Finite Group Databasebeta

← #27367561

Subgroups — #27367561

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286052C211yesmaximal, characteristic, complemented
2#20286041C211yesmaximal, characteristic
3#20285274C211yesmaximal, characteristic, complemented
4#20274221C211yesmaximal, characteristic, complemented
5#20274214C211yesmaximal, characteristic, complemented
6#20274187C211yesmaximal, characteristic, complemented
7#20274180C211yesmaximal, characteristic, complemented
8#20274173C211yesmaximal, characteristic, complemented
9#20274166C211yesmaximal, characteristic, complemented
10#20274159C211yesmaximal, characteristic, complemented
11#20274152C211yesmaximal, characteristic, complemented
12C2×#16810376C211yesmaximal, characteristic, complemented
13#20273416C211yesmaximal, characteristic, complemented
14#20273411C211yesmaximal, characteristic, complemented
15C53×#2966C211yesmaximal, characteristic, complemented
16#16811180C2211yescharacteristic
17#16811173C2211yescharacteristic
18#16811021C2211yescharacteristic
19#16810996C2211yescharacteristic
20#16810974C2211yescharacteristic
21#16810970C2211yescharacteristic
22#16810927—21no—
23#16810920—21no—
24#16810918C2211yescharacteristic
25#16810913C2211yescharacteristic
26#16810908C2211yescharacteristic
27#16810903C2211yescharacteristic
28#16810862—21no—
29#16810857—21no—
30#16810854—21no—
31#16810852—21no—
32#16810852—21no—
33#16810850—21no—
34#16810842—21no—
35#16810835—21no—
36#16810833C2211yescharacteristic
37#16810828C2211yescharacteristic
38#16810823C2211yescharacteristic
39#16810818C2211yescharacteristic
40C4×#11719913—21no—
41C4×#11719913—21no—
42C2×C4⋊#88322C2211yescharacteristic
43C2×C4⋊#88322C2211yescharacteristic
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810376C2211yescharacteristic
47#16810376C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C53×#529C2211yescharacteristic
57C53×#446C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C53×#436C2211yescharacteristic
60C53×#435—21no—
61C8×D4×C53—21no—
62C8×D4×C53C2211yescharacteristic
63C53×#433—21no—
64C53×#432C2211yescharacteristic
65C53×#405—21no—
66C2×C4×C8×C53C2211yescharacteristic
67C2×#11719945—21no—
68C2×#11719925—21no—
69C2×#11719913—21no—
70C2×#11719913—21no—
71C2×#11719923—21no—
72C2×#11719924—21no—
73C2×#11719922—21no—
74C22×#88322—21no—
75C22×#88322—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#13133149—21no—
94#13133149—21no—
95#13133148—21no—
96#13133148—21no—
97C4⋊#88322—21no—
98C4⋊#88322—21no—
99C4⋊#88322C2311yescharacteristic
100C4⋊#88322C2311yescharacteristic
101C4⋊#88322C2311yescharacteristic
102C4⋊#88322C2311yescharacteristic
103#13133142—21no—
104#13133142—21no—
105#13133141—21no—
106#13133141—21no—
107#13133140—21no—
108#13133140—21no—
109#13133139—21no—
110#13133139—21no—
111#13133138—21no—
112#13133138—21no—
113#13133137—21no—
114#13133137—21no—
115C2×M16×C53—21no—
116C22×C8×C53—21no—
117C22×C8×C53D411yescharacteristic
118C22×C8×C53D411yescharacteristic
119#126×C53—21no—
120#125×C53D411yescharacteristic
121#124×C53D411yescharacteristic
122#123×C53—21no—
123C22⋊Q8×C53D411yescharacteristic
124#121×C53D411yescharacteristic
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×C53—21no—
139C4×C8×C53—21no—
140C4×C8×C53D411yescharacteristic
141C4×C8×C53D411yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C2×D411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C2×D411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53#4111yescharacteristic
151C2×C22⋊D53—41no—
152#11719945—82no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×C4×D53—41no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160M16×C53—42no—
161C2×C8×C53—22no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—41no—
165C2×C8×C53—42no—
166C2×C8×C53C2×D411yescharacteristic
167C2×C8×C53C2×D411yescharacteristic
168C2×C8×C53C2×D411yescharacteristic
169C2×C8×C53C2×D411yescharacteristic
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53C2×D411yescharacteristic
