Finite Group Databasebeta

← #27369545

Subgroups — #27369545

Order 13568.

628 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285673C211yesmaximal, characteristic, complemented
2#20284866C211yesmaximal, characteristic
3#20284786C211yesmaximal, characteristic, complemented
4#20274159C211yesmaximal, characteristic, complemented
5#20274152C211yesmaximal, characteristic, complemented
6#20274145C211yesmaximal, characteristic, complemented
7#20274138C211yesmaximal, characteristic, complemented
8#20274085C211yesmaximal, characteristic, complemented
9#20274080C211yesmaximal, characteristic, complemented
10#20274063C211yesmaximal, characteristic, complemented
11#20274056C211yesmaximal, characteristic, complemented
12#20273495C211yesmaximal, characteristic, complemented
13#20273413C211yesmaximal, characteristic, complemented
14#20273365C211yesmaximal, characteristic, complemented
15C53×#2988C211yesmaximal, characteristic, complemented
16#16811180C2211yescharacteristic
17#16811173C2211yescharacteristic
18#16811003C2211yescharacteristic
19#16810996C2211yescharacteristic
20#16810974C2211yescharacteristic
21#16810970C2211yescharacteristic
22#16810964—21no—
23#16810947—21no—
24#16810908C2211yescharacteristic
25#16810908C2211yescharacteristic
26#16810903C2211yescharacteristic
27#16810903C2211yescharacteristic
28#16810846—21no—
29#16810846—21no—
30#16810833C2211yescharacteristic
31#16810828C2211yescharacteristic
32#16810823C2211yescharacteristic
33#16810818C2211yescharacteristic
34#16810787—21no—
35#16810787—21no—
36#16810785—21no—
37#16810768—21no—
38#16810733—21no—
39#16810721—21no—
40C4×#11719925—21no—
41C4×#11719923—21no—
42#16810680C2211yescharacteristic
43#16810676C2211yescharacteristic
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810370C2211yescharacteristic
49#16810370C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C4×#41×C53C2211yescharacteristic
57C8×Q8×C53C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C8×D4×C53—21no—
60C8×D4×C53—21no—
61C8×D4×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#431—21no—
64C53×#431C2211yescharacteristic
65C2×C4⋊C8×C53—21no—
66C2×C4⋊C8×C53C2211yescharacteristic
67C2×#11719945—21no—
68C2×Dic212—21no—
69C2×Dic212—21no—
70C2×#11719925—21no—
71C2×#11719913—21no—
72C2×#11719913—21no—
73C2×#11719923—21no—
74C2×#11719924—21no—
75C2×#11719922—21no—
76#13133217C2311yescharacteristic
77#13133217C2311yescharacteristic
78#13133217C2311yescharacteristic
79#13133214C2311yescharacteristic
80#13133209—21no—
81#13133205—21no—
82#13133196—21no—
83#13133196—21no—
84#13133191—21no—
85#13133191—21no—
86C4×Dic106—21no—
87C4×Dic106—21no—
88C4×D212—21no—
89C4×D212—21no—
90#13133176—21no—
91#13133150—21no—
92#13133150—21no—
93#13133148—21no—
94#13133148—21no—
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99#13133142—21no—
100#13133142—21no—
101#13133141—21no—
102#13133141—21no—
103#13133140—21no—
104#13133140—21no—
105#13133139—21no—
106#13133139—21no—
107#13133138—21no—
108#13133138—21no—
109#13133137—21no—
110#13133137—21no—
111C22⋊#88322—21no—
112C22⋊#88322—21no—
113#13133128—21no—
114C4×#88322—21no—
115C2×#41×C53D411yescharacteristic
116C22×C8×C53—21no—
117C22×C8×C53—21no—
118C22×C8×C53—21no—
119C4×Q8×C53D411yescharacteristic
120C4×Q8×C53C2311yescharacteristic
121C4×D4×C53—21no—
122C4×D4×C53—21no—
123C4×D4×C53D411yescharacteristic
124C4×D4×C53C2311yescharacteristic
125#117×C53—21no—
126#117×C53D411yescharacteristic
127C2×C42×C53—21no—
128C2×C42×C53C2311yescharacteristic
129C4⋊C8×C53—21no—
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53D411yescharacteristic
133C4⋊C8×C53D411yescharacteristic
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53D411yescharacteristic
139C22⋊C8×C53D411yescharacteristic
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144#41×C53—22no—
145#41×C53—22no—
146C2×Q8×C53C2×D411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C2×D411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53#4111yescharacteristic
155C2×C22⋊D53—41no—
156#11719945—81no—
157#11719945—81no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×C4×D53—41no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—81no—
168C2×C8×C53—81no—
