Finite Group Databasebeta

← #27367557

Subgroups — #27367557

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#20285270C211yesmaximal, characteristic, complemented
3#20283691C211yesmaximal, characteristic, complemented
4#20274223C211yesmaximal, characteristic, complemented
5#20274216C211yesmaximal, characteristic, complemented
6#20274185C211yesmaximal, characteristic, complemented
7#20274178C211yesmaximal, characteristic, complemented
8#20274175C211yesmaximal, characteristic, complemented
9#20274168C211yesmaximal, characteristic, complemented
10#20274157C211yesmaximal, characteristic, complemented
11#20274150C211yesmaximal, characteristic, complemented
12C2×#16810375C211yesmaximal, characteristic, complemented
13#20273413C211yesmaximal, characteristic, complemented
14#20273410C211yesmaximal, characteristic, complemented
15C53×#3247C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811019C2211yescharacteristic
18#16810995C2211yescharacteristic
19#16810975C2211yescharacteristic
20#16810969C2211yescharacteristic
21#16810927—21no—
22#16810908C2211yescharacteristic
23#16810903C2211yescharacteristic
24#16810862—21no—
25#16810850—21no—
26#16810850—21no—
27#16810850—21no—
28#16810842—21no—
29#16810833C2211yescharacteristic
30#16810828C2211yescharacteristic
31#16810754C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33C4×#11719923—21no—
34C4×#11719923—21no—
35#16810701C2211yescharacteristic
36#16810695—21no—
37D4×#88322—21no—
38D4×#88322C2211yescharacteristic
39#16810684—21no—
40#16810680C2211yescharacteristic
41C2×C4⋊#88322C2211yescharacteristic
42#16810579—21no—
43C2×C4×#88322C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810375C2211yescharacteristic
47#16810375C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810367C2211yescharacteristic
55#16810367C2211yescharacteristic
56C53×#532C2211yescharacteristic
57C53×#529C2211yescharacteristic
58C53×#478—21no—
59C53×#477C2211yescharacteristic
60C53×#476C2211yescharacteristic
61C53×#466—21no—
62C53×#465—21no—
63C53×#463—21no—
64C53×#460C2211yescharacteristic
65C53×#459C2211yescharacteristic
66C2×C4⋊C8×C53C2211yescharacteristic
67C2×#11719925—21no—
68C2×#11719923—21no—
69C2×#11719923—21no—
70C2×#11719923—21no—
71C2×#11719917—21no—
72C22×#88322—21no—
73C22×#88322—21no—
74C22×#88322D411yescharacteristic
75C22×#88322D411yescharacteristic
76#13133217C2311yescharacteristic
77#13133217C2311yescharacteristic
78#13133209—21no—
79#13133196—21no—
80#13133196—21no—
81C4×Dic106—21no—
82C4×Dic106—21no—
83#13133176—21no—
84#13133150—21no—
85#13133148—21no—
86#13133148—21no—
87#13133148—21no—
88C4⋊#88322—21no—
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133141—21no—
96#13133140—21no—
97#13133140—21no—
98#13133140—21no—
99#13133140—21no—
100#13133138—21no—
101#13133138—21no—
102#13133138—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106#13133128—21no—
107#13133128—21no—
108C4×#88322—21no—
109C4×#88322D411yescharacteristic
110C4×#88322D411yescharacteristic
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×#41×C53—21no—
114C2×SD16×C53—21no—
115C2×D8×C53—21no—
116C22×C8×C53—21no—
117C4⋊Q8×C53C2311yescharacteristic
118#127×C53C2311yescharacteristic
119#126×C53—21no—
120#125×C53D411yescharacteristic
121#124×C53—21no—
122#124×C53D411yescharacteristic
123#123×C53—21no—
124C22⋊Q8×C53D411yescharacteristic
125#121×C53—21no—
126#121×C53—21no—
127#121×C53D411yescharacteristic
128C4×Q8×C53C2311yescharacteristic
129C4×D4×C53—21no—
130C4×D4×C53—21no—
131C4×D4×C53—21no—
132C4×D4×C53C2311yescharacteristic
133#117×C53—21no—
134C2×C42×C53C2311yescharacteristic
135#107×C53—21no—
136#106×C53—21no—
137C4⋊C8×C53—21no—
138C4⋊C8×C53C2311yescharacteristic
139C4⋊C8×C53C2311yescharacteristic
140#103×C53—21no—
141#102×C53—21no—
142#102×C53—21no—
143#102×C53—21no—
144#41×C53—82no—
145C2×Q8×C53—21no—
146C2×Q8×C53C2×D411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—41no—
