Finite Group Databasebeta

← #27457202

Subgroups — #27457202

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279146C211yesmaximal, characteristic
2#20277747C211yesmaximal, characteristic
3#20276123C211yesmaximal, characteristic, complemented
4#20275910C211yesmaximal, characteristic
5#20275910C211yesmaximal, characteristic
6#20275479C211yesmaximal, characteristic, complemented
7#20275456C211yesmaximal, characteristic, complemented
8#20275327C211yesmaximal, characteristic
9#20275299C211yesmaximal, characteristic, complemented
10#20275297C211yesmaximal, characteristic, complemented
11#20274981C211yesmaximal, characteristic, complemented
12#20274911C211yesmaximal, characteristic
13#20274808C211yesmaximal, characteristic
14#20274547C211yesmaximal, characteristic, complemented
15C53×#2068C211yesmaximal, characteristic, complemented
16C2×#13133193C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133129C2211yescharacteristic
25C2×#13133129C2211yescharacteristic
26C2×#13133127C2211yescharacteristic
27#16810565—21no—
28#16810545—21no—
29#16810519C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810515—21no—
32#16810515—21no—
33#16810512—21no—
34#16810510—21no—
35#16810509—21no—
36#16810493C2211yescharacteristic
37#16810480C2211yescharacteristic
38#16810480C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41#16810474—21no—
42C4×#11719921C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#107×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C53×#391C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133193—22no—
63#13133193—22no—
64Q8×Dic53—22no—
65Q8×Dic53—41no—
66#13133175—22no—
67#13133175—41no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921D411yescharacteristic
83C2×#11719921D411yescharacteristic
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89#13133149—22no—
90#13133149—22no—
91#13133148—22no—
92#13133141—22no—
93#13133141—22no—
94#13133129—22no—
95#13133129—41no—
96#13133127—41no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×Q8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113#107×C53—41no—
114C4⋊C8×C53—22no—
115#103×C53—22no—
116#103×C53—41no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120C23×D53C2×D411yescharacteristic
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53#4112yes—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C2×C4×D53—21no—
142C2×C4×D53—21no—
143C2×C4×D53—21no—
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53#4112yes—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—42no—
160C42×C53—21no—
161C42×C53—21no—
162#11719921—24no—
163#11719921—22no—
164#11719921—41no—
165#11719921—42no—
166#11719921—81no—
167#11719921—81no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—42no—
175C4⋊Dic53—81no—
176C4⋊Dic53—81no—
177C4⋊Dic53#4112yes—
178C4⋊Dic53#4112yes—
179#11719919—24no—
180#11719919—42no—
181#11719919—81no—
182C4×Dic53—21no—
183C4×Dic53—21no—
184C4×Dic53—42no—
185C4×Dic53—42no—
186C4×Dic53—42no—
187C4×Dic53—81no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C23×C53C22×D411yescharacteristic
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202Q8×C53—22no—
203Q8×C53—22no—
204Q8×C53—41no—
205Q8×C53—41no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—22no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53#13712yes—
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53H(5,𝔽2)11yescharacteristic
229C2×C4×C532−1+411yescharacteristic
230C2×Dic53—21no—
231C2×Dic53—22no—
232C2×Dic53—21no—
233C2×Dic53—22no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C4×D53—41no—
252C4×D53—41no—
253C4×D53—41no—
254C4×D53—41no—
255C8×C53—22no—
256C8×C53—41no—
257#88322—41no—
258#88322—41no—
259#27060—531nomaximal, Sylow
260C22×C53#52511yescharacteristic
261C22×C53#54611yescharacteristic
262C22×C53#54711yescharacteristic
263C22×C53C2×#13711yescharacteristic
264C22×C53#57511yescharacteristic
265C22×C53#57711yescharacteristic
266C22×C53#57711yescharacteristic
267C2×D53—41no—
268C2×D53—41no—
269C2×D53—41no—
270C2×D53—41no—
271C2×D53—41no—
272C2×D53—41no—
273C2×D53—41no—
274C4×C53—22no—
275C4×C53—41no—
276C4×C53—42no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53#52911yescharacteristic
281C4×C53#53911yescharacteristic
282C4×C53#55311yescharacteristic
283C4×C53#55611yescharacteristic
284Dic53—41no—
285Dic53—42no—
286Dic53—42no—
287Dic53—41no—
288Dic53—41no—
289Dic53—81no—
290Dic53—81no—
291#2497—531no—
292#2362—531no—
293#2072—531no—
294#2068D5311yescharacteristic, complemented
295#2035—531no—
296#2035—531no—
297#1948—531no—
298#1944—531no—
299#1924—531no—
300#1920—531no—
301#1920—531no—
302#1865—531no—
303#1854—531no—
304#1825—531no—
305#1763—531no—
306C2×C53#246611yescharacteristic
307C2×C53#311611yescharacteristic
308C2×C53#312611yescharacteristic
