Finite Group Databasebeta

← #27447832

Subgroups — #27447832

Order 13568.

794 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278428C211yesmaximal, characteristic, complemented
2#20276278C211yesmaximal, characteristic, complemented
3#20276217C211yesmaximal, characteristic, complemented
4#20276089C211yesmaximal, characteristic, complemented
5#20275861C211yesmaximal, characteristic, complemented
6#20275672C211yesmaximal, characteristic, complemented
7#20275286C211yesmaximal, characteristic, complemented
8#20275144C211yesmaximal, characteristic
9#20275038C211yesmaximal, characteristic
10#20274962C211yesmaximal, characteristic, complemented
11#20274914C211yesmaximal, characteristic, complemented
12#20274895C211yesmaximal, characteristic, complemented
13#20274715C211yesmaximal, characteristic, complemented
14#20274431C211yesmaximal, characteristic, complemented
15C53×#2363C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810559—21no—
29#16810546—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810520—21no—
33#16810506—21no—
34#16810493C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#121×C53C2211yescharacteristic
49C2×C4×D4×C53C2211yescharacteristic
50C2×#117×C53C2211yescharacteristic
51C53×#390—21no—
52C53×#390C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#386—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C22⋊C4×C53—21no—
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×#88322—22no—
60C22×#88322—41no—
61C22×#88322—41no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133209—22no—
67#13133209—41no—
68#13133176—22no—
69#13133176—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83#13133150—22no—
84#13133148—41no—
85C4⋊#88322—41no—
86C4⋊#88322—41no—
87#13133138—22no—
88#13133138—41no—
89#13133138—41no—
90#13133138—41no—
91C22⋊#88322—22no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C22×D4×C53—21no—
96C22×D4×C53C2311yescharacteristic
97C23×C4×C53—21no—
98C23×C4×C53C2311yescharacteristic
99#121×C53—22no—
100#121×C53Q814yes—
101C4×D4×C53—22no—
102C4×D4×C53—41no—
103#117×C53—22no—
104#117×C53—41no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53—21no—
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53D411yescharacteristic
123#95×C53D411yescharacteristic
124C24×C53—21no—
125C24×C53C2×D411yescharacteristic
126C2×D4×C53—24no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—81no—
133C2×D4×C53—81no—
134C2×D4×C53C2×Q812yes—
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—42no—
143C22×C4×C53—42no—
144C22×C4×C53—81no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×Q812yes—
151C22×C4×C53#4112yes—
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—81no—
162C4⋊C4×C53C2×Q812yes—
163C4⋊C4×C53#4112yes—
164C22⋊C4×C53—24no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—41no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—81no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—42no—
176C42×C53—81no—
177C22⋊Dic53—42no—
178C22⋊Dic53—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—42no—
187C4⋊Dic53—81no—
188C4⋊Dic53#4112yes—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192#11719919—81no—
193C4×Dic53—42no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—44no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—22no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53C22⋊Q812yes—
221C23×C53C22×D411yescharacteristic
222D4×C53—22no—
223D4×C53—22no—
224D4×C53—41no—
225D4×C53—41no—
226D4×C53—84no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
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—41no—
253C2×C4×C53—84no—
254C2×C4×C53C22⋊Q812yes—
255C2×C4×C53#13512yes—
256C2×C4×C53C22×Q811yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53H(5,𝔽2)11yescharacteristic
261C2×C4×C53H(5,𝔽2)11yescharacteristic
262C2×Dic53—22no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274#88322—41no—
275#88322—41no—
276#88322—41no—
277#88322—41no—
278#26421—531nomaximal, Sylow
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—42no—
289C22×C53—81no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53C2×C22⋊Q811yescharacteristic
294C22×C53#54611yescharacteristic
295C22×C53#54611yescharacteristic
296C22×C53C2×#13511yescharacteristic
297C22×C53#57511yescharacteristic
298C22×C53#57611yescharacteristic
299C22×C53#57711yescharacteristic
300C4×C53—22no—
301C4×C53—22no—
302C4×C53—42no—
303C4×C53—42no—
304C4×C53—41no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53—81no—
308C4×C53#52911yescharacteristic
309C4×C53#53811yescharacteristic
310C4×C53#55411yescharacteristic
311C4×C53#55411yescharacteristic
312Dic53—42no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316#2441—531no—
317#2363D5311yescharacteristic
318#2100—531no—
319#2088—531no—
320#2065—531no—
321#2025—531no—
322#1989—531no—
323#1918—531no—
324#1893—531no—
325#1875—531no—
326#1862—531no—
327#1854—531no—
328#1850—531no—
329#1802—531no—
330#1735—531no—
331C2×C53—42no—
