Finite Group Databasebeta

← C5×D33×AGL(1,𝔽11) #49137422

Subgroups — C5×D33×AGL(1,𝔽11) #49137422

Order 36300.

328 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×C11⋊C5×D33C211yesmaximal, characteristic, complemented
2C5×C3⋊((C11⋊C5)⋊D11)C211yesmaximal, characteristic, complemented
3C3×C5×C11×AGL(1,𝔽11)C211yesmaximal, characteristic, complemented
4C5×D11×AGL(1,𝔽11)—31nomaximal, Hall
5C3×C5×C11×C11⋊C5C2211yescharacteristic, complemented, Hall
6C5×D11×D33C511yesmaximal, characteristic, complemented
7D33×AGL(1,𝔽11)C515yesmaximal, direct product factor
8C5×D11×C11⋊C5—31no—
9C5×(C11⋊C5)⋊D11—31no—
10C5×C11×AGL(1,𝔽11)D311yescharacteristic, complemented
11C5×C3⋊(C11⋊D11)C2×C511yescharacteristic, complemented
12C5×C11×D33C2×C511yescharacteristic, complemented
13C3×C5×C11×D11C2×C511yescharacteristic, complemented
14C11⋊C5×D33C2×C515yescomplemented
15C3⋊((C11⋊C5)⋊D11)C2×C515yescomplemented
16C3×C11×AGL(1,𝔽11)C2×C515yescomplemented
17C2×C52×D33—111nomaximal
18C5×D3×AGL(1,𝔽11)—111nomaximal
19C5×C11×C11⋊C5C2×D311yescharacteristic, complemented, Hall
20C5×D112—31no—
21D11×AGL(1,𝔽11)—35no—
22C3×C5×C112C22×C511yescharacteristic, complemented
23C3×C11×C11⋊C5C22×C515yescomplemented
24C2×C3×C52×C11—111no—
25C52×D33—111no—
26C52×D33—111no—
27C5×C3⋊AGL(1,𝔽11)—111no—
28C5×D3×C11⋊C5—111no—
29C3×C5×AGL(1,𝔽11)D1111yescharacteristic, complemented
30D11×D33C5211yescharacteristic, complemented, Hall
31C5×C11⋊D11—31no—
32C5×C11×D11—31no—
33C5×C11×D11C5×D311yescharacteristic, complemented
34D11×C11⋊C5—35no—
35(C11⋊C5)⋊D11—35no—
36C11×AGL(1,𝔽11)C5×D315yescomplemented
37C2×C52×D11—331no—
38C2×C5×AGL(1,𝔽11)—331no—
39C3×C52×C11—111no—
40C3×C5×C11⋊C5C2×D1111yescharacteristic, complemented
41C3⋊(C11⋊D11)C2×C5211yescharacteristic, complemented
42C11×D33C2×C5211yescharacteristic, complemented
43C3×C11×D11C2×C5211yescharacteristic, complemented
44C2×C5×D33—115no—
45C2×C5×D33—551no—
46C5×D3×D11—111no—
47D3×AGL(1,𝔽11)—115no—
48C5×C112C2×C5×D311yescharacteristic, complemented
49C11×C11⋊C5C2×C5×D315yescomplemented
50C2×C52×C11—111no—
51C52×D11—331no—
52C52×D11—331no—
53C2×C5×C11⋊C5—331no—
54C5×AGL(1,𝔽11)—331no—
55C5×AGL(1,𝔽11)D3311yescharacteristic, direct product factor
56D112—31noHall
57C3×C112C22×C5211yescharacteristic, complemented, Hall
58C2×C3×C5×C11—115no—
59C2×C3×C5×C11—111no—
60C5×D33—111no—
61C5×D33—115no—
62C5×D33—115no—
63C5×D33—551no—
64C5×D33—1101no—
65C5×D33AGL(1,𝔽11)11yescharacteristic, direct product factor
66C5×D3×C11—111no—
67C3×C5×D11C5×D1111yescharacteristic, complemented
68C3⋊AGL(1,𝔽11)—115no—
69D3×C11⋊C5—115no—
70C3×AGL(1,𝔽11)C5×D1115yescomplemented
