Finite Group Databasebeta

← #27465957

Subgroups — #27465957

Order 13568.

796 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810695C211yesmaximal, characteristic
2C2×#16810659C211yesmaximal, characteristic, complemented
3#20279012C211yesmaximal, characteristic
4#20276217C211yesmaximal, characteristic, complemented
5#20276216C211yesmaximal, characteristic, complemented
6#20276152C211yesmaximal, characteristic, complemented
7#20276151C211yesmaximal, characteristic, complemented
8#20275455C211yesmaximal, characteristic, complemented
9#20275455C211yesmaximal, characteristic, complemented
10#20274810C211yesmaximal, characteristic, complemented
11#20274807C211yesmaximal, characteristic, complemented
12#20274746C211yesmaximal, characteristic, complemented
13#20274746C211yesmaximal, characteristic, complemented
14#20274738C211yesmaximal, characteristic, complemented
15C53×#2485C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#16810695—24no—
20#16810659—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31#16810551—21no—
32#16810537—21no—
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810511—21no—
36#16810504—21no—
37#16810483C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#397—21no—
53C53×#394—21no—
54C53×#389C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#387—21no—
58C53×#380C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C53×#380C2211yescharacteristic
61C2×#11719917—22no—
62C2×#11719917—22no—
63C2×#11719917—22no—
64C22×#88322—22no—
65C22×#88322—22no—
66C22×#88322D414yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71#13133176—22no—
72#13133176—22no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C4×Dic53—21no—
84#13133150—22no—
85#13133150—22no—
86#13133148—22no—
87#13133148—22no—
88#13133148—22no—
89#13133148—22no—
90C4⋊#88322—22no—
91C22⋊#88322—22no—
92C22⋊#88322—22no—
93#13133128—22no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99C22×D4×C53D411yescharacteristic
100C23×C4×C53C2311yescharacteristic
101C23×C4×C53C2311yescharacteristic
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104#117×C53—22no—
105#117×C53—22no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
127C24×C53C2×D411yescharacteristic
128C2×D4×C53—22no—
129C2×D4×C53—41no—
130C2×D4×C53—41no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D412yes—
150C22×C4×C53#4112yes—
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53—81no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—42no—
171C22⋊C4×C53—81no—
172C22⋊C4×C53—81no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—42no—
176C22⋊Dic53—42no—
177C22⋊Dic53—42no—
178C22⋊Dic53—81no—
179C22⋊Dic53—81no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188#11719919—42no—
189#11719919—81no—
190C4×Dic53—42no—
191#11719917—44no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—28no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322C2×D412yes—
202C2×#88322C2×D412yes—
203C2×#88322#4112yes—
204C2×#88322#4112yes—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53#12112yes—
219C23×C53C22×D411yescharacteristic
220D4×C53—44no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
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—41no—
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×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53#12112yes—
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C532−1+411yescharacteristic
258C2×C4×C532−1+411yescharacteristic
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271#88322—22no—
272#88322—41no—
273#88322—41no—
274#88322#12314yes—
275#27665—531nomaximal, Sylow
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53C2×#12111yescharacteristic
291C22×C53#54711yescharacteristic
292C22×C53#54711yescharacteristic
293C22×C53#57511yescharacteristic
294C22×C53#57511yescharacteristic
295C22×C53#57711yescharacteristic
296C22×C53#57711yescharacteristic
297C4×C53—22no—
298C4×C53—41no—
299C4×C53—41no—
300C4×C53—41no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53C2×#12311yescharacteristic
306C4×C53#53611yescharacteristic
307C4×C53#55211yescharacteristic
308C4×C53#55211yescharacteristic
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313C2×#435—531no—
314C2×#427—531no—
315#2485—531no—
316#2485D5311yescharacteristic
317#2088—531no—
318#2088—531no—
319#2077—531no—
320#2077—531no—
321#1944—531no—
322#1944—531no—
323#1825—531no—
324#1825—531no—
325#1809—531no—
326#1809—531no—
327#1807—531no—
328C2×C53—42no—
329C2×C53—81no—
330C2×C53#248611yescharacteristic
331C2×C53#305411yescharacteristic
