Finite Group Databasebeta

← #27366573

Subgroups — #27366573

Order 13568.

620 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#20285297C211yesmaximal, characteristic, complemented
2#20284786C211yesmaximal, characteristic
3#41×#88322C211yesmaximal, characteristic
4#20274147C211yesmaximal, characteristic, complemented
5#20274143C211yesmaximal, characteristic, complemented
6#20274140C211yesmaximal, characteristic, complemented
7#20274136C211yesmaximal, characteristic, complemented
8#20274107C211yesmaximal, characteristic, complemented
9#20274102C211yesmaximal, characteristic, complemented
10#20274083C211yesmaximal, characteristic, complemented
11#20274078C211yesmaximal, characteristic, complemented
12#20273461C211yesmaximal, characteristic, complemented
13#20273380C211yesmaximal, characteristic, complemented
14#20273379C211yesmaximal, characteristic, complemented
15C53×#3107C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811026C2211yescharacteristic
18#16811002C2211yescharacteristic
19#16810975C2211yescharacteristic
20#16810973C2211yescharacteristic
21#16810964—21no—
22#16810908C2211yescharacteristic
23#16810903C2211yescharacteristic
24#16810862—21no—
25#16810862—21no—
26#16810833C2211yescharacteristic
27#16810828C2211yescharacteristic
28#16810785—21no—
29Q8×#88322C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31C4×#11719925—21no—
32C4×#11719925—21no—
33C4×#11719925—21no—
34C4×#11719923—21no—
35D4×#88322—21no—
36D4×#88322—21no—
37D4×#88322C2211yescharacteristic
38D4×#88322C2211yescharacteristic
39#16810676—21no—
40#16810676C2211yescharacteristic
41#16810676C2211yescharacteristic
42C2×C4×#88322—21no—
43C2×C4×#88322C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810370C2211yescharacteristic
49#16810370C2211yescharacteristic
50#16810369C2211yescharacteristic
51#16810369C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C4×#41×C53C2211yescharacteristic
58C53×#485—21no—
59C53×#475C2211yescharacteristic
60C53×#474C2211yescharacteristic
61C53×#462C2211yescharacteristic
62C53×#461C2211yescharacteristic
63C53×#452—21no—
64C4×Q16×C53—21no—
65C4×SD16×C53—21no—
66C53×#431C2211yescharacteristic
67C2×#11719925—21no—
68C2×#11719925—21no—
69C2×#11719925—21no—
70C2×#11719923—21no—
71C22×#88322—21no—
72C22×#88322—21no—
73C22×#88322D411yescharacteristic
74C22×#88322D411yescharacteristic
75#13133217C2311yescharacteristic
76#13133217C2311yescharacteristic
77#13133209—21no—
78#13133196—21no—
79#13133196—21no—
80C4×Dic106—21no—
81C4×Dic106—21no—
82#13133176—21no—
83#13133150—21no—
84#13133150—21no—
85#13133150—21no—
86#13133148—21no—
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322C2311yescharacteristic
90C4⋊#88322C2311yescharacteristic
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93#13133141—21no—
94#13133141—21no—
95#13133141—21no—
96#13133140—21no—
97#13133140—21no—
98#13133140—21no—
99#13133140—21no—
100#13133138—21no—
101C22⋊#88322—21no—
102C22⋊#88322—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C4×#88322—21no—
106C4×#88322—21no—
107C4×#88322—21no—
108C4×#88322—21no—
109C4×#88322D411yescharacteristic
110C4×#88322D411yescharacteristic
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×#41×C53D411yescharacteristic
114C2×Q16×C53—21no—
115C2×SD16×C53—21no—
116C4⋊Q8×C53C2311yescharacteristic
117C4⋊Q8×C53C2311yescharacteristic
118#125×C53—21no—
119C22⋊Q8×C53—21no—
120C22⋊Q8×C53—21no—
121C4×Q8×C53—21no—
122C4×Q8×C53—21no—
123C4×Q8×C53D411yescharacteristic
124C4×Q8×C53C2311yescharacteristic
125C4×D4×C53—21no—
126C4×D4×C53—21no—
127C4×D4×C53D411yescharacteristic
128C4×D4×C53C2311yescharacteristic
129#117×C53—21no—
130#117×C53—21no—
131#117×C53D411yescharacteristic
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—21no—
135#106×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53—21no—
139#103×C53—21no—
140#103×C53—21no—
141#102×C53—21no—
142C22⋊C8×C53—21no—
143C4×C8×C53—21no—
