Finite Group Databasebeta

← C2×C5×#3804 #13089335

Subgroups — C2×C5×#3804 #13089335

Order 1440.

710 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#638C211yesmaximal, characteristic, complemented
2C2×C3×C5×#67C211yesmaximal, characteristic, complemented
3C22×C3×C5×Dic3C211yesmaximal, characteristic, complemented
4C2×C5×#627C211yesmaximal, characteristic, complemented
5C2×C5×#626C211yesmaximal, characteristic, complemented
6C2×C5×#624C211yesmaximal, characteristic, complemented
7C5×#3804C218yesmaximal, direct product factor
8C2×C5×D3×Dic3C211yesmaximal, characteristic, complemented
9C2×C5×#237—31nomaximal
10C2×C5×#235—31nomaximal
11C23×C32×C5C2211yescharacteristic
12C22×C5×C3⋊D3C2211yescharacteristic
13C22×C3×C5×D3C2211yescharacteristic
14C5×#638C2214yescomplemented
15C2×C5×C3⋊Dic3C2211yescharacteristic
16C3×C5×#67C2214yescomplemented
17C2×C3×C5×Dic3C2211yescharacteristic
18C2×C3×C5×Dic3C2211yescharacteristic
19C2×C3×C5×Dic3C2211yescharacteristic
20C2×C3×C5×Dic3C2214yescomplemented
21C5×#627C2214yescomplemented
22C5×#626C2214yescomplemented
23C5×#624C2214yescomplemented
24C5×D3×Dic3C2214yescomplemented
25C2×#3804C511yesmaximal, characteristic, direct product factor, Hall
26C2×C3×C5×D4—31no—
27C22×C3×C4×C5—31no—
28C2×C5×#67—31no—
29C2×C5×#67—31no—
30C2×C5×#67—31no—
31C2×C5×#67—61no—
32C2×C5×#67D311yescharacteristic, complemented
33C22×C5×Dic3—31no—
34C22×C5×Dic3D311yescharacteristic, complemented
35C5×#237—38no—
36C5×#235—38no—
37C2×C5×D12—31no—
38C2×C4×C5×D3—31no—
39C2×C4×C5×D3—31no—
40C2×C4×C5×D3—31no—
41C2×C5×Dic6—31no—
42C2×C5×Dic6—31no—
43C22×C32×C5—22no—
44C22×C32×C5C2312yes—
45C22×C32×C5C2311yescharacteristic
46C2×C5×C3⋊D3—22no—
47C2×C5×C3⋊D3C2312yes—
48C2×C3×C5×D3—22no—
49C2×C3×C5×D3C2312yes—
50C5×C3⋊Dic3C2312yes—
51C3×C5×Dic3C2312yes—
52C3×C5×Dic3C2312yes—
53C3×C5×Dic3C2312yes—
54C2×C5×#41—91noHall
55C2×#638C2×C511yescharacteristic, complemented
56C2×C3×#67C2×C511yescharacteristic, complemented
57C22×C3×Dic3C2×C511yescharacteristic, complemented
58C2×#627C2×C511yescharacteristic, complemented
59C2×#626C2×C511yescharacteristic, complemented
60C2×#624C2×C511yescharacteristic, complemented
61#3804C2×C518yesdirect product factor
62C2×D3×Dic3C2×C511yescharacteristic, complemented
63C23×C3×C5—21no—
64C23×C3×C5—31no—
65C23×C3×C5C2×D311yescharacteristic
66C23×C3×C5C2×D311yescharacteristic
67C22×C5×D3—31no—
68C22×C5×D3—31no—
69C22×C5×D3—61no—
70C22×C5×D3C2×D311yescharacteristic
71C3×C5×D4—34no—
72C2×C3×C4×C5—31no—
73C2×C3×C4×C5—31no—
74C2×C3×C4×C5—31no—
75C2×C3×C4×C5—34no—
76C5×#67—34no—
77C5×#67—34no—
78C5×#67—34no—
79C5×#67—64no—
80C5×#67C2×D314yescomplemented
81C2×C5×Dic3—31no—
82C2×C5×Dic3—31no—
83C2×C5×Dic3—34no—
84C2×C5×Dic3—61no—
85C2×C5×Dic3C2×D311yescharacteristic
86C2×C5×Dic3C2×D311yescharacteristic
87C2×C5×Dic3C2×D311yescharacteristic
88C2×C5×Dic3C2×D314yescomplemented
89C5×D12—34no—
90C4×C5×D3—34no—
91C4×C5×D3—34no—
