Finite Group Databasebeta

← #27355035

Subgroups — #27355035

Order 13568.

616 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284727C211yesmaximal, characteristic
2#20283733C211yesmaximal, characteristic, complemented
3#20273605C211yesmaximal, characteristic, complemented
4#20273598C211yesmaximal, characteristic, complemented
5#20273563C211yesmaximal, characteristic, complemented
6#20273558C211yesmaximal, characteristic, complemented
7#20273556C211yesmaximal, characteristic, complemented
8#20273544C211yesmaximal, characteristic, complemented
9C8×#11719925C211yesmaximal, characteristic, complemented
10C8×#11719923C211yesmaximal, characteristic, complemented
11#20273493C211yesmaximal, characteristic, complemented
12#20273175C211yesmaximal, characteristic, complemented
13#20273095C211yesmaximal, characteristic, complemented
14M16×#88322C211yesmaximal, characteristic, complemented
15C53×#2969C211yesmaximal, characteristic, complemented
16C4×#11719945C2211yescharacteristic
17#16810961C411yescharacteristic
18#16810944C411yescharacteristic
19#16810927C411yescharacteristic
20#16810920C411yescharacteristic
21#16810842C411yescharacteristic
22#16810835C411yescharacteristic
23#16810783C411yescharacteristic
24#16810766C411yescharacteristic
25C8×Dic106C2211yescharacteristic
26C8×Dic106C2211yescharacteristic
27C4×#11719925C2211yescharacteristic
28C4×#11719924C2211yescharacteristic
29C4×#11719923C2211yescharacteristic
30C4×#11719922C2211yescharacteristic
31#16810702C2211yescharacteristic
32#16810702C2211yescharacteristic
33#16810699—21no—
34#16810699—21no—
35#16810680C2211yescharacteristic
36#16810678—21no—
37#16810676C2211yescharacteristic
38#16810575—21no—
39C2×C4×#88322C2211yescharacteristic
40#16810375C2211yescharacteristic
41#16810375C2211yescharacteristic
42#16810373C2211yescharacteristic
43#16810373C2211yescharacteristic
44Q8⋊#88322C2211yescharacteristic
45Q8⋊#88322C2211yescharacteristic
46#16810353C2211yescharacteristic
47#16810353C2211yescharacteristic
48D4⋊#88322C2211yescharacteristic
49D4⋊#88322C2211yescharacteristic
50#16810350C2211yescharacteristic
51#16810350C2211yescharacteristic
52#16810344C2211yescharacteristic
53#16810344C2211yescharacteristic
54C8×#88322C2211yescharacteristic
55C8×#88322C2211yescharacteristic
56C53×#529C2211yescharacteristic
57C53×#446C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C53×#436—21no—
60C53×#436C2211yescharacteristic
61C53×#435—21no—
62C8×D4×C53C2211yescharacteristic
63C53×#433—21no—
64C53×#431C2211yescharacteristic
65C53×#405—21no—
66C4×M16×C53C2211yescharacteristic
67C2×#11719945C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719923C2×C411yescharacteristic
70C2×#11719924C2×C411yescharacteristic
71C2×#11719922C2×C411yescharacteristic
72C2×#11719912—21no—
73C2×#11719912—21no—
74C22×#88322—21no—
75C4×Dic106C2×C411yescharacteristic
76C4×Dic106C2311yescharacteristic
77C4×D212C2×C411yescharacteristic
78C4×D212C2311yescharacteristic
79C4×C22⋊D53—21no—
80C4×C22⋊D53—21no—
81#13133176C2×C411yescharacteristic
82#13133174—21no—
83C42×D53—21no—
84#13133150C2×C411yescharacteristic
85#13133150C2×C411yescharacteristic
86#13133148C2×C411yescharacteristic
87#13133148C2×C411yescharacteristic
88C4⋊#88322C2×C411yescharacteristic
89C4⋊#88322C2×C411yescharacteristic
90C4⋊#88322C2311yescharacteristic
91C4⋊#88322C2311yescharacteristic
