Finite Group Databasebeta

← #27367115

Subgroups — #27367115

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285429C211yesmaximal, characteristic, complemented
2#20284503C211yesmaximal, characteristic, complemented
3#20283735C211yesmaximal, characteristic, complemented
4#20274292C211yesmaximal, characteristic, complemented
5#20274246C211yesmaximal, characteristic, complemented
6#20274236C211yesmaximal, characteristic, complemented
7#20274178C211yesmaximal, characteristic, complemented
8#20274175C211yesmaximal, characteristic, complemented
9#20274129C211yesmaximal, characteristic, complemented
10#20274117C211yesmaximal, characteristic, complemented
11#20274061C211yesmaximal, characteristic, complemented
12#20273468C211yesmaximal, characteristic, complemented
13#20273423C211yesmaximal, characteristic, complemented
14#20273410C211yesmaximal, characteristic, complemented
15C53×#3106C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811026C2211yescharacteristic
18#16810995C2211yescharacteristic
19#16810975C2211yescharacteristic
20#16810969C2211yescharacteristic
21#16810930—21no—
22#16810923—21no—
23#16810918C2211yescharacteristic
24#16810913C2211yescharacteristic
25#16810908C2211yescharacteristic
26#16810903C2211yescharacteristic
27#16810867—21no—
28#16810867—21no—
29Q8×#88322C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31#16810723—21no—
32#16810723—21no—
33#16810701—21no—
34#16810701—21no—
35#16810701C2211yescharacteristic
36#16810701C2211yescharacteristic
37#16810680—21no—
38#16810680C2211yescharacteristic
39#16810659—21no—
40C2×C4⋊#88322C2211yescharacteristic
41#16810619—21no—
42C4×#11719917—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810367C2211yescharacteristic
55#16810367C2211yescharacteristic
56C53×#532C2211yescharacteristic
57C4×#41×C53C2211yescharacteristic
58C53×#482—21no—
59C53×#477C2211yescharacteristic
60C53×#476C2211yescharacteristic
61C53×#460C2211yescharacteristic
62C53×#459C2211yescharacteristic
63C53×#449—21no—
64C4×SD16×C53—21no—
65C4×D8×C53—21no—
66C53×#431C2211yescharacteristic
67C2×#11719923—21no—
68C2×#11719923—21no—
69C2×#11719917—21no—
70C2×#11719917—21no—
71C2×#11719917D411yescharacteristic
72C2×#11719917D411yescharacteristic
73C22×#88322—21no—
74#13133217C2311yescharacteristic
75#13133217C2311yescharacteristic
76#13133209—21no—
77#13133196—21no—
78#13133196—21no—
79C4×Dic106—21no—
80C4×Dic106—21no—
81#13133176—21no—
82#13133150—21no—
83#13133150—21no—
84#13133150—21no—
85#13133150—21no—
86#13133148—21no—
87#13133148—21no—
88C4⋊#88322—21no—
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133141—21no—
96#13133141—21no—
97#13133140—21no—
98#13133140—21no—
99#13133138—21no—
100#13133138—21no—
101#13133138—21no—
102#13133138—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106#13133128—21no—
107#13133128D411yescharacteristic
108#13133128D411yescharacteristic
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×#41×C53—21no—
114C2×#41×C53D411yescharacteristic
115C2×SD16×C53—21no—
116C2×D8×C53—21no—
117C4⋊Q8×C53C2311yescharacteristic
118#127×C53C2311yescharacteristic
119#124×C53—21no—
120#121×C53—21no—
121#121×C53—21no—
122C4×Q8×C53D411yescharacteristic
123C4×Q8×C53C2311yescharacteristic
124C4×D4×C53—21no—
125C4×D4×C53—21no—
126C4×D4×C53—21no—
127C4×D4×C53—21no—
128C4×D4×C53D411yescharacteristic
129C4×D4×C53C2311yescharacteristic
130#117×C53—21no—
131#117×C53D411yescharacteristic
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—21no—
135#106×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53—21no—
139#102×C53—21no—
140#102×C53—21no—
141#102×C53—21no—
142C22⋊C8×C53—21no—
143C4×C8×C53—21no—
144#41×C53—22no—
145#41×C53—41no—
146#41×C53—82no—
147C2×Q8×C53—21no—
148C2×Q8×C53C2×D411yescharacteristic
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53—41no—
154C2×D4×C53C2×D411yescharacteristic
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—41no—
