Finite Group Databasebeta

← #27366808

Subgroups — #27366808

Order 13568.

596 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285297C211yesmaximal, characteristic, complemented
2#20284503C211yesmaximal, characteristic
3#20283682C211yesmaximal, characteristic
4#20274299C211yesmaximal, characteristic, complemented
5#20274295C211yesmaximal, characteristic, complemented
6#20274292C211yesmaximal, characteristic, complemented
7#20274288C211yesmaximal, characteristic, complemented
8#20274241C211yesmaximal, characteristic, complemented
9#20274236C211yesmaximal, characteristic, complemented
10#20274207C211yesmaximal, characteristic, complemented
11#20274202C211yesmaximal, characteristic, complemented
12C2×#16810373C211yesmaximal, characteristic, complemented
13#20273373C211yesmaximal, characteristic, complemented
14#20273372C211yesmaximal, characteristic, complemented
15C53×#3076C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811026C2211yescharacteristic
18#16811002C2211yescharacteristic
19#16810975C2211yescharacteristic
20#16810973C2211yescharacteristic
21#16810930—21no—
22#16810923—21no—
23#16810918C2211yescharacteristic
24#16810913C2211yescharacteristic
25#16810908C2211yescharacteristic
26#16810903C2211yescharacteristic
27#16810862—21no—
28#16810862—21no—
29#16810754C2211yescharacteristic
30#16810754C2211yescharacteristic
31C4×#11719925—21no—
32C4×#11719925—21no—
33#16810701C2211yescharacteristic
34#16810701C2211yescharacteristic
35#16810695—21no—
36#16810695—21no—
37#16810676—21no—
38#16810676C2211yescharacteristic
39#16810659—21no—
40C2×C4⋊#88322C2211yescharacteristic
41#16810619—21no—
42C4×#11719917—21no—
43C2×C4×#88322C2211yescharacteristic
44#16810373C2211yescharacteristic
45#16810373C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C4×#41×C53C2211yescharacteristic
58C53×#479—21no—
59C53×#478—21no—
60C53×#477C2211yescharacteristic
61C53×#476C2211yescharacteristic
62C53×#475C2211yescharacteristic
63C53×#474C2211yescharacteristic
64C53×#427—21no—
65C2×C4⋊C8×C53C2211yescharacteristic
66C53×#416—21no—
67C2×#11719925—21no—
68C2×#11719925—21no—
69C2×#11719917—21no—
70C2×#11719917—21no—
71C22×#88322—21no—
72C22×#88322D411yescharacteristic
73C22×#88322D411yescharacteristic
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#13133141—21no—
98#13133141—21no—
99#13133140—21no—
100#13133140—21no—
101#13133138—21no—
102#13133138—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106#13133128—21no—
107#13133128—21no—
108C4×#88322—21no—
109C4×#88322D411yescharacteristic
110C4×#88322D411yescharacteristic
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×#41×C53D411yescharacteristic
114C22×C8×C53—21no—
115C4⋊Q8×C53C2311yescharacteristic
116C4⋊Q8×C53C2311yescharacteristic
117#125×C53—21no—
118C22⋊Q8×C53—21no—
119C22⋊Q8×C53—21no—
120C4×Q8×C53—21no—
121C4×Q8×C53—21no—
122C4×Q8×C53D411yescharacteristic
123C4×Q8×C53C2311yescharacteristic
124C4×D4×C53—21no—
125C4×D4×C53—21no—
126C4×D4×C53D411yescharacteristic
127C4×D4×C53C2311yescharacteristic
128#117×C53—21no—
129#117×C53—21no—
130#117×C53D411yescharacteristic
131C2×C42×C53—21no—
132C2×C42×C53C2311yescharacteristic
133#107×C53—21no—
134#107×C53—21no—
135#106×C53—21no—
136#106×C53—21no—
137C4⋊C8×C53—21no—
138C4⋊C8×C53C2311yescharacteristic
139C4⋊C8×C53C2311yescharacteristic
140#103×C53—21no—
141#103×C53—21no—
142#102×C53—21no—
143#102×C53—21no—
144#41×C53—22no—
145#41×C53—22no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53C2×D411yescharacteristic
149C2×D4×C53—21no—
150C2×D4×C53C2×D411yescharacteristic
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53#4111yescharacteristic
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—42no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—41no—
