Finite Group Databasebeta

← C5×#71754 #14282755

Subgroups — C5×#71754 #14282755

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#529C211yesmaximal, characteristic
2C5×#4932C211yesmaximal, characteristic, complemented
3C5×#4925C211yesmaximal, characteristic, complemented
4C5×#4538C212yesmaximal, complemented
5C5×#4537C212yesmaximal
6C5×#1500—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×C4×C5×Dic3—21no—
19C5×C3⋊#95—21no—
20C5×C3⋊#95C2212yes—
21C5×#830C2212yes—
22C5×C4⋊(C3⋊C8)—21no—
23C4×C5×C3⋊C8—21no—
24#71754C511yesmaximal, characteristic, direct product factor, Hall
25C5×#529D311yescharacteristic
26C5×#431—31no—
27C5×#376—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
54C4×C5×Dic3—24no—
55C2×C5×C3⋊C8—22no—
56C3×#529C2×C511yescharacteristic
57#4932C2×C511yescharacteristic, complemented
58#4925C2×C511yescharacteristic, complemented
59#4538C2×C512yescomplemented
60#4537C2×C512yes—
61C5×#126—21no—
62C5×#125Dic311yescharacteristic, complemented
63C5×#124Dic311yescharacteristic, complemented
64C5×#123—21no—
65C5×C22⋊Q8C2×D311yescharacteristic
66C5×#121C2×D311yescharacteristic
67C4×C5×Q8Dic311yescharacteristic, complemented
68C4×C5×D4—21no—
69C4×C5×D4Dic311yescharacteristic, complemented
70C5×#117—21no—
71C2×C42×C5—61no—
72C2×C42×C5C2×D311yescharacteristic
73C5×C4⋊C8—61no—
74C5×C22⋊C8—32no—
75C4×C5×C8—61no—
76C5×#95—32no—
77C5×#95—61no—
78#1500—31noSylow
79C23×C3×C5—21no—
80C23×C3×C5C2×D411yescharacteristic
81C3×C5×Q8—41no—
82C3×C5×D4—41no—
83C3×C5×D4—42no—
84C2×C3×C4×C5—22no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5—42no—
91C2×C3×C4×C5C22⋊C411yescharacteristic
92C2×C3×C4×C5C22⋊C411yescharacteristic
93C2×C3×C4×C5C22⋊C411yescharacteristic
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C2×C3×C4×C5C22×C411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C5×Dic3—24no—
98C2×C5×Dic3—42no—
99C2×C5×Dic3—42no—
100C5×C3⋊C8—42no—
101C3×#126—21no—
102C3×#125C4×C511yescharacteristic, complemented
103C3×#124C4×C511yescharacteristic, complemented
104C3×#123—21no—
105C3×C22⋊Q8C22×C511yescharacteristic
106C3×#121C22×C511yescharacteristic
107C3×C4×Q8C4×C511yescharacteristic, complemented
108C3×C4×D4—21no—
109C3×C4×D4C4×C511yescharacteristic, complemented
110C3×#117—21no—
111C2×C3×C42C22×C511yescharacteristic
112C2×C4×Dic3—21no—
113C3⋊#95—21no—
114C3⋊#95C22×C512yes—
115#830C22×C512yes—
116C4⋊(C3⋊C8)—21no—
117C4×C3⋊C8—21no—
118C2×C5×Q8C2×Dic311yescharacteristic
119C2×C5×D4—21no—
120C2×C5×D4C2×Dic311yescharacteristic
121C22×C4×C5—21no—
122C22×C4×C5—62no—
123C22×C4×C5#6711yescharacteristic
124C22×C4×C5#6711yescharacteristic
125C22×C4×C5C22×D311yescharacteristic
126C2×C5×C8—62no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4—21no—
131C5×C4⋊C4C2×Dic311yescharacteristic
132C5×C4⋊C4C2×Dic311yescharacteristic
133C5×C22⋊C4—21no—
134C5×C22⋊C4—21no—
135C5×C22⋊C4—21no—
136C5×C22⋊C4—21no—
137C5×C22⋊C4—21no—
138C42×C5—21no—
139C42×C5—64no—
140C42×C5C2×Dic311yescharacteristic
141C42×C5C2×Dic311yescharacteristic