169C2×C8×C53—81no—
170C2×C8×C53C2×D411yescharacteristic
171C2×C8×C53C2×D411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53C2×D411yescharacteristic
177C4⋊C4×C53C2×D411yescharacteristic
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53C2×D411yescharacteristic
185C42×C53C2×D411yescharacteristic
186C42×C53#4111yescharacteristic
187C42×C53C2411yescharacteristic
188C22⋊Dic53—41no—
189#11719925—42no—
190#11719924—42no—
191#11719923—42no—
192#11719922—42no—
193#11719921—41no—
194#11719921—41no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201#11719919—41no—
202#11719919—41no—
203C4×Dic53—41no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208Dic212—81no—
209Dic212—81no—
210#11719913—81no—
211#11719913—81no—
212C23×C53—21no—
213C23×C53#12111yescharacteristic
214C22×D53—41no—
215Q8×C53—21no—
216Q8×C53#13512yes—
217D4×C53—24no—
218D4×C53—21no—
219D4×C53#13512yes—
220C2×C4×C53—24no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—81no—
232C2×C4×C53#12111yescharacteristic
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C22⋊D53—82no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242D212—41no—
243D212—41no—
244D212—41no—
245D212—41no—
246D212—81no—
247C4×D53—82no—
248Dic106—41no—
249Dic106—41no—
250Dic106—41no—
251Dic106—41no—
252Dic106—41no—
253Dic106—41no—
254C8×C53—21no—
255C8×C53—21no—
256C8×C53—21no—
257C8×C53—21no—
258C8×C53—41no—
259C8×C53—41no—
260#88322—41no—
261#88322—41no—
262#88322—41no—
263#88322—41no—
264#19413—531nomaximal, Sylow
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—22no—
268C22×C53—21no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53C2×#12111yescharacteristic
272C2×D53—81no—
273C2×D53—81no—
274C2×D53—81no—
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—22no—
278C4×C53—21no—
279C4×C53—21no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53C2×#13511yescharacteristic
283C4×C53C2×#13511yescharacteristic
284C4×C53#57611yescharacteristic
285C4×C53#57711yescharacteristic
286Dic53—81no—
287Dic53—81no—
288Dic53—81no—
289#3210—531no—
290#3115—531no—
291#3107—531no—
292#2988D5311yescharacteristic, complemented
293#1682—531no—
294#1681—531no—
295#1680—531no—
296#1679—531no—
297#1670—531no—
298#1669—531no—
299#1666—531no—
300#1665—531no—
301#1572—531no—
302#1552—531no—
303#1540—531no—
304C2×C53—41no—
305C2×C53—42no—
306C2×C53#304511yescharacteristic
307C2×C53#305011yescharacteristic
308C2×C53#306911yescharacteristic
309D53—161no—
310#533—531no—
311#532—531no—
312C4×#41C2×D5311yescharacteristic
313#494—531no—
314#492—531no—
315#487—531no—
316#486—531no—
317#485—1061no—
318#482—1061no—
319#475—531no—
320#475—531no—
321#474—531no—
322#474—531no—
323#464—1061no—
324#464—1061no—
325#462—531no—
326#461—531no—
327#460—531no—
328#459—531no—
329#452—1061no—
330#452—1061no—
331#452—1061no—
332#449—1061no—
333C8×Q8C2×D5311yescharacteristic
334C8×Q8C2×D5311yescharacteristic
335#442—1061no—
336#440—1061no—
337C4×Q16—1061no—
338C4×SD16—1061no—
339C8×D4—21no—
340C8×D4—21no—
341C8×D4C2×D5311yescharacteristic
342C8×D4C2×D5311yescharacteristic
343#432—531no—
344#431—21no—
345#431—531no—
346#431C2×D5311yescharacteristic
347C2×C4⋊C8—21no—
348C2×C4⋊C8C2×D5311yescharacteristic
349#335—531no—
350#335—531no—
351#334—531no—
352#334—531no—
353#332—531no—
354#332—531no—
355#332—531no—
356#332—531no—
357#332—531no—
358#332—531no—
359#331—531no—
360#331—531no—
361C53#1941311yescharacteristic, complemented, Sylow
362C2×#41—1061no—
363C2×#41D21211yescharacteristic
364C2×Q16—1061no—
365C2×Q16—1061no—
366C2×Q16—1061no—
367C2×SD16—1061no—
368C2×SD16—1061no—
369C2×SD16—1061no—
370C2×SD16—1061no—
371C2×D8—1061no—
372C22×C8—21no—
373C22×C8—21no—
374C22×C8—21no—
375C4⋊Q8—531no—
376C4⋊Q8—531no—
377C4⋊Q8—531no—
378#127—531no—
379#125—1061no—
380#124—1061no—
381C22⋊Q8—1061no—
382C22⋊Q8—1061no—
383#121—1061no—
384#121—1061no—
385C4×Q8—1061no—
386C4×Q8—1061no—
387C4×Q8D21211yescharacteristic
388C4×Q8C22×D5311yescharacteristic
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—1061no—
392C4×D4—1061no—
393C4×D4D21211yescharacteristic
394C4×D4C22×D5311yescharacteristic
395#117—21no—
396#117—1061no—
397#117D21211yescharacteristic
398C2×C42—21no—
399C2×C42C22×D5311yescharacteristic
