Finite Group Databasebeta

← #27275995

Subgroups — #27275995

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275635C211yesmaximal, characteristic, complemented
2#20274545C211yesmaximal, characteristic, complemented
3#20273326C211yesmaximal, characteristic, complemented
4#20273303C211yesmaximal, characteristic, complemented
5#20273240C211yesmaximal, characteristic, complemented
6#20273162C211yesmaximal, characteristic, complemented
7C53×#3522C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9C2×#13133144—22no—
10C2×#13133141—21no—
11C4×#11719917C2211yescharacteristic
12C2×#13133128C2211yescharacteristic
13C4×C4⋊Dic53C2211yescharacteristic
14#16810385—21no—
15#16810365—21no—
16#16810363—21no—
17#16810363—22no—
18#16810360—21no—
19#16810357—21no—
20Q8⋊#88322—21no—
21Q8⋊#88322—21no—
22C53×#559—21no—
23C53×#559C411yescharacteristic, complemented
24Q82×C53C411yescharacteristic, complemented
25C53×#554—21no—
26C53×#554C411yescharacteristic, complemented
27C53×#550C411yescharacteristic, complemented
28C53×#548—21no—
29C53×#547—22no—
30C53×#547—21no—
31C53×#542—22no—
32C53×#542—21no—
33C53×#532—21no—
34C53×#532C2211yescharacteristic
35C53×#529—21no—
36C53×#529C2211yescharacteristic
37C53×#520—21no—
38C2×C4×Q8×C53C2211yescharacteristic
39C2×#11719917—21no—
40C22×#88322—21no—
41C2×C4⋊Dic53—21no—
42C2×C4×Dic53—21no—
43C4⋊#88322—21no—
44#13133144—44no—
45#13133141—24no—
46C22⋊#88322—21no—
47#13133128—21no—
48#13133128—21no—
49#13133128—21no—
50#13133128—21no—
51C4×#88322—21no—
52#13133124—21no—
53C2×#41×C53—22no—
54C2×#41×C53—21no—
55C2×#41×C53D412yes—
56C22×Q8×C53D411yescharacteristic
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—22no—
60C4⋊Q8×C53C2×C411yescharacteristic
61#127×C53—21no—
62#127×C53C2×C411yescharacteristic
63#126×C53—22no—
64#126×C53—21no—
65#125×C53—21no—
66#125×C53D411yescharacteristic
67#124×C53—22no—
68#124×C53—22no—
69#124×C53—21no—
70#124×C53—21no—
71#124×C53D411yescharacteristic
72#124×C53D412yes—
73#123×C53—22no—
74#123×C53—22no—
75#123×C53—22no—
76#123×C53—22no—
77#123×C53—21no—
78C22⋊Q8×C53—22no—
79C22⋊Q8×C53—22no—
80C22⋊Q8×C53—22no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—21no—
83C22⋊Q8×C53D411yescharacteristic
84#121×C53—22no—
85#121×C53—21no—
86#121×C53—22no—
87#121×C53—21no—
88#121×C53—21no—
89#121×C53D411yescharacteristic
90C4×Q8×C53—21no—
91C4×Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53C2×C411yescharacteristic
95C4×Q8×C53C2×C411yescharacteristic
96C4×Q8×C53C2×C411yescharacteristic
97C4×Q8×C53D411yescharacteristic
98C4×Q8×C53D411yescharacteristic
99C4×D4×C53—22no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53—21no—
105C4×D4×C53—21no—
106C4×D4×C53—21no—
107C4×D4×C53—21no—
108C4×D4×C53—21no—
109C4×D4×C53C2×C411yescharacteristic
110#117×C53—22no—
111#117×C53—21no—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118#41×C53—42no—
119#41×C53—42no—
120#41×C53—42no—
121#41×C53—42no—
122#41×C53—82no—
123#41×C53—82no—
124C2×Q8×C53—22no—
125C2×Q8×C53—21no—
126C2×Q8×C53—22no—
127C2×Q8×C53—41no—
128C2×Q8×C53C22⋊C411yescharacteristic
129C2×Q8×C53C22⋊C411yescharacteristic
130C2×Q8×C53C22⋊C411yescharacteristic
131C2×Q8×C53C22⋊C412yes—
132C2×D4×C53—22no—
133C2×D4×C53—22no—
134C2×D4×C53—21no—
135C2×D4×C53—22no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53C22⋊C411yescharacteristic
139C2×D4×C53C22⋊C412yes—
140C22×C4×C53—22no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×Dic53—41no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
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×C53C22⋊C411yescharacteristic
167C4⋊C4×C53C22⋊C411yescharacteristic
168C4⋊C4×C53C22⋊C411yescharacteristic
169C4⋊C4×C53C22⋊C411yescharacteristic
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—21no—
