Finite Group Databasebeta

← #27369619

Subgroups — #27369619

Order 13568.

616 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285853C211yesmaximal, characteristic, complemented
2#20284877C211yesmaximal, characteristic, complemented
3#20284775C211yesmaximal, characteristic
4#20274159C211yesmaximal, characteristic, complemented
5#20274152C211yesmaximal, characteristic, complemented
6#20274131C211yesmaximal, characteristic, complemented
7#20274124C211yesmaximal, characteristic, complemented
8#20274119C211yesmaximal, characteristic, complemented
9#20274114C211yesmaximal, characteristic, complemented
10#20274097C211yesmaximal, characteristic, complemented
11#20274090C211yesmaximal, characteristic, complemented
12#20273495C211yesmaximal, characteristic, complemented
13#20273408C211yesmaximal, characteristic, complemented
14#20273373C211yesmaximal, characteristic, complemented
15C53×#2988C211yesmaximal, characteristic, complemented
16#16811180C2211yescharacteristic
17#16811173C2211yescharacteristic
18#16811028C2211yescharacteristic
19#16811021C2211yescharacteristic
20#16810970C2211yescharacteristic
21#16810970C2211yescharacteristic
22#16810964—21no—
23#16810948—21no—
24#16810948—21no—
25#16810947—21no—
26#16810924—21no—
27#16810924—21no—
28#16810918C2211yescharacteristic
29#16810918C2211yescharacteristic
30#16810913C2211yescharacteristic
31#16810913C2211yescharacteristic
32#16810833C2211yescharacteristic
33#16810828C2211yescharacteristic
34#16810823C2211yescharacteristic
35#16810818C2211yescharacteristic
36#16810785—21no—
37#16810768—21no—
38#16810728—21no—
39#16810723—21no—
40C4×#11719924—21no—
41C4×#11719922—21no—
42#16810680C2211yescharacteristic
43#16810676C2211yescharacteristic
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C4×#41×C53C2211yescharacteristic
57C8×Q8×C53C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C8×D4×C53—21no—
60C8×D4×C53—21no—
61C8×D4×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#431—21no—
64C53×#431C2211yescharacteristic
65C2×C4⋊C8×C53—21no—
66C2×C4⋊C8×C53C2211yescharacteristic
67C2×#11719945—21no—
68C2×#11719925—21no—
69C2×#11719923—21no—
70C2×#11719924—21no—
71C2×#11719922—21no—
72#13133217C2311yescharacteristic
73#13133217C2311yescharacteristic
74#13133217C2311yescharacteristic
75#13133214C2311yescharacteristic
76#13133209—21no—
77#13133205—21no—
78#13133196—21no—
79#13133196—21no—
80#13133191—21no—
81#13133191—21no—
82C4×Dic106—21no—
83C4×Dic106—21no—
84C4×D212—21no—
85C4×D212—21no—
86#13133176—21no—
87#13133151—21no—
88#13133151—21no—
89#13133150—21no—
90#13133150—21no—
91#13133149—21no—
92#13133149—21no—
93#13133148—21no—
94#13133148—21no—
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99#13133141—21no—
100#13133141—21no—
101#13133140—21no—
102#13133140—21no—
103#13133139—21no—
104#13133139—21no—
105#13133139—21no—
106#13133139—21no—
107#13133138—21no—
108#13133138—21no—
109#13133137—21no—
110#13133137—21no—
111C22⋊#88322—21no—
112C22⋊#88322—21no—
113#13133128—21no—
114C4×#88322—21no—
115C2×#41×C53D411yescharacteristic
116C22×C8×C53—21no—
117C22×C8×C53—21no—
118C22×C8×C53—21no—
119C4×Q8×C53D411yescharacteristic
120C4×Q8×C53C2311yescharacteristic
121C4×D4×C53—21no—
122C4×D4×C53—21no—
123C4×D4×C53D411yescharacteristic
124C4×D4×C53C2311yescharacteristic
125#117×C53—21no—
126#117×C53D411yescharacteristic
127C2×C42×C53—21no—
128C2×C42×C53C2311yescharacteristic
129C4⋊C8×C53—21no—
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53D411yescharacteristic
133C4⋊C8×C53D411yescharacteristic
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53D411yescharacteristic
139C22⋊C8×C53D411yescharacteristic
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144#41×C53—22no—
145#41×C53—22no—
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—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—81no—
167C2×C8×C53—81no—
168C2×C8×C53—81no—
169C2×C8×C53C2×D411yescharacteristic
170C2×C8×C53C2×D411yescharacteristic
