Finite Group Databasebeta

← #27590179

Subgroups — #27590179

Order 13568.

2130 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#20283096C213yesmaximal, complemented
2#20283095C211yesmaximal, characteristic, complemented
3#20283081C213yesmaximal, complemented
4#20283080C211yesmaximal, characteristic, complemented
5#20283033C213yesmaximal, complemented
6#20283031C211yesmaximal, characteristic, complemented
7#20283029C213yesmaximal, complemented
8#20283027C211yesmaximal, characteristic, complemented
9C4×#13133212C213yesmaximal, complemented
10C4×#13133211C211yesmaximal, characteristic, complemented
11C4×#13133210C213yesmaximal, complemented
12#20277201C213yesmaximal
13#20277195C213yesmaximal, complemented
14C43×D53C211yesmaximal, characteristic, complemented
15C4×#126×C53C211yesmaximal, characteristic, complemented
16C22×#11719945C2211yescharacteristic
17C2×#13133212C2213yes—
18C2×#13133211C2211yescharacteristic
19C2×#13133210C2213yes—
20C2×#13133174C2213yes—
21C2×C42×D53C2211yescharacteristic
22C2×C42×D53C2213yes—
23C2×C42×D53C2213yes—
24#16810527C2213yes—
25#16810527C2213yes—
26#16810523C2213yes—
27#16810523C2213yes—
28#16810520C2213yes—
29#16810520C2213yes—
30#16810518C2213yes—
31#16810518C2213yes—
32#16810517C2213yes—
33#16810516C2213yes—
34#16810515C2213yes—
35#16810514C2213yes—
36#16810513C2213yes—
37#16810512C2213yes—
38#16810511C2213yes—
39#16810510C2213yes—
40#16810509C2213yes—
41#16810508C2213yes—
42#16810507C2213yes—
43#16810506C2213yes—
44#16810505C2213yes—
45#16810504C2213yes—
46#16810501C2213yes—
47#16810501C2213yes—
48#16810499C2213yes—
49#16810499C2213yes—
50#16810497C2213yes—
51#16810497C2213yes—
52#16810494C2213yes—
53#16810494C2213yes—
54#16810491C2211yescharacteristic
55#16810490C2213yes—
56#16810490C2213yes—
57C4×#11719919C2213yes—
58C4×#11719919C2213yes—
59C4×C4⋊Dic53C2213yes—
60C4×#11719921C2213yes—
61C4×#11719921C2213yes—
62C4×C22⋊Dic53C2213yes—
63C42×Dic53C2213yes—
64C42×Dic53C2211yescharacteristic
65C42×Dic53C2213yes—
66C2×#126×C53C2211yescharacteristic
67C53×#388C2213yes—
68C53×#387C2213yes—
69C53×#383C2211yescharacteristic
70C4×C4⋊C4×C53C2213yes—
71C4×C22⋊C4×C53C2213yes—
72C43×C53C2211yescharacteristic
73C2×#11719945—212no—
74C22×Dic106C2311yescharacteristic
75C22×D212C2311yescharacteristic
76C22×C22⋊D53C2311yescharacteristic
77C22×C22⋊D53C2311yescharacteristic
78C22×C4×D53C2313yes—
79C22×C4×D53C2313yes—
80C22×C4×D53C2311yescharacteristic
81C22×C4×D53C2311yescharacteristic
82#13133212—212no—
83#13133211—24no—
84#13133210—212no—
85#13133174—212no—
