Finite Group Databasebeta

← #27591247

Subgroups — #27591247

Order 13568.

2214 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20282998C212yesmaximal, complemented
2#20281927C214yesmaximal, complemented
3#20281837C214yesmaximal, complemented
4#20281357C214yesmaximal, complemented
5#20280958C214yesmaximal, complemented
6#20280817C211yesmaximal, characteristic, complemented
7#20278454C214yesmaximal, complemented
8#20278249C214yesmaximal, complemented
9#20277415C212yesmaximal, complemented
10C2×#16810482C211yesmaximal, characteristic, complemented
11C53×#2646C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C2×#13133202C2214yescomplemented
14C2×#13133192C2212yes—
15C2×#13133192C2212yes—
16C2×#13133192C2214yes—
17C2×#13133191C2212yes—
18C2×#13133191C2214yes—
19C2×#13133187C2214yes—
20C2×#13133177C2214yes—
21C22×#11719919C2212yes—
22C22×#11719921C2212yes—
23#16810560C2214yescomplemented
24#16810558C2214yescomplemented
25#16810556C2214yescomplemented
26#16810552C2214yescomplemented
27#16810545C2214yes—
28#16810541C2214yes—
29#16810541C2214yescomplemented
30#16810540C2214yes—
31#16810533C2214yes—
32#16810533C2214yescomplemented
33#16810524C2214yes—
34#16810515C2214yes—
35#16810513C2214yescomplemented
36#16810512C2214yescomplemented
37#16810510C2214yescomplemented
38#16810506C2214yescomplemented
39#16810501C2214yes—
40#16810501C2214yes—
41#16810501C2214yes—
42#16810501C2214yescomplemented
43#16810500C2214yes—
44#16810500C2214yes—
45#16810494C2214yescomplemented
46#16810494C2214yescomplemented
47#16810482C2214yes—
48#16810482C2214yes—
49#16810480C2212yes—
50#16810480C2212yes—
51C2×#13133124C2211yescharacteristic
52C53×#400C2214yescomplemented
53C53×#398C2214yescomplemented
54C53×#385C2214yes—
55C53×#380C2212yes—
56C2×#95×C53C2211yescharacteristic
57C24×D53C2311yescharacteristic
58C22×D212C2314yes—
59C22×D212C2314yes—
60C22×D212C2312yes—
61C22×D212C2314yes—
62C22×C22⋊D53C2314yes—
63C23×Dic53C2311yescharacteristic
64C23×Dic53C2312yes—
65C22×C4×D53—24no—
66C22×C4×D53—44no—
67C22×C4×D53C2312yes—
68C22×C4×D53C2312yes—
69C22×C4×D53C2314yes—
70#13133202—48no—
71#13133192—44no—
72#13133192—44no—
73#13133192—48no—
74#13133191—28no—
75#13133191—48no—
76#13133187—48no—
77#13133177—48no—
78C2×C4⋊Dic53C2312yes—
79C2×C4⋊Dic53C2314yes—
80C2×#11719919—44no—
81C2×#11719919C2314yes—
82C2×#11719919C2314yes—
83C2×#11719919C2314yes—
84C2×#11719921—44no—
85C2×#11719921C2314yes—
86C2×#11719921C2314yes—
87C2×#11719921C2314yes—
88C2×#11719921C2314yes—
89C2×#11719921C2314yes—
90C2×#11719921C2314yes—
91C2×#11719921C2314yes—
92C2×#11719921C2314yes—
93C2×#11719921C2314yes—
94C2×#11719921C2314yes—
95C2×#11719921C2314yes—
96C2×#11719921C2314yes—
97C2×C4×Dic53C2314yes—
98#13133124C2314yes—
99#13133124C2314yes—
100#13133124C2314yes—
101#13133124C2314yes—
102#13133124C2314yes—
103C23×C4×C53C2312yes—
104C23×C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2312yes—
106C2×C4⋊C4×C53C2314yes—
107C2×C4⋊C4×C53C2314yes—
108C2×C4⋊C4×C53C2312yes—
109C2×C22⋊C4×C53C2314yes—
110#95×C53C2314yes—
111#95×C53C2314yes—
112#95×C53C2314yes—
113#95×C53C2314yes—
114C24×C53C2411yescharacteristic
115C23×D53—22no—
116C23×D53—22no—
117C23×D53—22no—
118C23×D53—24no—
119C23×D53—42no—
120C23×D53C2414yes—
121C23×D53C2411yescharacteristic
122C23×D53C2411yescharacteristic
123C22×C4×C53—24no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53#4114yes—
128C22×C4×C53C2414yes—
129C22×C4×C53C2414yes—
130C22×C4×C53C2414yes—
131C22×C4×C53C2412yes—
132C2×C22⋊D53—28no—
133C2×C22⋊D53—28no—
134C2×C22⋊D53—28no—
135C22×Dic53—22no—
136C22×Dic53—24no—
137C22×Dic53—24no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53C2414yes—
141C22×Dic53C2414yes—
142C22×Dic53C2412yes—
143C2×D212—28no—
144C2×D212—28no—
145C2×D212—24no—
146C2×D212—44no—
147C2×D212—44no—
148C2×D212—44no—
149C2×D212—42no—
150C2×D212—42no—
