Finite Group Databasebeta

← #12070441

Subgroups — #12070441

Order 1280.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2971C211yesmaximal, characteristic
2#10614265C211yesmaximal, characteristic
3#10610239C211yesmaximal, characteristic, complemented
4#10602432C211yesmaximal, characteristic
5#10601728C211yesmaximal, characteristic, complemented
6#10601727C211yesmaximal, characteristic, complemented
7#10601725C211yesmaximal, characteristic, complemented
8C8×#689C211yesmaximal, characteristic, complemented
9#10601659C211yesmaximal, characteristic, complemented
10#10601658C211yesmaximal, characteristic, complemented
11#10601657C211yesmaximal, characteristic, complemented
12#10601653C211yesmaximal, characteristic, complemented
13#10601596C211yesmaximal, characteristic, complemented
14#10601590C211yesmaximal, characteristic, complemented
15#10600641C211yesmaximal, characteristic, complemented
16C4×C5×#41C2211yescharacteristic
17#65804C2211yescharacteristic
18C5×C8×Q8C2211yescharacteristic
19C5×C8×Q8C2211yescharacteristic
20C5×#436—21no—
21C5×#436—21no—
22C5×#436C2211yescharacteristic
23C5×#436C2211yescharacteristic
24C5×#432—21no—
25C5×#432C2211yescharacteristic
26C4×C5×M16—21no—
27C4×C5×M16C2211yescharacteristic
28#65528C411yescharacteristic
29#65528C411yescharacteristic
30#65411—21no—
31#65408—21no—
32#65320C411yescharacteristic
33C4×#689C2211yescharacteristic
34C4×#689C2211yescharacteristic
35#65309C411yescharacteristic
36#65308C411yescharacteristic
37C4×#687C2211yescharacteristic
38C4×#687C2211yescharacteristic
39#65296C411yescharacteristic
40#65285C411yescharacteristic
41#65285C411yescharacteristic
42#65214C2211yescharacteristic
43#65214C2211yescharacteristic
44C2×C4⋊#168C2211yescharacteristic
45#64990—21no—
46#64990—21no—
47#64968C2211yescharacteristic
48#64968C2211yescharacteristic
49C8×Dic10C2211yescharacteristic
50C8×Dic10C2211yescharacteristic
51Q8⋊#168C2211yescharacteristic
52Q8⋊#168C2211yescharacteristic
53D4⋊#168C2211yescharacteristic
54D4⋊#168C2211yescharacteristic
55#64704C2211yescharacteristic
56#64704C2211yescharacteristic
57#64704C2211yescharacteristic
58#64704C2211yescharacteristic
59#64700C2211yescharacteristic
60#64700C2211yescharacteristic
61#64699C2211yescharacteristic
62#64699C2211yescharacteristic
63#64675C2211yescharacteristic
64#64674C2211yescharacteristic
65#64674C2211yescharacteristic
66C8×#168C2211yescharacteristic
67#18013—51nomaximal, Sylow
68C2×C5×#41C2×C411yescharacteristic
69C2×C5×M16—21no—
70C2×C5×M16—21no—
71C2×C5×M16D411yescharacteristic
72C2×C5×M16D411yescharacteristic
73C4×C5×Q8C2×C411yescharacteristic
74C4×C5×Q8C2311yescharacteristic
75C4×C5×D4—21no—
76C4×C5×D4—21no—
77C4×C5×D4C2×C411yescharacteristic
78C4×C5×D4C2311yescharacteristic
79C5×#117—21no—
80C5×#117C2×C411yescharacteristic
81C2×C42×C5—21no—
82C2×C42×C5C2311yescharacteristic
83C2×#689C2×C411yescharacteristic
84C2×#689C2×C411yescharacteristic
85C2×#687C2×C411yescharacteristic
86C2×#687C2×C411yescharacteristic
87#4072—21no—
88#4070—21no—
89#4070C2×C411yescharacteristic
90#4070C2×C411yescharacteristic
91C22×#168—21no—
92#4016—21no—
93#4015C2×C411yescharacteristic
94#4015C2×C411yescharacteristic
95C4×Dic10—21no—
96C4×Dic10C2311yescharacteristic
97C4×Dic10C2311yescharacteristic
98C5×C4⋊C8—21no—
99C5×C4⋊C8—21no—
100C5×C4⋊C8C2311yescharacteristic
101C5×C4⋊C8C2311yescharacteristic
