Finite Group Databasebeta

← C5×#71159 #14289379

Subgroups — C5×#71159 #14289379

Order 1920.

436 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5585C211yesmaximal, characteristic
2C3×C5×#477C211yesmaximal, characteristic, complemented
3C5×#5091C211yesmaximal, characteristic, complemented
4C5×#5085C211yesmaximal, characteristic
5C5×#4535C211yesmaximal, characteristic, complemented
6C5×#4485C211yesmaximal, characteristic
7C5×#4484C211yesmaximal, characteristic, complemented
8C5×#1630—31nomaximal, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C4×C5×Q8C2211yescharacteristic
11C5×#947—21no—
12C2×C5×#215—22no—
13C5×#894—22no—
14C5×#893—21no—
15C5×#875—21no—
16C5×#874C2211yescharacteristic
17C4×C5×D12—21no—
18C4×C5×Dic6C2211yescharacteristic
19C3×C5×#107—21no—
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C5×#103—21no—
22C5×#810—21no—
23C5×#807—21no—
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C5×C3⋊C8C2211yescharacteristic
26#71159C511yesmaximal, characteristic, direct product factor, Hall
27C5×#559—31no—
28C5×#492—31no—
29C5×#477—31no—
30C5×#477D311yescharacteristic, complemented
31C5×#331—31no—
32C5×Q8⋊C8—31no—
33C5×Q8⋊C8—31no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C5×D12—21no—
37C2×C5×D12—21no—
38C2×C5×D12D412yes—
39C2×C4×C5×D3—22no—
40C2×C5×Dic6D411yescharacteristic
41C2×C3×C5×C8—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4D411yescharacteristic
46C3×C42×C5—21no—
47C3×C42×C5C2311yescharacteristic
48C5×#215—44no—
49C5×(C2×D3)⋊C4—22no—
50C5×(C2×D3)⋊C4—22no—
51C5×(C2×D3)⋊C4—22no—
52C5×C4⋊Dic3D411yescharacteristic
53C5×Dic3⋊C4—21no—
54C4×C5×Dic3—21no—
55C2×C5×C3⋊C8—21no—
56C2×C5×C3⋊C8—21no—
57#5585C2×C511yescharacteristic
58C3×#477C2×C511yescharacteristic, complemented
59#5091C2×C511yescharacteristic, complemented
60#5085C2×C511yescharacteristic
61#4535C2×C511yescharacteristic, complemented
62#4485C2×C511yescharacteristic
63#4484C2×C511yescharacteristic, complemented
64C2×C5×SD16—62no—
65C5×C4⋊Q8C2×D311yescharacteristic
66C5×#127—31no—
67C5×#124—61no—
68C5×#124—61no—
69C5×#123—62no—
70C4×C5×Q8—31no—
71C4×C5×Q8C2×D311yescharacteristic
72C4×C5×D4—61no—
73C4×C5×D4—61no—
74C5×#107—21no—
75C5×#107—61no—
76C5×C4⋊C8—31no—
77C5×C4⋊C8C2×D311yescharacteristic
78C5×#103—21no—
79C5×#103—61no—
80C4×C5×C8—31no—
81#1630—31noSylow
82C22×C5×D3—22no—
83C3×C5×Q8—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—42no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C5×Dic3—21no—
93C2×C5×Dic3—21no—
94C5×D12—42no—
95C5×D12—41no—
96C5×D12—41no—
97C5×D12—41no—
98C5×D12—41no—
99C4×C5×D3—82no—
100C5×Dic6—21no—
101C5×Dic6—21no—
102C3×C5×C8—41no—
103C5×C3⋊C8—22no—
104C5×C3⋊C8—41no—
105C3×C4⋊Q8C22×C511yescharacteristic
106C3×C4×Q8C22×C511yescharacteristic
107#947—21no—
108C2×#215—22no—
109#894—22no—
110#893—21no—
111#875—21no—
112#874C22×C511yescharacteristic
113C4×D12—21no—
114C4×Dic6C22×C511yescharacteristic
115C3×#107—21no—
