Finite Group Databasebeta

← #27551436

Subgroups — #27551436

Order 13568.

1738 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280682C211yesmaximal, characteristic, complemented
2#20280679C213yesmaximal, complemented
3#20280677C211yesmaximal, characteristic, complemented
4#20280674C213yesmaximal, complemented
5#20280671C213yesmaximal, complemented
6#20280670C213yesmaximal, complemented
7#20280668C211yesmaximal, characteristic, complemented
8#20278018C211yesmaximal, characteristic, complemented
9#20278017C211yesmaximal, characteristic, complemented
10#20278014C213yesmaximal, complemented
11#20278012C213yesmaximal, complemented
12#20277200C213yesmaximal
13#20277194C213yesmaximal, complemented
14C2×#16810468C211yesmaximal, characteristic, complemented
15C53×#2643C211yesmaximal, characteristic, complemented
16C22×#11719947C2211yescharacteristic
17C2×#13133173C2213yes—
18C22×C4×Dic53C2213yes—
19C53⋊#401C2211yescharacteristic
20C53⋊#401C2213yes—
21C53⋊#401C2213yes—
22#16810565C2213yes—
23#16810565C2213yes—
24#16810564C2213yes—
25#16810564C2213yes—
26#16810563C2213yes—
27#16810562C2213yes—
28#16810561C2213yes—
29#16810552C2213yes—
30#16810552C2213yes—
31#16810551C2213yes—
32#16810551C2213yes—
33#16810550C2213yes—
34#16810549C2213yes—
35#16810548C2213yes—
36#16810547C2211yescharacteristic
37#16810546C2213yes—
38#16810545C2213yes—
39#16810527C2213yes—
40#16810527C2213yes—
41#16810523C2213yes—
42#16810523C2213yes—
43#16810520C2213yes—
44#16810520C2213yes—
45#16810518C2213yes—
46#16810518C2213yes—
47#16810503C2213yes—
48#16810502C2213yes—
49#16810500C2213yes—
50#16810498C2213yes—
51#16810496C2213yes—
52#16810495C2213yes—
53C4⋊C4×Dic53C2213yes—
54C4⋊C4×Dic53C2213yes—
55#16810474C2213yes—
56#16810474C2213yes—
57C22⋊C4×Dic53C2213yes—
58C22⋊C4×Dic53C2213yes—
59#16810470C2213yes—
60#16810470C2213yes—
61#16810468C2211yescharacteristic
62#16810468C2213yes—
63#16810468C2211yescharacteristic
64#16810468C2213yes—
65C2×#13133124C2213yes—
66C53×#401C2211yescharacteristic
67C53×#400C2213yes—
68C53×#397C2213yes—
69C53×#396C2211yescharacteristic
70C53×#386C2213yes—
71C53×#385C2213yes—
72C2×#95×C53C2211yescharacteristic
73C2×#11719947—212no—
74C22×Dic106C2311yescharacteristic
75C22×C22⋊D53C2311yescharacteristic
76C22×C22⋊D53C2311yescharacteristic
77C23×Dic53C2311yescharacteristic
78C23×Dic53C2313yes—
79C23×Dic53C2311yescharacteristic
80C22×C4×D53C2311yescharacteristic
81#13133173—212no—
82C2×C4⋊Dic53C2313yes—
83C2×C4⋊Dic53C2313yes—
84C2×#11719919C2313yes—
85C2×#11719919C2313yes—
86C2×#11719919C2313yes—
87C2×#11719919C2313yes—
88C2×#11719919C2313yes—
89C2×#11719919C2313yes—
90C2×C22⋊Dic53C2313yes—
91C2×C22⋊Dic53C2313yes—
92C2×C22⋊Dic53C2313yes—
93C2×#11719921C2313yes—
94C2×#11719921C2313yes—
95C2×#11719921C2313yes—
96C2×#11719921C2313yes—
97C2×C4×Dic53—212no—
98C2×C4×Dic53C2313yes—
99C2×C4×Dic53C2313yes—
100C2×C4×Dic53C2313yes—
101C2×C4×Dic53C2313yes—
102C2×C4×Dic53C2313yes—
103C2×C4×Dic53C2313yes—
104#13133124C2313yes—
105#13133124C2313yes—
