Abstract group 1597200.j: $C_{11}^3:(C_{120}:C_{10})$

• Label: 1597200.j
• Order: \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5^{2} \)
• $\card{Z(G)}$: 10
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5^{3} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5^{2} \)
• Perm deg.: $90$
• Trans deg.: $1320$
• Rank: $2$

Group information

Description:	$C_{11}^3:(C_{120}:C_{10})$
Order:	\(1597200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
Exponent:	\(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	Group of order \(127776000\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{3} \cdot 11^{3} \)
Composition factors:	$C_2$ x 4, $C_3$, $C_5$ x 2, $C_{11}$ x 3
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	5	6	8	10	11	12	15	20	22	24	30	33	40	44	55	60	66	110	120	132	165	220	330	660
Elements	1	1453	242	374	26624	242	5324	351872	1330	484	54208	374176	15850	10648	54208	2420	127776	18260	5320	108416	2420	63400	255552	4840	9680	73040	9680	19360	1597200
Conjugacy classes	1	2	1	2	24	1	2	48	18	2	24	48	20	4	24	1	48	25	72	48	1	80	96	2	4	100	4	8	710
Divisions	1	2	1	2	6	1	1	12	13	1	6	12	14	1	6	1	6	8	13	6	1	14	6	1	1	8	1	1	146
Autjugacy classes	1	2	1	2	5	1	1	10	5	2	5	10	6	2	5	1	5	4	5	10	1	6	10	2	1	4	1	2	110

Minimal presentations

Permutation degree:	$90$
Transitive degree:	$1320$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	60	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{10}=b^{120}=c^{11}=d^{11}=e^{11}=[c,d]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $90$ $\langle(86,87,88,89,90), (1,2,4,7,14,9,31,40,8,11,15)(3,16,19,32,18,20,5,12,10,29,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 10 & 2 & 8 & 3 \\ 9 & 5 & 6 & 0 \\ 9 & 5 & 7 & 2 \\ 1 & 9 & 1 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 2 & 9 & 6 \\ 1 & 0 & 2 & 3 \\ 1 & 8 & 3 & 9 \\ 10 & 7 & 7 & 0 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 5 & 9 & 4 & 0 \\ 2 & 2 & 10 & 0 \\ 2 & 2 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 9 & 5 & 3 & 8 \\ 10 & 6 & 4 & 5 \\ 8 & 8 & 6 & 4 \\ 9 & 10 & 6 & 6 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 9 & 8 \\ 10 & 6 & 4 & 9 \\ 6 & 4 & 1 & 9 \\ 0 & 6 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 9 & 0 & 0 & 0 \\ 7 & 5 & 10 & 0 \\ 6 & 6 & 5 & 0 \\ 3 & 6 & 4 & 1 \end{array}\right), \left(\begin{array}{rrrr} 5 & 3 & 8 & 0 \\ 0 & 0 & 1 & 6 \\ 5 & 1 & 4 & 8 \\ 8 & 7 & 5 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 0 & 10 & 3 & 5 \\ 10 & 8 & 3 & 3 \\ 6 & 7 & 4 & 1 \\ 0 & 6 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(C_{11}^3:(C_{120}:C_2))$
Semidirect product:	$C_{11}^3$ $\,\rtimes\,$ $(C_{120}:C_{10})$	$(C_{11}^3.C_6.D_4)$ $\,\rtimes\,$ $C_5^2$	$(C_5\times C_{11}^3:C_{15})$ $\,\rtimes\,$ $\SD_{16}$	$(C_{11}^3:C_5^2)$ $\,\rtimes\,$ $(C_{24}:C_2)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_{11}^3:C_{120})$ . $C_{10}$ (5)	$(C_{11}^3:C_{120})$ . $C_{10}$	$(C_5\times C_{11}^3:C_{40})$ . $S_3$	$(C_5\times C_{11}^3:C_{30})$ . $D_4$	all 59

Elements of the group are displayed as matrices in $\GL_{4}(\F_{11})$.

Homology

Abelianization:	$C_{10}^{2} \simeq C_{2}^{2} \times C_{5}^{2}$
Schur multiplier:	$C_{5}$
Commutator length:	$1$

Subgroups

There are 110 normal subgroups (55 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_{10}$
Commutator:	a subgroup isomorphic to $C_{11}^2:C_{132}$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 6655 (order at least 240) are shown, as well as all normal subgroups of any index.