86C2×C4⋊Dic53C2313yes—
87C2×C4⋊Dic53C2313yes—
88C2×#11719919C2313yes—
89C2×#11719919C2313yes—
90C2×#11719919C2313yes—
91C2×#11719919C2313yes—
92C2×#11719919C2313yes—
93C2×#11719919C2313yes—
94C2×C22⋊Dic53C2313yes—
95C2×C22⋊Dic53C2313yes—
96C2×#11719921C2313yes—
97C2×#11719921C2313yes—
98C2×#11719921C2313yes—
99C2×#11719921C2313yes—
100C2×#11719921C2313yes—
101C2×#11719921C2313yes—
102C2×C4×Dic53C2313yes—
103C2×C4×Dic53C2313yes—
104C2×C4×Dic53C2313yes—
105C2×C4×Dic53C2313yes—
106C2×C4×Dic53C2313yes—
107C2×C4×Dic53C2313yes—
108C2×C4×Dic53C2313yes—
109C2×C4×Dic53C2313yes—
110C42×D53—24no—
111C42×D53—212no—
112C42×D53—212no—
113#13133124C2313yes—
114#13133124C2313yes—
115#13133124C2313yes—
116#13133124C2313yes—
117#13133124C2313yes—
118#13133124C2313yes—
119#13133124C2313yes—
120#13133124C2313yes—
121#13133124C2313yes—
122#13133124C2313yes—
123#13133124C2313yes—
124#13133124C2313yes—
125C23×C4×C53C2311yescharacteristic
126#126×C53—24no—
127C2×C4⋊C4×C53C2313yes—
128C2×C4⋊C4×C53C2313yes—
129C2×C22⋊C4×C53C2313yes—
130C2×C22⋊C4×C53C2313yes—
131C2×C42×C53C2313yes—
132C2×C42×C53C2313yes—
133C2×C42×C53C2313yes—
134C2×C42×C53C2311yescharacteristic
135#95×C53C2313yes—
136#95×C53C2313yes—
137#95×C53C2313yes—
138#95×C53C2313yes—
139C24×C53C2411yescharacteristic
140C23×D53C2411yescharacteristic
141C23×D53C2411yescharacteristic
142C22×C4×C53—26no—
143C22×C4×C53C2411yescharacteristic
144C22×C4×C53C2413yes—
145C22×C4×C53C2413yes—
146C22×C4×C53C2413yes—
147C22×C4×C53C2413yes—
148C22×C4×C53C2411yescharacteristic
149C2×C22⋊D53—26no—
150C2×C22⋊D53—26no—
151C22×Dic53C2411yescharacteristic
152C22×Dic53C2413yes—
153C22×Dic53C2413yes—
154C22×Dic53C2413yes—
155C22×Dic53C2413yes—
156C22×Dic53C2411yescharacteristic
157#11719945—416no—
158C2×D212—26no—
159C2×C4×D53—26no—
160C2×C4×D53—26no—
161C2×C4×D53—26no—
162C2×C4×D53—26no—
163C2×C4×D53—26no—
164C2×C4×D53—26no—
165C2×C4×D53—26no—
166C2×C4×D53#41112yes—
167C2×Dic106—26no—
168C4⋊C4×C53—26no—
169C4⋊C4×C53—26no—
170C22⋊C4×C53—26no—
171C22⋊C4×C53—26no—
172C42×C53—26no—
173C42×C53—26no—
174C42×C53—26no—
175C42×C53#4114yescomplemented
176C22⋊Dic53—26no—
177C22⋊Dic53—26no—
178#11719921—26no—
179#11719921—26no—
180#11719921—26no—
181#11719921—26no—
182#11719921—26no—
183#11719921—26no—
184C4⋊Dic53—26no—
185C4⋊Dic53—26no—
186#11719919—26no—
187#11719919—26no—
188#11719919—26no—
189#11719919—26no—
190#11719919—26no—
191#11719919—26no—
192C4×Dic53—26no—
193C4×Dic53—26no—
194C4×Dic53—26no—
195C4×Dic53—26no—
196C4×Dic53—26no—
197C4×Dic53—26no—
198C4×Dic53—26no—
