Finite Group Databasebeta

← #27368395

Subgroups — #27368395

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284833C211yesmaximal, characteristic
2#20284516C211yesmaximal, characteristic, complemented
3M16×Dic106C211yesmaximal, characteristic, complemented
4#20274294C211yesmaximal, characteristic, complemented
5#20274252C211yesmaximal, characteristic, complemented
6#20274240C211yesmaximal, characteristic, complemented
7#20274184C211yesmaximal, characteristic, complemented
8#20274170C211yesmaximal, characteristic, complemented
9#20274128C211yesmaximal, characteristic, complemented
10#20274116C211yesmaximal, characteristic, complemented
11#20274060C211yesmaximal, characteristic, complemented
12#20273482C211yesmaximal, characteristic, complemented
13#20273450C211yesmaximal, characteristic
14#20273445C211yesmaximal, characteristic, complemented
15C53×#3182C211yesmaximal, characteristic
16C2×C4×Dic106C2211yescharacteristic
17#16810934—21no—
18#16810927—21no—
19#16810927—21no—
20#16810915C2211yescharacteristic
21#16810915C2211yescharacteristic
22#16810915C2211yescharacteristic
23#16810915C2211yescharacteristic
24#16810905C2211yescharacteristic
25#16810905C2211yescharacteristic
26#16810820C2211yescharacteristic
27#16810820C2211yescharacteristic
28#16810761—21no—
29#16810753C2211yescharacteristic
30#16810753C2211yescharacteristic
31#16810751—21no—
32#16810751—21no—
33C8×Dic106C2211yescharacteristic
34C8×Dic106C2211yescharacteristic
35#16810723—21no—
36#16810723—21no—
37#16810680C2211yescharacteristic
38#16810666—21no—
39#16810647C2211yescharacteristic
40#16810647C2211yescharacteristic
41#16810646—21no—
42#16810631—21no—
43M16×Dic53—21no—
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810372C2211yescharacteristic
51#16810372C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810367C2211yescharacteristic
55#16810367C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C53×#532C2211yescharacteristic
58C53×#500C2211yescharacteristic
59C53×#492C2211yescharacteristic
60C53×#487C2211yescharacteristic
61C53×#486C2211yescharacteristic
62C53×#468—21no—
63C53×#468—21no—
64C53×#440—21no—
65C53×#440—21no—
66C4×M16×C53C2211yescharacteristic
67C22×Dic106D411yescharacteristic
68C2×#11719923—21no—
69C2×#11719923—21no—
70C2×#11719917—21no—
71C2×#11719917—21no—
72C4×Dic106—21no—
73C4×Dic106—21no—
74C4×Dic106D411yescharacteristic
75C4×Dic106D411yescharacteristic
76C4×Dic106C2311yescharacteristic
77C4×Dic106C2311yescharacteristic
78C2×C4⋊Dic53D411yescharacteristic
79C2×#11719919—21no—
80C2×C4×Dic53—21no—
81#13133150—21no—
82#13133150—21no—
83#13133150—21no—
84#13133150—21no—
85#13133148—21no—
86#13133148—21no—
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322C2311yescharacteristic
90C4⋊#88322C2311yescharacteristic
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133146—21no—
96#13133146—21no—
97#13133146—21no—
98#13133146—21no—
99#13133140—21no—
100#13133140—21no—
101#13133140—21no—
102#13133140—21no—
103#13133140—21no—
104#13133140—21no—
105#13133138—21no—
106#13133138—21no—
107C22⋊#88322—21no—
108#13133128—21no—
109#13133127—21no—
110#13133127—21no—
111C8×Dic53—21no—
112C8×Dic53—21no—
113C2×#41×C53—21no—
114C2×SD16×C53—21no—
115C2×SD16×C53—21no—
116C2×M16×C53D411yescharacteristic
117C2×M16×C53D411yescharacteristic
118C4⋊Q8×C53C2311yescharacteristic
119C4⋊Q8×C53C2311yescharacteristic
120C4⋊Q8×C53C2311yescharacteristic
121#127×C53C2311yescharacteristic
122#125×C53—21no—
123#124×C53—21no—
124C22⋊Q8×C53—21no—
125C22⋊Q8×C53—21no—
126#121×C53—21no—
127#121×C53—21no—
128C4×Q8×C53—21no—
129C4×Q8×C53—21no—
130C4×D4×C53—21no—
131C4×D4×C53—21no—
132#117×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—21no—
135#107×C53—21no—
136#103×C53—21no—
137#103×C53—21no—
138#102×C53—21no—
139#102×C53—21no—
140#97×C53D411yescharacteristic
141#97×C53D411yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144#41×C53—82no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—41no—
