Finite Group Databasebeta

← #27367141

Subgroups — #27367141

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285460C211yesmaximal, characteristic, complemented
2#20284786C211yesmaximal, characteristic
3#20284503C211yesmaximal, characteristic
4#20274290C211yesmaximal, characteristic, complemented
5#20274262C211yesmaximal, characteristic, complemented
6#20274214C211yesmaximal, characteristic, complemented
7#20274204C211yesmaximal, characteristic, complemented
8#20274173C211yesmaximal, characteristic, complemented
9#20274145C211yesmaximal, characteristic, complemented
10#20274097C211yesmaximal, characteristic, complemented
11#20274085C211yesmaximal, characteristic, complemented
12#20273467C211yesmaximal, characteristic, complemented
13#20273426C211yesmaximal, characteristic, complemented
14#20273416C211yesmaximal, characteristic, complemented
15C53×#2971C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811180C2211yescharacteristic
18#16811021C2211yescharacteristic
19#16811003C2211yescharacteristic
20#16810974C2211yescharacteristic
21#16810974C2211yescharacteristic
22#16810964—21no—
23#16810930—21no—
24#16810923—21no—
25#16810918C2211yescharacteristic
26#16810913C2211yescharacteristic
27#16810908C2211yescharacteristic
28#16810908C2211yescharacteristic
29#16810903C2211yescharacteristic
30#16810903C2211yescharacteristic
31#16810881—21no—
32#16810881—21no—
33#16810833C2211yescharacteristic
34#16810828C2211yescharacteristic
35#16810785—21no—
36#16810731—21no—
37#16810731—21no—
38C4×#11719925—21no—
39C4×#11719923—21no—
40#16810676C2211yescharacteristic
41#16810659—21no—
42C2×C4⋊#88322C2211yescharacteristic
43#16810619—21no—
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810370C2211yescharacteristic
53#16810370C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C4×#41×C53C2211yescharacteristic
57C8×Q8×C53C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C53×#436—21no—
60C53×#436—21no—
61C53×#436C2211yescharacteristic
62C53×#436C2211yescharacteristic
63C53×#432—21no—
64C53×#432C2211yescharacteristic
65C4×M16×C53—21no—
66C4×M16×C53C2211yescharacteristic
67C2×Dic212—21no—
68C2×Dic212—21no—
69C2×#11719925—21no—
70C2×#11719923—21no—
71C22×#88322—21no—
72#13133217C2311yescharacteristic
73#13133217C2311yescharacteristic
74#13133217C2311yescharacteristic
75#13133217C2311yescharacteristic
76#13133209—21no—
77#13133209—21no—
78#13133196—21no—
79#13133196—21no—
80#13133196—21no—
81#13133196—21no—
82C4×Dic106—21no—
83C4×Dic106—21no—
84C4×Dic106—21no—
85C4×Dic106—21no—
86#13133176—21no—
87#13133176—21no—
88#13133150—21no—
89#13133150—21no—
90#13133150—21no—
91#13133149—21no—
92#13133149—21no—
93#13133148—21no—
94#13133148—21no—
95#13133148—21no—
96C4⋊#88322—21no—
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100C4⋊#88322C2311yescharacteristic
101#13133142—21no—
102#13133142—21no—
103#13133142—21no—
104#13133142—21no—
105#13133141—21no—
106#13133141—21no—
107#13133141—21no—
108#13133140—21no—
109#13133140—21no—
110#13133138—21no—
111#13133138—21no—
112#13133138—21no—
113C22⋊#88322—21no—
114C4×#88322—21no—
115C2×#41×C53D411yescharacteristic
116C2×M16×C53—21no—
117C2×M16×C53—21no—
118C2×M16×C53D411yescharacteristic
119C2×M16×C53D411yescharacteristic
120C4×Q8×C53D411yescharacteristic
121C4×Q8×C53C2311yescharacteristic
122C4×D4×C53—21no—
123C4×D4×C53—21no—
124C4×D4×C53D411yescharacteristic
125C4×D4×C53C2311yescharacteristic
126#117×C53—21no—
127#117×C53D411yescharacteristic
128C2×C42×C53—21no—
129C2×C42×C53C2311yescharacteristic
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53C2311yescharacteristic
133C4⋊C8×C53C2311yescharacteristic
134C22⋊C8×C53—21no—
135C22⋊C8×C53—21no—
136C22⋊C8×C53—21no—
137#97×C53—21no—
138#97×C53D411yescharacteristic
139#97×C53D411yescharacteristic
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144#41×C53—22no—
145#41×C53—41no—