142C42×C5#6711yescharacteristic
143C42×C5#6711yescharacteristic
144#529C5×D311yescharacteristic
145#431—31no—
146#376—31no—
147#328—32no—
148#327—32no—
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5—41no—
152C22×C3×C5—41no—
153C22×C3×C5—41no—
154C22×C3×C5C2×C22⋊C411yescharacteristic
155C22×C3×C5#13512yes—
156C3×C4×C5—21no—
157C3×C4×C5—21no—
158C3×C4×C5—21no—
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161C3×C4×C5—41no—
162C3×C4×C5—41no—
163C3×C4×C5—41no—
164C3×C4×C5#10412yes—
165C5×Dic3—44no—
166C2×C3×Q8C2×C4×C511yescharacteristic
167C2×C3×D4—21no—
168C2×C3×D4C2×C4×C511yescharacteristic
169C22×C3×C4—21no—
170C22×C3×C4C5×D411yescharacteristic
171C22×C3×C4C5×D411yescharacteristic
172C22×C3×C4C23×C511yescharacteristic
173C22×Dic3—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4C2×C4×C511yescharacteristic
179C3×C4⋊C4C2×C4×C511yescharacteristic
180C3×C22⋊C4—21no—
181C3×C22⋊C4—21no—
182C3×C22⋊C4—21no—
183C3×C22⋊C4—21no—
184C3×C22⋊C4—21no—
185C3×C42—21no—
186C3×C42C2×C4×C511yescharacteristic
187C3×C42C2×C4×C511yescharacteristic
188C3×C42C5×D411yescharacteristic
189C3×C42C5×D411yescharacteristic
190C4×Dic3—24no—
191C2×C3⋊C8—22no—
192C23×C5—21no—
193C23×C5C2×#6711yescharacteristic
194C5×Q8—41no—
195C5×D4—41no—
196C5×D4—42no—
197C2×C4×C5—22no—
198C2×C4×C5—22no—
199C2×C4×C5—21no—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—41no—
203C2×C4×C5—42no—
204C2×C4×C5—64no—
205C2×C4×C5—122no—
206C2×C4×C5—122no—
207C2×C4×C5#21711yescharacteristic
208C2×C4×C5#21711yescharacteristic
209C2×C4×C5#21711yescharacteristic
210C2×C4×C5#21711yescharacteristic
211C2×C4×C5C22×Dic311yescharacteristic
212C2×C4×C5C2×#6711yescharacteristic
213C5×C8—122no—
214#126—21no—
215#125C5×Dic311yescharacteristic, complemented
216#124C5×Dic311yescharacteristic, complemented
217#123—21no—
218C22⋊Q8C2×C5×D311yescharacteristic
219#121C2×C5×D311yescharacteristic
220C4×Q8C5×Dic311yescharacteristic, complemented
221C4×D4—21no—
222C4×D4C5×Dic311yescharacteristic, complemented
223#117—21no—
224C2×C42—61no—
225C2×C42C2×C5×D311yescharacteristic
226C4⋊C8—61no—
227C22⋊C8—32no—
228C4×C8—61no—
229#95—32no—
230#95—61no—
231C2×C3×C5—22no—
232C2×C3×C5—81no—
233C2×C3×C5#41011yescharacteristic
234C2×C3×C5#41611yescharacteristic
235C2×C3×C5C2×#10411yescharacteristic
236C23×C3—21no—
237C23×C3C2×C5×D411yescharacteristic
238C3×Q8—41no—
239C3×D4—41no—
240C3×D4—42no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4C5×C22⋊C411yescharacteristic
249C2×C3×C4C5×C22⋊C411yescharacteristic
250C2×C3×C4C5×C22⋊C411yescharacteristic
251C2×C3×C4C5×C22⋊C411yescharacteristic
252C2×C3×C4C22×C4×C511yescharacteristic
253C2×C3×C4C2×C5×D411yescharacteristic
254C2×Dic3—24no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C3⋊C8—42no—
258C22×C5—21no—
259C22×C5—21no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5—41no—
263C22×C5#95012yes—
264C22×C5C2×#21711yescharacteristic
265C4×C5—21no—
266C4×C5—21no—
267C4×C5—21no—
268C4×C5—21no—
269C4×C5—21no—
270C4×C5—41no—
271C4×C5—41no—
