Finite Group Databasebeta

← #12084514

Subgroups — #12084514

Order 1280.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3229C211yesmaximal, characteristic, complemented
2#10615081C211yesmaximal, characteristic
3C2×#65372C211yesmaximal, characteristic
4#10611017C211yesmaximal, characteristic
5#10602621C211yesmaximal, characteristic, complemented
6#10602566C211yesmaximal, characteristic, complemented
7#10602137C214yesmaximal, complemented
8#10602025C214yesmaximal, complemented
9#10601542C211yesmaximal, characteristic, complemented
10C5×#533C2211yescharacteristic
11C2×C5×C4⋊Q8C2211yescharacteristic
12C22×#688—21no—
13C2×C4×D20C2211yescharacteristic
14C5×#498C2212yescomplemented
15C5×#487C2212yescomplemented
16C5×#483—22no—
17C5×#475—22no—
18C5×#431C2211yescharacteristic
19C2×#3967—21no—
20#65376C2212yes—
21#65373C2212yes—
22#65372—22no—
23#65372C2214yes—
24#65355—22no—
25#65290—21no—
26#65284—21no—
27#65214C2211yescharacteristic
28#65213C2211yescharacteristic
29C2×C4⋊#168C2211yescharacteristic
30#65123—22no—
31D5×C22⋊C8—22no—
32C8×D20C2214yes—
33#64767C2212yescomplemented
34#64767C2214yescomplemented
35#64766C2212yescomplemented
36#64677C2214yescomplemented
37#19390—51nomaximal, Sylow
38C22×C5×Q8—21no—
39C22×D20D411yescharacteristic
40C22×C4×D5—21no—
41C5×C4⋊Q8—21no—
42C5×C4⋊Q8—21no—
43C5×C4⋊Q8—21no—
44C5×C4⋊Q8C2312yes—
45C5×C4⋊Q8C2311yescharacteristic
46C5×C4⋊Q8C2311yescharacteristic
47C5×#125—21no—
48C5×C22⋊Q8—21no—
49C5×C22⋊Q8—21no—
50C4×C5×Q8—21no—
51C4×C5×Q8—21no—
52C5×#117—21no—
53C2×C5×C4⋊C4—21no—
54C2×C5×C4⋊C4—21no—
55C2×C42×C5C2311yescharacteristic
56C2×#688—24no—
57C2×#688—44no—
58C2×#685—21no—
59C2×C4⋊Dic5D411yescharacteristic
60C2×#681—21no—
61C22×#168—21no—
62C2×C8×D5—24no—
63C4×D20—21no—
64C4×D20—21no—
65C4×D20D412yes—
66C4×D20C2312yes—
67C5×#106—24no—
68C5×C4⋊C8D412yes—
69C5×#103—24no—
70C5×C22⋊C8D412yes—
71C4×C5×C8C2312yes—
72#3967—24no—
73C22⋊#168—21no—
74#3952—24no—
75#3941—22no—
76#3941—22no—
77#3940—22no—
78#3939—22no—
79C4⋊#168—21no—
80C4⋊#168—21no—
81C4⋊#168C2312yes—
82C4⋊#168C2312yes—
83C4⋊#168C2312yes—
84C4×#168—21no—
85#3229D511yescharacteristic, complemented
86#3077—51no—
87C2×#461—51no—
88#2972—51no—
89#1679—54no—
90#1671—54no—
91#1570—51no—
92#1546—51no—
93#1542—51no—
94C23×D5—21no—
95C2×C5×Q8—21no—
96C2×C5×Q8—22no—
97C2×C5×Q8—21no—
98C2×C5×Q8—41no—
99C2×C5×Q8—42no—
100C2×C5×Q8—42no—
101C22×C4×C5—41no—
102C22×C4×C5C2×D411yescharacteristic
103C22×C4×C5C2×D411yescharacteristic
104C22×C4×C5#4111yescharacteristic
105C22×Dic5—21no—
106C2×D20—21no—
107C2×D20—22no—
108C2×D20SD1614yes—
109C2×D20C2×D412yes—
110C2×C4×D5—22no—
111C2×C4×D5—22no—
112C2×C4×D5—82no—
113C2×C5×C8C2×D414yes—
114C5×C4⋊C4—22no—
115C5×C4⋊C4—21no—
116C5×C4⋊C4—21no—
117C5×C4⋊C4—21no—
118C5×C4⋊C4—22no—
119C5×C4⋊C4—21no—
