Finite Group Databasebeta

← #27468212

Subgroups — #27468212

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279146C211yesmaximal, characteristic
2#20279135C211yesmaximal, characteristic
3#20276278C211yesmaximal, characteristic, complemented
4#20276273C211yesmaximal, characteristic, complemented
5#20276150C211yesmaximal, characteristic
6#20276150C211yesmaximal, characteristic
7#20275441C211yesmaximal, characteristic, complemented
8#20275299C211yesmaximal, characteristic, complemented
9#20275292C211yesmaximal, characteristic, complemented
10#20274872C211yesmaximal, characteristic, complemented
11#20274737C211yesmaximal, characteristic
12#20274737C211yesmaximal, characteristic
13#106×Dic53C211yesmaximal, characteristic, complemented
14C8×#11719921C211yesmaximal, characteristic
15C53×#1941C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C2×#13133212—21no—
18C2×#13133149C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133129C2211yescharacteristic
23C2×C8×Dic53C2211yescharacteristic
24#16810565—21no—
25#16810562—21no—
26#16810550—21no—
27#16810545—21no—
28#16810519C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810515—21no—
31#16810509—21no—
32#16810506—21no—
33#16810493C2211yescharacteristic
34#16810493C2211yescharacteristic
35#16810480C2211yescharacteristic
36#16810480C2211yescharacteristic
37#16810480C2211yescharacteristic
38#16810480C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×#11719921C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×C4×C8×C53C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C2×C8×D53—22no—
63C2×C8×D53—22no—
64C2×C8×D53—22no—
65#13133212—44no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921D411yescharacteristic
81C2×#11719921D411yescharacteristic
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2311yescharacteristic
85#13133149—22no—
86#13133149—22no—
87#13133149—22no—
88#13133149—22no—
89#13133129—22no—
90C8×Dic53—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124D411yescharacteristic
100#13133124D411yescharacteristic
101C22×C8×C53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C42×C53C2311yescharacteristic
110#106×C53—22no—
111#106×C53—22no—
112C4×C8×C53—22no—
113#95×C53—21no—
114#95×C53—21no—
115C23×D53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2411yescharacteristic
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×C4×D53—21no—
135C2×C4×D53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53#4112yes—
139C2×C8×C53#4112yes—
140C2×C8×C53#4112yes—
141C2×C8×C53#4112yes—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—42no—
151C4⋊C4×C53—42no—
152C42×C53—21no—
153C42×C53—21no—
154#11719921—22no—
155#11719921—42no—
156#11719921—42no—
157#11719921—81no—
158#11719921—81no—
159#11719921SD1614yes—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—42no—
169C4⋊Dic53—42no—
170C4⋊Dic53—81no—
171C4⋊Dic53—81no—
172#11719919—42no—
173#11719919—81no—
174C4×Dic53—21no—
175C4×Dic53—21no—
176C4×Dic53—42no—
177C4×Dic53—81no—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C8×D53—44no—
185C23×C53C22×D411yescharacteristic
186C22×D53—21no—
187C22×D53—21no—
188C22×D53—21no—
189C22×D53—21no—
190C22×D53—21no—
191C22×D53—21no—
192C22×D53C2×SD1612yes—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53C2×SD1612yes—
210C2×C4×C53#13512yes—
211C2×C4×C53C2×#4111yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C2×C4×C532−1+411yescharacteristic
