Finite Group Databasebeta

← #27367145

Subgroups — #27367145

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285456C211yesmaximal, characteristic, complemented
2#20284503C211yesmaximal, characteristic
3#20283735C211yesmaximal, characteristic
4#20274288C211yesmaximal, characteristic, complemented
5#20274264C211yesmaximal, characteristic, complemented
6#20274212C211yesmaximal, characteristic, complemented
7#20274202C211yesmaximal, characteristic, complemented
8#20274171C211yesmaximal, characteristic, complemented
9#20274147C211yesmaximal, characteristic, complemented
10#20274095C211yesmaximal, characteristic, complemented
11#20274083C211yesmaximal, characteristic, complemented
12#20273468C211yesmaximal, characteristic, complemented
13#20273423C211yesmaximal, characteristic, complemented
14#20273415C211yesmaximal, characteristic, complemented
15C53×#3107C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811019C2211yescharacteristic
18#16811002C2211yescharacteristic
19#16810975C2211yescharacteristic
20#16810973C2211yescharacteristic
21#16810930—21no—
22#16810923—21no—
23#16810918C2211yescharacteristic
24#16810913C2211yescharacteristic
25#16810908C2211yescharacteristic
26#16810903C2211yescharacteristic
27#16810879—21no—
28#16810879—21no—
29Q8×#88322C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31#16810728—21no—
32#16810728—21no—
33#16810701—21no—
34#16810701—21no—
35#16810701C2211yescharacteristic
36#16810701C2211yescharacteristic
37#16810680—21no—
38#16810680C2211yescharacteristic
39#16810659—21no—
40C2×C4⋊#88322C2211yescharacteristic
41#16810619—21no—
42C4×#11719917—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810369C2211yescharacteristic
51#16810369C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
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×#11719925—21no—
69C2×#11719917—21no—
70C2×#11719917—21no—
71C2×#11719917D411yescharacteristic
72C2×#11719917D411yescharacteristic
73C22×#88322—21no—
74#13133217C2311yescharacteristic
75#13133217C2311yescharacteristic
76#13133209—21no—
77#13133196—21no—
78#13133196—21no—
79C4×Dic106—21no—
80C4×Dic106—21no—
81#13133176—21no—
82#13133150—21no—
83#13133150—21no—
84#13133148—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#13133141—21no—
97#13133141—21no—
98#13133141—21no—
99#13133140—21no—
100#13133140—21no—
101#13133138—21no—
102#13133138—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106#13133128—21no—
107#13133128D411yescharacteristic
108#13133128D411yescharacteristic
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—41no—
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#11719925—42no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192#11719919—41no—
193#11719919—41no—
194C4×Dic53—41no—
195#11719917—22no—
196#11719917—41no—
197#11719917—42no—
198#11719917—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—42no—
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C23×C53—21no—
209C23×C53#12111yescharacteristic
210Q8×C53—21no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53—41no—
214Q8×C53—41no—
215Q8×C53#13712yes—
216D4×C53—21no—
217D4×C53—21no—
218D4×C53—42no—
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—42no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—81no—
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×Dic53—41no—
240C2×Dic53—41no—
241Dic106—41no—
242Dic106—41no—
243Dic106—81no—
244C8×C53—41no—
245C8×C53—41no—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—41no—
250#88322—41no—
251#88322—41no—
252#88322—41no—
253#88322#13512yes—
254#19185—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×#12111yescharacteristic
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—22no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—81no—
270C4×C53—81no—
271C4×C53C2×#13511yescharacteristic
272C4×C53C2×#13711yescharacteristic
273C4×C53#57511yescharacteristic
274C4×C53#57511yescharacteristic
275Dic53—81no—
276Dic53—81no—
277#3185—531no—
278#3107D5311yescharacteristic
279#3076—531no—
280#2971—531no—
281#1703—531no—
282#1699—531no—
283#1691—531no—
284#1689—531no—
285#1684—531no—
286#1680—531no—
287#1672—531no—
288#1670—531no—
289#1565—531no—
290#1554—531no—
291#1552—531no—
292C2×C53—41no—
293C2×C53—81no—
294C2×C53#303511yescharacteristic
295C2×C53#305611yescharacteristic
296C2×C53#306911yescharacteristic
297#533—531no—
298#533C2×D5311yescharacteristic
299C4×#41C2×D5311yescharacteristic
300#498—531no—
301#494—531no—
302#487—531no—
303#487—531no—
304#485—21no—
305#479—1061no—
306#478—1061no—
307#477—531no—
308#476—531no—
309#475—531no—
310#475C2×D5311yescharacteristic
311#474—531no—
312#474C2×D5311yescharacteristic
313#470—1061no—
314#470—1061no—
