Finite Group Databasebeta

← #27457210

Subgroups — #27457210

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277740C211yesmaximal, characteristic
2#20276134C211yesmaximal, characteristic, complemented
3#20276093C211yesmaximal, characteristic, complemented
4#20275918C211yesmaximal, characteristic
5#20275911C211yesmaximal, characteristic
6#20275481C211yesmaximal, characteristic, complemented
7#20275450C211yesmaximal, characteristic, complemented
8#20275330C211yesmaximal, characteristic
9#20275304C211yesmaximal, characteristic, complemented
10#20275300C211yesmaximal, characteristic, complemented
11#20274968C211yesmaximal, characteristic, complemented
12#20274912C211yesmaximal, characteristic
13#20274807C211yesmaximal, characteristic
14#20274548C211yesmaximal, characteristic, complemented
15C53×#2494C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810517—21no—
34#16810511—21no—
35#16810505—21no—
36C4×C4⋊Dic53C2211yescharacteristic
37#16810474—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C53×#401—21no—
49C53×#397—21no—
50C53×#389C2211yescharacteristic
51C53×#389C2211yescharacteristic
52C53×#388—21no—
53C53×#387—21no—
54C53×#382C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133196—22no—
65#13133196—22no—
66C4×Dic106—22no—
67C4×Dic106—41no—
68#13133176—22no—
69#13133176—41no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81#13133150—22no—
82#13133150—22no—
83#13133148—22no—
84#13133148—41no—
85C4⋊#88322—22no—
86#13133140—22no—
87#13133140—22no—
88#13133140—22no—
89#13133140—41no—
90C22⋊#88322—22no—
91C22⋊#88322—41no—
92#13133128—41no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C23×C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C42×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111#95×C53—21no—
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53D411yescharacteristic
119#95×C53D411yescharacteristic
120C24×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106#4112yes—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—42no—
156C4⋊C4×C53—81no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—81no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—81no—
166C22⋊Dic53—24no—
167C22⋊Dic53—42no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53#4112yes—
173C4⋊Dic53#4112yes—
174#11719919—24no—
175#11719919—42no—
176#11719919—42no—
177C4×Dic53—42no—
178C4×Dic53—42no—
179C4×Dic53—42no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322#4112yes—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53C22×D411yescharacteristic
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53H(5,𝔽2)11yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249Dic106—22no—
250Dic106—22no—
251Dic106—41no—
252Dic106—41no—
253#88322—22no—
254#88322—41no—
255#88322—41no—
256#88322—41no—
257#27061—531nomaximal, Sylow
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53#52511yescharacteristic
266C22×C53#54611yescharacteristic
267C22×C53#54711yescharacteristic
268C22×C53#57511yescharacteristic
269C22×C53#57511yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57811yescharacteristic
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53#52911yescharacteristic
282C4×C53#53911yescharacteristic
283C4×C53#55311yescharacteristic
284C4×C53#55611yescharacteristic
285Dic53—42no—
286Dic53—42no—
287Dic53—42no—
288Dic53—81no—
289#2494D5311yescharacteristic
290#2362—531no—
291#2074—531no—
292#2066—531no—
293#2036—531no—
294#2035—531no—
295#1949—531no—
296#1943—531no—
297#1924—531no—
298#1921—531no—
299#1920—531no—
300#1863—531no—
301#1854—531no—
302#1825—531no—
303#1763—531no—
304C2×C53—81no—
305C2×C53#246611yescharacteristic
306C2×C53#311111yescharacteristic
307C2×C53#312111yescharacteristic
308C2×C53#330711yescharacteristic
309C2×C53#332711yescharacteristic
310C2×C53#334911yescharacteristic
311C2×C53#335011yescharacteristic
312C2×C22⋊Q8—531no—
313C2×C4×Q8—531no—
314C2×#117—531no—
315C2×#107—531no—
316C2×#107—531no—
317C2×#106—531no—
318C2×#106—531no—
