Finite Group Databasebeta

← #12254637

Subgroups — #12254637

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×#3164C211yesmaximal, characteristic, complemented
2#10610296C211yesmaximal, characteristic
3#10602176C212yesmaximal, complemented
4#10602174C212yesmaximal, complemented
5#10602145C212yesmaximal, complemented
6#10602144C212yesmaximal, complemented
7#10601445C211yesmaximal, characteristic, complemented
8#10601444C211yesmaximal, characteristic, complemented
9#10601391C211yesmaximal, characteristic, complemented
10C2×#64711C211yesmaximal, characteristic, complemented
11#10600674C211yesmaximal, characteristic, complemented
12C5×#544—21no—
13C2×#4054—21no—
14C2×#4050C2211yescharacteristic
15C22×D40C2211yescharacteristic
16#65859—21no—
17C5×#501—22no—
18C5×#500C2212yescomplemented
19C5×#499C2212yescomplemented
20C5×#428—21no—
21C5×#427C2211yescharacteristic
22C2×C5×#107C2211yescharacteristic
23C5×#405C2211yescharacteristic
24C2×#3944C2211yescharacteristic
25C22×#672C2211yescharacteristic
26#65001—22no—
27#64985C2212yes—
28#64984C2212yes—
29#64712C2211yescharacteristic, complemented
30#64712C2212yescomplemented
31#64712C2211yescharacteristic, complemented
32#64711C2212yes—
33#64711C2212yes—
34#64711C2212yes—
35#64711C2212yes—
36#64711C2211yescharacteristic, complemented
37#64711C2212yescomplemented
38#64711C2211yescharacteristic, complemented
39#64681C2212yescomplemented
40C4×#672C2212yescomplemented
41#33702—51nomaximal, Sylow
42C2×#709—21no—
43C22×D20—21no—
44C2×C5×M16D411yescharacteristic
45C22×C5×C8C2311yescharacteristic
46C5×C4⋊Q8—22no—
47C5×C4⋊Q8—41no—
48C5×#125—22no—
49C5×#125—41no—
50C5×C22⋊Q8—42no—
51C5×#117—21no—
52C5×#117D411yescharacteristic
53C2×C5×C4⋊C4—21no—
54#4054—44no—
55C2×Dic20C2311yescharacteristic
56#4050—22no—
57#4050—22no—
58C2×D40—22no—
59C2×D40—22no—
60C2×D40—21no—
61C2×D40—22no—
62C2×D40C2312yes—
63C2×D40C2311yescharacteristic
64C2×#677—21no—
65C2×#677—22no—
66#4041—41no—
67#4041—41no—
68#4028—42no—
69#4021—22no—
70#4020—22no—
71C5×#107—21no—
72C5×#107—22no—
73C5×#107C2312yes—
74C5×#107C2311yescharacteristic
75C5×#107C2311yescharacteristic
76C5×#106—21no—
77C5×#106—22no—
78C5×#97D412yes—
79C4×C5×C8C2312yes—
80#3944—22no—
81C2×#672—22no—
82C2×#672C2311yescharacteristic
83C2×#672C2311yescharacteristic
84C2×#672C2312yes—
85#3164D511yescharacteristic, complemented
86#3155—51no—
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—
96C23×D5—41no—
97C2×C5×Q8—42no—
98C22×C4×C5—41no—
99C22×C4×C5C2×D411yescharacteristic
100C2×C22⋊D5—41no—
101#709—42no—
102#709—42no—
103C2×D20—22no—
104C2×D20—21no—
105C2×D20—42no—
106C2×D20—42no—
107C2×D20—41no—
108C2×D20—42no—
109C2×D20—42no—
110C2×C4×D5—41no—
111C2×Dic10—21no—
112C5×M16SD1614yes—
113C2×C5×C8C2×D411yescharacteristic
114C2×C5×C8C2×D412yes—
115C2×C5×C8#4112yes—
116C2×C5×C8#4112yes—
117C2×C5×C8C2411yescharacteristic
118C5×C4⋊C4—22no—
119C5×C4⋊C4—21no—
120C5×C4⋊C4—21no—
121C5×C4⋊C4—41no—
