Finite Group Databasebeta

← #28316500

Subgroups — #28316500

Order 13568.

660 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#20291527C211yesmaximal, characteristic
2Q8×D53⋊C8C211yesmaximal, characteristic, complemented
3#20290345C211yesmaximal, characteristic, complemented
4#20290344C211yesmaximal, characteristic, complemented
5#20290339C211yesmaximal, characteristic, complemented
6#20290338C211yesmaximal, characteristic, complemented
7#20290327C211yesmaximal, characteristic, complemented
8#20290326C211yesmaximal, characteristic, complemented
9#20290325C211yesmaximal, characteristic, complemented
10#20290324C211yesmaximal, characteristic, complemented
11#20290267C211yesmaximal, characteristic, complemented
12#20290018C211yesmaximal, characteristic, complemented
13#20290011C211yesmaximal, characteristic
14#20289923C211yesmaximal, characteristic, complemented
15#20283537C211yesmaximal, characteristic
16#16811673C2211yescharacteristic
17#16811673C2211yescharacteristic
18#16811672—21no—
19#16811672—21no—
20#16811672C2211yescharacteristic
21#16811672C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811658C2211yescharacteristic
25#16811658C2211yescharacteristic
26#16811655—21no—
27#16811655—21no—
28#16811642C2211yescharacteristic
29#16811641C2211yescharacteristic
30#16811639—21no—
31D53⋊(C4⋊C8)—21no—
32D53⋊(C4⋊C8)C2211yescharacteristic
33C4×#11719940—21no—
34C4×D53⋊C8—21no—
35C4×D53⋊C8C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42#16811473C2211yescharacteristic
43#16811473C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#88324⋊C8C2211yescharacteristic
51#88324⋊C8C2211yescharacteristic
52C2×#13133246C411yescharacteristic
53C4×#11719947C2211yescharacteristic
54C4×Q8×D53C2211yescharacteristic
55#16810731C411yescharacteristic
56#16810728C411yescharacteristic
57#16810723C411yescharacteristic
58#16810717C2211yescharacteristic
59#16810715C2211yescharacteristic
60#16810714C2211yescharacteristic
61SD16×Dic53C2211yescharacteristic
62#16810664C411yescharacteristic
63#16810630C411yescharacteristic
64#16810621C411yescharacteristic
65#16810575C2211yescharacteristic
66C53×#440C411yescharacteristic
67C2×#11719940—21no—
68C2×#11719940—21no—
69C2×D53⋊C8—21no—
70C2×D53⋊C8D411yescharacteristic
71C2×D53⋊C8D411yescharacteristic
72#13133284—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88#13133272—21no—
89#13133272—21no—
90C4×#88324—21no—
91C4×#88324—21no—
92C4×#88324—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324D411yescharacteristic
96C4×#88324D411yescharacteristic
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719947C2×C411yescharacteristic
102C2×Q8×D53C2×C411yescharacteristic
103#13133246—24no—
104C2×Dic212C2×C411yescharacteristic
105C2×#11719925C2×C411yescharacteristic
106C2×#11719923C2×C411yescharacteristic
107C2×#11719912C2×C411yescharacteristic
108C2×#11719912C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110C4×Dic106—21no—
111C4×Dic106C2×C411yescharacteristic
112C4×Dic106C2×C411yescharacteristic
113#13133182—21no—
114#13133182C2311yescharacteristic
115#13133182C2311yescharacteristic
116D4×Dic53C2311yescharacteristic
117C4×C22⋊D53—21no—
118#13133177—21no—
