Finite Group Databasebeta

← #12070738

Subgroups — #12070738

Order 1280.

716 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10610777C211yesmaximal, characteristic, complemented
2#10610774C211yesmaximal, characteristic
3C2×C4×#677C211yesmaximal, characteristic
4C5×#1566C211yesmaximal, characteristic, complemented
5#10601293C211yesmaximal, characteristic, complemented
6#10601289C211yesmaximal, characteristic
7C8×#681C211yesmaximal, characteristic, complemented
8#10599316C212yesmaximal, complemented
9#10599313C212yesmaximal, complemented
10#10599312C212yesmaximal, complemented
11#10599309C212yesmaximal, complemented
12C2×D5×M16—21no—
13C22×#677C411yescharacteristic
14C2×C4×D20C2211yescharacteristic
15C2×C4×Dic10C2211yescharacteristic
16C5×#423C2211yescharacteristic
17C5×#423C2211yescharacteristic
18C2×C4×C5×C8C2211yescharacteristic
19#65415—21no—
20#65407—21no—
21M16×Dic5—21no—
22C2×#3953C411yescharacteristic
23C2×#3949C411yescharacteristic
24C2×#3948C411yescharacteristic
25#65214C2211yescharacteristic
26C4×#681C2211yescharacteristic
27#65127C2212yes—
28#65125—22no—
29#65123C2212yes—
30#65120C2212yescomplemented
31#65119C2212yescomplemented
32#65117—22no—
33C4×#677C411yescharacteristic
34C4×#677C411yescharacteristic
35C4×#677C411yescharacteristic
36C4×#677C411yescharacteristic
37C4×#677C2211yescharacteristic
38C4×#677C2211yescharacteristic
39C4×#677C2211yescharacteristic
40C4×#677C2211yescharacteristic
41C5×#334C2212yescomplemented
42C5×#334C2212yescomplemented
43#64704C2212yescomplemented
44#64704C2212yescomplemented
45#64703C2212yes—
46#64703C2212yes—
47#64674C2212yescomplemented
48C8×#168C2212yescomplemented
49M16×SD16—51nomaximal, Sylow
50C22×D20C2×C411yescharacteristic
51C22×C4×D5—21no—
52C22×Dic10C2×C411yescharacteristic
53C2×C5×M16—21no—
54C2×C5×M16—21no—
55C22×C5×C8C2×C411yescharacteristic
56C22×C5×C8C2×C411yescharacteristic
57C2×C42×C5C2311yescharacteristic
58C2×#685—21no—
59C2×C4⋊Dic5C2×C411yescharacteristic
60C2×C4⋊Dic5C2×C411yescharacteristic
61C2×#683—21no—
62C2×C4×Dic5—21no—
63C2×#681D411yescharacteristic
64C2×#681D411yescharacteristic
65D5×M16—44no—
66C2×#677—24no—
67C2×#677C2×C411yescharacteristic
68C2×#677C2×C411yescharacteristic
69C2×#677C2×C411yescharacteristic
70C2×#677C2×C411yescharacteristic
71C2×#676—22no—
72C2×C8×D5—22no—
73C4×D20—21no—
74C4×D20—21no—
75C4×D20C2×C411yescharacteristic
76C4×D20C2×C411yescharacteristic
77C4×D20C2311yescharacteristic
78C4×D20C2311yescharacteristic
79C4×Dic10—21no—
80C4×Dic10—21no—
81C4×Dic10C2×C411yescharacteristic
82C4×Dic10C2×C411yescharacteristic
83C4×Dic10C2311yescharacteristic
84C4×Dic10C2311yescharacteristic
85C5×C4⋊C8—21no—
86C5×C4⋊C8—21no—
87C5×C4⋊C8C2312yes—
88C5×C4⋊C8C2312yes—
89C4×C5×C8C2×C411yescharacteristic
90C4×C5×C8C2×C411yescharacteristic
91C4×C5×C8C2311yescharacteristic
92C4×C5×C8C2311yescharacteristic
93C22⋊#168—21no—
94#3953C2×C411yescharacteristic
95#3953C2×C411yescharacteristic
96#3953C2×C411yescharacteristic
97#3953C2×C411yescharacteristic
98#3952—22no—
99#3952—22no—
100#3949C2×C411yescharacteristic
101#3949C2×C411yescharacteristic
102#3949C2×C411yescharacteristic
103#3949C2×C411yescharacteristic