102C5×C22⋊C8—21no—
103C5×C22⋊C8—21no—
104C5×C22⋊C8—21no—
105C5×#97—21no—
106C5×#97D411yescharacteristic
107C5×#97D411yescharacteristic
108C4×C5×C8—21no—
109C4×C5×C8—21no—
110C4×C5×C8C2311yescharacteristic
111C4×C5×C8C2311yescharacteristic
112#3967C2×C411yescharacteristic
113#3967C2×C411yescharacteristic
114#3964C2×C411yescharacteristic
115#3964C2×C411yescharacteristic
116C22⋊#168—21no—
117C22⋊#168C2×C411yescharacteristic
118C22⋊#168C2×C411yescharacteristic
119#3947—21no—
120#3947—21no—
121#3946—21no—
122#3946—21no—
123#3946—21no—
124#3946—21no—
125C8×Dic5—21no—
126C8×Dic5—21no—
127#3942C2×C411yescharacteristic
128#3942C2×C411yescharacteristic
129#3942C2×C411yescharacteristic
130#3942C2×C411yescharacteristic
131#3940C2×C411yescharacteristic
132#3940C2×C411yescharacteristic
133#3939C2×C411yescharacteristic
134#3939C2×C411yescharacteristic
135C4⋊#168—21no—
136C4⋊#168—21no—
137C4⋊#168C2×C411yescharacteristic
138C4⋊#168C2×C411yescharacteristic
139C4⋊#168C2311yescharacteristic
140C4⋊#168C2311yescharacteristic
141C4×#168C2311yescharacteristic
142C4×#168C2311yescharacteristic
143C4×#168C2311yescharacteristic
144C4×#168C2311yescharacteristic
145#3107—51no—
146#2971D511yescharacteristic
147#2962—51no—
148#1592—51no—
149#1590—51no—
150#1588—51no—
151#1587—51no—
152#1584—51no—
153#1583—51no—
154#1579—51no—
155C8×Q16—51no—
156#1573—51no—
157#1497—51no—
158#1477—51no—
159#1453—51no—
160C5×#41—22no—
161C5×#41—41no—
162C2×C5×Q8C22×C411yescharacteristic
163C2×C5×D4—21no—
164C2×C5×D4C22×C411yescharacteristic
165C22×C4×C5—21no—
166C22×C4×C5—21no—
167C22×C4×C5—21no—
168C22×C4×C5C22×C411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5#4111yescharacteristic
171C2×Dic10—21no—
172C2×Dic10C22×C411yescharacteristic
173C2×Dic10C22×C411yescharacteristic
174C5×M16—22no—
175C5×M16—41no—
176C5×M16—42no—
177C5×M16—42no—
178C2×C5×C8—21no—
179C2×C5×C8—21no—
180C2×C5×C8C2×D411yescharacteristic
181C2×C5×C8C2×D411yescharacteristic
182C2×C5×C8C2×D411yescharacteristic
183C2×C5×C8C2×D411yescharacteristic
184C5×C4⋊C4—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4C22×C411yescharacteristic
187C5×C4⋊C4C22×C411yescharacteristic
188C5×C22⋊C4—21no—
189C5×C22⋊C4—21no—
190C5×C22⋊C4—21no—
191C42×C5—21no—
192C42×C5—21no—
193C42×C5—21no—
194C42×C5C22×C411yescharacteristic
195C42×C5C2×D411yescharacteristic
196C42×C5#4111yescharacteristic
197C42×C5C2411yescharacteristic
198C22⋊Dic5—21no—
199C22⋊Dic5—21no—
200#689—22no—
201#689—22no—
202#687—41no—
203#687—41no—
204C4⋊Dic5—21no—
205C4⋊Dic5—21no—
206C4⋊Dic5—21no—
207C4⋊Dic5C22×C411yescharacteristic
208C4⋊Dic5C22×C411yescharacteristic
209#683—21no—
210#683—21no—
211#683—21no—
212#683—21no—
213C4×Dic5—21no—
214C4×Dic5—21no—
215C2×#168—21no—
216C2×#168—21no—
217C2×#168—21no—
218C2×#168—21no—
219C2×#168—42no—
220C2×#168C22×C411yescharacteristic
221C2×#168C22×C411yescharacteristic
222C2×#168C22×C411yescharacteristic
223C2×#168C22×C411yescharacteristic
224#533—51no—
225C4×#41C2×D511yescharacteristic
226#485—51no—
227#485—51no—
228#475—51no—
229#474—51no—
230#462—51no—
231#461—51no—
232#452—51no—
233#452—51no—
234#446—51no—
235#446—51no—
236#446—101no—
237#446—101no—
238C8×Q8—51no—
239C8×Q8—51no—
240C8×Q8C2×D511yescharacteristic
241C8×Q8C2×D511yescharacteristic
242C4×Q16—51no—
243C4×Q16—51no—
244C4×SD16—51no—
245C4×SD16—51no—
246#436—21no—
247#436—21no—