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53#4111yescharacteristic
162C2×Dic106—21no—
163C2×Dic106—21no—
164SD16×C53—42no—
165D8×C53—42no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C2×D411yescharacteristic
173C4⋊C4×C53C2×D411yescharacteristic
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—41no—
178C22⋊C4×C53—41no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53C2×D411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53#4111yescharacteristic
185C42×C53C2411yescharacteristic
186C22⋊Dic53—41no—
187#11719923—42no—
188#11719923—42no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193#11719919—41no—
194#11719919—41no—
195C4×Dic53—41no—
196#11719917—22no—
197#11719917—41no—
198#11719917—42no—
199#11719917—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—42no—
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C2×#88322C2×D411yescharacteristic
209C23×C53—21no—
210C23×C53—41no—
211C23×C53#12111yescharacteristic
212Q8×C53—21no—
213Q8×C53—81no—
214Q8×C53#13712yes—
215D4×C53—21no—
216D4×C53—21no—
217D4×C53—42no—
218D4×C53—41no—
219D4×C53—41no—
220D4×C53—41no—
221D4×C53—41no—
222D4×C53—81no—
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—41no—
232C2×C4×C53—42no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—81no—
237C2×C4×C53—82no—
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53#12111yescharacteristic
240C2×C4×C53#12111yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×Dic53—41no—
245C2×Dic53—41no—
246Dic106—41no—
247Dic106—41no—
248Dic106—81no—
249C8×C53—41no—
250C8×C53—41no—
251#88322—21no—
252#88322—21no—
253#88322—21no—
254#88322—41no—
255#88322—41no—
256#88322—41no—
257#88322—41no—
258#88322#13512yes—
259#19183—531nomaximal, Sylow
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53C2×#12111yescharacteristic
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—22no—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—81no—
278C4×C53C2×#13511yescharacteristic
279C4×C53C2×#13711yescharacteristic
280C4×C53#57511yescharacteristic
281C4×C53#57511yescharacteristic
282Dic53—81no—
283Dic53—81no—
284#3182—531no—
285#3106D5311yescharacteristic
286#3076—531no—
287#2971—531no—
288#1703—531no—
289#1697—531no—
290#1695—531no—
291#1685—531no—
292#1684—531no—
293#1678—531no—
294#1676—531no—
295#1666—531no—
296#1565—531no—
297#1554—531no—
298#1551—531no—
299C2×C53—41no—
300C2×C53—81no—
301C2×C53—161no—
302C2×C53#303511yescharacteristic
303C2×C53#305611yescharacteristic
304C2×C53#306811yescharacteristic
305#533—531no—
306#532C2×D5311yescharacteristic
307C4×#41C2×D5311yescharacteristic
308#500—531no—
309#492—531no—
310#487—531no—
311#486—531no—
312#482—21no—
313#479—1061no—
314#478—1061no—
315#477—531no—
316#477C2×D5311yescharacteristic
317#476—531no—
318#476C2×D5311yescharacteristic
319#475—531no—
320#474—531no—
321#468—1061no—
322#468—1061no—
323#460C2×D5311yescharacteristic
324#459C2×D5311yescharacteristic
325#449—21no—
326C8×Q8—531no—
327C8×Q8—531no—
328#440—1061no—
329#440—1061no—
330C4×SD16—21no—
331C4×D8—21no—
332#436—531no—
333#436—531no—
334#436—1061no—
335#436—1061no—
336#432—531no—
337#432—1061no—
338#431C2×D5311yescharacteristic
339#427—1061no—
340C2×C4⋊C8—531no—
341#416—1061no—
342C4×M16—531no—
343C4×M16—1061no—
344#335—531no—
345#335—531no—
346#334—531no—
347#334—531no—
348#333—531no—
349#333—531no—
350#333—531no—
351#333—531no—
352#332—531no—
353#332—531no—
354#331—531no—
355#331—531no—
356C53#1918311yescharacteristic, complemented, Sylow
357C2×#41—21no—
358C2×#41C22⋊D5311yescharacteristic
359C2×SD16—21no—
360C2×SD16—1061no—
361C2×SD16—1061no—
362C2×D8—21no—
363C2×M16—531no—
364C2×M16—531no—
365C2×M16—1061no—
366C2×M16—1061no—
367C22×C8—1061no—
368C4⋊Q8—531no—
369C4⋊Q8—531no—
370C4⋊Q8C22×D5311yescharacteristic
371#127C22×D5311yescharacteristic
372#125—1061no—
373#124—21no—
374C22⋊Q8—1061no—
375C22⋊Q8—1061no—
376#121—21no—
377#121—21no—
378C4×Q8—1061no—
379C4×Q8—1061no—
380C4×Q8C22⋊D5311yescharacteristic