151C2×D212—44no—
152C2×D212—44no—
153C2×D212—44no—
154C2×D212—44no—
155C2×C4×D53—24no—
156C2×C4×D53—24no—
157C2×C4×D53—24no—
158C2×C4×D53—24no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53—416no—
162C2×C4×D53—44no—
163C2×C4×D53—44no—
164C2×C4×D53—44no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53—44no—
168C4⋊C4×C53—44no—
169C22⋊C4×C53—28no—
170#11719921—28no—
171#11719921—28no—
172#11719921—28no—
173#11719921—28no—
174#11719921—44no—
175#11719921—44no—
176#11719921—44no—
177#11719921—44no—
178#11719921—44no—
179#11719921—44no—
180#11719921—44no—
181#11719921—44no—
182C4⋊Dic53—24no—
183C4⋊Dic53—44no—
184#11719919—28no—
185#11719919—44no—
186#11719919—44no—
187C4×Dic53—44no—
188C23×C53—22no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53C2×#4112yes—
192C23×C53H(5,𝔽2)14yes—
193C23×C53C2511yescharacteristic
194C22×D53—28no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—21no—
202C22×D53—21no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—48no—
211C22×D53—48no—
212C22×D53—44no—
213C22×D53—44no—
214C22×D53H(5,𝔽2)12yes—
215C22×D532−1+412yes—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—44no—
228C2×C4×C53—84no—
229C2×C4×C53C2×#4114yes—
230C22⋊D53—416no—
231C2×Dic53—28no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—24no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—84no—
241C2×Dic53H(5,𝔽2)14yes—
242D212—84no—
243D212—88no—
244D212—88no—
245D212—88no—
246C4×D53—48no—
247C4×D53—48no—
248C4×D53—88no—
249#36727—531nomaximal, Sylow
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—24no—
253C22×C53—24no—
254C22×C53#54014yes—
255C22×C53C22×#4111yescharacteristic
256C22×C53C2×H(5,𝔽2)12yes—
257C22×C53C2×H(5,𝔽2)11yescharacteristic
258C22×C53C2×H(5,𝔽2)12yes—
259C22×C53C2×2−1+411yescharacteristic
260C2×D53—22no—
261C2×D53—22no—
262C2×D53—24no—
263C2×D53—24no—
264C2×D53—416no—
265C2×D53—41no—
266C2×D53—41no—
267C2×D53—41no—
268C2×D53—41no—
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—44no—
272C2×D53—44no—
273C2×D53—44no—
274C2×D53—44no—
275C2×D53—44no—
276C2×D53—44no—
277C2×D53—44no—
278C4×C53—24no—
279C4×C53—44no—
280C4×C53—44no—
281C4×C53—44no—
282Dic53—24no—
283Dic53—44no—
284Dic53—44no—
285#2847—532no—
286#2748—534no—
287#2741—534no—
288#2703—534no—
289#2671—534no—
290#2656—531no—
291#2646D5311yescharacteristic, complemented
292#2443—534no—
293#2424—534no—
294#2325—532no—
295C2×#380—531no—
296C2×C53—24no—
297C2×C53C2×#54011yescharacteristic
298C2×C53#349011yescharacteristic
299C2×C53#349211yescharacteristic
300C2×C53#353612yes—
301C2×C53#353712yes—
302D53—42no—
303D53—42no—
304D53—84no—
305C24×C4—531no—
306C2×#123—534no—
307C2×#121—532no—
308C2×#121—534no—
309C2×#121—532no—
310C2×#121—532no—
311C2×#121—534no—
312C2×#120—534no—
313C2×C4×D4—534no—
314C22×C4⋊C4—532no—
315C22×C22⋊C4—532no—
316#400C2×D5314yescomplemented
317#399—534no—
318#399—534no—
319#398—534no—
320#398C2×D5314yescomplemented
321#397—534no—
322#396—534no—
323#394—534no—
324#394—534no—
325#394—534no—
326#392—534no—
327#392—534no—
328#390—534no—
329#388—534no—
330#388—534no—
331#388—534no—
332#387—534no—
333#387—534no—
334#386—534no—
335#386—534no—
336#386—534no—
337#386—534no—
338#386—534no—
339#386—534no—
340#385—534no—
341#385—534no—
342#385C2×D5314yes—
343#380—532no—
344#380—532no—
345#380—534no—
346#380—534no—
347#380C2×D5312yes—
348C2×#95—531no—
349C2×#95C2×D5311yescharacteristic
350C53#3672711yescharacteristic, complemented, Sylow
351C25—531no—
352C22×D4—534no—
353C22×D4—534no—
354C22×D4—534no—
355C22×D4—532no—
356C22×D4—534no—
357C23×C4—531no—
358C23×C4—532no—
359C23×C4—532no—
360C23×C4—532no—
361C23×C4—534no—
362C23×C4—1064no—
363C23×C4—2124no—
364C23×C4C22×D5312yes—
365C23×C4C22×D5311yescharacteristic
366#123—2128no—
367#121—1068no—
368#121—2124no—
369#121—2124no—
370#121—2128no—
371#121—2128no—
372#120—2128no—
