Finite Group Databasebeta

← #27370136

Subgroups — #27370136

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285870C211yesmaximal, characteristic, complemented
2#20284723C211yesmaximal, characteristic
3#20283750C211yesmaximal, characteristic
4#20274288C211yesmaximal, characteristic, complemented
5#20274260C211yesmaximal, characteristic, complemented
6#20274236C211yesmaximal, characteristic, complemented
7#20274216C211yesmaximal, characteristic, complemented
8#20274164C211yesmaximal, characteristic, complemented
9#20274136C211yesmaximal, characteristic, complemented
10#20274112C211yesmaximal, characteristic, complemented
11#20274092C211yesmaximal, characteristic, complemented
12#20273506C211yesmaximal, characteristic, complemented
13#20273443C211yesmaximal, characteristic, complemented
14#20273425C211yesmaximal, characteristic, complemented
15C53×#3077C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811026C2211yescharacteristic, complemented
18#16811019C2211yescharacteristic, complemented
19#16810973C2211yescharacteristic, complemented
20#16810973C2211yescharacteristic, complemented
21#16810961—21no—
22#16810961—21no—
23#16810953—21no—
24#16810930—21no—
25#16810930—21no—
26#16810913C2211yescharacteristic
27#16810913C2211yescharacteristic
28#16810828C2211yescharacteristic
29#16810828C2211yescharacteristic
30#16810774—21no—
31C4×#11719923—21no—
32C4×#11719923—21no—
33#16810701C2211yescharacteristic
34#16810701C2211yescharacteristic
35#16810695—21no—
36#16810695—21no—
37D4×#88322C2211yescharacteristic
38D4×#88322C2211yescharacteristic
39#16810680C2211yescharacteristic
40#16810676C2211yescharacteristic
41#16810676C2211yescharacteristic
42#16810587—21no—
43C2×C22⋊#88322—21no—
44#16810375C2211yescharacteristic, complemented
45#16810375C2211yescharacteristic, complemented
46#16810373C2211yescharacteristic, complemented
47#16810373C2211yescharacteristic, complemented
48#16810372C2211yescharacteristic, complemented
49#16810372C2211yescharacteristic, complemented
50#16810371C2211yescharacteristic, complemented
51#16810371C2211yescharacteristic, complemented
52#16810370C2211yescharacteristic, complemented
53#16810370C2211yescharacteristic, complemented
54#16810369C2211yescharacteristic, complemented
55#16810369C2211yescharacteristic, complemented
56C53×#533C2211yescharacteristic
57C2×C4×D4×C53C2211yescharacteristic
58C53×#478—21no—
59C53×#478—21no—
60C53×#476C2211yescharacteristic
61C53×#476C2211yescharacteristic
62C53×#474C2211yescharacteristic
63C53×#474C2211yescharacteristic
64C53×#428—21no—
65C53×#423C2211yescharacteristic
66C53×#418—21no—
67C2×#11719923—21no—
68C2×#11719923—21no—
69C2×#11719917—21no—
70C2×#11719917—21no—
71C22×#88322—21no—
72C22×#88322—21no—
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#13133148—21no—
84#13133148—21no—
85#13133148—21no—
86#13133148—21no—
87C4⋊#88322D411yescharacteristic
88C4⋊#88322D411yescharacteristic
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—
105C22⋊#88322—21no—
106C22⋊#88322—21no—
107C22⋊#88322D411yescharacteristic
108C22⋊#88322D411yescharacteristic
109#13133128—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C22×D4×C53D411yescharacteristic
114C23×C4×C53—21no—
115C2×M16×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×D4×C53—21no—
124C4×D4×C53—21no—
125C4×D4×C53D411yescharacteristic
126C4×D4×C53D411yescharacteristic
127C4×D4×C53C2311yescharacteristic
128C4×D4×C53C2311yescharacteristic
129#117×C53—21no—
130C2×C4⋊C4×C53D411yescharacteristic
131C2×C22⋊C4×C53—21no—
132C2×C42×C53C2311yescharacteristic
133#107×C53—21no—
134#107×C53—21no—
135#106×C53—21no—
136#106×C53—21no—
137C4⋊C8×C53—21no—
138C4⋊C8×C53C2311yescharacteristic
139C4⋊C8×C53C2311yescharacteristic
140#102×C53—21no—
141#102×C53—21no—
142#102×C53—21no—
143#102×C53—21no—
144C24×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×D4×C53—22no—
148C2×D4×C53—21no—
149C2×D4×C53—41no—
150C2×D4×C53C2×D411yescharacteristic
151C2×D4×C53C2×D411yescharacteristic
