Finite Group Databasebeta

← C2×C3×C5×#11925896 #49137635

Subgroups — C2×C3×C5×#11925896 #49137635

Order 36300.

328 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C2×C3×C5×#10597245C211yesmaximal, characteristic, complemented
2C3×C5×#11925896C211yesmaximal, characteristic, direct product factor
3C3×C5×#11925896C211yesmaximal, characteristic, direct product factor
4C2×C5×#11925896C311yesmaximal, characteristic, direct product factor, Hall
5C3×C5×#10597245C2211yescharacteristic, complemented, Hall
6C2×C3×C5×C11×D11C511yesmaximal, characteristic, complemented
7C2×C3×#11925896C515yesmaximal, direct product factor
8C2×C5×#10597245C2×C311yescharacteristic, complemented
9C5×#11925896C2×C311yescharacteristic, direct product factor
10C5×#11925896C2×C311yescharacteristic, direct product factor
11C3×C5×C11×D11C2×C511yescharacteristic, complemented
12C3×C5×C11×D11C2×C511yescharacteristic, complemented
13C2×C3×#10597245C2×C515yescomplemented
14C3×#11925896C2×C515yesdirect product factor
15C3×#11925896C2×C515yesdirect product factor
16C2×C3×C5×C112C2×C511yescharacteristic, complemented
17C22×C3×C5×C11⋊C5—111nomaximal
18C2×C3×C5×AGL(1,𝔽11)—111nomaximal
19C5×#10597245C22×C311yescharacteristic, complemented, Hall
20C2×C5×C11×D11C3×C511yescharacteristic, complemented
21C2×#11925896C3×C515yesdirect product factor
22C3×C5×C112C22×C511yescharacteristic, complemented
23C3×#10597245C22×C515yescomplemented
24C2×C3×C5×C11⋊C5—111no—
25C2×C3×C5×C11⋊C5—111no—
26C2×C3×C5×C11⋊C5—111no—
27C2×C3×C5×C11⋊C5—111no—
28C3×C5×AGL(1,𝔽11)—111no—
29C3×C5×AGL(1,𝔽11)—111no—
30C2×C3×C11×D11C5211yescharacteristic, complemented, Hall
31C2×C5×C112C2×C3×C511yescharacteristic, complemented
32C5×C11×D11C2×C3×C511yescharacteristic, complemented
33C5×C11×D11C2×C3×C511yescharacteristic, complemented
34C2×#10597245C2×C3×C515yescomplemented
35#11925896C2×C3×C515yesdirect product factor
36#11925896C2×C3×C515yesdirect product factor
37C22×C5×C11⋊C5—111no—
38C2×C5×AGL(1,𝔽11)—111no—
39C3×C5×C11⋊C5—111no—
40C3×C5×C11⋊C5—111no—
41C2×C3×C112C2×C5211yescharacteristic, complemented
42C3×C11×D11C2×C5211yescharacteristic, complemented
43C3×C11×D11C2×C5211yescharacteristic, complemented
44C22×C3×C5×C11—551no—
45C2×C3×C5×D11C11⋊C511yescharacteristic, complemented
46C22×C3×C11⋊C5—115no—
47C2×C3×AGL(1,𝔽11)—115no—
48C5×C112C22×C3×C511yescharacteristic, complemented
49#10597245C22×C3×C515yescomplemented
50C2×C5×C11⋊C5—111no—
51C2×C5×C11⋊C5—111no—
52C2×C5×C11⋊C5—111no—
53C2×C5×C11⋊C5—111no—
54C5×AGL(1,𝔽11)—111no—
55C5×AGL(1,𝔽11)—111no—
56C2×C11×D11C3×C5211yescharacteristic, complemented, Hall
57C3×C112C22×C5211yescharacteristic, complemented, Hall
58C2×C3×C5×C11—101no—
59C2×C3×C5×C11—551no—
60C2×C3×C5×C11—551no—
61C2×C3×C5×C11AGL(1,𝔽11)11yescharacteristic, complemented
62C2×C3×C5×C11C2×C11⋊C511yescharacteristic, complemented
63C3×C5×D11C2×C11⋊C511yescharacteristic, complemented
64C3×C5×D11C2×C11⋊C511yescharacteristic, complemented
65C2×C3×C11⋊C5—115no—
66C2×C3×C11⋊C5—115no—
67C2×C3×C11⋊C5—115no—
68C2×C3×C11⋊C5—115no—
