Finite Group Databasebeta

← #27369623

Subgroups — #27369623

Order 13568.

616 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285851C211yesmaximal, characteristic, complemented
2#20284775C211yesmaximal, characteristic
3#20283826C211yesmaximal, characteristic, complemented
4#20274161C211yesmaximal, characteristic, complemented
5#20274154C211yesmaximal, characteristic, complemented
6#20274133C211yesmaximal, characteristic, complemented
7#20274126C211yesmaximal, characteristic, complemented
8#20274117C211yesmaximal, characteristic, complemented
9#20274112C211yesmaximal, characteristic, complemented
10#20274095C211yesmaximal, characteristic, complemented
11#20274088C211yesmaximal, characteristic, complemented
12#20273496C211yesmaximal, characteristic, complemented
13#20273411C211yesmaximal, characteristic, complemented
14#20273372C211yesmaximal, characteristic, complemented
15C53×#3116C211yesmaximal, characteristic, complemented
16#16811173C2211yescharacteristic
17#16811026C2211yescharacteristic
18#16811019C2211yescharacteristic
19#16810971C2211yescharacteristic
20#16810971C2211yescharacteristic
21#16810947—21no—
22#16810947—21no—
23#16810947—21no—
24#16810920—21no—
25#16810920—21no—
26#16810918C2211yescharacteristic
27#16810913C2211yescharacteristic
28#16810823C2211yescharacteristic
29#16810818C2211yescharacteristic
30#16810768—21no—
31Q8×#88322C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33C4×#11719924—21no—
34C4×#11719922—21no—
35D4×#88322—21no—
36D4×#88322—21no—
37D4×#88322C2211yescharacteristic
38D4×#88322C2211yescharacteristic
39#16810676—21no—
40#16810676C2211yescharacteristic
41#16810676C2211yescharacteristic
42C2×C4⋊#88322—21no—
43C2×C4⋊#88322C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C4×#41×C53C2211yescharacteristic
58C53×#485—21no—
59C53×#477C2211yescharacteristic
60C53×#476C2211yescharacteristic
61C53×#462C2211yescharacteristic
62C53×#461C2211yescharacteristic
63C53×#452—21no—
64C53×#441—21no—
65C53×#440—21no—
66C53×#432C2211yescharacteristic
67C2×#11719945—21no—
68C2×#11719924—21no—
69C2×#11719922—21no—
70C22×#88322—21no—
71C22×#88322—21no—
72C22×#88322—21no—
73#13133217C2311yescharacteristic
74#13133214C2311yescharacteristic
75#13133205—21no—
76#13133191—21no—
77#13133191—21no—
78C4×D212—21no—
79C4×D212—21no—
80#13133150—21no—
81#13133150—21no—
82#13133150—21no—
83#13133148—21no—
84#13133148—21no—
85#13133148—21no—
86C4⋊#88322—21no—
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322D411yescharacteristic
90C4⋊#88322D411yescharacteristic
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133141—21no—
96#13133138—21no—
97#13133137—21no—
98#13133137—21no—
99#13133137—21no—
100#13133137—21no—
101#13133137—21no—
102#13133137—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106C22⋊#88322D411yescharacteristic
107C22⋊#88322D411yescharacteristic
108C4×#88322—21no—
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×#41×C53D411yescharacteristic
114C2×Q16×C53—21no—
115C2×SD16×C53—21no—
116C4⋊Q8×C53C2311yescharacteristic
117C4⋊Q8×C53C2311yescharacteristic
118#125×C53—21no—
119C22⋊Q8×C53—21no—
120C22⋊Q8×C53—21no—
121C4×Q8×C53—21no—
122C4×Q8×C53—21no—
123C4×Q8×C53D411yescharacteristic
124C4×Q8×C53C2311yescharacteristic
125C4×D4×C53—21no—
126C4×D4×C53—21no—
127C4×D4×C53D411yescharacteristic
128C4×D4×C53C2311yescharacteristic
129#117×C53—21no—
130#117×C53—21no—
131#117×C53D411yescharacteristic
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—21no—
135#106×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53—21no—
139#103×C53—21no—
140#103×C53—21no—
141#102×C53—21no—
142C22⋊C8×C53—21no—
143#97×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×C22⋊D53—41no—
158#11719945—82no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×C4×D53—41no—
163C2×Dic106—21no—
164Q16×C53—42no—
165SD16×C53—42no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—41no—
