Finite Group Databasebeta

← C5×#71031 #14288845

Subgroups — C5×#71031 #14288845

Order 1920.

480 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
1C5×#5585C211yesmaximal, characteristic, complemented
2C3×C5×#460C211yesmaximal, characteristic, complemented
3C5×#5075C211yesmaximal, characteristic, complemented
4C5×#5073C211yesmaximal, characteristic
5C5×#4535C211yesmaximal, characteristic, complemented
6C5×#4485C211yesmaximal, characteristic, complemented
7C5×#4484C211yesmaximal, characteristic, complemented
8C5×#1620—31nomaximal, Hall
9C2×C3×C5×SD16—21no—
10C3×C5×#127C2211yescharacteristic
11C3×C4×C5×Q8C2211yescharacteristic
12C5×#947—21no—
13C2×C5×#214—21no—
14C2×C5×#213—22no—
15C5×#894—22no—
16C5×#893—21no—
17C5×#875—21no—
18C5×#874C2211yescharacteristic
19C4×C5×D12—21no—
20C4×C5×Dic6C2211yescharacteristic
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C5×#102—21no—
23C5×#832—21no—
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C5×C3⋊C8C2211yescharacteristic
26#71031C511yesmaximal, characteristic, direct product factor, Hall
27C5×#559—31no—
28C5×#493—31no—
29C5×#460—31no—
30C5×#460D311yescharacteristic, complemented
31C5×#331—31no—
32C5×Q8⋊C8—31no—
33C5×Q8⋊C8—31no—
34C2×C3×C5×Q8D411yescharacteristic
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C5×D12—21no—
39C2×C5×D12—21no—
40C2×C5×D12D412yes—
41C2×C4×C5×D3—22no—
42C2×C5×Dic6D411yescharacteristic
43C3×C5×SD16—42no—
44C2×C3×C5×C8—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4D411yescharacteristic
47C3×C42×C5—21no—
48C3×C42×C5C2311yescharacteristic
49C5×#214—42no—
50C5×#213—44no—
51C5×(C2×D3)⋊C4—22no—
52C5×(C2×D3)⋊C4—22no—
53C5×(C2×D3)⋊C4—22no—
54C5×C4⋊Dic3D411yescharacteristic
55C5×Dic3⋊C4—21no—
56C4×C5×Dic3—21no—
57C2×C5×C3⋊C8—21no—
58C2×C5×C3⋊C8—21no—
59#5585C2×C511yescharacteristic, complemented
60C3×#460C2×C511yescharacteristic, complemented
61#5075C2×C511yescharacteristic, complemented
62#5073C2×C511yescharacteristic
63#4535C2×C511yescharacteristic, complemented
64#4485C2×C511yescharacteristic, complemented
65#4484C2×C511yescharacteristic, complemented
66C2×C5×SD16—21no—
67C2×C5×SD16—61no—
68C2×C5×D8—62no—
69C5×#127—31no—
70C5×#127C2×D311yescharacteristic
71C5×#124—61no—
72C5×#124—61no—
73C5×#123—62no—
74C4×C5×Q8—31no—
75C4×C5×Q8C2×D311yescharacteristic
76C4×C5×D4—61no—
77C4×C5×D4—61no—
78C5×C4⋊C8—31no—
79C5×C4⋊C8C2×D311yescharacteristic
80C5×#102—21no—
81C5×#102—61no—
82C4×C5×C8—31no—
83#1620—31noSylow
84C23×C3×C5—41no—
85C22×C5×D3—22no—
86C3×C5×Q8—21no—
87C3×C5×Q8—21no—
88C3×C5×D4—42no—
89C3×C5×D4—41no—
90C3×C5×D4—41no—
91C3×C5×D4—41no—
92C3×C5×D4—41no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5C2×D411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C5×Dic3—21no—
99C2×C5×Dic3—21no—
100C5×D12—42no—
101C5×D12—41no—
102C5×D12—41no—
103C5×D12—41no—
104C5×D12—41no—
105C4×C5×D3—82no—
106C5×Dic6—21no—
107C5×Dic6—21no—
108C3×C5×C8—41no—
109C5×C3⋊C8—22no—
110C5×C3⋊C8—41no—
111C2×C3×SD16—21no—
112C3×#127C22×C511yescharacteristic
113C3×C4×Q8C22×C511yescharacteristic
114#947—21no—
115C2×#214—21no—
116C2×#213—22no—
117#894—22no—
118#893—21no—
119#875—21no—
120#874C22×C511yescharacteristic
121C4×D12—21no—
122C4×Dic6C22×C511yescharacteristic
123C3×C4⋊C8C22×C511yescharacteristic