171C2×C8×C53C2×D411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53C2×D411yescharacteristic
176C4⋊C4×C53C2×D411yescharacteristic
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53C2×D411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C42×C53#4111yescharacteristic
186C42×C53C2411yescharacteristic
187C22⋊Dic53—41no—
188#11719925—42no—
189#11719924—42no—
190#11719923—42no—
191#11719922—42no—
192#11719921—41no—
193#11719921—41no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200#11719919—41no—
201#11719919—41no—
202C4×Dic53—41no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C23×C53—21no—
208C23×C53#12111yescharacteristic
209C22×D53—41no—
210Q8×C53—21no—
211Q8×C53#13512yes—
212D4×C53—24no—
213D4×C53—21no—
214D4×C53#13512yes—
215C2×C4×C53—24no—
216C2×C4×C53—21no—
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—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—81no—
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C22⋊D53—82no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237D212—41no—
238D212—41no—
239D212—41no—
240D212—41no—
241D212—81no—
242C4×D53—82no—
243Dic106—41no—
244Dic106—41no—
245Dic106—81no—
246Dic106—81no—
247C8×C53—21no—
248C8×C53—21no—
249C8×C53—21no—
250C8×C53—21no—
251C8×C53—41no—
252C8×C53—41no—
253#88322—41no—
254#88322—41no—
255#88322—41no—
256#88322—41no—
257#19419—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—22no—
261C22×C53—21no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53C2×#12111yescharacteristic
265C2×D53—81no—
266C2×D53—81no—
267C2×D53—81no—
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—22no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53C2×#13511yescharacteristic
276C4×C53C2×#13511yescharacteristic
277C4×C53#57711yescharacteristic
278C4×C53#57811yescharacteristic
279Dic53—81no—
280Dic53—81no—
281Dic53—81no—
282#3228—531no—
283#3116—531no—
284#3106—531no—
285#2988D5311yescharacteristic, complemented
286#1682—531no—
287#1681—531no—
288#1678—531no—
289#1677—531no—
290#1676—531no—
291#1675—531no—
292#1672—531no—
293#1671—531no—
294#1572—531no—
295#1551—531no—
296#1542—531no—
297C2×C53—41no—
298C2×C53—42no—
299C2×C53#304611yescharacteristic
300C2×C53#305011yescharacteristic
301C2×C53#306811yescharacteristic
302D53—161no—
303#533—531no—
304#532—531no—
305C4×#41C2×D5311yescharacteristic
306#500—531no—
307#498—531no—
308#486—531no—
309#486—531no—
310#485—1061no—
311#482—1061no—
312#482—1061no—
313#482—1061no—
314#478—1061no—
315#478—1061no—
316#477—531no—
317#477—531no—
318#476—531no—
319#476—531no—
320#462—531no—
321#461—531no—
322#460—531no—
323#459—531no—
324#452—1061no—
325#449—1061no—
326C8×Q8C2×D5311yescharacteristic
327C8×Q8C2×D5311yescharacteristic
328#441—1061no—
329#440—1061no—
330C4×SD16—1061no—
331C4×D8—1061no—
332C8×D4—21no—
333C8×D4—21no—
334C8×D4C2×D5311yescharacteristic
335C8×D4C2×D5311yescharacteristic
336#432—531no—
337#431—21no—
338#431—531no—
339#431C2×D5311yescharacteristic
340C2×C4⋊C8—21no—
341C2×C4⋊C8C2×D5311yescharacteristic
342#335—531no—
343#335—531no—
344#334—531no—
345#334—531no—
346#333—531no—
347#333—531no—
348#333—531no—
349#333—531no—
350#332—531no—
351#332—531no—
352#331—531no—
353#331—531no—
354C53#1941911yescharacteristic, complemented, Sylow
355C2×#41—1061no—
356C2×#41D21211yescharacteristic
357C2×Q16—1061no—
358C2×SD16—1061no—
359C2×SD16—1061no—
360C2×D8—1061no—
361C22×C8—21no—
362C22×C8—21no—
363C22×C8—21no—
364C4⋊Q8—531no—
365C4⋊Q8—531no—
366C4⋊Q8—531no—
367#127—531no—
368#125—1061no—
369#124—1061no—
370C22⋊Q8—1061no—
371C22⋊Q8—1061no—
372#121—1061no—
373#121—1061no—
374C4×Q8—1061no—
375C4×Q8—1061no—
376C4×Q8D21211yescharacteristic
377C4×Q8C22×D5311yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—1061no—
381C4×D4—1061no—
382C4×D4D21211yescharacteristic
383C4×D4C22×D5311yescharacteristic
384#117—21no—
385#117—1061no—
386#117D21211yescharacteristic
387C2×C42—21no—
388C2×C42C22×D5311yescharacteristic
389#107—1061no—
390#107—1061no—
391#107—1061no—
392#107—1061no—