Order 1597200: $C_{11}^3:(C_{120}:C_{10})$

Order 798600: $C_{11}^3.C_{60}.C_{10}$, $C_5\times C_{11}^3:C_{120}$, $C_{11}^3.C_{60}.C_{10}$

Order 532400: $C_{11}^3.C_4.C_{10}^2$

Order 399300: $C_{11}^2:C_{165}:C_{20}$, $C_5\times C_{11}^3:C_{60}$, $C_{11}\wr C_3:C_{10}^2$

Order 319440: $C_{11}^3:(C_{120}:C_2)$ x 5, $(C_{11}^2\times C_{110}).D_{12}$

Order 266200: $C_5\times C_{11}^3:C_{40}$, $C_{11}^3:(D_4\times C_5^2)$, $C_{11}^3:(Q_8\times C_5^2)$

Order 199650: $C_5\times C_{11}^3:C_{30}$, $C_{11}^3:(S_3\times C_5^2)$

Order 159720: $C_{11}^3:(C_5\times D_{12})$ x 5, $C_{11}^3:C_{120}$ x 5, $C_{11}^3:C_{15}:Q_8$ x 5, $C_{11}^2:C_{165}:Q_8$, $C_5\times C_{11}^2:D_{132}$, $C_{11}^3:C_{120}$

Order 145200: $C_{11}^2:C_{120}:C_{10}$

Order 133100: $C_5\times C_{11}^3:C_{20}$, $C_5\times C_{11}^3:C_{20}$, $C_{11}^3:C_{10}^2$

Order 106480: $C_{11}^3:(C_5\times \SD_{16})$ x 5, $C_5\times C_{11}^3:\SD_{16}$

Order 99825: $C_5\times C_{11}^3:C_{15}$

Order 79860: $C_{11}^3:C_{60}$ x 5, $C_{11}^3:(S_3\times C_{10})$ x 5, $C_{11}\wr C_3:C_{20}$ x 5, $C_{11}\wr C_3:C_{20}$, $C_{11}^2:C_{660}$, $C_5\times C_{11}^2:D_{66}$

Order 72600: $C_{11}^2:(D_{12}\times C_5^2)$, $C_{11}^2:C_{60}.C_{10}$, $C_5\times C_{11}^2:C_{120}$

Order 66550: $C_5\times C_{11}^3:C_{10}$, $C_5\times C_{11}^3:C_{10}$

Order 63888: $C_{11}^3.C_6.D_4$

Order 53240: $C_{11}^3:(C_5\times D_4)$ x 5, $C_{11}^3:(C_5\times Q_8)$ x 5, $C_{11}^3:C_{40}$ x 5, $C_{11}^3:C_{40}$, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$

Order 48400: $C_{11}^2:(\SD_{16}\times C_5^2)$

Order 39930: $C_{11}^3:(C_5\times S_3)$ x 5, $C_{11}^3:C_{30}$ x 5, $C_{11}^2:C_{330}$, $C_5\times C_{11}^2:D_{33}$

Order 36300: $C_{11}^2:(D_6\times C_5^2)$, $C_{11}^2:C_{15}:C_{20}$, $C_5\times C_{11}^2:C_{60}$

Order 33275: $C_{11}^3:C_5^2$

Order 31944: $C_{11}^3:C_{24}$, $C_{11}^2:D_{132}$, $C_{11}\wr C_3:Q_8$

Order 29040: $C_{11}^2:C_{120}:C_2$ x 5, $(C_{11}\times C_{110}).D_{12}$

Order 26620: $C_2\times C_{11}^3:C_{10}$ x 5, $C_{11}^3:C_{20}$ x 5, $C_{11}^3:C_{20}$ x 5, $C_{11}^2:C_{220}$, $C_{11}:D_{11}\times C_{110}$, $C_{11}^2:C_{220}$

Order 24200: $C_{220}.F_{11}$, $C_5\times D_{22}:F_{11}$, $C_{11}^2:(Q_8\times C_5^2)$, $C_5\times C_{11}^2:C_{40}$, $C_5\times D_{22}:F_{11}$

Order 21296: $C_{11}^3:\SD_{16}$

Order 19965: $C_{11}^3:C_{15}$ x 5, $C_{11}^2:C_{165}$

Order 18150: $C_{11}^2:(S_3\times C_5^2)$, $C_5\times C_{11}^2:C_{30}$

Order 15972: $C_{11}^2:D_{66}$, $C_{11}\wr C_3:C_4$, $C_{11}^2:C_{132}$

Order 14520: $C_{11}^2:C_{120}$ x 5, $C_{11}^2:C_{15}:Q_8$ x 5, $C_{11}^2:(C_5\times D_{12})$ x 5, $(C_{11}\times C_{110}).D_6$, $C_{11}^2:C_{120}$, $C_5\times C_{11}^2:D_{12}$