122C5×C4⋊C4—42no—
123C5×C4⋊C4—42no—
124C5×C4⋊C4—42no—
125C5×C4⋊C4D811yescharacteristic
126C5×C4⋊C4D811yescharacteristic
127C5×C22⋊C4—41no—
128C5×C22⋊C4D812yes—
129C42×C5—41no—
130C42×C5C2×D412yes—
131#685—42no—
132#685—42no—
133Dic20—21no—
134Dic20—21no—
135D40—22no—
136D40—22no—
137D40—21no—
138D40—21no—
139D40—42no—
140D40—44no—
141#677—42no—
142#677—44no—
143#672—21no—
144#672—21no—
145#672—22no—
146C2×AGL(1,ℤ/8)—101no—
147C2×#135—51no—
148C22×D8—51no—
149#544—21no—
150#535—101no—
151C2×M32—51no—
152C22×C16—51no—
153#501—22no—
154#500C2×D512yescomplemented
155#499C2×D512yescomplemented
156#496—102no—
157#495—52no—
158#493—52no—
159#428—21no—
160#427C2×D511yescharacteristic
161C2×#107C2×D511yescharacteristic
162#405C2×D511yescharacteristic
163#358—51no—
164#358—51no—
165#358—52no—
166#357—52no—
167#357—52no—
168#357—51no—
169#357—51no—
170#357—52no—
171#357—52no—
172#357—52no—
173#346—52no—
174C4×C16—52no—
175C23×C5C2×D811yescharacteristic
176C22×D5—42no—
177C22×D5—41no—
178C22×D5—82no—
179C22×D5—161no—
180C22×D5—161no—
181C5×Q8—162no—
182C2×C4×C5—41no—
183C2×C4×C5—41no—
184C2×C4×C5—42no—
185C2×C4×C5—161no—
186C2×C4×C5#12412yes—
187C2×C4×C5C2×D812yes—
188C2×C4×C5C2×D812yes—
189C2×C4×C5C2×D811yescharacteristic
190C2×C4×C5C2×SD1612yes—
191C2×C4×C5C22×D411yescharacteristic
192C22⋊D5—82no—
193C2×Dic5—41no—
194D20—41no—
195D20—42no—
196D20—42no—
197D20—41no—
198D20—82no—
199D20—162no—
200D20—162no—
201C4×D5—82no—
202Dic10—41no—
203Dic10—41no—
204C5×C8#12412yes—
205C5×C8C2×SD1614yes—
206C5×C8C2×#4111yescharacteristic
207C5×C8C2×#4111yescharacteristic
208C2×#41—101no—
209C22×D4—101no—
210AGL(1,ℤ/8)—204no—
211#135—102no—
212#135—102no—
213C2×Q16—51no—
214C2×SD16—101no—
215C2×SD16—102no—
216C2×D8—52no—
217C2×D8—51no—
218C2×D8—102no—
219C2×D8—102no—
220C2×D8—101no—
221C2×D8—102no—
222C2×M16C22⋊D511yescharacteristic
223C22×C8C22×D511yescharacteristic
224C4⋊Q8—22no—
225C4⋊Q8—41no—
226#127—102no—
227#125—22no—
228#125—41no—
229#124—102no—
230C22⋊Q8—42no—
231#121—201no—
232#121—201no—
233#120—202no—
234#117—21no—
235#117C22⋊D511yescharacteristic
236C2×C4⋊C4—21no—
237M32—102no—
238C2×C16—51no—
239C2×C16—52no—
240C2×C16—51no—
241C2×C16—102no—
242#107—21no—
243#107—22no—
244#107C22×D512yes—
245#107C22×D511yescharacteristic
246#107C22×D511yescharacteristic
247#106—21no—
248#106—22no—
249#97C22⋊D512yes—
250C4×C8C22×D512yes—
251C22×C5—21no—
252C22×C5—21no—
253C22×C5#48612yes—
254C22×C5C22×D811yescharacteristic
255C2×D5—81no—
256C2×D5—82no—
257C2×D5—81no—
258C2×D5—82no—
259C2×D5—81no—
260C2×D5—82no—
261C2×D5—164no—
262C4×C5—22no—
263C4×C5—22no—
264C4×C5—82no—
265C4×C5—81no—
266C4×C5—81no—
267C4×C5#48612yes—
268C4×C5C2×#12411yescharacteristic
269C4×C5C22×SD1611yescharacteristic
270Dic5—81no—
271C24—201no—
272#41—202no—
273#41—202no—
274C2×Q8—42no—
275C2×Q8—101no—
276C2×D4—102no—
277C2×D4—101no—
278C2×D4—201no—
279C2×D4—201no—
280C2×D4—202no—
281C2×D4—202no—
282C2×D4—202no—
283C2×D4—202no—
