Finite Group Databasebeta

← #27369179

Subgroups — #27369179

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#20285933C211yesmaximal, characteristic
3#20285261C211yesmaximal, characteristic, complemented
4#20274243C211yesmaximal, characteristic, complemented
5#20274238C211yesmaximal, characteristic, complemented
6#20274221C211yesmaximal, characteristic, complemented
7#20274214C211yesmaximal, characteristic, complemented
8#20274119C211yesmaximal, characteristic, complemented
9#20274114C211yesmaximal, characteristic, complemented
10#20274097C211yesmaximal, characteristic, complemented
11#20274090C211yesmaximal, characteristic, complemented
12#20273482C211yesmaximal, characteristic, complemented
13#20273423C211yesmaximal, characteristic, complemented
14#20273395C211yesmaximal, characteristic, complemented
15C53×#2969C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18#16811033—21no—
19#16811032—21no—
20#16811032—21no—
21#16811028C2211yescharacteristic
22#16811028C2211yescharacteristic
23#16811021C2211yescharacteristic
24#16811021C2211yescharacteristic
25#16810982—21no—
26#16810951—21no—
27#16810934—21no—
28#16810927—21no—
29#16810918C2211yescharacteristic
30#16810918C2211yescharacteristic
31#16810918C2211yescharacteristic
32#16810913C2211yescharacteristic
33#16810913C2211yescharacteristic
34#16810913C2211yescharacteristic
35#16810879—21no—
36#16810867—21no—
37#16810842—21no—
38#16810833C2211yescharacteristic
39#16810828C2211yescharacteristic
40#16810680C2211yescharacteristic
41#16810665—21no—
42#16810665—21no—
43#16810647C2211yescharacteristic
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810371C2211yescharacteristic
53#16810371C2211yescharacteristic
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
67C22×Dic106—21no—
68C2×#11719925—21no—
69C2×#11719923—21no—
70C2×#11719917—21no—
71#13133217—21no—
72#13133217—21no—
73#13133217—21no—
74#13133217C2311yescharacteristic
75#13133217C2311yescharacteristic
76#13133217C2311yescharacteristic
77#13133217C2311yescharacteristic
78#13133209—21no—
79#13133196—21no—
80#13133196—21no—
81C4×Dic106—21no—
82C4×Dic106—21no—
83#13133176—21no—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53—21no—
86#13133151—21no—
87#13133151—21no—
88#13133151—21no—
89#13133151—21no—
90#13133150—21no—
91#13133150—21no—
92#13133150—21no—
93#13133149—21no—
94#13133149—21no—
95#13133149—21no—
96#13133149—21no—
97#13133148—21no—
98#13133148—21no—
99#13133148—21no—
100C4⋊#88322—21no—
101C4⋊#88322C2311yescharacteristic
102C4⋊#88322C2311yescharacteristic
103C4⋊#88322C2311yescharacteristic
104C4⋊#88322C2311yescharacteristic
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—
114#13133128—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
151C22×Dic53—41no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—41no—
157C2×Dic106—82no—
158M16×C53—22no—
159M16×C53—22no—
160M16×C53—42no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—42no—
164C2×C8×C53C2×D411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C2×C8×C53C2×D411yescharacteristic
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C2×D411yescharacteristic
173C4⋊C4×C53C2×D411yescharacteristic
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C42×C53—21no—
180C42×C53C2×D411yescharacteristic
181C42×C53C2×D411yescharacteristic
182C42×C53#4111yescharacteristic
183C42×C53C2411yescharacteristic
184C22⋊Dic53—41no—
185#11719925—81no—
186#11719923—81no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—41no—
196C4⋊Dic53—41no—
197#11719919—41no—
198#11719919—41no—
199C4×Dic53—41no—
200#11719917—81no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C23×C53—21no—
206C23×C53#12111yescharacteristic
207Q8×C53—21no—
208Q8×C53—21no—
209D4×C53—21no—
210D4×C53—21no—
211D4×C53—81no—
212C2×C4×C53—21no—
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—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53#12111yescharacteristic
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—82no—
233Dic106—41no—
234Dic106—41no—
235Dic106—82no—
236Dic106—81no—
237Dic106—81no—
238Dic106—81no—
239C8×C53—21no—
240C8×C53—21no—
241C8×C53—41no—
242C8×C53—41no—
243C8×C53#13712yes—
244C8×C53#13712yes—
245#88322—41no—
246#88322—41no—
247#88322—41no—
248#88322—41no—
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×C53C2×#13711yescharacteristic
265C4×C53C2×#13711yescharacteristic
266C4×C53#57711yescharacteristic
267C4×C53#57811yescharacteristic
268Dic53—81no—
269Dic53—81no—
270Dic53—81no—
271Dic53—81no—
272#3252—531no—
273#3238—531no—
274#3165—531no—
275#2969D5311yescharacteristic
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⋊Q8—531no—
294#529C2×D5311yescharacteristic
295#501—1061no—
296#501—1061no—
