Finite Group Databasebeta

← #12254632

Subgroups — #12254632

Order 1280.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3155C211yesmaximal, characteristic
2#10610277C211yesmaximal, characteristic, complemented
3#10602871C211yesmaximal, characteristic, complemented
4#10602870C211yesmaximal, characteristic, complemented
5#10602826C211yesmaximal, characteristic, complemented
6C2×#64782C211yesmaximal, characteristic, complemented
7#10602112C212yesmaximal, complemented
8#10602111C212yesmaximal, complemented
9#10602082C212yesmaximal, complemented
10#10602081C212yesmaximal, complemented
11#10600674C211yesmaximal, characteristic, complemented
12C2×C5×AGL(1,ℤ/8)—21no—
13C2×C5×#135C2211yescharacteristic
14C22×C5×D8C2211yescharacteristic
15C5×#535—21no—
16#65853—21no—
17C5×#496—22no—
18C5×#495C2212yes—
19C5×#493C2212yes—
20C5×#405C2211yescharacteristic
21#65410—21no—
22#65395C2211yescharacteristic
23C2×#3950C2211yescharacteristic
24C2×#3944C2211yescharacteristic
25C22×#672C2211yescharacteristic
26#64991—22no—
27#64972C2212yescomplemented
28#64971C2212yescomplemented
29#64784C2211yescharacteristic, complemented
30#64784C2211yescharacteristic, complemented
31#64784C2212yescomplemented
32#64782C2212yes—
33#64782C2212yes—
34#64782C2212yes—
35#64782C2212yes—
36#64782C2211yescharacteristic, complemented
37#64782C2212yescomplemented
38#64782C2211yescharacteristic, complemented
39#64681C2212yescomplemented
40C4×#672C2212yescomplemented
41#33702—51nomaximal, Sylow
42C2×C5×#41—21no—
43C22×C5×D4—21no—
44C5×AGL(1,ℤ/8)—44no—
45C5×#135—22no—
46C5×#135—22no—
47C2×C5×Q16C2311yescharacteristic
48C2×C5×SD16—22no—
49C2×C5×SD16—21no—
50C2×C5×D8—22no—
51C2×C5×D8—21no—
52C2×C5×D8—22no—
53C2×C5×D8—22no—
54C2×C5×D8C2312yes—
55C2×C5×D8C2311yescharacteristic
56C2×C5×M16D411yescharacteristic
57C22×C5×C8C2311yescharacteristic
58C5×#127—22no—
59C5×#124—22no—
60C5×#121—41no—
61C5×#121—41no—
62C5×#120—42no—
63C5×#117D411yescharacteristic
64#4072—21no—
65C2×C4⋊Dic5—21no—
66#4036—41no—
67#4034—41no—
68#4024—42no—
69#4016—22no—
70#4015—22no—
71C5×#97D412yes—
72C4×C5×C8C2312yes—
73#3950—21no—
74#3950—22no—
75#3950C2312yes—
76#3950C2311yescharacteristic
77#3950C2311yescharacteristic
78#3949—21no—
79#3949—22no—
80#3944—22no—
81C2×#672—22no—
82C2×#672C2311yescharacteristic
83C2×#672C2311yescharacteristic
84C2×#672C2312yes—
85#3164—51no—
86#3155D511yescharacteristic
87#2244—52no—
88#2243—52no—
89#2242—52no—
90#2240—52no—
91#2140—51no—
92#2139—51no—
93#2138—51no—
94C2×#357—51no—
95#2108—51no—
96C24×C5—41no—
97C5×#41—42no—
98C5×#41—42no—
99C2×C5×Q8—21no—
100C2×C5×D4—22no—
101C2×C5×D4—21no—
102C2×C5×D4—41no—
103C2×C5×D4—42no—
104C2×C5×D4—41no—
105C2×C5×D4—42no—
106C2×C5×D4—42no—
107C2×C5×D4—42no—
108C22×C4×C5—41no—
109C22×C4×C5C2×D411yescharacteristic
110C22×Dic5—41no—
111C2×Dic10—42no—
112C5×Q16—21no—
113C5×Q16—21no—
114C5×SD16—42no—
115C5×SD16—44no—
116C5×D8—22no—
117C5×D8—22no—
118C5×D8—21no—
119C5×D8—21no—
120C5×D8—42no—
121C5×D8—44no—