119#13133175C2×C411yescharacteristic
120#13133173—21no—
121C4⋊C4×D53—21no—
122C4⋊C4×D53C2×C411yescharacteristic
123C2×C4×Dic53—21no—
124C42×D53—21no—
125C42×D53C2311yescharacteristic
126#13133150C2×C411yescharacteristic
127#13133149C2×C411yescharacteristic
128#13133148C2×C411yescharacteristic
129#13133142C2×C411yescharacteristic
130#13133142C2×C411yescharacteristic
131#13133141C2×C411yescharacteristic
132#13133140C2×C411yescharacteristic
133#13133140C2×C411yescharacteristic
134#13133138C2×C411yescharacteristic
135#13133128C2×C411yescharacteristic
136C8×Dic53C2311yescharacteristic
137C8×Dic53C2311yescharacteristic
138C2×SD16×C53C2×C411yescharacteristic
139C4×Q8×C53C2×C411yescharacteristic
140C4×D4×C53C2×C411yescharacteristic
141#107×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#102×C53C2×C411yescharacteristic
144#97×C53C2×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C2×C22⋊D53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151Q8×D53—22no—
152Q8×D53M1614yescomplemented
153#11719947—22no—
154#11719947—22no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53C22×C411yescharacteristic
158C2×C4×D53C2×D411yescharacteristic
159C2×C4×D53#4111yescharacteristic
160C2×Dic106—21no—
161C2×Dic106C22×C411yescharacteristic
162C2×Dic106C22×C411yescharacteristic
163#11719940—42no—
164#11719940—42no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324C2×D411yescharacteristic
172C2×#88324C2×D411yescharacteristic
173C2×#88324C2×D411yescharacteristic
174C2×#88324C2×D411yescharacteristic
175D53⋊C8—22no—
176D53⋊C8—41no—
177D53⋊C8—42no—
178SD16×C53—22no—
179C2×C8×C53C22×C411yescharacteristic
180C2×C8×C53C22×C411yescharacteristic
181C4⋊C4×C53—21no—
182C4⋊C4×C53C22×C411yescharacteristic
183C4⋊C4×C53C22×C411yescharacteristic
184C22⋊C4×C53—21no—
185C42×C53—21no—
186C42×C53C22×C411yescharacteristic
187C22⋊Dic53—21no—
188#11719925—22no—
189#11719923—22no—
190#11719921—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53C22×C411yescharacteristic
193C4⋊Dic53C22×C411yescharacteristic
194#11719919—21no—
195#11719919—21no—
196#11719919—21no—
197#11719919—21no—
198#11719919—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53C2×D411yescharacteristic
206C4×Dic53#4111yescharacteristic
207C4×Dic53C2411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209C2×#88322C22×C411yescharacteristic
210Dic212—22no—
211#11719912—22no—
212#11719912—22no—
213C23×C53—21no—
214C22×D53C4×D411yescharacteristic
215Q8×C53—21no—
216Q8×C53C2×M1612yes—
217D4×C53—21no—
218D4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—42no—
223C2×C4×C53C4×D411yescharacteristic
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C23×C411yescharacteristic
226C22⋊D53—42no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—42no—
234C2×Dic53C4×D411yescharacteristic
235C2×Dic53C22×D411yescharacteristic
236C2×Dic53C2×#4111yescharacteristic
237C4×D53—24no—
238C4×D53—21no—
239C4×D53—21no—
240C4×D53—41no—
241C4×D53—42no—
242C4×D53C2×M1612yes—
243C4×D53#13112yes—
244Dic106—21no—
245Dic106—24no—
246Dic106—21no—
247Dic106—21no—
248Dic106C2×M1612yes—
249#88324—21no—
250#88324—21no—
251#88324—21no—
252#88324—41no—
253#88324—41no—
254#88324—41no—
255#88324—41no—
