Abstract group 889344.a: $C_{24}\times F_{193}$

• Label: 889344.a
• Order: \( 2^{9} \cdot 3^{2} \cdot 193 \)
• Exponent: \( 2^{6} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{9} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: $4632$
• Rank: $2$

Group information

Description:	$C_{24}\times F_{193}$
Order:	\(889344\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 193 \)
Exponent:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Automorphism group:	$C_{2316}.C_{96}.C_2^5$, of order \(7114752\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 193 \)
Composition factors:	$C_2$ x 9, $C_3$ x 2, $C_{193}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	16	24	32	48	64	96	192	193	386	579	772	1158	1544	2316	4632
Elements	1	387	1160	1932	4248	8496	17760	12352	72576	24704	98816	49408	197632	395264	192	192	384	384	384	768	768	1536	889344
Conjugacy classes	1	3	8	12	24	48	96	64	384	128	512	256	1024	2048	1	1	2	2	2	4	4	8	4632
Divisions	1	3	4	6	12	12	24	8	48	8	32	8	32	32	1	1	1	1	1	1	1	1	238
Autjugacy classes	1	3	3	8	9	20	24	24	60	32	72	32	96	96	1	1	1	1	1	1	1	1	488

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$4632$
Rank:	$2$
Inequivalent generating pairs:	$24576$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	192	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{192}=b^{4632}=1, b^{a}=b^{1081} \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 186 & 0 \\ 0 & 55 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_8$ $\, \times\, $ $C_3$ $\, \times\, $ $F_{193}$
Semidirect product:	$C_{4632}$ $\,\rtimes\,$ $C_{192}$	$(C_{4632}:C_3)$ $\,\rtimes\,$ $C_{64}$	$(C_{579}:C_{24})$ $\,\rtimes\,$ $C_{64}$	$(C_{579}:C_{24})$ $\,\rtimes\,$ $C_{64}$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6\times F_{193})$ . $C_4$	$(C_4\times F_{193})$ . $C_6$	$C_6$ . $(C_4\times F_{193})$	$C_4$ . $(C_6\times F_{193})$	all 176

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

Homology

Abelianization:	$C_{24} \times C_{192} \simeq C_{8} \times C_{64} \times C_{3}^{2}$
Schur multiplier:	$C_{24}$
Commutator length:	$1$

Subgroups

There are 93912 subgroups in 984 conjugacy classes, 500 normal (110 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{24}$	$G/Z \simeq$ $F_{193}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_{24}\times C_{192}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_6\times F_{193}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{4632}$	$G/\operatorname{Fit} \simeq$ $C_{192}$
Radical:	$R \simeq$ $C_{24}\times F_{193}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1158}$	$G/\operatorname{soc} \simeq$ $C_4\times C_{192}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times C_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
193-Sylow subgroup:	$P_{ 193 } \simeq$ $C_{193}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 889344: $C_{24}\times F_{193}$

Order 444672: $C_{12}\times F_{193}$ x 2, $C_{4632}:C_{96}$

Order 296448: $C_8\times F_{193}$ x 3, $C_{1544}:C_{192}$

Order 222336: $C_6\times F_{193}$ x 4, $C_{2316}:C_{96}$ x 2, $C_{4632}:C_{48}$

Order 148224: $C_4\times F_{193}$ x 6, $C_{1544}:C_{96}$ x 3, $C_{2316}:C_{64}$ x 2, $C_{1544}:C_{96}$

Order 111168: $C_3\times F_{193}$ x 8, $C_{1158}:C_{96}$ x 4, $C_{2316}:C_{48}$ x 2, $C_{4632}:C_{24}$

Order 98816: $C_{1544}:C_{64}$

Order 74112: $C_2\times F_{193}$ x 12, $C_{772}:C_{96}$ x 6, $C_{1158}:C_{64}$ x 4, $C_{1544}:C_{48}$ x 3, $C_{2316}:C_{32}$ x 2, $C_{1544}:C_{48}$

Order 55584: $C_{579}:C_{96}$ x 8, $C_{1158}:C_{48}$ x 4, $C_{2316}:C_{24}$ x 2, $C_{4632}:C_{12}$

Order 49408: $C_{772}:C_{64}$ x 2, $C_{1544}:C_{32}$

Order 37056: $F_{193}$ x 24, $C_{386}:C_{96}$ x 12, $C_{193}:C_{192}$ x 8, $C_{772}:C_{48}$ x 6, $C_{1158}:C_{32}$ x 4, $C_{1544}:C_{24}$ x 3, $C_{2316}:C_{16}$ x 2, $C_{1544}:C_{24}$

Order 27792: $C_{579}:C_{48}$ x 8, $C_{1158}:C_{24}$ x 4, $C_{4632}:C_6$, $C_{2316}:C_{12}$, $C_{2316}.C_{12}$

Order 24704: $C_{386}:C_{64}$ x 4, $C_{772}:C_{32}$ x 2, $C_{1544}:C_{16}$

Order 18528: $C_{193}:C_{96}$ x 24, $C_{386}:C_{48}$ x 12, $C_{193}:C_{96}$ x 8, $C_{772}:C_{24}$ x 6, $C_{1158}:C_{16}$ x 4, $C_{1544}:C_{12}$ x 3, $C_{2316}:C_8$ x 2, $C_{1544}:C_{12}$

Order 13896: $C_{579}:C_{24}$ x 8, $C_{579}:C_{24}$ x 2, $C_{1158}:C_{12}$ x 2, $C_{4632}:C_3$, $C_{2316}:C_6$, $C_{579}:C_{24}$

Order 12352: $C_{193}:C_{64}$ x 8, $C_{386}:C_{32}$ x 4, $C_{772}:C_{16}$ x 2, $C_{1544}:C_8$

Order 9264: $C_{193}:C_{48}$ x 24, $C_{386}:C_{24}$ x 12, $C_{193}:C_{48}$ x 8, $C_{1158}:C_8$ x 4, $C_{1544}:C_6$ x 3, $D_{193}:C_{24}$ x 3, $C_{772}:C_{12}$ x 3, $C_{24}\times D_{193}$, $C_{2316}:C_4$, $D_{193}:C_{24}$