146C2×Q8×C53C2×D411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C2×D411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53#4111yescharacteristic
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159M16×C53—22no—
160M16×C53—41no—
161M16×C53—42no—
162M16×C53—42no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C2×C8×C53C2×D411yescharacteristic
168C2×C8×C53C2×D411yescharacteristic
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53C2×D411yescharacteristic
172C4⋊C4×C53C2×D411yescharacteristic
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53C2×D411yescharacteristic
180C42×C53C2×D411yescharacteristic
181C42×C53#4111yescharacteristic
182C42×C53C2411yescharacteristic
183C22⋊Dic53—41no—
184C22⋊Dic53—41no—
185#11719925—42no—
186#11719923—42no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195#11719919—41no—
196#11719919—41no—
197#11719919—41no—
198#11719919—41no—
199C4×Dic53—41no—
200C4×Dic53—41no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—42no—
206Dic212—42no—
207Dic212—42no—
208C23×C53—21no—
209C23×C53#12111yescharacteristic
210Q8×C53—21no—
211Q8×C53#13712yes—
212D4×C53—21no—
213D4×C53—21no—
214D4×C53—42no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—41no—
223C2×C4×C53—42no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—81no—
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237Dic106—41no—
238Dic106—41no—
239Dic106—41no—
240Dic106—41no—
241Dic106—41no—
242Dic106—41no—
243Dic106—81no—
244C8×C53—21no—
245C8×C53—21no—
246C8×C53—21no—
247C8×C53—41no—
248C8×C53—41no—
249C8×C53#13512yes—
250#88322—41no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#19185—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×#12111yescharacteristic
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—22no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53C2×#13511yescharacteristic
270C4×C53C2×#13711yescharacteristic
271C4×C53#57511yescharacteristic
272C4×C53#57511yescharacteristic
273Dic53—81no—
274Dic53—81no—
275Dic53—81no—
276Dic53—81no—
277#3185—531no—
278#3107—531no—
279#3076—531no—
280#2971D5311yescharacteristic
281#1703—531no—
282#1699—531no—
283#1691—531no—
284#1689—531no—
285#1684—531no—
286#1680—531no—
287#1672—531no—
288#1670—531no—
289#1565—531no—
290#1554—531no—
291#1552—531no—
292C2×C53—41no—
293C2×C53—81no—
294C2×C53#303511yescharacteristic
295C2×C53#305611yescharacteristic
296C2×C53#306911yescharacteristic
297#533—531no—
298#533—531no—
299C4×#41C2×D5311yescharacteristic
300#498—531no—
301#494—531no—
302#487—531no—
303#487—531no—
304#485—1061no—
305#479—1061no—
306#478—1061no—
307#477—531no—
308#476—531no—
309#475—531no—
310#475—531no—
311#474—531no—
312#474—531no—
313#470—1061no—
314#470—1061no—
315#462—531no—
316#461—531no—
317#452—1061no—
318C8×Q8C2×D5311yescharacteristic
319C8×Q8C2×D5311yescharacteristic
320#441—1061no—
321#441—1061no—
322C4×Q16—1061no—
323C4×SD16—1061no—
324#436—21no—
325#436—21no—
326#436C2×D5311yescharacteristic
327#436C2×D5311yescharacteristic
328#432—21no—
329#432C2×D5311yescharacteristic
330#431—531no—
331#427—1061no—
332C2×C4⋊C8—531no—
333#416—1061no—
334C4×M16—21no—
335C4×M16C2×D5311yescharacteristic
336#335—531no—
337#335—531no—
338#334—531no—
339#334—531no—
340#333—531no—
341#333—531no—
342#332—531no—
343#332—531no—
344#332—531no—
345#332—531no—
346#332—531no—
347#332—531no—
348C53#1918511yescharacteristic, complemented, Sylow
349C2×#41D21211yescharacteristic
350C2×Q16—1061no—
351C2×Q16—1061no—
352C2×Q16—1061no—
353C2×SD16—1061no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16C22⋊D5311yescharacteristic
357C2×M16C22⋊D5311yescharacteristic
358C22×C8—1061no—
359C4⋊Q8—531no—
360C4⋊Q8—531no—
361C4⋊Q8—531no—
362C4⋊Q8—531no—
363#125—1061no—
364#125—1061no—
365C22⋊Q8—1061no—
366C22⋊Q8—1061no—
367C22⋊Q8—1061no—
368C22⋊Q8—1061no—
369C4×Q8—1061no—
370C4×Q8—1061no—
371C4×Q8—1061no—
372C4×Q8—1061no—
373C4×Q8D21211yescharacteristic