315#462C2×D5311yescharacteristic
316#461C2×D5311yescharacteristic
317#452—21no—
318C8×Q8—531no—
319C8×Q8—531no—
320#441—1061no—
321#441—1061no—
322C4×Q16—21no—
323C4×SD16—21no—
324#436—531no—
325#436—531no—
326#436—1061no—
327#436—1061no—
328#432—531no—
329#432—1061no—
330#431C2×D5311yescharacteristic
331#427—1061no—
332C2×C4⋊C8—531no—
333#416—1061no—
334C4×M16—531no—
335C4×M16—1061no—
336#335—531no—
337#335—531no—
338#334—531no—
339#334—531no—
340#333—531no—
341#333—531no—
342#332—531no—
343#332—531no—
344#332—531no—
345#332—531no—
346#332—531no—
347#332—531no—
348C53#1918511yescharacteristic, complemented, Sylow
349C2×#41C22⋊D5311yescharacteristic
350C2×Q16—21no—
351C2×Q16—1061no—
352C2×Q16—1061no—
353C2×SD16—21no—
354C2×M16—531no—
355C2×M16—531no—
356C2×M16—1061no—
357C2×M16—1061no—
358C22×C8—1061no—
359C4⋊Q8—531no—
360C4⋊Q8—531no—
361C4⋊Q8C22×D5311yescharacteristic
362C4⋊Q8C22×D5311yescharacteristic
363#125—21no—
364#125—1061no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—1061no—
368C22⋊Q8—1061no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—1061no—
372C4×Q8—1061no—
373C4×Q8C22⋊D5311yescharacteristic
374C4×Q8C22×D5311yescharacteristic
375C4×D4—21no—
376C4×D4—21no—
377C4×D4C22⋊D5311yescharacteristic
378C4×D4C22×D5311yescharacteristic
379#117—21no—
380#117—21no—
381#117—1061no—
382#117C22⋊D5311yescharacteristic
383C2×C42—21no—
384C2×C42C22×D5311yescharacteristic
385#107—21no—
386#107—1061no—
387#107—1061no—
388#106—21no—
389#106—1061no—
390#106—1061no—
391#106—1061no—
392#106—1061no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8C22×D5311yescharacteristic
401C4⋊C8C22×D5311yescharacteristic
402#103—21no—
403#103—21no—
404#103—21no—
405#103—1061no—
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#102—21no—
412#102—1061no—
413#102—1061no—
414C22⋊C8—21no—
415C22⋊C8—1061no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418#97—531no—
419#97—531no—
420#97—1061no—
421C4×C8—21no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—1061no—
425C4×C8—1061no—
426#41—22no—
427#41—41no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×Q8C2×C22⋊D5311yescharacteristic
433C2×D4—21no—
434C2×D4C2×C22⋊D5311yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4D4×D5311yescharacteristic
439C22×C4#1171994711yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441Q16—42no—
442Q16—2122no—
443Q16—2122no—
444SD16—42no—
445M16—1062no—
446M16—2122no—
447M16—2121no—
448M16—2122no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—2122no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—2121no—
473C4⋊C4—2121no—
474C4⋊C4C2×C22⋊D5311yescharacteristic
475C4⋊C4C2×C22⋊D5311yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—41no—
480C22⋊C4—2121no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42—2121no—
486C42D4×D5311yescharacteristic
487C42#1171994711yescharacteristic
488C42C2×C22⋊D5311yescharacteristic
489C42C23×D5311yescharacteristic
490C23—21no—
491C23#1313318911yescharacteristic
492Q8—21no—
493Q8—41no—
494Q8—41no—
495Q8—41no—
496Q8—41no—
497Q8—4241no—
498Q8—2121no—
499Q8—2121no—
500Q8#1313324712yes—
501D4—21no—
502D4—21no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—81no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4#1313318911yescharacteristic
521C2×C4#1313318911yescharacteristic
522C2×C4#1313318911yescharacteristic
523C2×C4C2×D4×D5311yescharacteristic
524C2×C4C22×C22⋊D5311yescharacteristic
525C2×C4C2×#1171994711yescharacteristic
526C8—41no—
527C8—41no—
528C8—532no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×#1313318911yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—81no—
551C4—81no—
552C4—4241no—
553C4—4241no—
554C4#135×D5311yescharacteristic
555C4C2×#1313324711yescharacteristic
556C4#1681142011yescharacteristic
557C4#1681142111yescharacteristic
558C2—41no—
559C2—81no—
560C2#2028416611yescharacteristic
561C2#2028434711yescharacteristic
562C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3107C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenter#3107D53
hypocenterC53#19185
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. #20285456C2semidirect
4. #20274288C2semidirect
5. #20274264C2semidirect
6. #20274212C2semidirect
7. #20274202C2semidirect
8. #20274171C2semidirect
9. #20274147C2semidirect
10. #20274095C2semidirect
11. #20274083C2semidirect
12. #20273468C2semidirect
13. #20273423C2semidirect
14. #20273415C2semidirect
348. C53#19185semidirect

Maximal subgroups

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

subgroupindex
#202854562
#202845032
#202837352
#202742882
#202742642
#202742122
#202742022
#202741712
#202741472
#202740952
#202740832
#202734682
#202734232
#202734152
C53×#31072
#1918553