Order 6948: $C_{579}:C_{12}$ x 4, $C_{579}:C_{12}$, $C_{1158}:C_6$, $C_{2316}:C_3$

Order 6176: $C_{193}:C_{32}$ x 8, $C_{386}:C_{16}$ x 4, $C_{772}:C_8$ x 2, $C_{1544}:C_4$

Order 4632: $C_{193}:C_{24}$ x 24, $C_{193}:C_{24}$ x 8, $C_{193}:C_{24}$ x 6, $C_{386}:C_{12}$ x 6, $C_{772}:C_6$ x 3, $C_{1544}:C_3$ x 3, $C_{193}:C_{24}$ x 3, $C_{1158}:C_4$ x 2, $C_{193}:C_{24}$ x 2, $C_{4632}$, $C_{12}\times D_{193}$, $C_{193}:C_{24}$

Order 4608: $C_{24}\times C_{192}$

Order 3474: $C_{579}:C_6$ x 2, $C_{1158}:C_3$

Order 3088: $C_{193}:C_{16}$ x 8, $C_{386}:C_8$ x 4, $D_{193}:C_8$, $C_{772}:C_4$, $C_8\times D_{193}$

Order 2316: $C_{193}:C_{12}$ x 12, $C_{193}:C_{12}$ x 4, $C_{193}:C_{12}$ x 3, $C_{386}:C_6$ x 3, $C_{193}:C_{12}$ x 3, $C_3\times D_{386}$, $C_{193}:C_{12}$, $C_{2316}$

Order 2304: $C_{12}\times C_{192}$ x 2, $C_{24}\times C_{96}$

Order 1737: $C_{579}:C_3$

Order 1544: $C_{193}:C_8$ x 8, $C_{386}:C_4$ x 2, $C_{193}:C_8$ x 2, $C_4\times D_{193}$, $C_{193}:C_8$, $C_{1544}$

Order 1536: $C_8\times C_{192}$ x 4

Order 1158: $C_{193}:C_6$ x 6, $C_{193}:C_6$ x 3, $C_3\times D_{193}$ x 2, $C_{1158}$

Order 1152: $C_6\times C_{192}$ x 4, $C_{12}\times C_{96}$ x 2, $C_{24}\times C_{48}$

Order 772: $C_{193}:C_4$ x 4, $D_{386}$, $C_{772}$, $C_{193}:C_4$

Order 768: $C_4\times C_{192}$ x 8, $C_8\times C_{96}$ x 4

Order 579: $C_{193}:C_3$ x 3, $C_{579}$

Order 576: $C_3\times C_{192}$ x 8, $C_6\times C_{96}$ x 4, $C_{12}\times C_{48}$ x 2, $C_{24}^2$

Order 512: $C_8\times C_{64}$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 384: $C_2\times C_{192}$ x 16, $C_4\times C_{96}$ x 8, $C_8\times C_{48}$ x 4

Order 288: $C_3\times C_{96}$ x 8, $C_6\times C_{48}$ x 4, $C_{12}\times C_{24}$ x 3

Order 256: $C_4\times C_{64}$ x 2, $C_8\times C_{32}$

Order 193: $C_{193}$

Order 192: $C_{192}$ x 32, $C_2\times C_{96}$ x 16, $C_4\times C_{48}$ x 8, $C_8\times C_{24}$ x 4

Order 144: $C_3\times C_{48}$ x 8, $C_6\times C_{24}$ x 6, $C_{12}^2$

Order 128: $C_2\times C_{64}$ x 4, $C_4\times C_{32}$ x 2, $C_8\times C_{16}$

Order 96: $C_{96}$ x 32, $C_2\times C_{48}$ x 16, $C_4\times C_{24}$ x 12

Order 72: $C_3\times C_{24}$ x 12, $C_6\times C_{12}$ x 3

Order 64: $C_{64}$ x 8, $C_2\times C_{32}$ x 4, $C_4\times C_{16}$ x 2, $C_8^2$

Order 48: $C_{48}$ x 32, $C_2\times C_{24}$ x 24, $C_4\times C_{12}$ x 4

Order 36: $C_3\times C_{12}$ x 6, $C_6^2$

Order 32: $C_{32}$ x 8, $C_2\times C_{16}$ x 4, $C_4\times C_8$ x 3

Order 24: $C_{24}$ x 48, $C_2\times C_{12}$ x 12

Order 18: $C_3\times C_6$ x 3

Order 16: $C_{16}$ x 8, $C_2\times C_8$ x 6, $C_4^2$

Order 12: $C_{12}$ x 24, $C_2\times C_6$ x 4

Order 9: $C_3^2$

Order 8: $C_8$ x 12, $C_2\times C_4$ x 3

Order 6: $C_6$ x 12

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 889344: $C_{24}\times F_{193}$

Order 444672: $C_{12}\times F_{193}$, $C_{4632}:C_{96}$

Order 296448: $C_{1544}:C_{192}$, $C_8\times F_{193}$

Order 222336: $C_{2316}:C_{96}$ x 2, $C_{4632}:C_{48}$, $C_6\times F_{193}$

Order 148224: $C_{2316}:C_{64}$, $C_{1544}:C_{96}$, $C_{1544}:C_{96}$, $C_4\times F_{193}$

Order 111168: $C_{1158}:C_{96}$ x 2, $C_{2316}:C_{48}$ x 2, $C_{4632}:C_{24}$, $C_3\times F_{193}$

Order 98816: $C_{1544}:C_{64}$

Order 74112: $C_{2316}:C_{32}$ x 2, $C_{772}:C_{96}$ x 2, $C_{1158}:C_{64}$, $C_{1544}:C_{48}$, $C_{1544}:C_{48}$, $C_2\times F_{193}$

