Finite Group Databasebeta

← #27275988

Subgroups — #27275988

Order 13568.

772 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#20275660C211yesmaximal, characteristic, complemented
2#20274540C211yesmaximal, characteristic, complemented
3#20273330C211yesmaximal, characteristic, complemented
4#20273303C211yesmaximal, characteristic, complemented
5#20273240C211yesmaximal, characteristic, complemented
6#20273158C211yesmaximal, characteristic, complemented
7C53×#3520C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9C2×#13133138—21no—
10C4×#11719917C2211yescharacteristic
11C2×#13133128C2211yescharacteristic
12C4×C4⋊Dic53C2211yescharacteristic
13#16810385—21no—
14#16810382—22no—
15#16810365—21no—
16#16810365—22no—
17#16810363—21no—
18#16810360—21no—
19#16810357—21no—
20D4⋊#88322—21no—
21D4⋊#88322—21no—
22C53×#556C411yescharacteristic, complemented
23C53×#554—21no—
24C53×#552—21no—
25C53×#551—22no—
26D4×Q8×C53C411yescharacteristic, complemented
27C53×#548—21no—
28C53×#547C411yescharacteristic, complemented
29C53×#547C411yescharacteristic, complemented
30C53×#546—21no—
31C53×#542—21no—
32C53×#539—22no—
33C53×#533C2211yescharacteristic
34C53×#529C2211yescharacteristic
35C53×#525—21no—
36C2×C22⋊Q8×C53—21no—
37C53×#520—21no—
38C2×C4×D4×C53C2211yescharacteristic
39C2×#11719917—21no—
40C22×#88322—21no—
41C2×C4⋊Dic53—21no—
42C2×C4×Dic53—21no—
43C4⋊#88322—21no—
44#13133138—24no—
45C22⋊#88322—21no—
46#13133128—21no—
47#13133128—21no—
48#13133128—21no—
49#13133128—21no—
50C4×#88322—21no—
51#13133124—21no—
52C2×#41×C53—21no—
53C2×#41×C53—21no—
54C22×Q8×C53—21no—
55C22×D4×C53D411yescharacteristic
56C23×C4×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53C2×C411yescharacteristic
59C4⋊Q8×C53C2×C411yescharacteristic
60#126×C53—22no—
61#126×C53—22no—
62#126×C53—21no—
63#125×C53—22no—
64#125×C53D411yescharacteristic
65#125×C53D412yes—
66#124×C53—22no—
67#124×C53—21no—
68#124×C53D411yescharacteristic
69#123×C53—22no—
70#123×C53—22no—
71#123×C53—22no—
72#123×C53—22no—
73#123×C53—21no—
74C22⋊Q8×C53—22no—
75C22⋊Q8×C53—22no—
76C22⋊Q8×C53—22no—
77C22⋊Q8×C53—21no—
78C22⋊Q8×C53—21no—
79C22⋊Q8×C53—22no—
80C22⋊Q8×C53—21no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53—21no—
83C22⋊Q8×C53—21no—
84C22⋊Q8×C53D411yescharacteristic
85#121×C53—22no—
86#121×C53—21no—
87#121×C53—21no—
88#121×C53D411yescharacteristic
89#120×C53—22no—
90C4×Q8×C53—21no—
91C4×Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53C2×C411yescharacteristic
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53—22no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53C2×C411yescharacteristic
102C4×D4×C53C2×C411yescharacteristic
103C4×D4×C53C2×C411yescharacteristic
104C4×D4×C53D411yescharacteristic
105C4×D4×C53D411yescharacteristic
106#117×C53—22no—
107#117×C53—21no—
108#117×C53—21no—
109#117×C53D412yes—
110C2×C4⋊C4×C53—22no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53—22no—
115C2×C22⋊C4×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C24×C53—21no—
118#41×C53—42no—
119#41×C53—42no—
120#41×C53—82no—
121C2×Q8×C53—22no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—82no—
125C2×Q8×C53C22⋊C411yescharacteristic
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—21no—
129C2×D4×C53—22no—
130C2×D4×C53—21no—
131C2×D4×C53—41no—
132C2×D4×C53C22⋊C411yescharacteristic
133C2×D4×C53C22⋊C411yescharacteristic
134C2×D4×C53C22⋊C411yescharacteristic
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—82no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×Dic53—41no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53C22⋊C411yescharacteristic
163C4⋊C4×C53C22⋊C411yescharacteristic
164C4⋊C4×C53C22⋊C411yescharacteristic
165C4⋊C4×C53C22⋊C411yescharacteristic
