Finite Group Databasebeta

← #27370132

Subgroups — #27370132

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#20285876C211yesmaximal, characteristic, complemented
2#20284723C211yesmaximal, characteristic
3#20284512C211yesmaximal, characteristic
4#20274290C211yesmaximal, characteristic, complemented
5#20274262C211yesmaximal, characteristic, complemented
6#20274238C211yesmaximal, characteristic, complemented
7#20274214C211yesmaximal, characteristic, complemented
8#20274166C211yesmaximal, characteristic, complemented
9#20274138C211yesmaximal, characteristic, complemented
10#20274114C211yesmaximal, characteristic, complemented
11#20274090C211yesmaximal, characteristic, complemented
12#20273504C211yesmaximal, characteristic, complemented
13#20273446C211yesmaximal, characteristic, complemented
14#20273426C211yesmaximal, characteristic, complemented
15C53×#2974C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811180C2211yescharacteristic
18#16811028C2211yescharacteristic, complemented
19#16811021C2211yescharacteristic, complemented
20#16810974C2211yescharacteristic, complemented
21#16810974C2211yescharacteristic, complemented
22#16810965—21no—
23#16810965—21no—
24#16810953—21no—
25#16810931—21no—
26#16810931—21no—
27#16810923—21no—
28#16810923—21no—
29#16810913C2211yescharacteristic
30#16810913C2211yescharacteristic
31#16810913C2211yescharacteristic
32#16810913C2211yescharacteristic
33#16810903C2211yescharacteristic
34#16810903C2211yescharacteristic
35#16810828C2211yescharacteristic
36#16810828C2211yescharacteristic
37#16810774—21no—
38C4×#11719923—21no—
39C4×#11719923—21no—
40#16810676C2211yescharacteristic
41#16810663—21no—
42#16810647C2211yescharacteristic
43#16810628—21no—
44#16810376C2211yescharacteristic, complemented
45#16810376C2211yescharacteristic, complemented
46#16810374C2211yescharacteristic, complemented
47#16810374C2211yescharacteristic, complemented
48#16810371C2211yescharacteristic, complemented
49#16810371C2211yescharacteristic, complemented
50#16810371C2211yescharacteristic, complemented
51#16810371C2211yescharacteristic, complemented
52#16810370C2211yescharacteristic, complemented
53#16810370C2211yescharacteristic, complemented
54#16810368C2211yescharacteristic, complemented
55#16810368C2211yescharacteristic, complemented
56C2×C4×D4×C53C2211yescharacteristic
57C53×#436C2211yescharacteristic
58C53×#436C2211yescharacteristic
59C53×#435—21no—
60C53×#435—21no—
61C8×D4×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#432C2211yescharacteristic
64C53×#431C2211yescharacteristic
65C53×#407—21no—
66C2×C22⋊C8×C53—21no—
67C2×#11719923—21no—
68C2×#11719923—21no—
69C2×#11719917—21no—
70#13133217C2311yescharacteristic
71#13133217C2311yescharacteristic
72#13133217C2311yescharacteristic
73#13133217C2311yescharacteristic
74#13133209—21no—
75#13133209—21no—
76#13133196—21no—
77#13133196—21no—
78#13133196—21no—
79#13133196—21no—
80C4×Dic106—21no—
81C4×Dic106—21no—
82C4×Dic106—21no—
83C4×Dic106—21no—
84#13133176—21no—
85#13133176—21no—
86#13133151—21no—
87#13133151—21no—
88#13133150—21no—
89#13133150—21no—
90#13133149—21no—
91#13133149—21no—
92#13133148—21no—
93#13133148—21no—
94#13133148—21no—
95#13133148—21no—
96C4⋊#88322—21no—
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100C4⋊#88322C2311yescharacteristic
101#13133142—21no—
102#13133142—21no—
103#13133142—21no—
104#13133142—21no—
105#13133140—21no—
106#13133140—21no—
107#13133138—21no—
108#13133138—21no—
109#13133138—21no—
110#13133138—21no—
111#13133138—21no—
112#13133138—21no—
113C22⋊#88322—21no—
114C4×#88322—21no—
115C22×D4×C53D411yescharacteristic
116C23×C4×C53—21no—
117C2×M16×C53—21no—
118C2×M16×C53—21no—
119C22×C8×C53—21no—
120C22×C8×C53—21no—
121C4×D4×C53—21no—
122C4×D4×C53—21no—
123C4×D4×C53D411yescharacteristic
124C4×D4×C53D411yescharacteristic
125C4×D4×C53C2311yescharacteristic
126C4×D4×C53C2311yescharacteristic
127C2×C4⋊C4×C53D411yescharacteristic