Order 55584: $C_{1158}:C_{48}$ x 3, $C_{2316}:C_{24}$ x 2, $C_{579}:C_{96}$ x 2, $C_{4632}:C_{12}$

Order 49408: $C_{772}:C_{64}$, $C_{1544}:C_{32}$

Order 37056: $C_{2316}:C_{16}$ x 2, $C_{1158}:C_{32}$ x 2, $C_{386}:C_{96}$ x 2, $C_{772}:C_{48}$ x 2, $C_{1544}:C_{24}$, $C_{1544}:C_{24}$, $C_{193}:C_{192}$, $F_{193}$

Order 27792: $C_{1158}:C_{24}$ x 3, $C_{579}:C_{48}$ x 3, $C_{4632}:C_6$, $C_{2316}:C_{12}$, $C_{2316}.C_{12}$

Order 24704: $C_{772}:C_{32}$ x 2, $C_{386}:C_{64}$, $C_{1544}:C_{16}$

Order 18528: $C_{1158}:C_{16}$ x 3, $C_{386}:C_{48}$ x 3, $C_{2316}:C_8$ x 2, $C_{772}:C_{24}$ x 2, $C_{193}:C_{96}$ x 2, $C_{193}:C_{96}$ x 2, $C_{1544}:C_{12}$, $C_{1544}:C_{12}$

Order 13896: $C_{579}:C_{24}$ x 4, $C_{1158}:C_{12}$ x 2, $C_{4632}:C_3$, $C_{2316}:C_6$, $C_{579}:C_{24}$, $C_{579}:C_{24}$

Order 12352: $C_{386}:C_{32}$ x 2, $C_{772}:C_{16}$ x 2, $C_{193}:C_{64}$, $C_{1544}:C_8$

Order 9264: $C_{1158}:C_8$ x 3, $C_{386}:C_{24}$ x 3, $C_{193}:C_{48}$ x 3, $C_{193}:C_{48}$ x 3, $C_{24}\times D_{193}$, $C_{1544}:C_6$, $C_{2316}:C_4$, $D_{193}:C_{24}$, $D_{193}:C_{24}$, $C_{772}:C_{12}$

Order 6948: $C_{579}:C_{12}$ x 3, $C_{579}:C_{12}$, $C_{1158}:C_6$, $C_{2316}:C_3$

Order 6176: $C_{386}:C_{16}$ x 3, $C_{193}:C_{32}$ x 2, $C_{772}:C_8$ x 2, $C_{1544}:C_4$

Order 4632: $C_{193}:C_{24}$ x 4, $C_{193}:C_{24}$ x 4, $C_{1158}:C_4$ x 2, $C_{386}:C_{12}$ x 2, $C_{4632}$, $C_{12}\times D_{193}$, $C_{772}:C_6$, $C_{1544}:C_3$, $C_{193}:C_{24}$, $C_{193}:C_{24}$, $C_{193}:C_{24}$, $C_{193}:C_{24}$

Order 4608: $C_{24}\times C_{192}$

Order 3474: $C_{579}:C_6$ x 2, $C_{1158}:C_3$

Order 3088: $C_{193}:C_{16}$ x 3, $C_{386}:C_8$ x 3, $D_{193}:C_8$, $C_{772}:C_4$, $C_8\times D_{193}$

Order 2316: $C_{193}:C_{12}$ x 3, $C_{193}:C_{12}$ x 3, $C_3\times D_{386}$, $C_{193}:C_{12}$, $C_{193}:C_{12}$, $C_{386}:C_6$, $C_{193}:C_{12}$, $C_{2316}$

Order 2304: $C_{12}\times C_{192}$, $C_{24}\times C_{96}$

Order 1737: $C_{579}:C_3$

Order 1544: $C_{193}:C_8$ x 4, $C_{386}:C_4$ x 2, $C_4\times D_{193}$, $C_{193}:C_8$, $C_{193}:C_8$, $C_{1544}$

Order 1536: $C_8\times C_{192}$ x 2

Order 1158: $C_3\times D_{193}$ x 2, $C_{193}:C_6$ x 2, $C_{1158}$, $C_{193}:C_6$

Order 1152: $C_{12}\times C_{96}$ x 2, $C_6\times C_{192}$, $C_{24}\times C_{48}$

Order 772: $C_{193}:C_4$ x 3, $D_{386}$, $C_{772}$, $C_{193}:C_4$

Order 768: $C_4\times C_{192}$ x 2, $C_8\times C_{96}$ x 2

Order 579: $C_{579}$, $C_{193}:C_3$

Order 576: $C_6\times C_{96}$ x 2, $C_{12}\times C_{48}$ x 2, $C_3\times C_{192}$, $C_{24}^2$

Order 512: $C_8\times C_{64}$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 384: $C_4\times C_{96}$ x 4, $C_2\times C_{192}$ x 2, $C_8\times C_{48}$ x 2

Order 288: $C_6\times C_{48}$ x 3, $C_{12}\times C_{24}$ x 3, $C_3\times C_{96}$ x 2

Order 256: $C_4\times C_{64}$, $C_8\times C_{32}$

Order 193: $C_{193}$

Order 192: $C_2\times C_{96}$ x 4, $C_4\times C_{48}$ x 4, $C_{192}$ x 2, $C_8\times C_{24}$ x 2

Order 144: $C_6\times C_{24}$ x 5, $C_3\times C_{48}$ x 3, $C_{12}^2$

Order 128: $C_4\times C_{32}$ x 2, $C_8\times C_{16}$, $C_2\times C_{64}$

Order 96: $C_2\times C_{48}$ x 6, $C_4\times C_{24}$ x 6, $C_{96}$ x 4

Order 72: $C_3\times C_{24}$ x 7, $C_6\times C_{12}$ x 3

Order 64: $C_2\times C_{32}$ x 2, $C_4\times C_{16}$ x 2, $C_8^2$, $C_{64}$

Order 48: $C_2\times C_{24}$ x 10, $C_{48}$ x 6, $C_4\times C_{12}$ x 2

Order 36: $C_3\times C_{12}$ x 5, $C_6^2$

Order 32: $C_4\times C_8$ x 3, $C_2\times C_{16}$ x 3, $C_{32}$ x 2

Order 24: $C_{24}$ x 14, $C_2\times C_{12}$ x 6

Order 18: $C_3\times C_6$ x 3

Order 16: $C_2\times C_8$ x 5, $C_{16}$ x 3, $C_4^2$

Order 12: $C_{12}$ x 10, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $C_8$ x 7, $C_2\times C_4$ x 3