124C3×#102—21no—
125#832—21no—
126C4⋊(C3⋊C8)C22×C511yescharacteristic
127C4×C3⋊C8C22×C511yescharacteristic
128C2×C5×Q8—31no—
129C2×C5×Q8#6711yescharacteristic
130C2×C5×D4—21no—
131C2×C5×D4—21no—
132C2×C5×D4—21no—
133C2×C5×D4—32no—
134C2×C5×D4—61no—
135C2×C5×D4—61no—
136C22×C4×C5—62no—
137C5×SD16—42no—
138C5×SD16—122no—
139C5×D8—124no—
140C2×C5×C8—21no—
141C2×C5×C8—61no—
142C2×C5×C8—61no—
143C5×C4⋊C4—21no—
144C5×C4⋊C4—31no—
145C5×C4⋊C4—61no—
146C5×C4⋊C4#6711yescharacteristic
147C5×C22⋊C4—62no—
148C5×C22⋊C4—62no—
149C5×C22⋊C4—62no—
150C42×C5—21no—
151C42×C5—61no—
152C42×C5C22×D311yescharacteristic
153#559—31no—
154#493—31no—
155#460—31no—
156#460C5×D311yescharacteristic, complemented
157#331—31no—
158Q8⋊C8—31no—
159Q8⋊C8—31no—
160C22×C3×C5—81no—
161C22×C3×C5—81no—
162C22×C3×C5—81no—
163C22×C3×C5#12011yescharacteristic
164C2×C5×D3—42no—
165C2×C5×D3—42no—
166C2×C5×D3—42no—
167C3×C4×C5—21no—
168C3×C4×C5—41no—
169C3×C4×C5—41no—
170C3×C4×C5#13511yescharacteristic
171C3×C4×C5#13511yescharacteristic
172C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
173C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
174C5×Dic3—41no—
175C5×Dic3—41no—
176C2×C3×Q8C5×D411yescharacteristic
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12C5×D412yes—
183C2×C4×D3—22no—
184C2×Dic6C5×D411yescharacteristic
185C3×SD16—42no—
186C2×C3×C8—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4C5×D411yescharacteristic
189C3×C42—21no—
190C3×C42C23×C511yescharacteristic
191#214—42no—
192#213—44no—
193(C2×D3)⋊C4—22no—
194(C2×D3)⋊C4—22no—
195(C2×D3)⋊C4—22no—
196C4⋊Dic3C5×D411yescharacteristic
197Dic3⋊C4—21no—
198C4×Dic3—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C23×C5—41no—
202C23×C5—62no—
203C5×Q8—21no—
204C5×Q8—21no—
205C5×Q8—61no—
206C5×Q8—61no—
207C5×D4—42no—
208C5×D4—41no—
209C5×D4—41no—
210C5×D4—41no—
211C5×D4—41no—
212C5×D4—122no—
213C5×D4—121no—
214C5×D4—121no—
215C5×D4—121no—
216C5×D4—121no—
217C2×C4×C5—21no—
218C2×C4×C5—21no—
219C2×C4×C5—61no—
220C2×C4×C5—61no—
221C2×C4×C5—242no—
222C2×C4×C5D3×D411yescharacteristic
223C2×C4×C5D3×D411yescharacteristic
224C2×C4×C5C2×#6711yescharacteristic
225C5×C8—41no—
226C5×C8—62no—
227C5×C8—121no—
228C2×SD16—21no—
229C2×SD16—61no—
230C2×D8—62no—
231#127—31no—
232#127C2×C5×D311yescharacteristic
233#124—61no—
234#124—61no—
235#123—62no—
236C4×Q8—31no—
237C4×Q8C2×C5×D311yescharacteristic
238C4×D4—61no—
239C4×D4—61no—
240C4⋊C8—31no—
241C4⋊C8C2×C5×D311yescharacteristic
242#102—21no—
243#102—61no—
244C4×C8—31no—
245C2×C3×C5—161no—
246C2×C3×C5#44911yescharacteristic
247C2×C3×C5#44911yescharacteristic
248C2×C3×C5#45311yescharacteristic
249C5×D3—82no—
250C23×C3—41no—
251C22×D3—22no—
252C3×Q8—21no—
253C3×Q8—21no—
254C3×D4—42no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—41no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4C2×C5×D411yescharacteristic
262C2×C3×C4C2×C5×D411yescharacteristic
263C2×C3×C4C2×C5×D411yescharacteristic
264C2×Dic3—21no—
265C2×Dic3—21no—
266D12—42no—
267D12—41no—
268D12—41no—
269D12—41no—
270D12—41no—
271C4×D3—82no—
272Dic6—21no—
273Dic6—21no—
274C3×C8—41no—
275C3⋊C8—22no—
276C3⋊C8—41no—
277C22×C5—81no—
278C22×C5—81no—
279C22×C5—81no—
280C22×C5—122no—
281C22×C5—122no—
282C22×C5—122no—
283C22×C5#93711yescharacteristic