400#107—1061no—
401#107—1061no—
402#106—1061no—
403#106—1061no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406C4⋊C8—21no—
407C4⋊C8—531no—
408C4⋊C8—531no—
409C4⋊C8—531no—
410C4⋊C8—531no—
411C4⋊C8C22⋊D5311yescharacteristic
412C4⋊C8C22⋊D5311yescharacteristic
413C4⋊C8C22×D5311yescharacteristic
414C4⋊C8C22×D5311yescharacteristic
415#103—1061no—
416#103—1061no—
417#103—1061no—
418#103—1061no—
419#103—1061no—
420#103—1061no—
421#102—1061no—
422#102—1061no—
423#102—1061no—
424#102—1061no—
425#102—1061no—
426#102—1061no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—1061no—
430C22⋊C8—1061no—
431C22⋊C8C22⋊D5311yescharacteristic
432C22⋊C8C22⋊D5311yescharacteristic
433#97—1061no—
434C4×C8—21no—
435C4×C8—21no—
436C4×C8—1061no—
437C4×C8C22×D5311yescharacteristic
438C4×C8C22×D5311yescharacteristic
439#41—22no—
440#41—22no—
441#41—4241no—
442#41—4241no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8C2×D21211yescharacteristic
447C2×D4—21no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—2121no—
452C2×D4C2×D21211yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—2121no—
457C22×C4C2×D21211yescharacteristic
458C22×C4#1171994511yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460Q16—4241no—
461Q16—4241no—
462Q16—2122no—
463SD16—4241no—
464SD16—4241no—
465SD16—2122no—
466SD16—2122no—
467D8—2122no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—81no—
471C2×C8—81no—
472C2×C8—81no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8C2×C22⋊D5311yescharacteristic
478C2×C8C2×C22⋊D5311yescharacteristic
479C2×C8C2×C22⋊D5311yescharacteristic
480C2×C8C2×C22⋊D5311yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—2121no—
490C4⋊C4—2121no—
491C4⋊C4C2×D21211yescharacteristic
492C4⋊C4C2×D21211yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—2121no—
497C22⋊C4—2121no—
498C22⋊C4—2121no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—2121no—
503C42C2×D21211yescharacteristic
504C42#1171994511yescharacteristic
505C42C2×C22⋊D5311yescharacteristic
506C42C23×D5311yescharacteristic
507C23—21no—
508C23—2121no—
509C23#1313319111yescharacteristic
510Q8—21no—
511Q8—2121no—
512Q8—2121no—
513Q8—2121no—
514Q8—2121no—
515Q8—2121no—
516Q8—2121no—
517Q8#1313323912yes—
518D4—24no—
519D4—21no—
520D4—4241no—
521D4—4242no—
522D4—2121no—
523D4—2121no—
524D4—2121no—
525D4—2121no—
526D4#1313323912yes—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—81no—
539C2×C4—4242no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4#1313319111yescharacteristic
544C2×C4#1313319111yescharacteristic
545C2×C4#1313319111yescharacteristic
546C2×C4C22×C22⋊D5311yescharacteristic
547C2×C4C22×D21211yescharacteristic
548C2×C4C2×#1171994511yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—21no—
553C8—41no—
554C8—41no—
555C8—2121no—
556C8—2121no—
557C8—2121no—
558C8—2121no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—4241no—
566C22—4241no—
567C22—4241no—
568C22C2×#1313319111yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—4241no—
577C4—4241no—
578C4—4241no—
579C4C2×#1313323911yescharacteristic
580C4C2×#1313323911yescharacteristic
581C4#1681142611yescharacteristic
582C4#1681143111yescharacteristic
583C2—41no—
584C2—42no—
585C2—8481no—
586C2#2028425411yescharacteristic
587C2#2028429611yescharacteristic
588C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2988C2
socleC22×C53C2×#121
prime radical#20284866C2
hypercenter#2988D53
hypocenterC53#19413
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #20285673C2semidirect
3. #20284786C2semidirect
4. #20274159C2semidirect
5. #20274152C2semidirect
6. #20274145C2semidirect
7. #20274138C2semidirect
8. #20274085C2semidirect
9. #20274080C2semidirect
10. #20274063C2semidirect
11. #20274056C2semidirect
12. #20273495C2semidirect
13. #20273413C2semidirect
14. #20273365C2semidirect
15. C53×#2988C2semidirect
292. #2988D53semidirect
361. C53#19413semidirect

Maximal subgroups

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

subgroupindex
#202856732
#202848662
#202847862
#202741592
#202741522
#202741452
#202741382
#202740852
#202740802
#202740632
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C53×#29882
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