Order 13310: $C_{11}^3:C_{10}$ x 5, $C_{11}^3:C_{10}$ x 5, $C_{11}^2\times C_{110}$, $C_{11}^2:C_{110}$

Order 13200: $C_{132}.C_{10}^2$

Order 12100: $C_{110}.F_{11}$ x 9, $C_{110}:F_{11}$ x 7, $C_5\times C_{11}^2:C_{20}$, $C_{110}.F_{11}$, $C_{110}:F_{11}$

Order 10648: $C_{11}^3:C_8$, $C_{11}^2:D_{44}$, $C_{11}^3:Q_8$

Order 9680: $C_{11}^2:(C_5\times \SD_{16})$ x 5, $C_5\times C_{11}^2:\SD_{16}$

Order 9075: $C_5\times C_{11}^2:C_{15}$

Order 7986: $C_{11}^2:D_{33}$, $C_{11}^2:C_{66}$

Order 7260: $C_{11}^2:C_{60}$ x 5, $C_{11}^2:(S_3\times C_{10})$ x 5, $C_{11}^2:C_3:C_{20}$ x 5, $C_{10}\times C_{11}^2:S_3$, $(C_{11}\times C_{110}).S_3$, $C_{11}^2:C_{60}$

Order 6655: $C_{11}^3:C_5$ x 5, $C_{11}^2\times C_{55}$

Order 6600: $C_{660}.C_{10}$, $C_{660}:C_{10}$, $C_{55}:C_{120}$

Order 6050: $C_5\times C_{11}^2:C_{10}$ x 13, $C_{55}:F_{11}$ x 12, $C_{55}:F_{11}$

Order 5808: $(C_{11}\times C_{22}).D_{12}$

Order 5324: $C_{11}^2:D_{22}$, $C_{11}^2:C_{44}$, $C_{11}^2:C_{44}$

Order 4840: $D_{22}:F_{11}$ x 5, $C_{44}.F_{11}$ x 5, $C_{11}^2:(C_5\times Q_8)$ x 5, $D_{22}:F_{11}$ x 5, $C_{11}^2:C_{40}$ x 5, $C_{44}.C_{110}$, $C_{110}.D_{22}$, $C_{110}.D_{22}$, $C_{55}:D_{44}$, $C_{11}^2:C_{40}$

Order 4400: $C_{44}.C_{10}^2$

Order 3993: $C_{11}\wr C_3$

Order 3630: $C_{11}^2:(C_5\times S_3)$ x 5, $C_{11}^2:C_{30}$ x 5, $C_5\times C_{11}^2:S_3$, $C_{11}^2:C_{30}$

Order 3300: $C_{165}:C_{20}$, $C_{55}:C_{60}$, $C_{330}:C_{10}$

Order 3025: $C_{11}^2:C_5^2$ x 13

Order 2904: $C_{11}^2:D_{12}$, $(C_{11}\times C_{22}).D_6$, $C_{11}^2:C_{24}$

Order 2662: $C_{11}^2\times C_{22}$, $C_{11}^2:C_{22}$

Order 2640: $D_{132}.C_{10}$ x 5, $C_{165}:\SD_{16}$

Order 2420: $C_{11}^2:C_{20}$ x 45, $C_{22}:F_{11}$ x 35, $C_{11}:C_{220}$ x 9, $D_{11}\times C_{110}$ x 7, $C_{11}^2:C_{20}$ x 5, $C_{22}:F_{11}$ x 5, $C_{11}^2:C_{20}$ x 5, $C_{11}^2:C_{20}$, $C_{11}\times C_{220}$, $C_{110}:D_{11}$

Order 2200: $C_{220}.C_{10}$, $C_{220}.C_{10}$, $C_{220}:C_{10}$, $C_{22}.C_{10}^2$, $C_{55}:C_{40}$

Order 1936: $C_{11}^2:\SD_{16}$

Order 1815: $C_{11}^2:C_{15}$ x 5, $C_{11}^2:C_{15}$

Order 1650: $C_{110}:C_{15}$, $C_{165}:C_{10}$

Order 1452: $C_{11}^2:D_6$, $C_{11}^2:C_{12}$, $(C_{11}\times C_{22}).S_3$

Order 1331: $C_{11}^3$

Order 1320: $C_{132}:C_{10}$ x 5, $C_{132}.C_{10}$ x 5, $C_{11}:C_{120}$ x 5, $C_{11}:C_{120}$, $C_5\times D_{132}$, $C_{165}:Q_8$