92C4×C5×D3—34no—
93C5×Dic6—34no—
94C5×Dic6—34no—
95C2×#237—31no—
96C2×#235—31no—
97C2×C32×C5—22no—
98C2×C32×C5#4112yescomplemented
99C2×C32×C5C2411yescharacteristic
100C5×C3⋊D3—22no—
101C3×C5×D3—22no—
102C5×#41—98no—
103C2×C5×Q8—91no—
104C2×C5×D4—31no—
105C2×C5×D4—91no—
106C2×C5×D4—91no—
107C22×C4×C5—31no—
108C22×C4×C5—91no—
109C22×C4×C5—91no—
110C23×C32C22×C511yescharacteristic
111C22×C3⋊D3C22×C511yescharacteristic
112C22×C3×D3C22×C511yescharacteristic
113#638C22×C514yescomplemented
114C2×C3⋊Dic3C22×C511yescharacteristic
115C3×#67C22×C514yescomplemented
116C2×C3×Dic3C22×C511yescharacteristic
117C2×C3×Dic3C22×C511yescharacteristic
118C2×C3×Dic3C22×C511yescharacteristic
119C2×C3×Dic3C22×C514yescomplemented
120#627C22×C514yescomplemented
121#626C22×C514yescomplemented
122#624C22×C514yescomplemented
123D3×Dic3C22×C514yescomplemented
124C22×C3×C5—22no—
125C22×C3×C5—22no—
126C22×C3×C5—22no—
127C22×C3×C5—21no—
128C22×C3×C5—32no—
129C22×C3×C5—42no—
130C22×C3×C5—62no—
131C22×C3×C5C22×D312yes—
132C22×C3×C5C22×D312yes—
133C22×C3×C5C22×D311yescharacteristic
134C22×C3×C5C22×D311yescharacteristic
135C2×C5×D3—22no—
136C2×C5×D3—32no—
137C2×C5×D3—32no—
138C2×C5×D3—62no—
139C2×C5×D3—62no—
140C2×C5×D3—62no—
141C2×C5×D3—122no—
142C2×C5×D3C22×D312yes—
143C3×C4×C5—32no—
144C3×C4×C5—32no—
145C3×C4×C5—32no—
146C5×Dic3—32no—
147C5×Dic3—32no—
148C5×Dic3—62no—
149C5×Dic3C22×D312yes—
150C5×Dic3C22×D312yes—
151C5×Dic3C22×D312yes—
152C2×C3×D4—31no—
153C22×C3×C4—31no—
154C2×#67—31no—
155C2×#67—31no—
156C2×#67—31no—
157C2×#67—61no—
158C2×#67C5×D311yescharacteristic, complemented
159C22×Dic3—31no—
160C22×Dic3C5×D311yescharacteristic, complemented
161#237—38no—
162#235—38no—
163C2×D12—31no—
164C2×C4×D3—31no—
165C2×C4×D3—31no—
166C2×C4×D3—31no—
167C2×Dic6—31no—
168C2×Dic6—31no—
169C32×C5C2×#4111yescharacteristic, complemented, Hall
170C23×C5—31no—
171C23×C5—91no—
172C23×C5D3211yescharacteristic
173C5×Q8—94no—
174C5×D4—34no—
175C5×D4—94no—
176C5×D4—94no—
177C2×C4×C5—34no—
178C2×C4×C5—31no—
179C2×C4×C5—31no—
180C2×C4×C5—31no—
181C2×C4×C5—94no—
182C2×C4×C5—94no—
183C2×C4×C5—91no—
184C22×C32—22no—
185C22×C32C23×C512yes—
186C22×C32C23×C511yescharacteristic
187C2×C3⋊D3—22no—
188C2×C3⋊D3C23×C512yes—
189C2×C3×D3—22no—
190C2×C3×D3C23×C512yes—
191C3⋊Dic3C23×C512yes—
192C3×Dic3C23×C512yes—
193C3×Dic3C23×C512yes—
194C3×Dic3C23×C512yes—
195C2×#41—91noSylow
196C2×C3×C5—22no—
197C2×C3×C5—22no—
198C2×C3×C5—21no—
199C2×C3×C5—22no—
200C2×C3×C5—42no—
201C2×C3×C5—62no—
202C2×C3×C5#23512yescomplemented
203C2×C3×C5#23712yescomplemented
204C2×C3×C5C23×D311yescharacteristic
205C2×C3×C5C23×D311yescharacteristic
206C5×D3—22no—
207C5×D3—62no—
208C5×D3—62no—
209C5×D3—122no—
210C23×C3—21no—
211C23×C3—31no—
212C23×C3C2×C5×D311yescharacteristic
213C23×C3C2×C5×D311yescharacteristic
214C22×D3—31no—
215C22×D3—31no—
216C22×D3—61no—
217C22×D3C2×C5×D311yescharacteristic