175C4⋊C4×C53—41no—
176C4⋊C4×C53C2×D411yescharacteristic
177C4⋊C4×C53C2×D411yescharacteristic
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—41no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—41no—
186C42×C53C2×D411yescharacteristic
187C42×C53C2×D411yescharacteristic
188C42×C53#4111yescharacteristic
189C42×C53C2411yescharacteristic
190#11719924—42no—
191#11719922—42no—
192#11719921—41no—
193#11719921—41no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—81no—
201C2×#88322—81no—
202C2×#88322—81no—
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C23×C53—21no—
208C23×C53#12111yescharacteristic
209C22×D53—41no—
210Q8×C53—21no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53—81no—
214Q8×C53#13512yes—
215D4×C53—24no—
216D4×C53—21no—
217D4×C53#13512yes—
218C2×C4×C53—24no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—81no—
232C2×C4×C53#12111yescharacteristic
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C22⋊D53—82no—
239C2×Dic53—41no—
240D212—41no—
241D212—41no—
242D212—41no—
243D212—41no—
244D212—81no—
245C4×D53—82no—
246Dic106—81no—
247C8×C53—41no—
248C8×C53—41no—
249#88322—21no—
250#88322—21no—
251#88322—21no—
252#88322—21no—
253#88322—41no—
254#88322—41no—
255#88322—41no—
256#88322—41no—
257#19419—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—22no—
261C22×C53—21no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53C2×#12111yescharacteristic
265C2×D53—81no—
266C2×D53—81no—
267C2×D53—81no—
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—22no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53C2×#13511yescharacteristic
278C4×C53C2×#13511yescharacteristic
279C4×C53#57711yescharacteristic
280C4×C53#57811yescharacteristic
281Dic53—81no—
282#3228—531no—
283#3116D5311yescharacteristic, complemented
284#3106—531no—
285#2988—531no—
286#1682—531no—
287#1681—531no—
288#1678—531no—
289#1677—531no—
290#1676—531no—
291#1675—531no—
292#1672—531no—
293#1671—531no—
294#1572—531no—
295#1551—531no—
296#1542—531no—
297C2×C53—41no—
298C2×C53—42no—
299C2×C53#304611yescharacteristic
300C2×C53#305011yescharacteristic
301C2×C53#306811yescharacteristic
302D53—161no—
303#533C2×D5311yescharacteristic
304#532—531no—
305C4×#41C2×D5311yescharacteristic
306#500—531no—
307#498—531no—
308#486—531no—
309#486—531no—
310#485—21no—
311#482—1061no—
312#482—1061no—
313#482—1061no—
314#478—1061no—
315#478—1061no—
316#477—531no—
317#477C2×D5311yescharacteristic
318#476—531no—
319#476C2×D5311yescharacteristic
320#462C2×D5311yescharacteristic
321#461C2×D5311yescharacteristic
322#460—531no—
323#459—531no—
324#452—21no—
325#449—1061no—
326C8×Q8—531no—
327C8×Q8—531no—
328#441—21no—
329#440—21no—
330C4×SD16—1061no—
331C4×D8—1061no—
332C8×D4—531no—
333C8×D4—531no—
334C8×D4—1061no—
335C8×D4—1061no—
336#432C2×D5311yescharacteristic
337#431—531no—
338#431—531no—
339#431—1061no—
340C2×C4⋊C8—531no—
341C2×C4⋊C8—1061no—
342#335—531no—
343#335—531no—
344#334—531no—
345#334—531no—
346#333—531no—
347#333—531no—
348#333—531no—
349#333—531no—
350#332—531no—
351#332—531no—
352#331—531no—
353#331—531no—
354C53#1941911yescharacteristic, complemented, Sylow
355C2×#41—1061no—
356C2×#41C22⋊D5311yescharacteristic
357C2×Q16—21no—
358C2×SD16—21no—
359C2×SD16—1061no—
360C2×D8—1061no—
361C22×C8—1061no—
362C22×C8—1061no—
363C22×C8—1061no—
364C4⋊Q8—531no—
365C4⋊Q8C22×D5311yescharacteristic
366C4⋊Q8C22×D5311yescharacteristic
367#127—531no—
368#125—21no—
369#124—1061no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372#121—1061no—
373#121—1061no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8C22⋊D5311yescharacteristic
377C4×Q8C22×D5311yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—1061no—
381C4×D4—1061no—
382C4×D4C22⋊D5311yescharacteristic
383C4×D4C22×D5311yescharacteristic
384#117—21no—
385#117—21no—
386#117C22⋊D5311yescharacteristic