152C2×D4×C53C2×D411yescharacteristic
153C22×C4×C53—21no—
154C22×C4×C53—41no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53#4111yescharacteristic
158C2×Dic106—21no—
159C2×Dic106—21no—
160SD16×C53—81no—
161D8×C53—81no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—81no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53C2×D411yescharacteristic
172C4⋊C4×C53C2×D411yescharacteristic
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—41no—
179C22⋊C4×C53—41no—
180C42×C53—21no—
181C42×C53C2×D411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53#4111yescharacteristic
184C42×C53C2411yescharacteristic
185C22⋊Dic53—41no—
186#11719925—81no—
187#11719923—42no—
188#11719923—42no—
189#11719923—81no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194#11719919—41no—
195#11719919—41no—
196C4×Dic53—41no—
197#11719917—42no—
198C2×#88322—22no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—41no—
204C2×#88322—42no—
205C2×#88322—81no—
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C2×#88322C2×D411yescharacteristic
209C2×#88322C2×D411yescharacteristic
210C23×C53—21no—
211C23×C53—41no—
212C23×C53#12111yescharacteristic
213Q8×C53—21no—
214Q8×C53—21no—
215Q8×C53—81no—
216D4×C53—21no—
217D4×C53—21no—
218D4×C53—42no—
219D4×C53—41no—
220D4×C53—41no—
221D4×C53—41no—
222D4×C53—41no—
223D4×C53—81no—
224D4×C53—82no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—82no—
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53#12111yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×Dic53—41no—
244C2×Dic53—41no—
245Dic106—41no—
246Dic106—41no—
247Dic106—41no—
248Dic106—41no—
249C8×C53—41no—
250C8×C53—41no—
251#88322—21no—
252#88322—21no—
253#88322—21no—
254#88322—41no—
255#88322—41no—
256#88322—41no—
257#88322—41no—
258#88322#13512yes—
259#19229—531nomaximal, Sylow
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53#47112yes—
269C22×C53C2×#12111yescharacteristic
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—81no—
278C4×C53C2×#13511yescharacteristic
279C4×C53#57511yescharacteristic
280C4×C53#57611yescharacteristic
281C4×C53#57811yescharacteristic
282Dic53—81no—
283Dic53—81no—
284#3248—531no—
285#3247D5311yescharacteristic
286#3167—531no—
287#2966—531no—
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—161no—
302C2×C53#303511yescharacteristic
303C2×C53#306111yescharacteristic
304C2×C53C2×#47111yescharacteristic
305#533—531no—
306#532C2×D5311yescharacteristic
307#529C2×D5311yescharacteristic
308#498—531no—
309#492—531no—
310#487—531no—
311#486—531no—
312#479—1061no—
313#478—21no—
314#477C2×D5311yescharacteristic
315#476C2×D5311yescharacteristic
316#475—531no—
317#474—531no—
318#467—1061no—
319#466—21no—
320#465—21no—
321#465—1061no—
322#465—1061no—
323#465—1061no—
324#464—1061no—
325#463—21no—
326#462—531no—
327#461—531no—
328#460C2×D5311yescharacteristic
329#459C2×D5311yescharacteristic
330#446—531no—
331C8×Q8—531no—
332C4×SD16—1061no—
333C4×SD16—1061no—
334#436—531no—
335#435—1061no—
336C8×D4—531no—
337C8×D4—1061no—
338#433—1061no—
339#432—531no—
340C2×C4⋊C8—531no—
341C2×C4⋊C8C2×D5311yescharacteristic
342#405—1061no—
343C2×C4×C8—531no—
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—21no—
358C2×Q16—1061no—
359C2×SD16—21no—
360C2×SD16—1061no—
361C2×SD16—1061no—
362C2×SD16—1061no—
363C2×D8—21no—
364C2×M16—1061no—
365C22×C8—21no—
366C22×C8—531no—
367C22×C8—531no—
368C22×C8—1061no—
369C22×C8—1061no—
370C4⋊Q8—531no—
371C4⋊Q8—531no—
372C4⋊Q8C22×D5311yescharacteristic
373#127C22×D5311yescharacteristic
374#126—21no—
375#125—1061no—
376#125C22⋊D5311yescharacteristic
377#124—21no—
378#124C22⋊D5311yescharacteristic
379#123—21no—
380C22⋊Q8—1061no—
381C22⋊Q8—1061no—
382C22⋊Q8C22⋊D5311yescharacteristic
383#121—21no—
384#121—21no—
385#121C22⋊D5311yescharacteristic
386C4×Q8—1061no—
387C4×Q8—1061no—
388C4×Q8C22×D5311yescharacteristic
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4C22×D5311yescharacteristic
