Finite Group Databasebeta

← #27367524

Subgroups — #27367524

Order 13568.

562 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285890C211yesmaximal, characteristic
2#20285297C211yesmaximal, characteristic, complemented
3#20283691C211yesmaximal, characteristic
4#20274241C211yesmaximal, characteristic, complemented
5#20274236C211yesmaximal, characteristic, complemented
6#20274207C211yesmaximal, characteristic, complemented
7#20274202C211yesmaximal, characteristic, complemented
8#20274175C211yesmaximal, characteristic, complemented
9#20274171C211yesmaximal, characteristic, complemented
10#20274168C211yesmaximal, characteristic, complemented
11#20274164C211yesmaximal, characteristic, complemented
12#20273461C211yesmaximal, characteristic, complemented
13#20273425C211yesmaximal, characteristic, complemented
14#20273423C211yesmaximal, characteristic, complemented
15C53×#3233C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811026C2211yescharacteristic
18#16811022—21no—
19#16811002C2211yescharacteristic
20#16810977—21no—
21#16810975C2211yescharacteristic
22#16810973C2211yescharacteristic
23#16810961—21no—
24#16810944—21no—
25#16810918C2211yescharacteristic
26#16810913C2211yescharacteristic
27#16810908C2211yescharacteristic
28#16810903C2211yescharacteristic
29#16810862—21no—
30#16810862—21no—
31#16810754C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33C4×#11719925—21no—
34C4×#11719925—21no—
35#16810701C2211yescharacteristic
36#16810695—21no—
37D4×#88322—21no—
38D4×#88322C2211yescharacteristic
39#16810684—21no—
40#16810680C2211yescharacteristic
41#16810676C2211yescharacteristic
42#16810579—21no—
43C2×C4×#88322C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C53×#529C2211yescharacteristic
58C53×#499—21no—
59C53×#488—21no—
60C53×#485—21no—
61C53×#482—21no—
62C53×#477C2211yescharacteristic
63C53×#476C2211yescharacteristic
64C53×#475C2211yescharacteristic
65C53×#474C2211yescharacteristic
66C53×#431C2211yescharacteristic
67C2×#11719925—21no—
68C2×#11719925—21no—
69C2×#11719917—21no—
70C22×#88322—21no—
71C22×#88322D411yescharacteristic
72C22×#88322D411yescharacteristic
73#13133217C2311yescharacteristic
74#13133217C2311yescharacteristic
75#13133209—21no—
76#13133196—21no—
77#13133196—21no—
78C4×Dic106—21no—
79C4×Dic106—21no—
80#13133176—21no—
81#13133150—21no—
82#13133150—21no—
83#13133150—21no—
84#13133150—21no—
85#13133148—21no—
86#13133148—21no—
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322C2311yescharacteristic
90C4⋊#88322C2311yescharacteristic
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93#13133141—21no—
94#13133141—21no—
95#13133141—21no—
96#13133141—21no—
97#13133140—21no—
98#13133140—21no—
99#13133138—21no—
100#13133138—21no—
101C22⋊#88322—21no—
102C22⋊#88322—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105#13133128—21no—
106#13133128—21no—
107C4×#88322—21no—
108C4×#88322—21no—
109C4×#88322D411yescharacteristic
110C4×#88322D411yescharacteristic
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C4⋊Q8×C53C2311yescharacteristic
114C4⋊Q8×C53C2311yescharacteristic
115#126×C53—21no—
116#125×C53—21no—
117#125×C53D411yescharacteristic
118#124×C53D411yescharacteristic
119#123×C53—21no—
120C22⋊Q8×C53—21no—
121C22⋊Q8×C53—21no—
122C22⋊Q8×C53D411yescharacteristic
123#121×C53D411yescharacteristic
124C4×Q8×C53—21no—
125C4×Q8×C53—21no—
126C4×Q8×C53C2311yescharacteristic
127C4×D4×C53—21no—
128C4×D4×C53C2311yescharacteristic
129#117×C53—21no—
130#117×C53—21no—
131C2×C42×C53C2311yescharacteristic
132#107×C53—21no—
133#107×C53—21no—
134#106×C53—21no—
135#106×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53—21no—
139#103×C53—21no—
140#102×C53—21no—
141#102×C53—21no—
142C22⋊C8×C53—21no—
