Finite Group Databasebeta

← #27369600

Subgroups — #27369600

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285870C211yesmaximal, characteristic, complemented
2#20284877C211yesmaximal, characteristic
3#20283826C211yesmaximal, characteristic
4#20274171C211yesmaximal, characteristic, complemented
5#20274164C211yesmaximal, characteristic, complemented
6#20274143C211yesmaximal, characteristic, complemented
7#20274136C211yesmaximal, characteristic, complemented
8#20274107C211yesmaximal, characteristic, complemented
9#20274102C211yesmaximal, characteristic, complemented
10#20274095C211yesmaximal, characteristic, complemented
11#20274088C211yesmaximal, characteristic, complemented
12#20273493C211yesmaximal, characteristic, complemented
13#20273415C211yesmaximal, characteristic, complemented
14#20273370C211yesmaximal, characteristic, complemented
15C53×#3107C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811026C2211yescharacteristic
18#16811019C2211yescharacteristic
19#16810973C2211yescharacteristic
20#16810973C2211yescharacteristic
21#16810964—21no—
22#16810961—21no—
23#16810961—21no—
24#16810930—21no—
25#16810930—21no—
26#16810918C2211yescharacteristic
27#16810913C2211yescharacteristic
28#16810833C2211yescharacteristic
29#16810828C2211yescharacteristic
30#16810785—21no—
31Q8×#88322C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33#16810728—21no—
34#16810723—21no—
35D4×#88322—21no—
36D4×#88322—21no—
37D4×#88322C2211yescharacteristic
38D4×#88322C2211yescharacteristic
39#16810680C2211yescharacteristic
40#16810676—21no—
41#16810676C2211yescharacteristic
42C2×C4⋊#88322—21no—
43C2×C4⋊#88322C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810369C2211yescharacteristic
55#16810369C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C4×#41×C53C2211yescharacteristic
58C53×#485—21no—
59C53×#475C2211yescharacteristic
60C53×#474C2211yescharacteristic
61C53×#462C2211yescharacteristic
62C53×#461C2211yescharacteristic
63C53×#452—21no—
64C4×Q16×C53—21no—
65C4×SD16×C53—21no—
66C53×#431C2211yescharacteristic
67C2×#11719925—21no—
68C2×#11719923—21no—
69C22×#88322—21no—
70C22×#88322—21no—
71C22×#88322—21no—
72#13133217C2311yescharacteristic
73#13133217C2311yescharacteristic
74#13133209—21no—
75#13133196—21no—
76#13133196—21no—
77C4×Dic106—21no—
78C4×Dic106—21no—
79#13133176—21no—
80#13133150—21no—
81#13133150—21no—
82#13133150—21no—
83#13133148—21no—
84#13133148—21no—
85#13133148—21no—
86C4⋊#88322—21no—
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322D411yescharacteristic
90C4⋊#88322D411yescharacteristic
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133141—21no—
96#13133141—21no—
97#13133141—21no—
98#13133141—21no—
99#13133141—21no—
100#13133140—21no—
101#13133140—21no—
102#13133138—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106C22⋊#88322D411yescharacteristic
107C22⋊#88322D411yescharacteristic
108#13133128—21no—
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×#41×C53D411yescharacteristic
114C2×Q16×C53—21no—
115C2×SD16×C53—21no—
116C4⋊Q8×C53C2311yescharacteristic
117C4⋊Q8×C53C2311yescharacteristic
118#125×C53—21no—
119C22⋊Q8×C53—21no—
120C22⋊Q8×C53—21no—
121C4×Q8×C53—21no—
122C4×Q8×C53—21no—
123C4×Q8×C53D411yescharacteristic
124C4×Q8×C53C2311yescharacteristic
125C4×D4×C53—21no—
126C4×D4×C53—21no—
127C4×D4×C53D411yescharacteristic
128C4×D4×C53C2311yescharacteristic
129#117×C53—21no—
130#117×C53—21no—
131#117×C53D411yescharacteristic
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—21no—
135#106×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53—21no—
139#103×C53—21no—
140#103×C53—21no—
141#102×C53—21no—
142C22⋊C8×C53—21no—
143C4×C8×C53—21no—
144#41×C53—22no—
145#41×C53—22no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53C2×D411yescharacteristic
149C2×D4×C53—21no—