Order 6: $C_6$ x 6

Order 4: $C_4$ x 5, $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 889344: $C_{24}\times F_{193}$ ($C_1$)

Order 444672: $C_{12}\times F_{193}$ ($C_2$) x 2, $C_{4632}:C_{96}$ ($C_2$)

Order 296448: $C_8\times F_{193}$ ($C_3$) x 3, $C_{1544}:C_{192}$ ($C_3$)

Order 222336: $C_6\times F_{193}$ ($C_4$) x 4, $C_{2316}:C_{96}$ ($C_2^2$), $C_{2316}:C_{96}$ ($C_4$), $C_{4632}:C_{48}$ ($C_4$)

Order 148224: $C_4\times F_{193}$ ($C_6$) x 6, $C_{1544}:C_{96}$ ($C_6$) x 3, $C_{2316}:C_{64}$ ($C_6$) x 2, $C_{1544}:C_{96}$ ($C_6$)

Order 111168: $C_3\times F_{193}$ ($C_8$) x 8, $C_{1158}:C_{96}$ ($C_2\times C_4$) x 2, $C_{1158}:C_{96}$ ($C_8$) x 2, $C_{4632}:C_{24}$ ($C_8$), $C_{2316}:C_{48}$ ($C_2\times C_4$), $C_{2316}:C_{48}$ ($C_8$)

Order 98816: $C_{1544}:C_{64}$ ($C_3^2$)

Order 74112: $C_2\times F_{193}$ ($C_{12}$) x 12, $C_{1158}:C_{64}$ ($C_{12}$) x 4, $C_{772}:C_{96}$ ($C_2\times C_6$) x 3, $C_{772}:C_{96}$ ($C_{12}$) x 3, $C_{1544}:C_{48}$ ($C_{12}$) x 3, $C_{2316}:C_{32}$ ($C_2\times C_6$), $C_{2316}:C_{32}$ ($C_{12}$), $C_{1544}:C_{48}$ ($C_{12}$)

Order 55584: $C_{579}:C_{96}$ ($C_2\times C_8$) x 4, $C_{579}:C_{96}$ ($C_{16}$) x 4, $C_{1158}:C_{48}$ ($C_{16}$) x 2, $C_{2316}:C_{24}$ ($C_2\times C_8$), $C_{2316}:C_{24}$ ($C_{16}$), $C_{4632}:C_{12}$ ($C_{16}$), $C_{1158}:C_{48}$ ($C_2\times C_8$), $C_{1158}:C_{48}$ ($C_4^2$)

Order 49408: $C_{772}:C_{64}$ ($C_3\times C_6$) x 2, $C_{1544}:C_{32}$ ($C_3\times C_6$)

Order 37056: $F_{193}$ ($C_{24}$) x 24, $C_{193}:C_{192}$ ($C_{24}$) x 8, $C_{386}:C_{96}$ ($C_2\times C_{12}$) x 6, $C_{386}:C_{96}$ ($C_{24}$) x 6, $C_{1544}:C_{24}$ ($C_{24}$) x 3, $C_{772}:C_{48}$ ($C_2\times C_{12}$) x 3, $C_{772}:C_{48}$ ($C_{24}$) x 3, $C_{1158}:C_{32}$ ($C_2\times C_{12}$) x 2, $C_{1158}:C_{32}$ ($C_{24}$) x 2, $C_{1544}:C_{24}$ ($C_{24}$), $C_{2316}:C_{16}$ ($C_2\times C_{12}$), $C_{2316}:C_{16}$ ($C_{24}$)

Order 27792: $C_{579}:C_{48}$ ($C_{32}$) x 4, $C_{1158}:C_{24}$ ($C_{32}$) x 2, $C_{579}:C_{48}$ ($C_4\times C_8$) x 2, $C_{579}:C_{48}$ ($C_2\times C_{16}$) x 2, $C_{4632}:C_6$ ($C_{32}$), $C_{2316}:C_{12}$ ($C_2\times C_{16}$), $C_{2316}.C_{12}$ ($C_{32}$), $C_{1158}:C_{24}$ ($C_4\times C_8$), $C_{1158}:C_{24}$ ($C_2\times C_{16}$)

Order 24704: $C_{386}:C_{64}$ ($C_3\times C_{12}$) x 4, $C_{772}:C_{32}$ ($C_3\times C_{12}$), $C_{772}:C_{32}$ ($C_6^2$), $C_{1544}:C_{16}$ ($C_3\times C_{12}$)

Order 18528: $C_{193}:C_{96}$ ($C_2\times C_{24}$) x 12, $C_{193}:C_{96}$ ($C_{48}$) x 12, $C_{386}:C_{48}$ ($C_{48}$) x 6, $C_{193}:C_{96}$ ($C_2\times C_{24}$) x 4, $C_{193}:C_{96}$ ($C_{48}$) x 4, $C_{772}:C_{24}$ ($C_2\times C_{24}$) x 3, $C_{772}:C_{24}$ ($C_{48}$) x 3, $C_{1544}:C_{12}$ ($C_{48}$) x 3, $C_{386}:C_{48}$ ($C_2\times C_{24}$) x 3, $C_{386}:C_{48}$ ($C_4\times C_{12}$) x 3, $C_{1158}:C_{16}$ ($C_{48}$) x 2, $C_{1544}:C_{12}$ ($C_{48}$), $C_{2316}:C_8$ ($C_2\times C_{24}$), $C_{2316}:C_8$ ($C_{48}$), $C_{1158}:C_{16}$ ($C_2\times C_{24}$), $C_{1158}:C_{16}$ ($C_4\times C_{12}$)