332C2×C53—82no—
333C2×C53#245011yescharacteristic
334C2×C53#307611yescharacteristic
335C2×C53#308111yescharacteristic
336C2×C53#329711yescharacteristic
337C2×C53#330311yescharacteristic
338C2×C53#330911yescharacteristic
339C2×C53#331711yescharacteristic
340C23×C8—531no—
341C2×#125—531no—
342C2×#121C2×D5311yescharacteristic
343C2×C4×D4C2×D5311yescharacteristic
344C2×#117—531no—
345C2×#117C2×D5311yescharacteristic
346C2×#107—531no—
347C2×#106—531no—
348C2×C4⋊C8—531no—
349C2×C4⋊C8—531no—
350C2×#102—531no—
351C2×#102—531no—
352C2×#102—531no—
353C2×#102—531no—
354C2×C22⋊C8—531no—
355#399—1061no—
356#396—1061no—
357#390—21no—
358#390C2×D5311yescharacteristic
359#389—531no—
360#389—531no—
361#389—1061no—
362#389C2×D5311yescharacteristic
363#388—21no—
364#387—1061no—
365#386—21no—
366#384—531no—
367#380—531no—
368C4×C4⋊C4C2×D5311yescharacteristic
369C4×C22⋊C4—21no—
370C4×C22⋊C4C2×D5311yescharacteristic
371#340—531no—
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#336—531no—
382#336—531no—
383C53#2642111yescharacteristic, complemented, Sylow
384C22×D4—21no—
385C22×D4C22×D5311yescharacteristic
386C23×C4—21no—
387C23×C4C22×D5311yescharacteristic
388C22×C8—531no—
389C22×C8—531no—
390C22×C8—531no—
391C22×C8—531no—
392C22×C8—1062no—
393C22×C8—2121no—
394C22×C8—2121no—
395#125—1062no—
396#125—2121no—
397#121—22no—
398#121Dic10614yes—
399C4×D4—22no—
400C4×D4—41no—
401#117—22no—
402#117—41no—
403#117—1062no—
404#117—2121no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4C22⋊D5311yescharacteristic
427C2×C22⋊C4C22⋊D5311yescharacteristic
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—1061no—
431C2×C42C22×D5311yescharacteristic
432C2×C42C22×D5311yescharacteristic
433#107—1062no—
434#106—2121no—
435C4⋊C8—2121no—
436C4⋊C8—2121no—
437#102—1062no—
438#102—2121no—
439#102—2121no—
440#102—2121no—
441C22⋊C8—1062no—
442#95—21no—
443#95—21no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95C22⋊D5311yescharacteristic
448#95C22⋊D5311yescharacteristic
449C24—21no—
450C24C2×C22⋊D5311yescharacteristic
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—81no—
458C2×D4—81no—
459C2×D4C2×Dic10612yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—81no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4C2×Dic10612yes—
475C22×C4#1171994512yes—
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C23×D5311yescharacteristic
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—2124no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—81no—
500C4⋊C4—532no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4C2×Dic10612yes—
515C4⋊C4#1171994712yes—
516C22⋊C4—24no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C22⋊C4—4241no—
524C22⋊C4—2122no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—42no—
530C42—81no—
531C42—2122no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23#1313319612yes—
546C23C22×C22⋊D5311yescharacteristic
547D4—22no—
548D4—22no—
549D4—41no—
550D4—41no—
551D4—84no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—1062no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4#1313319612yes—
592C2×C4#1313323712yes—
593C2×C4C22×Dic10611yescharacteristic
594C2×C4#41×D5311yescharacteristic
595C2×C4C2×#1171994711yescharacteristic
596C2×C4C2×#1171994511yescharacteristic
597C2×C4#1313326211yescharacteristic
598C2×C4#1313326311yescharacteristic
599C8—2121no—
600C8—2121no—
601C8—2121no—
602C8—2121no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1313319611yescharacteristic
618C22#1681126111yescharacteristic
619C22#1681126711yescharacteristic
620C22C2×#1313323711yescharacteristic
621C22#1681142111yescharacteristic
622C22#1681142611yescharacteristic
623C22#1681143411yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—4241no—
633C4—4241no—
634C4—4241no—
635C4—2122no—
636C4#1681115311yescharacteristic
637C4#1681120611yescharacteristic
638C4#1681133011yescharacteristic
639C4#1681133211yescharacteristic
640C2—42no—
641C2—82no—
642C2#2027853311yescharacteristic
643C2#2028450311yescharacteristic
644C2#2028454611yescharacteristic
645C2#2028656511yescharacteristic
646C2#2028663711yescharacteristic
647C2#2028671511yescharacteristic
648C2#2028684111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2363C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2363D53
hypocenterC53#26421
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. #20278428C2semidirect
2. #20276278C2semidirect
3. #20276217C2semidirect
4. #20276089C2semidirect
5. #20275861C2semidirect
6. #20275672C2semidirect
7. #20275286C2semidirect
10. #20274962C2semidirect
11. #20274914C2semidirect
12. #20274895C2semidirect
13. #20274715C2semidirect
14. #20274431C2semidirect
383. C53#26421semidirect

Maximal subgroups

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

subgroupindex
#202784282
#202762782
#202762172
#202760892
#202758612
#202756722
#202752862
#202751442
#202750382
#202749622
#202749142
#202748952
#202747152
#202744312
C53×#23632
#2642153