122C5×M16SD1614yes—
123C2×C5×C8C2×D411yescharacteristic
124C2×C5×C8C2×D412yes—
125C2×C5×C8#4112yes—
126C2×C5×C8#4112yes—
127C2×C5×C8C2411yescharacteristic
128C5×C4⋊C4D811yescharacteristic
129C5×C4⋊C4D811yescharacteristic
130C5×C22⋊C4—42no—
131C5×C22⋊C4—42no—
132C5×C22⋊C4D812yes—
133C42×C5C2×D412yes—
134C22⋊Dic5—41no—
135C4⋊Dic5—21no—
136C4⋊Dic5—21no—
137C4⋊Dic5—22no—
138C4⋊Dic5—42no—
139C4⋊Dic5—41no—
140#683—42no—
141#683—42no—
142C4×Dic5—41no—
143#672—21no—
144#672—21no—
145#672—22no—
146C2×AGL(1,ℤ/8)—21no—
147C2×#135C2×D511yescharacteristic
148C22×D8C2×D511yescharacteristic
149#544—101no—
150#535—21no—
151C2×M32—51no—
152C22×C16—51no—
153#501—102no—
154#500—52no—
155#499—52no—
156#496—22no—
157#495C2×D512yes—
158#493C2×D512yes—
159#428—101no—
160#427—51no—
161C2×#107—51no—
162#405C2×D511yescharacteristic
163#358—51no—
164#358—51no—
165#358—52no—
166#357—52no—
167#357—52no—
168#357—52no—
169#357—52no—
170#357—51no—
171#357—51no—
172#357—52no—
173#346—52no—
174C4×C16—52no—
175C23×C5—42no—
176C23×C5—41no—
177C23×C5—82no—
178C23×C5—161no—
179C23×C5—161no—
180C23×C5C2×D811yescharacteristic
181C5×Q8—41no—
182C5×Q8—41no—
183C5×D4—42no—
184C5×D4—41no—
185C5×D4—41no—
186C5×D4—42no—
187C5×D4—82no—
188C5×D4—82no—
189C5×D4—162no—
190C5×D4—162no—
191C2×C4×C5—41no—
192C2×C4×C5—82no—
193C2×C4×C5#12412yes—
194C2×C4×C5C2×D812yes—
195C2×C4×C5C2×D812yes—
196C2×C4×C5C2×D811yescharacteristic
197C2×C4×C5C2×SD1612yes—
198C2×C4×C5C22×D411yescharacteristic
199C2×Dic5—41no—
200C2×Dic5—41no—
201C2×Dic5—42no—
202C2×Dic5—161no—
203Dic10—162no—
204C5×C8#12412yes—
205C5×C8C2×SD1614yes—
206C5×C8C2×#4111yescharacteristic
207C5×C8C2×#4111yescharacteristic
208C2×#41—21no—
209C22×D4—21no—
210AGL(1,ℤ/8)—44no—
211#135—22no—
212#135—22no—
213C2×Q16C22×D511yescharacteristic
214C2×SD16—21no—
215C2×SD16—22no—
216C2×D8—22no—
217C2×D8—22no—
218C2×D8—21no—
219C2×D8—22no—
220C2×D8C22×D512yes—
221C2×D8C22×D511yescharacteristic
222C2×M16C22⋊D511yescharacteristic
223C22×C8C22×D511yescharacteristic
224C4⋊Q8—102no—
225C4⋊Q8—201no—
226#127—22no—
227#125—102no—
228#125—201no—
229#124—22no—
230C22⋊Q8—202no—
231#121—41no—
232#121—41no—
233#120—42no—
234#117—101no—
235#117C22⋊D511yescharacteristic
236C2×C4⋊C4—101no—
237M32—102no—
238C2×C16—51no—
239C2×C16—52no—
240C2×C16—51no—
241C2×C16—102no—
242#107—52no—
243#107—51no—
244#107—51no—
245#107—101no—
246#107—102no—
247#106—101no—
248#106—102no—
249#97C22⋊D512yes—
250C4×C8C22×D512yes—
251C22×C5—21no—
252C22×C5—21no—
253C22×C5—81no—
254C22×C5—82no—
255C22×C5—81no—
256C22×C5—82no—
257C22×C5—81no—
258C22×C5—82no—
259C22×C5—164no—
260C22×C5#48612yes—
261C22×C5C22×D811yescharacteristic
262C4×C5—22no—
263C4×C5—22no—
264C4×C5—81no—
265C4×C5#48612yes—
266C4×C5C2×#12411yescharacteristic
267C4×C5C22×SD1611yescharacteristic
268Dic5—81no—
269Dic5—82no—
270Dic5—81no—
271C24—41no—
272#41—42no—
273#41—42no—
274C2×Q8—21no—
275C2×Q8—202no—
276C2×D4—22no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C22×C4—41no—
285C22×C4—201no—
