Finite Group Databasebeta

← #27367111

Subgroups — #27367111

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285433C211yesmaximal, characteristic, complemented
2#20284775C211yesmaximal, characteristic
3#20284503C211yesmaximal, characteristic, complemented
4#20274290C211yesmaximal, characteristic, complemented
5#20274248C211yesmaximal, characteristic, complemented
6#20274238C211yesmaximal, characteristic, complemented
7#20274180C211yesmaximal, characteristic, complemented
8#20274173C211yesmaximal, characteristic, complemented
9#20274131C211yesmaximal, characteristic, complemented
10#20274119C211yesmaximal, characteristic, complemented
11#20274063C211yesmaximal, characteristic, complemented
12#20273467C211yesmaximal, characteristic, complemented
13#20273426C211yesmaximal, characteristic, complemented
14#20273411C211yesmaximal, characteristic, complemented
15C53×#2971C211yesmaximal, characteristic, complemented
16#16811180C2211yescharacteristic
17#16811173C2211yescharacteristic
18#16811028C2211yescharacteristic
19#16810996C2211yescharacteristic
20#16810974C2211yescharacteristic
21#16810970C2211yescharacteristic
22#16810947—21no—
23#16810930—21no—
24#16810923—21no—
25#16810918C2211yescharacteristic
26#16810918C2211yescharacteristic
27#16810913C2211yescharacteristic
28#16810913C2211yescharacteristic
29#16810908C2211yescharacteristic
30#16810903C2211yescharacteristic
31#16810869—21no—
32#16810869—21no—
33#16810823C2211yescharacteristic
34#16810818C2211yescharacteristic
35#16810768—21no—
36#16810725—21no—
37#16810725—21no—
38C4×#11719924—21no—
39C4×#11719922—21no—
40#16810676C2211yescharacteristic
41#16810659—21no—
42C2×C4⋊#88322C2211yescharacteristic
43#16810619—21no—
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810370C2211yescharacteristic
53#16810370C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C4×#41×C53C2211yescharacteristic
57C8×Q8×C53C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C53×#436—21no—
60C53×#436—21no—
61C53×#436C2211yescharacteristic
62C53×#436C2211yescharacteristic
63C53×#432—21no—
64C53×#432C2211yescharacteristic
65C4×M16×C53—21no—
66C4×M16×C53C2211yescharacteristic
67C2×#11719945—21no—
68C2×#11719913—21no—
69C2×#11719913—21no—
70C2×#11719924—21no—
71C2×#11719922—21no—
72C22×#88322—21no—
73#13133217C2311yescharacteristic
74#13133217C2311yescharacteristic
75#13133217C2311yescharacteristic
76#13133214C2311yescharacteristic
77#13133209—21no—
78#13133205—21no—
79#13133196—21no—
80#13133196—21no—
81#13133191—21no—
82#13133191—21no—
83C4×Dic106—21no—
84C4×Dic106—21no—
85C4×D212—21no—
86C4×D212—21no—
87#13133176—21no—
88#13133151—21no—
89#13133151—21no—
90#13133150—21no—
91#13133150—21no—
92#13133150—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#13133141—21no—
104#13133141—21no—
105#13133141—21no—
106#13133139—21no—
107#13133139—21no—
108#13133138—21no—
109#13133138—21no—
110#13133138—21no—
111#13133137—21no—
112#13133137—21no—
113C22⋊#88322—21no—
114C4×#88322—21no—
115C2×#41×C53D411yescharacteristic
116C2×M16×C53—21no—
117C2×M16×C53—21no—
118C2×M16×C53D411yescharacteristic
119C2×M16×C53D411yescharacteristic
120C4×Q8×C53D411yescharacteristic
121C4×Q8×C53C2311yescharacteristic
122C4×D4×C53—21no—
123C4×D4×C53—21no—
124C4×D4×C53D411yescharacteristic
125C4×D4×C53C2311yescharacteristic
126#117×C53—21no—
127#117×C53D411yescharacteristic
128C2×C42×C53—21no—
129C2×C42×C53C2311yescharacteristic
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53C2311yescharacteristic
133C4⋊C8×C53C2311yescharacteristic
134C22⋊C8×C53—21no—
135C22⋊C8×C53—21no—
136C22⋊C8×C53—21no—
137#97×C53—21no—
138#97×C53D411yescharacteristic
139#97×C53D411yescharacteristic
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144#41×C53—22no—
145#41×C53—41no—
146C2×Q8×C53C2×D411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C2×D411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53#4111yescharacteristic
155C2×C22⋊D53—41no—
156#11719945—82no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×C4×D53—41no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164M16×C53—22no—
165M16×C53—41no—
166M16×C53—42no—