92#13133146—21no—
93#13133146—21no—
94#13133141C2×C411yescharacteristic
95#13133141C2×C411yescharacteristic
96#13133140C2×C411yescharacteristic
97#13133140C2×C411yescharacteristic
98#13133138C2×C411yescharacteristic
99#13133138C2×C411yescharacteristic
100#13133137C2×C411yescharacteristic
101#13133137C2×C411yescharacteristic
102C22⋊#88322—21no—
103C22⋊#88322C2×C411yescharacteristic
104C22⋊#88322C2×C411yescharacteristic
105#13133129—21no—
106#13133129—21no—
107#13133128—21no—
108C4×#88322—21no—
109C4×#88322C2311yescharacteristic
110C4×#88322C2311yescharacteristic
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C8×Dic53—21no—
114C8×Dic53—21no—
115C2×M16×C53—21no—
116C2×M16×C53D411yescharacteristic
117C2×M16×C53D411yescharacteristic
118C22×C8×C53—21no—
119#126×C53—21no—
120#125×C53C2×C411yescharacteristic
121#124×C53C2×C411yescharacteristic
122#123×C53—21no—
123C22⋊Q8×C53C2×C411yescharacteristic
124#121×C53C2×C411yescharacteristic
125C4×Q8×C53C2311yescharacteristic
126C4×D4×C53—21no—
127C4×D4×C53C2311yescharacteristic
128#117×C53—21no—
129C2×C42×C53C2311yescharacteristic
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53C2311yescharacteristic
133C4⋊C8×C53C2311yescharacteristic
134C22⋊C8×C53—21no—
135C22⋊C8×C53—21no—
136C22⋊C8×C53—21no—
137#97×C53—21no—
138#97×C53D411yescharacteristic
139#97×C53D411yescharacteristic
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53#4111yescharacteristic
151C2×C22⋊D53—21no—
152#11719945—22no—
153#11719945—41no—
154C2×D212C22×C411yescharacteristic
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×Dic106C22×C411yescharacteristic
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—42no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—42no—
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C2×C8×C53C2×D411yescharacteristic
168C2×C8×C53C2×D411yescharacteristic
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C22×C411yescharacteristic
174C4⋊C4×C53C22×C411yescharacteristic
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C42×C53—21no—
181C42×C53C22×C411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53#4111yescharacteristic
184C42×C53C2411yescharacteristic
185C22⋊Dic53—21no—
186#11719925—22no—
187#11719924—41no—
188#11719923—22no—
189#11719922—41no—
190#11719921—21no—
191#11719921—21no—
192C4⋊Dic53C22×C411yescharacteristic
193C4⋊Dic53C22×C411yescharacteristic
194#11719919—21no—
195#11719919—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—42no—
204C2×#88322C22×C411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206C2×#88322C22×C411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208#11719912—42no—
209#11719912—42no—
210C23×C53—21no—
211C23×C53C4×D411yescharacteristic
212C22×D53—21no—
213Q8×C53—21no—
214Q8×C53—21no—
215D4×C53—21no—
216D4×C53—21no—
217D4×C53—81no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C23×C411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C22⋊D53—42no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238D212—21no—
239D212—21no—
240C4×D53—42no—
241C4×D53—41no—
242C4×D53—41no—
243C4×D53—81no—
244Dic106—21no—
245Dic106#13112yes—
246C8×C53—21no—
247C8×C53—21no—
248C8×C53—41no—
249C8×C53—41no—
250C8×C53#13512yes—
251C8×C53#13512yes—
252#88322—22no—
253#88322—21no—
254#88322—21no—
255#88322—21no—
256#88322—21no—
257#88322—41no—
258#88322—41no—