152C22×C4×C53—41no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53#4111yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C2×Dic106—22no—
159C2×Dic106—21no—
160C2×Dic106—41no—
161C2×Dic106C2×D411yescharacteristic
162C2×Dic106C2×D411yescharacteristic
163SD16×C53—42no—
164SD16×C53—42no—
165M16×C53—22no—
166M16×C53—41no—
167C2×C8×C53C2×D411yescharacteristic
168C2×C8×C53C2×D411yescharacteristic
169C2×C8×C53C2×D411yescharacteristic
170C2×C8×C53C2×D411yescharacteristic
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—41no—
178C4⋊C4×C53—41no—
179C22⋊C4×C53—41no—
180C22⋊C4×C53—41no—
181C22⋊C4×C53—41no—
182C42×C53—41no—
183C42×C53C2×D411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C42×C53#4111yescharacteristic
186C42×C53C2411yescharacteristic
187#11719923—42no—
188#11719923—42no—
189C4⋊Dic53—21no—
190C4⋊Dic53C2×D411yescharacteristic
191C4⋊Dic53C2×D411yescharacteristic
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195#11719919—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200#11719917—42no—
201#11719917—42no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C23×C53—41no—
209C23×C53#12111yescharacteristic
210Q8×C53—41no—
211Q8×C53—41no—
212Q8×C53—81no—
213Q8×C53—81no—
214D4×C53—41no—
215D4×C53—41no—
216D4×C53—41no—
217D4×C53—41no—
218D4×C53—81no—
219D4×C53—82no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—82no—
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53#12111yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—42no—
240C2×Dic53—81no—
241Dic106—21no—
242Dic106—21no—
243Dic106—21no—
244Dic106—42no—
245Dic106#13712yes—
246C8×C53—21no—
247C8×C53—21no—
248C8×C53—21no—
249C8×C53#13712yes—
250#88322—41no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#88322—41no—
255#88322—41no—
256#19309—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53C2×#12111yescharacteristic
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—81no—
269C4×C53—81no—
270C4×C53—81no—
271C4×C53C2×#13711yescharacteristic
272C4×C53C2×#13711yescharacteristic
273C4×C53#57511yescharacteristic
274C4×C53#57511yescharacteristic
275Dic53—22no—
276Dic53—41no—
277Dic53—41no—
278Dic53—41no—
279#3182D5311yescharacteristic
280#3112—531no—
281#3078—531no—
282Q8×M16—531no—
283#1704—531no—
284#1698—531no—
285#1696—531no—
286#1686—531no—
287#1684—531no—
288#1678—531no—
289#1676—531no—
290#1666—531no—
291#1569—531no—
292#1560—531no—
293#1559—531no—
294C2×C53—41no—
295C2×C53—161no—
296C2×C53#303711yescharacteristic
297C2×C53#305611yescharacteristic
298C2×C53#306811yescharacteristic
299#533C2×D5311yescharacteristic
300#532C2×D5311yescharacteristic
301C2×C4×Q8—531no—
302#500C2×D5311yescharacteristic
303#492C2×D5311yescharacteristic
304#487C2×D5311yescharacteristic
305#486C2×D5311yescharacteristic
306#480—1061no—
307#479—1061no—
308#479—1061no—
309#477—531no—
310#477—531no—
311#477—531no—
312#477—531no—
313#475—531no—
314#475—531no—
315#468—21no—
316#468—21no—
317#460—531no—
318#460—531no—
319#448—1061no—
320#446—531no—
321#446—531no—
322#446—1061no—
323#446—1061no—
324C8×Q8—531no—
325C8×Q8—531no—
326#440—21no—
327#440—21no—
328#440—1061no—
329#440—1061no—
330#432—531no—
331#428—1061no—
332#423—531no—
333#423—531no—
334#423—1061no—
335#419—1061no—
336C4×M16—1061no—
337C4×M16C2×D5311yescharacteristic
338#335—531no—
339#335—531no—
340#334—531no—
341#334—531no—
342#333—531no—
343#333—531no—
344#333—531no—
345#333—531no—
346#332—531no—
347#332—531no—
348#331—531no—
349#331—531no—
350C53#1930911yescharacteristic, complemented, Sylow
351C2×#41—21no—
352C22×Q8—531no—
353C2×SD16—21no—
354C2×SD16—21no—
355C2×SD16—1061no—
356C2×SD16—1061no—
357C2×M16—1061no—
358C2×M16—1061no—
359C2×M16C22⋊D5311yescharacteristic
360C2×M16C22⋊D5311yescharacteristic
361C4⋊Q8C22×D5311yescharacteristic
362C4⋊Q8C22×D5311yescharacteristic
363C4⋊Q8C22×D5311yescharacteristic
364#127C22×D5311yescharacteristic
365#125—21no—
