Finite Group Databasebeta

← #27367546

Subgroups — #27367546

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286052C211yesmaximal, characteristic
2#20285297C211yesmaximal, characteristic, complemented
3#20283691C211yesmaximal, characteristic
4#20274231C211yesmaximal, characteristic, complemented
5#20274226C211yesmaximal, characteristic, complemented
6#20274207C211yesmaximal, characteristic, complemented
7#20274202C211yesmaximal, characteristic, complemented
8#20274147C211yesmaximal, characteristic, complemented
9#20274143C211yesmaximal, characteristic, complemented
10#20274140C211yesmaximal, characteristic, complemented
11#20274136C211yesmaximal, characteristic, complemented
12C2×#16810373C211yesmaximal, characteristic, complemented
13#20273415C211yesmaximal, characteristic, complemented
14#20273413C211yesmaximal, characteristic, complemented
15C53×#3248C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811026C2211yescharacteristic
18#16811002C2211yescharacteristic
19#16810975C2211yescharacteristic
20#16810973C2211yescharacteristic
21#16810927—21no—
22#16810908C2211yescharacteristic
23#16810903C2211yescharacteristic
24#16810862—21no—
25#16810862—21no—
26#16810862—21no—
27#16810850—21no—
28#16810842—21no—
29#16810833C2211yescharacteristic
30#16810828C2211yescharacteristic
31#16810754C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33C4×#11719925—21no—
34C4×#11719925—21no—
35#16810701C2211yescharacteristic
36#16810695—21no—
37D4×#88322—21no—
38D4×#88322C2211yescharacteristic
39#16810684—21no—
40#16810680C2211yescharacteristic
41C2×C4⋊#88322C2211yescharacteristic
42#16810579—21no—
43C2×C4×#88322C2211yescharacteristic
44#16810373C2211yescharacteristic
45#16810373C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810370C2211yescharacteristic
49#16810370C2211yescharacteristic
50#16810369C2211yescharacteristic
51#16810369C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C53×#529C2211yescharacteristic
58C53×#479—21no—
59C53×#475C2211yescharacteristic
60C53×#474C2211yescharacteristic
61C53×#467—21no—
62C53×#465—21no—
63C53×#464—21no—
64C53×#462C2211yescharacteristic
65C53×#461C2211yescharacteristic
66C2×C4⋊C8×C53C2211yescharacteristic
67C2×#11719925—21no—
68C2×#11719925—21no—
69C2×#11719925—21no—
70C2×#11719923—21no—
71C2×#11719917—21no—
72C22×#88322—21no—
73C22×#88322—21no—
74C22×#88322D411yescharacteristic
75C22×#88322D411yescharacteristic
76#13133217C2311yescharacteristic
77#13133217C2311yescharacteristic
78#13133209—21no—
79#13133196—21no—
80#13133196—21no—
81C4×Dic106—21no—
82C4×Dic106—21no—
83#13133176—21no—
84#13133150—21no—
85#13133150—21no—
86#13133150—21no—
87#13133148—21no—
88C4⋊#88322—21no—
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133141—21no—
96#13133141—21no—
97#13133141—21no—
98#13133140—21no—
99#13133140—21no—
100#13133140—21no—
101#13133140—21no—
102#13133138—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106#13133128—21no—
107#13133128—21no—
108C4×#88322—21no—
109C4×#88322D411yescharacteristic
110C4×#88322D411yescharacteristic
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×Q16×C53—21no—
114C2×SD16×C53—21no—
115C22×C8×C53—21no—
116C4⋊Q8×C53C2311yescharacteristic
117C4⋊Q8×C53C2311yescharacteristic
118#126×C53—21no—
119#125×C53—21no—
120#125×C53D411yescharacteristic
121#124×C53D411yescharacteristic
122#123×C53—21no—
123C22⋊Q8×C53—21no—
124C22⋊Q8×C53—21no—
125C22⋊Q8×C53D411yescharacteristic
126#121×C53D411yescharacteristic
127C4×Q8×C53—21no—
128C4×Q8×C53—21no—
129C4×Q8×C53C2311yescharacteristic
130C4×D4×C53—21no—
131C4×D4×C53C2311yescharacteristic
132#117×C53—21no—
133#117×C53—21no—
134C2×C42×C53C2311yescharacteristic
135#107×C53—21no—
136#106×C53—21no—
137C4⋊C8×C53—21no—
138C4⋊C8×C53C2311yescharacteristic
139C4⋊C8×C53C2311yescharacteristic
140#103×C53—21no—
141#103×C53—21no—
142#103×C53—21no—
