Finite Group Databasebeta

← #27367827

Subgroups — #27367827

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286052C211yesmaximal, characteristic
2#20285890C211yesmaximal, characteristic
3#20285460C211yesmaximal, characteristic, complemented
4#20274297C211yesmaximal, characteristic, complemented
5#20274269C211yesmaximal, characteristic, complemented
6#20274214C211yesmaximal, characteristic, complemented
7#20274204C211yesmaximal, characteristic, complemented
8#20274166C211yesmaximal, characteristic, complemented
9#20274138C211yesmaximal, characteristic, complemented
10#20274097C211yesmaximal, characteristic, complemented
11#20274085C211yesmaximal, characteristic, complemented
12#20273467C211yesmaximal, characteristic, complemented
13#20273379C211yesmaximal, characteristic, complemented
14#20273370C211yesmaximal, characteristic, complemented
15C53×#2969C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811180C2211yescharacteristic
18#16811022—21no—
19#16811021C2211yescharacteristic
20#16811003C2211yescharacteristic
21#16810977—21no—
22#16810974C2211yescharacteristic
23#16810974C2211yescharacteristic
24#16810961—21no—
25#16810944—21no—
26#16810927—21no—
27#16810918C2211yescharacteristic
28#16810913C2211yescharacteristic
29#16810908C2211yescharacteristic
30#16810908C2211yescharacteristic
31#16810903C2211yescharacteristic
32#16810903C2211yescharacteristic
33#16810881—21no—
34#16810881—21no—
35#16810862—21no—
36#16810850—21no—
37#16810842—21no—
38#16810833C2211yescharacteristic
39#16810828C2211yescharacteristic
40#16810731—21no—
41#16810731—21no—
42#16810676C2211yescharacteristic
43C2×C4⋊#88322C2211yescharacteristic
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810370C2211yescharacteristic
53#16810370C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C53×#529C2211yescharacteristic
57C53×#446C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C53×#436—21no—
60C53×#436C2211yescharacteristic
61C53×#435—21no—
62C8×D4×C53C2211yescharacteristic
63C53×#433—21no—
64C53×#431C2211yescharacteristic
65C53×#405—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×M16×C53—21no—
116C2×M16×C53D411yescharacteristic
117C2×M16×C53D411yescharacteristic
118C22×C8×C53—21no—
119#126×C53—21no—
120#125×C53D411yescharacteristic
121#124×C53D411yescharacteristic
122#123×C53—21no—
123C22⋊Q8×C53D411yescharacteristic
124#121×C53D411yescharacteristic
125C4×Q8×C53C2311yescharacteristic
126C4×D4×C53—21no—
127C4×D4×C53C2311yescharacteristic
128#117×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
144C2×Q8×C53C2×D411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C2×D411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53#4111yescharacteristic
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155M16×C53—22no—
156M16×C53—41no—
157M16×C53—42no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—42no—
161C2×C8×C53C2×D411yescharacteristic
162C2×C8×C53C2×D411yescharacteristic
163C2×C8×C53C2×D411yescharacteristic
164C2×C8×C53C2×D411yescharacteristic
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C2×D411yescharacteristic
170C4⋊C4×C53C2×D411yescharacteristic
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C42×C53—21no—
177C42×C53C2×D411yescharacteristic
178C42×C53C2×D411yescharacteristic
179C42×C53#4111yescharacteristic
180C42×C53C2411yescharacteristic
181C22⋊Dic53—41no—
182C22⋊Dic53—41no—
183#11719925—81no—
