Finite Group Databasebeta

← C2×C3×C5×#11925892 #49137630

Subgroups — C2×C3×C5×#11925892 #49137630

Order 36300.

1120 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 184 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×#10597243C211yesmaximal, characteristic, complemented
2C3×C5×#11925892C211yesmaximal, characteristic, direct product factor
3C3×C5×#11925892C211yesmaximal, characteristic, direct product factor
4C2×C5×#11925892C311yesmaximal, characteristic, direct product factor, Hall
5C3×C5×#10597243C2211yescharacteristic, complemented, Hall
6C2×C3×C5×C11⋊D11C511yesmaximal, characteristic, complemented
7C2×C3×#11925892C515yesmaximal, direct product factor
8C2×C5×#10597243C2×C311yescharacteristic, complemented
9C5×#11925892C2×C311yescharacteristic, direct product factor
10C5×#11925892C2×C311yescharacteristic, direct product factor
11C3×C5×C11⋊D11C2×C511yescharacteristic, complemented
12C3×C5×C11⋊D11C2×C511yescharacteristic, complemented
13C2×C3×#10597243C2×C515yescomplemented
14C3×#11925892C2×C515yesdirect product factor
15C3×#11925892C2×C515yesdirect product factor
16C2×C3×C5×C112C2×C511yescharacteristic, complemented
17C2×C3×C5×AGL(1,𝔽11)—111nomaximal
18C2×C3×C5×AGL(1,𝔽11)—111nomaximal
19C2×C3×C5×AGL(1,𝔽11)—111nomaximal
20C2×C3×C5×AGL(1,𝔽11)—111nomaximal
21C2×C3×C5×AGL(1,𝔽11)—118nomaximal
22C5×#10597243C22×C311yescharacteristic, complemented, Hall
23C2×C5×C11⋊D11C3×C511yescharacteristic, complemented
24C2×#11925892C3×C515yesdirect product factor
25C3×C5×C112C22×C511yescharacteristic, complemented
26C3×#10597243C22×C515yescomplemented
27C2×C3×C5×C11⋊C5—111no—
28C2×C3×C5×C11⋊C5—111no—
29C2×C3×C5×C11⋊C5—111no—
30C2×C3×C5×C11⋊C5—111no—
31C2×C3×C5×C11⋊C5—118no—
32C3×C5×AGL(1,𝔽11)—111no—
33C3×C5×AGL(1,𝔽11)—111no—
34C3×C5×AGL(1,𝔽11)—111no—
35C3×C5×AGL(1,𝔽11)—111no—
36C3×C5×AGL(1,𝔽11)—111no—
37C3×C5×AGL(1,𝔽11)—111no—
38C3×C5×AGL(1,𝔽11)—118no—
39C3×C5×AGL(1,𝔽11)—111no—
40C3×C5×AGL(1,𝔽11)—111no—
41C3×C5×AGL(1,𝔽11)—118no—
42C2×C3×C11⋊D11C5211yescharacteristic, complemented, Hall
43C2×C5×C112C2×C3×C511yescharacteristic, complemented
44C5×C11⋊D11C2×C3×C511yescharacteristic, complemented
45C5×C11⋊D11C2×C3×C511yescharacteristic, complemented
46C2×#10597243C2×C3×C515yescomplemented
47#11925892C2×C3×C515yesdirect product factor
48#11925892C2×C3×C515yesdirect product factor
49C2×C5×AGL(1,𝔽11)—111no—
50C2×C5×AGL(1,𝔽11)—111no—
51C2×C5×AGL(1,𝔽11)—111no—
52C2×C5×AGL(1,𝔽11)—111no—
53C2×C5×AGL(1,𝔽11)—118no—
54C3×C5×C11⋊C5—111no—
