Finite Group Databasebeta

← #27355229

Subgroups — #27355229

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284780C211yesmaximal, characteristic
2#20283733C211yesmaximal, characteristic
3#20273615C211yesmaximal, characteristic, complemented
4#20273605C211yesmaximal, characteristic, complemented
5#20273593C211yesmaximal, characteristic, complemented
6#20273584C211yesmaximal, characteristic, complemented
7#20273563C211yesmaximal, characteristic, complemented
8#20273558C211yesmaximal, characteristic, complemented
9#20273532C211yesmaximal, characteristic, complemented
10C8×#11719925C211yesmaximal, characteristic, complemented
11#20273498C211yesmaximal, characteristic, complemented
12#20273188C211yesmaximal, characteristic, complemented
13#20273100C211yesmaximal, characteristic
14#20273012C211yesmaximal, characteristic, complemented
15C53×#2962C211yesmaximal, characteristic, complemented
16C4×#11719945C2211yescharacteristic
17#16810961C411yescharacteristic
18#16810961C411yescharacteristic
19#16810905C411yescharacteristic
20#16810900C411yescharacteristic
21#16810830C411yescharacteristic
22#16810825C411yescharacteristic
23#16810783C411yescharacteristic
24#16810783C411yescharacteristic
25C8×Dic106C2211yescharacteristic
26C8×Dic106C2211yescharacteristic
27C4×#11719925C2211yescharacteristic
28C4×#11719925C2211yescharacteristic
29C4×#11719924C2211yescharacteristic
30C4×#11719924C2211yescharacteristic
31#16810702C2211yescharacteristic
32#16810702C2211yescharacteristic
33#16810699—21no—
34#16810699—21no—
35#16810680C2211yescharacteristic
36#16810678—21no—
37#16810676C2211yescharacteristic
38#16810676C2211yescharacteristic
39#16810575—21no—
40#16810375C2211yescharacteristic
41#16810375C2211yescharacteristic
42#16810373C2211yescharacteristic
43#16810373C2211yescharacteristic
44Q8⋊#88322C2211yescharacteristic
45Q8⋊#88322C2211yescharacteristic
46Q8⋊#88322C2211yescharacteristic
47Q8⋊#88322C2211yescharacteristic
48#16810353C2211yescharacteristic
49#16810353C2211yescharacteristic
50#16810350C2211yescharacteristic
51#16810350C2211yescharacteristic
52#16810345C2211yescharacteristic
53#16810345C2211yescharacteristic
54#16810344C2211yescharacteristic
55C8×#88322C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C53×#446—21no—
58C53×#446—21no—
59C53×#446C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C8×Q8×C53C2211yescharacteristic
62C8×Q8×C53C2211yescharacteristic
63C53×#424—21no—
64C2×C4⋊C8×C53C2211yescharacteristic
65C53×#405—21no—
66C4×M16×C53C2211yescharacteristic
67C2×#11719945C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719925C2×C411yescharacteristic
70C2×#11719924C2×C411yescharacteristic
71C2×#11719924C2×C411yescharacteristic
72C2×#11719912—21no—
73C2×#11719912—21no—
74C4×Dic106C2×C411yescharacteristic
75C4×Dic106C2311yescharacteristic
76C4×D212C2×C411yescharacteristic
77C4×D212C2311yescharacteristic
78C4×C22⋊D53—21no—
79C4×C22⋊D53—21no—
80#13133176C2×C411yescharacteristic
81#13133174—21no—
82C42×D53—21no—
83#13133150C2×C411yescharacteristic
84#13133150C2×C411yescharacteristic
85#13133148C2×C411yescharacteristic
86#13133148C2×C411yescharacteristic
87C4⋊#88322—21no—
88C4⋊#88322C2×C411yescharacteristic
89C4⋊#88322C2×C411yescharacteristic
90C4⋊#88322C2311yescharacteristic
91C4⋊#88322C2311yescharacteristic
92#13133146—21no—
93#13133146—21no—
94#13133141C2×C411yescharacteristic
95#13133141C2×C411yescharacteristic
96#13133141C2×C411yescharacteristic
97#13133141C2×C411yescharacteristic
98#13133140C2×C411yescharacteristic
99#13133140C2×C411yescharacteristic
100#13133137C2×C411yescharacteristic
101#13133137C2×C411yescharacteristic
102C22⋊#88322—21no—
103C22⋊#88322—21no—
104C22⋊#88322C2×C411yescharacteristic
105C22⋊#88322C2×C411yescharacteristic
106#13133129—21no—
107#13133129—21no—
108#13133128—21no—
109C4×#88322C2311yescharacteristic
110C4×#88322C2311yescharacteristic
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C8×Dic53—21no—
114C8×Dic53—21no—
115C2×M16×C53D411yescharacteristic
116C2×M16×C53D411yescharacteristic
117C22×C8×C53—21no—
118C4⋊Q8×C53C2×C411yescharacteristic
119C4⋊Q8×C53C2×C411yescharacteristic
120#125×C53—21no—
121C22⋊Q8×C53—21no—
122C22⋊Q8×C53C2×C411yescharacteristic
123C22⋊Q8×C53C2×C411yescharacteristic
124C4×Q8×C53—21no—
125C4×Q8×C53C2311yescharacteristic