199C4×Dic53#41112yescomplemented
200C23×C53—23no—
201C23×C53—23no—
202C23×C53—21no—
203C23×C53C2511yescharacteristic
204C22×D53—23no—
205C22×D53—21no—
206C22×D53—23no—
207C22×D53—23no—
208C22×D53—21no—
209C22×D53C2×#4116yes—
210C2×C4×C53—23no—
211C2×C4×C53—23no—
212C2×C4×C53—23no—
213C2×C4×C53—23no—
214C2×C4×C53—23no—
215C2×C4×C53—23no—
216C2×C4×C53—23no—
217C2×C4×C53—23no—
218C2×C4×C53—23no—
219C2×C4×C53—23no—
220C2×C4×C53—23no—
221C2×C4×C53—44no—
222C2×C4×C53C2×#4116yes—
223C2×C4×C53C2×#4116yes—
224C2×C4×C53C2×#4116yes—
225C22⋊D53—44no—
226C22⋊D53—44no—
227C2×Dic53—23no—
228C2×Dic53—23no—
229C2×Dic53—23no—
230C2×Dic53—23no—
231C2×Dic53—23no—
232C2×Dic53—23no—
233C2×Dic53—23no—
234C2×Dic53—23no—
235C2×Dic53—23no—
236C2×Dic53—23no—
237C2×Dic53—23no—
238C2×Dic53C2×#4116yes—
239C2×Dic53C2×#4116yes—
240C2×Dic53C2×#4116yes—
241D212—44no—
242C4×D53—224no—
243C4×D53—224no—
244C4×D53—44no—
245C4×D53#126116yescomplemented
246Dic106—44no—
247#36641—531nomaximal, Sylow
248C22×C53—41no—
249C22×C53—43no—
250C22×C53—43no—
251C22×C53C22×#4113yes—
252C22×C53C22×#4113yes—
253C22×C53C22×#4111yescharacteristic
254C2×D53—26no—
255C2×D53—26no—
256C2×D53—41no—
257C2×D53—43no—
258C2×D53—43no—
259C2×D53C2×#12614yes—
260C4×C53—26no—
261C4×C53—26no—
262C4×C53—41no—
263C4×C53—43no—
264C4×C53—43no—
265C4×C53C2×#12614yes—
266Dic53—26no—
267Dic53—26no—
268Dic53—41no—
269Dic53—43no—
270Dic53—43no—
271Dic53C2×#12614yes—
272#2863—533no—
273#2863—531no—
274#2860—533no—
275#2860—531no—
276#2852—533no—
277#2852—533no—
278#2852—531no—
279#2852—531no—
280C4×#126—533no—
281C4×#126—533no—
282C4×#126—531no—
283C4×#126D5311yescharacteristic, complemented
284#2297—533no—
285#2296—533no—
286C2×C43—531no—
287C2×C53—81no—
288C2×C53C22×#12611yescharacteristic
289C2×C53#344413yes—
290C2×C53#344413yes—
291D53—24no—
292D53—81no—
293C22×#41—531no—
294C2×#126—533no—
295C2×#126—533no—
296C2×#126—531no—
297C2×#126C2×D5311yescharacteristic
298C2×#117—533no—
299C22×C42—533no—
300C22×C42—531no—
301C22×C42—533no—
302#391—533no—
303#391—533no—
304#390—533no—
305#390—533no—
306#389—533no—
307#389—533no—
308#389—533no—
309#389—533no—
310#388—533no—
311#388—533no—
312#388—533no—
313#388—533no—
314#388—533no—
315#388—533no—
316#388—533no—
317#388C2×D5313yes—
318#387—533no—
319#387—533no—
320#387—533no—
321#387—533no—
322#387—533no—
323#387—533no—
324#387—533no—
325#387C2×D5313yes—
326#386—533no—
327#386—533no—
328#386—533no—
329#386—533no—
330#385—533no—
331#385—533no—
332#385—533no—
333#385—533no—
334#383—531no—
335#383—533no—
336#383—533no—