218C3×D4—34no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—34no—
223#67—34no—
224#67—34no—
225#67—34no—
226#67—64no—
227#67C2×C5×D314yescomplemented
228C2×Dic3—31no—
229C2×Dic3—31no—
230C2×Dic3—34no—
231C2×Dic3—61no—
232C2×Dic3C2×C5×D311yescharacteristic
233C2×Dic3C2×C5×D311yescharacteristic
234C2×Dic3C2×C5×D311yescharacteristic
235C2×Dic3C2×C5×D314yescomplemented
236D12—34no—
237C4×D3—34no—
238C4×D3—34no—
239C4×D3—34no—
240Dic6—34no—
241Dic6—34no—
242C22×C5—22no—
243C22×C5—32no—
244C22×C5—62no—
245C22×C5—92no—
246C22×C5—182no—
247C22×C5C2×D3212yes—
248C22×C5C2×D3211yescharacteristic
249C4×C5—32no—
250C4×C5—32no—
251C4×C5—32no—
252C4×C5—92no—
253C2×C32—22no—
254C2×C32C5×#4112yescomplemented
255C2×C32C24×C511yescharacteristic
256C3⋊D3—22no—
257C3×D3—22no—
258#41—98no—
259C2×Q8—91no—
260C2×D4—31no—
261C2×D4—91no—
262C2×D4—91no—
263C22×C4—31no—
264C22×C4—91no—
265C22×C4—91no—
266C3×C5—21no—
267C3×C5C2×#23511yescharacteristic, complemented
268C3×C5C2×#23711yescharacteristic, complemented
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—21no—
272C22×C3—22no—
273C22×C3—32no—
274C22×C3—42no—
275C22×C3—62no—
276C22×C3C22×C5×D312yes—
277C22×C3C22×C5×D312yes—
278C22×C3C22×C5×D311yescharacteristic
279C22×C3C22×C5×D311yescharacteristic
280C2×D3—22no—
281C2×D3—32no—
282C2×D3—32no—
283C2×D3—62no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—122no—
287C2×D3C22×C5×D312yes—
288C3×C4—32no—
289C3×C4—32no—
290C3×C4—32no—
291Dic3—32no—
292Dic3—32no—
293Dic3—62no—
294Dic3C22×C5×D312yes—
295Dic3C22×C5×D312yes—
296Dic3C22×C5×D312yes—
297C2×C5—22no—
298C2×C5—62no—
299C2×C5—182no—
300C2×C5#380412yesdirect product factor
301C2×C5C22×D3211yescharacteristic
302C32C2×C5×#4111yescharacteristic, complemented, Sylow
303C23—31no—
304C23—91no—
305C23C5×D3211yescharacteristic
306Q8—94no—
307D4—34no—
308D4—94no—
309D4—94no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—34no—
313C2×C4—31no—
314C2×C4—94no—
315C2×C4—94no—
316C2×C4—91no—
317C2×C3—22no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—42no—
322C2×C3—62no—
323C2×C3C5×#23512yescomplemented
324C2×C3C5×#23712yescomplemented
325C2×C3C23×C5×D311yescharacteristic
326C2×C3C23×C5×D311yescharacteristic
327D3—22no—
328D3—62no—
329D3—62no—
330D3—122no—
331C5C2×#380411yescharacteristic, direct product factor, Sylow
332C22—22no—
333C22—32no—
334C22—62no—
335C22—92no—
336C22—182no—
337C22C2×C5×D3212yes—
338C22C2×C5×D3211yescharacteristic
339C4—32no—
340C4—32no—
341C4—32no—
342C4—92no—
343C3—21no—
344C3C2×C5×#23511yescharacteristic, complemented
345C3C2×C5×#23711yescharacteristic, complemented
346C2—22no—
347C2—62no—
348C2—182no—
349C2C5×#380412yesdirect product factor
350C2C22×C5×D3211yescharacteristic
subgroupG/H
centerC22×C5C2×D32
commutator, cosocleC2×C32C24×C5
FrattiniC2C22×C5×D32
FittingC23×C32×C5C22
socleC22×C32×C5C23
hypercenterC23×C5D32
hypocenterC32C2×C5×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C23×C5

