Finite Group Databasebeta

← #27449501

Subgroups — #27449501

Order 13568.

752 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279746C211yesmaximal, characteristic, complemented
2#20276258C211yesmaximal, characteristic, complemented
3#20276248C211yesmaximal, characteristic
4#20276096C211yesmaximal, characteristic, complemented
5#20276089C211yesmaximal, characteristic
6#20275648C211yesmaximal, characteristic
7#20275532C211yesmaximal, characteristic
8#20275478C211yesmaximal, characteristic, complemented
9#20275439C211yesmaximal, characteristic, complemented
10#20274969C211yesmaximal, characteristic, complemented
11#20274962C211yesmaximal, characteristic
12#20274720C211yesmaximal, characteristic, complemented
13#20274715C211yesmaximal, characteristic
14#20274613C211yesmaximal, characteristic, complemented
15C53×#2367C211yesmaximal, characteristic
16C2×#13133208—21no—
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33#16810555—21no—
34#16810555—21no—
35#16810522—21no—
36#16810521C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810493C2211yescharacteristic
39#16810493C2211yescharacteristic
40#16810493C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#124×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#388—21no—
54C53×#388—21no—
55C53×#385—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53—21no—
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133208—44no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C4×Dic53—21no—
77#13133150—22no—
78#13133148—22no—
79#13133148—41no—
80#13133148—41no—
81C4⋊#88322—22no—
82C4⋊#88322—41no—
83C4⋊#88322—41no—
84#13133141—22no—
85#13133141—41no—
86#13133141—41no—
87#13133141—41no—
88#13133138—22no—
89#13133138—41no—
90#13133138—41no—
91#13133138—41no—
92#13133128—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96C22×Q8×C53C2311yescharacteristic
97C22×D4×C53C2311yescharacteristic
98C23×C4×C53—21no—
99#124×C53—22no—
100#124×C53Q814yes—
101C4×Q8×C53—22no—
102C4×Q8×C53—41no—
103C4×D4×C53—22no—
104C4×D4×C53—41no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53D411yescharacteristic
124#95×C53D411yescharacteristic
125C24×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C2×Q812yes—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53C2×Q812yes—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53—81no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53#4112yes—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—81no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—42no—
172C42×C53—81no—
173C42×C53C2×Q812yes—
174C42×C53#4112yes—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53—81no—
187C4⋊Dic53—81no—
188C4⋊Dic53—81no—
189#11719919—42no—
190#11719919—42no—
191C4×Dic53—42no—
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×#88322—21no—
202C2×#88322—21no—
203C2×#88322#4112yes—
204C23×C53—22no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53C22×D411yescharacteristic
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216D4×C53—22no—
217D4×C53—22no—
218D4×C53—41no—
219D4×C53—41no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
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—84no—
244C2×C4×C53C22⋊Q812yes—
245C2×C4×C53C22⋊Q812yes—
246C2×C4×C53AGL(1,ℤ/8)12yes—
247C2×C4×C53#13712yes—
248C2×C4×C53C22×Q811yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266#88322—22no—
267#88322—41no—
268#88322—41no—
269#88322—41no—
270#26544—531nomaximal, Sylow
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53C2×C22⋊Q811yescharacteristic
279C22×C53#54611yescharacteristic
280C22×C53#55011yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57711yescharacteristic
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—42no—
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—41no—
293C4×C53—81no—
294C4×C53#52911yescharacteristic
295C4×C53#54011yescharacteristic
296C4×C53#55411yescharacteristic
297C4×C53#55411yescharacteristic
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302#2552—531no—
303#2367D5311yescharacteristic
304#2096—531no—
305#2094—531no—
306#2066—531no—
307#2065—531no—
308#1983—531no—
309#1959—531no—
310#1948—531no—
311#1940—531no—
312#1863—531no—
313#1862—531no—
314#1803—531no—
315#1802—531no—
316#1776—531no—
317C2×C53—82no—
318C2×C53#245411yescharacteristic
319C2×C53#307911yescharacteristic
320C2×C53#308411yescharacteristic
321C2×C53#330611yescharacteristic
322C2×C53#331611yescharacteristic
323C2×C53#339711yescharacteristic
324C2×C53#340011yescharacteristic
325C2×#125—1061no—
326C2×#124C2×D5311yescharacteristic
327C2×C4×Q8C2×D5311yescharacteristic