182C42×C53—22no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53C22×C411yescharacteristic
189C42×C53C2×D411yescharacteristic
190C42×C53C2×D411yescharacteristic
191C42×C53C2×D411yescharacteristic
192C4⋊Dic53—22no—
193C4⋊Dic53—41no—
194C4×Dic53—42no—
195#11719917—22no—
196#11719917—41no—
197C2×#88322—22no—
198C2×#88322—41no—
199C2×#88322—41no—
200C2×#88322—41no—
201C2×#88322—41no—
202C23×C53—22no—
203C23×C53—21no—
204C23×C53#12011yescharacteristic
205Q8×C53—21no—
206Q8×C53—21no—
207Q8×C53—21no—
208Q8×C53—42no—
209Q8×C53—41no—
210Q8×C53—42no—
211Q8×C53—82no—
212Q8×C53#10412yescomplemented
213D4×C53—41no—
214D4×C53—42no—
215D4×C53—42no—
216D4×C53—44no—
217D4×C53—42no—
218D4×C53—41no—
219D4×C53—42no—
220D4×C53—41no—
221D4×C53—41no—
222D4×C53—41no—
223D4×C53—82no—
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—21no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—44no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—82no—
244C2×C4×C53—82no—
245C2×C4×C53—82no—
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C2×C22⋊C411yescharacteristic
248C2×C4×C53C2×C22⋊C411yescharacteristic
249C2×C4×C53#12011yescharacteristic
250C2×C4×C53#12011yescharacteristic
251C2×C4×C53#12011yescharacteristic
252C2×Dic53—42no—
253C2×Dic53—81no—
254C2×Dic53—81no—
255#88322—42no—
256#88322—42no—
257#88322—81no—
258#8419—531nomaximal, Sylow
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—42no—
262C22×C53—41no—
263C22×C53—42no—
264C22×C53—41no—
265C22×C53—42no—
266C22×C53—41no—
267C22×C53#37911yescharacteristic
268C22×C53#45311yescharacteristic
269C22×C53#45511yescharacteristic
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—22no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—42no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53C2×#10411yescharacteristic
284C4×C53#42111yescharacteristic
285Dic53—82no—
286#3522D5311yescharacteristic
287#1980—531no—
288#1763—531no—
289#1530—531no—
290#1524—531no—
291#1509—531no—
292#1491—531no—
293C2×C53—21no—
294C2×C53—21no—
295C2×C53—82no—
296C2×C53—81no—
297C2×C53#178911yescharacteristic
298C2×C53#179211yescharacteristic
299C2×C53#179611yescharacteristic
300#559—21no—
301#559Dic5311yescharacteristic, complemented
302Q82Dic5311yescharacteristic, complemented
303#554—21no—
304#554Dic5311yescharacteristic, complemented
305#550Dic5311yescharacteristic, complemented
306#548—21no—
307#547—22no—
308#547—21no—
309#542—22no—
310#542—21no—
311#532—21no—
312#532C2×D5311yescharacteristic
313#529—21no—
314#529C2×D5311yescharacteristic
315#520—21no—
316C2×C4×Q8C2×D5311yescharacteristic
317#432—531no—
318C2×#104—1062no—
319C2×#103—1061no—
320C4×M16—531no—
321C2×#97—531no—
322C4×C4⋊C4—531no—
323#340—1061no—
324#330—1061no—
325#329—1061no—
326#329—1062no—
327#328—1061no—
328#327—1061no—
329Q8⋊C8—1061no—
330Q8⋊C8—1061no—
331C53#841911yescharacteristic, complemented, Sylow
332C2×#41—22no—
333C2×#41—21no—
334C2×#41C22⋊D5312yes—
335C22×Q8C22⋊D5311yescharacteristic
336C2×M16—1061no—
337C22×C8—1061no—
338C4⋊Q8—22no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8C2×Dic5311yescharacteristic
342#127—21no—
343#127C2×Dic5311yescharacteristic
344#126—22no—
345#126—21no—
346#125—21no—
347#125C22⋊D5311yescharacteristic
348#124—22no—
349#124—22no—
350#124—21no—
351#124—21no—
352#124C22⋊D5311yescharacteristic
353#124C22⋊D5312yes—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358#123—21no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—21no—
364C22⋊Q8C22⋊D5311yescharacteristic
365#121—22no—
366#121—22no—
367#121—21no—
368#121—21no—
369#121—21no—
370#121C22⋊D5311yescharacteristic
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8C2×Dic5311yescharacteristic
376C4×Q8C2×Dic5311yescharacteristic