216C2×C4×C532−1+411yescharacteristic
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53C2×SD1612yes—
238C4×D53—41no—
239C4×D53—41no—
240C4×D53—41no—
241C4×D53—41no—
242C8×C53—22no—
243C8×C53#12614yescomplemented
244#88322—41no—
245#88322—41no—
246#27835—531nomaximal, Sylow
247C22×C53#53211yescharacteristic
248C22×C53#54711yescharacteristic
249C22×C53#54711yescharacteristic
250C22×C53C22×SD1611yescharacteristic
251C22×C53C2×#13511yescharacteristic
252C22×C53#57711yescharacteristic
253C22×C53#57711yescharacteristic
254C2×D53—22no—
255C2×D53—22no—
256C2×D53—22no—
257C2×D53—41no—
258C2×D53—41no—
259C2×D53—41no—
260C2×D53—41no—
261C4×C53—22no—
262C4×C53—22no—
263C4×C53—81no—
264C4×C53—81no—
265C4×C53—81no—
266C4×C53—81no—
267C4×C53C2×#12611yescharacteristic
268C4×C53#54111yescharacteristic
269C4×C53#55211yescharacteristic
270C4×C53#55611yescharacteristic
271Dic53—22no—
272Dic53—41no—
273Dic53—41no—
274Dic53—81no—
275Dic53—81no—
276Dic53—81no—
277Dic53—81no—
278#2497—531no—
279#2496—531no—
280#2100—531no—
281#2099—531no—
282#2077—531no—
283#2077—531no—
284#1941—531no—
285#1941D5311yescharacteristic, complemented
286#1920—531no—
287#1919—531no—
288#1844—531no—
289#1807—531no—
290#1807—531no—
291C4×#106—531no—
292C8×C22⋊C4—531no—
293C2×C53#249611yescharacteristic
294C2×C53#316711yescharacteristic
295C2×C53#317411yescharacteristic
296C2×C53#333511yescharacteristic
297C2×C53#333511yescharacteristic
298C2×C53#333711yescharacteristic
299C2×C53#334011yescharacteristic
300D53—42no—
301C23×C8—531no—
302C2×#126—1061no—
303C2×#106—531no—
304C2×#106—531no—
305C2×#106—531no—
306C2×#106—531no—
307C2×#106C2×D5311yescharacteristic
308C2×#106C2×D5311yescharacteristic
309C2×C22⋊C8—531no—
310C2×C4×C8—531no—
311C2×C4×C8C2×D5311yescharacteristic
312#400—1061no—
313#400—1061no—
314#397—1061no—
315#396—1061no—
316#389—531no—
317#389—531no—
318#389C2×D5311yescharacteristic
319#389C2×D5311yescharacteristic
320#388—1061no—
321#387—1061no—
322#387—1061no—
323#384—531no—
324#384—531no—
325#380—531no—
326#380—531no—
327#380—531no—
328#380—531no—
329C4×C4⋊C4—531no—
330C4×C4⋊C4—531no—
331C4×C22⋊C4—531no—
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340—531no—
337#340—531no—
338#340—531no—
339#340—531no—
340#340C2×D5311yescharacteristic
341#340C2×D5311yescharacteristic
342#336—531no—
343#336—531no—
344C53#2783511yescharacteristic, complemented, Sylow
345C23×C4—531no—
346C22×C8—531no—
347C22×C8—531no—
348C22×C8—1062no—
349C22×C8—1062no—
350C22×C8—1062no—
351C22×C8C22×D5311yescharacteristic
352C22×C8C22×D5311yescharacteristic
353#126—2124no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—531no—
357C2×C4⋊C4—531no—
358C2×C4⋊C4—531no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4C22×D5311yescharacteristic
367C2×C4⋊C4C22×D5311yescharacteristic
368C2×C4⋊C4C22×D5311yescharacteristic
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C22⋊C4—531no—
371C2×C22⋊C4—531no—
372C2×C22⋊C4—1061no—
373C2×C22⋊C4—1061no—
374C2×C22⋊C4—1061no—
375C2×C22⋊C4—1061no—
376C2×C42—531no—
377C2×C42—1061no—
378C2×C42—1061no—
379C2×C42C22×D5311yescharacteristic
380#106—22no—
381#106—22no—
382#106—1062no—
383#106—1062no—
384#106—1062no—
385#106—1062no—
386C22⋊C8—1062no—
387C4×C8—22no—
388C4×C8—1062no—
389#95—21no—
390#95—21no—
391#95—531no—