393#117—21no—
394#117—1061no—
395C2×C42C22×D5311yescharacteristic
396#107—21no—
397#107—1061no—
398#106—21no—
399#106—1061no—
400#106—1061no—
401#106—1061no—
402C4⋊C8—21no—
403C4⋊C8—531no—
404C4⋊C8—531no—
405C4⋊C8—531no—
406C4⋊C8—531no—
407C4⋊C8—1061no—
408C4⋊C8—1061no—
409C4⋊C8—1061no—
410C4⋊C8C22×D5311yescharacteristic
411C4⋊C8C22×D5311yescharacteristic
412#103—21no—
413#103—1061no—
414#103—1061no—
415#103—1061no—
416#103—1061no—
417#103—1061no—
418#102—21no—
419#102—21no—
420#102—21no—
421#102—1061no—
422#102—1061no—
423#102—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426C22⋊C8—1061no—
427#97—1061no—
428#97—1061no—
429C4×C8—531no—
430C4×C8—531no—
431C4×C8—531no—
432C4×C8—531no—
433C4×C8—1061no—
434#41—82no—
435C2×Q8—21no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8C2×C22⋊D5311yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—41no—
444C2×D4C2×C22⋊D5311yescharacteristic
445C22×C4—21no—
446C22×C4—41no—
447C22×C4D4×D5311yescharacteristic
448C22×C4#1171994711yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450Q16—4241no—
451SD16—81no—
452SD16—4241no—
453SD16—2122no—
454SD16—2122no—
455D8—81no—
456M16—2122no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—81no—
460C2×C8—531no—
461C2×C8—531no—
462C2×C8—531no—
463C2×C8—531no—
464C2×C8—1062no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—4241no—
470C2×C8—2121no—
471C2×C8—2122no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—2121no—
483C4⋊C4—2121no—
484C4⋊C4C2×C22⋊D5311yescharacteristic
485C4⋊C4C2×C22⋊D5311yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—2121no—
494C42—21no—
495C42—2121no—
496C42D4×D5311yescharacteristic
497C42#1171994711yescharacteristic
498C42C2×C22⋊D5311yescharacteristic
499C42C23×D5311yescharacteristic
500C23—21no—
501C23—41no—
502C23#1313318911yescharacteristic
503Q8—21no—
504Q8—21no—
505Q8—81no—
506Q8—2121no—
507Q8—2121no—
508Q8—2121no—
509Q8—2121no—
510D4—21no—
511D4—21no—
512D4—42no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—81no—
518D4—82no—
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—41no—
530C2×C4—82no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4#1313318911yescharacteristic
534C2×C4#1313318911yescharacteristic
535C2×C4#1313318911yescharacteristic
536C2×C4C2×D4×D5311yescharacteristic
537C2×C4C22×C22⋊D5311yescharacteristic
538C2×C4C2×#1171994711yescharacteristic
539C8—41no—
540C8—41no—
541C8—532no—
542C8—1061no—
543C8—1061no—
544C8—1061no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C22—21no—
550C22—21no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#1681088412yes—
558C22C2×#1313318911yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—4241no—
568C4—4241no—
569C4#135×D5311yescharacteristic
570C4#1681142011yescharacteristic
571C4#1681142611yescharacteristic
572C4#1681143811yescharacteristic
573C2—22no—
574C2—81no—
575C2—161no—
576C2#2028416611yescharacteristic
577C2#2028438211yescharacteristic
578C2C2×#1681088411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3247C2
socleC22×C53C2×#121
hypercenter#3247D53
hypocenterC53#19229
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3247

Complements

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

NH ≅ G/Ntype
1. #20286052C2semidirect
2. #20285270C2semidirect
3. #20283691C2semidirect
4. #20274223C2semidirect
5. #20274216C2semidirect
6. #20274185C2semidirect
7. #20274178C2semidirect
8. #20274175C2semidirect
9. #20274168C2semidirect
10. #20274157C2semidirect
11. #20274150C2semidirect
12. C2×#16810375C2semidirect
13. #20273413C2semidirect
14. #20273410C2semidirect
356. C53#19229semidirect

Maximal subgroups

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

subgroupindex
#202860522
#202852702
#202836912
#202742232
#202742162
#202741852
#202741782
#202741752
#202741682
#202741572
#202741502
C2×#168103752
#202734132
#202734102
C53×#32472
#1922953