126C4×Q8×C53C2311yescharacteristic
127#117×C53—21no—
128C2×C42×C53C2311yescharacteristic
129C4⋊C8×C53—21no—
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53—21no—
133C4⋊C8×C53—21no—
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136#97×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
144C2×Q8×C53—21no—
145C2×Q8×C53C22×C411yescharacteristic
146C2×Q8×C53C22×C411yescharacteristic
147C22×C4×C53C22×C411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53#4111yescharacteristic
150C2×C22⋊D53—21no—
151#11719945—22no—
152#11719945—41no—
153C2×D212C22×C411yescharacteristic
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×Dic106C22×C411yescharacteristic
158M16×C53—22no—
159M16×C53—41no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—42no—
163C2×C8×C53C2×D411yescharacteristic
164C2×C8×C53C2×D411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53C22×C411yescharacteristic
175C4⋊C4×C53C22×C411yescharacteristic
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53C22×C411yescharacteristic
181C42×C53C2×D411yescharacteristic
182C42×C53#4111yescharacteristic
183C42×C53C2411yescharacteristic
184C22⋊Dic53—21no—
185#11719925—22no—
186#11719925—22no—
187#11719924—41no—
188#11719924—41no—
189#11719921—21no—
190#11719921—21no—
191C4⋊Dic53C22×C411yescharacteristic
192C4⋊Dic53C22×C411yescharacteristic
193#11719919—21no—
194#11719919—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322C22×C411yescharacteristic
203C2×#88322C22×C411yescharacteristic
204C2×#88322C22×C411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206#11719912—42no—
207#11719912—42no—
208C23×C53C4×D411yescharacteristic
209C22×D53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214Q8×C53—81no—
215C2×C4×C53—21no—
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×C53C4×D411yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C23×C411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C22⋊D53—42no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234D212—21no—
235D212—21no—
236C4×D53—42no—
237C4×D53—41no—
238C4×D53—41no—
239C4×D53—81no—
240Dic106—21no—
241Dic106#13112yes—
242C8×C53—21no—
243C8×C53—21no—
244C8×C53—21no—
245C8×C53—41no—
246C8×C53—41no—
247C8×C53#13712yes—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—21no—
252#88322—41no—
253#88322—41no—
254#88322—41no—
255#88322—41no—
256#18013—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53C2×C4×D411yescharacteristic
261C2×D53—41no—
262C2×D53—41no—
263C2×D53—41no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53C2×#13111yescharacteristic
273C4×C53#56811yescharacteristic
274C4×C53C2×#13711yescharacteristic
275C4×C53#57511yescharacteristic
276Dic53—22no—
277Dic53—41no—
278Dic53—41no—
279#3107—531no—
280#2971—531no—
281#2962D5311yescharacteristic, complemented
282#1592—531no—
283#1590—531no—
284#1588—531no—
285#1587—531no—
286#1584—531no—
287#1583—531no—
288#1579—531no—
289C8×Q16—531no—
290#1573—531no—
291#1497—531no—
292#1477—531no—
293#1453—531no—
294C2×C53—41no—
295C2×C53#293311yescharacteristic
296C2×C53C4×#13711yescharacteristic
297C2×C53#295711yescharacteristic
298D53—81no—
299#533C2×D5311yescharacteristic
300C4×#41—531no—
301#485—531no—
302#485—531no—
303#475—531no—
304#474—531no—
305#462—531no—
306#461—531no—
307#452—531no—
308#452—531no—
309#446—21no—
310#446—21no—
311#446C2×D5311yescharacteristic
312#446C2×D5311yescharacteristic
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8C2×D5311yescharacteristic
316C8×Q8C2×D5311yescharacteristic
317C4×Q16—531no—
318C4×Q16—531no—
319C4×SD16—531no—
320C4×SD16—531no—
321#436—531no—
322#436—531no—
323#436—1061no—
324#436—1061no—
325#432—531no—
326#432—1061no—
327#431—531no—
328#431—531no—
329#424—21no—
330C2×C4⋊C8C2×D5311yescharacteristic
331#405—21no—
332C4×M16—1061no—
333C4×M16C2×D5311yescharacteristic
334#335—531no—
335#335—531no—
336#334—531no—
337#334—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342Q8⋊C8—531no—
343Q8⋊C8—531no—
344D4⋊C8—531no—
345D4⋊C8—531no—
346#322—531no—
347#322—531no—
348#322—531no—
349C82—531no—
350C53#1801311yescharacteristic, complemented, Sylow
351C2×#41—531no—
352C2×Q16—531no—
353C2×Q16—531no—
354C2×SD16—531no—
355C2×SD16—531no—
356C2×M16—1061no—
357C2×M16—1061no—
358C2×M16C22⋊D5311yescharacteristic
359C2×M16C22⋊D5311yescharacteristic
360C22×C8—21no—