106#13133124C2313yes—
107#13133124C2313yes—
108#13133124C2313yes—
109#13133124C2313yes—
110#13133124C2313yes—
111#13133124C2313yes—
112#13133124C2313yes—
113#13133124C2313yes—
114#13133124C2313yes—
115#13133124C2313yes—
116#13133124C2313yes—
117#13133124C2313yes—
118#13133124C2313yes—
119#13133124C2313yes—
120C22×D4×C53C2311yescharacteristic
121C23×C4×C53C2313yes—
122C2×C4⋊C4×C53C2313yes—
123C2×C4⋊C4×C53C2313yes—
124C2×C22⋊C4×C53C2313yes—
125C2×C22⋊C4×C53C2313yes—
126C2×C22⋊C4×C53C2313yes—
127#95×C53C2313yes—
128#95×C53C2313yes—
129#95×C53C2313yes—
130#95×C53C2313yes—
131#95×C53C2311yescharacteristic
132#95×C53C2313yes—
133C24×C53C2411yescharacteristic
134C24×C53C2411yescharacteristic
135C23×D53C2411yescharacteristic
136C2×D4×C53—26no—
137C22×C4×C53—26no—
138C22×C4×C53—43no—
139C22×C4×C53—43no—
140C22×C4×C53C2411yescharacteristic
141C22×C4×C53C2413yes—
142C22×C4×C53C2413yes—
143C22×C4×C53C2413yes—
144C22×C4×C53C2413yes—
145C2×C22⋊D53—26no—
146C2×C22⋊D53—26no—
147C22×Dic53—26no—
148C22×Dic53—26no—
149C22×Dic53—43no—
150C22×Dic53—43no—
151C22×Dic53#41112yes—
152C22×Dic53C2411yescharacteristic
153C22×Dic53C2413yes—
154C22×Dic53C2413yes—
155C22×Dic53C2413yes—
156C22×Dic53C2413yes—
157C22×Dic53C2411yescharacteristic
158C22×Dic53C2411yescharacteristic
159#11719947—416no—
160C2×C4×D53—26no—
161C2×Dic106—26no—
162C4⋊C4×C53—43no—
163C4⋊C4×C53—43no—
164C22⋊C4×C53—26no—
165C22⋊C4×C53—43no—
166C22⋊C4×C53—43no—
167C22⋊Dic53—26no—
168C22⋊Dic53—43no—
169C22⋊Dic53—43no—
170#11719921—43no—
171#11719921—43no—
172#11719921—43no—
173#11719921—43no—
174C4⋊Dic53—43no—
175C4⋊Dic53—43no—
176#11719919—26no—
177#11719919—26no—
178#11719919—43no—
179#11719919—43no—
180#11719919—43no—
181#11719919—43no—
182C4×Dic53—26no—
183C4×Dic53—26no—
184C4×Dic53—26no—
185C4×Dic53—26no—
186C4×Dic53—26no—
187C4×Dic53—26no—
188C23×C53—21no—
189C23×C53—23no—
190C23×C53—23no—
191C23×C53—21no—
192C23×C53—23no—
193C23×C53C2×#4116yes—
194C23×C53C2511yescharacteristic
195C22×D53—23no—
196C22×D53—23no—
197C22×D53—21no—
198D4×C53—44no—
199C2×C4×C53—23no—
200C2×C4×C53—23no—
201C2×C4×C53—23no—
202C2×C4×C53—23no—
203C2×C4×C53—23no—
204C2×C4×C53—23no—
205C2×C4×C53—23no—
206C2×C4×C53—23no—
207C2×C4×C53—23no—
208C2×C4×C53—23no—
209C2×C4×C53—23no—
210C2×C4×C53—23no—
211C2×C4×C53—23no—
212C2×C4×C53—412no—
213C22⋊D53—44no—
214C22⋊D53—44no—
215C2×Dic53—23no—
216C2×Dic53—23no—
217C2×Dic53—23no—
218C2×Dic53—23no—
219C2×Dic53—23no—
220C2×Dic53—23no—
221C2×Dic53—23no—
222C2×Dic53—23no—
223C2×Dic53—23no—
224C2×Dic53—23no—
225C2×Dic53—23no—
226C2×Dic53—23no—
227C2×Dic53—23no—
228C2×Dic53—412no—
229C2×Dic53—44no—
230C2×Dic53#126116yes—
231C2×Dic53C2×#4116yes—
232C2×Dic53C2×#4116yes—
233C4×D53—44no—
234Dic106—44no—
235#34639—531nomaximal, Sylow
236C22×C53—26no—
237C22×C53—26no—
238C22×C53—41no—
239C22×C53—43no—
240C22×C53—43no—
241C22×C53C2×#12614yes—
242C22×C53C22×#4113yes—
243C22×C53#58513yes—
244C22×C53#58511yescharacteristic
245C2×D53—41no—
246C2×D53—43no—
247C2×D53—43no—
248C4×C53—41no—