Order 13896: $C_{579}:C_{24}$ ($C_{64}$) x 4, $C_{579}:C_{24}$ ($C_2\times C_{32}$) x 2, $C_{579}:C_{24}$ ($C_{64}$) x 2, $C_{4632}:C_3$ ($C_{64}$), $C_{2316}:C_6$ ($C_2\times C_{32}$), $C_{579}:C_{24}$ ($C_{64}$), $C_{579}:C_{24}$ ($C_4\times C_{16}$), $C_{579}:C_{24}$ ($C_8^2$), $C_{1158}:C_{12}$ ($C_2\times C_{32}$), $C_{1158}:C_{12}$ ($C_4\times C_{16}$)

Order 12352: $C_{193}:C_{64}$ ($C_3\times C_{24}$) x 8, $C_{386}:C_{32}$ ($C_6\times C_{12}$) x 2, $C_{386}:C_{32}$ ($C_3\times C_{24}$) x 2, $C_{772}:C_{16}$ ($C_6\times C_{12}$), $C_{772}:C_{16}$ ($C_3\times C_{24}$), $C_{1544}:C_8$ ($C_3\times C_{24}$)

Order 9264: $C_{193}:C_{48}$ ($C_{96}$) x 12, $C_{386}:C_{24}$ ($C_{96}$) x 6, $C_{193}:C_{48}$ ($C_2\times C_{48}$) x 6, $C_{193}:C_{48}$ ($C_4\times C_{24}$) x 6, $C_{193}:C_{48}$ ($C_{96}$) x 4, $C_{1544}:C_6$ ($C_{96}$) x 3, $C_{386}:C_{24}$ ($C_2\times C_{48}$) x 3, $C_{386}:C_{24}$ ($C_4\times C_{24}$) x 3, $D_{193}:C_{24}$ ($C_{96}$) x 3, $C_{772}:C_{12}$ ($C_2\times C_{48}$) x 3, $C_{1158}:C_8$ ($C_{96}$) x 2, $C_{193}:C_{48}$ ($C_2\times C_{48}$) x 2, $C_{193}:C_{48}$ ($C_4\times C_{24}$) x 2, $C_{24}\times D_{193}$ ($C_{96}$), $C_{2316}:C_4$ ($C_2\times C_{48}$), $C_{1158}:C_8$ ($C_2\times C_{48}$), $C_{1158}:C_8$ ($C_4\times C_{24}$), $D_{193}:C_{24}$ ($C_{96}$)

Order 6948: $C_{579}:C_{12}$ ($C_2\times C_{64}$) x 2, $C_{579}:C_{12}$ ($C_2\times C_{64}$), $C_{1158}:C_6$ ($C_4\times C_{32}$), $C_{2316}:C_3$ ($C_2\times C_{64}$), $C_{579}:C_{12}$ ($C_8\times C_{16}$), $C_{579}:C_{12}$ ($C_4\times C_{32}$)

Order 6176: $C_{193}:C_{32}$ ($C_3\times C_{48}$) x 4, $C_{193}:C_{32}$ ($C_6\times C_{24}$) x 4, $C_{386}:C_{16}$ ($C_3\times C_{48}$) x 2, $C_{386}:C_{16}$ ($C_6\times C_{24}$), $C_{386}:C_{16}$ ($C_{12}^2$), $C_{772}:C_8$ ($C_3\times C_{48}$), $C_{772}:C_8$ ($C_6\times C_{24}$), $C_{1544}:C_4$ ($C_3\times C_{48}$)

Order 4632: $C_{193}:C_{24}$ ($C_{192}$) x 12, $C_{193}:C_{24}$ ($C_2\times C_{96}$) x 6, $C_{193}:C_{24}$ ($C_{192}$) x 6, $C_{193}:C_{24}$ ($C_{192}$) x 4, $C_{772}:C_6$ ($C_2\times C_{96}$) x 3, $C_{1544}:C_3$ ($C_{192}$) x 3, $C_{193}:C_{24}$ ($C_{192}$) x 3, $C_{193}:C_{24}$ ($C_4\times C_{48}$) x 3, $C_{193}:C_{24}$ ($C_8\times C_{24}$) x 3, $C_{386}:C_{12}$ ($C_2\times C_{96}$) x 3, $C_{386}:C_{12}$ ($C_4\times C_{48}$) x 3, $C_{193}:C_{24}$ ($C_2\times C_{96}$) x 2, $C_{193}:C_{24}$ ($C_{192}$) x 2, $C_{4632}$ ($C_{192}$), $C_{12}\times D_{193}$ ($C_2\times C_{96}$), $C_{1158}:C_4$ ($C_2\times C_{96}$), $C_{1158}:C_4$ ($C_4\times C_{48}$), $C_{193}:C_{24}$ ($C_{192}$), $C_{193}:C_{24}$ ($C_4\times C_{48}$), $C_{193}:C_{24}$ ($C_8\times C_{24}$)

Order 3474: $C_{1158}:C_3$ ($C_4\times C_{64}$), $C_{579}:C_6$ ($C_4\times C_{64}$), $C_{579}:C_6$ ($C_8\times C_{32}$)

Order 3088: $C_{193}:C_{16}$ ($C_3\times C_{96}$) x 4, $C_{193}:C_{16}$ ($C_6\times C_{48}$) x 2, $C_{193}:C_{16}$ ($C_{12}\times C_{24}$) x 2, $C_{386}:C_8$ ($C_3\times C_{96}$) x 2, $C_{386}:C_8$ ($C_6\times C_{48}$), $C_{386}:C_8$ ($C_{12}\times C_{24}$), $D_{193}:C_8$ ($C_3\times C_{96}$), $C_{772}:C_4$ ($C_6\times C_{48}$), $C_8\times D_{193}$ ($C_3\times C_{96}$)

