Finite Group Databasebeta

← #27276121

Subgroups — #27276121

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274735C211yesmaximal, characteristic, complemented
2C4×#13133141C211yesmaximal, characteristic, complemented
3#20273210C211yesmaximal, characteristic, complemented
4#20273166C211yesmaximal, characteristic, complemented
5#20273144C211yesmaximal, characteristic, complemented
6#20273097C211yesmaximal, characteristic, complemented
7C53×#3480C211yesmaximal, characteristic
8#16810676C2211yescharacteristic
9#16810647C2211yescharacteristic
10C2×#13133144—21no—
11C2×#13133141—21no—
12C2×C4×#88322C2211yescharacteristic
13C4×C4⋊Dic53C2211yescharacteristic
14#16810385—21no—
15#16810382—21no—
16#16810357—22no—
17Q8⋊#88322—21no—
18Q8⋊#88322—21no—
19Q8⋊#88322—21no—
20D4⋊#88322—22no—
21D4⋊#88322—21no—
22C53×#559—21no—
23C53×#559C411yescharacteristic, complemented
24C53×#557C411yescharacteristic, complemented
25C53×#556—21no—
26C53×#556C411yescharacteristic, complemented
27C53×#553—22no—
28C53×#553—21no—
29C53×#550C411yescharacteristic, complemented
30C53×#548—21no—
31C53×#532—21no—
32C53×#532C2211yescharacteristic
33C53×#529—21no—
34C53×#529—22no—
35C4×#41×C53—21no—
36C4×#41×C53—21no—
37C4×#41×C53C2211yescharacteristic
38C2×C4×Q8×C53C2211yescharacteristic
39C2×#11719917—21no—
40C22×#88322—21no—
41C2×C4⋊Dic53—21no—
42C2×C4×Dic53—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46#13133144—42no—
47#13133141—24no—
48C22⋊#88322—21no—
49C4×#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52#13133124—21no—
53C2×#41×C53—21no—
54C2×#41×C53—21no—
55C2×#41×C53D411yescharacteristic
56C22×Q8×C53D411yescharacteristic
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53C2×C411yescharacteristic
59#127×C53—21no—
60#127×C53C2×C411yescharacteristic
61#126×C53—22no—
62#126×C53—22no—
63#126×C53—22no—
64#125×C53—22no—
65#125×C53—21no—
66#124×C53—22no—
67#124×C53—21no—
68#124×C53—21no—
69#124×C53—21no—
70#124×C53—21no—
71#123×C53—22no—
72#123×C53—22no—
73#123×C53—22no—
74C22⋊Q8×C53—22no—
75C22⋊Q8×C53—21no—
76#121×C53—22no—
77#121×C53—22no—
78#121×C53—21no—
79#121×C53—22no—
80#121×C53—21no—
81#121×C53D412yes—
82C4×Q8×C53—21no—
83C4×Q8×C53—21no—
84C4×Q8×C53—21no—
85C4×Q8×C53—21no—
86C4×Q8×C53—21no—
87C4×Q8×C53C2×C411yescharacteristic
88C4×Q8×C53C2×C411yescharacteristic
89C4×Q8×C53C2×C411yescharacteristic
90C4×Q8×C53D411yescharacteristic
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53D411yescharacteristic
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53—22no—
97C4×D4×C53—22no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53C2×C411yescharacteristic
105C4×D4×C53D411yescharacteristic
106C4×D4×C53D412yes—
107#117×C53—22no—
108#117×C53—22no—
109#117×C53—21no—
110#117×C53—21no—
111#117×C53D411yescharacteristic
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C42×C53—22no—
115C2×C42×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118#41×C53—21no—
119#41×C53—21no—
120#41×C53—21no—
121#41×C53—21no—
122#41×C53—42no—
123#41×C53—42no—
124#41×C53—41no—
125#41×C53—41no—
126#41×C53—41no—
127#41×C53—41no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—22no—
131C2×Q8×C53—41no—
132C2×Q8×C53C22⋊C411yescharacteristic
133C2×Q8×C53C22⋊C411yescharacteristic
134C2×Q8×C53C22⋊C411yescharacteristic
135C2×D4×C53—22no—
136C2×D4×C53—21no—
137C2×D4×C53—22no—
138C2×D4×C53—22no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53C22⋊C411yescharacteristic
142C2×D4×C53C22⋊C412yes—
143C22×C4×C53—22no—
144C22×C4×C53—21no—
145C22×C4×C53—22no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—22no—
150C22×C4×C53—21no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×Dic53—41no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53C22⋊C411yescharacteristic
168C4⋊C4×C53C22⋊C411yescharacteristic
169C4⋊C4×C53C22⋊C411yescharacteristic
170C4⋊C4×C53C22⋊C411yescharacteristic
171C4⋊C4×C53C22⋊C412yes—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—22no—
183C42×C53—22no—
184C42×C53—22no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—21no—
192C42×C53C22×C411yescharacteristic