284C22×C4—41no—
285C22×C4—201no—
286C22×C4C2×C22⋊D511yescharacteristic
287Q16—101no—
288Q16—101no—
289SD16—204no—
290SD16—202no—
291D8—102no—
292D8—102no—
293D8—101no—
294D8—101no—
295D8—204no—
296D8—202no—
297M16#68814yes—
298C2×C8#71312yes—
299C2×C8#71312yes—
300C2×C8C2×C22⋊D512yes—
301C2×C8C2×C22⋊D511yescharacteristic
302C2×C8C23×D511yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4#68611yescharacteristic
311C4⋊C4#68611yescharacteristic
312C22⋊C4—41no—
313C22⋊C4—202no—
314C22⋊C4—202no—
315C22⋊C4#68612yes—
316C42—41no—
317C42C2×C22⋊D512yes—
318C16—101no—
319C16—101no—
320C16—102no—
321C2×C5—22no—
322C2×C5C2×#48611yescharacteristic
323C2×C5#341511yescharacteristic
324C2×C5#341611yescharacteristic
325D5—161no—
326D5—162no—
327C23—202no—
328C23—201no—
329C23—402no—
330C23—801no—
331C23—801no—
332C23C2×#68611yescharacteristic
333Q8—162no—
334Q8—201no—
335Q8—201no—
336D4—201no—
337D4—202no—
338D4—202no—
339D4—201no—
340D4—402no—
341D4—402no—
342D4—802no—
343D4—802no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—161no—
348C2×C4—201no—
349C2×C4—402no—
350C2×C4C2×#68612yes—
351C2×C4C2×#68612yes—
352C2×C4C2×#68611yescharacteristic
353C2×C4C2×#68812yes—
354C2×C4#409312yes—
355C2×C4C22×C22⋊D511yescharacteristic
356C8C2×#68814yes—
357C8#409312yes—
358C8C2×#71311yescharacteristic
359C8C2×#71311yescharacteristic
360C5#3370211yescharacteristic, complemented, Sylow
361C22—21no—
362C22—21no—
363C22—401no—
364C22—402no—
365C22—401no—
366C22—402no—
367C22—401no—
368C22—402no—
369C22—804no—
370C22#6537712yes—
371C22C22×#68611yescharacteristic
372C4—22no—
373C4—22no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4—401no—
378C4#6537712yes—
379C4C22×#68811yescharacteristic
380C4C2×#409311yescharacteristic
381C2—22no—
382C2—802no—
383C2—801no—
384C2C2×#6537711yescharacteristic
385C2#1061573811yescharacteristic
386C2#1061574711yescharacteristic
subgroupG/H
centerC22C22×#686
commutator, cosocleC2×C5×C8C24
FrattiniC2×C8C23×D5
FittingC5×#3164C2
socleC22×C5C22×D8
prime radical#10610296C2
hypercenter#3164D5
hypocenterC5#33702
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
1. C5×#3164C2semidirect
3. #10602176C2semidirect
4. #10602174C2semidirect
5. #10602145C2semidirect
6. #10602144C2semidirect
7. #10601445C2semidirect
8. #10601444C2semidirect
9. #10601391C2semidirect
10. C2×#64711C2semidirect
11. #10600674C2semidirect
18. C5×#500C22semidirect
19. C5×#499C22semidirect
29. #64712C22semidirect
30. #64712C22semidirect
31. #64712C22semidirect
36. #64711C22semidirect
37. #64711C22semidirect
38. #64711C22semidirect
39. #64681C22semidirect
40. C4×#672C22semidirect
85. #3164D5semidirect
154. #500C2×D5semidirect
155. #499C2×D5semidirect
360. C5#33702semidirect

Maximal subgroups

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

subgroupindex
C5×#31642
#106102962
#106021762
#106021742
#106021452
#106021442
#106014452
#106014442
#106013912
C2×#647112
#106006742
#337025