166C4⋊C4×C53C22⋊C412yes—
167C4⋊C4×C53C22⋊C412yes—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—21no—
183C42×C53—22no—
184C42×C53—21no—
185C42×C53—22no—
186C42×C53C22×C411yescharacteristic
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53C2×D411yescharacteristic
190C4⋊Dic53—22no—
191C4⋊Dic53—41no—
192C4×Dic53—42no—
193#11719917—22no—
194#11719917—41no—
195C2×#88322—22no—
196C2×#88322—41no—
197C2×#88322—41no—
198C2×#88322—41no—
199C2×#88322—41no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—42no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—81no—
207C23×C53#12011yescharacteristic
208Q8×C53—42no—
209Q8×C53—41no—
210Q8×C53—41no—
211Q8×C53—41no—
212Q8×C53—82no—
213D4×C53—21no—
214D4×C53—21no—
215D4×C53—21no—
216D4×C53—42no—
217D4×C53—41no—
218D4×C53—44no—
219D4×C53—41no—
220D4×C53—41no—
221D4×C53—82no—
222D4×C53#10412yes—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—41no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—44no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—81no—
241C2×C4×C53—81no—
242C2×C4×C53—81no—
243C2×C4×C53—82no—
244C2×C4×C53—82no—
245C2×C4×C53C2×C22⋊C411yescharacteristic
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C2×C22⋊C411yescharacteristic
248C2×C4×C53#12011yescharacteristic
249C2×C4×C53#12011yescharacteristic
250C2×C4×C53#12011yescharacteristic
251C2×Dic53—42no—
252C2×Dic53—81no—
253C2×Dic53—81no—
254#88322—42no—
255#88322—42no—
256#88322—81no—
257#8418—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—22no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—82no—
270C22×C53—82no—
271C22×C53#37911yescharacteristic
272C22×C53#45411yescharacteristic
273C22×C53#45611yescharacteristic
274C4×C53—21no—
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—21no—
278C4×C53—21no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—42no—
283C4×C53—41no—
284C4×C53—41no—
285C4×C53—42no—
286C4×C53C2×#10411yescharacteristic
287C4×C53#42111yescharacteristic
288Dic53—82no—
289#3520D5311yescharacteristic
290#1986—531no—
291#1762—531no—
292#1531—531no—
293#1524—531no—
294#1509—531no—
295#1490—531no—
296C2×C53—21no—
297C2×C53—21no—
298C2×C53—42no—
299C2×C53—81no—
300C2×C53—81no—
301C2×C53#178911yescharacteristic
302C2×C53#179311yescharacteristic
303C2×C53#179511yescharacteristic
304#556Dic5311yescharacteristic, complemented
305#554—21no—
306#552—21no—
307#551—22no—
308D4×Q8Dic5311yescharacteristic, complemented
309#548—21no—
310#547Dic5311yescharacteristic, complemented
311#547Dic5311yescharacteristic, complemented
312#546—21no—
313#542—21no—
314#539—22no—
315#533C2×D5311yescharacteristic
316#529C2×D5311yescharacteristic
317#525—21no—
318C2×C22⋊Q8—21no—
319#520—21no—
320C2×C4×D4C2×D5311yescharacteristic
321#432—531no—
322C2×#102—1061no—
323C4×M16—531no—
324C2×#97—531no—
325C4×C4⋊C4—531no—
326#340—1061no—
327#339—1062no—
328#330—1061no—
329#330—1062no—
330#329—1061no—
331#328—1061no—
332#327—1061no—
333D4⋊C8—1061no—
334D4⋊C8—1061no—
335C53#841811yescharacteristic, complemented, Sylow
336C2×#41—21no—
337C2×#41—21no—
338C22×Q8—21no—
339C22×D4C22⋊D5311yescharacteristic
340C23×C4—21no—
341C2×M16—1061no—
342C22×C8—1061no—
343C4⋊Q8—21no—
344C4⋊Q8C2×Dic5311yescharacteristic
345C4⋊Q8C2×Dic5311yescharacteristic
346#126—22no—
347#126—22no—
348#126—21no—
349#125—22no—
350#125C22⋊D5311yescharacteristic
351#125C22⋊D5312yes—
352#124—22no—
353#124—21no—
354#124C22⋊D5311yescharacteristic
355#123—22no—
356#123—22no—
357#123—21no—
358#123—22no—
359#123—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—22no—
369C22⋊Q8—21no—
370C22⋊Q8C22⋊D5311yescharacteristic
371#121—22no—
372#121—21no—
373#121—21no—
374#121C22⋊D5311yescharacteristic
375#120—22no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8C2×Dic5311yescharacteristic