55C3×C5×C11⋊C5—111no—
56C3×C5×C11⋊C5—111no—
57C3×C5×C11⋊C5—111no—
58C3×C5×C11⋊C5—118no—
59C2×C3×C112C2×C5211yescharacteristic, complemented
60C3×C11⋊D11C2×C5211yescharacteristic, complemented
61C3×C11⋊D11C2×C5211yescharacteristic, complemented
62C2×C3×C5×D11—111no—
63C2×C3×C5×D11—551no—
64C2×C3×C5×D11—551no—
65C2×C3×C5×D11—551no—
66C2×C3×C5×D11—558no—
67C2×C3×AGL(1,𝔽11)—115no—
68C2×C3×AGL(1,𝔽11)—115no—
69C2×C3×AGL(1,𝔽11)—115no—
70C2×C3×AGL(1,𝔽11)—115no—
71C2×C3×AGL(1,𝔽11)—1140no—
72C5×C112C22×C3×C511yescharacteristic, complemented
73#10597243C22×C3×C515yescomplemented
74C2×C5×C11⋊C5—111no—
75C2×C5×C11⋊C5—111no—
76C2×C5×C11⋊C5—111no—
77C2×C5×C11⋊C5—111no—
78C2×C5×C11⋊C5—118no—
79C5×AGL(1,𝔽11)—111no—
80C5×AGL(1,𝔽11)—111no—
81C5×AGL(1,𝔽11)—111no—
82C5×AGL(1,𝔽11)—111no—
83C5×AGL(1,𝔽11)—111no—
84C5×AGL(1,𝔽11)—118no—
85C5×AGL(1,𝔽11)—111no—
86C5×AGL(1,𝔽11)—111no—
87C5×AGL(1,𝔽11)—111no—
88C5×AGL(1,𝔽11)—118no—
89C2×C11⋊D11C3×C5211yescharacteristic, complemented, Hall
90C3×C112C22×C5211yescharacteristic, complemented, Hall
91C2×C3×C5×C11AGL(1,𝔽11)11yescharacteristic, complemented
92C2×C3×C5×C11AGL(1,𝔽11)11yescharacteristic, complemented
93C2×C3×C5×C11AGL(1,𝔽11)11yescharacteristic, complemented
94C2×C3×C5×C11AGL(1,𝔽11)11yescharacteristic, complemented
95C2×C3×C5×C11AGL(1,𝔽11)18yescomplemented
96C3×C5×D11—111no—
97C3×C5×D11—551no—
98C3×C5×D11—551no—
99C3×C5×D11—551no—
100C3×C5×D11—551no—
101C3×C5×D11—551no—
102C3×C5×D11—558no—
103C3×C5×D11—551no—
104C3×C5×D11—551no—
105C3×C5×D11—558no—
106C2×C3×C11⋊C5—115no—
107C2×C3×C11⋊C5—115no—
108C2×C3×C11⋊C5—115no—
109C2×C3×C11⋊C5—115no—
110C2×C3×C11⋊C5—1140no—
111C3×AGL(1,𝔽11)—115no—
112C3×AGL(1,𝔽11)—115no—
113C3×AGL(1,𝔽11)—115no—
114C3×AGL(1,𝔽11)—115no—
115C3×AGL(1,𝔽11)—115no—
116C3×AGL(1,𝔽11)—115no—
117C3×AGL(1,𝔽11)—1140no—
118C3×AGL(1,𝔽11)—115no—
119C3×AGL(1,𝔽11)—115no—
120C3×AGL(1,𝔽11)—1140no—
121C22×C3×C52—1211noHall
122C5×C11⋊C5—111no—
123C5×C11⋊C5—111no—
124C5×C11⋊C5—111no—
125C5×C11⋊C5—111no—
126C5×C11⋊C5—118no—
127C2×C112C2×C3×C5211yescharacteristic, complemented
128C11⋊D11C2×C3×C5211yescharacteristic, complemented
129C11⋊D11C2×C3×C5211yescharacteristic, complemented
130C2×C5×D11—551no—
131C2×C5×D11—551no—
132C2×C5×D11—551no—
133C2×C5×D11—551no—
134C2×C5×D11—558no—
135C2×AGL(1,𝔽11)—115no—
136C2×AGL(1,𝔽11)—115no—
137C2×AGL(1,𝔽11)—115no—
138C2×AGL(1,𝔽11)—115no—
139C2×AGL(1,𝔽11)—1140no—
140C3×C5×C11C2×AGL(1,𝔽11)11yescharacteristic, complemented
141C3×C5×C11C2×AGL(1,𝔽11)11yescharacteristic, complemented