193C42×C53C2×D411yescharacteristic
194C42×C53C2×D411yescharacteristic
195C42×C53C2×D411yescharacteristic
196C4⋊Dic53—22no—
197C4⋊Dic53—41no—
198C4×Dic53—41no—
199C4×Dic53—41no—
200#11719917—41no—
201#11719917—41no—
202C2×#88322—22no—
203C2×#88322—41no—
204C2×#88322—41no—
205C2×#88322—41no—
206C2×#88322—41no—
207C23×C53—22no—
208C23×C53—21no—
209C23×C53#12011yescharacteristic
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214Q8×C53—21no—
215Q8×C53—42no—
216Q8×C53—41no—
217Q8×C53—41no—
218Q8×C53#10412yescomplemented
219D4×C53—21no—
220D4×C53—21no—
221D4×C53—42no—
222D4×C53—44no—
223D4×C53—41no—
224D4×C53—42no—
225D4×C53—41no—
226D4×C53—42no—
227D4×C53—41no—
228D4×C53—41no—
229D4×C53—41no—
230D4×C53—41no—
231D4×C53—82no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—41no—
243C2×C4×C53—42no—
244C2×C4×C53—44no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—42no—
249C2×C4×C53—41no—
250C2×C4×C53—42no—
251C2×C4×C53—81no—
252C2×C4×C53—81no—
253C2×C4×C53—82no—
254C2×C4×C53C2×C22⋊C411yescharacteristic
255C2×C4×C53C2×C22⋊C411yescharacteristic
256C2×C4×C53C2×C22⋊C411yescharacteristic
257C2×C4×C53#12011yescharacteristic
258C2×C4×C53#12011yescharacteristic
259C2×C4×C53#12011yescharacteristic
260C2×C4×C53#13511yescharacteristic
261C2×C4×C53#13511yescharacteristic
262C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
263C2×C4×C53#13711yescharacteristic
264C2×Dic53—42no—
265C2×Dic53—81no—
266C2×Dic53—81no—
267#88322—42no—
268#88322—81no—
269#88322—81no—
270#8437—531nomaximal, Sylow
271C22×C53—21no—
272C22×C53—21no—
273C22×C53—42no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53#37911yescharacteristic
280C22×C53#44911yescharacteristic
281C22×C53#45211yescharacteristic
282C4×C53—21no—
283C4×C53—21no—
284C4×C53—21no—
285C4×C53—22no—
286C4×C53—41no—
287C4×C53—41no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—42no—
291C4×C53—41no—
292C4×C53—41no—
293C4×C53#41611yescharacteristic
294C4×C53#41711yescharacteristic
295C4×C53C2×#10411yescharacteristic
296C4×C53#42111yescharacteristic
297C4×C53#44911yescharacteristic
298C4×C53#45211yescharacteristic
299Dic53—82no—
300#3480D5311yescharacteristic
301#1806—531no—
302C4×#103—531no—
303#1502—531no—
304#1492—531no—
305#1487—531no—
306#1476—531no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—82no—
310C2×C53—81no—
311C2×C53#178811yescharacteristic
312C2×C53#178911yescharacteristic
313C2×C53#179411yescharacteristic
314#559—21no—
315#559Dic5311yescharacteristic, complemented
316#557Dic5311yescharacteristic, complemented
317#556—21no—
318#556Dic5311yescharacteristic, complemented
319#553—22no—
320#553—21no—
321#550Dic5311yescharacteristic, complemented
322#548—21no—
323#532—21no—
324#532C2×D5311yescharacteristic
325#529—21no—
326#529—22no—
327C4×#41—21no—
328C4×#41—21no—
329C4×#41C2×D5311yescharacteristic
330C2×C4×Q8C2×D5311yescharacteristic
331#431—531no—
332#423—531no—
333C2×#104—1061no—
334C2×#103—1061no—
335C2×C4×C8—531no—
336C4×C4⋊C4—531no—
337#340—1061no—
338#339—1061no—
339#327—1062no—
340Q8⋊C8—1061no—
341Q8⋊C8—1061no—
342Q8⋊C8—1061no—
343D4⋊C8—1061no—
344D4⋊C8—1062no—
345C53#843711yescharacteristic, complemented, Sylow
346C2×#41—21no—
347C2×#41—21no—
348C2×#41C22⋊D5311yescharacteristic
349C22×Q8C22⋊D5311yescharacteristic
350C2×M16—1061no—
351C22×C8—1061no—
352C4⋊Q8—21no—
353C4⋊Q8C2×Dic5311yescharacteristic
354#127—21no—
355#127C2×Dic5311yescharacteristic
356#126—22no—
357#126—22no—
358#126—22no—
359#125—22no—
360#125—21no—
361#124—22no—
362#124—21no—
363#124—21no—
364#124—21no—
365#124—21no—
366#123—22no—
367#123—22no—
368#123—22no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—21no—
375#121—21no—
376#121C22⋊D5312yes—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C2×Dic5311yescharacteristic
383C4×Q8C2×Dic5311yescharacteristic
384C4×Q8C2×Dic5311yescharacteristic
385C4×Q8C22⋊D5311yescharacteristic
386C4×Q8C22⋊D5311yescharacteristic
387C4×Q8C22⋊D5311yescharacteristic
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4C2×Dic5311yescharacteristic
400C4×D4C22⋊D5311yescharacteristic
401C4×D4C22⋊D5312yes—