Order 2316: $C_{193}:C_{12}$ ($C_2\times C_{192}$) x 6, $C_{193}:C_{12}$ ($C_2\times C_{192}$) x 3, $C_{386}:C_6$ ($C_4\times C_{96}$) x 3, $C_{193}:C_{12}$ ($C_2\times C_{192}$) x 3, $C_{193}:C_{12}$ ($C_4\times C_{96}$) x 3, $C_{193}:C_{12}$ ($C_8\times C_{48}$) x 3, $C_{193}:C_{12}$ ($C_2\times C_{192}$) x 2, $C_3\times D_{386}$ ($C_4\times C_{96}$), $C_{193}:C_{12}$ ($C_2\times C_{192}$), $C_{193}:C_{12}$ ($C_4\times C_{96}$), $C_{193}:C_{12}$ ($C_8\times C_{48}$), $C_{2316}$ ($C_2\times C_{192}$)

Order 1737: $C_{579}:C_3$ ($C_8\times C_{64}$)

Order 1544: $C_{193}:C_8$ ($C_3\times C_{192}$) x 4, $C_{193}:C_8$ ($C_6\times C_{96}$) x 2, $C_{193}:C_8$ ($C_3\times C_{192}$) x 2, $C_4\times D_{193}$ ($C_6\times C_{96}$), $C_{193}:C_8$ ($C_3\times C_{192}$), $C_{193}:C_8$ ($C_{12}\times C_{48}$), $C_{193}:C_8$ ($C_{24}^2$), $C_{386}:C_4$ ($C_6\times C_{96}$), $C_{386}:C_4$ ($C_{12}\times C_{48}$), $C_{1544}$ ($C_3\times C_{192}$)

Order 1158: $C_{193}:C_6$ ($C_4\times C_{192}$) x 3, $C_{193}:C_6$ ($C_4\times C_{192}$) x 3, $C_{193}:C_6$ ($C_8\times C_{96}$) x 3, $C_{1158}$ ($C_4\times C_{192}$), $C_3\times D_{193}$ ($C_4\times C_{192}$), $C_3\times D_{193}$ ($C_8\times C_{96}$)

Order 772: $C_{193}:C_4$ ($C_6\times C_{192}$) x 2, $D_{386}$ ($C_{12}\times C_{96}$), $C_{193}:C_4$ ($C_{12}\times C_{96}$), $C_{193}:C_4$ ($C_{24}\times C_{48}$), $C_{772}$ ($C_6\times C_{192}$), $C_{193}:C_4$ ($C_6\times C_{192}$)

Order 579: $C_{193}:C_3$ ($C_8\times C_{192}$) x 3, $C_{579}$ ($C_8\times C_{192}$)

Order 386: $C_{386}$ ($C_{12}\times C_{192}$), $D_{193}$ ($C_{12}\times C_{192}$), $D_{193}$ ($C_{24}\times C_{96}$)

Order 193: $C_{193}$ ($C_{24}\times C_{192}$)

Order 24: $C_{24}$ ($F_{193}$)

Order 12: $C_{12}$ ($C_2\times F_{193}$)

Order 8: $C_8$ ($C_3\times F_{193}$)

Order 6: $C_6$ ($C_4\times F_{193}$)

Order 4: $C_4$ ($C_6\times F_{193}$)

Order 3: $C_3$ ($C_8\times F_{193}$)

Order 2: $C_2$ ($C_{12}\times F_{193}$)

Order 1: $C_1$ ($C_{24}\times F_{193}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 889344: $C_{24}\times F_{193}$ ($C_1$)

Order 444672: $C_{12}\times F_{193}$ ($C_2$), $C_{4632}:C_{96}$ ($C_2$)

Order 296448: $C_{1544}:C_{192}$ ($C_3$), $C_8\times F_{193}$ ($C_3$)

Order 222336: $C_{2316}:C_{96}$ ($C_2^2$), $C_{2316}:C_{96}$ ($C_4$), $C_{4632}:C_{48}$ ($C_4$), $C_6\times F_{193}$ ($C_4$)

Order 148224: $C_{2316}:C_{64}$ ($C_6$), $C_{1544}:C_{96}$ ($C_6$), $C_{1544}:C_{96}$ ($C_6$), $C_4\times F_{193}$ ($C_6$)

Order 111168: $C_{4632}:C_{24}$ ($C_8$), $C_{1158}:C_{96}$ ($C_2\times C_4$), $C_{1158}:C_{96}$ ($C_8$), $C_{2316}:C_{48}$ ($C_2\times C_4$), $C_{2316}:C_{48}$ ($C_8$), $C_3\times F_{193}$ ($C_8$)

Order 98816: $C_{1544}:C_{64}$ ($C_3^2$)

Order 74112: $C_{2316}:C_{32}$ ($C_2\times C_6$), $C_{2316}:C_{32}$ ($C_{12}$), $C_{1158}:C_{64}$ ($C_{12}$), $C_{1544}:C_{48}$ ($C_{12}$), $C_{772}:C_{96}$ ($C_2\times C_6$), $C_{772}:C_{96}$ ($C_{12}$), $C_{1544}:C_{48}$ ($C_{12}$), $C_2\times F_{193}$ ($C_{12}$)

Order 55584: $C_{2316}:C_{24}$ ($C_2\times C_8$), $C_{2316}:C_{24}$ ($C_{16}$), $C_{579}:C_{96}$ ($C_2\times C_8$), $C_{579}:C_{96}$ ($C_{16}$), $C_{4632}:C_{12}$ ($C_{16}$), $C_{1158}:C_{48}$ ($C_2\times C_8$), $C_{1158}:C_{48}$ ($C_4^2$), $C_{1158}:C_{48}$ ($C_{16}$)

Order 49408: $C_{772}:C_{64}$ ($C_3\times C_6$), $C_{1544}:C_{32}$ ($C_3\times C_6$)