328C2×C4×D4C2×D5311yescharacteristic
329C2×#107—531no—
330C2×#106—531no—
331C2×#106—531no—
332C2×#106—531no—
333C2×C4⋊C8—531no—
334C2×C4⋊C8—531no—
335C2×C4⋊C8—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#102—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×#97—531no—
345#398—1061no—
346#398—1061no—
347#391C2×D5311yescharacteristic
348#390C2×D5311yescharacteristic
349#389—531no—
350#389—531no—
351#389—1061no—
352#388—21no—
353#388—21no—
354#385—21no—
355#384—531no—
356#384—531no—
357#384—531no—
358C4×C4⋊C4—531no—
359C4×C4⋊C4C2×D5311yescharacteristic
360C4×C4⋊C4C2×D5311yescharacteristic
361C4×C22⋊C4—21no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#336—531no—
367#336—531no—
368C53#2654411yescharacteristic, complemented, Sylow
369C22×Q8C22×D5311yescharacteristic
370C22×D4C22×D5311yescharacteristic
371C23×C4—21no—
372C22×C8—531no—
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—531no—
376#125—2124no—
377#124—22no—
378#124Dic10614yes—
379C4×Q8—22no—
380C4×Q8—41no—
381C4×D4—22no—
382C4×D4—41no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4C22⋊D5311yescharacteristic
397C2×C4⋊C4C22⋊D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—1061no—
408C2×C42C22×D5311yescharacteristic
409C2×C42C22×D5311yescharacteristic
410C2×C42C22×D5311yescharacteristic
411#107—1062no—
412#106—1062no—
413#106—2121no—
414#106—2121no—
415C4⋊C8—1062no—
416C4⋊C8—2121no—
417C4⋊C8—2121no—
418#103—1062no—
419#103—2121no—
420#103—2121no—
421#103—2121no—
422#102—1062no—
423#102—2121no—
424#102—2121no—
425#102—2121no—
426#97—1062no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95C22⋊D5311yescharacteristic
434#95C22⋊D5311yescharacteristic
435C24—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C2×Dic10612yes—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4C2×Dic10612yes—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—81no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C23×D5311yescharacteristic
465C2×C8—532no—
466C2×C8—1061no—
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—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—4241no—
492C4⋊C4—4241no—
493C4⋊C4—4241no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4#1171994712yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—81no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—81no—
511C42—2122no—
512C42C2×Dic10612yes—
513C42#1171994512yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C22×C22⋊D5311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
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—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—84no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
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×C4#1313319612yes—
567C2×C4#1313319612yes—
568C2×C4#1313324212yes—
569C2×C4#1313324712yes—
570C2×C4C22×Dic10611yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C2×C4C2×#1171994711yescharacteristic
573C2×C4C2×#1171994511yescharacteristic
574C2×C4#1313326211yescharacteristic
575C2×C4#1313326311yescharacteristic
576C8—1062no—
577C8—2121no—
578C8—2121no—
579C8—2121no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×#1313319611yescharacteristic
588C22#1681126711yescharacteristic
589C22#1681129211yescharacteristic
590C22C2×#1313324211yescharacteristic
591C22C2×#1313324711yescharacteristic
592C22#1681143111yescharacteristic
593C22#1681143411yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—81no—
603C4—4241no—
604C4—4241no—
605C4—4241no—
606C4—4241no—
607C4#1681115511yescharacteristic
608C4#1681122411yescharacteristic
609C4#1681133011yescharacteristic
610C4#1681133211yescharacteristic
611C2—82no—
612C2#2027858111yescharacteristic
613C2#2028453011yescharacteristic
614C2#2028457711yescharacteristic
615C2#2028667611yescharacteristic
616C2#2028682611yescharacteristic
617C2#2028773511yescharacteristic
618C2#2028776811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2367C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2367D53
hypocenterC53#26544
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #20279746C2semidirect
2. #20276258C2semidirect
4. #20276096C2semidirect
8. #20275478C2semidirect
9. #20275439C2semidirect
10. #20274969C2semidirect
12. #20274720C2semidirect
14. #20274613C2semidirect
368. C53#26544semidirect

Maximal subgroups

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

subgroupindex
#202797462
#202762582
#202762482
#202760962
#202760892
#202756482
#202755322
#202754782
#202754392
#202749692
#202749622
#202747202
#202747152
#202746132
C53×#23672
#2654453