402#117—22no—
403#117—22no—
404#117—21no—
405#117—21no—
406#117C22⋊D5311yescharacteristic
407C2×C4⋊C4—21no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4C22⋊D5311yescharacteristic
410C2×C42—22no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—1061no—
414C2×C42C22×D5311yescharacteristic
415C4⋊C8—1061no—
416C4⋊C8—1061no—
417C4⋊C8—1061no—
418#104—2122no—
419#103—1064no—
420C22⋊C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8—1061no—
424#95—1061no—
425#41—21no—
426#41—21no—
427#41—21no—
428#41—21no—
429#41—42no—
430#41—42no—
431#41—41no—
432#41—41no—
433#41—41no—
434#41—41no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8C22⋊Dic5311yescharacteristic
440C2×Q8C22⋊Dic5311yescharacteristic
441C2×Q8C22⋊Dic5311yescharacteristic
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C22⋊Dic5311yescharacteristic
449C2×D4C22⋊Dic5312yes—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—2121no—
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462M16—2121no—
463M16—2121no—
464C2×C8—1062no—
465C2×C8—2121no—
466C2×C8—2121no—
467C2×C8—2121no—
468C2×C8—2121no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—1062no—
482C4⋊C4—2121no—
483C4⋊C4C22⋊Dic5311yescharacteristic
484C4⋊C4C22⋊Dic5311yescharacteristic
485C4⋊C4C22⋊Dic5311yescharacteristic
486C4⋊C4C22⋊Dic5311yescharacteristic
487C4⋊C4C22⋊Dic5312yes—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C42—22no—
500C42—22no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—2121no—
509C42—2121no—
510C42C22×Dic5311yescharacteristic
511C42C2×C22⋊D5311yescharacteristic
512C42C2×C22⋊D5311yescharacteristic
513C42C2×C22⋊D5311yescharacteristic
514C23—21no—
515C23—22no—
516C23#1313318611yescharacteristic
517Q8—21no—
518Q8—21no—
519Q8—21no—
520Q8—21no—
521Q8—21no—
522Q8—42no—
523Q8—41no—
524Q8—41no—
525Q8#1313314412yescomplemented
526D4—21no—
527D4—21no—
528D4—42no—
529D4—44no—
530D4—41no—
531D4—42no—
532D4—41no—
533D4—42no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—82no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—82no—
561C2×C4—4241no—
562C2×C4—4241no—
563C2×C4—2122no—
564C2×C4C2×C22⋊Dic5311yescharacteristic
565C2×C4C2×C22⋊Dic5311yescharacteristic
566C2×C4C2×C22⋊Dic5311yescharacteristic
567C2×C4#1313318611yescharacteristic
568C2×C4#1313318611yescharacteristic
569C2×C4#1313318611yescharacteristic
570C2×C4#1313323711yescharacteristic
571C2×C4#1313323711yescharacteristic
572C2×C4#1313324211yescharacteristic
573C2×C4#1313324711yescharacteristic
574C8—4241no—
575C8—4241no—
576C8—2122no—
577C22—21no—
578C22—21no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—42no—
584C22—41no—
585C22#1681047911yescharacteristic
586C22#1681076811yescharacteristic
587C22#1681078511yescharacteristic
588C4—21no—
589C4—22no—
590C4—21no—
591C4—21no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—4242no—
600C4#1681061911yescharacteristic
601C4#1681062411yescharacteristic
602C4C2×#1313314411yescharacteristic
603C4#1681064011yescharacteristic
604C4#1681076811yescharacteristic
605C4#1681078511yescharacteristic
606C2—21no—
607C2—21no—
608C2—81no—
609C2—82no—
610C2#2027465511yescharacteristic
611C2#2027465911yescharacteristic
612C2#2027467911yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3480C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3480D53
hypocenterC53#8437
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3480

Complements

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

NH ≅ G/Ntype
1. #20274735C2semidirect
2. C4×#13133141C2semidirect
3. #20273210C2semidirect
4. #20273166C2semidirect
5. #20273144C2semidirect
6. #20273097C2semidirect
23. C53×#559C4semidirect
24. C53×#557C4semidirect
26. C53×#556C4semidirect
29. C53×#550C4semidirect
218. Q8×C53#104semidirect
315. #559Dic53semidirect
316. #557Dic53semidirect
318. #556Dic53semidirect
321. #550Dic53semidirect
345. C53#8437semidirect
525. Q8#13133144semidirect

Maximal subgroups

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

subgroupindex
#202747352
C4×#131331412
#202732102
#202731662
#202731442
#202730972
C53×#34802
#843753