Order 37056: $C_{1544}:C_{24}$ ($C_{24}$), $C_{2316}:C_{16}$ ($C_2\times C_{12}$), $C_{2316}:C_{16}$ ($C_{24}$), $C_{1158}:C_{32}$ ($C_2\times C_{12}$), $C_{1158}:C_{32}$ ($C_{24}$), $C_{1544}:C_{24}$ ($C_{24}$), $C_{193}:C_{192}$ ($C_{24}$), $C_{386}:C_{96}$ ($C_2\times C_{12}$), $C_{386}:C_{96}$ ($C_{24}$), $C_{772}:C_{48}$ ($C_2\times C_{12}$), $C_{772}:C_{48}$ ($C_{24}$), $F_{193}$ ($C_{24}$)

Order 27792: $C_{4632}:C_6$ ($C_{32}$), $C_{2316}:C_{12}$ ($C_2\times C_{16}$), $C_{2316}.C_{12}$ ($C_{32}$), $C_{1158}:C_{24}$ ($C_4\times C_8$), $C_{1158}:C_{24}$ ($C_2\times C_{16}$), $C_{1158}:C_{24}$ ($C_{32}$), $C_{579}:C_{48}$ ($C_4\times C_8$), $C_{579}:C_{48}$ ($C_2\times C_{16}$), $C_{579}:C_{48}$ ($C_{32}$)

Order 24704: $C_{386}:C_{64}$ ($C_3\times C_{12}$), $C_{772}:C_{32}$ ($C_3\times C_{12}$), $C_{772}:C_{32}$ ($C_6^2$), $C_{1544}:C_{16}$ ($C_3\times C_{12}$)

Order 18528: $C_{1544}:C_{12}$ ($C_{48}$), $C_{2316}:C_8$ ($C_2\times C_{24}$), $C_{2316}:C_8$ ($C_{48}$), $C_{772}:C_{24}$ ($C_2\times C_{24}$), $C_{772}:C_{24}$ ($C_{48}$), $C_{1158}:C_{16}$ ($C_2\times C_{24}$), $C_{1158}:C_{16}$ ($C_4\times C_{12}$), $C_{1158}:C_{16}$ ($C_{48}$), $C_{193}:C_{96}$ ($C_2\times C_{24}$), $C_{193}:C_{96}$ ($C_{48}$), $C_{193}:C_{96}$ ($C_2\times C_{24}$), $C_{193}:C_{96}$ ($C_{48}$), $C_{1544}:C_{12}$ ($C_{48}$), $C_{386}:C_{48}$ ($C_2\times C_{24}$), $C_{386}:C_{48}$ ($C_4\times C_{12}$), $C_{386}:C_{48}$ ($C_{48}$)

Order 13896: $C_{4632}:C_3$ ($C_{64}$), $C_{2316}:C_6$ ($C_2\times C_{32}$), $C_{579}:C_{24}$ ($C_{64}$), $C_{579}:C_{24}$ ($C_2\times C_{32}$), $C_{579}:C_{24}$ ($C_4\times C_{16}$), $C_{579}:C_{24}$ ($C_8^2$), $C_{579}:C_{24}$ ($C_{64}$), $C_{579}:C_{24}$ ($C_{64}$), $C_{1158}:C_{12}$ ($C_2\times C_{32}$), $C_{1158}:C_{12}$ ($C_4\times C_{16}$)

Order 12352: $C_{193}:C_{64}$ ($C_3\times C_{24}$), $C_{386}:C_{32}$ ($C_6\times C_{12}$), $C_{386}:C_{32}$ ($C_3\times C_{24}$), $C_{772}:C_{16}$ ($C_6\times C_{12}$), $C_{772}:C_{16}$ ($C_3\times C_{24}$), $C_{1544}:C_8$ ($C_3\times C_{24}$)

Order 9264: $C_{24}\times D_{193}$ ($C_{96}$), $C_{1544}:C_6$ ($C_{96}$), $C_{2316}:C_4$ ($C_2\times C_{48}$), $C_{1158}:C_8$ ($C_2\times C_{48}$), $C_{1158}:C_8$ ($C_4\times C_{24}$), $C_{1158}:C_8$ ($C_{96}$), $C_{386}:C_{24}$ ($C_2\times C_{48}$), $C_{386}:C_{24}$ ($C_4\times C_{24}$), $C_{386}:C_{24}$ ($C_{96}$), $D_{193}:C_{24}$ ($C_{96}$), $C_{193}:C_{48}$ ($C_2\times C_{48}$), $C_{193}:C_{48}$ ($C_4\times C_{24}$), $C_{193}:C_{48}$ ($C_{96}$), $D_{193}:C_{24}$ ($C_{96}$), $C_{772}:C_{12}$ ($C_2\times C_{48}$), $C_{193}:C_{48}$ ($C_2\times C_{48}$), $C_{193}:C_{48}$ ($C_4\times C_{24}$), $C_{193}:C_{48}$ ($C_{96}$)

Order 6948: $C_{579}:C_{12}$ ($C_2\times C_{64}$), $C_{1158}:C_6$ ($C_4\times C_{32}$), $C_{2316}:C_3$ ($C_2\times C_{64}$), $C_{579}:C_{12}$ ($C_8\times C_{16}$), $C_{579}:C_{12}$ ($C_2\times C_{64}$), $C_{579}:C_{12}$ ($C_4\times C_{32}$)

Order 6176: $C_{193}:C_{32}$ ($C_3\times C_{48}$), $C_{193}:C_{32}$ ($C_6\times C_{24}$), $C_{386}:C_{16}$ ($C_3\times C_{48}$), $C_{386}:C_{16}$ ($C_6\times C_{24}$), $C_{386}:C_{16}$ ($C_{12}^2$), $C_{772}:C_8$ ($C_3\times C_{48}$), $C_{772}:C_8$ ($C_6\times C_{24}$), $C_{1544}:C_4$ ($C_3\times C_{48}$)