337#383C2×D5311yescharacteristic
338C4×C4⋊C4—533no—
339C4×C4⋊C4—533no—
340C4×C4⋊C4—533no—
341C4×C4⋊C4C2×D5313yes—
342C4×C22⋊C4—533no—
343C4×C22⋊C4—533no—
344C4×C22⋊C4—533no—
345C4×C22⋊C4C2×D5313yes—
346C43—533no—
347C43—533no—
348C43—531no—
349C43C2×D5311yescharacteristic
350C53#3664111yescharacteristic, complemented, Sylow
351C2×#41—10612no—
352C22×Q8—531no—
353C22×D4—531no—
354C22×D4—531no—
355C22×D4—531no—
356C23×C4—531no—
357C23×C4—533no—
358C23×C4—533no—
359C23×C4—531no—
360C23×C4C22×D5311yescharacteristic
361#126—24no—
362#126—10612no—
363#126—10612no—
364#126—1064no—
365#117—10612no—
366C2×C4⋊C4—533no—
367C2×C4⋊C4—533no—
368C2×C4⋊C4—533no—
369C2×C4⋊C4—533no—
370C2×C4⋊C4—533no—
371C2×C4⋊C4—533no—
372C2×C4⋊C4—533no—
373C2×C4⋊C4—533no—
374C2×C4⋊C4C22×D5313yes—
375C2×C4⋊C4C22×D5313yes—
376C2×C22⋊C4—533no—
377C2×C22⋊C4—533no—
378C2×C22⋊C4—533no—
379C2×C22⋊C4—533no—
380C2×C22⋊C4—533no—
381C2×C22⋊C4—533no—
382C2×C22⋊C4—533no—
383C2×C22⋊C4—533no—
384C2×C22⋊C4C22×D5313yes—
385C2×C22⋊C4C22×D5313yes—
386C2×C42—533no—
387C2×C42—533no—
388C2×C42—533no—
389C2×C42—533no—
390C2×C42—533no—
391C2×C42—533no—
392C2×C42—533no—
393C2×C42—533no—
394C2×C42—10612no—
395C2×C42—10612no—
396C2×C42—1064no—
397C2×C42C22×D5313yes—
398C2×C42C22×D5313yes—
399C2×C42C22×D5313yes—
400C2×C42C22×D5311yescharacteristic
401#95—533no—
402#95—533no—
403#95—533no—
404#95—533no—
405#95—533no—
406#95—533no—
407#95—533no—
408#95—533no—
409#95—533no—
410#95—533no—
411#95—533no—
412#95—533no—
413#95C22×D5313yes—
414#95C22×D5313yes—
415#95C22×D5313yes—
416#95C22×D5313yes—
417C24—531no—
418C24—531no—
419C24C23×D5311yescharacteristic
420#41—21216no—
421C2×Q8—1066no—
422C2×D4—1066no—
423C2×D4—1066no—
424C2×D4—1066no—
425C22×C4—26no—
426C22×C4—531no—
427C22×C4—5312no—
428C22×C4—533no—
429C22×C4—533no—
430C22×C4—531no—
431C22×C4—533no—
432C22×C4—533no—
433C22×C4—1066no—
434C22×C4—1066no—
435C22×C4—1066no—
436C22×C4—1066no—
437C22×C4—1066no—
438C22×C4—1066no—
439C22×C4—1066no—
440C22×C4C23×D5311yescharacteristic
441C22×C4C23×D5313yes—
442C22×C4C23×D5313yes—
443C22×C4C23×D5313yes—
444C22×C4C23×D5313yes—
445C22×C4C23×D5311yescharacteristic
446C4⋊C4—26no—
447C4⋊C4—26no—
448C4⋊C4—1066no—
449C4⋊C4—1066no—
450C4⋊C4—1066no—
451C4⋊C4—1066no—
452C4⋊C4—1066no—
453C4⋊C4—1066no—
454C4⋊C4—1066no—
455C4⋊C4—1066no—
456C22⋊C4—26no—
457C22⋊C4—26no—
458C22⋊C4—1066no—
459C22⋊C4—1066no—
460C22⋊C4—1066no—
461C22⋊C4—1066no—
462C22⋊C4—1066no—
463C22⋊C4—1066no—
464C22⋊C4—1066no—
465C22⋊C4—1066no—
466C42—26no—
467C42—26no—
468C42—26no—
469C42—5312no—
