Finite Group Databasebeta

← C5×#71126 #14290570

Subgroups — C5×#71126 #14290570

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5584C211yesmaximal, characteristic
2C3×C5×#475C211yesmaximal, characteristic, complemented
3C5×#5083C211yesmaximal, characteristic
4C5×#5078C211yesmaximal, characteristic, complemented
5C5×#4535C211yesmaximal, characteristic, complemented
6C5×#4486C211yesmaximal, characteristic, complemented
7C5×#4483C211yesmaximal, characteristic
8C5×#1650—31nomaximal, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C4×C5×Q8C2211yescharacteristic
11C5×#946—21no—
12C2×C5×#215—21no—
13C5×#898—22no—
14C5×#892—21no—
15C5×#875—21no—
16C4×C5×D12C2211yescharacteristic
17C5×#869C2211yescharacteristic
18C4×C5×Dic6—21no—
19C3×C5×#106—21no—
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C5×#103—21no—
22C5×#835—21no—
23C5×#807—22no—
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C5×C3⋊C8C2211yescharacteristic
26#71126C511yesmaximal, characteristic, direct product factor, Hall
27C5×#556—31no—
28C5×#500—31no—
29C5×#475D311yescharacteristic, complemented
30C5×#461—31no—
31C5×#332—31no—
32C5×Q8⋊C8—31no—
33C5×D4⋊C8—31no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C5×D12D411yescharacteristic
37C2×C4×C5×D3—21no—
38C2×C5×Dic6—21no—
39C2×C3×C5×C8—21no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4D411yescharacteristic
44C3×C42×C5—21no—
45C3×C42×C5C2311yescharacteristic
46C5×#215—42no—
47C5×(C2×D3)⋊C4—22no—
48C5×(C2×D3)⋊C4—22no—
49C5×(C2×D3)⋊C4—21no—
50C5×C4⋊Dic3—21no—
51C5×C4⋊Dic3D411yescharacteristic
52C5×C4⋊Dic3D412yes—
53C5×Dic3⋊C4—22no—
54C4×C5×Dic3—22no—
55C2×C5×C3⋊C8—21no—
56C2×C5×C3⋊C8—21no—
57#5584C2×C511yescharacteristic
58C3×#475C2×C511yescharacteristic, complemented
59#5083C2×C511yescharacteristic
60#5078C2×C511yescharacteristic, complemented
61#4535C2×C511yescharacteristic, complemented
62#4486C2×C511yescharacteristic, complemented
63#4483C2×C511yescharacteristic
64C2×C5×SD16—61no—
65C5×C4⋊Q8—31no—
66C5×C4⋊Q8C2×D311yescharacteristic
67C5×#126—62no—
68C5×#124—61no—
69C5×C22⋊Q8—61no—
70C4×C5×Q8—61no—
71C4×C5×Q8C2×D311yescharacteristic
72C4×C5×D4—31no—
73C5×#117—61no—
74C5×#107—62no—
75C5×#106—21no—
76C5×C4⋊C8—31no—
77C5×C4⋊C8C2×D311yescharacteristic
78C5×#103—21no—
79C5×#103—61no—
80C4×C5×C8—31no—
81#1650—31noSylow
82C22×C5×D3—21no—
83C3×C5×Q8—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—41no—
86C3×C5×Q8—41no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—21no—
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—22no—
95C5×D12—21no—
96C5×D12—21no—
97C4×C5×D3—41no—
98C4×C5×D3—41no—
99C5×Dic6—41no—
100C5×Dic6—41no—
101C3×C5×C8—41no—
102C5×C3⋊C8—22no—
103C5×C3⋊C8—41no—
104C3×C4⋊Q8C22×C511yescharacteristic
105C3×C4×Q8C22×C511yescharacteristic
106#946—21no—
107C2×#215—21no—
108#898—22no—
109#892—21no—
110#875—21no—
111C4×D12C22×C511yescharacteristic
112#869C22×C511yescharacteristic
113C4×Dic6—21no—
114C3×#106—21no—
115C3×C4⋊C8C22×C511yescharacteristic