144#41×C53—22no—
145#41×C53—22no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53C2×D411yescharacteristic
149C2×D4×C53—21no—
150C2×D4×C53C2×D411yescharacteristic
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53#4111yescharacteristic
157C2×Dic106—21no—
158C2×Dic106—21no—
159Q16×C53—42no—
160SD16×C53—42no—
161C2×C8×C53—21no—
162C2×C8×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—21no—
169C4⋊C4×C53—41no—
170C4⋊C4×C53—41no—
171C4⋊C4×C53C2×D411yescharacteristic
172C4⋊C4×C53C2×D411yescharacteristic
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—41no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—41no—
181C42×C53C2×D411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53#4111yescharacteristic
184C42×C53C2411yescharacteristic
185C22⋊Dic53—41no—
186#11719925—42no—
187#11719925—42no—
188#11719925—42no—
189#11719923—42no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194#11719919—41no—
195#11719919—41no—
196C4×Dic53—41no—
197C2×#88322—22no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—41no—
203C2×#88322—42no—
204C2×#88322—42no—
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C2×#88322C2×D411yescharacteristic
209C23×C53—21no—
210C23×C53#12111yescharacteristic
211Q8×C53—21no—
212Q8×C53—41no—
213Q8×C53—41no—
214Q8×C53—41no—
215Q8×C53—41no—
216Q8×C53#13512yes—
217D4×C53—24no—
218D4×C53—21no—
219D4×C53#13512yes—
220C2×C4×C53—24no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×Dic53—41no—
241C2×Dic53—41no—
242Dic106—41no—
243Dic106—41no—
244Dic106—41no—
245Dic106—41no—
246C8×C53—41no—
247C8×C53—41no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#88322—41no—
255#88322#13512yes—
256#19129—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—22no—
260C22×C53—21no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53C2×#12111yescharacteristic
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—22no—
267C4×C53—21no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53—81no—
271C4×C53—81no—
272C4×C53#47312yes—
273C4×C53C2×#13511yescharacteristic
274C4×C53C2×#13511yescharacteristic
275C4×C53C2×#13511yescharacteristic
276C4×C53#57511yescharacteristic
277Dic53—81no—
278Dic53—81no—
279#3170—531no—
280#3107—531no—
281#3107D5311yescharacteristic
282C8×#41—531no—
283#1680—531no—
284#1680—531no—
285#1679—531no—
286#1679—531no—
287#1674—531no—
288#1673—531no—
289#1670—531no—
290#1669—531no—
291#1564—531no—
292#1544—531no—
293#1544—531no—
294C2×C53—41no—
295C2×C53—42no—
296C2×C53#303511yescharacteristic
297C2×C53#305011yescharacteristic
298C2×C53C2×#47311yescharacteristic
299#533—531no—
300#533C2×D5311yescharacteristic
301C4×#41C2×D5311yescharacteristic
302#500—531no—
303#494—531no—
304#487—531no—
305#487—531no—
306#485—21no—
307#485—1061no—
308#475—531no—
309#475C2×D5311yescharacteristic
310#474—531no—
311#474C2×D5311yescharacteristic
312#467—1061no—
313#467—1061no—
314#462—531no—
315#462C2×D5311yescharacteristic
316#461—531no—
317#461C2×D5311yescharacteristic
318#452—21no—
319#452—1061no—
320C8×Q8—531no—
321C8×Q8—531no—
322C4×Q16—21no—
323C4×Q16—1061no—
324C4×Q16—1061no—
325C4×Q16—1061no—
326C4×SD16—21no—
327C4×SD16—1061no—
328C8×D4—531no—
329C8×D4—531no—
330C8×D4—1061no—
331C8×D4—1061no—
332#431—531no—
333#431—531no—
334#431—1061no—
335#431C2×D5311yescharacteristic
336C2×C4×C8—531no—
337C2×C4×C8—1061no—
338#335—531no—
339#335—531no—
340#334—531no—
341#334—531no—
342#332—531no—
343#332—531no—
344#332—531no—
345#332—531no—
346#332—531no—
347#332—531no—
348#332—531no—
349#332—531no—
350C53#1912911yescharacteristic, complemented, Sylow
351C2×#41C22⋊D5311yescharacteristic
352C2×Q16—21no—
353C2×Q16—1061no—
354C2×Q16—1061no—
355C2×Q16—1061no—
356C2×SD16—21no—
357C2×SD16—1061no—
358C22×C8—531no—
359C22×C8—531no—
360C22×C8—1061no—
361C22×C8—1061no—
362C4⋊Q8—531no—
363C4⋊Q8—531no—
364C4⋊Q8C22×D5311yescharacteristic
365C4⋊Q8C22×D5311yescharacteristic
366#125—21no—