Order 4632: $C_{4632}$ ($C_{192}$), $C_{12}\times D_{193}$ ($C_2\times C_{96}$), $C_{1158}:C_4$ ($C_2\times C_{96}$), $C_{1158}:C_4$ ($C_4\times C_{48}$), $C_{772}:C_6$ ($C_2\times C_{96}$), $C_{1544}:C_3$ ($C_{192}$), $C_{193}:C_{24}$ ($C_{192}$), $C_{193}:C_{24}$ ($C_{192}$), $C_{193}:C_{24}$ ($C_{192}$), $C_{193}:C_{24}$ ($C_2\times C_{96}$), $C_{193}:C_{24}$ ($C_4\times C_{48}$), $C_{193}:C_{24}$ ($C_8\times C_{24}$), $C_{193}:C_{24}$ ($C_{192}$), $C_{193}:C_{24}$ ($C_{192}$), $C_{193}:C_{24}$ ($C_2\times C_{96}$), $C_{193}:C_{24}$ ($C_4\times C_{48}$), $C_{193}:C_{24}$ ($C_8\times C_{24}$), $C_{193}:C_{24}$ ($C_{192}$), $C_{386}:C_{12}$ ($C_2\times C_{96}$), $C_{386}:C_{12}$ ($C_4\times C_{48}$)

Order 3474: $C_{1158}:C_3$ ($C_4\times C_{64}$), $C_{579}:C_6$ ($C_4\times C_{64}$), $C_{579}:C_6$ ($C_8\times C_{32}$)

Order 3088: $C_{193}:C_{16}$ ($C_3\times C_{96}$), $C_{193}:C_{16}$ ($C_6\times C_{48}$), $C_{193}:C_{16}$ ($C_{12}\times C_{24}$), $C_{386}:C_8$ ($C_3\times C_{96}$), $C_{386}:C_8$ ($C_6\times C_{48}$), $C_{386}:C_8$ ($C_{12}\times C_{24}$), $D_{193}:C_8$ ($C_3\times C_{96}$), $C_{772}:C_4$ ($C_6\times C_{48}$), $C_8\times D_{193}$ ($C_3\times C_{96}$)

Order 2316: $C_3\times D_{386}$ ($C_4\times C_{96}$), $C_{193}:C_{12}$ ($C_2\times C_{192}$), $C_{193}:C_{12}$ ($C_2\times C_{192}$), $C_{386}:C_6$ ($C_4\times C_{96}$), $C_{193}:C_{12}$ ($C_2\times C_{192}$), $C_{193}:C_{12}$ ($C_2\times C_{192}$), $C_{193}:C_{12}$ ($C_4\times C_{96}$), $C_{193}:C_{12}$ ($C_8\times C_{48}$), $C_{193}:C_{12}$ ($C_2\times C_{192}$), $C_{193}:C_{12}$ ($C_4\times C_{96}$), $C_{193}:C_{12}$ ($C_8\times C_{48}$), $C_{2316}$ ($C_2\times C_{192}$)

Order 1737: $C_{579}:C_3$ ($C_8\times C_{64}$)

Order 1544: $C_4\times D_{193}$ ($C_6\times C_{96}$), $C_{193}:C_8$ ($C_3\times C_{192}$), $C_{193}:C_8$ ($C_3\times C_{192}$), $C_{193}:C_8$ ($C_6\times C_{96}$), $C_{193}:C_8$ ($C_{12}\times C_{48}$), $C_{193}:C_8$ ($C_{24}^2$), $C_{386}:C_4$ ($C_6\times C_{96}$), $C_{386}:C_4$ ($C_{12}\times C_{48}$), $C_{193}:C_8$ ($C_3\times C_{192}$), $C_{1544}$ ($C_3\times C_{192}$)

Order 1158: $C_{1158}$ ($C_4\times C_{192}$), $C_3\times D_{193}$ ($C_4\times C_{192}$), $C_3\times D_{193}$ ($C_8\times C_{96}$), $C_{193}:C_6$ ($C_4\times C_{192}$), $C_{193}:C_6$ ($C_4\times C_{192}$), $C_{193}:C_6$ ($C_8\times C_{96}$)

Order 772: $D_{386}$ ($C_{12}\times C_{96}$), $C_{193}:C_4$ ($C_6\times C_{192}$), $C_{193}:C_4$ ($C_{12}\times C_{96}$), $C_{193}:C_4$ ($C_{24}\times C_{48}$), $C_{772}$ ($C_6\times C_{192}$), $C_{193}:C_4$ ($C_6\times C_{192}$)

Order 579: $C_{579}$ ($C_8\times C_{192}$), $C_{193}:C_3$ ($C_8\times C_{192}$)

Order 386: $C_{386}$ ($C_{12}\times C_{192}$), $D_{193}$ ($C_{12}\times C_{192}$), $D_{193}$ ($C_{24}\times C_{96}$)

Order 193: $C_{193}$ ($C_{24}\times C_{192}$)

Order 24: $C_{24}$ ($F_{193}$)

Order 12: $C_{12}$ ($C_2\times F_{193}$)

Order 8: $C_8$ ($C_3\times F_{193}$)

Order 6: $C_6$ ($C_4\times F_{193}$)

Order 4: $C_4$ ($C_6\times F_{193}$)

Order 3: $C_3$ ($C_8\times F_{193}$)

Order 2: $C_2$ ($C_{12}\times F_{193}$)

Order 1: $C_1$ ($C_{24}\times F_{193}$)

Series

Derived series

$C_{24}\times F_{193}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{24}\times F_{193}$

$\rhd$

$C_{4632}:C_{96}$

$\rhd$

$C_{4632}:C_{48}$

$\rhd$

$C_{4632}:C_{24}$

$\rhd$

$C_{4632}:C_{12}$

$\rhd$

$C_{4632}:C_6$

$\rhd$

$C_{4632}:C_3$

$\rhd$

$C_{4632}$

$\rhd$

$C_{2316}$

$\rhd$

$C_{1158}$

$\rhd$

$C_{579}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{24}\times F_{193}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_{24}$

Character theory

Complex character table

The $4632 \times 4632$ character table is not available for this group.

Rational character table

The $238 \times 238$ rational character table is not available for this group.