Finite Group Databasebeta

← C5×#71228 #14289619

Subgroups — C5×#71228 #14289619

Order 1920.

468 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5726C211yesmaximal, characteristic
2C3×C5×#487C211yesmaximal, characteristic, complemented
3C5×#5084C211yesmaximal, characteristic, complemented
4C5×#5084C211yesmaximal, characteristic, complemented
5C5×#4547C211yesmaximal, characteristic, complemented
6C5×#4459C211yesmaximal, characteristic, complemented
7C5×#4459C211yesmaximal, characteristic, complemented
8C5×#1635—31nomaximal, Hall
9C2×C5×#239—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×C4⋊Q8C2211yescharacteristic
12C5×#946—21no—
13C5×#895—21no—
14C5×#894—22no—
15C5×D3×C4⋊C4—21no—
16C4×C5×D12C2211yescharacteristic
17C4×C5×D12C2211yescharacteristic
18C3×C5×#103—22no—
19C3×C4×C5×C8C2211yescharacteristic
20C5×#809—21no—
21C5×#809—21no—
22C5×#808—21no—
23C5×#808—21no—
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C5×C4⋊(C3⋊C8)C2211yescharacteristic
26#71228C511yesmaximal, characteristic, direct product factor, Hall
27C5×#547—31no—
28C5×#487D311yescharacteristic, complemented
29C5×#476—31no—
30C5×#476—31no—
31C5×#332—31no—
32C5×D4⋊C8—31no—
33C5×D4⋊C8—31no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8—21no—
36C5×#239—42no—
37C5×#239—42no—
38C2×C5×D12—22no—
39C2×C5×D12D411yescharacteristic
40C2×C5×D12D411yescharacteristic
41C2×C4×C5×D3—22no—
42C2×C4×C5×D3—21no—
43C2×C4×C5×D3—21no—
44C2×C3×C5×C8—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4D412yes—
49C3×C42×C5C2311yescharacteristic
50C5×(C2×D3)⋊C4—22no—
51C5×(C2×D3)⋊C4—21no—
52C5×(C2×D3)⋊C4—21no—
53C5×C4⋊Dic3D411yescharacteristic
54C5×C4⋊Dic3D411yescharacteristic
55C5×Dic3⋊C4—22no—
56C2×C5×C3⋊C8—21no—
57C2×C5×C3⋊C8—21no—
58#5726C2×C511yescharacteristic
59C3×#487C2×C511yescharacteristic, complemented
60#5084C2×C511yescharacteristic, complemented
61#5084C2×C511yescharacteristic, complemented
62#4547C2×C511yescharacteristic, complemented
63#4459C2×C511yescharacteristic, complemented
64#4459C2×C511yescharacteristic, complemented
65C2×C5×#41—61no—
66C5×C4⋊Q8C2×D311yescharacteristic
67C5×C4⋊Q8C2×D311yescharacteristic
68C5×#123—62no—
69C5×C22⋊Q8—61no—
70C5×#121—61no—
71C4×C5×D4—31no—
72C4×C5×D4—31no—
73C2×C5×C4⋊C4—61no—
74C5×#106—61no—
75C5×#106—61no—
76C5×C4⋊C8—31no—
77C5×C4⋊C8—31no—
78C5×#103—22no—
79C5×#102—61no—
80C5×#102—61no—
81C4×C5×C8C2×D311yescharacteristic
82#1635—31noSylow
83C22×C5×D3—21no—
84C22×C5×D3—21no—
85C3×C5×Q8—42no—
86C3×C5×Q8—41no—
87C3×C5×Q8—41no—
88C2×C3×C4×C5—22no—
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×C5×Dic3—21no—
94C2×C5×Dic3—21no—
95C5×D12—21no—
96C5×D12—21no—
97C5×D12—21no—
98C5×D12—44no—
99C5×D12SD1612yes—
100C4×C5×D3—44no—
101C4×C5×D3—41no—
102C4×C5×D3—41no—
103C4×C5×D3—41no—
104C4×C5×D3—41no—
105C3×C5×C8—41no—
106C5×C3⋊C8—41no—
107C5×C3⋊C8—41no—
108C2×#239—21no—
109C3×C4⋊Q8C22×C511yescharacteristic
110C3×C4⋊Q8C22×C511yescharacteristic
111#946—21no—
112#895—21no—
113#894—22no—
114D3×C4⋊C4—21no—
115C4×D12C22×C511yescharacteristic
116C4×D12C22×C511yescharacteristic
117C3×#103—22no—
118C3×C4×C8C22×C511yescharacteristic
119#809—21no—
