Finite Group Databasebeta

← #27369194

Subgroups — #27369194

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286088C211yesmaximal, characteristic, complemented
2#20285730C211yesmaximal, characteristic
3#20285090C211yesmaximal, characteristic, complemented
4#20274283C211yesmaximal, characteristic, complemented
5#20274276C211yesmaximal, characteristic, complemented
6#20274255C211yesmaximal, characteristic, complemented
7#20274248C211yesmaximal, characteristic, complemented
8#20274159C211yesmaximal, characteristic, complemented
9#20274152C211yesmaximal, characteristic, complemented
10#20274131C211yesmaximal, characteristic, complemented
11#20274124C211yesmaximal, characteristic, complemented
12#20273482C211yesmaximal, characteristic, complemented
13#20273423C211yesmaximal, characteristic, complemented
14#20273382C211yesmaximal, characteristic, complemented
15C53×#2969C211yesmaximal, characteristic, complemented
16#16811180C2211yescharacteristic
17C2×#13133214C2211yescharacteristic
18#16811013—21no—
19#16810989—21no—
20#16810988—21no—
21#16810988—21no—
22#16810970C2211yescharacteristic
23#16810970C2211yescharacteristic
24#16810970C2211yescharacteristic
25#16810970C2211yescharacteristic
26#16810927—21no—
27#16810918C2211yescharacteristic, complemented
28#16810913C2211yescharacteristic, complemented
29#16810879—21no—
30#16810867—21no—
31#16810842—21no—
32#16810833C2211yescharacteristic, complemented
33#16810828C2211yescharacteristic, complemented
34#16810823C2211yescharacteristic
35#16810823C2211yescharacteristic
36#16810818C2211yescharacteristic
37#16810818C2211yescharacteristic
38#16810773—21no—
39#16810756—21no—
40#16810680C2211yescharacteristic
41#16810647C2211yescharacteristic
42#16810629—21no—
43#16810629—21no—
44#16810376C2211yescharacteristic, complemented
45#16810376C2211yescharacteristic, complemented
46#16810374C2211yescharacteristic, complemented
47#16810374C2211yescharacteristic, complemented
48#16810371C2211yescharacteristic, complemented
49#16810371C2211yescharacteristic, complemented
50#16810368C2211yescharacteristic, complemented
51#16810368C2211yescharacteristic, complemented
52#16810366C2211yescharacteristic
53#16810366C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C53×#529C2211yescharacteristic
57C53×#446C2211yescharacteristic, complemented
58C8×Q8×C53C2211yescharacteristic, complemented
59C53×#436—21no—
60C53×#436C2211yescharacteristic, complemented
61C53×#435—21no—
62C8×D4×C53C2211yescharacteristic, complemented
63C53×#433—21no—
64C53×#431C2211yescharacteristic
65C53×#405—21no—
66C4×M16×C53C2211yescharacteristic
67C22×D212—21no—
68C22×D212—21no—
69C22×D212—21no—
70C2×#11719925—21no—
71C2×#11719923—21no—
72C2×#11719924—21no—
73C2×#11719924—21no—
74C2×#11719922—21no—
75C2×#11719922—21no—
76C2×#11719917—21no—
77#13133217C2311yescharacteristic
78#13133217C2311yescharacteristic
79#13133214—21no—
80#13133214—21no—
81#13133214—21no—
82#13133214C2311yescharacteristic
83#13133214C2311yescharacteristic
84#13133209—21no—
85#13133196—21no—
86#13133196—21no—
87C4×Dic106—21no—
88C4×Dic106—21no—
89#13133176—21no—
90#13133150—21no—
91#13133148—21no—
92C4⋊#88322—21no—
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133141—21no—
98#13133140—21no—
99#13133140—21no—
100#13133139—21no—