142C3×C5×C11C2×AGL(1,𝔽11)11yescharacteristic, complemented
143C3×C5×C11C2×AGL(1,𝔽11)11yescharacteristic, complemented
144C3×C5×C11C2×AGL(1,𝔽11)18yescomplemented
145C3×C11⋊C5—115no—
146C3×C11⋊C5—115no—
147C3×C11⋊C5—115no—
148C3×C11⋊C5—115no—
149C3×C11⋊C5—1140no—
150C2×C3×C52—1211no—
151C2×C3×C52—1211no—
152C2×C3×C52—1211no—
153C2×C3×D11—111no—
154C2×C3×D11—551no—
155C2×C3×D11—551no—
156C2×C3×D11—551no—
157C2×C3×D11—558no—
158C112C22×C3×C5211yescharacteristic, complemented, Sylow
159C2×C5×C11C3×AGL(1,𝔽11)11yescharacteristic, complemented
160C2×C5×C11C3×AGL(1,𝔽11)11yescharacteristic, complemented
161C2×C5×C11C3×AGL(1,𝔽11)11yescharacteristic, complemented
162C2×C5×C11C3×AGL(1,𝔽11)11yescharacteristic, complemented
163C2×C5×C11C3×AGL(1,𝔽11)18yescomplemented
164C5×D11—111no—
165C5×D11—111no—
166C5×D11—111no—
167C5×D11—551no—
168C5×D11—551no—
169C5×D11—558no—
170C5×D11—551no—
171C5×D11—551no—
172C5×D11—551no—
173C5×D11—558no—
174C2×C11⋊C5—115no—
175C2×C11⋊C5—115no—
176C2×C11⋊C5—115no—
177C2×C11⋊C5—115no—
178C2×C11⋊C5—1140no—
179AGL(1,𝔽11)—115no—
180AGL(1,𝔽11)—115no—
181AGL(1,𝔽11)—115no—
182AGL(1,𝔽11)—115no—
183AGL(1,𝔽11)—115no—
184AGL(1,𝔽11)—1140no—
185AGL(1,𝔽11)—115no—
186AGL(1,𝔽11)—115no—
187AGL(1,𝔽11)—115no—
188AGL(1,𝔽11)—1140no—
189C22×C52—1211noHall
190C3×C52—1211noHall
191C2×C3×C11C5×AGL(1,𝔽11)11yescharacteristic, complemented
192C2×C3×C11C5×AGL(1,𝔽11)11yescharacteristic, complemented
193C2×C3×C11C5×AGL(1,𝔽11)11yescharacteristic, complemented
194C2×C3×C11C5×AGL(1,𝔽11)11yescharacteristic, complemented
195C2×C3×C11C5×AGL(1,𝔽11)18yescomplemented
196C3×D11—111no—
197C3×D11—551no—
198C3×D11—551no—
199C3×D11—551no—
200C3×D11—551no—
201C3×D11—551no—
202C3×D11—558no—
203C3×D11—551no—
204C3×D11—551no—
205C3×D11—558no—
206C22×C3×C5—6051no—
207C22×C3×C5—1215no—
208C5×C11C2×C3×AGL(1,𝔽11)11yescharacteristic, complemented
209C5×C11C2×C3×AGL(1,𝔽11)11yescharacteristic, complemented
210C5×C11C2×C3×AGL(1,𝔽11)11yescharacteristic, complemented
211C5×C11C2×C3×AGL(1,𝔽11)11yescharacteristic, complemented
212C5×C11C2×C3×AGL(1,𝔽11)18yescomplemented
213C11⋊C5—115no—
214C11⋊C5—115no—
215C11⋊C5—115no—
216C11⋊C5—115no—
217C11⋊C5—1140no—
218C2×C52—1211no—
219C2×C52—1211no—
220C2×C52—1211no—
221C2×D11—551no—
222C2×D11—551no—
223C2×D11—551no—
224C2×D11—551no—
225C2×D11—558no—
226C3×C11C2×C5×AGL(1,𝔽11)11yescharacteristic, complemented
227C3×C11C2×C5×AGL(1,𝔽11)11yescharacteristic, complemented
228C3×C11C2×C5×AGL(1,𝔽11)11yescharacteristic, complemented