319C2×C4⋊C8—531no—
320C2×#103—531no—
321C2×#103—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×C22⋊C8—531no—
325C2×C22⋊C8—531no—
326C2×#97—531no—
327#401—21no—
328#397—21no—
329#391—531no—
330#389—531no—
331#389C2×D5311yescharacteristic
332#389C2×D5311yescharacteristic
333#388—21no—
334#388—1061no—
335#388—1061no—
336#387—21no—
337#387—1061no—
338#382C2×D5311yescharacteristic
339#380C2×D5311yescharacteristic
340#380C2×D5311yescharacteristic
341C4×C4⋊C4—531no—
342C4×C4⋊C4—1061no—
343C4×C4⋊C4C2×D5311yescharacteristic
344C4×C22⋊C4C2×D5311yescharacteristic
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#336—531no—
354#336—531no—
355C53#2706111yescharacteristic, complemented, Sylow
356C22×Q8—531no—
357C23×C4C22×D5311yescharacteristic
358C22×C8—531no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8—531no—
362C22⋊Q8—1062no—
363C22⋊Q8—1062no—
364C4×Q8—1062no—
365C4×Q8—2121no—
366#117—1062no—
367#117—2121no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4C22⋊D5311yescharacteristic
385C2×C22⋊C4C22⋊D5311yescharacteristic
386C2×C42—21no—
387C2×C42—1061no—
388C2×C42—1061no—
389C2×C42—1061no—
390C2×C42C22×D5311yescharacteristic
391C2×C42C22×D5311yescharacteristic
392#107—1062no—
393#107—1062no—
394#106—1062no—
395#106—2121no—
396C4⋊C8—1062no—
397#103—1062no—
398#103—1062no—
399#103—1062no—
400#103—2121no—
401C22⋊C8—1062no—
402C22⋊C8—2121no—
403#97—2121no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95C22⋊D5311yescharacteristic
416#95C22⋊D5311yescharacteristic
417C24C2×C22⋊D5311yescharacteristic
418C2×Q8—532no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—532no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
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—42no—
465C4⋊C4—81no—
466C4⋊C4—532no—
467C4⋊C4—532no—
468C4⋊C4—1064no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—1064no—
480C22⋊C4—2122no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—81no—
486C42—2122no—
487C42—2122no—
488C42—2122no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C22×C22⋊D5311yescharacteristic
497Q8—1062no—
498Q8—1062no—
499Q8—2121no—
500Q8—2121no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4#41×D5311yescharacteristic
545C2×C4#41×D5311yescharacteristic
546C2×C4#41×D5311yescharacteristic
547C2×C4#41×D5311yescharacteristic
548C2×C4#1313326311yescharacteristic
549C2×C4#1313326411yescharacteristic
550C8—1062no—
551C8—2121no—
552C8—2121no—
553C8—2121no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#1681113011yescharacteristic
562C22#1681126111yescharacteristic
563C22#1681127211yescharacteristic
564C22#1681142111yescharacteristic
565C22#1681142111yescharacteristic
566C22#1681143411yescharacteristic
567C22#1681143811yescharacteristic
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—4241no—
578C4—2122no—
579C4—2122no—
580C4—2122no—
581C4#1681115611yescharacteristic
582C4#1681121411yescharacteristic
583C4#1681132411yescharacteristic
584C4#1681134011yescharacteristic
585C2—81no—
586C2#2027874711yescharacteristic
587C2#2028481811yescharacteristic
588C2#2028493811yescharacteristic
589C2#2028668511yescharacteristic
590C2#2028697711yescharacteristic
591C2#2028723911yescharacteristic
592C2#2028725411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2494C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2494D53
hypocenterC53#27061
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2494

Complements

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

NH ≅ G/Ntype
2. #20276134C2semidirect
3. #20276093C2semidirect
6. #20275481C2semidirect
7. #20275450C2semidirect
9. #20275304C2semidirect
10. #20275300C2semidirect
11. #20274968C2semidirect
14. #20274548C2semidirect
355. C53#27061semidirect

Maximal subgroups

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

subgroupindex
#202777402
#202761342
#202760932
#202759182
#202759112
#202754812
#202754502
#202753302
#202753042
#202753002
#202749682
#202749122
#202748072
#202745482
C53×#24942
#2706153