167M16×C53—42no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53C2×D411yescharacteristic
171C2×C8×C53C2×D411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53C2×D411yescharacteristic
177C4⋊C4×C53C2×D411yescharacteristic
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53C2×D411yescharacteristic
185C42×C53C2×D411yescharacteristic
186C42×C53#4111yescharacteristic
187C42×C53C2411yescharacteristic
188C22⋊Dic53—41no—
189#11719924—42no—
190#11719922—42no—
191#11719921—41no—
192#11719921—41no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199#11719919—41no—
200#11719919—41no—
201C4×Dic53—41no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—42no—
207#11719913—42no—
208#11719913—42no—
209C23×C53—21no—
210C23×C53#12111yescharacteristic
211C22×D53—41no—
212Q8×C53—21no—
213Q8×C53#13712yes—
214D4×C53—21no—
215D4×C53—21no—
216D4×C53—42no—
217C2×C4×C53—21no—
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—41no—
225C2×C4×C53—42no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—81no—
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53#12111yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C22⋊D53—82no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239D212—41no—
240D212—41no—
241D212—41no—
242D212—41no—
243D212—81no—
244C4×D53—82no—
245Dic106—41no—
246Dic106—41no—
247Dic106—81no—
248Dic106—81no—
249C8×C53—21no—
250C8×C53—21no—
251C8×C53—21no—
252C8×C53—41no—
253C8×C53—41no—
254C8×C53#13512yes—
255#88322—41no—
256#88322—41no—
257#88322—41no—
258#88322—41no—
259#19183—531nomaximal, Sylow
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53C2×#12111yescharacteristic
267C2×D53—81no—
268C2×D53—81no—
269C2×D53—81no—
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—22no—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53C2×#13511yescharacteristic
278C4×C53C2×#13711yescharacteristic
279C4×C53#57511yescharacteristic
280C4×C53#57511yescharacteristic
281Dic53—81no—
282Dic53—81no—
283Dic53—81no—
284#3182—531no—
285#3106—531no—
286#3076—531no—
287#2971D5311yescharacteristic, complemented
288#1703—531no—
289#1697—531no—
290#1695—531no—
291#1685—531no—
292#1684—531no—
293#1678—531no—
294#1676—531no—
295#1666—531no—
296#1565—531no—
297#1554—531no—
298#1551—531no—
299C2×C53—41no—
300C2×C53—81no—
301C2×C53#303511yescharacteristic
302C2×C53#305611yescharacteristic
303C2×C53#306811yescharacteristic
304D53—161no—
305#533—531no—
306#532—531no—
307C4×#41C2×D5311yescharacteristic
308#500—531no—
309#492—531no—
310#487—531no—
311#486—531no—
312#482—1061no—
313#479—1061no—
314#478—1061no—
315#477—531no—
316#477—531no—
317#476—531no—
318#476—531no—
319#475—531no—
320#474—531no—
321#468—1061no—
322#468—1061no—
323#460—531no—
324#459—531no—
325#449—1061no—
326C8×Q8C2×D5311yescharacteristic
327C8×Q8C2×D5311yescharacteristic
328#440—1061no—
329#440—1061no—
330C4×SD16—1061no—
331C4×D8—1061no—
332#436—21no—
333#436—21no—
334#436C2×D5311yescharacteristic
335#436C2×D5311yescharacteristic
336#432—21no—
337#432C2×D5311yescharacteristic
338#431—531no—
339#427—1061no—
340C2×C4⋊C8—531no—
341#416—1061no—
342C4×M16—21no—
343C4×M16C2×D5311yescharacteristic
344#335—531no—
345#335—531no—
346#334—531no—
347#334—531no—
348#333—531no—
349#333—531no—
350#333—531no—
351#333—531no—
352#332—531no—
353#332—531no—
354#331—531no—
355#331—531no—
356C53#1918311yescharacteristic, complemented, Sylow
357C2×#41—1061no—
358C2×#41D21211yescharacteristic
359C2×SD16—1061no—
360C2×SD16—1061no—
361C2×SD16—1061no—
362C2×D8—1061no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16C22⋊D5311yescharacteristic
366C2×M16C22⋊D5311yescharacteristic
367C22×C8—1061no—
368C4⋊Q8—531no—
369C4⋊Q8—531no—
370C4⋊Q8—531no—
371#127—531no—
372#125—1061no—
373#124—1061no—
374C22⋊Q8—1061no—
375C22⋊Q8—1061no—
376#121—1061no—
377#121—1061no—
378C4×Q8—1061no—
379C4×Q8—1061no—
380C4×Q8D21211yescharacteristic
381C4×Q8C22×D5311yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—1061no—
385C4×D4—1061no—
386C4×D4D21211yescharacteristic
387C4×D4C22×D5311yescharacteristic
388#117—21no—
389#117—1061no—
390#117D21211yescharacteristic
391C2×C42—21no—
392C2×C42C22×D5311yescharacteristic
