Finite Group Databasebeta

← #27370058

Subgroups — #27370058

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285673C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3#20284512C211yesmaximal, characteristic, complemented
4#20274290C211yesmaximal, characteristic, complemented
5#20274248C211yesmaximal, characteristic, complemented
6#20274204C211yesmaximal, characteristic, complemented
7#20274180C211yesmaximal, characteristic, complemented
8#20274166C211yesmaximal, characteristic, complemented
9#20274124C211yesmaximal, characteristic, complemented
10#20274080C211yesmaximal, characteristic, complemented
11#20274056C211yesmaximal, characteristic, complemented
12#20273504C211yesmaximal, characteristic, complemented
13#20273438C211yesmaximal, characteristic, complemented
14#20273426C211yesmaximal, characteristic, complemented
15C53×#2974C211yesmaximal, characteristic, complemented
16#16811180C2211yescharacteristic
17#16811173C2211yescharacteristic
18#16811003C2211yescharacteristic, complemented
19#16810996C2211yescharacteristic, complemented
20#16810974C2211yescharacteristic, complemented
21#16810970C2211yescharacteristic, complemented
22#16810936—21no—
23#16810923—21no—
24#16810923—21no—
25#16810913C2211yescharacteristic
26#16810913C2211yescharacteristic
27#16810903C2211yescharacteristic
28#16810903C2211yescharacteristic
29#16810903C2211yescharacteristic
30#16810903C2211yescharacteristic
31#16810846—21no—
32#16810846—21no—
33#16810818C2211yescharacteristic
34#16810818C2211yescharacteristic
35#16810787—21no—
36#16810787—21no—
37#16810757—21no—
38C4×#11719922—21no—
39C4×#11719922—21no—
40#16810676C2211yescharacteristic
41#16810663—21no—
42#16810647C2211yescharacteristic
43#16810628—21no—
44#16810376C2211yescharacteristic, complemented
45#16810376C2211yescharacteristic, complemented
46#16810374C2211yescharacteristic, complemented
47#16810374C2211yescharacteristic, complemented
48#16810371C2211yescharacteristic, complemented
49#16810371C2211yescharacteristic, complemented
50#16810370C2211yescharacteristic, complemented
51#16810370C2211yescharacteristic, complemented
52#16810370C2211yescharacteristic, complemented
53#16810370C2211yescharacteristic, complemented
54#16810366C2211yescharacteristic, complemented
55#16810366C2211yescharacteristic, complemented
56C2×C4×D4×C53C2211yescharacteristic
57C53×#436C2211yescharacteristic
58C53×#436C2211yescharacteristic
59C53×#435—21no—
60C53×#435—21no—
61C8×D4×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#432C2211yescharacteristic
64C53×#431C2211yescharacteristic
65C53×#407—21no—
66C2×C22⋊C8×C53—21no—
67C2×#11719945—21no—
68C2×Dic212—21no—
69C2×Dic212—21no—
70C2×#11719913—21no—
71C2×#11719913—21no—
72C2×#11719922—21no—
73C2×#11719922—21no—
74C2×#11719917—21no—
75#13133217C2311yescharacteristic
76#13133217C2311yescharacteristic
77#13133217C2311yescharacteristic
78#13133214C2311yescharacteristic
79#13133209—21no—
80#13133205—21no—
81#13133196—21no—
82#13133196—21no—
83#13133191—21no—
84#13133191—21no—
85C4×Dic106—21no—
86C4×Dic106—21no—
87C4×D212—21no—
88C4×D212—21no—
89#13133176—21no—
90#13133150—21no—
91#13133150—21no—
92#13133150—21no—
93#13133150—21no—
94#13133148—21no—
95#13133148—21no—
96C4⋊#88322—21no—
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100C4⋊#88322C2311yescharacteristic
101#13133142—21no—
102#13133142—21no—
103#13133139—21no—
104#13133139—21no—
105#13133138—21no—
106#13133138—21no—
107#13133138—21no—
108#13133138—21no—