104#3948C2×C411yescharacteristic
105#3948C2×C411yescharacteristic
106#3948C2×C411yescharacteristic
107#3948C2×C411yescharacteristic
108#3947—22no—
109#3946—22no—
110#3946—22no—
111C8×Dic5—22no—
112C4⋊#168—21no—
113#3935D412yes—
114C4×#168C2312yes—
115C4×#168C2312yes—
116Q8×M16—51no—
117C2×C4×SD16—51no—
118D4×M16—51no—
119#1590—52no—
120#1583—52no—
121#1582—52no—
122C8×SD16—52no—
123#1566D511yescharacteristic, complemented
124#1487—51no—
125#1486—51no—
126C8×M16—51no—
127C23×D5—21no—
128C22×C4×C5C22×C411yescharacteristic
129C22×C4×C5C2×D411yescharacteristic
130C22×C4×C5#4111yescharacteristic
131C22×Dic5—21no—
132C22×Dic5—21no—
133C22×Dic5—21no—
134C2×D20—22no—
135C2×D20—21no—
136C2×D20—41no—
137C2×D20C22×C411yescharacteristic
138C2×D20C22×C411yescharacteristic
139C2×C4×D5—21no—
140C2×C4×D5—21no—
141C2×C4×D5—21no—
142C2×C4×D5—21no—
143C2×C4×D5—42no—
144C2×C4×D5—42no—
145C2×Dic10—22no—
146C2×Dic10—21no—
147C2×Dic10—41no—
148C2×Dic10C22×C411yescharacteristic
149C2×Dic10C22×C411yescharacteristic
150C5×M16—42no—
151C5×M16—42no—
152C2×C5×C8—22no—
153C2×C5×C8—22no—
154C2×C5×C8—22no—
155C2×C5×C8—41no—
156C2×C5×C8C22×C411yescharacteristic
157C2×C5×C8C22×C411yescharacteristic
158C2×C5×C8C22×C411yescharacteristic
159C2×C5×C8C22×C411yescharacteristic
160C42×C5C22×C411yescharacteristic
161C42×C5C2×D411yescharacteristic
162C42×C5#4111yescharacteristic
163C42×C5C2411yescharacteristic
164#685—21no—
165#685—21no—
166#685—21no—
167#685—21no—
168C4⋊Dic5—21no—
169C4⋊Dic5—21no—
170C4⋊Dic5C22×C411yescharacteristic
171C4⋊Dic5C22×C411yescharacteristic
172C4⋊Dic5C22×C411yescharacteristic
173C4⋊Dic5C22×C411yescharacteristic
174#683—21no—
175#683—21no—
176#683—21no—
177#683—21no—
178C4×Dic5—21no—
179C4×Dic5—21no—
180C4×Dic5—21no—
181C4×Dic5—21no—
182#681—22no—
183#681SD1614yescomplemented
184C2×#168—22no—
185C2×#168C2×D412yes—
186C2×#168C2×D412yes—
187#677—22no—
188#677—22no—
189#677—22no—
190#677M1614yescomplemented
191#676—44no—
192C8×D5—44no—
193C22×SD16—51no—
194C22×M16—101no—
195C2×C4×Q8—51no—
196C2×C4×D4—51no—
197#446—52no—
198#446—102no—
199C8×Q8—52no—
200C4×SD16—51no—
201C4×SD16—51no—
202C4×SD16—51no—
203C4×SD16—51no—
204C4×SD16—51no—
205C4×SD16—51no—
206C4×SD16—51no—
207C4×SD16—51no—
208#436—52no—
209#435—102no—
210C8×D4—52no—
211#431—51no—
212C2×#106—51no—
213#423—101no—
214#423C2×D511yescharacteristic
215#423C2×D511yescharacteristic
216C2×#103—51no—
217C2×#102—51no—
218#407—101no—
219C4×M16—51no—
220C4×M16—101no—
221C2×C4×C8C2×D511yescharacteristic
222#334C2×D512yescomplemented
223#334C2×D512yescomplemented
224Q8⋊C8—52no—
225Q8⋊C8—52no—
226D4⋊C8—52no—
227D4⋊C8—52no—
228#322—52no—
229C82—52no—
230C23×C5C4×D411yescharacteristic
231C22×D5—21no—
232C22×D5—21no—
233C22×D5—42no—
234C22×D5—42no—
235C22×D5—81no—
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—41no—
239C2×C4×C5C4×D411yescharacteristic
240C2×C4×C5C4×D411yescharacteristic
241C2×C4×C5C4×D411yescharacteristic
242C2×C4×C5C2×SD1612yes—
243C2×C4×C5#13512yes—
244C2×C4×C5C23×C411yescharacteristic
245C2×C4×C5C22×D411yescharacteristic
246C2×C4×C5C2×#4111yescharacteristic
247C2×Dic5—21no—
248C2×Dic5—21no—
249C2×Dic5—21no—