256#88324—41no—
257#88324—41no—
258#88324#13712yescomplemented
259C8×C53—21no—
260C8×C53—21no—
261#88322—21no—
262#88322—21no—
263#17870—531nomaximal, Sylow
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53C2×C4×D411yescharacteristic
268C2×D53—21no—
269C2×D53—21no—
270C2×D53#43512yes—
271C4×C53—22no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53C22×M1611yescharacteristic
277C4×C53C2×#13111yescharacteristic
278Dic53—22no—
279Dic53—21no—
280Dic53—41no—
281Dic53—41no—
282Dic53—41no—
283Dic53#43512yes—
284Dic53C2×#13711yescharacteristic
285Dic53#57511yescharacteristic
286#2965—531no—
287C2×C8×Q8—531no—
288C4×#137—531no—
289#1588—531no—
290#1588—531no—
291#1587—531no—
292#1587—531no—
293#1584—531no—
294#1584—531no—
295#1583—531no—
296#1583—531no—
297#1565—531no—
298#1479—531no—
299#1477—531no—
300#1451—531no—
301C2×C53—81no—
302C2×C53C2×#43511yescharacteristic
303C2×C53C4×#13711yescharacteristic
304C2×C53#295511yescharacteristic
305D53—22no—
306C2×#137—531no—
307C4×#41—531no—
308C2×C4×Q8—531no—
309#446—531no—
310#446—531no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—1061no—
316C8×Q8—1061no—
317#441—531no—
318#441—531no—
319#440—531no—
320#440#615711yescharacteristic
321C4×Q16—531no—
322C4×Q16—531no—
323C4×SD16—531no—
324C4×SD16—531no—
325#436—531no—
326#436—531no—
327#435—1061no—
328#435—1061no—
329#431—531no—
330#431—531no—
331#431—1061no—
332#428—531no—
333C2×C4⋊C8—531no—
334C2×C4⋊C8—1061no—
335#419—531no—
336#417—531no—
337C4×M16—531no—
338C4×M16—1061no—
339C2×C4×C8—531no—
340C2×C4×C8—1061no—
341#335—531no—
342#335—531no—
343#334—531no—
344#334—531no—
345Q8⋊C8—531no—
346Q8⋊C8—531no—
347Q8⋊C8—531no—
348Q8⋊C8—531no—
349Q8⋊C8—531no—
350Q8⋊C8—531no—
351D4⋊C8—531no—
352D4⋊C8—531no—
353#322—531no—
354#322—531no—
355#322—531no—
356#322—531no—
357C53#1787011yescharacteristic, complemented, Sylow
358C2×#41—531no—
359C22×Q8—531no—
360#137—1064no—
361C2×Q16—531no—
362C2×Q16—531no—
363C2×SD16—531no—
364C2×SD16C2×#615711yescharacteristic
365C2×M16—531no—
366C2×M16—531no—
367C2×M16—1061no—
368C2×M16—1061no—
369C22×C8—531no—
370C22×C8—531no—
371C22×C8—1061no—
372C4×Q8—531no—
373C4×Q8—531no—
374C4×Q8—531no—
375C4×Q8—531no—
376C4×Q8—531no—
377C4×Q8—1061no—
378C4×Q8—1061no—
379C4×Q8C2×#615711yescharacteristic
380C4×D4—531no—
381C4×D4—1061no—
382C4×D4—1061no—
383C4×D4C2×#615711yescharacteristic
384#117—531no—
385#117—1061no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—1061no—
388C2×C42—531no—
389C2×C42—1061no—
390C2×C42—1061no—
391#107—531no—
392#107C2×#615711yescharacteristic
393#106—531no—
394#106—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407#103—531no—
408#103—531no—
409#103—531no—
410#103—531no—
411#103—531no—
412#103C2×#615711yescharacteristic
413#102—531no—
414#102C2×#615711yescharacteristic
415C22⋊C8—1061no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419#97—531no—
420#97—1061no—
421#97—1061no—
422#97C2×#615711yescharacteristic
423C4×C8—531no—
424C4×C8—531no—
425C4×C8—531no—
426C4×C8—531no—
427C4×C8—531no—
428C4×C8—531no—
429C4×C8—531no—
430C4×C8—531no—
431C4×C8—1061no—
432C4×C8—1061no—
433C4×C8—1061no—
434C4×C8—1061no—
435C4×C8—1061no—
436#41—1062no—
437#41—1062no—