248#436C2×D511yescharacteristic
249#436C2×D511yescharacteristic
250#432—21no—
251#432C2×D511yescharacteristic
252#431—51no—
253#431—51no—
254#424—101no—
255C2×C4⋊C8—51no—
256#405—101no—
257C4×M16—21no—
258C4×M16C2×D511yescharacteristic
259#335—51no—
260#335—51no—
261#334—51no—
262#334—51no—
263Q8⋊C8—51no—
264Q8⋊C8—51no—
265Q8⋊C8—51no—
266Q8⋊C8—51no—
267Q8⋊C8—51no—
268Q8⋊C8—51no—
269D4⋊C8—51no—
270D4⋊C8—51no—
271#322—51no—
272#322—51no—
273#322—51no—
274C82—51no—
275C23×C5—21no—
276C23×C5C4×D411yescharacteristic
277C5×Q8—21no—
278C5×Q8#13112yes—
279C5×D4—21no—
280C5×D4—21no—
281C5×D4—42no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—41no—
290C2×C4×C5—42no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—81no—
294C2×C4×C5C4×D411yescharacteristic
295C2×C4×C5C4×D411yescharacteristic
296C2×C4×C5C4×D411yescharacteristic
297C2×C4×C5C23×C411yescharacteristic
298C2×C4×C5C22×D411yescharacteristic
299C2×C4×C5C2×#4111yescharacteristic
300C2×Dic5—21no—
301C2×Dic5—21no—
302C2×Dic5—21no—
303C2×Dic5—21no—
304Dic10—21no—
305Dic10—21no—
306Dic10—21no—
307Dic10—21no—
308Dic10—81no—
309C5×C8—21no—
310C5×C8—21no—
311C5×C8—21no—
312C5×C8—41no—
313C5×C8—41no—
314C5×C8#13712yes—
315#168—21no—
316#168—21no—
317#168—21no—
318#168—21no—
319#168—41no—
320#168—41no—
321#168—41no—
322#168—41no—
323C2×#41C4×D511yescharacteristic
324C2×Q16—51no—
325C2×Q16—51no—
326C2×SD16—51no—
327C2×SD16—51no—
328C2×M16—21no—
329C2×M16—21no—
330C2×M16C22⋊D511yescharacteristic
331C2×M16C22⋊D511yescharacteristic
332C22×C8—101no—
333C4⋊Q8—51no—
334C4⋊Q8—51no—
335#125—101no—
336C22⋊Q8—51no—
337C22⋊Q8—51no—
338C22⋊Q8—101no—
339C4×Q8—51no—
340C4×Q8—51no—
341C4×Q8—101no—
342C4×Q8C4×D511yescharacteristic
343C4×Q8C22×D511yescharacteristic
344C4×D4—21no—
345C4×D4—21no—
346C4×D4C4×D511yescharacteristic
347C4×D4C22×D511yescharacteristic
348#117—21no—
349#117—101no—
350#117C4×D511yescharacteristic
351C2×C42—21no—
352C2×C42C22×D511yescharacteristic
353#107—51no—
354#107—51no—
355#106—51no—
356#106—51no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—51no—
360C4⋊C8—51no—
361C4⋊C8—51no—
362C4⋊C8—51no—
363C4⋊C8—101no—
364C4⋊C8—101no—
365C4⋊C8—101no—
366C4⋊C8—101no—
367C4⋊C8—101no—
368C4⋊C8—101no—
369C4⋊C8C22×D511yescharacteristic
370C4⋊C8C22×D511yescharacteristic
371#103—51no—
372#103—51no—
373#103—51no—
374#103—51no—
375#103—51no—
376#103—51no—
377#102—51no—
378#102—51no—
379C22⋊C8—21no—
380C22⋊C8—21no—
381C22⋊C8—21no—
382C22⋊C8—51no—
383C22⋊C8—51no—
384C22⋊C8—101no—
385#97—21no—
386#97—101no—
387#97—101no—
388#97C22⋊D511yescharacteristic
389#97C22⋊D511yescharacteristic
390C4×C8—21no—
391C4×C8—21no—
392C4×C8—51no—
393C4×C8—51no—
394C4×C8—51no—
395C4×C8—51no—
396C4×C8—101no—
397C4×C8—101no—
398C4×C8C22×D511yescharacteristic
399C4×C8C22×D511yescharacteristic
400C22×C5—21no—
401C22×C5—21no—
402C22×C5—21no—
403C22×C5—41no—
404C22×C5—41no—
405C22×C5—41no—
406C22×C5C2×C4×D411yescharacteristic
407C4×C5—21no—
408C4×C5—22no—
409C4×C5—21no—
410C4×C5—21no—
411C4×C5—21no—
412C4×C5—41no—
413C4×C5—41no—
414C4×C5C2×#13111yescharacteristic
415C4×C5#56811yescharacteristic
416C4×C5C2×#13711yescharacteristic
417C4×C5#57511yescharacteristic
418Dic5—41no—
419Dic5—41no—
420Dic5—41no—
421Dic5—41no—
422#41—22no—
423#41—41no—
424C2×Q8—51no—
425C2×Q8—51no—
426C2×Q8—101no—
427C2×Q8C2×C4×D511yescharacteristic
428C2×D4—21no—
429C2×D4C2×C4×D511yescharacteristic
