Finite Group Databasebeta

← #27369175

Subgroups — #27369175

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286088C211yesmaximal, characteristic
2#20285257C211yesmaximal, characteristic, complemented
3#20283720C211yesmaximal, characteristic
4#20274241C211yesmaximal, characteristic, complemented
5#20274236C211yesmaximal, characteristic, complemented
6#20274223C211yesmaximal, characteristic, complemented
7#20274216C211yesmaximal, characteristic, complemented
8#20274117C211yesmaximal, characteristic, complemented
9#20274112C211yesmaximal, characteristic, complemented
10#20274099C211yesmaximal, characteristic, complemented
11#20274092C211yesmaximal, characteristic, complemented
12#20273483C211yesmaximal, characteristic, complemented
13#20273426C211yesmaximal, characteristic, complemented
14#20273396C211yesmaximal, characteristic, complemented
15C53×#3238C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811029—21no—
18#16811029—21no—
19#16811026C2211yescharacteristic
20#16811026C2211yescharacteristic
21#16811019C2211yescharacteristic
22#16811019C2211yescharacteristic
23#16810927—21no—
24#16810918C2211yescharacteristic
25#16810913C2211yescharacteristic
26#16810879—21no—
27#16810867—21no—
28#16810842—21no—
29#16810833C2211yescharacteristic
30#16810828C2211yescharacteristic
31#16810754C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33#16810701—21no—
34#16810701C2211yescharacteristic
35#16810695—21no—
36D4×#88322C2211yescharacteristic
37#16810684—21no—
38#16810676C2211yescharacteristic
39#16810663—21no—
40#16810663—21no—
41#16810647C2211yescharacteristic
42#16810579—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810372C2211yescharacteristic
51#16810372C2211yescharacteristic
52#16810371C2211yescharacteristic
53#16810371C2211yescharacteristic
54#16810370C2211yescharacteristic
55#16810370C2211yescharacteristic
56C2×C4⋊Q8×C53C2211yescharacteristic
57C53×#529C2211yescharacteristic
58C53×#501—21no—
59C53×#489—21no—
60C53×#483—21no—
61C53×#480—21no—
62C53×#477C2211yescharacteristic
63C53×#477C2211yescharacteristic
64C53×#476C2211yescharacteristic
65C53×#476C2211yescharacteristic
66C53×#432C2211yescharacteristic
67C2×#11719925—21no—
68C2×#11719923—21no—
69C2×#11719917—21no—
70C2×#11719917—21no—
71C2×#11719917D411yescharacteristic
72C2×#11719917D411yescharacteristic
73C22×#88322—21no—
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#13133150—21no—
87#13133148—21no—
88#13133148—21no—
89#13133148—21no—
90#13133148—21no—
91#13133148—21no—
92C4⋊#88322—21no—
93C4⋊#88322—21no—
94C4⋊#88322—21no—
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99#13133141—21no—
100#13133140—21no—
101#13133140—21no—
102#13133138—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106#13133128—21no—
107#13133128D411yescharacteristic
108#13133128D411yescharacteristic
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C22×Q8×C53—21no—
114C4⋊Q8×C53—21no—
115C4⋊Q8×C53—21no—
116C4⋊Q8×C53—21no—
117C4⋊Q8×C53C2311yescharacteristic
118C4⋊Q8×C53C2311yescharacteristic
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×C4⋊C4×C53—21no—
130C2×C4⋊C4×C53—21no—
131C2×C42×C53C2311yescharacteristic
132#107×C53—21no—
133#107×C53—21no—
134#106×C53—21no—
135#106×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53—21no—
139#103×C53—21no—
140#102×C53—21no—
141#102×C53—21no—
142C22⋊C8×C53—21no—
143#97×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—41no—
147C2×Q8×C53—82no—
148C2×Q8×C53C2×D411yescharacteristic
149C2×D4×C53—21no—
150C2×D4×C53C2×D411yescharacteristic
151C22×C4×C53—21no—
152C22×C4×C53—41no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53#4111yescharacteristic