381C4×Q8C22×D5311yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4C22⋊D5311yescharacteristic
387C4×D4C22×D5311yescharacteristic
388#117—21no—
389#117—1061no—
390#117C22⋊D5311yescharacteristic
391C2×C42—21no—
392C2×C42C22×D5311yescharacteristic
393#107—21no—
394#107—1061no—
395#107—1061no—
396#107—1061no—
397#107—1061no—
398#106—21no—
399#106—1061no—
400#106—1061no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—531no—
404C4⋊C8—531no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407C4⋊C8—1061no—
408C4⋊C8C22×D5311yescharacteristic
409C4⋊C8C22×D5311yescharacteristic
410#103—21no—
411#103—1061no—
412#103—1061no—
413#103—1061no—
414#103—1061no—
415#102—21no—
416#102—21no—
417#102—21no—
418#102—1061no—
419#102—1061no—
420#102—1061no—
421#102—1061no—
422C22⋊C8—21no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426#97—531no—
427#97—531no—
428#97—1061no—
429C4×C8—21no—
430C4×C8—531no—
431C4×C8—531no—
432C4×C8—1061no—
433C4×C8—1061no—
434#41—22no—
435#41—41no—
436#41—82no—
437C2×Q8—21no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8C2×C22⋊D5311yescharacteristic
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—41no—
446C2×D4C2×C22⋊D5311yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4D4×D5311yescharacteristic
452C22×C4#1171994711yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454SD16—42no—
455SD16—2122no—
456SD16—2122no—
457D8—42no—
458M16—1062no—
459M16—2122no—
460M16—2121no—
461M16—2122no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—531no—
465C2×C8—531no—
466C2×C8—531no—
467C2×C8—531no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—2122no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—2121no—
482C4⋊C4—2121no—
483C4⋊C4C2×C22⋊D5311yescharacteristic
484C4⋊C4C2×C22⋊D5311yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—2121no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—2121no—
495C42D4×D5311yescharacteristic
496C42#1171994711yescharacteristic
497C42C2×C22⋊D5311yescharacteristic
498C42C23×D5311yescharacteristic
499C23—21no—
500C23—41no—
501C23#1313318911yescharacteristic
502Q8—21no—
503Q8—81no—
504Q8—4241no—
505Q8—2121no—
506Q8—2121no—
507Q8#1313324712yes—
508D4—21no—
509D4—21no—
510D4—42no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—81no—
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—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—81no—
530C2×C4—82no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4#1313318911yescharacteristic
534C2×C4#1313318911yescharacteristic
535C2×C4#1313318911yescharacteristic
536C2×C4C2×D4×D5311yescharacteristic
537C2×C4C22×C22⋊D5311yescharacteristic
538C2×C4C2×#1171994711yescharacteristic
539C8—41no—
540C8—41no—
541C8—532no—
542C8—1061no—
543C8—1061no—
544C8—1061no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22C2×#1313318911yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—4241no—
568C4—4241no—
569C4#135×D5311yescharacteristic
570C4C2×#1313324711yescharacteristic
571C4#1681142011yescharacteristic
572C4#1681142111yescharacteristic
573C2—41no—
574C2—81no—
575C2—161no—
576C2#2028416611yescharacteristic
577C2#2028434711yescharacteristic
578C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3106C2
socleC22×C53C2×#121
hypercenter#3106D53
hypocenterC53#19183
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #20285429C2semidirect
2. #20284503C2semidirect
3. #20283735C2semidirect
4. #20274292C2semidirect
5. #20274246C2semidirect
6. #20274236C2semidirect
7. #20274178C2semidirect
8. #20274175C2semidirect
9. #20274129C2semidirect
10. #20274117C2semidirect
11. #20274061C2semidirect
12. #20273468C2semidirect
13. #20273423C2semidirect
14. #20273410C2semidirect
356. C53#19183semidirect

Maximal subgroups

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

subgroupindex
#202854292
#202845032
#202837352
#202742922
#202742462
#202742362
#202741782
#202741752
#202741292
#202741172
#202740612
#202734682
#202734232
#202734102
C53×#31062
#1918353