101#13133139—21no—
102#13133139—21no—
103#13133139—21no—
104#13133139—21no—
105#13133139—21no—
106#13133139—21no—
107#13133139—21no—
108#13133138—21no—
109#13133137—21no—
110#13133137—21no—
111#13133137—21no—
112#13133137—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
144C23×D53—41no—
145C2×Q8×C53C2×D411yescharacteristic
146C2×D4×C53—21no—
147C2×D4×C53C2×D411yescharacteristic
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53#4111yescharacteristic
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—42no—
159C2×D212—42no—
160C2×D212—41no—
161C2×D212—41no—
162C2×D212—41no—
163C2×D212—81no—
164C2×D212—81no—
165C2×D212—82no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168M16×C53—41no—
169M16×C53—41no—
170M16×C53—42no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—42no—
174C2×C8×C53C2×D411yescharacteristic
175C2×C8×C53C2×D411yescharacteristic
176C2×C8×C53C2×D411yescharacteristic
177C2×C8×C53C2×D411yescharacteristic
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53C2×D411yescharacteristic
183C4⋊C4×C53C2×D411yescharacteristic
184C22⋊C4×C53—21no—
185C22⋊C4×C53—21no—
186C22⋊C4×C53—21no—
187C22⋊C4×C53—21no—
188C22⋊C4×C53—21no—
189C42×C53—21no—
190C42×C53C2×D411yescharacteristic
191C42×C53C2×D411yescharacteristic
192C42×C53#4111yescharacteristic
193C42×C53C2411yescharacteristic
194C22⋊Dic53—41no—
195#11719925—81no—
196#11719924—81no—
197#11719924—81no—
198#11719923—81no—
199#11719922—81no—
200#11719922—81no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205#11719919—41no—
206#11719919—41no—
207C4×Dic53—41no—
208#11719917—81no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C23×C53—21no—
214C23×C53#12111yescharacteristic
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—82no—
218C22×D53—82no—
219C22×D53—82no—
220Q8×C53—21no—
221Q8×C53—21no—
222D4×C53—21no—
223D4×C53—21no—
224D4×C53—81no—
225C2×C4×C53—21no—
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#12111yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×Dic53—41no—
242C2×Dic53—41no—
243D212—41no—
244D212—41no—
245D212—41no—
246D212—41no—
247D212—41no—
248D212—41no—
249D212—41no—
250D212—41no—
251D212—81no—
252D212—81no—
253D212—82no—
254D212—82no—
255D212—82no—
256Dic106—41no—
257Dic106—41no—
258Dic106—81no—
259C8×C53—21no—
260C8×C53—21no—
261C8×C53—21no—
262C8×C53—21no—
263C8×C53—41no—
264C8×C53—41no—
265#88322—41no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#19380—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53—21no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53C2×#12111yescharacteristic
277C2×D53—81no—
278C2×D53—81no—
279C2×D53—81no—
280C2×D53—81no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—164no—
284C4×C53—21no—
285C4×C53—21no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—41no—
291C4×C53#57511yescharacteristic
292C4×C53#57511yescharacteristic
293C4×C53#57611yescharacteristic
294C4×C53#57711yescharacteristic
295Dic53—81no—
296Dic53—81no—
297#3252—531no—
298#3217—531no—
299#3138—531no—
300#2969D5311yescharacteristic, complemented
301#1702—531no—
302#1701—531no—
303#1698—531no—
304#1697—531no—
305#1682—531no—
306#1681—531no—
307#1678—531no—
308#1677—531no—
309#1569—531no—
310#1554—531no—
311#1545—531no—
312C2×C53—41no—
313C2×C53—81no—
314C2×C53#304011yescharacteristic