393#107—1061no—
394#107—1061no—
395#107—1061no—
396#107—1061no—
397#107—1061no—
398#106—1061no—
399#106—1061no—
400#106—1061no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403C4⋊C8—531no—
404C4⋊C8—531no—
405C4⋊C8—531no—
406C4⋊C8—531no—
407C4⋊C8—1061no—
408C4⋊C8C22×D5311yescharacteristic
409C4⋊C8C22×D5311yescharacteristic
410#103—1061no—
411#103—1061no—
412#103—1061no—
413#103—1061no—
414#103—1061no—
415#102—1061no—
416#102—1061no—
417#102—1061no—
418#102—1061no—
419#102—1061no—
420#102—1061no—
421#102—1061no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424C22⋊C8—21no—
425C22⋊C8—1061no—
426#97—21no—
427#97C22⋊D5311yescharacteristic
428#97C22⋊D5311yescharacteristic
429C4×C8—21no—
430C4×C8—21no—
431C4×C8—1061no—
432C4×C8C22×D5311yescharacteristic
433C4×C8C22×D5311yescharacteristic
434#41—22no—
435#41—41no—
436#41—4242no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8C2×D21211yescharacteristic
441C2×D4—21no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—2121no—
446C2×D4C2×D21211yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—2121no—
451C22×C4C2×D21211yescharacteristic
452C22×C4#1171994511yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454SD16—2122no—
455SD16—2122no—
456SD16—2122no—
457D8—2122no—
458M16—22no—
459M16—41no—
460M16—42no—
461M16—42no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—2122no—
469C2×C8C2×C22⋊D5311yescharacteristic
470C2×C8C2×C22⋊D5311yescharacteristic
471C2×C8C2×C22⋊D5311yescharacteristic
472C2×C8C2×C22⋊D5311yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
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⋊C4C2×D21211yescharacteristic
484C4⋊C4C2×D21211yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—2121no—
489C22⋊C4—2121no—
490C22⋊C4—2121no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—2121no—
495C42C2×D21211yescharacteristic
496C42#1171994511yescharacteristic
497C42C2×C22⋊D5311yescharacteristic
498C42C23×D5311yescharacteristic
499C23—21no—
500C23—2121no—
501C23#1313319111yescharacteristic
502Q8—21no—
503Q8—4241no—
504Q8—4241no—
505Q8—2121no—
506Q8—2121no—
507Q8#1313325012yes—
508D4—21no—
509D4—21no—
510D4—42no—
511D4—4241no—
512D4—4242no—
513D4—2121no—
514D4—2121no—
515D4—2121no—
516D4—2121no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—81no—
529C2×C4—4242no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4#1313319111yescharacteristic
534C2×C4#1313319111yescharacteristic
535C2×C4#1313319111yescharacteristic
536C2×C4C22×C22⋊D5311yescharacteristic
537C2×C4C22×D21211yescharacteristic
538C2×C4C2×#1171994511yescharacteristic
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—2121no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C8#1313323712yes—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—4241no—
556C22—4241no—
557C22—4241no—
558C22C2×#1313319111yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—4241no—
567C4—4241no—
568C4—4241no—
569C4C2×#1313323711yescharacteristic
570C4C2×#1313325011yescharacteristic
571C4#1681142111yescharacteristic
572C4#1681142311yescharacteristic
573C2—41no—
574C2—81no—
575C2—8481no—
576C2#2028417011yescharacteristic
577C2#2028434511yescharacteristic
578C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2971C2
socleC22×C53C2×#121
prime radical#20284775C2
hypercenter#2971D53
hypocenterC53#19183
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2971

Complements

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

NH ≅ G/Ntype
1. #20285433C2semidirect
3. #20284503C2semidirect
4. #20274290C2semidirect
5. #20274248C2semidirect
6. #20274238C2semidirect
7. #20274180C2semidirect
8. #20274173C2semidirect
9. #20274131C2semidirect
10. #20274119C2semidirect
11. #20274063C2semidirect
12. #20273467C2semidirect
13. #20273426C2semidirect
14. #20273411C2semidirect
15. C53×#2971C2semidirect
287. #2971D53semidirect
356. C53#19183semidirect

Maximal subgroups

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

subgroupindex
#202854332
#202847752
#202845032
#202742902
#202742482
#202742382
#202741802
#202741732
#202741312
#202741192
#202740632
#202734672
#202734262
#202734112
C53×#29712
#1918353