184#11719923—81no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193#11719919—41no—
194#11719919—41no—
195#11719919—41no—
196#11719919—41no—
197C4×Dic53—41no—
198C4×Dic53—41no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—81no—
204Dic212—42no—
205Dic212—42no—
206C23×C53—21no—
207C23×C53#12111yescharacteristic
208Q8×C53—21no—
209Q8×C53—21no—
210D4×C53—21no—
211D4×C53—21no—
212D4×C53—81no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233Dic106—41no—
234Dic106—41no—
235Dic106—41no—
236Dic106—41no—
237Dic106—41no—
238Dic106—41no—
239Dic106—81no—
240C8×C53—21no—
241C8×C53—21no—
242C8×C53—21no—
243C8×C53—41no—
244C8×C53—41no—
245C8×C53#13512yes—
246#88322—41no—
247#88322—41no—
248#88322—41no—
249#88322—41no—
250#19251—531nomaximal, Sylow
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53C2×#12111yescharacteristic
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53C2×#13511yescharacteristic
266C4×C53#57511yescharacteristic
267C4×C53#57711yescharacteristic
268C4×C53#57811yescharacteristic
269Dic53—81no—
270Dic53—81no—
271Dic53—81no—
272Dic53—81no—
273#3248—531no—
274#3233—531no—
275#3185—531no—
276#2969D5311yescharacteristic
277#1704—531no—
278#1700—531no—
279#1691—531no—
280#1689—531no—
281#1683—531no—
282#1679—531no—
283#1672—531no—
284#1670—531no—
285#1565—531no—
286#1544—531no—
287#1542—531no—
288C2×C53—41no—
289C2×C53—81no—
290C2×C53#303511yescharacteristic
291C2×C53#306311yescharacteristic
292C2×C53#306911yescharacteristic
293#533—531no—
294#533—531no—
295#529C2×D5311yescharacteristic
296#499—1061no—
297#498—531no—
298#494—531no—
299#488—1061no—
300#487—531no—
301#487—531no—
302#485—1061no—
303#482—1061no—
304#479—1061no—
305#477—531no—
306#476—531no—
307#475—531no—
308#475—531no—
309#474—531no—
310#474—531no—
311#470—1061no—
312#470—1061no—
313#467—1061no—
314#465—1061no—
315#464—1061no—
316#462—531no—
317#461—531no—
318#446C2×D5311yescharacteristic
319C8×Q8C2×D5311yescharacteristic
320#441—1061no—
321#441—1061no—
322#436—21no—
323#436C2×D5311yescharacteristic
324#435—21no—
325C8×D4C2×D5311yescharacteristic
326#433—21no—
327#431—531no—
328#431C2×D5311yescharacteristic
329C2×C4⋊C8—531no—
330#405—21no—
331C4×M16C2×D5311yescharacteristic
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
336#333—531no—
337#333—531no—
338#332—531no—
339#332—531no—
340#332—531no—
341#332—531no—
342#332—531no—
343#332—531no—
344C53#1925111yescharacteristic, complemented, Sylow
345C2×Q16—1061no—
346C2×Q16—1061no—
347C2×Q16—1061no—
348C2×SD16—1061no—
349C2×M16—21no—
350C2×M16C22⋊D5311yescharacteristic
351C2×M16C22⋊D5311yescharacteristic
352C22×C8—21no—
353C22×C8—1061no—
354C4⋊Q8—531no—
355C4⋊Q8—531no—
356C4⋊Q8—531no—
357C4⋊Q8—531no—
358#126—21no—
359#125—1061no—
360#125—1061no—
361#125D21211yescharacteristic
362#124D21211yescharacteristic
363#123—21no—
364C22⋊Q8—1061no—
365C22⋊Q8—1061no—
366C22⋊Q8—1061no—
367C22⋊Q8—1061no—
368C22⋊Q8D21211yescharacteristic
369#121D21211yescharacteristic
370C4×Q8—1061no—
371C4×Q8—1061no—
372C4×Q8—1061no—
373C4×Q8—1061no—
374C4×Q8C22×D5311yescharacteristic
375C4×D4—21no—
376C4×D4C22×D5311yescharacteristic
377#117—21no—
378#117—1061no—
379#117—1061no—
380C2×C42C22×D5311yescharacteristic
381#107—1061no—
382#107—1061no—
383#107—1061no—
384#106—1061no—
385#106—1061no—
386#106—1061no—
387#106—1061no—
388#106—1061no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—531no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—1061no—
396C4⋊C8C22×D5311yescharacteristic