109#13133138—21no—
110#13133138—21no—
111#13133137—21no—
112#13133137—21no—
113C22⋊#88322—21no—
114C4×#88322—21no—
115C22×D4×C53D411yescharacteristic
116C23×C4×C53—21no—
117C2×M16×C53—21no—
118C2×M16×C53—21no—
119C22×C8×C53—21no—
120C22×C8×C53—21no—
121C4×D4×C53—21no—
122C4×D4×C53—21no—
123C4×D4×C53D411yescharacteristic
124C4×D4×C53D411yescharacteristic
125C4×D4×C53C2311yescharacteristic
126C4×D4×C53C2311yescharacteristic
127C2×C4⋊C4×C53D411yescharacteristic
128C2×C22⋊C4×C53—21no—
129C2×C42×C53C2311yescharacteristic
130C4⋊C8×C53D411yescharacteristic
131C4⋊C8×C53D411yescharacteristic
132C4⋊C8×C53C2311yescharacteristic
133C4⋊C8×C53C2311yescharacteristic
134C22⋊C8×C53—21no—
135C22⋊C8×C53—21no—
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53—21no—
139C22⋊C8×C53D411yescharacteristic
140C22⋊C8×C53D411yescharacteristic
141#97×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C24×C53—21no—
145C2×D4×C53—22no—
146C2×D4×C53—21no—
147C2×D4×C53—41no—
148C2×D4×C53C2×D411yescharacteristic
149C2×D4×C53C2×D411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—81no—
156C22×C4×C53—81no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53#4111yescharacteristic
160C2×C22⋊D53—41no—
161#11719945—81no—
162#11719945—81no—
163C2×D212—21no—
164C2×D212—21no—
165C2×D212—21no—
166C2×C4×D53—41no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170M16×C53—81no—
171M16×C53—81no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—81no—
175C2×C8×C53—81no—
176C2×C8×C53C2×D411yescharacteristic
177C2×C8×C53C2×D411yescharacteristic
178C2×C8×C53C2×D411yescharacteristic
179C2×C8×C53C2×D411yescharacteristic
180C4⋊C4×C53—21no—
181C4⋊C4×C53C2×D411yescharacteristic
182C4⋊C4×C53C2×D411yescharacteristic
183C22⋊C4×C53—21no—
184C22⋊C4×C53—21no—
185C22⋊C4×C53—21no—
186C22⋊C4×C53—21no—
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53#4111yescharacteristic
190C42×C53C2411yescharacteristic
191C22⋊Dic53—41no—
192#11719922—42no—
193#11719922—42no—
194#11719921—41no—
195#11719921—41no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202#11719919—41no—
203#11719919—41no—
204C4×Dic53—41no—
205#11719917—42no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210Dic212—81no—
211Dic212—81no—
212#11719913—81no—
213#11719913—81no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53—81no—
219C23×C53#12111yescharacteristic
220C22×D53—41no—
221D4×C53—21no—
222D4×C53—21no—
223D4×C53—21no—
224D4×C53—42no—
225D4×C53#13712yes—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—81no—
238C2×C4×C53—81no—
239C2×C4×C53—81no—
240C2×C4×C53#12111yescharacteristic
241C2×C4×C53#12111yescharacteristic
242C2×C4×C53#12111yescharacteristic
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C22⋊D53—82no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250D212—41no—
251D212—41no—
252D212—41no—
253D212—41no—
254D212—81no—
255C4×D53—82no—
256Dic106—41no—
257Dic106—41no—
258Dic106—41no—
259Dic106—41no—
260Dic106—81no—
261C8×C53—21no—
262C8×C53—21no—
263C8×C53—21no—
264C8×C53—21no—
265C8×C53—41no—
266C8×C53—41no—
267#88322—41no—
268#88322—41no—
269#88322—41no—
270#88322—41no—
271#19460—531nomaximal, Sylow
272C22×C53—21no—
273C22×C53—21no—
274C22×C53—22no—
275C22×C53—21no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—84no—
282C22×C53C2×#12111yescharacteristic
283C2×D53—81no—
284C2×D53—81no—
285C2×D53—81no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—21no—
289C4×C53—21no—
290C4×C53—41no—
291C4×C53—41no—