361C4⋊Q8C4×D5311yescharacteristic
362C4⋊Q8C4×D5311yescharacteristic
363#125—21no—
364C22⋊Q8—21no—
365C22⋊Q8C4×D5311yescharacteristic
366C22⋊Q8C4×D5311yescharacteristic
367C4×Q8—21no—
368C4×Q8—531no—
369C4×Q8—531no—
370C4×Q8C22×D5311yescharacteristic
371C4×Q8C22×D5311yescharacteristic
372C4×D4—531no—
373C4×D4—531no—
374C4×D4—1061no—
375C4×D4—1061no—
376#117—21no—
377#117—531no—
378#117—1061no—
379C2×C42—1061no—
380C2×C42C22×D5311yescharacteristic
381#107—531no—
382#107—531no—
383#106—531no—
384#106—531no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—531no—
391C4⋊C8—531no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8C22×D5311yescharacteristic
398C4⋊C8C22×D5311yescharacteristic
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#103—531no—
404#103—531no—
405#102—531no—
406#102—531no—
407C22⋊C8—531no—
408C22⋊C8—531no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—21no—
414#97—21no—
415#97—1061no—
416#97C22⋊D5311yescharacteristic
417#97C22⋊D5311yescharacteristic
418C4×C8—21no—
419C4×C8—21no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8C22×D5311yescharacteristic
427C4×C8C22×D5311yescharacteristic
428#41—1062no—
429#41—2121no—
430C2×Q8—21no—
431C2×Q8—531no—
432C2×Q8C2×C4×D5311yescharacteristic
433C2×Q8C2×C4×D5311yescharacteristic
434C2×D4—531no—
435C2×D4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4C2×C4×D5311yescharacteristic
440C22×C4#1171994511yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442Q16—1062no—
443Q16—1062no—
444SD16—2121no—
445SD16—2121no—
446M16—22no—
447M16—41no—
448M16—2122no—
449M16—2122no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8C2×C22⋊D5311yescharacteristic
462C2×C8C2×C22⋊D5311yescharacteristic
463C2×C8C2×C22⋊D5311yescharacteristic
464C2×C8C2×C22⋊D5311yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—531no—
473C4⋊C4—531no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4C2×C4×D5311yescharacteristic
477C4⋊C4C2×C4×D5311yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—1061no—
481C22⋊C4—1061no—
482C22⋊C4—1061no—
483C42—21no—
484C42—21no—
485C42—1061no—
486C42—1061no—
487C42—1061no—
488C42C2×C4×D5311yescharacteristic
489C42#1171994511yescharacteristic
490C42C2×C22⋊D5311yescharacteristic
491C42C23×D5311yescharacteristic
492C23—1061no—
493C23C4×C22⋊D5311yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—81no—
499Q8—532no—
500Q8—1061no—
501D4—1061no—
502D4—1061no—
503D4—2122no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—4241no—
517C2×C4—2122no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4C4×C22⋊D5311yescharacteristic
521C2×C4C4×C22⋊D5311yescharacteristic
522C2×C4C4×C22⋊D5311yescharacteristic
523C2×C4C22×C4×D5311yescharacteristic
524C2×C4C22×C22⋊D5311yescharacteristic
525C2×C4C2×#1171994511yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—1061no—
535C8—2121no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8#1313324812yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—2121no—
544C22—2121no—
545C22—2121no—
546C22C2×C4×C22⋊D5311yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—1062no—
556C4—2121no—
557C4—2121no—
558C4#131×D5311yescharacteristic
559C4#1681138811yescharacteristic
560C4C2×#1313324811yescharacteristic
561C4#1681142111yescharacteristic
562C2—41no—
563C2—4241no—
564C2#2028343711yescharacteristic
565C2C4×#1313324811yescharacteristic
566C2#2028363011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#2962C2
socleC22×C53C2×C4×D4
prime radicalC2×#11719945C2×C4
cosocleC42×C53C24
hypercenter#2962D53
hypocenterC53#18013
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2962

Complements

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

NH ≅ G/Ntype
3. #20273615C2semidirect
4. #20273605C2semidirect
5. #20273593C2semidirect
6. #20273584C2semidirect
7. #20273563C2semidirect
8. #20273558C2semidirect
9. #20273532C2semidirect
10. C8×#11719925C2semidirect
11. #20273498C2semidirect
12. #20273188C2semidirect
14. #20273012C2semidirect
15. C53×#2962C2semidirect
281. #2962D53semidirect
350. C53#18013semidirect

Maximal subgroups

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

subgroupindex
#202847802
#202837332
#202736152
#202736052
#202735932
#202735842
#202735632
#202735582
#202735322
C8×#117199252
#202734982
#202731882
#202731002
#202730122
C53×#29622
#1801353