120#809—21no—
121#808—21no—
122#808—21no—
123C4⋊(C3⋊C8)C22×C511yescharacteristic
124C4⋊(C3⋊C8)C22×C511yescharacteristic
125C5×#41—122no—
126C5×#41—122no—
127C2×C5×Q8—21no—
128C2×C5×Q8—21no—
129C2×C5×D4—31no—
130C2×C5×D4—31no—
131C2×C5×D4—62no—
132C22×C4×C5—62no—
133C22×C4×C5—61no—
134C22×C4×C5—61no—
135C2×C5×C8—21no—
136C2×C5×C8—61no—
137C2×C5×C8—61no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—21no—
140C5×C4⋊C4—21no—
141C5×C4⋊C4—31no—
142C5×C4⋊C4—31no—
143C5×C4⋊C4—62no—
144C5×C4⋊C4#6712yes—
145C5×C22⋊C4—62no—
146C5×C22⋊C4—61no—
147C5×C22⋊C4—61no—
148C42×C5C22×D311yescharacteristic
149#547—31no—
150#487C5×D311yescharacteristic, complemented
151#476—31no—
152#476—31no—
153#332—31no—
154D4⋊C8—31no—
155D4⋊C8—31no—
156C22×C3×C5#12011yescharacteristic
157C2×C5×D3—22no—
158C2×C5×D3—41no—
159C2×C5×D3—41no—
160C2×C5×D3—41no—
161C2×C5×D3—41no—
162C2×C5×D3—41no—
163C3×C4×C5—21no—
164C3×C4×C5—42no—
165C3×C4×C5—81no—
166C3×C4×C5C2×SD1611yescharacteristic
167C3×C4×C5#13511yescharacteristic
168C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
169C3×C4×C5#13711yescharacteristic
170C5×Dic3—41no—
171C5×Dic3—41no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174#239—42no—
175#239—42no—
176C2×D12—22no—
177C2×D12C5×D411yescharacteristic
178C2×D12C5×D411yescharacteristic
179C2×C4×D3—22no—
180C2×C4×D3—21no—
181C2×C4×D3—21no—
182C2×C3×C8—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4C5×D412yes—
187C3×C42C23×C511yescharacteristic
188(C2×D3)⋊C4—22no—
189(C2×D3)⋊C4—21no—
190(C2×D3)⋊C4—21no—
191C4⋊Dic3C5×D411yescharacteristic
192C4⋊Dic3C5×D411yescharacteristic
193Dic3⋊C4—22no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C23×C5—61no—
197C23×C5—61no—
198C5×Q8—42no—
199C5×Q8—41no—
200C5×Q8—41no—
201C5×D4—32no—
202C5×D4—61no—
203C5×D4—61no—
204C5×D4—61no—
205C5×D4—124no—
206C2×C4×C5—22no—
207C2×C4×C5—41no—
208C2×C4×C5—61no—
209C2×C4×C5—61no—
210C2×C4×C5—124no—
211C2×C4×C5—121no—
212C2×C4×C5—121no—
213C2×C4×C5—121no—
214C2×C4×C5—121no—
215C2×C4×C5D3×D411yescharacteristic
216C2×C4×C5D3×D411yescharacteristic
217C2×C4×C5C2×#6711yescharacteristic
218C5×C8—41no—
219C5×C8—121no—
220C5×C8—121no—
221C2×#41—61no—
222C4⋊Q8C2×C5×D311yescharacteristic
223C4⋊Q8C2×C5×D311yescharacteristic
224#123—62no—
225C22⋊Q8—61no—
226#121—61no—
227C4×D4—31no—
228C4×D4—31no—
229C2×C4⋊C4—61no—
230#106—61no—
231#106—61no—
232C4⋊C8—31no—
233C4⋊C8—31no—
234#103—22no—
235#102—61no—
236#102—61no—
237C4×C8C2×C5×D311yescharacteristic
238C2×C3×C5#45011yescharacteristic
239C2×C3×C5#45211yescharacteristic
240C2×C3×C5#45411yescharacteristic
241C5×D3—42no—
242C5×D3—81no—
243C22×D3—21no—
244C22×D3—21no—
245C3×Q8—42no—
246C3×Q8—41no—
247C3×Q8—41no—
248C2×C3×C4—22no—
249C2×C3×C4—41no—
250C2×C3×C4C2×C5×D411yescharacteristic
251C2×C3×C4C2×C5×D411yescharacteristic
252C2×C3×C4C2×C5×D411yescharacteristic
253C2×Dic3—21no—
254C2×Dic3—21no—
255D12—21no—
256D12—21no—
257D12—21no—
258D12—44no—
259D12C5×SD1612yes—
260C4×D3—44no—
261C4×D3—41no—
262C4×D3—41no—
263C4×D3—41no—
264C4×D3—41no—
265C3×C8—41no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C22×C5—62no—
269C22×C5—121no—
270C22×C5—121no—