143#102×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53C2×D411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C2×D411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53#4111yescharacteristic
153C2×Dic106—21no—
154C2×Dic106—21no—
155Q16×C53—81no—
156SD16×C53—81no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—81no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—41no—
169C4⋊C4×C53—41no—
170C4⋊C4×C53C2×D411yescharacteristic
171C4⋊C4×C53C2×D411yescharacteristic
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—41no—
178C42×C53—21no—
179C42×C53—41no—
180C42×C53C2×D411yescharacteristic
181C42×C53C2×D411yescharacteristic
182C42×C53#4111yescharacteristic
183C42×C53C2411yescharacteristic
184C22⋊Dic53—41no—
185#11719925—42no—
186#11719925—42no—
187#11719925—81no—
188#11719923—81no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193#11719919—41no—
194#11719919—41no—
195C4×Dic53—41no—
196#11719917—42no—
197C2×#88322—22no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—41no—
203C2×#88322—42no—
204C2×#88322—81no—
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C2×#88322C2×D411yescharacteristic
209C23×C53—21no—
210C23×C53#12111yescharacteristic
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—41no—
214Q8×C53—41no—
215Q8×C53—41no—
216Q8×C53—41no—
217D4×C53—21no—
218D4×C53—21no—
219D4×C53—42no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53#12111yescharacteristic
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×Dic53—41no—
239C2×Dic53—41no—
240Dic106—41no—
241Dic106—41no—
242Dic106—41no—
243Dic106—41no—
244C8×C53—41no—
245C8×C53—41no—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—41no—
250#88322—41no—
251#88322—41no—
252#88322—41no—
253#88322#13512yes—
254#19228—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53#47312yes—
261C22×C53C2×#12111yescharacteristic
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—81no—
270C4×C53—81no—
271C4×C53C2×#13511yescharacteristic
272C4×C53#57511yescharacteristic
273C4×C53#57711yescharacteristic
274C4×C53#57711yescharacteristic
275Dic53—81no—
276Dic53—81no—
277#3248—531no—
278#3248D5311yescharacteristic
279#3170—531no—
280#2966—531no—
281#1694—531no—
282#1693—531no—
283#1690—531no—
284#1689—531no—
285#1680—531no—
286#1680—531no—
287#1679—531no—
288#1679—531no—
289C2×#334—531no—
290#1552—531no—
291#1552—531no—
292C2×C53—22no—
293C2×C53—81no—
294C2×C53#303511yescharacteristic
295C2×C53#306011yescharacteristic
296C2×C53C2×#47311yescharacteristic
297#533—531no—
298#533C2×D5311yescharacteristic
299#529C2×D5311yescharacteristic
300#500—531no—
301#494—531no—
302#487—531no—
303#487—531no—
304#479—21no—
305#479—1061no—
306#475—531no—
307#475C2×D5311yescharacteristic
308#474—531no—
309#474C2×D5311yescharacteristic
310#467—21no—
311#467—1061no—
312#467—1061no—
313#467—1061no—
314#465—21no—
315#465—1061no—
316#464—21no—
317#464—1061no—
318#462—531no—
319#462C2×D5311yescharacteristic
320#461—531no—
321#461C2×D5311yescharacteristic
322#446—531no—
323C8×Q8—531no—
324C4×Q16—1061no—
325C4×Q16—1061no—
326#436—531no—
327#435—1061no—
328C8×D4—531no—
329C8×D4—1061no—
330#433—1061no—
331#432—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8C2×D5311yescharacteristic
334#405—1061no—
335C2×C4×C8—531no—
336#334—531no—
337#334—531no—
338#334—531no—
339#334—531no—
340#332—531no—
341#332—531no—
342#332—531no—
343#332—531no—
344#332—531no—
345#332—531no—
346#332—531no—
347#332—531no—
348C53#1922811yescharacteristic, complemented, Sylow
349C2×Q16—21no—
350C2×Q16—1061no—
351C2×Q16—1061no—
352C2×Q16—1061no—
353C2×SD16—21no—
354C2×SD16—1061no—
355C2×M16—1061no—
356C22×C8—21no—
357C22×C8—531no—
358C22×C8—531no—
359C22×C8—1061no—
360C22×C8—1061no—
361C4⋊Q8—531no—
362C4⋊Q8—531no—
363C4⋊Q8C22×D5311yescharacteristic
364C4⋊Q8C22×D5311yescharacteristic
365#126—21no—
366#125—21no—
367#125—1061no—
368#125C22⋊D5311yescharacteristic
369#124C22⋊D5311yescharacteristic