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4C2×C4×D511yescharacteristic
434C22×C4#70911yescharacteristic
435C22×C4C2×C22⋊D511yescharacteristic
436Q16—102no—
437Q16—102no—
438SD16—201no—
439SD16—201no—
440M16—22no—
441M16—41no—
442M16—42no—
443M16—42no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—51no—
447C2×C8—51no—
448C2×C8—51no—
449C2×C8—51no—
450C2×C8—101no—
451C2×C8—101no—
452C2×C8—101no—
453C2×C8—101no—
454C2×C8—202no—
455C2×C8C2×C22⋊D511yescharacteristic
456C2×C8C2×C22⋊D511yescharacteristic
457C2×C8C2×C22⋊D511yescharacteristic
458C2×C8C2×C22⋊D511yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—51no—
462C4⋊C4—51no—
463C4⋊C4—101no—
464C4⋊C4—101no—
465C4⋊C4—101no—
466C4⋊C4—101no—
467C4⋊C4—101no—
468C4⋊C4—101no—
469C4⋊C4—101no—
470C4⋊C4C2×C4×D511yescharacteristic
471C4⋊C4C2×C4×D511yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—101no—
476C22⋊C4—101no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—101no—
481C42—101no—
482C42C2×C4×D511yescharacteristic
483C42#70911yescharacteristic
484C42C2×C22⋊D511yescharacteristic
485C42C23×D511yescharacteristic
486C2×C5—41no—
487C2×C5—81no—
488C2×C5#293311yescharacteristic
489C2×C5C4×#13711yescharacteristic
490C2×C5#295711yescharacteristic
491C23—21no—
492C23C4×C22⋊D511yescharacteristic
493Q8—21no—
494Q8—101no—
495Q8—101no—
496Q8—101no—
497Q8—101no—
498Q8—401no—
499Q8#405312yes—
500D4—21no—
501D4—21no—
502D4—42no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—81no—
515C2×C4—101no—
516C2×C4—101no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4C4×C22⋊D511yescharacteristic
520C2×C4C4×C22⋊D511yescharacteristic
521C2×C4C4×C22⋊D511yescharacteristic
522C2×C4C22×C4×D511yescharacteristic
523C2×C4C2×#70911yescharacteristic
524C2×C4C22×C22⋊D511yescharacteristic
525C8—21no—
526C8—21no—
527C8—21no—
528C8—41no—
529C8—41no—
530C8—101no—
531C8—101no—
532C8—101no—
533C8—101no—
534C8—201no—
535C8—201no—
536C8—201no—
537C8—201no—
538C8#409712yes—
539C5#1801311yescharacteristic, complemented, Sylow
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×C4×C22⋊D511yescharacteristic
547C4—21no—
548C4—22no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—201no—
555C4—201no—
556C4—201no—
557C4—201no—
558C4C2×#405311yescharacteristic
559C4#6607911yescharacteristic
560C4#6615611yescharacteristic
561C4C2×#409711yescharacteristic
562C2—41no—
563C2—81no—
564C2C4×#409711yescharacteristic
565C2#1061548811yescharacteristic
566C2#1061554311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#2971C2
socleC22×C5C2×C4×D4
prime radicalC2×C5×#41C2×C4
cosocleC42×C5C24
hypercenter#2971D5
hypocenterC5#18013
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2971

Complements

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

NH ≅ G/Ntype
3. #10610239C2semidirect
5. #10601728C2semidirect
6. #10601727C2semidirect
7. #10601725C2semidirect
8. C8×#689C2semidirect
9. #10601659C2semidirect
10. #10601658C2semidirect
11. #10601657C2semidirect
12. #10601653C2semidirect
13. #10601596C2semidirect
14. #10601590C2semidirect
15. #10600641C2semidirect
539. C5#18013semidirect

Maximal subgroups

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

subgroupindex
C5×#29712
#106142652
#106102392
#106024322
#106017282
#106017272
#106017252
C8×#6892
#106016592
#106016582
#106016572
#106016532
#106015962
#106015902
#106006412
#180135