297#501—1061no—
298#500—531no—
299#500—531no—
300#498—531no—
301#498—531no—
302#489—1061no—
303#483—1061no—
304#480—1061no—
305#479—1061no—
306#477—531no—
307#477—531no—
308#477—531no—
309#476—531no—
310#476—531no—
311#476—531no—
312#470—1061no—
313#468—1061no—
314#464—1061no—
315#462—531no—
316#461—531no—
317#446C2×D5311yescharacteristic
318C8×Q8C2×D5311yescharacteristic
319#436—21no—
320#436C2×D5311yescharacteristic
321#435—21no—
322C8×D4C2×D5311yescharacteristic
323#433—21no—
324#432—531no—
325#431C2×D5311yescharacteristic
326#428—1061no—
327#428—1061no—
328#423—531no—
329#405—21no—
330C4×M16C2×D5311yescharacteristic
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—1061no—
345C2×Q16—1061no—
346C2×SD16—1061no—
347C2×M16—21no—
348C2×M16—1061no—
349C2×M16C22⋊D5311yescharacteristic
350C2×M16C22⋊D5311yescharacteristic
351C22×C8—21no—
352C4⋊Q8—531no—
353C4⋊Q8—531no—
354C4⋊Q8—531no—
355C4⋊Q8—531no—
356C4⋊Q8—1061no—
357C4⋊Q8—1061no—
358C4⋊Q8—1061no—
359#126—21no—
360#125—1061no—
361#125D21211yescharacteristic
362#124D21211yescharacteristic
363#123—21no—
364C22⋊Q8—1061no—
365C22⋊Q8—1061no—
366C22⋊Q8D21211yescharacteristic
367#121D21211yescharacteristic
368C4×Q8—1061no—
369C4×Q8—1061no—
370C4×Q8C22×D5311yescharacteristic
371C4×D4—21no—
372C4×D4C22×D5311yescharacteristic
373#117—21no—
374#117—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C42C22×D5311yescharacteristic
378#107—1061no—
379#107—1061no—
380#107—1061no—
381#107—1061no—
382#107—1061no—
383#107—1061no—
384#107—1061no—
385#106—1061no—
386#106—1061no—
387#106—1061no—
388#106—1061no—
389#106—1061no—
390#106—1061no—
391#106—1061no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—1061no—
399C4⋊C8C22×D5311yescharacteristic
400C4⋊C8C22×D5311yescharacteristic
401#103—1061no—
402#103—1061no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#102—1061no—
407#102—1061no—
408#102—1061no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8—1061no—
413#97—21no—
414#97—1061no—
415#97C22⋊D5311yescharacteristic
416#97C22⋊D5311yescharacteristic
417C4×C8—21no—
418C4×C8—21no—
419C4×C8C22×D5311yescharacteristic
420C4×C8C22×D5311yescharacteristic
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—4242no—
426C2×Q8—2121no—
427C2×Q8C2×D21211yescharacteristic
428C2×D4—21no—
429C2×D4C2×D21211yescharacteristic
430C22×C4—21no—
431C22×C4—2121no—
432C22×C4C2×D21211yescharacteristic
433C22×C4#1171994511yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435Q16—4241no—
436SD16—4241no—
437M16—22no—
438M16—22no—
439M16—42no—
440M16—4241no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—42no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
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—
476C42—21no—
477C42—2121no—
478C42C2×D21211yescharacteristic
479C42#1171994511yescharacteristic
480C42C2×C22⋊D5311yescharacteristic
481C42C23×D5311yescharacteristic
482C23—21no—
483C23#1313319111yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—4241no—
487Q8—4241no—
488Q8—4241no—
489Q8—4242no—
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—4242no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4#1313319111yescharacteristic
511C2×C4#1313319111yescharacteristic
512C2×C4#1313319111yescharacteristic
513C2×C4C22×C22⋊D5311yescharacteristic
514C2×C4C22×D21211yescharacteristic
515C2×C4C2×#1171994511yescharacteristic
516C8—21no—
517C8—21no—
518C8—41no—
519C8—41no—
520C8—2121no—
521C8—2121no—
522C8—2121no—
523C8—2121no—
524C8#1313324712yes—
525C8#1313324712yes—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22C2×#1313319111yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—4241no—
541C4—4241no—
542C4—4241no—
543C4—4241no—
544C4C2×#1313324711yescharacteristic
545C4C2×#1313324711yescharacteristic
546C4#1681143511yescharacteristic
547C4#1681144011yescharacteristic
548C2—41no—
549C2—81no—
550C2#2028420211yescharacteristic
551C2#2028440611yescharacteristic
552C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2969C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#2969D53
hypocenterC53#19379
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. #20285261C2semidirect
4. #20274243C2semidirect
5. #20274238C2semidirect
6. #20274221C2semidirect
7. #20274214C2semidirect
8. #20274119C2semidirect
9. #20274114C2semidirect
10. #20274097C2semidirect
11. #20274090C2semidirect
12. #20273482C2semidirect
13. #20273423C2semidirect
14. #20273395C2semidirect
343. C53#19379semidirect

Maximal subgroups

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

subgroupindex
#202860882
#202859332
#202852612
#202742432
#202742382
#202742212
#202742142
#202741192
#202741142
#202740972
#202740902
#202734822
#202734232
#202733952
C53×#29692
#1937953