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—81no—
158C22×C4×C53—81no—
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53#4111yescharacteristic
162C2×Dic106—21no—
163C2×Dic106—21no—
164M16×C53—42no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—41no—
173C4⋊C4×C53—41no—
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—41no—
181C42×C53—41no—
182C42×C53C2×D411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53#4111yescharacteristic
185C42×C53C2411yescharacteristic
186C22⋊Dic53—41no—
187#11719923—42no—
188#11719923—42no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193#11719919—41no—
194#11719919—41no—
195C4×Dic53—41no—
196#11719917—81no—
197#11719917—81no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—81no—
203C2×#88322—81no—
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53—81no—
213C23×C53#12111yescharacteristic
214Q8×C53—41no—
215Q8×C53—41no—
216Q8×C53—81no—
217D4×C53—21no—
218D4×C53—21no—
219D4×C53—21no—
220D4×C53—42no—
221D4×C53#13712yes—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—81no—
236C2×C4×C53—81no—
237C2×C4×C53—81no—
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53#12111yescharacteristic
240C2×C4×C53#12111yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×Dic53—41no—
245C2×Dic53—41no—
246Dic106—41no—
247Dic106—41no—
248Dic106—81no—
249C8×C53—41no—
250C8×C53—41no—
251#88322—21no—
252#88322—21no—
253#88322—21no—
254#88322—21no—
255#88322—41no—
256#88322—41no—
257#88322—41no—
258#88322—41no—
259#19466—531nomaximal, Sylow
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—22no—
263C22×C53—21no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—84no—
270C22×C53C2×#12111yescharacteristic
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53C2×#13711yescharacteristic
280C4×C53#57511yescharacteristic
281C4×C53#57711yescharacteristic
282C4×C53#57711yescharacteristic
283Dic53—81no—
284Dic53—81no—
285#3231—531no—
286#3101—531no—
287#3077D5311yescharacteristic
288#2974—531no—
289#1703—531no—
290#1699—531no—
291#1695—531no—
292#1691—531no—
293#1683—531no—
294#1679—531no—
295#1675—531no—
296#1671—531no—
297#1574—531no—
298#1559—531no—
299#1554—531no—
300C2×C53—41no—
301C2×C53—42no—
302C2×C53—81no—
303C2×C53#304211yescharacteristic
304C2×C53#305611yescharacteristic
305C2×C53#306711yescharacteristic
306#533—531no—
307#533C2×D5311yescharacteristic
308C2×C4×D4C2×D5311yescharacteristic
309#500—531no—
310#498—531no—
311#487—531no—
312#487—531no—
313#485—1061no—
314#485—1061no—
315#483—1061no—
316#479—1061no—
317#479—1061no—
318#478—21no—
319#478—21no—
320#476—531no—
321#476—531no—
322#476C2×D5311yescharacteristic
323#476C2×D5311yescharacteristic
324#474C2×D5311yescharacteristic
325#474C2×D5311yescharacteristic
326#461—531no—
327#461—531no—
328#450—1061no—
329C4×SD16—1061no—
330C4×SD16—1061no—
331#436—531no—
332#436—531no—
333#435—1061no—
334#435—1061no—
335C8×D4—531no—
336C8×D4—531no—
337#432—531no—
338#431—531no—
339#431—531no—
340#428—21no—
341#423C2×D5311yescharacteristic
342#418—21no—
343#407—1061no—
344C2×C22⋊C8—1061no—
345#335—531no—
346#335—531no—
347#334—531no—
348#334—531no—
349#333—531no—
350#333—531no—
351#333—531no—
352#333—531no—
353#332—531no—
354#332—531no—
355#332—531no—
356#332—531no—
357C53#1946611yescharacteristic, complemented, Sylow
358C22×D4C22⋊D5311yescharacteristic
359C23×C4—21no—
360C2×SD16—1061no—
361C2×SD16—1061no—
362C2×M16—21no—
363C2×M16—1061no—
364C2×M16—1061no—
365C22×C8—1061no—
366C22×C8—1061no—
367C4⋊Q8—531no—
368C4⋊Q8—531no—
369C4⋊Q8C22×D5311yescharacteristic
370C4⋊Q8C22×D5311yescharacteristic
371#125—21no—
372#125—1061no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—1061no—
376C22⋊Q8—1061no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—1061no—
380C4×Q8—1061no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4C22⋊D5311yescharacteristic
384C4×D4C22⋊D5311yescharacteristic
385C4×D4C22×D5311yescharacteristic
386C4×D4C22×D5311yescharacteristic
387#117—21no—
388#117—1061no—