366#124—21no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369#121—21no—
370#121—21no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—531no—
374C4×Q8—531no—
375C4×Q8—531no—
376C4×Q8—531no—
377C4×Q8—1061no—
378C4×Q8—1061no—
379C4×D4—21no—
380C4×D4—21no—
381#117—21no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C42—1061no—
385C2×C42C22×D5311yescharacteristic
386#107—21no—
387#107—21no—
388#107—1061no—
389#107—1061no—
390#107—1061no—
391#107—1061no—
392#106—1061no—
393#106—1061no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406#103—21no—
407#103—21no—
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#103—1061no—
412#103—1061no—
413#103—1061no—
414#102—21no—
415#102—21no—
416#102—1061no—
417#102—1061no—
418C22⋊C8—1061no—
419#97—1061no—
420#97—1061no—
421#97—1061no—
422#97C22⋊D5311yescharacteristic
423#97C22⋊D5311yescharacteristic
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8C22×D5311yescharacteristic
427C4×C8C22×D5311yescharacteristic
428#41—82no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—531no—
433C2×Q8—531no—
434C2×Q8—1062no—
435C2×Q8—1061no—
436C2×Q8—2121no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C22×C4—41no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4D4×D5311yescharacteristic
445C22×C4#1171994711yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447SD16—42no—
448SD16—42no—
449SD16—2122no—
450SD16—2122no—
451M16—22no—
452M16—41no—
453M16—2122no—
454M16—2122no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8C2×C22⋊D5311yescharacteristic
462C2×C8C2×C22⋊D5311yescharacteristic
463C2×C8C2×C22⋊D5311yescharacteristic
464C2×C8C2×C22⋊D5311yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—531no—
474C4⋊C4—531no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C42—41no—
484C42—1061no—
485C42—1061no—
486C42—1061no—
487C42—1061no—
488C42D4×D5311yescharacteristic
489C42#1171994711yescharacteristic
490C42C2×C22⋊D5311yescharacteristic
491C42C23×D5311yescharacteristic
492C23—41no—
493C23#1313318811yescharacteristic
494Q8—41no—
495Q8—41no—
496Q8—81no—
497Q8—81no—
498Q8—532no—
499Q8—1061no—
500Q8—1061no—
501Q8—1061no—
502Q8—2122no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—81no—
508D4—82no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—4241no—
523C2×C4—2122no—
524C2×C4#1313318811yescharacteristic
525C2×C4#1313318811yescharacteristic
526C2×C4#1313318811yescharacteristic
527C2×C4C2×D4×D5311yescharacteristic
528C2×C4C22×C22⋊D5311yescharacteristic
529C2×C4C2×#1171994711yescharacteristic
530C8—21no—
531C8—21no—
532C8—21no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8#1313324712yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22C2×#1313318811yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—1062no—
555C4—2121no—
556C4—2121no—
557C4—2121no—
558C4#137×D5311yescharacteristic
559C4C2×#1313324711yescharacteristic
560C4#1681142111yescharacteristic
561C4#1681142211yescharacteristic
562C2—41no—
563C2—161no—
564C2#2028418311yescharacteristic
565C2#2028433911yescharacteristic
566C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3182C2
socleC22×C53C2×#121
prime radicalC53×#532C22
hypercenter#3182D53
hypocenterC53#19309
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3182

Complements

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

NH ≅ G/Ntype
2. #20284516C2semidirect
3. M16×Dic106C2semidirect
4. #20274294C2semidirect
5. #20274252C2semidirect
6. #20274240C2semidirect
7. #20274184C2semidirect
8. #20274170C2semidirect
9. #20274128C2semidirect
10. #20274116C2semidirect
11. #20274060C2semidirect
12. #20273482C2semidirect
14. #20273445C2semidirect
350. C53#19309semidirect

Maximal subgroups

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

subgroupindex
#202848332
#202845162
M16×Dic1062
#202742942
#202742522
#202742402
#202741842
#202741702
#202741282
#202741162
#202740602
#202734822
#202734502
#202734452
C53×#31822
#1930953