229C3×C11C2×C5×AGL(1,𝔽11)11yescharacteristic, complemented
230C3×C11C2×C5×AGL(1,𝔽11)18yescomplemented
231C2×C3×C5—6051no—
232C2×C3×C5—6051no—
233C2×C3×C5—1215no—
234C2×C3×C5—1215no—
235C2×C3×C5—1215no—
236C2×C3×C5#1192589211yescharacteristic, direct product factor
237C52—1211noSylow
238C2×C11C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
239C2×C11C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
240C2×C11C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
241C2×C11C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
242C2×C11C3×C5×AGL(1,𝔽11)18yescomplemented
243D11—111no—
244D11—111no—
245D11—111no—
246D11—551no—
247D11—551no—
248D11—558no—
249D11—551no—
250D11—551no—
251D11—551no—
252D11—558no—
253C22×C5—6051no—
254C22×C5—1215no—
255C3×C5—1215no—
256C3×C5C2×#1192589211yescharacteristic, direct product factor
257C22×C3—6051noHall
258C11C2×C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
259C11C2×C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
260C11C2×C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
261C11C2×C3×C5×AGL(1,𝔽11)11yescharacteristic, complemented
262C11C2×C3×C5×AGL(1,𝔽11)18yescomplemented
263C2×C5—6051no—
264C2×C5—6051no—
265C2×C5—1215no—
266C2×C5—1215no—
267C2×C5—1215no—
268C2×C5C3×#1192589211yescharacteristic, direct product factor
269C2×C3—6051no—
270C2×C3—6051no—
271C2×C3C5×#1192589211yescharacteristic, direct product factor
272C5—1215no—
273C5C2×C3×#1192589211yescharacteristic, direct product factor
274C22—6051noSylow
275C3C2×C5×#1192589211yescharacteristic, direct product factor, Sylow
276C2—6051no—
277C2—6051no—
278C2C3×C5×#1192589211yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5#11925892
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×#10597243C2semidirect
2. C3×C5×#11925892C2direct (N × H)
3. C3×C5×#11925892C2direct (N × H)
4. C2×C5×#11925892C3direct (N × H)
5. C3×C5×#10597243C22semidirect
6. C2×C3×C5×C11⋊D11C5semidirect
7. C2×C3×#11925892C5direct (N × H)
8. C2×C5×#10597243C2×C3semidirect
9. C5×#11925892C2×C3direct (N × H)
10. C5×#11925892C2×C3direct (N × H)
11. C3×C5×C11⋊D11C2×C5semidirect
12. C3×C5×C11⋊D11C2×C5semidirect
13. C2×C3×#10597243C2×C5semidirect
14. C3×#11925892C2×C5direct (N × H)
15. C3×#11925892C2×C5direct (N × H)
16. C2×C3×C5×C112C2×C5semidirect
22. C5×#10597243C22×C3semidirect
23. C2×C5×C11⋊D11C3×C5semidirect
24. C2×#11925892C3×C5direct (N × H)
25. C3×C5×C112C22×C5semidirect
26. C3×#10597243C22×C5semidirect
42. C2×C3×C11⋊D11C52semidirect