249C4×C53—43no—
250C4×C53—43no—
251C4×C53—43no—
252C4×C53—43no—
253Dic53—41no—
254Dic53—43no—
255Dic53—43no—
256Dic53—43no—
257Dic53—43no—
258Dic53C2×#12614yes—
259Dic53C2×#12614yes—
260#2645—531no—
261#2645—533no—
262#2644—533no—
263#2644—531no—
264#2643—533no—
265#2643—533no—
266#2643—531no—
267#2643D5311yescharacteristic, complemented
268#2401—531no—
269#2401—533no—
270#2401—533no—
271#2401—531no—
272#2297—533no—
273#2296—533no—
274C2×#376—531no—
275C2×C53—24no—
276C2×C53—81no—
277C2×C53C22×#12611yescharacteristic
278C2×C53#345513yes—
279C2×C53#345513yes—
280D53—81no—
281C22×#41—531no—
282C2×#117—533no—
283C22×C42—533no—
284#401—533no—
285#401—533no—
286#401—531no—
287#401C2×D5311yescharacteristic
288#400—533no—
289#400—533no—
290#400—533no—
291#400—533no—
292#400—533no—
293#400—533no—
294#400—533no—
295#400C2×D5313yes—
296#397—533no—
297#397—533no—
298#397—533no—
299#397—533no—
300#397—533no—
301#397—533no—
302#397—533no—
303#397C2×D5313yes—
304#396—533no—
305#396—533no—
306#396—531no—
307#396C2×D5311yescharacteristic
308#391—533no—
309#391—533no—
310#390—533no—
311#390—533no—
312#389—533no—
313#389—533no—
314#389—533no—
315#389—533no—
316#386—533no—
317#386—533no—
318#386—533no—
319#386C2×D5313yes—
320#385—533no—
321#385—533no—
322#385—533no—
323#385C2×D5313yes—
324C4×C4⋊C4—533no—
325C4×C4⋊C4—533no—
326C4×C4⋊C4—533no—
327C4×C4⋊C4—533no—
328C4×C22⋊C4—533no—
329C4×C22⋊C4—533no—
330C4×C22⋊C4—533no—
331C4×C22⋊C4—533no—
332#376—533no—
333#376—531no—
334#376—533no—
335#376—531no—
336C2×#95—533no—
337C2×#95C2×D5311yescharacteristic
338C53#3463911yescharacteristic, complemented, Sylow
339C2×#41—10612no—
340C22×Q8—531no—
341C22×D4—531no—
342C22×D4—531no—
343C22×D4C22×D5311yescharacteristic
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—533no—
347C23×C4—531no—
348C23×C4C22×D5313yes—
349#117—10612no—
350C2×C4⋊C4—533no—
351C2×C4⋊C4—533no—
352C2×C4⋊C4—533no—
353C2×C4⋊C4—533no—
354C2×C4⋊C4—533no—
355C2×C4⋊C4—533no—
356C2×C4⋊C4—533no—
357C2×C4⋊C4—533no—
358C2×C4⋊C4C22×D5313yes—
359C2×C4⋊C4C22×D5313yes—
360C2×C22⋊C4—533no—
361C2×C22⋊C4—533no—
362C2×C22⋊C4—533no—
363C2×C22⋊C4—533no—
364C2×C22⋊C4—533no—
365C2×C22⋊C4—533no—
366C2×C22⋊C4—533no—
367C2×C22⋊C4C22×D5313yes—
368C2×C22⋊C4C22×D5313yes—
369C2×C22⋊C4C22×D5313yes—
370C2×C42—533no—
371C2×C42—533no—
372C2×C42—533no—
373C2×C42—533no—
374C2×C42—533no—
375C2×C42—533no—
376C2×C42—10612no—
377#95—533no—
378#95—533no—
379#95—533no—
380#95—533no—
381#95—533no—
382#95—533no—
383#95—533no—
384#95—533no—
385#95—533no—
386#95—533no—
387#95—533no—
388#95—533no—
389#95—533no—
390#95—533no—
391#95—533no—
392#95—533no—
393#95C22×D5311yescharacteristic
394#95C22×D5313yes—
395#95C22×D5313yes—
396#95C22×D5313yes—
397#95C22×D5313yes—
398#95C22×D5313yes—
399C24—531no—
400C24C23×D5311yescharacteristic
401C24C23×D5311yescharacteristic
402#41—21216no—
403C2×Q8—1066no—
404C2×D4—26no—
405C2×D4—1066no—
406C2×D4—1066no—
407C22×C4—26no—
408C22×C4—43no—
409C22×C4—43no—
410C22×C4—5312no—
411C22×C4—531no—