69C3×AGL(1,𝔽11)—115no—
70C3×AGL(1,𝔽11)—115no—
71C22×C3×C52—1211noHall
72C5×C11⋊C5—111no—
73C5×C11⋊C5—111no—
74C2×C112C2×C3×C5211yescharacteristic, complemented
75C11×D11C2×C3×C5211yescharacteristic, complemented
76C11×D11C2×C3×C5211yescharacteristic, complemented
77C22×C5×C11—551no—
78C2×C5×D11C3×C11⋊C511yescharacteristic, complemented
79C22×C11⋊C5—115no—
80C2×AGL(1,𝔽11)—115no—
81C3×C5×C11—101no—
82C3×C5×C11C2×AGL(1,𝔽11)11yescharacteristic, complemented
83C3×C5×C11C22×C11⋊C511yescharacteristic, complemented
84C3×C11⋊C5—115no—
85C3×C11⋊C5—115no—
86C2×C3×C52—1211no—
87C2×C3×C52—1211no—
88C2×C3×C52—1211no—
89C22×C3×C11—551no—
90C2×C3×D11C5×C11⋊C511yescharacteristic, complemented
91C112C22×C3×C5211yescharacteristic, complemented, Sylow
92C2×C5×C11—101no—
93C2×C5×C11—551no—
94C2×C5×C11—551no—
95C2×C5×C11C3×AGL(1,𝔽11)11yescharacteristic, complemented
96C2×C5×C11C2×C3×C11⋊C511yescharacteristic, complemented
97C5×D11C2×C3×C11⋊C511yescharacteristic, complemented
98C5×D11C2×C3×C11⋊C511yescharacteristic, complemented
99C2×C11⋊C5—115no—
100C2×C11⋊C5—115no—
101C2×C11⋊C5—115no—
102C2×C11⋊C5—115no—
103AGL(1,𝔽11)—115no—
104AGL(1,𝔽11)—115no—
105C22×C52—1211noHall
106C3×C52—1211noHall
107C2×C3×C11—101no—
108C2×C3×C11—551no—
109C2×C3×C11—551no—
110C2×C3×C11C5×AGL(1,𝔽11)11yescharacteristic, complemented
111C2×C3×C11C2×C5×C11⋊C511yescharacteristic, complemented
112C3×D11C2×C5×C11⋊C511yescharacteristic, complemented
113C3×D11C2×C5×C11⋊C511yescharacteristic, complemented
114C22×C3×C5—551no—
115C22×C3×C5—1215no—
116C5×C11—101no—
117C5×C11C2×C3×AGL(1,𝔽11)11yescharacteristic, complemented
118C5×C11C22×C3×C11⋊C511yescharacteristic, complemented
119C11⋊C5—115no—
120C11⋊C5—115no—
121C2×C52—1211no—
122C2×C52—1211no—
123C2×C52—1211no—
124C22×C11—551no—
125C2×D11C3×C5×C11⋊C511yescharacteristic, complemented
126C3×C11—101no—
127C3×C11C2×C5×AGL(1,𝔽11)11yescharacteristic, complemented
128C3×C11C22×C5×C11⋊C511yescharacteristic, complemented
129C2×C3×C5—551no—
130C2×C3×C5—551no—
131C2×C3×C5—1215no—
132C2×C3×C5—1215no—
133C2×C3×C5—1215no—
134C2×C3×C5#1192589611yescharacteristic, direct product factor
135C52—1211noSylow
136C2×C11—101no—
137C2×C11—551no—
138C2×C11—551no—
139C2×C11C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
140C2×C11C2×C3×C5×C11⋊C511yescharacteristic, complemented
141D11C2×C3×C5×C11⋊C511yescharacteristic, complemented
142D11C2×C3×C5×C11⋊C511yescharacteristic, complemented
143C22×C5—551no—
144C22×C5—1215no—
145C3×C5—1215no—
146C3×C5C2×#1192589611yescharacteristic, direct product factor
147C22×C3—551noHall
148C11—101no—
149C11C2×C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
150C11C22×C3×C5×C11⋊C511yescharacteristic, complemented
151C2×C5—111no—
152C2×C5—551no—
153C2×C5—1215no—
154C2×C5—1215no—
155C2×C5—1215no—
156C2×C5C3×#1192589611yescharacteristic, direct product factor
157C2×C3—551no—
158C2×C3—551no—
159C2×C3C5×#1192589611yescharacteristic, direct product factor
160C5—1215no—
161C5C2×C3×#1192589611yescharacteristic, direct product factor
162C22—551noSylow
