Finite Group Databasebeta

← C3×C5×D11×AGL(1,𝔽11) #49137368

Subgroups — C3×C5×D11×AGL(1,𝔽11) #49137368

Order 36300.

328 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 124 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
1C3×C5×D11×C11⋊C5C211yesmaximal, characteristic, complemented
2C3×C5×(C11⋊C5)⋊D11C211yesmaximal, characteristic, complemented
3C3×C5×C11×AGL(1,𝔽11)C211yesmaximal, characteristic, complemented
4C5×D11×AGL(1,𝔽11)C311yesmaximal, characteristic, direct product factor, Hall
5C3×C5×C11×C11⋊C5C2211yescharacteristic, complemented, Hall
6C3×C5×D112C511yesmaximal, characteristic, complemented
7C3×D11×AGL(1,𝔽11)C515yesmaximal, direct product factor
8C5×D11×C11⋊C5C2×C311yescharacteristic, complemented
9C5×(C11⋊C5)⋊D11C2×C311yescharacteristic, complemented
10C5×C11×AGL(1,𝔽11)C2×C311yescharacteristic, complemented
11C3×C5×C11⋊D11C2×C511yescharacteristic, complemented
12C3×C5×C11×D11C2×C511yescharacteristic, complemented
13C3×C5×C11×D11C2×C511yescharacteristic, complemented
14C3×D11×C11⋊C5C2×C515yescomplemented
15C3×(C11⋊C5)⋊D11C2×C515yescomplemented
16C3×C11×AGL(1,𝔽11)C2×C515yescomplemented
17C2×C3×C52×D11—111nomaximal
18C2×C3×C5×AGL(1,𝔽11)—111nomaximal
19C5×C11×C11⋊C5C22×C311yescharacteristic, complemented, Hall
20C5×D112C3×C511yescharacteristic, complemented
21D11×AGL(1,𝔽11)C3×C515yesdirect product factor
22C3×C5×C112C22×C511yescharacteristic, complemented
23C3×C11×C11⋊C5C22×C515yescomplemented
24C2×C3×C52×C11—111no—
25C3×C52×D11—111no—
26C3×C52×D11—111no—
27C2×C3×C5×C11⋊C5—111no—
28C3×C5×AGL(1,𝔽11)—111no—
29C3×C5×AGL(1,𝔽11)D1111yescharacteristic, direct product factor
30C3×D112C5211yescharacteristic, complemented, Hall
31C5×C11⋊D11C2×C3×C511yescharacteristic, complemented
32C5×C11×D11C2×C3×C511yescharacteristic, complemented
33C5×C11×D11C2×C3×C511yescharacteristic, complemented
34D11×C11⋊C5C2×C3×C515yescomplemented
35(C11⋊C5)⋊D11C2×C3×C515yescomplemented
36C11×AGL(1,𝔽11)C2×C3×C515yescomplemented
37C2×C52×D11—111no—
38C2×C5×AGL(1,𝔽11)—111no—
39C3×C52×C11—111no—
40C3×C5×C11⋊C5C2×D1111yescharacteristic, complemented
41C3×C11⋊D11C2×C5211yescharacteristic, complemented
42C3×C11×D11C2×C5211yescharacteristic, complemented
43C3×C11×D11C2×C5211yescharacteristic, complemented
44C2×C3×C5×D11—111no—
45C2×C3×C5×D11—115no—
46C2×C3×C5×D11—551no—
47C2×C3×AGL(1,𝔽11)—115no—
48C5×C112C22×C3×C511yescharacteristic, complemented
49C11×C11⋊C5C22×C3×C515yescomplemented
50C2×C52×C11—111no—
51C52×D11—111no—
52C52×D11—111no—
53C2×C5×C11⋊C5—111no—
54C5×AGL(1,𝔽11)—111no—
55C5×AGL(1,𝔽11)C3×D1111yescharacteristic, direct product factor
56D112C3×C5211yescharacteristic, complemented, Hall
57C3×C112C22×C5211yescharacteristic, complemented, Hall
58C2×C3×C5×C11—111no—
59C2×C3×C5×C11—115no—
60C2×C3×C5×C11—551no—
61C3×C5×D11—111no—
62C3×C5×D11—115no—
63C3×C5×D11—115no—
64C3×C5×D11—551no—
65C3×C5×D11—1101no—