120C5×C4⋊C4—41no—
121C5×C4⋊C4—41no—
122C5×C4⋊C4—41no—
123C5×C4⋊C4—41no—
124C5×C22⋊C4—41no—
125C42×C5—41no—
126C42×C5C2×D411yescharacteristic
127C42×C5C2×D411yescharacteristic
128C42×C5#4111yescharacteristic
129C42×C5C2411yescharacteristic
130#688—48no—
131#685—22no—
132#685—22no—
133C4⋊Dic5—21no—
134C4⋊Dic5C2×D412yes—
135#681—42no—
136C2×#168—22no—
137C2×#168—22no—
138C2×#168—22no—
139C2×#168—42no—
140C8×D5—84no—
141C22×SD16—101no—
142#533C2×D511yescharacteristic
143C2×C4⋊Q8C2×D511yescharacteristic
144C2×C4×D4—51no—
145#498C2×D512yescomplemented
146#487C2×D512yescomplemented
147#483—22no—
148#478—102no—
149#476—52no—
150#475—22no—
151#474—52no—
152#461—54no—
153#461—102no—
154C8×D4—54no—
155C8×D4—102no—
156#431—51no—
157#431C2×D511yescharacteristic
158#428—101no—
159#423—51no—
160C2×C4⋊C8—51no—
161#418—101no—
162C2×#103—101no—
163C2×C22⋊C8—102no—
164#334—54no—
165#333—52no—
166#332—52no—
167#332—54no—
168C23×C5#12111yescharacteristic
169C22×D5—22no—
170C22×D5—24no—
171C22×D5—81no—
172C22×D5—81no—
173C5×Q8—42no—
174C5×Q8—42no—
175C5×Q8—42no—
176C5×Q8—81no—
177C5×Q8—82no—
178C2×C4×C5—22no—
179C2×C4×C5—41no—
180C2×C4×C5—41no—
181C2×C4×C5—41no—
182C2×C4×C5—42no—
183C2×C4×C5—82no—
184C2×C4×C5#12111yescharacteristic
185C2×C4×C5#12111yescharacteristic
186C2×C4×C5#12111yescharacteristic
187C2×C4×C5C2×SD1612yes—
188C2×C4×C5#13512yes—
189C2×C4×C5C22×D411yescharacteristic
190C2×C4×C5C22×D411yescharacteristic
191C2×C4×C5C2×#4111yescharacteristic
192C2×Dic5—22no—
193C2×Dic5—81no—
194D20—22no—
195D20—24no—
196D20C2×SD1614yes—
197C4×D5—44no—
198C4×D5—82no—
199C5×C8—24no—
200#168—42no—
201#168—42no—
202#168—42no—
203C22×Q8—21no—
204C22×D4—51no—
205C23×C4—101no—
206C2×SD16—104no—
207C2×SD16—204no—
208C2×M16—101no—
209C22×C8—101no—
210C22×C8—104no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8C22×D512yes—
215C4⋊Q8C22×D511yescharacteristic
216C4⋊Q8C22×D511yescharacteristic
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×D4—52no—
223C4×D4—52no—
224C4×D4—101no—
225C4×D4—101no—
226#117—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—51no—
230C2×C22⋊C4—101no—
231C2×C42C22×D511yescharacteristic
232#107—102no—
233#106—24no—
234#106—102no—
235C4⋊C8—52no—
236C4⋊C8—52no—
237C4⋊C8—52no—
238C4⋊C8—101no—
239C4⋊C8—101no—
240C4⋊C8C22⋊D512yes—
241#103—24no—
242#103—104no—
243#102—102no—
244#102—102no—
245C22⋊C8—101no—
246C22⋊C8—104no—
247C22⋊C8C22⋊D512yes—
248C4×C8—101no—
249C4×C8C22×D512yes—
250C22×C5—21no—
251C22×C5—21no—
252C22×C5#46512yes—
253C22×C5C2×#12111yescharacteristic
254C2×D5—24no—
255C2×D5—42no—
256C2×D5—42no—
257C2×D5—84no—
258C4×C5—21no—
259C4×C5—21no—
260C4×C5—21no—
261C4×C5—81no—
262C4×C5—82no—
263C4×C5—81no—
264C4×C5#46512yes—
265C4×C5C22×SD1611yescharacteristic
266C4×C5C2×#13511yescharacteristic