Complements

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

NH ≅ G/Ntype
1. C2×C5×#638C2semidirect
2. C2×C3×C5×#67C2semidirect
3. C22×C3×C5×Dic3C2semidirect
4. C2×C5×#627C2semidirect
5. C2×C5×#626C2semidirect
6. C2×C5×#624C2semidirect
7. C5×#3804C2direct (N × H)
8. C2×C5×D3×Dic3C2semidirect
14. C5×#638C22semidirect
16. C3×C5×#67C22semidirect
20. C2×C3×C5×Dic3C22semidirect
21. C5×#627C22semidirect
22. C5×#626C22semidirect
23. C5×#624C22semidirect
24. C5×D3×Dic3C22semidirect
25. C2×#3804C5direct (N × H)
32. C2×C5×#67D3semidirect
34. C22×C5×Dic3D3semidirect
55. C2×#638C2×C5semidirect
56. C2×C3×#67C2×C5semidirect
57. C22×C3×Dic3C2×C5semidirect
58. C2×#627C2×C5semidirect
59. C2×#626C2×C5semidirect
60. C2×#624C2×C5semidirect
61. #3804C2×C5direct (N × H)
62. C2×D3×Dic3C2×C5semidirect
80. C5×#67C2×D3semidirect
88. C2×C5×Dic3C2×D3semidirect
98. C2×C32×C5#41semidirect
113. #638C22×C5semidirect
115. C3×#67C22×C5semidirect
119. C2×C3×Dic3C22×C5semidirect
120. #627C22×C5semidirect
121. #626C22×C5semidirect
122. #624C22×C5semidirect
123. D3×Dic3C22×C5semidirect
158. C2×#67C5×D3semidirect
160. C22×Dic3C5×D3semidirect
169. C32×C5C2×#41semidirect
202. C2×C3×C5#235semidirect
203. C2×C3×C5#237semidirect
227. #67C2×C5×D3semidirect
235. C2×Dic3C2×C5×D3semidirect
254. C2×C32C5×#41semidirect
267. C3×C5C2×#235semidirect
268. C3×C5C2×#237semidirect
300. C2×C5#3804direct (N × H)
302. C32C2×C5×#41semidirect
323. C2×C3C5×#235semidirect
324. C2×C3C5×#237semidirect
331. C5C2×#3804direct (N × H)
344. C3C2×C5×#235semidirect
345. C3C2×C5×#237semidirect
349. C2C5×#3804direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#6382
C2×C3×C5×#672
C22×C3×C5×Dic32
C2×C5×#6272
C2×C5×#6262
C2×C5×#6242
C5×#38042
C2×C5×D3×Dic32
C2×C5×#2373
C2×C5×#2353
C2×#38045