71C2×C52×D3—1211noHall
72C52×C11—111no—
73C5×C11⋊C5C2×D3311yescharacteristic, complemented
74C11⋊D11—31no—
75C11×D11—31no—
76C11×D11C52×D311yescharacteristic, complemented
77C2×C5×D11—331no—
78C2×C5×D11—335no—
79C2×C5×D11—1651no—
80C2×AGL(1,𝔽11)—335no—
81C3×C5×C11—101no—
82C3×C5×C11—115no—
83C3×C5×C11C2×AGL(1,𝔽11)11yescharacteristic, complemented
84C3×C5×C11C2×C5×D1111yescharacteristic, complemented
85C3×C11⋊C5C2×C5×D1115yescomplemented
86C2×C3×C52—111no—
87C52×D3—1211no—
88C52×D3—1211no—
89C2×D33—551no—
90D3×D11—111no—
91C112C2×C52×D311yescharacteristic, complemented, Sylow
92C2×C5×C11—115no—
93C2×C5×C11—331no—
94C2×C5×C11—551no—
95C5×D11—31no—
96C5×D11—331no—
97C5×D11—331no—
98C5×D11—335no—
99C5×D11—335no—
100C5×D11—3301no—
101C5×D11C5×D3311yescharacteristic, complemented
102C2×C11⋊C5—335no—
103AGL(1,𝔽11)—335no—
104AGL(1,𝔽11)C5×D3315yesdirect product factor
105C22×C52—3631noHall
106C3×C52—111noHall
107C2×C3×C11—111no—
108D33—111no—
109D33—551no—
110D33—1101no—
111D33C5×AGL(1,𝔽11)11yescharacteristic, direct product factor
112C3×D11C52×D1111yescharacteristic, complemented
113D3×C11—111no—
114C2×C5×D3—6051no—
115C2×C5×D3—1215no—
116C5×C11—101no—
117C5×C11—115no—
118C5×C11D3×AGL(1,𝔽11)11yescharacteristic, complemented
119C5×C11C2×C5×D3311yescharacteristic, complemented
120C11⋊C5C2×C5×D3315yescomplemented
121C2×C52—111no—
122C2×C52—3631no—
123C2×C52—3631no—
124C2×D11—331no—
125C2×D11—1651no—
126C3×C11—101no—
127C3×C11C2×C5×AGL(1,𝔽11)11yescharacteristic, complemented
128C3×C11C2×C52×D1111yescharacteristic, complemented
129C2×C3×C5—115no—
130C2×C3×C5—551no—
131C5×D3—111no—
132C5×D3—6051no—
133C5×D3—1215no—
134C5×D3—1215no—
135C52—111noSylow
136C2×C11—331no—
137C2×C11—551no—
138D11—31no—
139D11—331no—
140D11—331no—
141D11—3301no—
142D11C52×D3311yescharacteristic, complemented
143C22×C5—18151no—
144C22×C5—3635no—
145C3×C5—115no—
146C3×C5D11×AGL(1,𝔽11)11yescharacteristic, complemented
147C2×D3—6051noHall
148C11—101no—
149C11C5×D3×AGL(1,𝔽11)11yescharacteristic, complemented
150C11C2×C52×D3311yescharacteristic, complemented
151C2×C5—115no—
152C2×C5—18151no—
153C2×C5—331no—
154C2×C5—551no—
155C2×C5—3635no—
156C2×C5—3635no—
157C2×C3—551no—
158D3—111no—
159D3—6051no—
160C5—115no—
161C5D33×AGL(1,𝔽11)11yescharacteristic, direct product factor
162C22—18151noSylow
163C3C5×D11×AGL(1,𝔽11)11yescharacteristic, complemented, Sylow
164C2—18151no—
165C2—331no—
166C2—551no—
subgroupG/H
center, hypercenterC5D33×AGL(1,𝔽11)
commutator, cosocle, hypocenterC3×C112C22×C52
FrattiniC1G
Fitting, socleC3×C5×C112C22×C5
LCSG ▷ C3×C112
UCSC1 ◁ C5

