Finite Group Databasebeta

← #27369213

Subgroups — #27369213

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285942C211yesmaximal, characteristic
2#20285097C211yesmaximal, characteristic, complemented
3#20283720C211yesmaximal, characteristic
4#20274299C211yesmaximal, characteristic, complemented
5#20274292C211yesmaximal, characteristic, complemented
6#20274271C211yesmaximal, characteristic, complemented
7#20274264C211yesmaximal, characteristic, complemented
8#20274175C211yesmaximal, characteristic, complemented
9#20274168C211yesmaximal, characteristic, complemented
10#20274147C211yesmaximal, characteristic, complemented
11#20274140C211yesmaximal, characteristic, complemented
12#20273480C211yesmaximal, characteristic, complemented
13#20273425C211yesmaximal, characteristic, complemented
14#20273401C211yesmaximal, characteristic, complemented
15C53×#3252C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17#16811033—21no—
18#16810993—21no—
19#16810993—21no—
20#16810982—21no—
21#16810975C2211yescharacteristic
22#16810975C2211yescharacteristic
23#16810975C2211yescharacteristic
24#16810975C2211yescharacteristic
25#16810956—21no—
26#16810939—21no—
27#16810908C2211yescharacteristic
28#16810908C2211yescharacteristic
29#16810903C2211yescharacteristic
30#16810903C2211yescharacteristic
31#16810754C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33#16810701—21no—
34#16810701C2211yescharacteristic
35#16810695—21no—
36D4×#88322C2211yescharacteristic
37#16810684—21no—
38#16810680C2211yescharacteristic
39#16810676C2211yescharacteristic
40#16810631—21no—
41#16810631—21no—
42#16810579—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810369C2211yescharacteristic
51#16810369C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C53×#529C2211yescharacteristic
58C53×#479—21no—
59C53×#477C2211yescharacteristic
60C53×#476C2211yescharacteristic
61C53×#470—21no—
62C53×#468—21no—
63C53×#464—21no—
64C53×#462C2211yescharacteristic
65C53×#461C2211yescharacteristic
66C53×#423C2211yescharacteristic
67C22×Dic106—21no—
68C2×#11719917—21no—
69C2×#11719917D411yescharacteristic
70C2×#11719917D411yescharacteristic
71C22×#88322—21no—
72#13133217—21no—
73#13133217—21no—
74#13133217—21no—
75#13133217C2311yescharacteristic
76#13133217C2311yescharacteristic
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79#13133150—21no—
80#13133150—21no—
81#13133148—21no—
82#13133148—21no—
83C4⋊#88322—21no—
84C4⋊#88322—21no—
85C4⋊#88322C2311yescharacteristic
86C4⋊#88322C2311yescharacteristic
87C4⋊#88322C2311yescharacteristic
88C4⋊#88322C2311yescharacteristic
89#13133141—21no—
90#13133141—21no—
91#13133140—21no—
92#13133140—21no—
93#13133140—21no—
94#13133140—21no—
95#13133140—21no—
96#13133140—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#13133128—21no—
107#13133128D411yescharacteristic
108#13133128D411yescharacteristic
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×Q16×C53—21no—
114C2×SD16×C53—21no—
115C2×M16×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
153C22×Dic53—41no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—42no—
157C2×Dic106—42no—
158C2×Dic106—41no—
159Q16×C53—81no—
160SD16×C53—81no—
161M16×C53—81no—
162C2×C8×C53—21no—
163C2×C8×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—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—41no—
173C4⋊C4×C53—41no—
174C4⋊C4×C53C2×D411yescharacteristic
175C4⋊C4×C53C2×D411yescharacteristic
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—41no—
182C42×C53—21no—
183C42×C53—41no—
184C42×C53C2×D411yescharacteristic
185C42×C53C2×D411yescharacteristic
186C42×C53#4111yescharacteristic
187C42×C53C2411yescharacteristic
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—41no—
193C4⋊Dic53—41no—
194#11719917—41no—
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—81no—
213D4×C53—21no—
214D4×C53—21no—
215D4×C53—81no—
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×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—82no—
237Dic106—41no—
238Dic106—41no—
239Dic106—41no—
240Dic106—41no—
241Dic106—82no—
242C8×C53—41no—
243C8×C53—41no—
244#88322—21no—
245#88322—21no—
246#88322—21no—
247#88322—21no—
248#88322—41no—
249#88322—41no—
250#88322—41no—
251#88322—41no—
252#19381—531nomaximal, Sylow
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53C2×#12111yescharacteristic
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53—81no—
268C4×C53—81no—
269C4×C53#57511yescharacteristic
270C4×C53#57511yescharacteristic
271C4×C53#57711yescharacteristic
272C4×C53#57811yescharacteristic
273Dic53—81no—
274Dic53—81no—
275#3252D5311yescharacteristic
276#3239—531no—
277#3139—531no—
278#2969—531no—
279#1704—531no—
280#1703—531no—
281#1700—531no—
282#1699—531no—
283#1684—531no—
284#1683—531no—
285#1680—531no—
286#1679—531no—
287#1569—531no—
288#1554—531no—
289#1549—531no—
290C2×C53—41no—
291C2×C53—81no—
292C2×C53#304011yescharacteristic
293C2×C53#306311yescharacteristic
294C2×C53#306911yescharacteristic
295#533C2×D5311yescharacteristic
296C2×C4⋊Q8—531no—
297#529C2×D5311yescharacteristic
298#501—1061no—
299#491—1061no—
300#491—1061no—