169C4⋊C4×C53—41no—
170C4⋊C4×C53C2×D411yescharacteristic
171C4⋊C4×C53C2×D411yescharacteristic
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—41no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—41no—
180C42×C53C2×D411yescharacteristic
181C42×C53C2×D411yescharacteristic
182C42×C53#4111yescharacteristic
183C42×C53C2411yescharacteristic
184C22⋊Dic53—41no—
185#11719925—42no—
186#11719925—42no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191#11719919—41no—
192#11719919—41no—
193C4×Dic53—41no—
194#11719917—42no—
195#11719917—42no—
196C2×#88322—22no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—41no—
202C2×#88322—42no—
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C23×C53—21no—
208C23×C53#12111yescharacteristic
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53—81no—
214D4×C53—21no—
215D4×C53—21no—
216D4×C53—42no—
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—42no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53#12111yescharacteristic
232C2×C4×C53#13712yes—
233C2×C4×C53#13712yes—
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×Dic53—41no—
238C2×Dic53—41no—
239Dic106—41no—
240Dic106—41no—
241Dic106—81no—
242C8×C53—41no—
243C8×C53—41no—
244#88322—21no—
245#88322—21no—
246#88322—21no—
247#88322—41no—
248#88322—41no—
249#88322—41no—
250#88322—41no—
251#88322#13512yes—
252#19158—531nomaximal, Sylow
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53#47012yes—
259C22×C53C2×#12111yescharacteristic
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—21no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53—81no—
267C4×C53—81no—
268C4×C53#47012yes—
269C4×C53C2×#13511yescharacteristic
270C4×C53C2×#13711yescharacteristic
271C4×C53C2×#13711yescharacteristic
272C4×C53#57511yescharacteristic
273Dic53—81no—
274Dic53—81no—
275#3170—531no—
276#3076—531no—
277#3076D5311yescharacteristic
278#2965—531no—
279#1704—531no—
280#1704—531no—
281#1703—531no—
282#1703—531no—
283#1696—531no—
284#1695—531no—
285#1690—531no—
286#1689—531no—
287C2×#334—531no—
288#1542—531no—
289#1542—531no—
290C2×C53—22no—
291C2×C53—81no—
292C2×C53#303511yescharacteristic
293C2×C53C2×#47011yescharacteristic
294C2×C53#306711yescharacteristic
295#533—531no—
296#533C2×D5311yescharacteristic
297C4×#41C2×D5311yescharacteristic
298#500—531no—
299#494—531no—
300#487—531no—
301#487—531no—
302#479—21no—
303#479—1061no—
304#478—21no—
305#478—1061no—
306#477—531no—
307#477C2×D5311yescharacteristic
308#476—531no—
309#476C2×D5311yescharacteristic
310#475—531no—
311#475C2×D5311yescharacteristic
312#474—531no—
313#474C2×D5311yescharacteristic
314#467—1061no—
315#467—1061no—
316#446—531no—
317#446—531no—
318C4×Q16—1061no—
319C4×Q16—1061no—
320#436—531no—
321#436—531no—
322#435—1061no—
323#435—1061no—
324#431—531no—
325#431—1061no—
326#427—21no—
327#427—1061no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8C2×D5311yescharacteristic
330#416—21no—
331#416—1061no—
332C4×M16—1061no—
333C2×C4×C8—531no—
334#334—531no—
335#334—531no—
336#334—531no—
337#334—531no—
338#333—531no—
339#333—531no—
340#333—531no—
341#333—531no—
342#332—531no—
343#332—531no—
344#332—531no—
345#332—531no—
346C53#1915811yescharacteristic, complemented, Sylow
347C2×#41C22⋊D5311yescharacteristic
348C2×Q16—1061no—
349C2×Q16—1061no—
350C2×M16—1061no—
351C2×M16—1061no—
352C22×C8—21no—
353C22×C8—531no—
354C22×C8—531no—
355C22×C8—1061no—
356C4⋊Q8—531no—
357C4⋊Q8—531no—
358C4⋊Q8C22×D5311yescharacteristic
359C4⋊Q8C22×D5311yescharacteristic
360#125—21no—
361#125—1061no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—1061no—
365C22⋊Q8—1061no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—1061no—
369C4×Q8—1061no—
370C4×Q8C22⋊D5311yescharacteristic