143C4×C8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53C2×D411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C2×D411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53#4111yescharacteristic
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—41no—
166C4⋊C4×C53—41no—
167C4⋊C4×C53C2×D411yescharacteristic
168C4⋊C4×C53C2×D411yescharacteristic
169C22⋊C4×C53—21no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—41no—
175C42×C53—21no—
176C42×C53—41no—
177C42×C53C2×D411yescharacteristic
178C42×C53C2×D411yescharacteristic
179C42×C53#4111yescharacteristic
180C42×C53C2411yescharacteristic
181C22⋊Dic53—41no—
182#11719925—42no—
183#11719925—42no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188#11719919—41no—
189#11719919—41no—
190C4×Dic53—41no—
191#11719917—42no—
192C2×#88322—22no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—41no—
198C2×#88322—42no—
199C2×#88322C2×D411yescharacteristic
200C2×#88322C2×D411yescharacteristic
201C2×#88322C2×D411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C23×C53—21no—
204C23×C53#12111yescharacteristic
205Q8×C53—21no—
206Q8×C53—21no—
207Q8×C53—41no—
208Q8×C53—41no—
209Q8×C53—81no—
210D4×C53—21no—
211D4×C53—21no—
212D4×C53—42no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×Dic53—41no—
232C2×Dic53—41no—
233Dic106—41no—
234Dic106—41no—
235Dic106—81no—
236C8×C53—41no—
237C8×C53—41no—
238#88322—21no—
239#88322—21no—
240#88322—21no—
241#88322—41no—
242#88322—41no—
243#88322—41no—
244#88322—41no—
245#88322#13512yes—
246#19226—531nomaximal, Sylow
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53C2×#12111yescharacteristic
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—21no—
257C4×C53—21no—
258C4×C53—41no—
259C4×C53—41no—
260C4×C53—41no—
261C4×C53—81no—
262C4×C53—81no—
263C4×C53C2×#13511yescharacteristic
264C4×C53#57511yescharacteristic
265C4×C53#57811yescharacteristic
266C4×C53#57811yescharacteristic
267Dic53—81no—
268Dic53—81no—
269#3233—531no—
270#3233D5311yescharacteristic
271#3170—531no—
272#2966—531no—
273#1696—531no—
274#1695—531no—
275#1690—531no—
276#1689—531no—
277#1684—531no—
278#1684—531no—
279#1683—531no—
280#1683—531no—
281#1564—531no—
282#1554—531no—
283#1554—531no—
284C2×C53—41no—
285C2×C53—81no—
286C2×C53#303511yescharacteristic
287C2×C53#305911yescharacteristic
288C2×C53#306711yescharacteristic
289#533—531no—
290#533C2×D5311yescharacteristic
291#529C2×D5311yescharacteristic
292#500—531no—
293#499—21no—
294#499—1061no—
295#494—531no—
296#488—21no—
297#488—1061no—
298#487—531no—
299#487—531no—
300#485—21no—
301#485—1061no—
302#482—21no—
303#482—1061no—
304#477—531no—
305#477C2×D5311yescharacteristic
306#476—531no—
307#476C2×D5311yescharacteristic
308#475—531no—
309#475C2×D5311yescharacteristic
310#474—531no—
311#474C2×D5311yescharacteristic
312#467—1061no—
313#467—1061no—
314#446—531no—
315C8×Q8—531no—
316C4×Q16—1061no—
317C4×Q16—1061no—
318#436—531no—
319#435—1061no—
320C8×D4—531no—
321C8×D4—1061no—
322#433—1061no—
323#432—531no—
324#431—531no—
325#431C2×D5311yescharacteristic
326#405—1061no—
327C2×C4×C8—531no—
328#335—531no—
329#335—531no—
330#334—531no—
331#334—531no—
332#333—531no—
333#333—531no—
334#333—531no—
335#333—531no—
336#332—531no—
337#332—531no—
338#332—531no—
339#332—531no—
340C53#1922611yescharacteristic, complemented, Sylow
341C2×Q16—1061no—
342C2×Q16—1061no—
343C2×M16—1061no—
344C22×C8—531no—
345C22×C8—531no—
346C22×C8—1061no—
347C4⋊Q8—531no—
348C4⋊Q8—531no—
349C4⋊Q8C22×D5311yescharacteristic
350C4⋊Q8C22×D5311yescharacteristic
351#126—21no—
352#125—21no—
353#125—1061no—
354#125C22⋊D5311yescharacteristic
355#124C22⋊D5311yescharacteristic
356#123—21no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—1061no—
360C22⋊Q8—1061no—
361C22⋊Q8C22⋊D5311yescharacteristic
362#121C22⋊D5311yescharacteristic