315C2×C53#306211yescharacteristic
316C2×C53#306811yescharacteristic
317D53—161no—
318D53—161no—
319#533—531no—
320C2×#127—531no—
321#529C2×D5311yescharacteristic
322#496—1061no—
323#490—1061no—
324#490—1061no—
325#490—1061no—
326#486—531no—
327#486—531no—
328#486—531no—
329#486—531no—
330#479—1061no—
331#477—531no—
332#476—531no—
333#470—1061no—
334#468—1061no—
335#464—1061no—
336#462—531no—
337#461—531no—
338#460—531no—
339#460—531no—
340#459—531no—
341#459—531no—
342#450—1061no—
343#447—1061no—
344#446C2×D5311yescharacteristic, complemented
345C8×Q8C2×D5311yescharacteristic, complemented
346#436—21no—
347#436C2×D5311yescharacteristic, complemented
348#435—21no—
349C8×D4C2×D5311yescharacteristic, complemented
350#433—21no—
351#432—531no—
352#431C2×D5311yescharacteristic
353#423—531no—
354#418—1061no—
355#418—1061no—
356#405—21no—
357C4×M16C2×D5311yescharacteristic
358#335—531no—
359#335—531no—
360#334—531no—
361#334—531no—
362#333—531no—
363#333—531no—
364#332—531no—
365#332—531no—
366#331—531no—
367#331—531no—
368#331—531no—
369#331—531no—
370C53#1938011yescharacteristic, complemented, Sylow
371C22×D4—1061no—
372C22×D4—1061no—
373C22×D4—1061no—
374C2×Q16—1061no—
375C2×SD16—1061no—
376C2×SD16—1061no—
377C2×SD16—1061no—
378C2×D8—1061no—
379C2×D8—1061no—
380C2×M16—21no—
381C2×M16—1061no—
382C2×M16C22⋊D5311yescharacteristic
383C2×M16C22⋊D5311yescharacteristic
384C22×C8—21no—
385C4⋊Q8—531no—
386C4⋊Q8—531no—
387#127—531no—
388#127—531no—
389#127—1061no—
390#127—1061no—
391#127—1061no—
392#126—21no—
393#125—1061no—
394#125D21211yescharacteristic
395#124D21211yescharacteristic
396#123—21no—
397C22⋊Q8—1061no—
398C22⋊Q8—1061no—
399C22⋊Q8D21211yescharacteristic
400#121D21211yescharacteristic
401C4×Q8—1061no—
402C4×Q8—1061no—
403C4×Q8C22×D5311yescharacteristic
404C4×D4—21no—
405C4×D4C22×D5311yescharacteristic
406#117—21no—
407#117—1061no—
408C2×C42C22×D5311yescharacteristic
409#107—1061no—
410#106—1061no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8—531no—
414C4⋊C8—531no—
415C4⋊C8—531no—
416C4⋊C8—531no—
417C4⋊C8—1061no—
418C4⋊C8C22×D5311yescharacteristic
419C4⋊C8C22×D5311yescharacteristic
420#103—1061no—
421#103—1061no—
422#103—1061no—
423#102—1061no—
424#102—1061no—
425#102—1061no—
426#102—1061no—
427#102—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—
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8—21no—
439C22⋊C8—1061no—
440#97—21no—
441#97—1061no—
442#97C22⋊D5311yescharacteristic
443#97C22⋊D5311yescharacteristic
444C4×C8—21no—
445C4×C8—21no—
446C4×C8C22×D5311yescharacteristic
447C4×C8C22×D5311yescharacteristic
448C24—2121no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8C2×D21211yescharacteristic
452C2×D4—21no—
453C2×D4—1061no—
454C2×D4—1061no—
455C2×D4—1061no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4—1061no—
459C2×D4—4241no—
460C2×D4—4241no—
461C2×D4—4242no—
462C2×D4—2122no—
463C2×D4—2122no—
464C2×D4—2121no—
465C2×D4—2121no—
466C2×D4—2121no—
467C2×D4C2×D21211yescharacteristic
468C22×C4—21no—
469C22×C4C2×D21211yescharacteristic
470C22×C4#1171994511yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472Q16—4241no—
473SD16—4241no—
474SD16—4241no—
475SD16—4241no—
476D8—4241no—
477D8—4241no—
478M16—41no—
479M16—41no—
480M16—42no—