374C4×Q8C22×D5311yescharacteristic
375C4×D4—21no—
376C4×D4—21no—
377C4×D4D21211yescharacteristic
378C4×D4C22×D5311yescharacteristic
379#117—21no—
380#117—1061no—
381#117—1061no—
382#117D21211yescharacteristic
383C2×C42—21no—
384C2×C42C22×D5311yescharacteristic
385#107—1061no—
386#107—1061no—
387#107—1061no—
388#106—1061no—
389#106—1061no—
390#106—1061no—
391#106—1061no—
392#106—1061no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—1061no—
400C4⋊C8C22×D5311yescharacteristic
401C4⋊C8C22×D5311yescharacteristic
402#103—1061no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#102—1061no—
412#102—1061no—
413#102—1061no—
414C22⋊C8—21no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—1061no—
418#97—21no—
419#97C22⋊D5311yescharacteristic
420#97C22⋊D5311yescharacteristic
421C4×C8—21no—
422C4×C8—21no—
423C4×C8—1061no—
424C4×C8C22×D5311yescharacteristic
425C4×C8C22×D5311yescharacteristic
426#41—22no—
427#41—41no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×Q8C2×D21211yescharacteristic
433C2×D4—21no—
434C2×D4C2×D21211yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4C2×D21211yescharacteristic
439C22×C4#1171994511yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441Q16—2122no—
442Q16—2122no—
443Q16—2122no—
444SD16—2122no—
445M16—22no—
446M16—41no—
447M16—42no—
448M16—42no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—2122no—
456C2×C8C2×C22⋊D5311yescharacteristic
457C2×C8C2×C22⋊D5311yescharacteristic
458C2×C8C2×C22⋊D5311yescharacteristic
459C2×C8C2×C22⋊D5311yescharacteristic
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—2121no—
471C4⋊C4—2121no—
472C4⋊C4—2121no—
473C4⋊C4—2121no—
474C4⋊C4C2×D21211yescharacteristic
475C4⋊C4C2×D21211yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—2121no—
480C22⋊C4—2121no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—2121no—
485C42—2121no—
486C42C2×D21211yescharacteristic
487C42#1171994511yescharacteristic
488C42C2×C22⋊D5311yescharacteristic
489C42C23×D5311yescharacteristic
490C23—21no—
491C23#1313319111yescharacteristic
492Q8—21no—
493Q8—4241no—
494Q8—2121no—
495Q8—2121no—
496Q8—2121no—
497Q8—2121no—
498Q8—2121no—
499Q8—2121no—
500Q8#1313325012yes—
501D4—21no—
502D4—21no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4#1313319111yescharacteristic
521C2×C4#1313319111yescharacteristic
522C2×C4#1313319111yescharacteristic
523C2×C4C22×C22⋊D5311yescharacteristic
524C2×C4C22×D21211yescharacteristic
525C2×C4C2×#1171994511yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—2121no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8#1313323712yes—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×#1313319111yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—4241no—
551C4—4241no—
552C4—4241no—
553C4—4241no—
554C4C2×#1313323711yescharacteristic
555C4C2×#1313325011yescharacteristic
556C4#1681142111yescharacteristic
557C4#1681142311yescharacteristic
558C2—41no—
559C2—81no—
560C2#2028417011yescharacteristic
561C2#2028434511yescharacteristic
562C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2971C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenter#2971D53
hypocenterC53#19185
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2971

Complements

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

NH ≅ G/Ntype
1. #20285460C2semidirect
4. #20274290C2semidirect
5. #20274262C2semidirect
6. #20274214C2semidirect
7. #20274204C2semidirect
8. #20274173C2semidirect
9. #20274145C2semidirect
10. #20274097C2semidirect
11. #20274085C2semidirect
12. #20273467C2semidirect
13. #20273426C2semidirect
14. #20273416C2semidirect
348. C53#19185semidirect

Maximal subgroups

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

subgroupindex
#202854602
#202847862
#202845032
#202742902
#202742622
#202742142
#202742042
#202741732
#202741452
#202740972
#202740852
#202734672
#202734262
#202734162
C53×#29712
#1918553