128C2×C22⋊C4×C53—21no—
129C2×C42×C53C2311yescharacteristic
130C4⋊C8×C53D411yescharacteristic
131C4⋊C8×C53D411yescharacteristic
132C4⋊C8×C53C2311yescharacteristic
133C4⋊C8×C53C2311yescharacteristic
134C22⋊C8×C53—21no—
135C22⋊C8×C53—21no—
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53—21no—
139C22⋊C8×C53D411yescharacteristic
140C22⋊C8×C53D411yescharacteristic
141#97×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C24×C53—21no—
145C2×D4×C53—22no—
146C2×D4×C53—21no—
147C2×D4×C53—41no—
148C2×D4×C53C2×D411yescharacteristic
149C2×D4×C53C2×D411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—81no—
156C22×C4×C53—81no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53#4111yescharacteristic
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164M16×C53—81no—
165M16×C53—81no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—81no—
169C2×C8×C53—81no—
170C2×C8×C53C2×D411yescharacteristic
171C2×C8×C53C2×D411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C4⋊C4×C53—21no—
175C4⋊C4×C53C2×D411yescharacteristic
176C4⋊C4×C53C2×D411yescharacteristic
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C42×C53C2×D411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53#4111yescharacteristic
184C42×C53C2411yescharacteristic
185C22⋊Dic53—41no—
186C22⋊Dic53—41no—
187#11719923—42no—
188#11719923—42no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197#11719919—41no—
198#11719919—41no—
199#11719919—41no—
200#11719919—41no—
201C4×Dic53—41no—
202C4×Dic53—41no—
203#11719917—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53—81no—
213C23×C53#12111yescharacteristic
214D4×C53—21no—
215D4×C53—21no—
216D4×C53—21no—
217D4×C53—42no—
218D4×C53#13712yes—
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—81no—
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—
241C2×Dic53—41no—
242C2×Dic53—41no—
243Dic106—41no—
244Dic106—41no—
245Dic106—41no—
246Dic106—41no—
247Dic106—81no—
248Dic106—81no—
249C8×C53—21no—
250C8×C53—21no—
251C8×C53—21no—
252C8×C53—21no—
253C8×C53—41no—
254C8×C53—41no—
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×C53C2×#13711yescharacteristic
278C4×C53#57511yescharacteristic
279C4×C53#57711yescharacteristic
280C4×C53#57711yescharacteristic
281Dic53—81no—
282Dic53—81no—
283Dic53—81no—
284Dic53—81no—
285#3231—531no—
286#3101—531no—
287#3077—531no—
288#2974D5311yescharacteristic
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#533—531no—
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—1061no—
319#478—1061no—
320#476—531no—
321#476—531no—
322#476—531no—
323#476—531no—
324#474—531no—
325#474—531no—
326#461—531no—
327#461—531no—
328#450—1061no—
329C4×SD16—1061no—
330C4×SD16—1061no—
331#436C2×D5311yescharacteristic
332#436C2×D5311yescharacteristic
333#435—21no—
334#435—21no—
335C8×D4C2×D5311yescharacteristic
336C8×D4C2×D5311yescharacteristic
337#432C2×D5311yescharacteristic
338#431—531no—
339#431C2×D5311yescharacteristic
340#428—1061no—
341#423—531no—
342#418—1061no—
343#407—21no—
344C2×C22⋊C8—21no—
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×D4D21211yescharacteristic
359C23×C4—21no—
360C2×SD16—1061no—
361C2×SD16—1061no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—1061no—
365C22×C8—21no—
366C22×C8—21no—
367C4⋊Q8—531no—
368C4⋊Q8—531no—
369C4⋊Q8—531no—
370C4⋊Q8—531no—
371#125—1061no—
372#125—1061no—
373C22⋊Q8—1061no—
374C22⋊Q8—1061no—
375C22⋊Q8—1061no—
376C22⋊Q8—1061no—
377C4×Q8—1061no—
378C4×Q8—1061no—
379C4×Q8—1061no—
380C4×Q8—1061no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4D21211yescharacteristic
384C4×D4D21211yescharacteristic
385C4×D4C22×D5311yescharacteristic
386C4×D4C22×D5311yescharacteristic
387#117—1061no—
388#117—1061no—
389C2×C4⋊C4D21211yescharacteristic
390C2×C22⋊C4—21no—
391C2×C42C22×D5311yescharacteristic
392#107—1061no—
393#107—1061no—
394#107—1061no—
395#107—1061no—
396#106—1061no—
397#106—1061no—
398#106—1061no—
399#106—1061no—
400#106—1061no—
401#106—1061no—