292C4×C53C2×#13711yescharacteristic
293C4×C53#57511yescharacteristic
294C4×C53#57611yescharacteristic
295C4×C53#57611yescharacteristic
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299#3210—531no—
300#3100—531no—
301#3077—531no—
302#2974D5311yescharacteristic, complemented
303#1703—531no—
304#1697—531no—
305#1689—531no—
306#1685—531no—
307#1683—531no—
308#1677—531no—
309#1669—531no—
310#1665—531no—
311#1574—531no—
312#1557—531no—
313#1554—531no—
314C2×C53—41no—
315C2×C53—42no—
316C2×C53—81no—
317C2×C53#304111yescharacteristic
318C2×C53#305611yescharacteristic
319C2×C53#306711yescharacteristic
320D53—161no—
321#533—531no—
322#532—531no—
323C2×C4×D4C2×D5311yescharacteristic
324#494—531no—
325#492—531no—
326#487—531no—
327#486—531no—
328#480—1061no—
329#478—1061no—
330#478—1061no—
331#476—531no—
332#476—531no—
333#474—531no—
334#474—531no—
335#474—531no—
336#474—531no—
337#464—1061no—
338#464—1061no—
339#459—531no—
340#459—531no—
341#452—1061no—
342#452—1061no—
343#447—1061no—
344C4×D8—1061no—
345C4×D8—1061no—
346#436C2×D5311yescharacteristic
347#436C2×D5311yescharacteristic
348#435—21no—
349#435—21no—
350C8×D4C2×D5311yescharacteristic
351C8×D4C2×D5311yescharacteristic
352#432C2×D5311yescharacteristic
353#431—531no—
354#431C2×D5311yescharacteristic
355#428—1061no—
356#423—531no—
357#418—1061no—
358#407—21no—
359C2×C22⋊C8—21no—
360#335—531no—
361#335—531no—
362#334—531no—
363#334—531no—
364#333—531no—
365#333—531no—
366#332—531no—
367#332—531no—
368#332—531no—
369#332—531no—
370#331—531no—
371#331—531no—
372C53#1946011yescharacteristic, complemented, Sylow
373C2×#41—1061no—
374C22×D4D21211yescharacteristic
375C23×C4—21no—
376C2×Q16—1061no—
377C2×Q16—1061no—
378C2×SD16—1061no—
379C2×SD16—1061no—
380C2×D8—1061no—
381C2×D8—1061no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—1061no—
385C22×C8—21no—
386C22×C8—21no—
387C4⋊Q8—531no—
388C4⋊Q8—531no—
389C4⋊Q8—531no—
390#127—531no—
391#125—1061no—
392#124—1061no—
393C22⋊Q8—1061no—
394C22⋊Q8—1061no—
395#121—1061no—
396#121—1061no—
397C4×Q8—1061no—
398C4×Q8—1061no—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—1061no—
402C4×D4—1061no—
403C4×D4D21211yescharacteristic
404C4×D4D21211yescharacteristic
405C4×D4C22×D5311yescharacteristic
406C4×D4C22×D5311yescharacteristic
407#117—1061no—
408C2×C4⋊C4D21211yescharacteristic
409C2×C22⋊C4—21no—
410C2×C42C22×D5311yescharacteristic
411#107—1061no—
412#107—1061no—
413#107—1061no—
414#107—1061no—
415#106—1061no—
416#106—1061no—
417C4⋊C8—531no—
418C4⋊C8—531no—
419C4⋊C8—531no—
420C4⋊C8—531no—
421C4⋊C8—1061no—
422C4⋊C8C22⋊D5311yescharacteristic
423C4⋊C8C22⋊D5311yescharacteristic
424C4⋊C8C22×D5311yescharacteristic
425C4⋊C8C22×D5311yescharacteristic
426#103—1061no—
427#103—1061no—
428#102—1061no—
429#102—1061no—
430#102—1061no—
431#102—1061no—
432#102—1061no—
433#102—1061no—
434#102—1061no—
435#102—1061no—
436#102—1061no—
437#102—1061no—
438C22⋊C8—21no—
439C22⋊C8—21no—
440C22⋊C8—21no—
441C22⋊C8—21no—
442C22⋊C8—21no—
443C22⋊C8—1061no—
444C22⋊C8C22⋊D5311yescharacteristic
445C22⋊C8C22⋊D5311yescharacteristic
446#97—21no—
447C4×C8—1061no—
448C4×C8C22×D5311yescharacteristic
449C4×C8C22×D5311yescharacteristic
450C24—21no—
451#41—4241no—
452#41—4241no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—41no—
459C2×D4—1061no—
460C2×D4—1061no—
461C2×D4—1061no—
462C2×D4—2121no—
463C2×D4C2×D21211yescharacteristic
464C2×D4C2×D21211yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—2121no—
473C22×C4C2×D21211yescharacteristic
