Finite Group Databasebeta

← #27367831

Subgroups — #27367831

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285890C211yesmaximal, characteristic
2#20285456C211yesmaximal, characteristic, complemented
3#20283720C211yesmaximal, characteristic
4#20274295C211yesmaximal, characteristic, complemented
5#20274271C211yesmaximal, characteristic, complemented
6#20274212C211yesmaximal, characteristic, complemented
7#20274202C211yesmaximal, characteristic, complemented
8#20274164C211yesmaximal, characteristic, complemented
9#20274140C211yesmaximal, characteristic, complemented
10#20274095C211yesmaximal, characteristic, complemented
11#20274083C211yesmaximal, characteristic, complemented
12#20273465C211yesmaximal, characteristic, complemented
13#20273380C211yesmaximal, characteristic, complemented
14#20273373C211yesmaximal, characteristic, complemented
15C53×#3248C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811022—21no—
18#16811019C2211yescharacteristic
19#16811002C2211yescharacteristic
20#16810977—21no—
21#16810975C2211yescharacteristic
22#16810973C2211yescharacteristic
23#16810961—21no—
24#16810944—21no—
25#16810918C2211yescharacteristic
26#16810913C2211yescharacteristic
27#16810908C2211yescharacteristic
28#16810903C2211yescharacteristic
29#16810879—21no—
30#16810879—21no—
31#16810754C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33#16810728—21no—
34#16810728—21no—
35#16810701—21no—
36#16810701C2211yescharacteristic
37#16810695—21no—
38D4×#88322C2211yescharacteristic
39#16810684—21no—
40#16810676C2211yescharacteristic
41#16810676C2211yescharacteristic
42#16810579—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
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×#11719917—21no—
70C2×#11719917D411yescharacteristic
71C2×#11719917D411yescharacteristic
72C22×#88322—21no—
73#13133217C2311yescharacteristic
74#13133217C2311yescharacteristic
75#13133209—21no—
76#13133196—21no—
77#13133196—21no—
78C4×Dic106—21no—
79C4×Dic106—21no—
80#13133176—21no—
81#13133150—21no—
82#13133150—21no—
83#13133148—21no—
84#13133148—21no—
85#13133148—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#13133141—21no—
97#13133140—21no—
98#13133140—21no—
99#13133138—21no—
100#13133138—21no—
101C22⋊#88322—21no—
102C22⋊#88322—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105#13133128—21no—
106#13133128D411yescharacteristic
107#13133128D411yescharacteristic
108C4×#88322—21no—
109C4×#88322—21no—
110C4×#88322—21no—
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—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191#11719919—41no—
192#11719919—41no—
193C4×Dic53—41no—
194#11719917—22no—
195#11719917—41no—
196#11719917—42no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—42no—
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C23×C53—21no—
207C23×C53#12111yescharacteristic
208Q8×C53—21no—
209Q8×C53—21no—
210Q8×C53—41no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53—41no—
214D4×C53—21no—
215D4×C53—21no—
216D4×C53—81no—
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—21no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53#12111yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×Dic53—41no—
236C2×Dic53—41no—
237Dic106—41no—
238Dic106—41no—
239Dic106—81no—
240C8×C53—41no—
241C8×C53—41no—
242#88322—21no—
243#88322—21no—
244#88322—21no—
245#88322—41no—
246#88322—41no—
247#88322—41no—
248#88322—41no—
249#88322#13512yes—
250#19251—531nomaximal, Sylow
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53C2×#12111yescharacteristic
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—81no—
266C4×C53—81no—
267C4×C53C2×#13511yescharacteristic
268C4×C53#57511yescharacteristic
269C4×C53#57711yescharacteristic
270C4×C53#57811yescharacteristic
271Dic53—81no—
272Dic53—81no—
273#3248D5311yescharacteristic
274#3233—531no—
275#3185—531no—
276#2969—531no—
277#1704—531no—
278#1700—531no—
279#1691—531no—
280#1689—531no—
281#1683—531no—
282#1679—531no—
283#1672—531no—
284#1670—531no—
285#1565—531no—
286#1544—531no—
287#1542—531no—
288C2×C53—41no—
289C2×C53—81no—
290C2×C53#303511yescharacteristic
291C2×C53#306311yescharacteristic
292C2×C53#306911yescharacteristic
293#533—531no—
294#533C2×D5311yescharacteristic
295#529C2×D5311yescharacteristic
296#499—1061no—
297#498—531no—
298#494—531no—
299#488—1061no—
300#487—531no—
301#487—531no—
302#485—1061no—
303#482—1061no—
304#479—21no—
305#477—531no—
306#476—531no—
307#475—531no—
308#475C2×D5311yescharacteristic
309#474—531no—
310#474C2×D5311yescharacteristic
311#470—1061no—
312#470—1061no—
313#467—21no—
314#465—21no—
315#464—21no—
316#462C2×D5311yescharacteristic