470C42—1066no—
471C42—1066no—
472C42—1066no—
473C42—1066no—
474C42—1066no—
475C42—1066no—
476C42—1066no—
477C42#1171994514yescomplemented
478C23—23no—
479C23—23no—
480C23—21no—
481C23—536no—
482C23—1063no—
483C23—1061no—
484C23—1063no—
485C23—1063no—
486C23—1061no—
487C23C24×D5311yescharacteristic
488Q8—2124no—
489D4—2124no—
490D4—2124no—
491D4—2124no—
492C2×C4—23no—
493C2×C4—23no—
494C2×C4—23no—
495C2×C4—23no—
496C2×C4—23no—
497C2×C4—23no—
498C2×C4—23no—
499C2×C4—23no—
500C2×C4—23no—
501C2×C4—23no—
502C2×C4—23no—
503C2×C4—44no—
504C2×C4—536no—
505C2×C4—5316no—
506C2×C4—536no—
507C2×C4—536no—
508C2×C4—10624no—
509C2×C4—10624no—
510C2×C4—1063no—
511C2×C4—1063no—
512C2×C4—1063no—
513C2×C4—1063no—
514C2×C4—1063no—
515C2×C4—1063no—
516C2×C4—1063no—
517C2×C4—1063no—
518C2×C4—1063no—
519C2×C4—1063no—
520C2×C4—1063no—
521C2×C4—2124no—
522C2×C4#41×D5316yes—
523C2×C4#41×D5316yes—
524C2×C4C2×#1171994516yes—
525C22—43no—
526C22—43no—
527C22—41no—
528C22—534no—
529C22—1066no—
530C22—1066no—
531C22—2123no—
532C22—2123no—
533C22—2121no—
534C22C2×#41×D5313yes—
535C22C2×#41×D5313yes—
536C22C22×#1171994511yescharacteristic
537C4—26no—
538C4—26no—
539C4—43no—
540C4—43no—
541C4—41no—
542C4—534no—
543C4—1066no—
544C4—1066no—
545C4—2123no—
546C4—2123no—
547C4—2121no—
548C4#126×D5314yes—
549C2—81no—
550C2—1064no—
551C2—4241no—
552C2C2×#126×D5311yescharacteristic
553C2#2028800813yes—
554C2#2028800913yes—
subgroupG/H
centerC22×C4C23×D53
commutator, socle, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC4×#126×C53C2
prime radicalC22×#11719945C22
hypercenterC4×#126D53
hypocenterC53#36641
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C4×#126

Complements

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

NH ≅ G/Ntype
1. #20283096C2semidirect
2. #20283095C2semidirect
3. #20283081C2semidirect
4. #20283080C2semidirect
5. #20283033C2semidirect
6. #20283031C2semidirect
7. #20283029C2semidirect
8. #20283027C2semidirect
9. C4×#13133212C2semidirect
10. C4×#13133211C2semidirect
11. C4×#13133210C2semidirect
13. #20277195C2semidirect
14. C43×D53C2semidirect
15. C4×#126×C53C2semidirect
175. C42×C53#41semidirect
199. C4×Dic53#41semidirect
245. C4×D53#126semidirect
283. C4×#126D53semidirect
350. C53#36641semidirect
477. C42#11719945semidirect

Maximal subgroups

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

subgroupindex
#202830962
#202830952
#202830812
#202830802
#202830332
#202830312
#202830292
#202830272
C4×#131332122
C4×#131332112
C4×#131332102
#202772012
#202771952
C43×D532
C4×#126×C532
#3664153