367#125—1061no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—1061no—
371C22⋊Q8—1061no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—1061no—
375C4×Q8—1061no—
376C4×Q8C22⋊D5311yescharacteristic
377C4×Q8C22×D5311yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C22⋊D5311yescharacteristic
381C4×D4C22×D5311yescharacteristic
382#117—21no—
383#117—21no—
384#117—1061no—
385#117C22⋊D5311yescharacteristic
386C2×C42—21no—
387C2×C42C22×D5311yescharacteristic
388#107—21no—
389#107—1061no—
390#107—1061no—
391#107—1061no—
392#106—21no—
393#106—1061no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8C22×D5311yescharacteristic
401C4⋊C8C22×D5311yescharacteristic
402#103—21no—
403#103—21no—
404#103—21no—
405#103—1061no—
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#103—1061no—
412#102—21no—
413#102—1061no—
414C22⋊C8—21no—
415C22⋊C8—1061no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419C4×C8—21no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8—1061no—
427C4×C8—1061no—
428#41—22no—
429#41—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—1061no—
433C2×Q8—1061no—
434C2×Q8C2×C22⋊D5311yescharacteristic
435C2×D4—21no—
436C2×D4C2×C22⋊D5311yescharacteristic
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4D4×D5311yescharacteristic
441C22×C4#1171994711yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443Q16—42no—
444Q16—2122no—
445Q16—2122no—
446Q16—2122no—
447SD16—42no—
448SD16—2122no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—1062no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—2122no—
461C2×C8—2121no—
462C2×C8—2122no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—2121no—
476C4⋊C4—2121no—
477C4⋊C4C2×C22⋊D5311yescharacteristic
478C4⋊C4C2×C22⋊D5311yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—41no—
483C22⋊C4—2121no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—41no—
488C42—2121no—
489C42D4×D5311yescharacteristic
490C42#1171994711yescharacteristic
491C42C2×C22⋊D5311yescharacteristic
492C42C23×D5311yescharacteristic
493C23—21no—
494C23#1313318911yescharacteristic
495Q8—21no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—2121no—
501Q8—2121no—
502Q8—2121no—
503Q8—2121no—
504Q8#1313323712yes—
505D4—24no—
506D4—21no—
507D4#1313323712yes—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4#1313318911yescharacteristic
525C2×C4#1313318911yescharacteristic
526C2×C4#1313318911yescharacteristic
527C2×C4C2×D4×D5311yescharacteristic
528C2×C4C22×C22⋊D5311yescharacteristic
529C2×C4C2×#1171994711yescharacteristic
530C8—41no—
531C8—41no—
532C8—532no—
533C8—1061no—
534C8—1061no—
535C8—1061no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8—2121no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—41no—
545C22—41no—
546C22C2×#1313318911yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—4241no—
556C4—4241no—
557C4#1681089612yes—
558C4#135×D5311yescharacteristic
559C4C2×#1313323711yescharacteristic
560C4C2×#1313323711yescharacteristic
561C4#1681142011yescharacteristic
562C2—41no—
563C2—42no—
564C2#2028416611yescharacteristic
565C2#2028429311yescharacteristic
566C2C2×#1681089611yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3107C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenter#3107D53
hypocenterC53#19129
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
1. #20285297C2semidirect
4. #20274147C2semidirect
5. #20274143C2semidirect
6. #20274140C2semidirect
7. #20274136C2semidirect
8. #20274107C2semidirect
9. #20274102C2semidirect
10. #20274083C2semidirect
11. #20274078C2semidirect
12. #20273461C2semidirect
13. #20273380C2semidirect
14. #20273379C2semidirect
350. C53#19129semidirect

Maximal subgroups

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

subgroupindex
#202852972
#202847862
#41×#883222
#202741472
#202741432
#202741402
#202741362
#202741072
#202741022
#202740832
#202740782
#202734612
#202733802
#202733792
C53×#31072
#1912953