373C4×D4—2128no—
374C2×C4⋊C4—534no—
375C2×C4⋊C4—532no—
376C2×C4⋊C4—534no—
377C2×C4⋊C4—534no—
378C2×C4⋊C4—534no—
379C2×C4⋊C4—2124no—
380C2×C4⋊C4C22×D5312yes—
381C2×C4⋊C4C22×D5312yes—
382C2×C4⋊C4C22×D5314yes—
383C2×C4⋊C4C22×D5314yes—
384C2×C22⋊C4—534no—
385C2×C22⋊C4—534no—
386C2×C22⋊C4—534no—
387C2×C22⋊C4—534no—
388C2×C22⋊C4—534no—
389C2×C22⋊C4—534no—
390C2×C22⋊C4—534no—
391C2×C22⋊C4—534no—
392C2×C22⋊C4—534no—
393C2×C22⋊C4—534no—
394C2×C22⋊C4—534no—
395C2×C22⋊C4—534no—
396C2×C22⋊C4—2124no—
397C2×C22⋊C4C22×D5314yes—
398C2×C42—534no—
399#95—534no—
400#95—534no—
401#95—534no—
402#95—534no—
403#95—534no—
404#95C22×D5314yes—
405#95C22×D5314yes—
406#95C22×D5314yes—
407#95C22×D5314yes—
408C24—531no—
409C24—531no—
410C24—534no—
411C24—1062no—
412C24—1062no—
413C24—1062no—
414C24—1064no—
415C24—2122no—
416C24C23×D5311yescharacteristic
417C2×D4—1068no—
418C2×D4—1068no—
419C2×D4—1068no—
420C2×D4—1068no—
421C2×D4—1068no—
422C2×D4—1064no—
423C2×D4—2124no—
424C2×D4—2124no—
425C2×D4—2122no—
426C2×D4—2122no—
427C2×D4—2124no—
428C2×D4—2124no—
429C2×D4—2124no—
430C2×D4—2124no—
431C2×D4—2124no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—532no—
437C22×C4—534no—
438C22×C4—534no—
439C22×C4—1064no—
440C22×C4—1064no—
441C22×C4—1064no—
442C22×C4—1062no—
443C22×C4—1064no—
444C22×C4—1064no—
445C22×C4—1064no—
446C22×C4—2124no—
447C22×C4—2124no—
448C22×C4—2124no—
449C22×C4—2122no—
450C22×C4—2122no—
451C22×C4—21216no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4#1171994914yes—
455C22×C4C23×D5314yes—
456C22×C4C23×D5314yes—
457C22×C4C23×D5314yes—
458C22×C4C23×D5312yes—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—1068no—
464C4⋊C4—1064no—
465C4⋊C4—2124no—
466C4⋊C4—2124no—
467C4⋊C4—2124no—
468C22⋊C4—28no—
469C22⋊C4—1068no—
470C22⋊C4—1068no—
471C22⋊C4—1068no—
472C22⋊C4—1068no—
473C22⋊C4—2124no—
474C22⋊C4—2124no—
475C22⋊C4—2124no—
476C22⋊C4—2124no—
477C22⋊C4—2124no—
478C22⋊C4—2124no—
479C22⋊C4—2124no—
480C22⋊C4—2124no—
481C42—2124no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—532no—
486C23—532no—
487C23—1068no—
488C23—1061no—
489C23—1061no—
490C23—1061no—
491C23—1062no—
492C23—1062no—
493C23—1062no—
494C23—1062no—
495C23—1061no—
496C23—1064no—
497C23—1064no—
498C23—1064no—
499C23—1064no—
500C23—1064no—
501C23—1064no—
502C23—1064no—
503C23—2128no—
504C23—2128no—
505C23—2124no—
506C23—2124no—
507C23C2×#1171994912yes—
508C23#1313326314yes—
509C23C24×D5311yescharacteristic
510D4—4244no—
511D4—4248no—
512D4—4248no—
513D4—4248no—
514D4—21216no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—44no—
527C2×C4—84no—
528C2×C4—534no—
529C2×C4—1068no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1064no—
533C2×C4—1064no—
534C2×C4—1064no—
535C2×C4—1064no—
536C2×C4—1064no—
537C2×C4—1064no—
538C2×C4—4248no—
539C2×C4—4244no—
540C2×C4—2128no—
541C2×C4—2128no—
542C2×C4C2×#1171994914yes—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—1062no—
548C22—1062no—
549C22—1064no—
550C22—1064no—
551C22—21216no—
552C22—2124no—
553C22—2124no—
554C22—2124no—
555C22—2124no—
556C22—2124no—
557C22—2124no—
558C22—2124no—
559C22—2121no—
560C22—2121no—
561C22—2121no—
562C22—2121no—
563C22—2122no—
564C22—2122no—
565C22#1681122914yes—
566C22C22×#1171994911yescharacteristic
567C22H(5,𝔽2)×D5312yes—
568C22H(5,𝔽2)×D5311yescharacteristic
569C22C2×#1313326312yes—
570C222−1+4×D5311yescharacteristic
571C4—24no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—1064no—
576C4—2124no—
577C4—2124no—
578C2—24no—
579C2—4244no—
580C2—2122no—
581C2—2122no—
582C2C2×#1681122911yescharacteristic
583C2#2028840511yescharacteristic
584C2#2028842511yescharacteristic
585C2#2028893912yes—
586C2#2028894712yes—
subgroupG/H
center, FrattiniC23C24×D53
commutator, socle, cosocleC23×C53C25
FittingC53×#2646C2
hypercenter#2646D53
hypocenterC53#36727
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2646