116C3×#103—21no—
117#835—21no—
118#807—22no—
119C4⋊(C3⋊C8)C22×C511yescharacteristic
120C4×C3⋊C8C22×C511yescharacteristic
121C2×C5×Q8—21no—
122C2×C5×Q8—61no—
123C2×C5×Q8#6711yescharacteristic
124C2×C5×D4—31no—
125C22×C4×C5—61no—
126C5×SD16—122no—
127C2×C5×C8—21no—
128C2×C5×C8—61no—
129C2×C5×C8—61no—
130C5×C4⋊C4—21no—
131C5×C4⋊C4—21no—
132C5×C4⋊C4—21no—
133C5×C4⋊C4—31no—
134C5×C4⋊C4—32no—
135C5×C4⋊C4—62no—
136C5×C4⋊C4—61no—
137C5×C4⋊C4#6711yescharacteristic
138C5×C22⋊C4—62no—
139C5×C22⋊C4—62no—
140C5×C22⋊C4—61no—
141C42×C5—21no—
142C42×C5—62no—
143C42×C5C22×D311yescharacteristic
144#556—31no—
145#500—31no—
146#475C5×D311yescharacteristic, complemented
147#461—31no—
148#332—31no—
149Q8⋊C8—31no—
150D4⋊C8—31no—
151C22×C3×C5#12011yescharacteristic
152C2×C5×D3—41no—
153C2×C5×D3—41no—
154C2×C5×D3—41no—
155C3×C4×C5—21no—
156C3×C4×C5—41no—
157C3×C4×C5—41no—
158C3×C4×C5—81no—
159C3×C4×C5#13511yescharacteristic
160C3×C4×C5#13511yescharacteristic
161C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
162C3×C4×C5#13711yescharacteristic
163C5×Dic3—42no—
164C5×Dic3—41no—
165C2×C3×Q8—21no—
166C2×C3×Q8C5×D411yescharacteristic
167C2×D12C5×D411yescharacteristic
168C2×C4×D3—21no—
169C2×Dic6—21no—
170C2×C3×C8—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4C5×D411yescharacteristic
175C3×C42—21no—
176C3×C42C23×C511yescharacteristic
177#215—42no—
178(C2×D3)⋊C4—22no—
179(C2×D3)⋊C4—22no—
180(C2×D3)⋊C4—21no—
181C4⋊Dic3—21no—
182C4⋊Dic3C5×D412yes—
183C4⋊Dic3C5×D411yescharacteristic
184Dic3⋊C4—22no—
185C4×Dic3—22no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188C23×C5—61no—
189C5×Q8—21no—
190C5×Q8—21no—
191C5×Q8—41no—
192C5×Q8—41no—
193C5×Q8—121no—
194C5×Q8—121no—
195C5×D4—61no—
196C5×D4—61no—
197C2×C4×C5—21no—
198C2×C4×C5—21no—
199C2×C4×C5—41no—
200C2×C4×C5—61no—
201C2×C4×C5—62no—
202C2×C4×C5—121no—
203C2×C4×C5—121no—
204C2×C4×C5D3×D411yescharacteristic
205C2×C4×C5D3×D411yescharacteristic
206C2×C4×C5C2×#6711yescharacteristic
207C5×C8—41no—
208C5×C8—62no—
209C5×C8—121no—
210C2×SD16—61no—
211C4⋊Q8—31no—
212C4⋊Q8C2×C5×D311yescharacteristic
213#126—62no—
214#124—61no—
215C22⋊Q8—61no—
216C4×Q8—61no—
217C4×Q8C2×C5×D311yescharacteristic
218C4×D4—31no—
219#117—61no—
220#107—62no—
221#106—21no—
222C4⋊C8—31no—
223C4⋊C8C2×C5×D311yescharacteristic
224#103—21no—
225#103—61no—
226C4×C8—31no—
227C2×C3×C5#44911yescharacteristic
228C2×C3×C5#45211yescharacteristic
229C2×C3×C5#45611yescharacteristic
230C5×D3—81no—
231C22×D3—21no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—41no—
235C3×Q8—41no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4C2×C5×D411yescharacteristic
240C2×C3×C4C2×C5×D411yescharacteristic
241C2×C3×C4C2×C5×D411yescharacteristic
242C2×Dic3—21no—
243C2×Dic3—22no—
244D12—21no—
245D12—21no—
246C4×D3—41no—
247C4×D3—41no—
248Dic6—41no—
249Dic6—41no—
250C3×C8—41no—
251C3⋊C8—22no—
252C3⋊C8—41no—
253C22×C5—121no—
254C22×C5—121no—
255C22×C5—121no—
256C22×C5#93711yescharacteristic