301#489—1061no—
302#487—531no—
303#487—531no—
304#487—531no—
305#487—531no—
306#484—1061no—
307#481—1061no—
308#479—21no—
309#477C2×D5311yescharacteristic
310#476C2×D5311yescharacteristic
311#475—531no—
312#475—531no—
313#474—531no—
314#474—531no—
315#470—21no—
316#468—21no—
317#464—21no—
318#462C2×D5311yescharacteristic
319#461C2×D5311yescharacteristic
320#446—531no—
321C8×Q8—531no—
322#436—531no—
323#436—1061no—
324#435—1061no—
325C8×D4—531no—
326#433—1061no—
327#432—531no—
328#431—531no—
329#423C2×D5311yescharacteristic
330#419—1061no—
331#419—1061no—
332#405—1061no—
333C4×M16—531no—
334#335—531no—
335#335—531no—
336#334—531no—
337#334—531no—
338#333—531no—
339#333—531no—
340#332—531no—
341#332—531no—
342#332—531no—
343#332—531no—
344#332—531no—
345#332—531no—
346C53#1938111yescharacteristic, complemented, Sylow
347C22×Q8—1061no—
348C2×Q16—21no—
349C2×SD16—21no—
350C2×M16—21no—
351C2×M16—531no—
352C2×M16—531no—
353C2×M16—1061no—
354C22×C8—1061no—
355C4⋊Q8—531no—
356C4⋊Q8—531no—
357C4⋊Q8—1061no—
358C4⋊Q8—1061no—
359C4⋊Q8—1061no—
360C4⋊Q8C22×D5311yescharacteristic
361C4⋊Q8C22×D5311yescharacteristic
362#126—21no—
363#125—21no—
364#125C22⋊D5311yescharacteristic
365#124C22⋊D5311yescharacteristic
366#123—21no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8C22⋊D5311yescharacteristic
370#121C22⋊D5311yescharacteristic
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8C22×D5311yescharacteristic
374C4×D4—21no—
375C4×D4C22×D5311yescharacteristic
376#117—21no—
377#117—21no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C42C22×D5311yescharacteristic
381#107—21no—
382#107—1061no—
383#107—1061no—
384#106—21no—
385#106—1061no—
386#106—1061no—
387C4⋊C8—21no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—531no—
391C4⋊C8—531no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8C22×D5311yescharacteristic
395C4⋊C8C22×D5311yescharacteristic
396#103—21no—
397#103—21no—
398#103—21no—
399#103—1061no—
400#103—1061no—
401#103—1061no—
402#103—1061no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#102—21no—
410#102—1061no—
411#102—1061no—
412C22⋊C8—1061no—
413C22⋊C8—1061no—
414C22⋊C8—1061no—
415C22⋊C8—1061no—
416#97—531no—
417#97—531no—
418#97—1061no—
419#97—1061no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—1061no—
423C4×C8—1061no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—1061no—
427C2×Q8—1061no—
428C2×Q8—2122no—
429C2×Q8—2122no—
430C2×Q8—2121no—
431C2×Q8C2×C22⋊D5311yescharacteristic
432C2×D4—21no—
433C2×D4C2×C22⋊D5311yescharacteristic
434C22×C4—21no—
435C22×C4—2121no—
436C22×C4D4×D5311yescharacteristic
437C22×C4#1171994711yescharacteristic
438C22×C4C2×C22⋊D5311yescharacteristic
439Q16—81no—
440SD16—81no—
441M16—81no—
442M16—2121no—
443M16—2121no—
444M16—2122no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—2122no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—2121no—
471C4⋊C4—2121no—
472C4⋊C4C2×C22⋊D5311yescharacteristic
473C4⋊C4C2×C22⋊D5311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—41no—
480C42—21no—
481C42—41no—
482C42D4×D5311yescharacteristic
483C42#1171994711yescharacteristic
484C42C2×C22⋊D5311yescharacteristic
485C42C23×D5311yescharacteristic
486C23—21no—
487C23#1313318911yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493Q8—4242no—
494Q8—2121no—
495Q8—2121no—
496Q8—2121no—
497Q8—2121no—
498D4—21no—
499D4—21no—
500D4—81no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—4242no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4#1313318911yescharacteristic
517C2×C4#1313318911yescharacteristic
518C2×C4#1313318911yescharacteristic
519C2×C4C2×D4×D5311yescharacteristic
520C2×C4C22×C22⋊D5311yescharacteristic
521C2×C4C2×#1171994711yescharacteristic
522C8—41no—
523C8—41no—
524C8—1061no—
525C8—1061no—
526C8—1061no—
527C8—1061no—
528C8—2121no—
529C8—2121no—
530C8—2121no—
531C8—2121no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22C2×#1313318911yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—4241no—
549C4—4241no—
550C4#1681142211yescharacteristic
551C4#1681142211yescharacteristic
552C4#1681143411yescharacteristic
553C4#1681143811yescharacteristic
554C2—41no—
555C2—81no—
556C2#2028421111yescharacteristic
557C2#2028440811yescharacteristic
558C2#2028445411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3252C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#3252D53
hypocenterC53#19381
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3252

Complements

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

NH ≅ G/Ntype
2. #20285097C2semidirect
4. #20274299C2semidirect
5. #20274292C2semidirect
6. #20274271C2semidirect
7. #20274264C2semidirect
8. #20274175C2semidirect
9. #20274168C2semidirect
10. #20274147C2semidirect
11. #20274140C2semidirect
12. #20273480C2semidirect
13. #20273425C2semidirect
14. #20273401C2semidirect
346. C53#19381semidirect

Maximal subgroups

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

subgroupindex
#202859422
#202850972
#202837202
#202742992
#202742922
#202742712
#202742642
#202741752
#202741682
#202741472
#202741402
#202734802
#202734252
#202734012
C53×#32522
#1938153