397C4⋊C8C22×D5311yescharacteristic
398#103—1061no—
399#103—1061no—
400#103—1061no—
401#103—1061no—
402#103—1061no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#103—1061no—
407#102—1061no—
408#102—1061no—
409#102—1061no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—1061no—
414#97—21no—
415#97C22⋊D5311yescharacteristic
416#97C22⋊D5311yescharacteristic
417C4×C8—21no—
418C4×C8—21no—
419C4×C8—1061no—
420C4×C8C22×D5311yescharacteristic
421C4×C8C22×D5311yescharacteristic
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8C2×D21211yescharacteristic
427C2×D4—21no—
428C2×D4C2×D21211yescharacteristic
429C22×C4—21no—
430C22×C4C2×D21211yescharacteristic
431C22×C4#1171994511yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433Q16—4241no—
434Q16—2122no—
435Q16—2122no—
436SD16—4241no—
437M16—22no—
438M16—41no—
439M16—42no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—42no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—4241no—
448C2×C8C2×C22⋊D5311yescharacteristic
449C2×C8C2×C22⋊D5311yescharacteristic
450C2×C8C2×C22⋊D5311yescharacteristic
451C2×C8C2×C22⋊D5311yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—2121no—
465C4⋊C4—2121no—
466C4⋊C4—2121no—
467C4⋊C4—2121no—
468C4⋊C4C2×D21211yescharacteristic
469C4⋊C4C2×D21211yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—2121no—
476C22⋊C4—2121no—
477C42—21no—
478C42—2121no—
479C42—2121no—
480C42C2×D21211yescharacteristic
481C42#1171994511yescharacteristic
482C42C2×C22⋊D5311yescharacteristic
483C42C23×D5311yescharacteristic
484C23—21no—
485C23#1313319111yescharacteristic
486Q8—21no—
487Q8—21no—
488Q8—4241no—
489Q8—2121no—
490Q8—2121no—
491Q8—2121no—
492Q8—2121no—
493Q8—2121no—
494Q8—2121no—
495D4—21no—
496D4—21no—
497D4—81no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4#1313319111yescharacteristic
513C2×C4#1313319111yescharacteristic
514C2×C4#1313319111yescharacteristic
515C2×C4C22×C22⋊D5311yescharacteristic
516C2×C4C22×D21211yescharacteristic
517C2×C4C2×#1171994511yescharacteristic
518C8—21no—
519C8—21no—
520C8—21no—
521C8—41no—
522C8—41no—
523C8—2121no—
524C8—2121no—
525C8—2121no—
526C8—2121no—
527C8#1313323712yes—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22C2×#1313319111yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—4241no—
543C4—4241no—
544C4—4241no—
545C4—4241no—
546C4C2×#1313323711yescharacteristic
547C4#1681142111yescharacteristic
548C4#1681143511yescharacteristic
549C4#1681144011yescharacteristic
550C2—41no—
551C2—81no—
552C2#2028417011yescharacteristic
553C2#2028440611yescharacteristic
554C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2969C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#2969D53
hypocenterC53#19251
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2969

Complements

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

NH ≅ G/Ntype
3. #20285460C2semidirect
4. #20274297C2semidirect
5. #20274269C2semidirect
6. #20274214C2semidirect
7. #20274204C2semidirect
8. #20274166C2semidirect
9. #20274138C2semidirect
10. #20274097C2semidirect
11. #20274085C2semidirect
12. #20273467C2semidirect
13. #20273379C2semidirect
14. #20273370C2semidirect
344. C53#19251semidirect

Maximal subgroups

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

subgroupindex
#202860522
#202858902
#202854602
#202742972
#202742692
#202742142
#202742042
#202741662
#202741382
#202740972
#202740852
#202734672
#202733792
#202733702
C53×#29692
#1925153