393#106—1061no—
394#106—1061no—
395#106—1061no—
396#106—1061no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—531no—
404C4⋊C8C22⋊D5311yescharacteristic
405C4⋊C8C22⋊D5311yescharacteristic
406C4⋊C8C22×D5311yescharacteristic
407C4⋊C8C22×D5311yescharacteristic
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#103—1061no—
412#102—1061no—
413#102—1061no—
414#102—1061no—
415#102—1061no—
416#102—1061no—
417#102—1061no—
418#102—1061no—
419#102—1061no—
420C22⋊C8—21no—
421C22⋊C8—21no—
422C22⋊C8—1061no—
423C22⋊C8—1061no—
424C22⋊C8C22⋊D5311yescharacteristic
425C22⋊C8C22⋊D5311yescharacteristic
426#97—1061no—
427C4×C8—21no—
428C4×C8—21no—
429C4×C8—1061no—
430C4×C8C22×D5311yescharacteristic
431C4×C8C22×D5311yescharacteristic
432#41—22no—
433#41—22no—
434#41—4242no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8C2×D21211yescharacteristic
439C2×D4—21no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—1061no—
443C2×D4—2121no—
444C2×D4C2×D21211yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—2121no—
449C22×C4C2×D21211yescharacteristic
450C22×C4#1171994511yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452Q16—2122no—
453SD16—2122no—
454SD16—2122no—
455D8—2122no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—81no—
459C2×C8—81no—
460C2×C8—81no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8C2×C22⋊D5311yescharacteristic
466C2×C8C2×C22⋊D5311yescharacteristic
467C2×C8C2×C22⋊D5311yescharacteristic
468C2×C8C2×C22⋊D5311yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—2121no—
478C4⋊C4—2121no—
479C4⋊C4C2×D21211yescharacteristic
480C4⋊C4C2×D21211yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—2121no—
485C22⋊C4—2121no—
486C22⋊C4—2121no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—2121no—
491C42C2×D21211yescharacteristic
492C42#1171994511yescharacteristic
493C42C2×C22⋊D5311yescharacteristic
494C42C23×D5311yescharacteristic
495C23—21no—
496C23—2121no—
497C23#1313319111yescharacteristic
498Q8—21no—
499Q8—4241no—
500Q8—4241no—
501Q8—2121no—
502Q8—2121no—
503Q8#1313323912yes—
504D4—24no—
505D4—21no—
506D4—4241no—
507D4—4242no—
508D4—2121no—
509D4—2121no—
510D4—2121no—
511D4—2121no—
512D4#1313323912yes—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—81no—
525C2×C4—4242no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4#1313319111yescharacteristic
530C2×C4#1313319111yescharacteristic
531C2×C4#1313319111yescharacteristic
532C2×C4C22×C22⋊D5311yescharacteristic
533C2×C4C22×D21211yescharacteristic
534C2×C4C2×#1171994511yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—2121no—
542C8—2121no—
543C8—2121no—
544C8—2121no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—4241no—
552C22—4241no—
553C22—4241no—
554C22C2×#1313319111yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—22no—
560C4—41no—
561C4—41no—
562C4—4241no—
563C4—4241no—
564C4—4241no—
565C4C2×#1313323911yescharacteristic
566C4C2×#1313323911yescharacteristic
567C4#1681143411yescharacteristic
568C4#1681143811yescharacteristic
569C2—41no—
570C2—42no—
571C2—8481no—
572C2#2028426511yescharacteristic
573C2#2028429611yescharacteristic
574C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2988C2
socleC22×C53C2×#121
prime radical#20284775C2
hypercenter#2988D53
hypocenterC53#19419
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #20285853C2semidirect
2. #20284877C2semidirect
4. #20274159C2semidirect
5. #20274152C2semidirect
6. #20274131C2semidirect
7. #20274124C2semidirect
8. #20274119C2semidirect
9. #20274114C2semidirect
10. #20274097C2semidirect
11. #20274090C2semidirect
12. #20273495C2semidirect
13. #20273408C2semidirect
14. #20273373C2semidirect
15. C53×#2988C2semidirect
285. #2988D53semidirect
354. C53#19419semidirect

Maximal subgroups

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

subgroupindex
#202858532
#202848772
#202847752
#202741592
#202741522
#202741312
#202741242
#202741192
#202741142
#202740972
#202740902
#202734952
#202734082
#202733732
C53×#29882
#1941953