371C4×Q8C22×D5311yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C22⋊D5311yescharacteristic
375C4×D4C22×D5311yescharacteristic
376#117—21no—
377#117—21no—
378#117—1061no—
379#117C22⋊D5311yescharacteristic
380C2×C42—21no—
381C2×C42C22×D5311yescharacteristic
382#107—21no—
383#107—21no—
384#107—1061no—
385#107—1061no—
386#107—1061no—
387#107—1061no—
388#106—21no—
389#106—21no—
390#106—1061no—
391#106—1061no—
392C4⋊C8—21no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8C22×D5311yescharacteristic
401C4⋊C8C22×D5311yescharacteristic
402#103—21no—
403#103—21no—
404#103—1061no—
405#103—1061no—
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410#102—21no—
411#102—21no—
412#102—1061no—
413#102—1061no—
414C22⋊C8—1061no—
415C22⋊C8—1061no—
416C22⋊C8—1061no—
417#97—1061no—
418#97—1061no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—1061no—
424#41—22no—
425#41—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8C2×C22⋊D5311yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D5311yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4D4×D5311yescharacteristic
437C22×C4#1171994711yescharacteristic
438C22×C4C2×C22⋊D5311yescharacteristic
439Q16—2122no—
440Q16—2122no—
441M16—2122no—
442M16—2122no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—42no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—1062no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—2121no—
456C2×C8—2122no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—2121no—
470C4⋊C4—2121no—
471C4⋊C4C2×C22⋊D5311yescharacteristic
472C4⋊C4C2×C22⋊D5311yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—41no—
477C22⋊C4—2121no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—2121no—
483C42D4×D5311yescharacteristic
484C42#1171994711yescharacteristic
485C42C2×C22⋊D5311yescharacteristic
486C42C23×D5311yescharacteristic
487C23—21no—
488C23#1313318911yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—41no—
492Q8—41no—
493Q8—81no—
494Q8—4241no—
495Q8—2121no—
496Q8—2121no—
497D4—21no—
498D4—21no—
499D4—42no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4#1313318911yescharacteristic
515C2×C4#1313318911yescharacteristic
516C2×C4#1313318911yescharacteristic
517C2×C4#1313324712yes—
518C2×C4#1313324812yes—
519C2×C4C2×D4×D5311yescharacteristic
520C2×C4C22×C22⋊D5311yescharacteristic
521C2×C4C2×#1171994711yescharacteristic
522C8—41no—
523C8—41no—
524C8—532no—
525C8—1061no—
526C8—1061no—
527C8—1061no—
528C8—2121no—
529C8—2121no—
530C8—2121no—
531C8—2121no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22#1681087912yes—
538C22C2×#1313318911yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—4241no—
548C4—4241no—
549C4#1681087912yes—
550C4#135×D5311yescharacteristic
551C4C2×#1313324711yescharacteristic
552C4C2×#1313324811yescharacteristic
553C4#1681142011yescharacteristic
554C2—22no—
555C2—81no—
556C2#2028416611yescharacteristic
557C2C2×#1681087911yescharacteristic
558C2#2028443211yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3076C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenter#3076D53
hypocenterC53#19158
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #20285297C2semidirect
4. #20274299C2semidirect
5. #20274295C2semidirect
6. #20274292C2semidirect
7. #20274288C2semidirect
8. #20274241C2semidirect
9. #20274236C2semidirect
10. #20274207C2semidirect
11. #20274202C2semidirect
12. C2×#16810373C2semidirect
13. #20273373C2semidirect
14. #20273372C2semidirect
346. C53#19158semidirect

Maximal subgroups

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

subgroupindex
#202852972
#202845032
#202836822
#202742992
#202742952
#202742922
#202742882
#202742412
#202742362
#202742072
#202742022
C2×#168103732
#202733732
#202733722
C53×#30762
#1915853