175C4⋊C4×C53C2×D411yescharacteristic
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C42×C53—21no—
182C42×C53C2×D411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53#4111yescharacteristic
185C42×C53C2411yescharacteristic
186C22⋊Dic53—41no—
187#11719925—81no—
188#11719924—81no—
189#11719923—81no—
190#11719922—81no—
191#11719921—41no—
192#11719921—41no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199#11719919—41no—
200#11719919—41no—
201C4×Dic53—41no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—81no—
207C2×#88322—81no—
208#11719913—42no—
209#11719913—42no—
210C23×C53—21no—
211C23×C53#12111yescharacteristic
212C22×D53—41no—
213Q8×C53—21no—
214Q8×C53—21no—
215D4×C53—21no—
216D4×C53—21no—
217D4×C53—42no—
218C2×C4×C53—22no—
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—21no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C22⋊D53—82no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238D212—41no—
239D212—41no—
240D212—41no—
241D212—41no—
242D212—81no—
243C4×D53—82no—
244Dic106—41no—
245Dic106—41no—
246Dic106—41no—
247Dic106—41no—
248Dic106—81no—
249C8×C53—21no—
250C8×C53—21no—
251C8×C53—21no—
252C8×C53—41no—
253C8×C53—41no—
254C8×C53#13512yes—
255#88322—41no—
256#88322—41no—
257#88322—41no—
258#88322—41no—
259#19229—531nomaximal, Sylow
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53#47112yes—
266C22×C53C2×#12111yescharacteristic
267C2×D53—81no—
268C2×D53—81no—
269C2×D53—81no—
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53C2×#13511yescharacteristic
278C4×C53#57511yescharacteristic
279C4×C53#57611yescharacteristic
280C4×C53#57811yescharacteristic
281Dic53—81no—
282Dic53—81no—
283Dic53—81no—
284#3248—531no—
285#3247—531no—
286#3167—531no—
287#2966D5311yescharacteristic, complemented
288#1692—531no—
289#1691—531no—
290#1686—531no—
291#1685—531no—
292#1684—531no—
293#1683—531no—
294#1682—531no—
295#1681—531no—
296C2×#335—531no—
297#1552—531no—
298#1551—531no—
299C2×C53—22no—
300C2×C53—81no—
301C2×C53#303511yescharacteristic
302C2×C53#306111yescharacteristic
303C2×C53C2×#47111yescharacteristic
304D53—161no—
305#533—531no—
306#532—531no—
307#529C2×D5311yescharacteristic
308#498—531no—
309#492—531no—
310#487—531no—
311#486—531no—
312#479—1061no—
313#478—1061no—
314#477—531no—
315#476—531no—
316#475—531no—
317#474—531no—
318#467—1061no—
319#466—1061no—
320#465—1061no—
321#465—1061no—
322#465—1061no—
323#465—1061no—
324#464—1061no—
325#463—1061no—
326#462—531no—
327#461—531no—
328#460—531no—
329#459—531no—
330#446C2×D5311yescharacteristic
331C8×Q8C2×D5311yescharacteristic
332C4×SD16—1061no—
333C4×SD16—1061no—
334#436C2×D5311yescharacteristic
335#435—21no—
336C8×D4—21no—
337C8×D4C2×D5311yescharacteristic
338#433—21no—
339#432C2×D5311yescharacteristic
340C2×C4⋊C8—531no—
341C2×C4⋊C8—531no—
342#405—21no—
343C2×C4×C8C2×D5311yescharacteristic
344#335—531no—
345#335—531no—
346#335—531no—
347#335—531no—
348#333—531no—
349#333—531no—
350#332—531no—
351#332—531no—
352#332—531no—
353#332—531no—
354#331—531no—
355#331—531no—
356C53#1922911yescharacteristic, complemented, Sylow
357C2×#41—1061no—
358C2×Q16—1061no—
359C2×SD16—1061no—
360C2×SD16—1061no—
361C2×SD16—1061no—
362C2×SD16—1061no—
363C2×D8—1061no—
364C2×M16—21no—
365C22×C8—21no—
366C22×C8—1061no—
367C22×C8—1061no—
368C22×C8C22⋊D5311yescharacteristic
369C22×C8C22⋊D5311yescharacteristic
370C4⋊Q8—531no—
371C4⋊Q8—531no—
372C4⋊Q8—531no—
373#127—531no—
374#126—21no—
375#125—1061no—
376#125D21211yescharacteristic
377#124—1061no—
378#124D21211yescharacteristic
379#123—21no—
380C22⋊Q8—1061no—
381C22⋊Q8—1061no—
382C22⋊Q8D21211yescharacteristic
383#121—1061no—
384#121—1061no—
385#121D21211yescharacteristic
386C4×Q8—1061no—
387C4×Q8—1061no—
388C4×Q8C22×D5311yescharacteristic
389C4×D4—21no—
390C4×D4—1061no—
391C4×D4—1061no—
392C4×D4C22×D5311yescharacteristic
393#117—21no—
394#117—1061no—
395C2×C42C22×D5311yescharacteristic
396#107—1061no—