163C3C2×C5×#1192589611yescharacteristic, direct product factor, Sylow
164C2—111no—
165C2—551no—
166C2C3×C5×#1192589611yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5#11925896
commutator, cosocle, hypocenterC112C22×C3×C52
FrattiniC1G
Fitting, socleC2×C3×C5×C112C2×C5
LCSG ▷ C112
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#10597245C2semidirect
2. C3×C5×#11925896C2direct (N × H)
3. C3×C5×#11925896C2direct (N × H)
4. C2×C5×#11925896C3direct (N × H)
5. C3×C5×#10597245C22semidirect
6. C2×C3×C5×C11×D11C5semidirect
7. C2×C3×#11925896C5direct (N × H)
8. C2×C5×#10597245C2×C3semidirect
9. C5×#11925896C2×C3direct (N × H)
10. C5×#11925896C2×C3direct (N × H)
11. C3×C5×C11×D11C2×C5semidirect
12. C3×C5×C11×D11C2×C5semidirect
13. C2×C3×#10597245C2×C5semidirect
14. C3×#11925896C2×C5direct (N × H)
15. C3×#11925896C2×C5direct (N × H)
16. C2×C3×C5×C112C2×C5semidirect
19. C5×#10597245C22×C3semidirect
20. C2×C5×C11×D11C3×C5semidirect
21. C2×#11925896C3×C5direct (N × H)
22. C3×C5×C112C22×C5semidirect
23. C3×#10597245C22×C5semidirect
30. C2×C3×C11×D11C52semidirect
31. C2×C5×C112C2×C3×C5semidirect
32. C5×C11×D11C2×C3×C5semidirect
33. C5×C11×D11C2×C3×C5semidirect
34. C2×#10597245C2×C3×C5semidirect
35. #11925896C2×C3×C5direct (N × H)
36. #11925896C2×C3×C5direct (N × H)
41. C2×C3×C112C2×C52semidirect
42. C3×C11×D11C2×C52semidirect
43. C3×C11×D11C2×C52semidirect
45. C2×C3×C5×D11C11⋊C5semidirect
48. C5×C112C22×C3×C5semidirect
49. #10597245C22×C3×C5semidirect
56. C2×C11×D11C3×C52semidirect
57. C3×C112C22×C52semidirect
61. C2×C3×C5×C11AGL(1,𝔽11)semidirect
62. C2×C3×C5×C11C2×C11⋊C5semidirect
63. C3×C5×D11C2×C11⋊C5semidirect
64. C3×C5×D11C2×C11⋊C5semidirect
74. C2×C112C2×C3×C52semidirect
75. C11×D11C2×C3×C52semidirect
76. C11×D11C2×C3×C52semidirect
78. C2×C5×D11C3×C11⋊C5semidirect
82. C3×C5×C11C2×AGL(1,𝔽11)semidirect
83. C3×C5×C11C22×C11⋊C5semidirect
90. C2×C3×D11C5×C11⋊C5semidirect
91. C112C22×C3×C52semidirect
95. C2×C5×C11C3×AGL(1,𝔽11)semidirect
96. C2×C5×C11C2×C3×C11⋊C5semidirect
97. C5×D11C2×C3×C11⋊C5semidirect
98. C5×D11C2×C3×C11⋊C5semidirect
110. C2×C3×C11C5×AGL(1,𝔽11)semidirect
111. C2×C3×C11C2×C5×C11⋊C5semidirect
112. C3×D11C2×C5×C11⋊C5semidirect
113. C3×D11C2×C5×C11⋊C5semidirect
117. C5×C11C2×C3×AGL(1,𝔽11)semidirect
118. C5×C11C22×C3×C11⋊C5semidirect
125. C2×D11C3×C5×C11⋊C5semidirect
127. C3×C11C2×C5×AGL(1,𝔽11)semidirect
128. C3×C11C22×C5×C11⋊C5semidirect
134. C2×C3×C5#11925896direct (N × H)
139. C2×C11C3×C5×AGL(1,𝔽11)semidirect
140. C2×C11C2×C3×C5×C11⋊C5semidirect
141. D11C2×C3×C5×C11⋊C5semidirect
142. D11C2×C3×C5×C11⋊C5semidirect
146. C3×C5C2×#11925896direct (N × H)
149. C11C2×C3×C5×AGL(1,𝔽11)semidirect
150. C11C22×C3×C5×C11⋊C5semidirect
156. C2×C5C3×#11925896direct (N × H)
159. C2×C3C5×#11925896direct (N × H)
161. C5C2×C3×#11925896direct (N × H)
163. C3C2×C5×#11925896direct (N × H)
166. C2C3×C5×#11925896direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#105972452
C3×C5×#119258962
C2×C5×#119258963
C2×C3×C5×C11×D115
C2×C3×#119258965
C22×C3×C5×C11⋊C511
C2×C3×C5×AGL(1,𝔽11)11