402C4⋊C8—531no—
403C4⋊C8—531no—
404C4⋊C8—531no—
405C4⋊C8—531no—
406C4⋊C8—1061no—
407C4⋊C8C22⋊D5311yescharacteristic
408C4⋊C8C22⋊D5311yescharacteristic
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—1061no—
418#102—1061no—
419#102—1061no—
420#102—1061no—
421#102—1061no—
422#102—1061no—
423C22⋊C8—21no—
424C22⋊C8—21no—
425C22⋊C8—21no—
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8—1061no—
429C22⋊C8C22⋊D5311yescharacteristic
430C22⋊C8C22⋊D5311yescharacteristic
431#97—21no—
432C4×C8—1061no—
433C4×C8C22×D5311yescharacteristic
434C4×C8C22×D5311yescharacteristic
435C24—21no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4C2×D21211yescharacteristic
444C2×D4C2×D21211yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4C2×D21211yescharacteristic
453C22×C4#1171994511yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455SD16—2122no—
456SD16—2122no—
457M16—81no—
458M16—81no—
459M16—2122no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—81no—
463C2×C8—81no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8C2×C22⋊D5311yescharacteristic
469C2×C8C2×C22⋊D5311yescharacteristic
470C2×C8C2×C22⋊D5311yescharacteristic
471C2×C8C2×C22⋊D5311yescharacteristic
472C4⋊C4—21no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—2121no—
482C4⋊C4—2121no—
483C4⋊C4—2121no—
484C4⋊C4—2121no—
485C4⋊C4C2×D21211yescharacteristic
486C4⋊C4C2×D21211yescharacteristic
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—2121no—
492C22⋊C4—2121no—
493C42—2121no—
494C42—2121no—
495C42C2×D21211yescharacteristic
496C42#1171994511yescharacteristic
497C42C2×C22⋊D5311yescharacteristic
498C42C23×D5311yescharacteristic
499C23—21no—
500C23—21no—
501C23—42no—
502C23—42no—
503C23—81no—
504C23#1313319111yescharacteristic
505Q8—4241no—
506Q8—4241no—
507Q8—2121no—
508Q8—2121no—
509Q8—2121no—
510Q8—2121no—
511D4—21no—
512D4—21no—
513D4—21no—
514D4—42no—
515D4#1313325012yes—
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—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4#1313319111yescharacteristic
535C2×C4#1313319111yescharacteristic
536C2×C4#1313319111yescharacteristic
537C2×C4C22×C22⋊D5311yescharacteristic
538C2×C4C22×D21211yescharacteristic
539C2×C4C2×#1171994511yescharacteristic
540C8—21no—
541C8—21no—
542C8—21no—
543C8—21no—
544C8—41no—
545C8—41no—
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×#1313319111yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—4241no—
568C4—4241no—
569C4—4241no—
570C4—4241no—
571C4C2×#1313325011yescharacteristic
572C4#1681142311yescharacteristic
573C4#1681143411yescharacteristic
574C4#1681143411yescharacteristic
575C2—41no—
576C2—42no—
577C2—81no—
578C2#2028422711yescharacteristic
579C2#2028434511yescharacteristic
580C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2974C2
socleC22×C53C2×#121
prime radicalC22×D4×C53D4
hypercenter#2974D53
hypocenterC53#19466
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. #20285876C2semidirect
4. #20274290C2semidirect
5. #20274262C2semidirect
6. #20274238C2semidirect
7. #20274214C2semidirect
8. #20274166C2semidirect
9. #20274138C2semidirect
10. #20274114C2semidirect
11. #20274090C2semidirect
12. #20273504C2semidirect
13. #20273446C2semidirect
14. #20273426C2semidirect
18. #16811028C22semidirect
19. #16811021C22semidirect
20. #16810974C22semidirect
21. #16810974C22semidirect
44. #16810376C22semidirect
45. #16810376C22semidirect
46. #16810374C22semidirect
47. #16810374C22semidirect
48. #16810371C22semidirect
49. #16810371C22semidirect
50. #16810371C22semidirect
51. #16810371C22semidirect
52. #16810370C22semidirect
53. #16810370C22semidirect
54. #16810368C22semidirect
55. #16810368C22semidirect
357. C53#19466semidirect

Maximal subgroups

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

subgroupindex
#202858762
#202847232
#202845122
#202742902
#202742622
#202742382
#202742142
#202741662
#202741382
#202741142
#202740902
#202735042
#202734462
#202734262
C53×#29742
#1946653