267C4×C5#57712yes—
268Dic5—42no—
269C24—101no—
270C2×Q8—22no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×D4—52no—
277C2×D4—54no—
278C2×D4—102no—
279C2×D4—101no—
280C22×C4—41no—
281C22×C4—101no—
282C22×C4—102no—
283C22×C4—102no—
284C22×C4—402no—
285C22×C4D4×D511yescharacteristic
286C22×C4#71111yescharacteristic
287C22×C4C2×C22⋊D511yescharacteristic
288SD16—208no—
289M16—202no—
290C2×C8—102no—
291C2×C8—102no—
292C2×C8—102no—
293C2×C8—202no—
294C2×C8—404no—
295C2×C8C2×C22⋊D514yes—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—52no—
307C4⋊C4—101no—
308C22⋊C4—41no—
309C22⋊C4—102no—
310C22⋊C4—102no—
311C42—41no—
312C42D4×D511yescharacteristic
313C42#71111yescharacteristic
314C42C2×C22⋊D511yescharacteristic
315C42C23×D511yescharacteristic
316C2×C5—22no—
317C2×C5#304211yescharacteristic
318C2×C5C2×#46511yescharacteristic
319C2×C5#306911yescharacteristic
320D5—44no—
321C23—102no—
322C23—104no—
323C23—401no—
324C23—401no—
325C23#408411yescharacteristic
326Q8—42no—
327Q8—42no—
328Q8—42no—
329Q8—81no—
330Q8—82no—
331D4—54no—
332D4—102no—
333D4—104no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—102no—
341C2×C4—204no—
342C2×C4—402no—
343C2×C4—401no—
344C2×C4D5×SD1612yes—
345C2×C4#406212yes—
346C2×C4#408411yescharacteristic
347C2×C4#408411yescharacteristic
348C2×C4#408411yescharacteristic
349C2×C4C2×D4×D511yescharacteristic
350C2×C4C2×#71111yescharacteristic
351C2×C4C22×C22⋊D511yescharacteristic
352C8—24no—
353C8—202no—
354C8—202no—
355C8—202no—
356C5#1939011yescharacteristic, complemented, Sylow
357C22—21no—
358C22—21no—
359C22—104no—
360C22—202no—
361C22—202no—
362C22—404no—
363C22#6546012yes—
364C22C2×#408411yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—202no—
372C4#6546012yes—
373C4C2×D5×SD1611yescharacteristic
374C4C2×#406211yescharacteristic
375C4#6617112yes—
376C2—22no—
377C2—204no—
378C2#1061513611yescharacteristic
379C2C2×#6546011yescharacteristic
380C2#1061572911yescharacteristic
subgroupG/H
centerC22C2×#4084
commutator, cosocleC42×C5C24
FrattiniC42C23×D5
FittingC5×#3229C2
socleC22×C5C2×#121
prime radicalC22×D20D4
hypercenter#3229D5
hypocenterC5#19390
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
1. C5×#3229C2semidirect
5. #10602621C2semidirect
6. #10602566C2semidirect
7. #10602137C2semidirect
8. #10602025C2semidirect
9. #10601542C2semidirect
14. C5×#498C22semidirect
15. C5×#487C22semidirect
33. #64767C22semidirect
34. #64767C22semidirect
35. #64766C22semidirect
36. #64677C22semidirect
85. #3229D5semidirect
145. #498C2×D5semidirect
146. #487C2×D5semidirect
356. C5#19390semidirect

Maximal subgroups

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

subgroupindex
C5×#32292
#106150812
C2×#653722
#106110172
#106026212
#106025662
#106021372
#106020252
#106015422
#193905