Finite Group Databasebeta

← C5×#71755 #14283202

Subgroups — C5×#71755 #14283202

Order 1920.

484 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#529C211yesmaximal, characteristic
2C4×C5×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C5×#4926C211yesmaximal, characteristic, complemented
4C5×#4538C212yesmaximal, complemented
5C5×#4537C212yesmaximal
6C5×#1509—31nomaximal, Hall
7C3×C5×#126—21no—
8C3×C5×#125C411yescharacteristic, complemented
9C3×C5×#124C411yescharacteristic, complemented
10C3×C5×#123—21no—
11C3×C5×C22⋊Q8C2211yescharacteristic
12C3×C5×#121C2211yescharacteristic
13C3×C4×C5×Q8C411yescharacteristic, complemented
14C3×C4×C5×D4—21no—
15C3×C4×C5×D4C411yescharacteristic, complemented
16C3×C5×#117—21no—
17C2×C3×C42×C5C2211yescharacteristic
18C2×C5×C4⋊Dic3—21no—
19C2×C4×C5×Dic3—21no—
20C5×C3⋊#95C2212yes—
21C5×#830C2212yes—
22C5×C4⋊(C3⋊C8)—21no—
23C5×#803—21no—
24#71755C511yesmaximal, characteristic, direct product factor, Hall
25C5×#529D311yescharacteristic
26C5×#432—31no—
27C4×C5×C4⋊C4—31no—
28C5×#328—32no—
29C5×#327—32no—
30C2×C3×C5×Q8C2×C411yescharacteristic
31C2×C3×C5×D4—21no—
32C2×C3×C5×D4C2×C411yescharacteristic
33C22×C3×C4×C5—21no—
34C22×C3×C4×C5D411yescharacteristic
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5C2311yescharacteristic
37C22×C5×Dic3—22no—
38C3×C5×C4⋊C4—21no—
39C3×C5×C4⋊C4—21no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4C2×C411yescharacteristic
43C3×C5×C4⋊C4C2×C411yescharacteristic
44C3×C5×C22⋊C4—21no—
45C3×C5×C22⋊C4—21no—
46C3×C5×C22⋊C4—21no—
47C3×C5×C22⋊C4—21no—
48C3×C5×C22⋊C4—21no—
49C3×C42×C5—21no—
50C3×C42×C5C2×C411yescharacteristic
51C3×C42×C5C2×C411yescharacteristic
52C3×C42×C5D411yescharacteristic
53C3×C42×C5D411yescharacteristic
54C5×C4⋊Dic3—24no—
55C4×C5×Dic3—42no—
56C2×C5×C3⋊C8—22no—
57C3×#529C2×C511yescharacteristic
58C4×C4⋊Dic3C2×C511yescharacteristic, complemented
59#4926C2×C511yescharacteristic, complemented
60#4538C2×C512yescomplemented
61#4537C2×C512yes—
62C5×#126—21no—
63C5×#125Dic311yescharacteristic, complemented
64C5×#124Dic311yescharacteristic, complemented
65C5×#123—21no—
66C5×C22⋊Q8C2×D311yescharacteristic
67C5×#121C2×D311yescharacteristic
68C4×C5×Q8Dic311yescharacteristic, complemented
69C4×C5×D4—21no—
70C4×C5×D4Dic311yescharacteristic, complemented
71C5×#117—21no—
72C2×C5×C4⋊C4—61no—
73C2×C42×C5—61no—
74C2×C42×C5C2×D311yescharacteristic
75C5×C4⋊C8—61no—
76C5×C22⋊C8—32no—
77C5×#97—61no—
78C5×#95—32no—
79#1509—31noSylow
80C23×C3×C5—21no—
81C23×C3×C5C2×D411yescharacteristic
82C3×C5×Q8—41no—
83C3×C5×D4—41no—
84C3×C5×D4—41no—
85C3×C5×D4—41no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—22no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—41no—
93C2×C3×C4×C5—42no—
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C2×C3×C4×C5C22⋊C411yescharacteristic
96C2×C3×C4×C5C22⋊C411yescharacteristic
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C5×Dic3—24no—
101C2×C5×Dic3—42no—
102C2×C5×Dic3—42no—
103C5×C3⋊C8—42no—
104C3×#126—21no—
105C3×#125C4×C511yescharacteristic, complemented
106C3×#124C4×C511yescharacteristic, complemented
107C3×#123—21no—
108C3×C22⋊Q8C22×C511yescharacteristic
109C3×#121C22×C511yescharacteristic
110C3×C4×Q8C4×C511yescharacteristic, complemented
111C3×C4×D4—21no—
112C3×C4×D4C4×C511yescharacteristic, complemented
113C3×#117—21no—
114C2×C3×C42C22×C511yescharacteristic
115C2×C4⋊Dic3—21no—
116C2×C4×Dic3—21no—
117C3⋊#95C22×C512yes—
118#830C22×C512yes—
119C4⋊(C3⋊C8)—21no—
120#803—21no—
121C2×C5×Q8C2×Dic311yescharacteristic
122C2×C5×D4—21no—
123C2×C5×D4C2×Dic311yescharacteristic
124C22×C4×C5—21no—
125C22×C4×C5—62no—
126C22×C4×C5#6711yescharacteristic
127C22×C4×C5#6711yescharacteristic
128C22×C4×C5C22×D311yescharacteristic
129C2×C5×C8—62no—
130C5×C4⋊C4—21no—
131C5×C4⋊C4—21no—
132C5×C4⋊C4—21no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4—64no—
135C5×C4⋊C4C2×Dic311yescharacteristic
136C5×C4⋊C4C2×Dic311yescharacteristic
137C5×C22⋊C4—21no—
138C5×C22⋊C4—21no—
139C5×C22⋊C4—21no—
140C5×C22⋊C4—21no—
141C5×C22⋊C4—21no—
142C42×C5—21no—
143C42×C5—122no—
144C42×C5C2×Dic311yescharacteristic
145C42×C5C2×Dic311yescharacteristic
146C42×C5#6711yescharacteristic
147C42×C5#6711yescharacteristic
148#529C5×D311yescharacteristic
149#432—31no—
150C4×C4⋊C4—31no—
151#328—32no—
152#327—32no—
153C22×C3×C5—21no—
154C22×C3×C5—21no—
155C22×C3×C5—41no—
156C22×C3×C5—41no—
157C22×C3×C5—41no—
158C22×C3×C5C2×C22⋊C411yescharacteristic
159C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
160C22×C3×C5#13711yescharacteristic
161C3×C4×C5—21no—
162C3×C4×C5—21no—
163C3×C4×C5—21no—
164C3×C4×C5—21no—
165C3×C4×C5—21no—
166C3×C4×C5—41no—
167C3×C4×C5—41no—
168C3×C4×C5—41no—
169C3×C4×C5#10412yes—
170C5×Dic3—44no—
171C2×C3×Q8C2×C4×C511yescharacteristic
172C2×C3×D4—21no—
173C2×C3×D4C2×C4×C511yescharacteristic
174C22×C3×C4—21no—
175C22×C3×C4C5×D411yescharacteristic
176C22×C3×C4C5×D411yescharacteristic