Complements

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

NH ≅ G/Ntype
1. #20282998C2semidirect
2. #20281927C2semidirect
3. #20281837C2semidirect
4. #20281357C2semidirect
5. #20280958C2semidirect
6. #20280817C2semidirect
7. #20278454C2semidirect
8. #20278249C2semidirect
9. #20277415C2semidirect
10. C2×#16810482C2semidirect
11. C53×#2646C2semidirect
13. C2×#13133202C22semidirect
23. #16810560C22semidirect
24. #16810558C22semidirect
25. #16810556C22semidirect
26. #16810552C22semidirect
29. #16810541C22semidirect
32. #16810533C22semidirect
35. #16810513C22semidirect
36. #16810512C22semidirect
37. #16810510C22semidirect
38. #16810506C22semidirect
42. #16810501C22semidirect
45. #16810494C22semidirect
46. #16810494C22semidirect
52. C53×#400C22semidirect
53. C53×#398C22semidirect
291. #2646D53semidirect
316. #400C2×D53semidirect
320. #398C2×D53semidirect
350. C53#36727semidirect

Maximal subgroups

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

subgroupindex
#202829982
#202819272
#202818372
#202813572
#202809582
#202808172
#202784542
#202782492
#202774152
C2×#168104822
C53×#26462
#3672753