250C2×Dic5—21no—
251C2×Dic5—21no—
252C2×Dic5—21no—
253C2×Dic5—42no—
254C2×Dic5—42no—
255C2×Dic5—81no—
256D20—21no—
257D20—21no—
258D20—21no—
259D20—42no—
260D20C2×M1612yes—
261C4×D5—41no—
262C4×D5—41no—
263C4×D5—41no—
264C4×D5—41no—
265C4×D5—42no—
266C4×D5—42no—
267Dic10—21no—
268Dic10—21no—
269Dic10—21no—
270Dic10—42no—
271Dic10C2×M1612yescomplemented
272C5×C8—21no—
273C5×C8—21no—
274C5×C8—21no—
275C5×C8—42no—
276C5×C8—42no—
277C5×C8C2×M1612yescomplemented
278#168—22no—
279#168—42no—
280#168C2×SD1614yescomplemented
281C22×Q8—51no—
282C22×D4—51no—
283C23×C4—101no—
284C2×SD16—51no—
285C2×SD16—51no—
286C2×SD16—51no—
287C2×SD16—51no—
288C2×SD16—104no—
289C2×M16—21no—
290C2×M16—21no—
291C2×M16—51no—
292C2×M16—51no—
293C2×M16—102no—
294C2×M16—204no—
295C22×C8—102no—
296C22×C8C4×D511yescharacteristic
297C22×C8C4×D511yescharacteristic
298C4×Q8—51no—
299C4×Q8—51no—
300C4×Q8—51no—
301C4×Q8—51no—
302C4×Q8—101no—
303C4×Q8—101no—
304C4×D4—51no—
305C4×D4—51no—
306C4×D4—51no—
307C4×D4—51no—
308C4×D4—101no—
309C4×D4—101no—
310C2×C4⋊C4—51no—
311C2×C4⋊C4—51no—
312C2×C4⋊C4—101no—
313C2×C22⋊C4—101no—
314C2×C42—101no—
315C2×C42C22×D511yescharacteristic
316#106—51no—
317#106—51no—
318#106—51no—
319#106—51no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—102no—
323C4⋊C8—102no—
324C4⋊C8—101no—
325C4⋊C8C22×D512yes—
326C4⋊C8C22×D512yes—
327#103—51no—
328#103—51no—
329#103—51no—
330#103—51no—
331#102—51no—
332#102—51no—
333#102—51no—
334#102—51no—
335C22⋊C8—102no—
336C22⋊C8—102no—
337C22⋊C8—101no—
338#97—52no—
339#97—102no—
340C4×C8—52no—
341C4×C8—52no—
342C4×C8—102no—
343C4×C8C4×D511yescharacteristic
344C4×C8C4×D511yescharacteristic
345C4×C8C22×D511yescharacteristic
346C4×C8C22×D511yescharacteristic
347C22×C5—21no—
348C22×C5—21no—
349C22×C5C4×SD1612yes—
350C22×C5C2×C4×D411yescharacteristic
351C2×D5—22no—
352C2×D5—41no—
353C2×D5—41no—
354C2×D5—41no—
355C2×D5—41no—
356C2×D5—41no—
357C2×D5—84no—
358C4×C5—21no—
359C4×C5—21no—
360C4×C5—21no—
361C4×C5C4×SD1612yes—
362C4×C5C22×M1611yescharacteristic
363C4×C5#56811yescharacteristic
364C4×C5C22×SD1611yescharacteristic
365C4×C5C2×#13511yescharacteristic
366Dic5—22no—
367Dic5—41no—
368Dic5—41no—
369Dic5—41no—
370Dic5—41no—
371Dic5—41no—
372C24—101no—
373C2×Q8—51no—
374C2×Q8—51no—
375C2×Q8—102no—
376C2×Q8—101no—
377C2×Q8—201no—
378C2×D4—51no—
379C2×D4—51no—
380C2×D4—102no—
381C2×D4—101no—
382C2×D4—201no—
383C22×C4—101no—
384C22×C4—101no—
385C22×C4—101no—
386C22×C4—101no—
387C22×C4—101no—
388C22×C4—101no—
389C22×C4—101no—
390C22×C4—202no—
391C22×C4—202no—
392C22×C4C2×C4×D511yescharacteristic
393C22×C4D4×D511yescharacteristic
394C22×C4#71311yescharacteristic
395SD16—54no—
396SD16—102no—
397SD16—102no—
398SD16—102no—
399M16—42no—
400M16—42no—
401M16—54no—
402M16—102no—
403M16—204no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—41no—
408C2×C8—52no—
409C2×C8—52no—
410C2×C8—102no—
411C2×C8—204no—
412C2×C8C2×C4×D511yescharacteristic
413C2×C8C2×C4×D511yescharacteristic
414C2×C8C2×C4×D511yescharacteristic
415C2×C8C2×C4×D511yescharacteristic
416C4⋊C4—51no—
417C4⋊C4—51no—
418C4⋊C4—51no—
419C4⋊C4—51no—
420C4⋊C4—101no—
421C4⋊C4—101no—
422C4⋊C4—101no—