272C4×C5—41no—
273C4×C5—124no—
274C4×C5#83712yes—
275C2×Q8C2×C5×Dic311yescharacteristic
276C2×D4—21no—
277C2×D4C2×C5×Dic311yescharacteristic
278C22×C4—21no—
279C22×C4—62no—
280C22×C4C5×#6711yescharacteristic
281C22×C4C5×#6711yescharacteristic
282C22×C4C22×C5×D311yescharacteristic
283C2×C8—62no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4C2×C5×Dic311yescharacteristic
289C4⋊C4C2×C5×Dic311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C42—21no—
296C42—64no—
297C42C2×C5×Dic311yescharacteristic
298C42C2×C5×Dic311yescharacteristic
299C42C5×#6711yescharacteristic
300C42C5×#6711yescharacteristic
301C3×C5#150011yescharacteristic, complemented, Hall
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3C2×C5×C22⋊C411yescharacteristic
308C22×C3C5×#13512yes—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4C5×#10412yes—
318Dic3—44no—
319C2×C5—22no—
320C2×C5—81no—
321C2×C5#523311yescharacteristic
322C2×C5C2×#83711yescharacteristic
323C2×C5#524011yescharacteristic
324C23—21no—
325C23C2×C5×#6711yescharacteristic
326Q8—41no—
327D4—41no—
328D4—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—64no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4C5×#21711yescharacteristic
340C2×C4C5×#21711yescharacteristic
341C2×C4C5×#21711yescharacteristic
342C2×C4C5×#21711yescharacteristic
343C2×C4C22×C5×Dic311yescharacteristic
344C2×C4C2×C5×#6711yescharacteristic
345C8—122no—
346C2×C3—22no—
347C2×C3—81no—
348C2×C3C5×#41011yescharacteristic
349C2×C3C5×#41611yescharacteristic
350C2×C3C2×C5×#10411yescharacteristic
351C5#7175411yescharacteristic, direct product factor, Sylow
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22C5×#95012yes—
358C22C2×C5×#21711yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—124no—
368C4C5×#83712yes—
369C3C5×#150011yescharacteristic, complemented, Sylow
370C2—22no—
371C2—81no—
372C2C5×#523311yescharacteristic
373C2C2×C5×#83711yescharacteristic
374C2C5×#524011yescharacteristic
subgroupG/H
centerC2×C4×C5#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×#1500
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C5×#529

Complements

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

NH ≅ G/Ntype
2. C5×#4932C2semidirect
3. C5×#4925C2semidirect
4. C5×#4538C2semidirect
8. C3×C5×#125C4semidirect
9. C3×C5×#124C4semidirect
13. C3×C4×C5×Q8C4semidirect
15. C3×C4×C5×D4C4semidirect
24. #71754C5direct (N × H)
57. #4932C2×C5semidirect
58. #4925C2×C5semidirect
59. #4538C2×C5semidirect
62. C5×#125Dic3semidirect
63. C5×#124Dic3semidirect
67. C4×C5×Q8Dic3semidirect
69. C4×C5×D4Dic3semidirect
102. C3×#125C4×C5semidirect
103. C3×#124C4×C5semidirect
107. C3×C4×Q8C4×C5semidirect
109. C3×C4×D4C4×C5semidirect
215. #125C5×Dic3semidirect
216. #124C5×Dic3semidirect
220. C4×Q8C5×Dic3semidirect
222. C4×D4C5×Dic3semidirect
301. C3×C5#1500semidirect
351. C5#71754direct (N × H)
369. C3C5×#1500semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5292
C5×#49322
C5×#49252
C5×#45382
C5×#45372
C5×#15003
#717545