43. C2×C5×C112C2×C3×C5semidirect
44. C5×C11⋊D11C2×C3×C5semidirect
45. C5×C11⋊D11C2×C3×C5semidirect
46. C2×#10597243C2×C3×C5semidirect
47. #11925892C2×C3×C5direct (N × H)
48. #11925892C2×C3×C5direct (N × H)
59. C2×C3×C112C2×C52semidirect
60. C3×C11⋊D11C2×C52semidirect
61. C3×C11⋊D11C2×C52semidirect
72. C5×C112C22×C3×C5semidirect
73. #10597243C22×C3×C5semidirect
89. C2×C11⋊D11C3×C52semidirect
90. C3×C112C22×C52semidirect
91. C2×C3×C5×C11AGL(1,𝔽11)semidirect
92. C2×C3×C5×C11AGL(1,𝔽11)semidirect
93. C2×C3×C5×C11AGL(1,𝔽11)semidirect
94. C2×C3×C5×C11AGL(1,𝔽11)semidirect
95. C2×C3×C5×C11AGL(1,𝔽11)semidirect
127. C2×C112C2×C3×C52semidirect
128. C11⋊D11C2×C3×C52semidirect
129. C11⋊D11C2×C3×C52semidirect
140. C3×C5×C11C2×AGL(1,𝔽11)semidirect
141. C3×C5×C11C2×AGL(1,𝔽11)semidirect
142. C3×C5×C11C2×AGL(1,𝔽11)semidirect
143. C3×C5×C11C2×AGL(1,𝔽11)semidirect
144. C3×C5×C11C2×AGL(1,𝔽11)semidirect
158. C112C22×C3×C52semidirect
159. C2×C5×C11C3×AGL(1,𝔽11)semidirect
160. C2×C5×C11C3×AGL(1,𝔽11)semidirect
161. C2×C5×C11C3×AGL(1,𝔽11)semidirect
162. C2×C5×C11C3×AGL(1,𝔽11)semidirect
163. C2×C5×C11C3×AGL(1,𝔽11)semidirect
191. C2×C3×C11C5×AGL(1,𝔽11)semidirect
192. C2×C3×C11C5×AGL(1,𝔽11)semidirect
193. C2×C3×C11C5×AGL(1,𝔽11)semidirect
194. C2×C3×C11C5×AGL(1,𝔽11)semidirect
195. C2×C3×C11C5×AGL(1,𝔽11)semidirect
208. C5×C11C2×C3×AGL(1,𝔽11)semidirect
209. C5×C11C2×C3×AGL(1,𝔽11)semidirect
210. C5×C11C2×C3×AGL(1,𝔽11)semidirect
211. C5×C11C2×C3×AGL(1,𝔽11)semidirect
212. C5×C11C2×C3×AGL(1,𝔽11)semidirect
226. C3×C11C2×C5×AGL(1,𝔽11)semidirect
227. C3×C11C2×C5×AGL(1,𝔽11)semidirect
228. C3×C11C2×C5×AGL(1,𝔽11)semidirect
229. C3×C11C2×C5×AGL(1,𝔽11)semidirect
230. C3×C11C2×C5×AGL(1,𝔽11)semidirect
236. C2×C3×C5#11925892direct (N × H)
238. C2×C11C3×C5×AGL(1,𝔽11)semidirect
239. C2×C11C3×C5×AGL(1,𝔽11)semidirect
240. C2×C11C3×C5×AGL(1,𝔽11)semidirect
241. C2×C11C3×C5×AGL(1,𝔽11)semidirect
242. C2×C11C3×C5×AGL(1,𝔽11)semidirect
256. C3×C5C2×#11925892direct (N × H)
258. C11C2×C3×C5×AGL(1,𝔽11)semidirect
259. C11C2×C3×C5×AGL(1,𝔽11)semidirect
260. C11C2×C3×C5×AGL(1,𝔽11)semidirect
261. C11C2×C3×C5×AGL(1,𝔽11)semidirect
262. C11C2×C3×C5×AGL(1,𝔽11)semidirect
268. C2×C5C3×#11925892direct (N × H)
271. C2×C3C5×#11925892direct (N × H)
273. C5C2×C3×#11925892direct (N × H)
275. C3C2×C5×#11925892direct (N × H)
278. C2C3×C5×#11925892direct (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×#105972432
C3×C5×#119258922
C2×C5×#119258923
C2×C3×C5×C11⋊D115
C2×C3×#119258925
C2×C3×C5×AGL(1,𝔽11)11