2, $D_{28}:C_6$ ($C_2\times D_4$) x 2, $(C_2\times C_{14}):C_{12}$ ($C_2\times D_4$), $C_{28}:C_{12}$ ($C_2\times D_4$), $C_2^2\times C_{14}:C_6$ ($C_2\times D_4$), $C_2^3.F_7$ ($C_2\times D_4$), $C_2\times C_{28}:C_6$ ($C_2^4$), $C_2\times C_{28}:C_6$ ($C_2\times D_4$), $C_{28}:C_{12}$ ($C_2\times D_4$), $C_2^3:F_7$ ($C_2\times D_4$), $D_{14}:C_{12}$ ($C_2\times D_4$)

Order 224: $C_2^2:D_{28}$ ($C_3\times D_4$), $D_{14}.D_4$ ($C_3\times D_4$), $C_4:D_{28}$ ($C_3\times D_4$), $D_7\times C_4^2$ ($C_3\times D_4$), $\OD_{16}:C_{14}$ ($C_2^2\times C_6$), $C_2^3:C_{28}$ ($C_2^2\times C_6$), $C_2^3.D_{14}$ ($C_2^2\times C_6$), $C_{28}.D_4$ ($C_2^2\times C_6$), $C_2^3\times D_{14}$ ($C_3\times D_4$), $C_{14}.C_2^4$ ($C_2^2\times C_6$), $D_4:D_{14}$ ($C_2^2\times C_6$), $D_4\times D_{14}$ ($C_2^2\times C_6$), $D_4\times D_{14}$ ($C_3\times D_4$), $C_{28}:D_4$ ($C_2^2\times C_6$), $C_2^4:C_{14}$ ($C_2^2\times C_6$), $C_2^4:D_7$ ($C_3\times D_4$), $C_{28}:D_4$ ($C_2^2\times C_6$), $C_{28}:D_4$ ($C_3\times D_4$), $D_{14}:D_4$ ($C_2^2\times C_6$)

Order 168: $C_2^2\times F_7$ ($C_2^2\wr C_2$) x 2, $C_2\times C_{14}:C_6$ ($C_2^2\times D_4$), $C_{14}:C_{12}$ ($C_2^2\times D_4$), $C_{14}:C_{12}$ ($C_2^2\wr C_2$)

Order 128: $C_2\wr D_4$ ($F_7$)

Order 112: $C_2^2\times D_{14}$ ($C_6\times D_4$) x 2, $C_2\times D_{28}$ ($C_6\times D_4$) x 2, $C_2^3\times C_{14}$ ($C_6\times D_4$), $D_4\times C_{14}$ ($C_2^3\times C_6$), $D_4\times C_{14}$ ($C_6\times D_4$), $C_{14}:D_4$ ($C_6\times D_4$), $C_4\times D_{14}$ ($C_6\times D_4$), $C_2^2:C_{28}$ ($C_6\times D_4$), $C_4\times C_{28}$ ($C_6\times D_4$), $C_{14}.D_4$ ($C_6\times D_4$), $C_{28}:C_4$ ($C_6\times D_4$)

Order 84: $C_2\times F_7$ ($C_2\wr C_2^2$) x 2, $C_{14}:C_6$ ($C_2^3:D_4$)

Order 64: $C_4^2:C_4$ ($C_2\times F_7$), $C_2\wr C_4$ ($C_2\times F_7$), $D_4^2$ ($C_2\times F_7$), $C_2\wr C_2^2$ ($C_2\times F_7$), $D_4:D_4$ ($C_2\times F_7$)

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

Order 42: $C_7:C_6$ ($C_2^4:D_4$), $F_7$ ($C_2\wr D_4$)

Order 32: $\OD_{16}:C_2$ ($C_2^2\times F_7$), $C_2^3:C_4$ ($C_2^2\times F_7$), $D_4:C_2^2$ ($C_2^2\times F_7$), $C_4:D_4$ ($C_2^2\times F_7$), $C_2^2\wr C_2$ ($C_2^2\times F_7$)

Order 28: $D_{14}$ ($C_3\times C_2\wr C_2^2$) x 2, $C_2\times C_{14}$ ($C_2^5:C_6$)

Order 21: $C_7:C_3$ ($D_4^2:C_2^2$)

Order 16: $C_2^2:C_4$ ($D_4\times F_7$), $C_4^2$ ($D_4\times F_7$), $C_2^4$ ($D_4\times F_7$), $C_2\times D_4$ ($C_2^3\times F_7$), $C_2\times D_4$ ($D_4\times F_7$)

Order 14: $C_{14}$ ($C_2\wr C_2^2\times C_6$), $D_7$ ($D_4^2:C_6$)

Order 8: $C_2^3$ ($C_2\times D_4\times F_7$), $C_2\times C_4$ ($C_2\times D_4\times F_7$)

Order 7: $C_7$ ($C_2\wr D_4\times C_6$)

Order 4: $C_2^2$ ($C_2^2\wr C_2\times F_7$)

Order 2: $C_2$ ($C_2\wr C_2^2\times F_7$)

Order 1: $C_1$ ($C_2\wr D_4\times F_7$)

Series

Derived series

$C_2\wr D_4\times F_7$

$\rhd$

$C_2\wr D_4\times F_7$

$\rhd$

$D_4\times C_{14}$

$\rhd$

$D_4\times C_{14}$

$\rhd$

$C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_2\wr D_4\times F_7$

$\rhd$

$C_2\wr D_4\times F_7$

$\rhd$

$C_2\wr C_4\times F_7$

$\rhd$

$C_2\wr C_4\times F_7$

$\rhd$

$(C_2^3\times C_{14}):C_{12}$

$\rhd$

$(C_2^3\times C_{14}):C_{12}$

$\rhd$

$(D_4\times C_{14}):C_6$

$\rhd$

$(D_4\times C_{14}):C_6$

$\rhd$

$C_2\times C_{28}:C_6$

$\rhd$

$C_2\times C_{28}:C_6$

$\rhd$

$D_4\times C_{14}$

$\rhd$

$D_4\times C_{14}$

$\rhd$

$C_2^2\times C_{14}$

$\rhd$

$C_2^2\times C_{14}$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2\wr D_4\times F_7$

$\rhd$

$C_2\wr D_4\times F_7$

$\rhd$

$D_4\times C_{14}$

$\rhd$

$D_4\times C_{14}$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

$\lhd$

$C_2\times D_4$

$\lhd$

$C_2\wr D_4$

$\lhd$

$C_2\wr D_4$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $140 \times 140$ 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 $100 \times 100$ rational character table.