392#95—531no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397#95—1061no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401C24—531no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—531no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4—1061no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4D4×D5311yescharacteristic
420C22×C4D4×D5311yescharacteristic
421C22×C4C23×D5311yescharacteristic
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—1061no—
425C2×C8—1061no—
426C2×C8—1061no—
427C2×C8—1061no—
428C2×C8—1061no—
429C2×C8—1061no—
430C2×C8—2124no—
431C2×C8#1171994512yes—
432C2×C8#1171994712yes—
433C2×C8#1171994912yes—
434C2×C8#1171994912yes—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—1061no—
446C4⋊C4—1061no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1061no—
450C4⋊C4—1061no—
451C4⋊C4—1061no—
452C4⋊C4—1061no—
453C4⋊C4—4241no—
454C4⋊C4—4241no—
455C4⋊C4—4241no—
456C4⋊C4—2122no—
457C4⋊C4—2122no—
458C4⋊C4—2122no—
459C22⋊C4—534no—
460C22⋊C4—1062no—
461C22⋊C4—4241no—
462C22⋊C4—4241no—
463C22⋊C4—2122no—
464C22⋊C4—2122no—
465C42—21no—
466C42—21no—
467C42—1061no—
468C42—1061no—
469C42—4241no—
470C42—2122no—
471C23—532no—
472C23—1061no—
473C23—1061no—
474C23—1061no—
475C23—1061no—
476C23—1061no—
477C23—1061no—
478C23C2×D4×D5311yescharacteristic
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—532no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4—1061no—
504C2×C4—2121no—
505C2×C4—2121no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4SD16×D5312yes—
521C2×C4#1313323812yes—
522C2×C4C2×#1171994711yescharacteristic
523C2×C4C2×#1171994511yescharacteristic
524C2×C4C2×#1171994911yescharacteristic
525C2×C4C2×#1171994911yescharacteristic
526C2×C4#1313326411yescharacteristic
527C2×C4#1313326411yescharacteristic
528C8—22no—
529C8—2121no—
530C8—2121no—
531C8#1313321214yescomplemented
532C22—1062no—
533C22—1062no—
534C22—1062no—
535C22—2121no—
536C22—2121no—
537C22—2121no—
538C22—2121no—
539C22#1681117411yescharacteristic
540C22#1681127311yescharacteristic
541C22#1681127811yescharacteristic
542C22C2×SD16×D5311yescharacteristic
543C22C2×#1313323811yescharacteristic
544C22#1681143011yescharacteristic
545C22#1681143011yescharacteristic
546C4—22no—
547C4—22no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—1062no—
553C4—4241no—
554C4—4241no—
555C4—4241no—
556C4—4241no—
557C4—2121no—
558C4—2121no—
559C4C2×#1313321211yescharacteristic
560C4#1681123611yescharacteristic
561C4#1681131511yescharacteristic
562C4#1681134611yescharacteristic
563C2—2122no—
564C2#2027913511yescharacteristic
565C2#2028527611yescharacteristic
566C2#2028534511yescharacteristic
567C2#2028706711yescharacteristic
568C2#2028707411yescharacteristic
569C2#2028710011yescharacteristic
570C2#2028713011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1941C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1941D53
hypocenterC53#27835
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
3. #20276278C2semidirect
4. #20276273C2semidirect
7. #20275441C2semidirect
8. #20275299C2semidirect
9. #20275292C2semidirect
10. #20274872C2semidirect
13. #106×Dic53C2semidirect
15. C53×#1941C2semidirect
243. C8×C53#126semidirect
285. #1941D53semidirect
344. C53#27835semidirect
531. C8#13133212semidirect

Maximal subgroups

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

subgroupindex
#202791462
#202791352
#202762782
#202762732
#202761502
#202754412
#202752992
#202752922
#202748722
#202747372
#106×Dic532
C8×#117199212
C53×#19412
#2783553