116C3×C4⋊C8C22×C511yescharacteristic
117C3×#103—21no—
118#810—21no—
119#807—21no—
120C4⋊(C3⋊C8)C22×C511yescharacteristic
121C4×C3⋊C8C22×C511yescharacteristic
122C2×C5×Q8—21no—
123C2×C5×Q8—31no—
124C2×C5×Q8#6711yescharacteristic
125C2×C5×D4—32no—
126C2×C5×D4—61no—
127C2×C5×D4—61no—
128C22×C4×C5—62no—
129C5×SD16—124no—
130C2×C5×C8—21no—
131C2×C5×C8—61no—
132C2×C5×C8—61no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4—21no—
135C5×C4⋊C4—21no—
136C5×C4⋊C4—31no—
137C5×C4⋊C4—61no—
138C5×C4⋊C4#6711yescharacteristic
139C5×C22⋊C4—62no—
140C5×C22⋊C4—62no—
141C5×C22⋊C4—62no—
142C42×C5—21no—
143C42×C5—61no—
144C42×C5C22×D311yescharacteristic
145#559—31no—
146#492—31no—
147#477—31no—
148#477C5×D311yescharacteristic, complemented
149#331—31no—
150Q8⋊C8—31no—
151Q8⋊C8—31no—
152C22×C3×C5#12011yescharacteristic
153C2×C5×D3—42no—
154C2×C5×D3—42no—
155C2×C5×D3—42no—
156C3×C4×C5—21no—
157C3×C4×C5—41no—
158C3×C4×C5—41no—
159C3×C4×C5—81no—
160C3×C4×C5#13511yescharacteristic
161C3×C4×C5#13511yescharacteristic
162C3×C4×C5#13711yescharacteristic
163C3×C4×C5#13711yescharacteristic
164C5×Dic3—41no—
165C5×Dic3—41no—
166C2×C3×Q8—21no—
167C2×C3×Q8C5×D411yescharacteristic
168C2×D12—21no—
169C2×D12—21no—
170C2×D12C5×D412yes—
171C2×C4×D3—22no—
172C2×Dic6C5×D411yescharacteristic
173C2×C3×C8—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4C5×D411yescharacteristic
178C3×C42—21no—
179C3×C42C23×C511yescharacteristic
180#215—44no—
181(C2×D3)⋊C4—22no—
182(C2×D3)⋊C4—22no—
183(C2×D3)⋊C4—22no—
184C4⋊Dic3C5×D411yescharacteristic
185Dic3⋊C4—21no—
186C4×Dic3—21no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—21no—
189C23×C5—62no—
190C5×Q8—21no—
191C5×Q8—21no—
192C5×Q8—42no—
193C5×Q8—61no—
194C5×Q8—61no—
195C5×D4—122no—
196C5×D4—121no—
197C5×D4—121no—
198C5×D4—121no—
199C5×D4—121no—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—41no—
203C2×C4×C5—61no—
204C2×C4×C5—61no—
205C2×C4×C5—242no—
206C2×C4×C5D3×D411yescharacteristic
207C2×C4×C5D3×D411yescharacteristic
208C2×C4×C5C2×#6711yescharacteristic
209C5×C8—41no—
210C5×C8—62no—
211C5×C8—121no—
212C2×SD16—62no—
213C4⋊Q8C2×C5×D311yescharacteristic
214#127—31no—
215#124—61no—
216#124—61no—
217#123—62no—
218C4×Q8—31no—
219C4×Q8C2×C5×D311yescharacteristic
220C4×D4—61no—
221C4×D4—61no—
222#107—21no—
223#107—61no—
224C4⋊C8—31no—
225C4⋊C8C2×C5×D311yescharacteristic
226#103—21no—
227#103—61no—
228C4×C8—31no—
229C2×C3×C5#45211yescharacteristic
230C2×C3×C5#45211yescharacteristic
231C2×C3×C5#45311yescharacteristic
232C5×D3—82no—
233C22×D3—22no—
234C3×Q8—21no—
235C3×Q8—21no—
236C3×Q8—42no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4C2×C5×D411yescharacteristic
241C2×C3×C4C2×C5×D411yescharacteristic
242C2×C3×C4C2×C5×D411yescharacteristic
243C2×Dic3—21no—
244C2×Dic3—21no—
245D12—42no—
246D12—41no—
247D12—41no—
248D12—41no—
249D12—41no—
250C4×D3—82no—
251Dic6—21no—
252Dic6—21no—
253C3×C8—41no—
254C3⋊C8—22no—
255C3⋊C8—41no—
256C22×C5—122no—