150C2×D4×C53C2×D411yescharacteristic
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53#4111yescharacteristic
157C2×Dic106—21no—
158C2×Dic106—21no—
159Q16×C53—42no—
160SD16×C53—42no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—41no—
170C4⋊C4×C53—41no—
171C4⋊C4×C53C2×D411yescharacteristic
172C4⋊C4×C53C2×D411yescharacteristic
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—41no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—41no—
181C42×C53C2×D411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53#4111yescharacteristic
184C42×C53C2411yescharacteristic
185C22⋊Dic53—41no—
186#11719925—42no—
187#11719923—42no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192#11719919—41no—
193#11719919—41no—
194C4×Dic53—41no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—81no—
200C2×#88322—81no—
201C2×#88322—81no—
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C23×C53—21no—
207C23×C53#12111yescharacteristic
208Q8×C53—21no—
209Q8×C53—41no—
210Q8×C53—41no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53#13512yes—
214D4×C53—24no—
215D4×C53—21no—
216D4×C53#13512yes—
217C2×C4×C53—24no—
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×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—81no—
231C2×C4×C53#12111yescharacteristic
232C2×C4×C53#12111yescharacteristic
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×Dic53—41no—
238C2×Dic53—41no—
239Dic106—41no—
240Dic106—41no—
241Dic106—81no—
242C8×C53—41no—
243C8×C53—41no—
244#88322—21no—
245#88322—21no—
246#88322—21no—
247#88322—21no—
248#88322—41no—
249#88322—41no—
250#88322—41no—
251#88322—41no—
252#19418—531nomaximal, Sylow
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—22no—
256C22×C53—21no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53C2×#12111yescharacteristic
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—22no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53—81no—
268C4×C53—81no—
269C4×C53C2×#13511yescharacteristic
270C4×C53C2×#13511yescharacteristic
271C4×C53#57711yescharacteristic
272C4×C53#57811yescharacteristic
273Dic53—81no—
274Dic53—81no—
275#3231—531no—
276#3116—531no—
277#3107D5311yescharacteristic
278#2988—531no—
279#1684—531no—
280#1683—531no—
281#1680—531no—
282#1679—531no—
283#1674—531no—
284#1673—531no—
285#1672—531no—
286#1671—531no—
287#1572—531no—
288#1552—531no—
289#1542—531no—
290C2×C53—41no—
291C2×C53—42no—
292C2×C53#304611yescharacteristic
293C2×C53#305011yescharacteristic
294C2×C53#306911yescharacteristic
295#533—531no—
296#533C2×D5311yescharacteristic
297C4×#41C2×D5311yescharacteristic
298#500—531no—
299#498—531no—
300#487—531no—
301#487—531no—
302#485—21no—
303#485—1061no—
304#485—1061no—
305#485—1061no—
306#479—1061no—
307#479—1061no—
308#477—531no—
309#476—531no—
310#475C2×D5311yescharacteristic
311#474C2×D5311yescharacteristic
312#462—531no—
313#462C2×D5311yescharacteristic
314#461—531no—
315#461C2×D5311yescharacteristic
316#452—21no—
317#452—1061no—
318C8×Q8—531no—
319C8×Q8—531no—
320#441—1061no—
321#440—1061no—
322C4×Q16—21no—
323C4×SD16—21no—
324C8×D4—531no—
325C8×D4—531no—
326C8×D4—1061no—
327C8×D4—1061no—
328#432—531no—
329#431—531no—
330#431—1061no—
331#431C2×D5311yescharacteristic
332C2×C4⋊C8—531no—
333C2×C4⋊C8—1061no—
334#335—531no—
335#335—531no—
336#334—531no—
337#334—531no—
338#333—531no—
339#333—531no—
340#332—531no—
341#332—531no—
342#332—531no—
343#332—531no—
344#332—531no—
345#332—531no—
346C53#1941811yescharacteristic, complemented, Sylow
347C2×#41C22⋊D5311yescharacteristic
348C2×Q16—21no—
349C2×Q16—1061no—
350C2×SD16—21no—
351C2×SD16—1061no—
352C22×C8—1061no—
353C22×C8—1061no—
354C22×C8—1061no—
355C4⋊Q8—531no—
356C4⋊Q8—531no—
357C4⋊Q8C22×D5311yescharacteristic
358C4⋊Q8C22×D5311yescharacteristic
359#125—21no—
360#125—1061no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—1061no—