481M16—4241no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—42no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8C2×C22⋊D5311yescharacteristic
490C2×C8C2×C22⋊D5311yescharacteristic
491C2×C8C2×C22⋊D5311yescharacteristic
492C2×C8C2×C22⋊D5311yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—2121no—
502C4⋊C4—2121no—
503C4⋊C4C2×D21211yescharacteristic
504C4⋊C4C2×D21211yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—2121no—
511C42—21no—
512C42—2121no—
513C42C2×D21211yescharacteristic
514C42#1171994511yescharacteristic
515C42C2×C22⋊D5311yescharacteristic
516C42C23×D5311yescharacteristic
517C23—21no—
518C23—4242no—
519C23—4242no—
520C23—4242no—
521C23—2121no—
522C23—2121no—
523C23#1313319111yescharacteristic
524Q8—21no—
525Q8—21no—
526Q8—4241no—
527Q8—2121no—
528Q8—2121no—
529D4—21no—
530D4—21no—
531D4—81no—
532D4—4241no—
533D4—4241no—
534D4—4242no—
535D4—4242no—
536D4—4242no—
537D4—2121no—
538D4—2121no—
539D4—2121no—
540D4—2121no—
541D4—2121no—
542D4—2121no—
543D4—2121no—
544D4—2121no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4#1313319111yescharacteristic
558C2×C4#1313319111yescharacteristic
559C2×C4#1313319111yescharacteristic
560C2×C4C22×C22⋊D5311yescharacteristic
561C2×C4C22×D21211yescharacteristic
562C2×C4C2×#1171994511yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—41no—
568C8—41no—
569C8—2121no—
570C8—2121no—
571C8—2121no—
572C8—2121no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—8484no—
580C22—4241no—
581C22—4241no—
582C22—4241no—
583C22—4241no—
584C22—4241no—
585C22—4241no—
586C22C2×#1313319111yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—4241no—
595C4—4241no—
596C4#1681142111yescharacteristic
597C4#1681142111yescharacteristic
598C4#1681142811yescharacteristic
599C4#1681143511yescharacteristic
600C2—41no—
601C2—81no—
602C2—8481no—
603C2—8481no—
604C2#2028420911yescharacteristic
605C2#2028439511yescharacteristic
606C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2969C2
socleC22×C53C2×#121
prime radical#20285730C2
hypercenter#2969D53
hypocenterC53#19380
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
1. #20286088C2semidirect
3. #20285090C2semidirect
4. #20274283C2semidirect
5. #20274276C2semidirect
6. #20274255C2semidirect
7. #20274248C2semidirect
8. #20274159C2semidirect
9. #20274152C2semidirect
10. #20274131C2semidirect
11. #20274124C2semidirect
12. #20273482C2semidirect
13. #20273423C2semidirect
14. #20273382C2semidirect
15. C53×#2969C2semidirect
27. #16810918C22semidirect
28. #16810913C22semidirect
32. #16810833C22semidirect
33. #16810828C22semidirect
44. #16810376C22semidirect
45. #16810376C22semidirect
46. #16810374C22semidirect
47. #16810374C22semidirect
48. #16810371C22semidirect
49. #16810371C22semidirect
50. #16810368C22semidirect
51. #16810368C22semidirect
57. C53×#446C22semidirect
58. C8×Q8×C53C22semidirect
60. C53×#436C22semidirect
62. C8×D4×C53C22semidirect
300. #2969D53semidirect
344. #446C2×D53semidirect
345. C8×Q8C2×D53semidirect
347. #436C2×D53semidirect
349. C8×D4C2×D53semidirect
370. C53#19380semidirect

Maximal subgroups

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

subgroupindex
#202860882
#202857302
#202850902
#202742832
#202742762
#202742552
#202742482
#202741592
#202741522
#202741312
#202741242
#202734822
#202734232
#202733822
C53×#29692
#1938053