423C4⋊C4—101no—
424C4⋊C4—101no—
425C4⋊C4—101no—
426C22⋊C4—101no—
427C22⋊C4—101no—
428C22⋊C4—101no—
429C22⋊C4—101no—
430C42—101no—
431C42—101no—
432C42—101no—
433C42—101no—
434C42C2×C4×D511yescharacteristic
435C42D4×D511yescharacteristic
436C42#71311yescharacteristic
437C42C23×D511yescharacteristic
438C2×C5—22no—
439C2×C5D4×M1611yescharacteristic
440C2×C5C2×C4×SD1611yescharacteristic
441C2×C5#295711yescharacteristic
442D5—42no—
443D5—81no—
444C23—101no—
445C23—101no—
446C23—202no—
447C23—202no—
448C23—401no—
449C23#403911yescharacteristic
450Q8—52no—
451Q8—101no—
452Q8—101no—
453Q8—101no—
454Q8—202no—
455D4—52no—
456D4—101no—
457D4—101no—
458D4—101no—
459D4—202no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—101no—
464C2×C4—101no—
465C2×C4—101no—
466C2×C4—101no—
467C2×C4—101no—
468C2×C4—101no—
469C2×C4—202no—
470C2×C4—202no—
471C2×C4—201no—
472C2×C4—201no—
473C2×C4—201no—
474C2×C4—201no—
475C2×C4—202no—
476C2×C4—202no—
477C2×C4—401no—
478C2×C4#403911yescharacteristic
479C2×C4#403911yescharacteristic
480C2×C4#403911yescharacteristic
481C2×C4D5×SD1612yes—
482C2×C4#406212yes—
483C2×C4C22×C4×D511yescharacteristic
484C2×C4C2×D4×D511yescharacteristic
485C2×C4C2×#71311yescharacteristic
486C8—21no—
487C8—21no—
488C8—21no—
489C8—42no—
490C8—42no—
491C8—54no—
492C8—102no—
493C8—202no—
494C8D5×M1612yescomplemented
495C5M16×SD1611yescharacteristic, complemented, Sylow
496C22—21no—
497C22—21no—
498C22—102no—
499C22—201no—
500C22—201no—
501C22—201no—
502C22—201no—
503C22—201no—
504C22—404no—
505C22#6514112yes—
506C22C2×#403911yescharacteristic
507C4—21no—
508C4—21no—
509C4—21no—
510C4—102no—
511C4—201no—
512C4—201no—
513C4—201no—
514C4—201no—
515C4—201no—
516C4#6514112yes—
517C4C2×D5×M1611yescharacteristic
518C4#6607911yescharacteristic
519C4C2×D5×SD1611yescharacteristic
520C4C2×#406211yescharacteristic
521C2—22no—
522C2—202no—
523C2—401no—
524C2#1061077411yescharacteristic
525C2C2×#6514111yescharacteristic
526C2#1061098011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#1566C2
socleC22×C5C2×C4×D4
prime radicalC22×D20C2×C4
cosocleC42×C5C24
hypercenter#1566D5
hypocenterC5M16×SD16
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1566

Complements

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

NH ≅ G/Ntype
1. #10610777C2semidirect
4. C5×#1566C2semidirect
5. #10601293C2semidirect
7. C8×#681C2semidirect
8. #10599316C2semidirect
9. #10599313C2semidirect
10. #10599312C2semidirect
11. #10599309C2semidirect
30. #65120C22semidirect
31. #65119C22semidirect
41. C5×#334C22semidirect
42. C5×#334C22semidirect
43. #64704C22semidirect
44. #64704C22semidirect
47. #64674C22semidirect
48. C8×#168C22semidirect
123. #1566D5semidirect
183. #681SD16semidirect
190. #677M16semidirect
222. #334C2×D5semidirect
223. #334C2×D5semidirect
271. Dic10C2×M16semidirect
277. C5×C8C2×M16semidirect
280. #168C2×SD16semidirect
494. C8D5×M16semidirect
495. C5M16×SD16semidirect

Maximal subgroups

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

subgroupindex
#106107772
#106107742
C2×C4×#6772
C5×#15662
#106012932
#106012892
C8×#6812
#105993162
#105993132
#105993122
#105993092
M16×SD165