387C2×C42—21no—
388C2×C42C22×D5311yescharacteristic
389#107—21no—
390#107—1061no—
391#107—1061no—
392#107—1061no—
393#106—21no—
394#106—1061no—
395#106—1061no—
396#106—1061no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8C22×D5311yescharacteristic
407C4⋊C8C22×D5311yescharacteristic
408#103—21no—
409#103—21no—
410#103—21no—
411#103—1061no—
412#102—21no—
413#102—1061no—
414#102—1061no—
415#102—1061no—
416#102—1061no—
417#102—1061no—
418#102—1061no—
419#102—1061no—
420C22⋊C8—21no—
421C22⋊C8—531no—
422C22⋊C8—531no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426#97—21no—
427C4×C8—531no—
428C4×C8—531no—
429C4×C8—1061no—
430C4×C8—1061no—
431C4×C8—1061no—
432#41—22no—
433#41—22no—
434#41—4242no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—1061no—
438C2×Q8C2×C22⋊D5311yescharacteristic
439C2×D4—21no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—1061no—
443C2×D4—2121no—
444C2×D4C2×C22⋊D5311yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—2121no—
449C22×C4D4×D5311yescharacteristic
450C22×C4#1171994711yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452Q16—42no—
453SD16—42no—
454SD16—2122no—
455D8—2122no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—531no—
459C2×C8—531no—
460C2×C8—531no—
461C2×C8—531no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—4241no—
467C2×C8—4241no—
468C2×C8—4241no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4C2×C22⋊D5311yescharacteristic
480C4⋊C4C2×C22⋊D5311yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—41no—
485C22⋊C4—2121no—
486C22⋊C4—2121no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42D4×D5311yescharacteristic
492C42#1171994711yescharacteristic
493C42C2×C22⋊D5311yescharacteristic
494C42C23×D5311yescharacteristic
495C23—21no—
496C23—2121no—
497C23#1313318911yescharacteristic
498Q8—21no—
499Q8—41no—
500Q8—41no—
501Q8—81no—
502Q8—4241no—
503Q8#1313323712yes—
504D4—24no—
505D4—21no—
506D4—4241no—
507D4—4242no—
508D4—2121no—
509D4—2121no—
510D4—2121no—
511D4—2121no—
512D4#1313323712yes—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—81no—
527C2×C4—4242no—
528C2×C4—2121no—
529C2×C4#1313318911yescharacteristic
530C2×C4#1313318911yescharacteristic
531C2×C4#1313318911yescharacteristic
532C2×C4C2×D4×D5311yescharacteristic
533C2×C4C22×C22⋊D5311yescharacteristic
534C2×C4C2×#1171994711yescharacteristic
535C8—41no—
536C8—41no—
537C8—1061no—
538C8—1061no—
539C8—1061no—
540C8—1061no—
541C8—2121no—
542C8—2121no—
543C8—2121no—
544C8—2121no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—4241no—
552C22—4241no—
553C22—4241no—
554C22C2×#1313318911yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—22no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—4241no—
565C4C2×#1313323711yescharacteristic
566C4C2×#1313323711yescharacteristic
567C4#1681143211yescharacteristic
568C4#1681143911yescharacteristic
569C2—41no—
570C2—42no—
571C2—8481no—
572C2#2028426911yescharacteristic
573C2#2028429311yescharacteristic
574C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3116C2
socleC22×C53C2×#121
prime radical#20284775C2
hypercenter#3116D53
hypocenterC53#19419
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
1. #20285851C2semidirect
3. #20283826C2semidirect
4. #20274161C2semidirect
5. #20274154C2semidirect
6. #20274133C2semidirect
7. #20274126C2semidirect
8. #20274117C2semidirect
9. #20274112C2semidirect
10. #20274095C2semidirect
11. #20274088C2semidirect
12. #20273496C2semidirect
13. #20273411C2semidirect
14. #20273372C2semidirect
15. C53×#3116C2semidirect
283. #3116D53semidirect
354. C53#19419semidirect

Maximal subgroups

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

subgroupindex
#202858512
#202847752
#202838262
#202741612
#202741542
#202741332
#202741262
#202741172
#202741122
#202740952
#202740882
#202734962
#202734112
#202733722
C53×#31162
#1941953