259#88322—41no—
260#17999—531nomaximal, Sylow
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53C2×C4×D411yescharacteristic
268C2×D53—41no—
269C2×D53—41no—
270C2×D53—41no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53C2×#13111yescharacteristic
279C4×C53#56811yescharacteristic
280C4×C53C2×#13511yescharacteristic
281C4×C53C2×#13511yescharacteristic
282Dic53—22no—
283Dic53—41no—
284Dic53—41no—
285#3102—531no—
286#2971—531no—
287#2969D5311yescharacteristic, complemented
288#1590—531no—
289#1589—531no—
290#1584—531no—
291#1583—531no—
292#1583—531no—
293#1581—531no—
294C8×Q16—531no—
295C8×SD16—531no—
296#1572—531no—
297#1494—531no—
298#1476—531no—
299C8×M16—531no—
300C2×C53—41no—
301C2×C53—81no—
302C2×C53#293311yescharacteristic
303C2×C53C4×#13511yescharacteristic
304C2×C53#295611yescharacteristic
305D53—81no—
306#529C2×D5311yescharacteristic
307C4×#41—531no—
308#485—531no—
309#482—531no—
310#479—531no—
311#478—531no—
312#464—531no—
313#463—531no—
314#452—531no—
315#449—531no—
316#446C2×D5311yescharacteristic
317C8×Q8—531no—
318C8×Q8—531no—
319C8×Q8C2×D5311yescharacteristic
320C4×Q16—531no—
321C4×SD16—531no—
322C4×SD16—531no—
323C4×D8—531no—
324#436—21no—
325#436—531no—
326#436—531no—
327#436—1061no—
328#436—1061no—
329#436C2×D5311yescharacteristic
330#435—21no—
331C8×D4C2×D5311yescharacteristic
332#433—21no—
333#432—531no—
334#432—1061no—
335#431—531no—
336#431C2×D5311yescharacteristic
337#405—21no—
338C4×M16—1061no—
339C4×M16C2×D5311yescharacteristic
340C2×C4×C8—531no—
341#335—531no—
342#335—531no—
343#334—531no—
344#334—531no—
345Q8⋊C8—531no—
346Q8⋊C8—531no—
347Q8⋊C8—531no—
348Q8⋊C8—531no—
349D4⋊C8—531no—
350D4⋊C8—531no—
351D4⋊C8—531no—
352D4⋊C8—531no—
353#322—531no—
354#322—531no—
355C82—531no—
356C82—531no—
357C53#1799911yescharacteristic, complemented, Sylow
358C2×#41—531no—
359C2×Q16—531no—
360C2×SD16—531no—
361C2×SD16—531no—
362C2×D8—531no—
363C2×M16—21no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16C22⋊D5311yescharacteristic
367C2×M16C22⋊D5311yescharacteristic
368C22×C8—21no—
369C22×C8—1061no—
370#126—21no—
371#125C4×D5311yescharacteristic
372#124C4×D5311yescharacteristic
373#123—21no—
374C22⋊Q8C4×D5311yescharacteristic
375#121C4×D5311yescharacteristic
376C4×Q8—531no—
377C4×Q8—531no—
378C4×Q8C22×D5311yescharacteristic
379C4×D4—21no—
380C4×D4—531no—
381C4×D4—531no—
382C4×D4—1061no—
383C4×D4—1061no—
384C4×D4C22×D5311yescharacteristic
385#117—21no—
386#117—531no—
387#117—1061no—
388C2×C42—1061no—
389C2×C42C22×D5311yescharacteristic
390#107—531no—
391#107—531no—
392#106—531no—
393#106—531no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8C22×D5311yescharacteristic
403C4⋊C8C22×D5311yescharacteristic
404#103—531no—
405#103—531no—
406#103—531no—
407#103—531no—
408#102—531no—
409#102—531no—
410#102—531no—
411#102—531no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8—531no—
416C22⋊C8—531no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419C22⋊C8—1061no—
420#97—21no—
421#97—1061no—
422#97C22⋊D5311yescharacteristic
423#97C22⋊D5311yescharacteristic
424C4×C8—21no—
425C4×C8—21no—
426C4×C8—531no—
427C4×C8—531no—
428C4×C8—531no—
429C4×C8—531no—
430C4×C8—1061no—
431C4×C8—1061no—
432C4×C8—1061no—
433C4×C8C22×D5311yescharacteristic
434C4×C8C22×D5311yescharacteristic
435#41—1062no—
436#41—2121no—
437C2×Q8—531no—
438C2×Q8C2×C4×D5311yescharacteristic