332C2×C53#306011yescharacteristic
333C2×C53#334811yescharacteristic
334C2×C53#334811yescharacteristic
335C2×C53#335011yescharacteristic
336C2×C53#335011yescharacteristic
337C22×M16—531no—
338C23×C8—531no—
339C2×C4×D4C2×D5311yescharacteristic
340C2×#117—531no—
341C2×#117C2×D5311yescharacteristic
342#435—1064no—
343#427—1064no—
344C2×#107—531no—
345C2×#107—531no—
346C2×#106—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×#106—531no—
350C2×C4⋊C8—531no—
351C2×C22⋊C8—531no—
352C2×C22⋊C8—531no—
353C2×#97—531no—
354#397—21no—
355#397—1061no—
356#394—21no—
357#394—1061no—
358#389—531no—
359#389—531no—
360#389C2×D5311yescharacteristic
361#389C2×D5311yescharacteristic
362#388—21no—
363#388—1061no—
364#387—21no—
365#387—1061no—
366#380—531no—
367#380—531no—
368#380—531no—
369#380C2×D5311yescharacteristic
370#380C2×D5311yescharacteristic
371#380C2×D5311yescharacteristic
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382C53#2766511yescharacteristic, complemented, Sylow
383C22×D4C22⋊D5311yescharacteristic
384C23×C4C22×D5311yescharacteristic
385C23×C4C22×D5311yescharacteristic
386C2×M16—1062no—
387C2×M16—1062no—
388C2×M16—1062no—
389C22×C8—531no—
390C22×C8—534no—
391C22×C8—531no—
392C22×C8—531no—
393C22×C8—531no—
394C22×C8—1062no—
395C22×C8—1062no—
396C4×D4—22no—
397C4×D4—22no—
398#117—22no—
399#117—22no—
400#117—1062no—
401#117—1062no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4C22⋊D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4C22⋊D5311yescharacteristic
424C2×C22⋊C4C22⋊D5311yescharacteristic
425C2×C42—21no—
426C2×C42—1061no—
427C2×C42C22×D5311yescharacteristic
428#107—1062no—
429#107—1062no—
430#106—1062no—
431#106—1062no—
432#106—1062no—
433#106—1062no—
434C4⋊C8—1062no—
435C22⋊C8—1062no—
436C22⋊C8—1062no—
437#97—1062no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448C24C2×C22⋊D5311yescharacteristic
449C24C2×C22⋊D5311yescharacteristic
450C2×D4—22no—
451C2×D4—41no—
452C2×D4—41no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4D4×D5312yes—
472C22×C4#1171994712yes—
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478M16—2124no—
479C2×C8—532no—
480C2×C8—532no—
481C2×C8—532no—
482C2×C8—532no—
483C2×C8—1068no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—4241no—
512C4⋊C4—2122no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—4241no—
520C22⋊C4—4241no—
521C22⋊C4—2122no—
522C22⋊C4—2122no—
523C42—21no—
524C42—21no—
525C42—42no—
526C42—2122no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#1313318912yes—
541C23C22×C22⋊D5311yescharacteristic
542D4—44no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4#1313318912yes—
587C2×C4C2×D4×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4C2×#1171994711yescharacteristic
591C2×C4#1313326411yescharacteristic
592C2×C4#1313326411yescharacteristic
593C8—534no—
594C8—1062no—
595C8—2121no—
596C8—2121no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22C2×#1313318911yescharacteristic
612C22#1681127211yescharacteristic
613C22#1681127211yescharacteristic
614C22#1681142111yescharacteristic
615C22#1681142111yescharacteristic
616C22#1681143111yescharacteristic
617C22#1681143411yescharacteristic
618C4—22no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—4241no—
627C4—4241no—
628C4—4241no—
629C4—4241no—
630C4#123×D5311yescharacteristic
631C4#1681119511yescharacteristic
632C4#1681131011yescharacteristic
633C4#1681131011yescharacteristic
634C2—42no—
635C2—81no—
636C2#2027902011yescharacteristic
637C2#2028432111yescharacteristic
638C2#2028437411yescharacteristic
639C2#2028722411yescharacteristic
640C2#2028723011yescharacteristic
641C2#2028725411yescharacteristic
642C2#2028726011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2485C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2485D53
hypocenterC53#27665
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2485

Complements

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

NH ≅ G/Ntype
2. C2×#16810659C2semidirect
4. #20276217C2semidirect
5. #20276216C2semidirect
6. #20276152C2semidirect
7. #20276151C2semidirect
8. #20275455C2semidirect
9. #20275455C2semidirect
10. #20274810C2semidirect
11. #20274807C2semidirect
12. #20274746C2semidirect
13. #20274746C2semidirect
14. #20274738C2semidirect
382. C53#27665semidirect

Maximal subgroups

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

subgroupindex
C2×#168106952
C2×#168106592
#202790122
#202762172
#202762162
#202761522
#202761512
#202754552
#202748102
#202748072
#202747462
#202747382
C53×#24852
#2766553