Finite Group Databasebeta

← #27387213

Subgroups — #27387213

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810721C211yesmaximal, characteristic
2#20277359C211yesmaximal, characteristic
3#20275342C211yesmaximal, characteristic
4#20275335C211yesmaximal, characteristic
5#20275307C211yesmaximal, characteristic, complemented
6#20275286C211yesmaximal, characteristic, complemented
7#20275070C211yesmaximal, characteristic
8#20275070C211yesmaximal, characteristic
9#20274983C211yesmaximal, characteristic, complemented
10#20274969C211yesmaximal, characteristic, complemented
11#20274613C211yesmaximal, characteristic, complemented
12C4×#13133141C211yesmaximal, characteristic, complemented
13C4×#13133137C211yesmaximal, characteristic
14#20274467C211yesmaximal, characteristic, complemented
15C53×#2276C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18#16810721C418yescomplemented
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30#16810525C2211yescharacteristic
31#16810525C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810512—21no—
35#16810512—21no—
36#16810485—21no—
37#16810482—21no—
38C4×C4⋊Dic53C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×C4×Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C4×C4⋊C4×C53—21no—
53C4×C4⋊C4×C53—21no—
54C4×C4⋊C4×C53—21no—
55C4×C4⋊C4×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#376—21no—
59C53×#376C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C2×#11719924—22no—
63C2×#11719924—22no—
64C2×#11719924C2×C414yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C4×D212—22no—
70C4×D212C2×C414yes—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×C4×Dic53—21no—
82#13133150—22no—
83#13133150C2×C414yes—
84#13133148—22no—
85#13133141—22no—
86#13133141—22no—
87#13133141C2×C414yes—
88#13133137—22no—
89#13133137—22no—
90#13133137C2×C414yes—
91#13133128C2×C414yes—
92C4×#88322—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96C22×Q8×C53C2311yescharacteristic
97C4×Q8×C53—22no—
98C4×Q8×C53—22no—
99C4×Q8×C53—22no—
100C4×Q8×C53—22no—
101C4×Q8×C53—22no—
102C4×Q8×C53C2×C414yes—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C42×C53—21no—
110C2×C42×C53—21no—
111C2×C42×C53—21no—
112C2×C42×C53—21no—
113C2×C42×C53—21no—
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121#95×C53D411yescharacteristic
122#95×C53D411yescharacteristic
123#95×C53D411yescharacteristic
124#95×C53D411yescharacteristic
125C23×D53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53C22×C412yes—
130C2×Q8×C53#4112yes—
131C2×Q8×C53#4112yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212C22×C412yes—
152C2×C4×D53—24no—
153C2×C4×D53—81no—
154C2×C4×D53—81no—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53C22×C412yes—
162C4⋊C4×C53#4112yes—
163C4⋊C4×C53#4112yes—
164C42×C53—24no—
165C42×C53—24no—
166C42×C53—24no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—81no—
172C42×C53—81no—
173C42×C53—81no—
174C42×C53C22×C412yes—
175C42×C53#4112yes—
176#11719924—28no—
177#11719921—24no—
178#11719921—81no—
179#11719921—81no—
180#11719921—81no—
181#11719921—81no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53C22×C412yes—
188#11719919—81no—
189#11719919—81no—
190C4×Dic53—81no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322C22×C412yes—
202C2×#88322C22×C412yes—
203C23×C53C22×D411yescharacteristic
204C22×D53—22no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211Q8×C53—22no—
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53#11714yes—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
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—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53C4×D412yes—
244C2×C4×C53C4×D412yes—
245C2×C4×C53C23×C411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×Dic53—22no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260D212—22no—
261D212—22no—
262D212—41no—
263D212—41no—
264C4×D53—84no—
265#88322—22no—
266#88322—22no—
267#88322—41no—
268#88322—41no—
269#21128—531nomaximal, Sylow
270C22×C53C2×C4×D411yescharacteristic
271C22×C53#52511yescharacteristic
272C22×C53#52511yescharacteristic
273C22×C53#57611yescharacteristic
274C22×C53#57711yescharacteristic
275C22×C53#57711yescharacteristic
276C22×C53#57811yescharacteristic
277C2×D53—42no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—81no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—81no—
284C4×C53—22no—
285C4×C53—22no—
286C4×C53—24no—
287C4×C53—41no—
288C4×C53—42no—
289C4×C53—41no—
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—41no—
293C4×C53—42no—
294C4×C53C2×#11711yescharacteristic
295C4×C53C4×#4111yescharacteristic
296C4×C53#52911yescharacteristic
297C4×C53#52911yescharacteristic
298Dic53—42no—
299Dic53—81no—
300Dic53—81no—
301C2×#440—531no—
302#2319—531no—
303#2276D5311yescharacteristic, complemented
304#1926—531no—
305#1925—531no—
306#1921—531no—
307#1918—531no—
308#1880—531no—
309#1880—531no—
310#1865—531no—
311#1863—531no—
312#1776—531no—
313C4×#103—531no—
314C4×#102—531no—
315#1746—531no—
316C2×C53#229011yescharacteristic
317C2×C53#294611yescharacteristic
318C2×C53#294711yescharacteristic