Order 1210: $C_{11}^2:C_{10}$ x 65, $C_{11}:F_{11}$ x 60, $C_{11}\times C_{110}$ x 13, $D_{11}\times C_{55}$ x 12, $C_{11}:F_{11}$ x 5, $C_{55}:D_{11}$

Order 1200: $C_{120}:C_{10}$

Order 1100: $C_{220}:C_5$ x 8, $C_{10}\times F_{11}$ x 7, $C_{55}:C_{20}$ x 3, $C_{110}:C_{10}$

Order 968: $C_{11}^2:Q_8$, $C_{11}^2:Q_8$, $C_{11}:D_{44}$, $D_{22}:D_{11}$, $C_{11}^2:C_8$

Order 880: $Q_8:F_{11}$ x 5, $C_{55}:\SD_{16}$

Order 825: $C_{165}:C_5$

Order 726: $C_{11}^2:C_6$, $C_{11}^2:S_3$

Order 660: $C_{11}:C_{60}$ x 5, $C_{33}:C_{20}$ x 5, $C_{66}:C_{10}$ x 5, $C_3:C_{220}$, $C_5\times D_{66}$, $C_{660}$

Order 605: $C_{11}^2:C_5$ x 65, $C_{11}\times C_{55}$ x 13

Order 600: $C_5\times C_{120}$, $D_{12}\times C_5^2$, $C_{60}.C_{10}$

Order 550: $C_{110}:C_5$ x 14, $C_5\times F_{11}$ x 12

Order 528: $C_{33}:\SD_{16}$

Order 484: $C_{11}:C_{44}$ x 9, $C_{11}\times D_{22}$ x 7, $C_{11}\times C_{44}$, $C_{11}^2:C_4$, $C_{11}:D_{22}$

Order 440: $C_{44}:C_{10}$ x 5, $C_{44}.C_{10}$ x 5, $C_{44}.C_{10}$ x 5, $D_{22}:C_{10}$ x 5, $C_{11}:C_{40}$ x 5, $Q_8\times C_{55}$, $C_{11}:C_{40}$, $C_{55}:D_4$, $C_5\times D_{44}$, $C_{55}:Q_8$

Order 400: $\SD_{16}\times C_5^2$

Order 363: $C_{11}^2:C_3$

Order 330: $C_{11}:C_{30}$ x 5, $C_{33}:C_{10}$ x 5, $C_5\times D_{33}$, $C_{330}$

Order 300: $D_6\times C_5^2$, $C_5\times C_{60}$, $C_{15}:C_{20}$

Order 275: $C_{55}:C_5$ x 13

Order 264: $D_{132}$, $C_{11}:C_{24}$, $C_{33}:Q_8$

Order 242: $C_{11}\times C_{22}$ x 13, $C_{11}\times D_{11}$ x 12, $C_{11}:D_{11}$

Order 240: $C_{120}:C_2$ x 6

Order 121: $C_{11}^2$

Order 110: $C_{110}$

Order 55: $C_{55}$

Order 22: $C_{22}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1597200: $C_{11}^3:(C_{120}:C_{10})$ ($C_1$)

Order 798600: $C_{11}^3.C_{60}.C_{10}$ ($C_2$), $C_5\times C_{11}^3:C_{120}$ ($C_2$), $C_{11}^3.C_{60}.C_{10}$ ($C_2$)

Order 399300: $C_5\times C_{11}^3:C_{60}$ ($C_2^2$)

Order 319440: $C_{11}^3:(C_{120}:C_2)$ ($C_5$) x 5, $(C_{11}^2\times C_{110}).D_{12}$ ($C_5$)

Order 266200: $C_5\times C_{11}^3:C_{40}$ ($S_3$)

Order 199650: $C_5\times C_{11}^3:C_{30}$ ($D_4$)

Order 159720: $C_{11}^3:(C_5\times D_{12})$ ($C_{10}$) x 5, $C_{11}^3:C_{120}$ ($C_{10}$) x 5, $C_{11}^3:C_{15}:Q_8$ ($C_{10}$) x 5, $C_{11}^2:C_{165}:Q_8$ ($C_{10}$), $C_5\times C_{11}^2:D_{132}$ ($C_{10}$), $C_{11}^3:C_{120}$ ($C_{10}$)

Order 133100: $C_5\times C_{11}^3:C_{20}$ ($D_6$)

Order 99825: $C_5\times C_{11}^3:C_{15}$ ($\SD_{16}$)