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
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—21no—
176C22⋊C4×C53—21no—
177C42×C53—21no—
178C42×C53C2×D411yescharacteristic
179C42×C53C2×D411yescharacteristic
180C42×C53#4111yescharacteristic
181C42×C53C2411yescharacteristic
182C22⋊Dic53—41no—
183#11719925—81no—
184#11719923—81no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189#11719919—41no—
190#11719919—41no—
191C4×Dic53—41no—
192#11719917—22no—
193#11719917—22no—
194#11719917—42no—
195#11719917—81no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—42no—
201C2×#88322C2×D411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C23×C53—21no—
206C23×C53#12111yescharacteristic
207Q8×C53—21no—
208Q8×C53—21no—
209Q8×C53—82no—
210Q8×C53—81no—
211Q8×C53—81no—
212D4×C53—21no—
213D4×C53—21no—
214D4×C53—81no—
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—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—82no—
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×Dic53—41no—
235C2×Dic53—41no—
236Dic106—41no—
237Dic106—41no—
238Dic106—81no—
239C8×C53—41no—
240C8×C53—41no—
241#88322—21no—
242#88322—21no—
243#88322—41no—
244#88322—41no—
245#88322—41no—
246#88322—41no—
247#88322#13712yes—
248#88322#13712yes—
249#19379—531nomaximal, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53C2×#12111yescharacteristic
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—81no—
265C4×C53—81no—
266C4×C53C2×#13711yescharacteristic
267C4×C53C2×#13711yescharacteristic
268C4×C53#57711yescharacteristic
269C4×C53#57811yescharacteristic
270Dic53—81no—
271Dic53—81no—
272#3252—531no—
273#3238D5311yescharacteristic
274#3165—531no—
275#2969—531no—
276#1696—531no—
277#1695—531no—
278#1692—531no—
279#1691—531no—
280#1676—531no—
281#1675—531no—
282#1672—531no—
283#1671—531no—
284#1569—531no—
285#1554—531no—
286#1548—531no—
287C2×C53—41no—
288C2×C53—81no—
289C2×C53#303911yescharacteristic
290C2×C53#306311yescharacteristic
291C2×C53#306811yescharacteristic
292#533—531no—
293C2×C4⋊Q8C2×D5311yescharacteristic
294#529C2×D5311yescharacteristic
295#501—21no—
296#501—1061no—
297#501—1061no—
298#500—531no—
299#500—531no—
300#498—531no—
301#498—531no—
302#489—21no—
303#483—21no—
304#480—21no—
305#479—1061no—
306#477—531no—
307#477C2×D5311yescharacteristic
308#477C2×D5311yescharacteristic
309#476—531no—
310#476C2×D5311yescharacteristic
311#476C2×D5311yescharacteristic
312#470—1061no—
313#468—1061no—
314#464—1061no—
315#462—531no—
316#461—531no—
317#446—531no—
318C8×Q8—531no—
319#436—531no—
320#436—1061no—
321#435—1061no—
322C8×D4—531no—
323#433—1061no—
324#432C2×D5311yescharacteristic
325#431—531no—
326#428—1061no—
327#428—1061no—
328#423—531no—
329#405—1061no—
330C4×M16—531no—
331#335—531no—
332#335—531no—
333#334—531no—
334#334—531no—
335#333—531no—
336#333—531no—
337#333—531no—
338#333—531no—
339#333—531no—
340#333—531no—
341#332—531no—
342#332—531no—
343C53#1937911yescharacteristic, complemented, Sylow
344C22×Q8—21no—
345C2×Q16—1061no—
346C2×SD16—1061no—
347C2×M16—531no—
348C2×M16—531no—
349C2×M16—1061no—
350C2×M16—1061no—
351C22×C8—1061no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—531no—
356C4⋊Q8—531no—
357C4⋊Q8C22×D5311yescharacteristic
358C4⋊Q8C22×D5311yescharacteristic
359#126—21no—
360#125—1061no—
361#125C22⋊D5311yescharacteristic
362#124C22⋊D5311yescharacteristic
363#123—21no—
364C22⋊Q8—1061no—
365C22⋊Q8—1061no—
366C22⋊Q8C22⋊D5311yescharacteristic
367#121C22⋊D5311yescharacteristic
368C4×Q8—1061no—
369C4×Q8—1061no—
370C4×Q8C22×D5311yescharacteristic
371C4×D4—21no—
372C4×D4C22×D5311yescharacteristic