319C2×C53#311911yescharacteristic
320C2×C53#312111yescharacteristic
321C2×C53#312411yescharacteristic
322C2×C53#312611yescharacteristic
323D53—82no—
324C22×SD16—531no—
325C2×C4×Q8C2×D5311yescharacteristic
326C2×C4×Q8C2×D5311yescharacteristic
327C2×C4×Q8C2×D5311yescharacteristic
328C2×C4×D4—531no—
329#440—538no—
330C2×#107—531no—
331C2×#107—531no—
332C2×#106—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#102—531no—
337C2×#102—531no—
338C2×#102—531no—
339C2×#97—531no—
340C2×C4×C8—531no—
341#390—531no—
342#390—531no—
343#389—531no—
344#389—531no—
345#388—1061no—
346#388—1061no—
347#381—1061no—
348#380—1061no—
349C4×C4⋊C4—21no—
350C4×C4⋊C4—21no—
351C4×C4⋊C4—21no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4C2×D5311yescharacteristic
354C4×C4⋊C4C2×D5311yescharacteristic
355C4×C4⋊C4C2×D5311yescharacteristic
356#376—21no—
357#376C2×D5311yescharacteristic
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#336—531no—
365#336—531no—
366#336—531no—
367#336—531no—
368C53#2112811yescharacteristic, complemented, Sylow
369C22×Q8C22×D5311yescharacteristic
370C22×D4—531no—
371C23×C4—1061no—
372C2×SD16—534no—
373C2×SD16—1062no—
374C2×SD16—1062no—
375C22×C8—531no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8—531no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8C4×D5314yes—
385C4×D4—534no—
386C4×D4—1062no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—1061no—
410C2×C42C22×D5311yescharacteristic
411C2×C42C22×D5311yescharacteristic
412C2×C42C22×D5311yescharacteristic
413#107—534no—
414#107—1062no—
415#106—1062no—
416#103—534no—
417#103—1062no—
418#103—1062no—
419#102—534no—
420#102—1062no—
421#102—1062no—
422#97—534no—
423C4×C8—1062no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95C22⋊D5311yescharacteristic
431#95C22⋊D5311yescharacteristic
432#95C22⋊D5311yescharacteristic
433#95C22⋊D5311yescharacteristic
434C24—1061no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D5312yes—
439C2×Q8#1171994512yes—
440C2×Q8#1171994512yes—
441C2×D4—532no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—1061no—
452C22×C4—1064no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—4241no—
456C22×C4—4241no—
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C23×D5311yescharacteristic
464SD16—1068no—
465C2×C8—532no—
466C2×C8—532no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—532no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—4241no—
490C4⋊C4—4241no—
491C4⋊C4C2×C4×D5312yes—
492C4⋊C4#1171994512yes—
493C4⋊C4#1171994512yes—
494C22⋊C4—1064no—
495C22⋊C4—4241no—
496C22⋊C4—4241no—
497C22⋊C4—4241no—
498C22⋊C4—4241no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—81no—
507C42—81no—
508C42—81no—
509C42—4241no—
510C42C2×C4×D5312yes—
511C42#1171994512yes—
512C23—1062no—
513C23—2121no—
514C23—2121no—
515C23—2121no—
516C23—2121no—
517C23—2121no—
518C23—2121no—
519C23C22×C22⋊D5311yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#1313317414yes—
524D4—1062no—
525D4—1062no—
526D4—2121no—
527D4—2121no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—1062no—
557C2×C4—4244no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4C4×C22⋊D5312yes—
567C2×C4C4×C22⋊D5312yes—
568C2×C4C22×C4×D5311yescharacteristic
569C2×C4C2×#1171994511yescharacteristic
570C2×C4C2×#1171994511yescharacteristic
571C2×C4C2×#1171994511yescharacteristic
572C2×C4C2×#1171994511yescharacteristic
573C2×C4C2×#1171994511yescharacteristic
574C8—1062no—
575C8—1062no—
576C8—2121no—
577C8—2121no—
578C22—4241no—
579C22—4241no—
580C22—4241no—
581C22—4241no—
582C22—2122no—
583C22—2122no—
584C22—2122no—
585C22C2×C4×C22⋊D5311yescharacteristic
586C22#1681113411yescharacteristic
587C22#1681113411yescharacteristic
588C22#1681142611yescharacteristic
589C22#1681143111yescharacteristic
590C22#1681143111yescharacteristic
591C22#1681143811yescharacteristic
592C4—22no—
593C4—24no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—42no—
602C4—4241no—
603C4—4241no—
604C4—2122no—
605C4C2×#1313317411yescharacteristic
606C4C4×#1171994511yescharacteristic
607C4#1681115711yescharacteristic
608C4#1681115711yescharacteristic
609C2—4242no—
610C2#2027715811yescharacteristic
611C2#2028354311yescharacteristic
612C2#2028355811yescharacteristic
613C2#2028491411yescharacteristic
614C2#2028494411yescharacteristic
615C2#2028497711yescharacteristic
616C2#2028500611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2276C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#2276D53
hypocenterC53#21128
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ #376 ◁ #2276

Complements

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

NH ≅ G/Ntype
5. #20275307C2semidirect
6. #20275286C2semidirect
9. #20274983C2semidirect
10. #20274969C2semidirect
11. #20274613C2semidirect
12. C4×#13133141C2semidirect
14. #20274467C2semidirect
15. C53×#2276C2semidirect
18. #16810721C4semidirect
303. #2276D53semidirect
368. C53#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#168107212
#202773592
#202753422
#202753352
#202753072
#202752862
#202750702
#202749832
#202749692
#202746132
C4×#131331412
C4×#131331372
#202744672
C53×#22762
#2112853