439C2×D4—21no—
440C2×D4—531no—
441C2×D4—1061no—
442C2×D4C2×C4×D5311yescharacteristic
443C22×C4—21no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4C2×C4×D5311yescharacteristic
448C22×C4#1171994511yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450Q16—1062no—
451SD16—1062no—
452SD16—2121no—
453D8—2121no—
454M16—22no—
455M16—22no—
456M16—42no—
457M16—2122no—
458M16—2122no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—42no—
462C2×C8—531no—
463C2×C8—531no—
464C2×C8—531no—
465C2×C8—531no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—2122no—
471C2×C8C2×C22⋊D5311yescharacteristic
472C2×C8C2×C22⋊D5311yescharacteristic
473C2×C8C2×C22⋊D5311yescharacteristic
474C2×C8C2×C22⋊D5311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—531no—
480C4⋊C4—531no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4C2×C4×D5311yescharacteristic
484C4⋊C4C2×C4×D5311yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—1061no—
491C22⋊C4—1061no—
492C22⋊C4—1061no—
493C42—21no—
494C42—1061no—
495C42—1061no—
496C42—1061no—
497C42C2×C4×D5311yescharacteristic
498C42#1171994511yescharacteristic
499C42C2×C22⋊D5311yescharacteristic
500C42C23×D5311yescharacteristic
501C23—21no—
502C23—1061no—
503C23C4×C22⋊D5311yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—532no—
507Q8—1061no—
508D4—21no—
509D4—21no—
510D4—81no—
511D4—1061no—
512D4—1061no—
513D4—2122no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—4241no—
528C2×C4—2122no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4C4×C22⋊D5311yescharacteristic
532C2×C4C4×C22⋊D5311yescharacteristic
533C2×C4C4×C22⋊D5311yescharacteristic
534C2×C4C22×C4×D5311yescharacteristic
535C2×C4C22×C22⋊D5311yescharacteristic
536C2×C4C2×#1171994511yescharacteristic
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—1062no—
542C8—1061no—
543C8—1061no—
544C8—1061no—
545C8—1061no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C8#1313323712yes—
550C8#1313323712yes—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—2121no—
558C22—2121no—
559C22—2121no—
560C22C2×C4×C22⋊D5311yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—1062no—
569C4—2121no—
570C4—2121no—
571C4#131×D5311yescharacteristic
572C4#1681138811yescharacteristic
573C4C2×#1313323711yescharacteristic
574C4C2×#1313323711yescharacteristic
575C2—41no—
576C2—81no—
577C2—4241no—
578C2#2028343711yescharacteristic
579C2C4×#1313323711yescharacteristic
580C2#2028362211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#2969C2
socleC22×C53C2×C4×D4
prime radical#16810766C4
cosocleC42×C53C24
hypercenter#2969D53
hypocenterC53#17999
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. #20283733C2semidirect
3. #20273605C2semidirect
4. #20273598C2semidirect
5. #20273563C2semidirect
6. #20273558C2semidirect
7. #20273556C2semidirect
8. #20273544C2semidirect
9. C8×#11719925C2semidirect
10. C8×#11719923C2semidirect
11. #20273493C2semidirect
12. #20273175C2semidirect
13. #20273095C2semidirect
14. M16×#88322C2semidirect
15. C53×#2969C2semidirect
287. #2969D53semidirect
357. C53#17999semidirect

Maximal subgroups

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

subgroupindex
#202847272
#202837332
#202736052
#202735982
#202735632
#202735582
#202735562
#202735442
C8×#117199252
C8×#117199232
#202734932
#202731752
#202730952
M16×#883222
C53×#29692
#1799953