373#117—21no—
374#117—1061no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C42C22×D5311yescharacteristic
378#107—21no—
379#107—21no—
380#107—1061no—
381#107—1061no—
382#107—1061no—
383#107—1061no—
384#107—1061no—
385#106—21no—
386#106—21no—
387#106—1061no—
388#106—1061no—
389#106—1061no—
390#106—1061no—
391#106—1061no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8C22×D5311yescharacteristic
400C4⋊C8C22×D5311yescharacteristic
401#103—21no—
402#103—21no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#102—21no—
407#102—21no—
408#102—1061no—
409C22⋊C8—21no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—21no—
414#97—531no—
415#97—531no—
416#97—1061no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—1061no—
420C4×C8—1061no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—41no—
424C2×Q8—82no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8C2×C22⋊D5311yescharacteristic
428C2×D4—21no—
429C2×D4C2×C22⋊D5311yescharacteristic
430C22×C4—21no—
431C22×C4—41no—
432C22×C4D4×D5311yescharacteristic
433C22×C4#1171994711yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435Q16—4241no—
436SD16—4241no—
437M16—1062no—
438M16—1062no—
439M16—4241no—
440M16—2122no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—531no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—2122no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—2121no—
467C4⋊C4—2121no—
468C4⋊C4C2×C22⋊D5311yescharacteristic
469C4⋊C4C2×C22⋊D5311yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—2121no—
476C42—21no—
477C42—2121no—
478C42D4×D5311yescharacteristic
479C42#1171994711yescharacteristic
480C42C2×C22⋊D5311yescharacteristic
481C42C23×D5311yescharacteristic
482C23—21no—
483C23#1313318911yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—81no—
487Q8—81no—
488Q8—82no—
489Q8—4241no—
490Q8—2121no—
491Q8—2121no—
492D4—21no—
493D4—21no—
494D4—81no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—82no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4#1313318911yescharacteristic
511C2×C4#1313318911yescharacteristic
512C2×C4#1313318911yescharacteristic
513C2×C4C2×D4×D5311yescharacteristic
514C2×C4C22×C22⋊D5311yescharacteristic
515C2×C4C2×#1171994711yescharacteristic
516C8—41no—
517C8—41no—
518C8—532no—
519C8—532no—
520C8—1061no—
521C8—1061no—
522C8—2121no—
523C8—2121no—
524C8—2121no—
525C8—2121no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22C2×#1313318911yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—81no—
541C4—81no—
542C4—4241no—
543C4—4241no—
544C4#137×D5311yescharacteristic
545C4#137×D5311yescharacteristic
546C4#1681143111yescharacteristic
547C4#1681143811yescharacteristic
548C2—41no—
549C2—81no—
550C2#2028420011yescharacteristic
551C2#2028440211yescharacteristic
552C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3238C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#3238D53
hypocenterC53#19379
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3238

Complements

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

NH ≅ G/Ntype
2. #20285257C2semidirect
4. #20274241C2semidirect
5. #20274236C2semidirect
6. #20274223C2semidirect
7. #20274216C2semidirect
8. #20274117C2semidirect
9. #20274112C2semidirect
10. #20274099C2semidirect
11. #20274092C2semidirect
12. #20273483C2semidirect
13. #20273426C2semidirect
14. #20273396C2semidirect
343. C53#19379semidirect

Maximal subgroups

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

subgroupindex
#202860882
#202852572
#202837202
#202742412
#202742362
#202742232
#202742162
#202741172
#202741122
#202740992
#202740922
#202734832
#202734262
#202733962
C53×#32382
#1937953