286C22×C4C2×C22⋊D511yescharacteristic
287Q16—21no—
288Q16—21no—
289SD16—44no—
290SD16—42no—
291D8—22no—
292D8—22no—
293D8—21no—
294D8—21no—
295D8—44no—
296D8—42no—
297M16#68714yes—
298C2×C8#71112yes—
299C2×C8#71112yes—
300C2×C8C2×C22⋊D512yes—
301C2×C8C2×C22⋊D511yescharacteristic
302C2×C8C23×D511yescharacteristic
303C4⋊C4—101no—
304C4⋊C4—102no—
305C4⋊C4—101no—
306C4⋊C4—201no—
307C4⋊C4—202no—
308C4⋊C4—202no—
309C4⋊C4—202no—
310C4⋊C4#68611yescharacteristic
311C4⋊C4#68611yescharacteristic
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—201no—
315C22⋊C4#68612yes—
316C42—201no—
317C42C2×C22⋊D512yes—
318C16—101no—
319C16—101no—
320C16—102no—
321C2×C5—22no—
322C2×C5—161no—
323C2×C5—162no—
324C2×C5C2×#48611yescharacteristic
325C2×C5#341511yescharacteristic
326C2×C5#341611yescharacteristic
327C23—42no—
328C23—41no—
329C23—82no—
330C23—161no—
331C23—161no—
332C23C2×#68611yescharacteristic
333Q8—41no—
334Q8—41no—
335Q8—802no—
336D4—41no—
337D4—42no—
338D4—42no—
339D4—41no—
340D4—82no—
341D4—82no—
342D4—162no—
343D4—162no—
344C2×C4—41no—
345C2×C4—82no—
346C2×C4—201no—
347C2×C4—201no—
348C2×C4—202no—
349C2×C4—801no—
350C2×C4C2×#68612yes—
351C2×C4C2×#68612yes—
352C2×C4C2×#68611yescharacteristic
353C2×C4C2×#68712yes—
354C2×C4#408212yes—
355C2×C4C22×C22⋊D511yescharacteristic
356C8C2×#68714yes—
357C8#408212yes—
358C8C2×#71111yescharacteristic
359C8C2×#71111yescharacteristic
360C5#3370211yescharacteristic, complemented, Sylow
361C22—21no—
362C22—21no—
363C22—81no—
364C22—82no—
365C22—81no—
366C22—82no—
367C22—81no—
368C22—82no—
369C22—164no—
370C22#6535912yes—
371C22C22×#68611yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—402no—
376C4—401no—
377C4—401no—
378C4#6535912yes—
379C4C22×#68711yescharacteristic
380C4C2×#408211yescharacteristic
381C2—22no—
382C2—162no—
383C2—161no—
384C2C2×#6535911yescharacteristic
385C2#1061573811yescharacteristic
386C2#1061573911yescharacteristic
subgroupG/H
centerC22C22×#686
commutator, cosocleC2×C5×C8C24
FrattiniC2×C8C23×D5
Fitting, prime radicalC5×#3155C2
socleC22×C5C22×D8
hypercenter#3155D5
hypocenterC5#33702
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3155

Complements

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

NH ≅ G/Ntype
2. #10610277C2semidirect
3. #10602871C2semidirect
4. #10602870C2semidirect
5. #10602826C2semidirect
6. C2×#64782C2semidirect
7. #10602112C2semidirect
8. #10602111C2semidirect
9. #10602082C2semidirect
10. #10602081C2semidirect
11. #10600674C2semidirect
27. #64972C22semidirect
28. #64971C22semidirect
29. #64784C22semidirect
30. #64784C22semidirect
31. #64784C22semidirect
36. #64782C22semidirect
37. #64782C22semidirect
38. #64782C22semidirect
39. #64681C22semidirect
40. C4×#672C22semidirect
360. C5#33702semidirect

Maximal subgroups

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

subgroupindex
C5×#31552
#106102772
#106028712
#106028702
#106028262
C2×#647822
#106021122
#106021112
#106020822
#106020812
#106006742
#337025