309C2×C53#332111yescharacteristic
310C2×C53#333111yescharacteristic
311C2×C53#334411yescharacteristic
312C2×C53#335011yescharacteristic
313D53—81no—
314C2×C22⋊Q8—531no—
315C2×C4×Q8—531no—
316C2×#117—531no—
317C2×#107C2×D5311yescharacteristic
318C2×#106—531no—
319C2×#106—531no—
320C2×#106—531no—
321C2×C4⋊C8C2×D5311yescharacteristic
322C2×#103—531no—
323C2×#103—531no—
324C2×#103C2×D5311yescharacteristic
325C2×#103C2×D5311yescharacteristic
326C2×C22⋊C8—531no—
327C2×C22⋊C8—531no—
328C2×#97—531no—
329#400—1061no—
330#396—1061no—
331#391C2×D5311yescharacteristic
332#389—531no—
333#389—531no—
334#389C2×D5311yescharacteristic
335#388—1061no—
336#388—1061no—
337#388—1061no—
338#387—1061no—
339#387—1061no—
340#384—531no—
341#380—531no—
342#380—531no—
343C4×C4⋊C4—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—1061no—
346C4×C22⋊C4—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340C2×D5311yescharacteristic
355#336—531no—
356#336—531no—
357C53#2706011yescharacteristic, complemented, Sylow
358C22×Q8C22×D5311yescharacteristic
359C23×C4—531no—
360C22×C8—531no—
361C22×C8—531no—
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364C22⋊Q8—1062no—
365C22⋊Q8—1062no—
366C4×Q8—1062no—
367C4×Q8—2121no—
368#117—1062no—
369#117—2121no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—531no—
385C2×C22⋊C4—531no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C42—531no—
391C2×C42—1061no—
392C2×C42—1061no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42C22×D5311yescharacteristic
396#107—41no—
397#106—1062no—
398#106—1062no—
399#106—1062no—
400C4⋊C8—22no—
401#103—22no—
402#103—41no—
403#103—1062no—
404#103—1062no—
405C22⋊C8—1062no—
406C22⋊C8—2121no—
407#97—2121no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—531no—
412#95—531no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419C24—531no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#1171994912yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4D4×D5311yescharacteristic
444C22×C4D4×D5311yescharacteristic
445C22×C4C23×D5311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8#1171994712yes—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—532no—
466C4⋊C4—532no—
467C4⋊C4—1064no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—4241no—
475C4⋊C4—4241no—
476C4⋊C4—4241no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C22⋊C4—1064no—
480C22⋊C4—1062no—
481C22⋊C4—4241no—
482C22⋊C4—4241no—
483C22⋊C4—2121no—
484C22⋊C4—2122no—
485C42—21no—
486C42—21no—
487C42—1061no—
488C42—1061no—
489C42—4241no—
490C42—2122no—
491C42—2122no—
492C42—2122no—
493C23—1061no—
494C23—1061no—
495C23—1061no—
496C23—1061no—
497C23—1061no—
498C23—1061no—
499C23—1061no—
500C23C2×D4×D5311yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1062no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1062no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4#1313324912yes—
548C2×C4#41×D5311yescharacteristic
549C2×C4#41×D5311yescharacteristic
550C2×C4C2×#1171994711yescharacteristic
551C2×C4C2×#1171994911yescharacteristic
552C2×C4#1313326211yescharacteristic
553C2×C4#1313326411yescharacteristic
554C8—22no—
555C8—41no—
556C8—2121no—
557C8—2121no—
558C22—2121no—
559C22—2121no—
560C22—2121no—
561C22—2121no—
562C22—2121no—
563C22—2121no—
564C22—2121no—
565C22#1681113611yescharacteristic
566C22#1681127011yescharacteristic
567C22#1681127311yescharacteristic
568C22C2×#1313324911yescharacteristic
569C22#1681141911yescharacteristic
570C22#1681143011yescharacteristic
571C22#1681143211yescharacteristic
572C4—22no—
573C4—41no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—4241no—
579C4—4241no—
580C4—2121no—
581C4—2122no—
582C4—2121no—
583C4—2122no—
584C4—2121no—
585C4#1681115911yescharacteristic
586C4#1681122211yescharacteristic
587C4#1681131611yescharacteristic
588C4#1681134011yescharacteristic
589C2—4241no—
590C2#2027875411yescharacteristic
591C2#2028488011yescharacteristic
592C2#2028500811yescharacteristic
593C2#2028690811yescharacteristic
594C2#2028703411yescharacteristic
595C2#2028717611yescharacteristic
596C2#2028725511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2068C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2068D53
hypocenterC53#27060
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
3. #20276123C2semidirect
6. #20275479C2semidirect
7. #20275456C2semidirect
9. #20275299C2semidirect
10. #20275297C2semidirect
11. #20274981C2semidirect
14. #20274547C2semidirect
15. C53×#2068C2semidirect
294. #2068D53semidirect
357. C53#27060semidirect

Maximal subgroups

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

subgroupindex
#202791462
#202777472
#202761232
#202759102
#202754792
#202754562
#202753272
#202752992
#202752972
#202749812
#202749112
#202748082
#202745472
C53×#20682
#2706053