389C2×C4⋊C4C22⋊D5311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C42C22×D5311yescharacteristic
392#107—21no—
393#107—21no—
394#107—1061no—
395#107—1061no—
396#106—21no—
397#106—21no—
398#106—1061no—
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—531no—
408C4⋊C8—531no—
409C4⋊C8C22×D5311yescharacteristic
410C4⋊C8C22×D5311yescharacteristic
411#103—1061no—
412#103—1061no—
413#103—1061no—
414#103—1061no—
415#103—1061no—
416#103—1061no—
417#102—21no—
418#102—21no—
419#102—21no—
420#102—21no—
421#102—1061no—
422#102—1061no—
423C22⋊C8—531no—
424C22⋊C8—531no—
425C22⋊C8—1061no—
426C22⋊C8—1061no—
427C22⋊C8—1061no—
428C22⋊C8—1061no—
429C22⋊C8—1061no—
430C22⋊C8—1061no—
431#97—1061no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—1061no—
435C24—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4C2×C22⋊D5311yescharacteristic
444C2×D4C2×C22⋊D5311yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4D4×D5311yescharacteristic
453C22×C4#1171994711yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455SD16—2122no—
456SD16—2122no—
457M16—42no—
458M16—4241no—
459M16—4241no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—531no—
463C2×C8—531no—
464C2×C8—531no—
465C2×C8—531no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—4241no—
471C2×C8—4241no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—2121no—
484C4⋊C4—2121no—
485C4⋊C4C2×C22⋊D5311yescharacteristic
486C4⋊C4C2×C22⋊D5311yescharacteristic
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—41no—
492C22⋊C4—2121no—
493C42—41no—
494C42—2121no—
495C42D4×D5311yescharacteristic
496C42#1171994711yescharacteristic
497C42C2×C22⋊D5311yescharacteristic
498C42C23×D5311yescharacteristic
499C23—21no—
500C23—21no—
501C23—42no—
502C23—42no—
503C23—81no—
504C23#1313318911yescharacteristic
505Q8—41no—
506Q8—41no—
507Q8—81no—
508Q8—4241no—
509Q8—2121no—
510Q8—2121no—
511D4—21no—
512D4—21no—
513D4—21no—
514D4—42no—
515D4#1313324812yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4#1313318911yescharacteristic
535C2×C4#1313318911yescharacteristic
536C2×C4#1313318911yescharacteristic
537C2×C4C2×D4×D5311yescharacteristic
538C2×C4C22×C22⋊D5311yescharacteristic
539C2×C4C2×#1171994711yescharacteristic
540C8—41no—
541C8—41no—
542C8—1061no—
543C8—1061no—
544C8—1061no—
545C8—1061no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C8—2121no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—84no—
560C22C2×#1313318911yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—4241no—
570C4—4241no—
571C4C2×#1313324811yescharacteristic
572C4#1681142111yescharacteristic
573C4#1681143311yescharacteristic
574C4#1681143311yescharacteristic
575C2—41no—
576C2—42no—
577C2—81no—
578C2#2028422911yescharacteristic
579C2#2028434111yescharacteristic
580C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3077C2
socleC22×C53C2×#121
prime radicalC22×D4×C53D4
hypercenter#3077D53
hypocenterC53#19466
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #20285870C2semidirect
4. #20274288C2semidirect
5. #20274260C2semidirect
6. #20274236C2semidirect
7. #20274216C2semidirect
8. #20274164C2semidirect
9. #20274136C2semidirect
10. #20274112C2semidirect
11. #20274092C2semidirect
12. #20273506C2semidirect
13. #20273443C2semidirect
14. #20273425C2semidirect
17. #16811026C22semidirect
18. #16811019C22semidirect
19. #16810973C22semidirect
20. #16810973C22semidirect
44. #16810375C22semidirect
45. #16810375C22semidirect
46. #16810373C22semidirect
47. #16810373C22semidirect
48. #16810372C22semidirect
49. #16810372C22semidirect
50. #16810371C22semidirect
51. #16810371C22semidirect
52. #16810370C22semidirect
53. #16810370C22semidirect
54. #16810369C22semidirect
55. #16810369C22semidirect
357. C53#19466semidirect

Maximal subgroups

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

subgroupindex
#202858702
#202847232
#202837502
#202742882
#202742602
#202742362
#202742162
#202741642
#202741362
#202741122
#202740922
#202735062
#202734432
#202734252
C53×#30772
#1946653