177C22×C3×C4C23×C511yescharacteristic
178C22×Dic3—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4C2×C4×C511yescharacteristic
184C3×C4⋊C4C2×C4×C511yescharacteristic
185C3×C22⋊C4—21no—
186C3×C22⋊C4—21no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—21no—
189C3×C22⋊C4—21no—
190C3×C42—21no—
191C3×C42C2×C4×C511yescharacteristic
192C3×C42C2×C4×C511yescharacteristic
193C3×C42C5×D411yescharacteristic
194C3×C42C5×D411yescharacteristic
195C4⋊Dic3—24no—
196C4×Dic3—42no—
197C2×C3⋊C8—22no—
198C23×C5—21no—
199C23×C5C2×#6711yescharacteristic
200C5×Q8—41no—
201C5×D4—41no—
202C5×D4—41no—
203C5×D4—41no—
204C2×C4×C5—21no—
205C2×C4×C5—21no—
206C2×C4×C5—22no—
207C2×C4×C5—21no—
208C2×C4×C5—21no—
209C2×C4×C5—21no—
210C2×C4×C5—41no—
211C2×C4×C5—42no—
212C2×C4×C5—64no—
213C2×C4×C5—122no—
214C2×C4×C5—122no—
215C2×C4×C5#21711yescharacteristic
216C2×C4×C5#21711yescharacteristic
217C2×C4×C5#21711yescharacteristic
218C2×C4×C5#21711yescharacteristic
219C2×C4×C5C22×Dic311yescharacteristic
220C2×C4×C5C2×#6711yescharacteristic
221C5×C8—122no—
222#126—21no—
223#125C5×Dic311yescharacteristic, complemented
224#124C5×Dic311yescharacteristic, complemented
225#123—21no—
226C22⋊Q8C2×C5×D311yescharacteristic
227#121C2×C5×D311yescharacteristic
228C4×Q8C5×Dic311yescharacteristic, complemented
229C4×D4—21no—
230C4×D4C5×Dic311yescharacteristic, complemented
231#117—21no—
232C2×C4⋊C4—61no—
233C2×C42—61no—
234C2×C42C2×C5×D311yescharacteristic
235C4⋊C8—61no—
236C22⋊C8—32no—
237#97—61no—
238#95—32no—
239C2×C3×C5—21no—
240C2×C3×C5—21no—
241C2×C3×C5—81no—
242C2×C3×C5#41011yescharacteristic
243C2×C3×C5#41711yescharacteristic
244C2×C3×C5C2×#10411yescharacteristic
245C23×C3—21no—
246C23×C3C2×C5×D411yescharacteristic
247C3×Q8—41no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—41no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4C5×C22⋊C411yescharacteristic
260C2×C3×C4C5×C22⋊C411yescharacteristic
261C2×C3×C4C5×C22⋊C411yescharacteristic
262C2×C3×C4C5×C22⋊C411yescharacteristic
263C2×C3×C4C22×C4×C511yescharacteristic
264C2×C3×C4C2×C5×D411yescharacteristic
265C2×Dic3—24no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C3⋊C8—42no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—41no—
272C22×C5—41no—
273C22×C5—41no—
274C22×C5#94911yescharacteristic
275C22×C5#95111yescharacteristic
276C22×C5C2×#21711yescharacteristic
277C4×C5—21no—
278C4×C5—21no—
279C4×C5—21no—
280C4×C5—21no—
281C4×C5—21no—
282C4×C5—41no—
283C4×C5—41no—
284C4×C5—41no—
285C4×C5—124no—
286C4×C5#83712yes—
287C2×Q8C2×C5×Dic311yescharacteristic
288C2×D4—21no—
289C2×D4C2×C5×Dic311yescharacteristic
290C22×C4—21no—
291C22×C4—62no—
292C22×C4C5×#6711yescharacteristic
293C22×C4C5×#6711yescharacteristic
294C22×C4C22×C5×D311yescharacteristic
295C2×C8—62no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—64no—
301C4⋊C4C2×C5×Dic311yescharacteristic
302C4⋊C4C2×C5×Dic311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C42—21no—
309C42—122no—
310C42C2×C5×Dic311yescharacteristic
311C42C2×C5×Dic311yescharacteristic
312C42C5×#6711yescharacteristic
313C42C5×#6711yescharacteristic
314C3×C5#150911yescharacteristic, complemented, Hall
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3C2×C5×C22⋊C411yescharacteristic
321C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
322C22×C3C5×#13711yescharacteristic
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4C5×#10412yes—
332Dic3—44no—
333C2×C5—21no—
334C2×C5—21no—
335C2×C5—81no—
336C2×C5#523211yescharacteristic
337C2×C5C2×#83711yescharacteristic
338C2×C5#524011yescharacteristic
339C23—21no—
340C23C2×C5×#6711yescharacteristic
341Q8—41no—
342D4—41no—
343D4—41no—
344D4—41no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—64no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4C5×#21711yescharacteristic
357C2×C4C5×#21711yescharacteristic
358C2×C4C5×#21711yescharacteristic
359C2×C4C5×#21711yescharacteristic
360C2×C4C22×C5×Dic311yescharacteristic
361C2×C4C2×C5×#6711yescharacteristic
362C8—122no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—81no—
366C2×C3C5×#41011yescharacteristic
367C2×C3C5×#41711yescharacteristic
368C2×C3C2×C5×#10411yescharacteristic
369C5#7175511yescharacteristic, direct product factor, Sylow
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22C5×#94911yescharacteristic
376C22C5×#95111yescharacteristic
377C22C2×C5×#21711yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—124no—
387C4C5×#83712yes—
388C3C5×#150911yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—81no—
392C2C5×#523211yescharacteristic
393C2C2×C5×#83711yescharacteristic
394C2C5×#524011yescharacteristic
subgroupG/H
centerC22×C5C2×#217
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#529C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×C5×#121C22
cosocleC22×C3×C4C23×C5
hypercenterC5×#529D3
hypocenterC3C5×#1509
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C5×#529