257C22×C5—122no—
258C22×C5—122no—
259C22×C5#93711yescharacteristic
260C4×C5—21no—
261C4×C5—41no—
262C4×C5—41no—
263C4×C5—81no—
264C4×C5—121no—
265C4×C5—121no—
266C4×C5#91811yescharacteristic
267C4×C5#91911yescharacteristic
268C4×C5#94211yescharacteristic
269C4×C5#95011yescharacteristic
270C2×Q8—21no—
271C2×Q8—31no—
272C2×Q8C5×#6711yescharacteristic
273C2×D4—32no—
274C2×D4—61no—
275C2×D4—61no—
276C22×C4—62no—
277SD16—124no—
278C2×C8—21no—
279C2×C8—61no—
280C2×C8—61no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—31no—
285C4⋊C4—61no—
286C4⋊C4C5×#6711yescharacteristic
287C22⋊C4—62no—
288C22⋊C4—62no—
289C22⋊C4—62no—
290C42—21no—
291C42—61no—
292C42C22×C5×D311yescharacteristic
293C3×C5#163011yescharacteristic, complemented, Hall
294C22×C3C5×#12011yescharacteristic
295C2×D3—42no—
296C2×D3—42no—
297C2×D3—42no—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—81no—
302C3×C4C5×#13511yescharacteristic
303C3×C4C5×#13511yescharacteristic
304C3×C4C5×#13711yescharacteristic
305C3×C4C5×#13711yescharacteristic
306Dic3—41no—
307Dic3—41no—
308C2×C5—242no—
309C2×C5#504711yescharacteristic
310C2×C5#518711yescharacteristic
311C2×C5#519811yescharacteristic
312C23—62no—
313Q8—21no—
314Q8—21no—
315Q8—42no—
316Q8—61no—
317Q8—61no—
318D4—122no—
319D4—121no—
320D4—121no—
321D4—121no—
322D4—121no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—242no—
329C2×C4C5×D3×D411yescharacteristic
330C2×C4C5×D3×D411yescharacteristic
331C2×C4C2×C5×#6711yescharacteristic
332C8—41no—
333C8—62no—
334C8—121no—
335C2×C3C5×#45211yescharacteristic
336C2×C3C5×#45211yescharacteristic
337C2×C3C5×#45311yescharacteristic
338D3—82no—
339C5#7115911yescharacteristic, direct product factor, Sylow
340C22—122no—
341C22—122no—
342C22—122no—
343C22C5×#93711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—121no—
349C4—121no—
350C4C5×#91811yescharacteristic
351C4C5×#91911yescharacteristic
352C4C5×#94211yescharacteristic
353C4C5×#95011yescharacteristic
354C3C5×#163011yescharacteristic, complemented, Sylow
355C2—242no—
356C2C5×#504711yescharacteristic
357C2C5×#518711yescharacteristic
358C2C5×#519811yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×#477C2
socleC22×C3×C5#120
prime radicalC5×#874C22
hypercenterC5×#477D3
hypocenterC3C5×#1630
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×#477

Complements

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

NH ≅ G/Ntype
2. C3×C5×#477C2semidirect
3. C5×#5091C2semidirect
5. C5×#4535C2semidirect
7. C5×#4484C2semidirect
26. #71159C5direct (N × H)
30. C5×#477D3semidirect
58. C3×#477C2×C5semidirect
59. #5091C2×C5semidirect
61. #4535C2×C5semidirect
63. #4484C2×C5semidirect
148. #477C5×D3semidirect
293. C3×C5#1630semidirect
339. C5#71159direct (N × H)
354. C3C5×#1630semidirect

Maximal subgroups

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

subgroupindex
C5×#55852
C3×C5×#4772
C5×#50912
C5×#50852
C5×#45352
C5×#44852
C5×#44842
C5×#16303
#711595