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—1061no—
366C4×Q8—1061no—
367C4×Q8C22×D5311yescharacteristic
368C4×D4—21no—
369C4×D4C22×D5311yescharacteristic
370#117—21no—
371#117—21no—
372#117—1061no—
373C2×C42C22×D5311yescharacteristic
374#107—21no—
375#107—21no—
376#107—1061no—
377#107—1061no—
378#107—1061no—
379#107—1061no—
380#106—21no—
381#106—21no—
382#106—1061no—
383#106—1061no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—1061no—
389C4⋊C8—1061no—
390C4⋊C8C22×D5311yescharacteristic
391C4⋊C8C22×D5311yescharacteristic
392#103—21no—
393#103—21no—
394#103—1061no—
395#103—1061no—
396#103—1061no—
397#103—1061no—
398#103—1061no—
399#103—1061no—
400#102—21no—
401#102—21no—
402#102—1061no—
403#102—1061no—
404C22⋊C8—21no—
405C22⋊C8—1061no—
406C22⋊C8—1061no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409#97—1061no—
410#97—1061no—
411C4×C8—21no—
412C4×C8—531no—
413C4×C8—531no—
414C4×C8—531no—
415C4×C8—531no—
416C4×C8—1061no—
417C4×C8—1061no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×Q8C2×C22⋊D5311yescharacteristic
423C2×D4—21no—
424C2×D4C2×C22⋊D5311yescharacteristic
425C22×C4—21no—
426C22×C4D4×D5311yescharacteristic
427C22×C4#1171994711yescharacteristic
428C22×C4C2×C22⋊D5311yescharacteristic
429Q16—2122no—
430Q16—2122no—
431M16—2122no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—531no—
435C2×C8—531no—
436C2×C8—531no—
437C2×C8—531no—
438C2×C8—1062no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—2121no—
444C2×C8—2122no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—2121no—
460C4⋊C4—2121no—
461C4⋊C4C2×C22⋊D5311yescharacteristic
462C4⋊C4C2×C22⋊D5311yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—41no—
469C22⋊C4—2121no—
470C42—21no—
471C42—41no—
472C42—2121no—
473C42D4×D5311yescharacteristic
474C42#1171994711yescharacteristic
475C42C2×C22⋊D5311yescharacteristic
476C42C23×D5311yescharacteristic
477C23—21no—
478C23#1313318911yescharacteristic
479Q8—21no—
480Q8—21no—
481Q8—41no—
482Q8—41no—
483Q8—81no—
484Q8—4241no—
485Q8—2121no—
486Q8—2121no—
487D4—21no—
488D4—21no—
489D4—42no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—2121no—
503C2×C4—2121no—
504C2×C4#1313318911yescharacteristic
505C2×C4#1313318911yescharacteristic
506C2×C4#1313318911yescharacteristic
507C2×C4C2×D4×D5311yescharacteristic
508C2×C4C22×C22⋊D5311yescharacteristic
509C2×C4C2×#1171994711yescharacteristic
510C8—41no—
511C8—41no—
512C8—532no—
513C8—1061no—
514C8—1061no—
515C8—1061no—
516C8—2121no—
517C8—2121no—
518C8—2121no—
519C8—2121no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22C2×#1313318911yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—81no—
535C4—81no—
536C4—4241no—
537C4—4241no—
538C4#135×D5311yescharacteristic
539C4#1681142011yescharacteristic
540C4#1681143811yescharacteristic
541C4#1681143811yescharacteristic
542C2—41no—
543C2—81no—
544C2#2028416611yescharacteristic
545C2#2028436711yescharacteristic
546C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3233C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#3233D53
hypocenterC53#19226
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3233

Complements

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

NH ≅ G/Ntype
2. #20285297C2semidirect
4. #20274241C2semidirect
5. #20274236C2semidirect
6. #20274207C2semidirect
7. #20274202C2semidirect
8. #20274175C2semidirect
9. #20274171C2semidirect
10. #20274168C2semidirect
11. #20274164C2semidirect
12. #20273461C2semidirect
13. #20273425C2semidirect
14. #20273423C2semidirect
340. C53#19226semidirect

Maximal subgroups

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

subgroupindex
#202858902
#202852972
#202836912
#202742412
#202742362
#202742072
#202742022
#202741752
#202741712
#202741682
#202741642
#202734612
#202734252
#202734232
C53×#32332
#1922653