474C22×C4#1171994511yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476Q16—4241no—
477Q16—4241no—
478SD16—4241no—
479SD16—4241no—
480D8—2122no—
481D8—2122no—
482M16—81no—
483M16—81no—
484M16—2122no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—81no—
488C2×C8—81no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8C2×C22⋊D5311yescharacteristic
494C2×C8C2×C22⋊D5311yescharacteristic
495C2×C8C2×C22⋊D5311yescharacteristic
496C2×C8C2×C22⋊D5311yescharacteristic
497C4⋊C4—21no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—2121no—
505C4⋊C4—2121no—
506C4⋊C4C2×D21211yescharacteristic
507C4⋊C4C2×D21211yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—2121no—
513C22⋊C4—2121no—
514C22⋊C4—2121no—
515C42—2121no—
516C42C2×D21211yescharacteristic
517C42#1171994511yescharacteristic
518C42C2×C22⋊D5311yescharacteristic
519C42C23×D5311yescharacteristic
520C23—21no—
521C23—21no—
522C23—42no—
523C23—42no—
524C23—81no—
525C23—2121no—
526C23#1313319111yescharacteristic
527Q8—4241no—
528Q8—2121no—
529Q8—2121no—
530Q8—2121no—
531Q8—2121no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—42no—
536D4—4241no—
537D4—4242no—
538D4—2121no—
539D4—2121no—
540D4—2121no—
541D4—2121no—
542D4#1313325012yes—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—4242no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4#1313319111yescharacteristic
562C2×C4#1313319111yescharacteristic
563C2×C4#1313319111yescharacteristic
564C2×C4C22×C22⋊D5311yescharacteristic
565C2×C4C22×D21211yescharacteristic
566C2×C4C2×#1171994511yescharacteristic
567C8—21no—
568C8—21no—
569C8—21no—
570C8—21no—
571C8—41no—
572C8—41no—
573C8—2121no—
574C8—2121no—
575C8—2121no—
576C8—2121no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—84no—
587C22—4241no—
588C22—4241no—
589C22—4241no—
590C22C2×#1313319111yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—4241no—
598C4—4241no—
599C4—4241no—
600C4C2×#1313325011yescharacteristic
601C4#1681142311yescharacteristic
602C4#1681142611yescharacteristic
603C4#1681142611yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—8481no—
608C2#2028421811yescharacteristic
609C2#2028434511yescharacteristic
610C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2974C2
socleC22×C53C2×#121
prime radical#20284715C2
hypercenter#2974D53
hypocenterC53#19460
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. #20285673C2semidirect
3. #20284512C2semidirect
4. #20274290C2semidirect
5. #20274248C2semidirect
6. #20274204C2semidirect
7. #20274180C2semidirect
8. #20274166C2semidirect
9. #20274124C2semidirect
10. #20274080C2semidirect
11. #20274056C2semidirect
12. #20273504C2semidirect
13. #20273438C2semidirect
14. #20273426C2semidirect
15. C53×#2974C2semidirect
18. #16811003C22semidirect
19. #16810996C22semidirect
20. #16810974C22semidirect
21. #16810970C22semidirect
44. #16810376C22semidirect
45. #16810376C22semidirect
46. #16810374C22semidirect
47. #16810374C22semidirect
48. #16810371C22semidirect
49. #16810371C22semidirect
50. #16810370C22semidirect
51. #16810370C22semidirect
52. #16810370C22semidirect
53. #16810370C22semidirect
54. #16810366C22semidirect
55. #16810366C22semidirect
302. #2974D53semidirect
372. C53#19460semidirect

Maximal subgroups

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

subgroupindex
#202856732
#202847152
#202845122
#202742902
#202742482
#202742042
#202741802
#202741662
#202741242
#202740802
#202740562
#202735042
#202734382
#202734262
C53×#29742
#1946053