Order 79860: $C_{11}^3:C_{60}$ ($C_2\times C_{10}$) x 5, $C_{11}^2:C_{660}$ ($C_2\times C_{10}$)

Order 66550: $C_5\times C_{11}^3:C_{10}$ ($D_{12}$)

Order 63888: $C_{11}^3.C_6.D_4$ ($C_5^2$)

Order 53240: $C_{11}^3:C_{40}$ ($C_5\times S_3$) x 5, $C_{11}^3:C_{40}$ ($C_5\times S_3$)

Order 39930: $C_{11}^3:C_{30}$ ($C_5\times D_4$) x 5, $C_{11}^2:C_{330}$ ($C_5\times D_4$)

Order 33275: $C_{11}^3:C_5^2$ ($C_{24}:C_2$)

Order 31944: $C_{11}^3:C_{24}$ ($C_5\times C_{10}$), $C_{11}^2:D_{132}$ ($C_5\times C_{10}$), $C_{11}\wr C_3:Q_8$ ($C_5\times C_{10}$)

Order 26620: $C_{11}^3:C_{20}$ ($S_3\times C_{10}$) x 5, $C_{11}^2:C_{220}$ ($S_3\times C_{10}$)

Order 19965: $C_{11}^3:C_{15}$ ($C_5\times \SD_{16}$) x 5, $C_{11}^2:C_{165}$ ($C_5\times \SD_{16}$)

Order 15972: $C_{11}^2:C_{132}$ ($C_{10}^2$)

Order 14520: $(C_{11}\times C_{110}).D_6$ ($F_{11}$)

Order 13310: $C_{11}^3:C_{10}$ ($C_5\times D_{12}$) x 5, $C_{11}^2\times C_{110}$ ($C_5\times D_{12}$)

Order 10648: $C_{11}^3:C_8$ ($S_3\times C_5^2$)

Order 7986: $C_{11}^2:C_{66}$ ($D_4\times C_5^2$)

Order 7260: $C_{11}^2:C_{60}$ ($C_2\times F_{11}$)

Order 6655: $C_{11}^3:C_5$ ($C_{120}:C_2$) x 5, $C_{11}^2\times C_{55}$ ($C_{120}:C_2$)

Order 5324: $C_{11}^2:C_{44}$ ($D_6\times C_5^2$)

Order 3993: $C_{11}\wr C_3$ ($\SD_{16}\times C_5^2$)

Order 3630: $C_{11}^2:C_{30}$ ($D_{22}:C_{10}$)

Order 2904: $(C_{11}\times C_{22}).D_6$ ($C_5\times F_{11}$)

Order 2662: $C_{11}^2\times C_{22}$ ($D_{12}\times C_5^2$)

Order 2420: $C_{11}^2:C_{20}$ ($S_3\times F_{11}$)

Order 1815: $C_{11}^2:C_{15}$ ($Q_8:F_{11}$)

Order 1452: $C_{11}^2:C_{12}$ ($C_{10}\times F_{11}$)

Order 1331: $C_{11}^3$ ($C_{120}:C_{10}$)

Order 1210: $C_{11}\times C_{110}$ ($D_{66}:C_{10}$)

Order 726: $C_{11}^2:C_6$ ($C_{22}.C_{10}^2$)

Order 605: $C_{11}\times C_{55}$ ($D_{132}.C_{10}$)

Order 484: $C_{11}^2:C_4$ ($C_{33}:C_{10}^2$)

Order 363: $C_{11}^2:C_3$ ($C_{44}.C_{10}^2$)

Order 242: $C_{11}\times C_{22}$ ($C_{66}.C_{10}^2$)

Order 121: $C_{11}^2$ ($C_{132}.C_{10}^2$)

Order 110: $C_{110}$ ($C_{11}^2:(C_5\times D_{12})$)

Order 55: $C_{55}$ ($C_{11}^2:C_{120}:C_2$)

Order 22: $C_{22}$ ($C_{11}^2.C_6.C_{10}^2$)

Order 11: $C_{11}$ ($C_{11}^2:C_{120}:C_{10}$)

Order 10: $C_{10}$ ($C_{11}^3.C_{30}.C_2^2$)

Order 5: $C_5$ ($C_{11}^3:(C_{120}:C_2)$)

Order 2: $C_2$ ($C_{11}^3.C_6.C_{10}^2$)

Order 1: $C_1$ ($C_{11}^3:(C_{120}:C_{10})$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

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

Character theory

Complex character table

The $710 \times 710$ character table is not available for this group.

Rational character table

The $146 \times 146$ rational character table is not available for this group.