370#123—21no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—1061no—
374C22⋊Q8—1061no—
375C22⋊Q8C22⋊D5311yescharacteristic
376#121C22⋊D5311yescharacteristic
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—1061no—
380C4×Q8—1061no—
381C4×Q8C22×D5311yescharacteristic
382C4×D4—21no—
383C4×D4C22×D5311yescharacteristic
384#117—21no—
385#117—21no—
386#117—1061no—
387C2×C42C22×D5311yescharacteristic
388#107—21no—
389#107—1061no—
390#107—1061no—
391#107—1061no—
392#106—21no—
393#106—1061no—
394C4⋊C8—21no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8C22×D5311yescharacteristic
403C4⋊C8C22×D5311yescharacteristic
404#103—21no—
405#103—21no—
406#103—21no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#103—1061no—
412#103—1061no—
413#103—1061no—
414#102—21no—
415#102—1061no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419#97—1061no—
420#97—1061no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—531no—
425C4×C8—1061no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8C2×C22⋊D5311yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D5311yescharacteristic
433C22×C4—21no—
434C22×C4D4×D5311yescharacteristic
435C22×C4#1171994711yescharacteristic
436C22×C4C2×C22⋊D5311yescharacteristic
437Q16—81no—
438Q16—4241no—
439Q16—2122no—
440Q16—2122no—
441SD16—81no—
442SD16—4241no—
443M16—2122no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—81no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—1062no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—4241no—
457C2×C8—2121no—
458C2×C8—2122no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—2121no—
474C4⋊C4—2121no—
475C4⋊C4C2×C22⋊D5311yescharacteristic
476C4⋊C4C2×C22⋊D5311yescharacteristic
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—41no—
483C22⋊C4—2121no—
484C42—21no—
485C42—41no—
486C42—2121no—
487C42D4×D5311yescharacteristic
488C42#1171994711yescharacteristic
489C42C2×C22⋊D5311yescharacteristic
490C42C23×D5311yescharacteristic
491C23—21no—
492C23#1313318911yescharacteristic
493Q8—21no—
494Q8—21no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—2121no—
500Q8—2121no—
501Q8—2121no—
502Q8—2121no—
503D4—21no—
504D4—21no—
505D4—42no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4#1313318911yescharacteristic
521C2×C4#1313318911yescharacteristic
522C2×C4#1313318911yescharacteristic
523C2×C4C2×D4×D5311yescharacteristic
524C2×C4C22×C22⋊D5311yescharacteristic
525C2×C4C2×#1171994711yescharacteristic
526C8—41no—
527C8—41no—
528C8—532no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—21no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22#1681089612yes—
542C22C2×#1313318911yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—81no—
551C4—81no—
552C4—4241no—
553C4—4241no—
554C4#135×D5311yescharacteristic
555C4#1681142011yescharacteristic
556C4#1681143111yescharacteristic
557C4#1681143411yescharacteristic
558C2—22no—
559C2—81no—
560C2#2028416611yescharacteristic
561C2#2028437411yescharacteristic
562C2C2×#1681089611yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3248C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#3248D53
hypocenterC53#19228
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3248

Complements

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

NH ≅ G/Ntype
2. #20285297C2semidirect
4. #20274231C2semidirect
5. #20274226C2semidirect
6. #20274207C2semidirect
7. #20274202C2semidirect
8. #20274147C2semidirect
9. #20274143C2semidirect
10. #20274140C2semidirect
11. #20274136C2semidirect
12. C2×#16810373C2semidirect
13. #20273415C2semidirect
14. #20273413C2semidirect
348. C53#19228semidirect

Maximal subgroups

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

subgroupindex
#202860522
#202852972
#202836912
#202742312
#202742262
#202742072
#202742022
#202741472
#202741432
#202741402
#202741362
C2×#168103732
#202734152
#202734132
C53×#32482
#1922853