66C3×C5×D11AGL(1,𝔽11)11yescharacteristic, direct product factor
67C3×C5×D11C5×D1111yescharacteristic, complemented
68C2×C3×C11⋊C5—115no—
69C3×AGL(1,𝔽11)—115no—
70C3×AGL(1,𝔽11)C5×D1115yesdirect product factor
71C22×C3×C52—1211noHall
72C52×C11—111no—
73C5×C11⋊C5C2×C3×D1111yescharacteristic, complemented
74C11⋊D11C2×C3×C5211yescharacteristic, complemented
75C11×D11C2×C3×C5211yescharacteristic, complemented
76C11×D11C2×C3×C5211yescharacteristic, complemented
77C2×C5×D11—111no—
78C2×C5×D11—115no—
79C2×C5×D11—551no—
80C2×AGL(1,𝔽11)—115no—
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—
87C2×C3×C52—1211no—
88C2×C3×C52—1211no—
89C2×C3×D11—111no—
90C2×C3×D11—551no—
91C112C22×C3×C5211yescharacteristic, complemented, Sylow
92C2×C5×C11—115no—
93C2×C5×C11—111no—
94C2×C5×C11—551no—
95C5×D11—111no—
96C5×D11—115no—
97C5×D11—115no—
98C5×D11—551no—
99C5×D11—1101no—
100C5×D11C3×AGL(1,𝔽11)11yescharacteristic, direct product factor
101C5×D11C3×C5×D1111yescharacteristic, complemented
102C2×C11⋊C5—115no—
103AGL(1,𝔽11)—115no—
104AGL(1,𝔽11)C3×C5×D1115yesdirect product factor
105C22×C52—1211noHall
106C3×C52—111noHall
107C2×C3×C11—111no—
108C2×C3×C11—551no—
109C3×D11—111no—
110C3×D11—551no—
111C3×D11—1101no—
112C3×D11C5×AGL(1,𝔽11)11yescharacteristic, direct product factor
113C3×D11C52×D1111yescharacteristic, complemented
114C22×C3×C5—6051no—
115C22×C3×C5—1215no—
116C5×C11—101no—
117C5×C11—115no—
118C5×C11C2×C3×AGL(1,𝔽11)11yescharacteristic, complemented
119C5×C11C2×C3×C5×D1111yescharacteristic, complemented
120C11⋊C5C2×C3×C5×D1115yescomplemented
121C2×C52—111no—
122C2×C52—1211no—
123C2×C52—1211no—
124C2×D11—111no—
125C2×D11—551no—
126C3×C11—101no—
127C3×C11C2×C5×AGL(1,𝔽11)11yescharacteristic, complemented
128C3×C11C2×C52×D1111yescharacteristic, complemented
129C2×C3×C5—111no—
130C2×C3×C5—115no—
131C2×C3×C5—551no—
132C2×C3×C5—6051no—
133C2×C3×C5—1215no—
134C2×C3×C5—1215no—
135C52—111noSylow
136C2×C11—111no—
137C2×C11—551no—
138D11—111no—
139D11—551no—
140D11—1101no—
141D11C3×C5×AGL(1,𝔽11)11yescharacteristic, direct product factor
142D11C3×C52×D1111yescharacteristic, complemented
143C22×C5—6051no—
144C22×C5—1215no—
145C3×C5—115no—
146C3×C5D11×AGL(1,𝔽11)11yescharacteristic, direct product factor
147C22×C3—6051noHall
148C11—101no—
149C11C2×C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
150C11C2×C3×C52×D1111yescharacteristic, complemented
151C2×C5—111no—
152C2×C5—115no—
153C2×C5—551no—
154C2×C5—6051no—
155C2×C5—1215no—
156C2×C5—1215no—
157C2×C3—111no—
158C2×C3—551no—
159C2×C3—6051no—
160C5—115no—
161C5C3×D11×AGL(1,𝔽11)11yescharacteristic, direct product factor
162C22—6051noSylow
163C3C5×D11×AGL(1,𝔽11)11yescharacteristic, direct product factor, Sylow
164C2—111no—
165C2—551no—
166C2—6051no—
subgroupG/H
center, hypercenterC3×C5D11×AGL(1,𝔽11)
commutator, cosocle, hypocenterC112C22×C3×C52
FrattiniC1G
Fitting, socleC3×C5×C112C22×C5
LCSG ▷ C112
UCSC1 ◁ C3×C5