271C22×C5—121no—
272C22×C5—121no—
273C22×C5—121no—
274C22×C5#93711yescharacteristic
275C4×C5—21no—
276C4×C5—42no—
277C4×C5—81no—
278C4×C5—121no—
279C4×C5—121no—
280C4×C5D3×SD1611yescharacteristic
281C4×C5#91411yescharacteristic
282C4×C5#91811yescharacteristic
283C4×C5#91911yescharacteristic
284#41—122no—
285#41—122no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×D4—31no—
289C2×D4—31no—
290C2×D4—62no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4—61no—
294C2×C8—21no—
295C2×C8—61no—
296C2×C8—61no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4—62no—
303C4⋊C4C5×#6712yes—
304C22⋊C4—62no—
305C22⋊C4—61no—
306C22⋊C4—61no—
307C42C22×C5×D311yescharacteristic
308C3×C5#163511yescharacteristic, complemented, Hall
309C22×C3C5×#12011yescharacteristic
310C2×D3—22no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C3×C4—21no—
317C3×C4—42no—
318C3×C4—81no—
319C3×C4C2×C5×SD1611yescharacteristic
320C3×C4C5×#13511yescharacteristic
321C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
322C3×C4C5×#13711yescharacteristic
323Dic3—41no—
324Dic3—41no—
325C2×C5—122no—
326C2×C5—241no—
327C2×C5#506211yescharacteristic
328C2×C5#516911yescharacteristic
329C2×C5#518711yescharacteristic
330C23—61no—
331C23—61no—
332Q8—42no—
333Q8—41no—
334Q8—41no—
335D4—32no—
336D4—61no—
337D4—61no—
338D4—61no—
339D4—124no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—124no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4C5×D3×D411yescharacteristic
350C2×C4C5×D3×D411yescharacteristic
351C2×C4C2×C5×#6711yescharacteristic
352C8—41no—
353C8—121no—
354C8—121no—
355C2×C3C5×#45011yescharacteristic
356C2×C3C5×#45211yescharacteristic
357C2×C3C5×#45411yescharacteristic
358D3—42no—
359D3—81no—
360C5#7122811yescharacteristic, direct product factor, Sylow
361C22—62no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22C5×#93711yescharacteristic
368C4—21no—
369C4—42no—
370C4—81no—
371C4—121no—
372C4—121no—
373C4C5×D3×SD1611yescharacteristic
374C4C5×#91411yescharacteristic
375C4C5×#91811yescharacteristic
376C4C5×#91911yescharacteristic
377C3C5×#163511yescharacteristic, complemented, Sylow
378C2—122no—
379C2—241no—
380C2C5×#506211yescharacteristic
381C2C5×#516911yescharacteristic
382C2C5×#518711yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×#487C2
socleC22×C3×C5#120
prime radicalC5×#5726C2
hypercenterC5×#487D3
hypocenterC3C5×#1635
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×#487

Complements

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

NH ≅ G/Ntype
2. C3×C5×#487C2semidirect
3. C5×#5084C2semidirect
4. C5×#5084C2semidirect
5. C5×#4547C2semidirect
6. C5×#4459C2semidirect
7. C5×#4459C2semidirect
26. #71228C5direct (N × H)
28. C5×#487D3semidirect
59. C3×#487C2×C5semidirect
60. #5084C2×C5semidirect
61. #5084C2×C5semidirect
62. #4547C2×C5semidirect
63. #4459C2×C5semidirect
64. #4459C2×C5semidirect
150. #487C5×D3semidirect
308. C3×C5#1635semidirect
360. C5#71228direct (N × H)
377. C3C5×#1635semidirect

Maximal subgroups

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

subgroupindex
C5×#57262
C3×C5×#4872
C5×#50842
C5×#45472
C5×#44592
C5×#16353
#712285