284C4×C5—21no—
285C4×C5—41no—
286C4×C5—41no—
287C4×C5—121no—
288C4×C5—121no—
289C4×C5#91411yescharacteristic
290C4×C5#91611yescharacteristic
291C4×C5#93211yescharacteristic
292C4×C5#95011yescharacteristic
293C2×Q8—31no—
294C2×Q8C5×#6711yescharacteristic
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—32no—
299C2×D4—61no—
300C2×D4—61no—
301C22×C4—62no—
302SD16—42no—
303SD16—122no—
304D8—124no—
305C2×C8—21no—
306C2×C8—61no—
307C2×C8—61no—
308C4⋊C4—21no—
309C4⋊C4—31no—
310C4⋊C4—61no—
311C4⋊C4C5×#6711yescharacteristic
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C42—21no—
316C42—61no—
317C42C22×C5×D311yescharacteristic
318C3×C5#162011yescharacteristic, complemented, Hall
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3C5×#12011yescharacteristic
323C2×D3—42no—
324C2×D3—42no—
325C2×D3—42no—
326C3×C4—21no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4C5×#13511yescharacteristic
330C3×C4C5×#13511yescharacteristic
331C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
332C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
333Dic3—41no—
334Dic3—41no—
335C2×C5—161no—
336C2×C5—242no—
337C2×C5#503711yescharacteristic
338C2×C5#517211yescharacteristic
339C2×C5#519811yescharacteristic
340C23—41no—
341C23—62no—
342Q8—21no—
343Q8—21no—
344Q8—61no—
345Q8—61no—
346D4—42no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—41no—
351D4—122no—
352D4—121no—
353D4—121no—
354D4—121no—
355D4—121no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—242no—
361C2×C4C5×D3×D411yescharacteristic
362C2×C4C5×D3×D411yescharacteristic
363C2×C4C2×C5×#6711yescharacteristic
364C8—41no—
365C8—62no—
366C8—121no—
367C2×C3—161no—
368C2×C3C5×#44911yescharacteristic
369C2×C3C5×#44911yescharacteristic
370C2×C3C5×#45311yescharacteristic
371D3—82no—
372C5#7103111yescharacteristic, direct product factor, Sylow
373C22—81no—
374C22—81no—
375C22—81no—
376C22—122no—
377C22—122no—
378C22—122no—
379C22C5×#93711yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—121no—
384C4—121no—
385C4C5×#91411yescharacteristic
386C4C5×#91611yescharacteristic
387C4C5×#93211yescharacteristic
388C4C5×#95011yescharacteristic
389C3C5×#162011yescharacteristic, complemented, Sylow
390C2—161no—
391C2—242no—
392C2C5×#503711yescharacteristic
393C2C5×#517211yescharacteristic
394C2C5×#519811yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×#460C2
socleC22×C3×C5#120
prime radicalC5×#5073C2
hypercenterC5×#460D3
hypocenterC3C5×#1620
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×#460

Complements

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

NH ≅ G/Ntype
1. C5×#5585C2semidirect
2. C3×C5×#460C2semidirect
3. C5×#5075C2semidirect
5. C5×#4535C2semidirect
6. C5×#4485C2semidirect
7. C5×#4484C2semidirect
26. #71031C5direct (N × H)
30. C5×#460D3semidirect
59. #5585C2×C5semidirect
60. C3×#460C2×C5semidirect
61. #5075C2×C5semidirect
63. #4535C2×C5semidirect
64. #4485C2×C5semidirect
65. #4484C2×C5semidirect
156. #460C5×D3semidirect
318. C3×C5#1620semidirect
372. C5#71031direct (N × H)
389. C3C5×#1620semidirect

Maximal subgroups

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

subgroupindex
C5×#55852
C3×C5×#4602
C5×#50752
C5×#50732
C5×#45352
C5×#44852
C5×#44842
C5×#16203
#710315