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C5×C11⋊C5×D33C2semidirect
2. C5×C3⋊((C11⋊C5)⋊D11)C2semidirect
3. C3×C5×C11×AGL(1,𝔽11)C2semidirect
5. C3×C5×C11×C11⋊C5C22semidirect
6. C5×D11×D33C5semidirect
7. D33×AGL(1,𝔽11)C5direct (N × H)
10. C5×C11×AGL(1,𝔽11)D3semidirect
11. C5×C3⋊(C11⋊D11)C2×C5semidirect
12. C5×C11×D33C2×C5semidirect
13. C3×C5×C11×D11C2×C5semidirect
14. C11⋊C5×D33C2×C5semidirect
15. C3⋊((C11⋊C5)⋊D11)C2×C5semidirect
16. C3×C11×AGL(1,𝔽11)C2×C5semidirect
19. C5×C11×C11⋊C5C2×D3semidirect
22. C3×C5×C112C22×C5semidirect
23. C3×C11×C11⋊C5C22×C5semidirect
29. C3×C5×AGL(1,𝔽11)D11semidirect
30. D11×D33C52semidirect
33. C5×C11×D11C5×D3semidirect
36. C11×AGL(1,𝔽11)C5×D3semidirect
40. C3×C5×C11⋊C5C2×D11semidirect
41. C3⋊(C11⋊D11)C2×C52semidirect
42. C11×D33C2×C52semidirect
43. C3×C11×D11C2×C52semidirect
48. C5×C112C2×C5×D3semidirect
49. C11×C11⋊C5C2×C5×D3semidirect
55. C5×AGL(1,𝔽11)D33direct (N × H)
57. C3×C112C22×C52semidirect
65. C5×D33AGL(1,𝔽11)direct (N × H)
67. C3×C5×D11C5×D11semidirect
70. C3×AGL(1,𝔽11)C5×D11semidirect
73. C5×C11⋊C5C2×D33semidirect
76. C11×D11C52×D3semidirect
83. C3×C5×C11C2×AGL(1,𝔽11)semidirect
84. C3×C5×C11C2×C5×D11semidirect
85. C3×C11⋊C5C2×C5×D11semidirect
91. C112C2×C52×D3semidirect
101. C5×D11C5×D33semidirect
104. AGL(1,𝔽11)C5×D33direct (N × H)
111. D33C5×AGL(1,𝔽11)direct (N × H)
112. C3×D11C52×D11semidirect
118. C5×C11D3×AGL(1,𝔽11)semidirect
119. C5×C11C2×C5×D33semidirect
120. C11⋊C5C2×C5×D33semidirect
127. C3×C11C2×C5×AGL(1,𝔽11)semidirect
128. C3×C11C2×C52×D11semidirect
142. D11C52×D33semidirect
146. C3×C5D11×AGL(1,𝔽11)semidirect
149. C11C5×D3×AGL(1,𝔽11)semidirect
150. C11C2×C52×D33semidirect
161. C5D33×AGL(1,𝔽11)direct (N × H)
163. C3C5×D11×AGL(1,𝔽11)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×C11⋊C5×D332
C5×C3⋊((C11⋊C5)⋊D11)2
C3×C5×C11×AGL(1,𝔽11)2
C5×D11×AGL(1,𝔽11)3
C5×D11×D335
D33×AGL(1,𝔽11)5
C2×C52×D3311
C5×D3×AGL(1,𝔽11)11