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4C2×Dic5311yescharacteristic
388C4×D4C2×Dic5311yescharacteristic
389C4×D4C2×Dic5311yescharacteristic
390C4×D4C22⋊D5311yescharacteristic
391C4×D4C22⋊D5311yescharacteristic
392#117—22no—
393#117—21no—
394#117—21no—
395#117C22⋊D5312yes—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22⋊D5311yescharacteristic
401C2×C22⋊C4—22no—
402C2×C22⋊C4—21no—
403C2×C42—1061no—
404C2×C42C22×D5311yescharacteristic
405C4⋊C8—1061no—
406#102—1064no—
407C22⋊C8—1061no—
408#97—1061no—
409#97—1061no—
410#97—1061no—
411#97—1061no—
412C4×C8—1061no—
413#95—1061no—
414C24—21no—
415#41—42no—
416#41—42no—
417#41—82no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—82no—
422C2×Q8C22⋊Dic5311yescharacteristic
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—41no—
429C2×D4C22⋊Dic5311yescharacteristic
430C2×D4C22⋊Dic5311yescharacteristic
431C2×D4C22⋊Dic5311yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—82no—
442C22×C4—2121no—
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446M16—1062no—
447M16—2121no—
448C2×C8—1062no—
449C2×C8—2121no—
450C2×C8—2121no—
451C2×C8—2121no—
452C2×C8—2121no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—1062no—
467C4⋊C4—2121no—
468C4⋊C4C22⋊Dic5311yescharacteristic
469C4⋊C4C22⋊Dic5311yescharacteristic
470C4⋊C4C22⋊Dic5311yescharacteristic
471C4⋊C4C22⋊Dic5311yescharacteristic
472C4⋊C4C22⋊Dic5312yes—
473C4⋊C4C22⋊Dic5312yes—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—22no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C42—22no—
490C42—21no—
491C42—22no—
492C42—2122no—
493C42C22×Dic5311yescharacteristic
494C42C2×C22⋊D5311yescharacteristic
495C42C2×C22⋊D5311yescharacteristic
496C42C2×C22⋊D5311yescharacteristic
497C23—21no—
498C23—21no—
499C23—21no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—81no—
504C23#1313318611yescharacteristic
505Q8—41no—
506Q8—42no—
507Q8—41no—
508Q8—41no—
509Q8—82no—
510D4—21no—
511D4—21no—
512D4—21no—
513D4—42no—
514D4—41no—
515D4—44no—
516D4—41no—
517D4—41no—
518D4—82no—
519D4#1313314412yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—4241no—
543C2×C4—4241no—
544C2×C4—2122no—
545C2×C4C2×C22⋊Dic5311yescharacteristic
546C2×C4C2×C22⋊Dic5311yescharacteristic
547C2×C4C2×C22⋊Dic5311yescharacteristic
548C2×C4#1313318611yescharacteristic
549C2×C4#1313318611yescharacteristic
550C2×C4#1313318611yescharacteristic
551C8—4241no—
552C8—2122no—
553C8—2122no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—82no—
566C22—82no—
567C22#1681047911yescharacteristic
568C22#1681079611yescharacteristic
569C22#1681080611yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—41no—
582C4—4242no—
583C4C2×#1313314411yescharacteristic
584C4#1681064011yescharacteristic
585C2—21no—
586C2—21no—
587C2—42no—
588C2—81no—
589C2—81no—
590C2#2027465911yescharacteristic
591C2#2027467411yescharacteristic
592C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3520C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3520D53
hypocenterC53#8418
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3520

Complements

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

NH ≅ G/Ntype
1. #20275660C2semidirect
2. #20274540C2semidirect
3. #20273330C2semidirect
4. #20273303C2semidirect
5. #20273240C2semidirect
6. #20273158C2semidirect
22. C53×#556C4semidirect
26. D4×Q8×C53C4semidirect
28. C53×#547C4semidirect
29. C53×#547C4semidirect
304. #556Dic53semidirect
308. D4×Q8Dic53semidirect
310. #547Dic53semidirect
311. #547Dic53semidirect
335. C53#8418semidirect

Maximal subgroups

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

subgroupindex
#202756602
#202745402
#202733302
#202733032
#202732402
#202731582
C53×#35202
#841853