Complements

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

NH ≅ G/Ntype
1. C3×C5×D11×C11⋊C5C2semidirect
2. C3×C5×(C11⋊C5)⋊D11C2semidirect
3. C3×C5×C11×AGL(1,𝔽11)C2semidirect
4. C5×D11×AGL(1,𝔽11)C3direct (N × H)
5. C3×C5×C11×C11⋊C5C22semidirect
6. C3×C5×D112C5semidirect
7. C3×D11×AGL(1,𝔽11)C5direct (N × H)
8. C5×D11×C11⋊C5C2×C3semidirect
9. C5×(C11⋊C5)⋊D11C2×C3semidirect
10. C5×C11×AGL(1,𝔽11)C2×C3semidirect
11. C3×C5×C11⋊D11C2×C5semidirect
12. C3×C5×C11×D11C2×C5semidirect
13. C3×C5×C11×D11C2×C5semidirect
14. C3×D11×C11⋊C5C2×C5semidirect
15. C3×(C11⋊C5)⋊D11C2×C5semidirect
16. C3×C11×AGL(1,𝔽11)C2×C5semidirect
19. C5×C11×C11⋊C5C22×C3semidirect
20. C5×D112C3×C5semidirect
21. D11×AGL(1,𝔽11)C3×C5direct (N × H)
22. C3×C5×C112C22×C5semidirect
23. C3×C11×C11⋊C5C22×C5semidirect
29. C3×C5×AGL(1,𝔽11)D11direct (N × H)
30. C3×D112C52semidirect
31. C5×C11⋊D11C2×C3×C5semidirect
32. C5×C11×D11C2×C3×C5semidirect
33. C5×C11×D11C2×C3×C5semidirect
34. D11×C11⋊C5C2×C3×C5semidirect
35. (C11⋊C5)⋊D11C2×C3×C5semidirect
36. C11×AGL(1,𝔽11)C2×C3×C5semidirect
40. C3×C5×C11⋊C5C2×D11semidirect
41. C3×C11⋊D11C2×C52semidirect
42. C3×C11×D11C2×C52semidirect
43. C3×C11×D11C2×C52semidirect
48. C5×C112C22×C3×C5semidirect
49. C11×C11⋊C5C22×C3×C5semidirect
55. C5×AGL(1,𝔽11)C3×D11direct (N × H)
56. D112C3×C52semidirect
57. C3×C112C22×C52semidirect
66. C3×C5×D11AGL(1,𝔽11)direct (N × H)
67. C3×C5×D11C5×D11semidirect
70. C3×AGL(1,𝔽11)C5×D11direct (N × H)
73. C5×C11⋊C5C2×C3×D11semidirect
74. C11⋊D11C2×C3×C52semidirect
75. C11×D11C2×C3×C52semidirect
76. C11×D11C2×C3×C52semidirect
83. C3×C5×C11C2×AGL(1,𝔽11)semidirect
84. C3×C5×C11C2×C5×D11semidirect
85. C3×C11⋊C5C2×C5×D11semidirect
91. C112C22×C3×C52semidirect
100. C5×D11C3×AGL(1,𝔽11)direct (N × H)
101. C5×D11C3×C5×D11semidirect
104. AGL(1,𝔽11)C3×C5×D11direct (N × H)
112. C3×D11C5×AGL(1,𝔽11)direct (N × H)
113. C3×D11C52×D11semidirect
118. C5×C11C2×C3×AGL(1,𝔽11)semidirect
119. C5×C11C2×C3×C5×D11semidirect
120. C11⋊C5C2×C3×C5×D11semidirect
127. C3×C11C2×C5×AGL(1,𝔽11)semidirect
128. C3×C11C2×C52×D11semidirect
141. D11C3×C5×AGL(1,𝔽11)direct (N × H)
142. D11C3×C52×D11semidirect
146. C3×C5D11×AGL(1,𝔽11)direct (N × H)
149. C11C2×C3×C5×AGL(1,𝔽11)semidirect
150. C11C2×C3×C52×D11semidirect
161. C5C3×D11×AGL(1,𝔽11)direct (N × H)
163. C3C5×D11×AGL(1,𝔽11)direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×D11×C11⋊C52
C3×C5×(C11⋊C5)⋊D112
C3×C5×C11×AGL(1,𝔽11)2
C5×D11×AGL(1,𝔽11)3
C3×C5×D1125
C3×D11×AGL(1,𝔽11)5
C2×C3×C52×D1111
C2×C3×C5×AGL(1,𝔽11)11