Complements

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

NH ≅ G/Ntype
2. C4×C5×C4⋊Dic3C2semidirect
3. C5×#4926C2semidirect
4. C5×#4538C2semidirect
8. C3×C5×#125C4semidirect
9. C3×C5×#124C4semidirect
13. C3×C4×C5×Q8C4semidirect
15. C3×C4×C5×D4C4semidirect
24. #71755C5direct (N × H)
58. C4×C4⋊Dic3C2×C5semidirect
59. #4926C2×C5semidirect
60. #4538C2×C5semidirect
63. C5×#125Dic3semidirect
64. C5×#124Dic3semidirect
68. C4×C5×Q8Dic3semidirect
70. C4×C5×D4Dic3semidirect
105. C3×#125C4×C5semidirect
106. C3×#124C4×C5semidirect
110. C3×C4×Q8C4×C5semidirect
112. C3×C4×D4C4×C5semidirect
223. #125C5×Dic3semidirect
224. #124C5×Dic3semidirect
228. C4×Q8C5×Dic3semidirect
230. C4×D4C5×Dic3semidirect
314. C3×C5#1509semidirect
369. C5#71755direct (N × H)
388. C3C5×#1509semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5292
C4×C5×C4⋊Dic32
C5×#49262
C5×#45382
C5×#45372
C5×#15093
#717555