317#461C2×D5311yescharacteristic
318#446—531no—
319C8×Q8—531no—
320#441—1061no—
321#441—1061no—
322#436—531no—
323#436—1061no—
324#435—1061no—
325C8×D4—531no—
326#433—1061no—
327#431—531no—
328#431—531no—
329C2×C4⋊C8C2×D5311yescharacteristic
330#405—1061no—
331C4×M16—531no—
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
336#333—531no—
337#333—531no—
338#332—531no—
339#332—531no—
340#332—531no—
341#332—531no—
342#332—531no—
343#332—531no—
344C53#1925111yescharacteristic, complemented, Sylow
345C2×Q16—21no—
346C2×Q16—1061no—
347C2×Q16—1061no—
348C2×SD16—21no—
349C2×M16—531no—
350C2×M16—531no—
351C2×M16—1061no—
352C22×C8—21no—
353C22×C8—1061no—
354C4⋊Q8—531no—
355C4⋊Q8—531no—
356C4⋊Q8C22×D5311yescharacteristic
357C4⋊Q8C22×D5311yescharacteristic
358#126—21no—
359#125—21no—
360#125—1061no—
361#125C22⋊D5311yescharacteristic
362#124C22⋊D5311yescharacteristic
363#123—21no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—1061no—
367C22⋊Q8—1061no—
368C22⋊Q8C22⋊D5311yescharacteristic
369#121C22⋊D5311yescharacteristic
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—1061no—
373C4×Q8—1061no—
374C4×Q8C22×D5311yescharacteristic
375C4×D4—21no—
376C4×D4C22×D5311yescharacteristic
377#117—21no—
378#117—21no—
379#117—1061no—
380C2×C42C22×D5311yescharacteristic
381#107—21no—
382#107—1061no—
383#107—1061no—
384#106—21no—
385#106—1061no—
386#106—1061no—
387#106—1061no—
388#106—1061no—
389C4⋊C8—21no—
390C4⋊C8—531no—
391C4⋊C8—531no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8C22×D5311yescharacteristic
397C4⋊C8C22×D5311yescharacteristic
398#103—21no—
399#103—21no—
400#103—21no—
401#103—1061no—
402#103—1061no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#103—1061no—
407#102—21no—
408#102—1061no—
409#102—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413C22⋊C8—1061no—
414#97—531no—
415#97—531no—
416#97—1061no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—1061no—
420C4×C8—1061no—
421C4×C8—1061no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8C2×C22⋊D5311yescharacteristic
427C2×D4—21no—
428C2×D4C2×C22⋊D5311yescharacteristic
429C22×C4—21no—
430C22×C4D4×D5311yescharacteristic
431C22×C4#1171994711yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433Q16—81no—
434Q16—2122no—
435Q16—2122no—
436SD16—81no—
437M16—1062no—
438M16—2122no—
439M16—2121no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—81no—
443C2×C8—531no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—2122no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—2121no—
467C4⋊C4—2121no—
468C4⋊C4C2×C22⋊D5311yescharacteristic
469C4⋊C4C2×C22⋊D5311yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—41no—
476C22⋊C4—2121no—
477C42—21no—
478C42—41no—
479C42—2121no—
480C42D4×D5311yescharacteristic
481C42#1171994711yescharacteristic
482C42C2×C22⋊D5311yescharacteristic
483C42C23×D5311yescharacteristic
484C23—21no—
485C23#1313318911yescharacteristic
486Q8—21no—
487Q8—21no—
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—4241no—
493Q8—2121no—
494Q8—2121no—
495D4—21no—
496D4—21no—
497D4—81no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4#1313318911yescharacteristic
513C2×C4#1313318911yescharacteristic
514C2×C4#1313318911yescharacteristic
515C2×C4C2×D4×D5311yescharacteristic
516C2×C4C22×C22⋊D5311yescharacteristic
517C2×C4C2×#1171994711yescharacteristic
518C8—41no—
519C8—41no—
520C8—532no—
521C8—1061no—
522C8—1061no—
523C8—1061no—
524C8—2121no—
525C8—2121no—
526C8—2121no—
527C8—2121no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22C2×#1313318911yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—81no—
543C4—81no—
544C4—4241no—
545C4—4241no—
546C4#135×D5311yescharacteristic
547C4#1681142011yescharacteristic
548C4#1681143411yescharacteristic
549C4#1681143811yescharacteristic
550C2—41no—
551C2—81no—
552C2#2028416611yescharacteristic
553C2#2028440811yescharacteristic
554C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3248C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#3248D53
hypocenterC53#19251
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. #20285456C2semidirect
4. #20274295C2semidirect
5. #20274271C2semidirect
6. #20274212C2semidirect
7. #20274202C2semidirect
8. #20274164C2semidirect
9. #20274140C2semidirect
10. #20274095C2semidirect
11. #20274083C2semidirect
12. #20273465C2semidirect
13. #20273380C2semidirect
14. #20273373C2semidirect
344. C53#19251semidirect

Maximal subgroups

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

subgroupindex
#202858902
#202854562
#202837202
#202742952
#202742712
#202742122
#202742022
#202741642
#202741402
#202740952
#202740832
#202734652
#202733802
#202733732
C53×#32482
#1925153