364C22⋊Q8—1061no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—1061no—
368C4×Q8—1061no—
369C4×Q8C22⋊D5311yescharacteristic
370C4×Q8C22×D5311yescharacteristic
371C4×D4—21no—
372C4×D4—21no—
373C4×D4C22⋊D5311yescharacteristic
374C4×D4C22×D5311yescharacteristic
375#117—21no—
376#117—21no—
377#117—1061no—
378#117C22⋊D5311yescharacteristic
379C2×C42—21no—
380C2×C42C22×D5311yescharacteristic
381#107—21no—
382#107—1061no—
383#107—1061no—
384#107—1061no—
385#106—21no—
386#106—1061no—
387#106—1061no—
388#106—1061no—
389C4⋊C8—531no—
390C4⋊C8—531no—
391C4⋊C8—531no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398C4⋊C8C22×D5311yescharacteristic
399C4⋊C8C22×D5311yescharacteristic
400#103—21no—
401#103—21no—
402#103—21no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410#102—21no—
411#102—1061no—
412C22⋊C8—21no—
413C22⋊C8—531no—
414C22⋊C8—531no—
415C22⋊C8—1061no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418#97—1061no—
419C4×C8—21no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—1061no—
423C4×C8—1061no—
424#41—22no—
425#41—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8C2×C22⋊D5311yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D5311yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4D4×D5311yescharacteristic
437C22×C4#1171994711yescharacteristic
438C22×C4C2×C22⋊D5311yescharacteristic
439Q16—42no—
440Q16—2122no—
441SD16—42no—
442SD16—2122no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—4241no—
454C2×C8—4241no—
455C2×C8—4241no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—2121no—
469C4⋊C4—2121no—
470C4⋊C4C2×C22⋊D5311yescharacteristic
471C4⋊C4C2×C22⋊D5311yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—41no—
476C22⋊C4—2121no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—41no—
481C42—2121no—
482C42D4×D5311yescharacteristic
483C42#1171994711yescharacteristic
484C42C2×C22⋊D5311yescharacteristic
485C42C23×D5311yescharacteristic
486C23—21no—
487C23#1313318911yescharacteristic
488Q8—21no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—4241no—
494Q8—2121no—
495Q8—2121no—
496Q8#1313323712yes—
497D4—24no—
498D4—21no—
499D4#1313323712yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—81no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4#1313318911yescharacteristic
517C2×C4#1313318911yescharacteristic
518C2×C4#1313318911yescharacteristic
519C2×C4C2×D4×D5311yescharacteristic
520C2×C4C22×C22⋊D5311yescharacteristic
521C2×C4C2×#1171994711yescharacteristic
522C8—41no—
523C8—41no—
524C8—1061no—
525C8—1061no—
526C8—1061no—
527C8—1061no—
528C8—2121no—
529C8—2121no—
530C8—2121no—
531C8—2121no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—41no—
537C22—41no—
538C22C2×#1313318911yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—4241no—
549C4—4241no—
550C4C2×#1313323711yescharacteristic
551C4C2×#1313323711yescharacteristic
552C4#1681143011yescharacteristic
553C4#1681143711yescharacteristic
554C2—41no—
555C2—42no—
556C2#2028426311yescharacteristic
557C2#2028429311yescharacteristic
558C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3107C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenter#3107D53
hypocenterC53#19418
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
1. #20285870C2semidirect
4. #20274171C2semidirect
5. #20274164C2semidirect
6. #20274143C2semidirect
7. #20274136C2semidirect
8. #20274107C2semidirect
9. #20274102C2semidirect
10. #20274095C2semidirect
11. #20274088C2semidirect
12. #20273493C2semidirect
13. #20273415C2semidirect
14. #20273370C2semidirect
346. C53#19418semidirect

Maximal subgroups

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

subgroupindex
#202858702
#202848772
#202838262
#202741712
#202741642
#202741432
#202741362
#202741072
#202741022
#202740952
#202740882
#202734932
#202734152
#202733702
C53×#31072
#1941853