412C22×C4—531no—
413C22×C4—531no—
414C22×C4—533no—
415C22×C4—533no—
416C22×C4—533no—
417C22×C4—533no—
418C22×C4—1066no—
419C22×C4—1066no—
420C22×C4—1066no—
421C22×C4—2123no—
422C22×C4—2123no—
423C22×C4C23×D5311yescharacteristic
424C22×C4C23×D5313yes—
425C22×C4C23×D5313yes—
426C22×C4C23×D5313yes—
427C22×C4C23×D5313yes—
428C4⋊C4—43no—
429C4⋊C4—43no—
430C4⋊C4—1066no—
431C4⋊C4—1066no—
432C4⋊C4—2123no—
433C4⋊C4—2123no—
434C4⋊C4—2123no—
435C4⋊C4—2123no—
436C4⋊C4—2123no—
437C4⋊C4—2123no—
438C22⋊C4—26no—
439C22⋊C4—43no—
440C22⋊C4—43no—
441C22⋊C4—1066no—
442C22⋊C4—2123no—
443C22⋊C4—2123no—
444C22⋊C4—2123no—
445C22⋊C4—2123no—
446C22⋊C4—2123no—
447C22⋊C4—2123no—
448C42—1066no—
449C42—1066no—
450C42—1066no—
451C42—1066no—
452C42—1066no—
453C42—1066no—
454C23—23no—
455C23—21no—
456C23—23no—
457C23—23no—
458C23—21no—
459C23—1063no—
460C23—1063no—
461C23—1061no—
462C23#41×D5316yes—
463C23C24×D5311yescharacteristic
464Q8—2124no—
465D4—44no—
466D4—2124no—
467D4—2124no—
468C2×C4—23no—
469C2×C4—23no—
470C2×C4—23no—
471C2×C4—23no—
472C2×C4—23no—
473C2×C4—23no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—23no—
477C2×C4—23no—
478C2×C4—23no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—412no—
482C2×C4—5316no—
483C2×C4—536no—
484C2×C4—536no—
485C2×C4—1063no—
486C2×C4—1063no—
487C2×C4—1063no—
488C2×C4—1063no—
489C2×C4—1063no—
490C2×C4—1063no—
491C2×C4—1063no—
492C2×C4—1063no—
493C2×C4—1063no—
494C2×C4—1063no—
495C2×C4—1063no—
496C2×C4—1063no—
497C2×C4—1063no—
498C2×C4—2124no—
499C2×C4—2124no—
500C2×C4—21212no—
501C22—26no—
502C22—26no—
503C22—43no—
504C22—43no—
505C22—41no—
506C22—2123no—
507C22—2123no—
508C22—2121no—
509C22#126×D5314yes—
510C22C2×#41×D5313yes—
511C22#1681145813yes—
512C22#1681146011yescharacteristic
513C4—43no—
514C4—43no—
515C4—43no—
516C4—43no—
517C4—41no—
518C4—534no—
519C4—534no—
520C4—2123no—
521C4—2123no—
522C4—2123no—
523C4—2123no—
524C4—2121no—
525C2—24no—
526C2—81no—
527C2—4241no—
528C2C2×#126×D5311yescharacteristic
529C2#2028809713yes—
530C2#2028810113yes—
subgroupG/H
center, FrattiniC23C24×D53
commutator, socle, cosocleC23×C53C25
FittingC53×#2643C2
prime radicalC22×#11719947C22
hypercenter#2643D53
hypocenterC53#34639
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2643

Complements

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

NH ≅ G/Ntype
1. #20280682C2semidirect
2. #20280679C2semidirect
3. #20280677C2semidirect
4. #20280674C2semidirect
5. #20280671C2semidirect
6. #20280670C2semidirect
7. #20280668C2semidirect
8. #20278018C2semidirect
9. #20278017C2semidirect
10. #20278014C2semidirect
11. #20278012C2semidirect
13. #20277194C2semidirect
14. C2×#16810468C2semidirect
15. C53×#2643C2semidirect
267. #2643D53semidirect
338. C53#34639semidirect

Maximal subgroups

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

subgroupindex
#202806822
#202806792
#202806772
#202806742
#202806712
#202806702
#202806682
#202780182
#202780172
#202780142
#202780122
#202772002
#202771942
C2×#168104682
C53×#26432
#3463953