257C4×C5—21no—
258C4×C5—41no—
259C4×C5—41no—
260C4×C5—81no—
261C4×C5—122no—
262C4×C5—121no—
263C4×C5#91411yescharacteristic
264C4×C5#91611yescharacteristic
265C4×C5#94211yescharacteristic
266C4×C5#95011yescharacteristic
267C2×Q8—21no—
268C2×Q8—61no—
269C2×Q8C5×#6711yescharacteristic
270C2×D4—31no—
271C22×C4—61no—
272SD16—122no—
273C2×C8—21no—
274C2×C8—61no—
275C2×C8—61no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—31no—
280C4⋊C4—32no—
281C4⋊C4—62no—
282C4⋊C4—61no—
283C4⋊C4C5×#6711yescharacteristic
284C22⋊C4—62no—
285C22⋊C4—62no—
286C22⋊C4—61no—
287C42—21no—
288C42—62no—
289C42C22×C5×D311yescharacteristic
290C3×C5#165011yescharacteristic, complemented, Hall
291C22×C3C5×#12011yescharacteristic
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C3×C4—21no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—81no—
299C3×C4C5×#13511yescharacteristic
300C3×C4C5×#13511yescharacteristic
301C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
302C3×C4C5×#13711yescharacteristic
303Dic3—42no—
304Dic3—41no—
305C2×C5—241no—
306C2×C5#504711yescharacteristic
307C2×C5#517211yescharacteristic
308C2×C5#520511yescharacteristic
309C23—61no—
310Q8—21no—
311Q8—21no—
312Q8—41no—
313Q8—41no—
314Q8—121no—
315Q8—121no—
316D4—61no—
317D4—61no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—61no—
322C2×C4—62no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4C5×D3×D411yescharacteristic
326C2×C4C5×D3×D411yescharacteristic
327C2×C4C2×C5×#6711yescharacteristic
328C8—41no—
329C8—62no—
330C8—121no—
331C2×C3C5×#44911yescharacteristic
332C2×C3C5×#45211yescharacteristic
333C2×C3C5×#45611yescharacteristic
334D3—81no—
335C5#7112611yescharacteristic, direct product factor, Sylow
336C22—121no—
337C22—121no—
338C22—121no—
339C22C5×#93711yescharacteristic
340C4—21no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—122no—
345C4—121no—
346C4C5×#91411yescharacteristic
347C4C5×#91611yescharacteristic
348C4C5×#94211yescharacteristic
349C4C5×#95011yescharacteristic
350C3C5×#165011yescharacteristic, complemented, Sylow
351C2—241no—
352C2C5×#504711yescharacteristic
353C2C5×#517211yescharacteristic
354C2C5×#520511yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×#475C2
socleC22×C3×C5#120
prime radicalC2×C5×D12D4
hypercenterC5×#475D3
hypocenterC3C5×#1650
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×#475

Complements

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

NH ≅ G/Ntype
2. C3×C5×#475C2semidirect
4. C5×#5078C2semidirect
5. C5×#4535C2semidirect
6. C5×#4486C2semidirect
26. #71126C5direct (N × H)
29. C5×#475D3semidirect
58. C3×#475C2×C5semidirect
60. #5078C2×C5semidirect
61. #4535C2×C5semidirect
62. #4486C2×C5semidirect
146. #475C5×D3semidirect
290. C3×C5#1650semidirect
335. C5#71126direct (N × H)
350. C3C5×#1650semidirect

Maximal subgroups

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

subgroupindex
C5×#55842
C3×C5×#4752
C5×#50832
C5×#50782
C5×#45352
C5×#44862
C5×#44832
C5×#16503
#711265