377C4×Q8C2×Dic5311yescharacteristic
378C4×Q8C22⋊D5311yescharacteristic
379C4×Q8C22⋊D5311yescharacteristic
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4—21no—
390C4×D4C2×Dic5311yescharacteristic
391#117—22no—
392#117—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4C22⋊D5311yescharacteristic
398C2×C42—21no—
399C2×C42—1061no—
400C2×C42C22×D5311yescharacteristic
401C4⋊C8—1061no—
402#104—2124no—
403#103—1064no—
404C22⋊C8—1061no—
405#97—1061no—
406#97—1061no—
407#97—1061no—
408#97—1061no—
409C4×C8—1061no—
410#95—1061no—
411#41—42no—
412#41—42no—
413#41—42no—
414#41—42no—
415#41—82no—
416#41—82no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8C22⋊Dic5311yescharacteristic
422C2×Q8C22⋊Dic5311yescharacteristic
423C2×Q8C22⋊Dic5311yescharacteristic
424C2×Q8C22⋊Dic5312yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4C22⋊Dic5311yescharacteristic
432C2×D4C22⋊Dic5312yes—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—2121no—
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444M16—1062no—
445M16—2121no—
446C2×C8—1062no—
447C2×C8—2121no—
448C2×C8—2121no—
449C2×C8—2121no—
450C2×C8—2121no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
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—21no—
465C4⋊C4—21no—
466C4⋊C4—1062no—
467C4⋊C4—2121no—
468C4⋊C4C22⋊Dic5311yescharacteristic
469C4⋊C4C22⋊Dic5311yescharacteristic
470C4⋊C4C22⋊Dic5311yescharacteristic
471C4⋊C4C22⋊Dic5311yescharacteristic
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—2122no—
491C42C22×Dic5311yescharacteristic
492C42C2×C22⋊D5311yescharacteristic
493C42C2×C22⋊D5311yescharacteristic
494C42C2×C22⋊D5311yescharacteristic
495C23—22no—
496C23—21no—
497C23#1313318611yescharacteristic
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—42no—
502Q8—41no—
503Q8—42no—
504Q8—82no—
505Q8#1313314412yescomplemented
506D4—41no—
507D4—42no—
508D4—42no—
509D4—44no—
510D4—42no—
511D4—41no—
512D4—42no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—82no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—4241no—
540C2×C4—4241no—
541C2×C4—2122no—
542C2×C4C2×C22⋊Dic5311yescharacteristic
543C2×C4C2×C22⋊Dic5311yescharacteristic
544C2×C4C2×C22⋊Dic5311yescharacteristic
545C2×C4#1313318611yescharacteristic
546C2×C4#1313318611yescharacteristic
547C2×C4#1313318611yescharacteristic
548C8—4241no—
549C8—2122no—
550C8—2122no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—41no—
559C22#1681047911yescharacteristic
560C22#1681079111yescharacteristic
561C22#1681080111yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—22no—
566C4—21no—
567C4—21no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—4242no—
576C4C2×#1313314411yescharacteristic
577C4#1681064011yescharacteristic
578C2—21no—
579C2—21no—
580C2—82no—
581C2—81no—
582C2#2027465911yescharacteristic
583C2#2027466911yescharacteristic
584C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3522C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3522D53
hypocenterC53#8419
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3522

Complements

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

NH ≅ G/Ntype
1. #20275635C2semidirect
2. #20274545C2semidirect
3. #20273326C2semidirect
4. #20273303C2semidirect
5. #20273240C2semidirect
6. #20273162C2semidirect
23. C53×#559C4semidirect
24. Q82×C53C4semidirect
26. C53×#554C4semidirect
27. C53×#550C4semidirect
212. Q8×C53#104semidirect
301. #559Dic53semidirect
302. Q82Dic53semidirect
304. #554Dic53semidirect
305. #550Dic53semidirect
331. C53#8419semidirect
505. Q8#13133144semidirect

Maximal subgroups

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

subgroupindex
#202756352
#202745452
#202733262
#202733032
#202732402
#202731622
C53×#35222
#841953