438C2×Q8—534no—
439C2×Q8—531no—
440C2×Q8—531no—
441C2×Q8—1061no—
442C2×Q8—1062no—
443C2×Q8C22×#615711yescharacteristic
444C2×D4—1061no—
445C2×D4C22×#615711yescharacteristic
446C22×C4—21no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454Q16—1062no—
455Q16—1062no—
456SD16—22no—
457SD16—1062no—
458M16—1062no—
459M16—1062no—
460M16—2122no—
461M16—2122no—
462C2×C8—531no—
463C2×C8—531no—
464C2×C8—531no—
465C2×C8—531no—
466C2×C8—531no—
467C2×C8—531no—
468C2×C8—1062no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—2121no—
476C2×C8—2122no—
477C2×C8C22×#615711yescharacteristic
478C2×C8C22×#615711yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—531no—
481C4⋊C4—531no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4C22×#615711yescharacteristic
489C4⋊C4C22×#615711yescharacteristic
490C22⋊C4—21no—
491C22⋊C4—1061no—
492C22⋊C4—1061no—
493C42—21no—
494C42—531no—
495C42—531no—
496C42—531no—
497C42—1061no—
498C42—1061no—
499C42—1061no—
500C42—1061no—
501C42—1061no—
502C42—1061no—
503C42C22×#615711yescharacteristic
504C23—21no—
505C23—531no—
506Q8—21no—
507Q8—532no—
508Q8—1064no—
509Q8—1061no—
510Q8—1061no—
511Q8—1061no—
512Q8D53⋊M1612yes—
513D4—21no—
514D4—21no—
515D4—2122no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—531no—
521C2×C4—531no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—531no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1064no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—2121no—
533C2×C4—2122no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2122no—
537C2×C4D4×#615711yescharacteristic
538C2×C4D4×#615711yescharacteristic
539C2×C4C23×#615711yescharacteristic
540C8—21no—
541C8—21no—
542C8—532no—
543C8—1061no—
544C8—1061no—
545C8—1061no—
546C8—1061no—
547C8—1061no—
548C8—2121no—
549C8—2121no—
550C8—2121no—
551C8—2121no—
552C8—2121no—
553C8—2121no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—532no—
558C22—1061no—
559C22—1061no—
560C22C2×D4×#615711yescharacteristic
561C4—22no—
562C4—21no—
563C4—21no—
564C4—41no—
565C4—41no—
566C4—532no—
567C4—531no—
568C4—531no—
569C4—1062no—
570C4—1061no—
571C4—2121no—
572C4—2121no—
573C4—2121no—
574C4C2×D53⋊M1611yescharacteristic
575C4D53⋊#13111yescharacteristic
576C2—81no—
577C2—1062no—
578C2D53⋊#43511yescharacteristic
579C2#137×#615711yescharacteristic
580C2#2029148711yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#440C4
socleC22×C53C2×C4×D4
prime radicalC2×#11719947C2×C4
cosocleC4×Dic53C24
hypercenter#440#6157
hypocenterC53#17870
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. Q8×D53⋊C8C2semidirect
3. #20290345C2semidirect
4. #20290344C2semidirect
5. #20290339C2semidirect
6. #20290338C2semidirect
7. #20290327C2semidirect
8. #20290326C2semidirect
9. #20290325C2semidirect
10. #20290324C2semidirect
11. #20290267C2semidirect
12. #20290018C2semidirect
14. #20289923C2semidirect
152. Q8×D53M16semidirect
258. #88324#137semidirect
357. C53#17870semidirect

Maximal subgroups

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

subgroupindex
#202915272
Q8×D53⋊C82
#202903452
#202903442
#202903392
#202903382
#202903272
#202903262
#202903252
#202903242
#202902672
#202900182
#202900112
#202899232
#202835372
#1787053