397#107—1061no—
398#106—1061no—
399#106—1061no—
400#106—1061no—
401#106—1061no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8—531no—
405C4⋊C8—531no—
406C4⋊C8—531no—
407C4⋊C8—531no—
408C4⋊C8—1061no—
409C4⋊C8—1061no—
410C4⋊C8C22×D5311yescharacteristic
411C4⋊C8C22×D5311yescharacteristic
412#103—1061no—
413#103—1061no—
414#103—1061no—
415#103—1061no—
416#103—1061no—
417#103—1061no—
418#102—1061no—
419#102—1061no—
420#102—1061no—
421#102—1061no—
422#102—1061no—
423#102—1061no—
424C22⋊C8—21no—
425C22⋊C8—21no—
426C22⋊C8—21no—
427#97—21no—
428#97—21no—
429C4×C8—21no—
430C4×C8C22⋊D5311yescharacteristic
431C4×C8C22⋊D5311yescharacteristic
432C4×C8C22×D5311yescharacteristic
433C4×C8C22×D5311yescharacteristic
434#41—4242no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8C2×D21211yescharacteristic
439C2×D4—21no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—1061no—
443C2×D4—2121no—
444C2×D4C2×D21211yescharacteristic
445C22×C4—21no—
446C22×C4—2121no—
447C22×C4C2×D21211yescharacteristic
448C22×C4#1171994511yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450Q16—4241no—
451SD16—4241no—
452SD16—4241no—
453SD16—2122no—
454SD16—2122no—
455D8—4241no—
456M16—42no—
457C2×C8—22no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—4241no—
467C2×C8—4241no—
468C2×C8C2×C22⋊D5311yescharacteristic
469C2×C8C2×C22⋊D5311yescharacteristic
470C2×C8C2×C22⋊D5311yescharacteristic
471C2×C8C2×C22⋊D5311yescharacteristic
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—2121no—
483C4⋊C4—2121no—
484C4⋊C4C2×D21211yescharacteristic
485C4⋊C4C2×D21211yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—2121no—
492C22⋊C4—2121no—
493C22⋊C4—2121no—
494C42—21no—
495C42—2121no—
496C42C2×D21211yescharacteristic
497C42#1171994511yescharacteristic
498C42C2×C22⋊D5311yescharacteristic
499C42C23×D5311yescharacteristic
500C23—21no—
501C23—2121no—
502C23#1313319111yescharacteristic
503Q8—21no—
504Q8—21no—
505Q8—4241no—
506Q8—2121no—
507Q8—2121no—
508Q8—2121no—
509Q8—2121no—
510D4—21no—
511D4—21no—
512D4—42no—
513D4—4241no—
514D4—4242no—
515D4—2121no—
516D4—2121no—
517D4—2121no—
518D4—2121no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—4242no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4#1313319111yescharacteristic
534C2×C4#1313319111yescharacteristic
535C2×C4#1313319111yescharacteristic
536C2×C4C22×C22⋊D5311yescharacteristic
537C2×C4C22×D21211yescharacteristic
538C2×C4C2×#1171994511yescharacteristic
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—2121no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C8#1313323712yes—
549C22—21no—
550C22—21no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—4241no—
555C22—4241no—
556C22—4241no—
557C22#1681088612yes—
558C22C2×#1313319111yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—4241no—
567C4—4241no—
568C4—4241no—
569C4C2×#1313323711yescharacteristic
570C4#1681142111yescharacteristic
571C4#1681142811yescharacteristic
572C4#1681144011yescharacteristic
573C2—22no—
574C2—81no—
575C2—8481no—
576C2#2028417011yescharacteristic
577C2#2028438511yescharacteristic
578C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2966C2
socleC22×C53C2×#121
prime radical#20286041C2
hypercenter#2966D53
hypocenterC53#19229
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
1. #20286052C2semidirect
3. #20285274C2semidirect
4. #20274221C2semidirect
5. #20274214C2semidirect
6. #20274187C2semidirect
7. #20274180C2semidirect
8. #20274173C2semidirect
9. #20274166C2semidirect
10. #20274159C2semidirect
11. #20274152C2semidirect
12. C2×#16810376C2semidirect
13. #20273416C2semidirect
14. #20273411C2semidirect
15. C53×#2966C2semidirect
287. #2966D53semidirect
356. C53#19229semidirect

Maximal subgroups

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

subgroupindex
#202860522
#202860412
#202852742
#202742212
#202742142
#202741872
#202741802
#202741732
#202741662
#202741592
#202741522
C2×#168103762
#202734162
#202734112
C53×#29662
#1922953