Finite Group Databasebeta

← #27487096

Subgroups — #27487096

Order 13568.

724 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277397C211yesmaximal, characteristic
2#20276141C211yesmaximal, characteristic
3#20275918C211yesmaximal, characteristic
4#20275915C211yesmaximal, characteristic
5#20275723C211yesmaximal, characteristic
6#20275356C211yesmaximal, characteristic, complemented
7#20275353C211yesmaximal, characteristic, complemented
8#20275286C211yesmaximal, characteristic, complemented
9#20275283C211yesmaximal, characteristic, complemented
10#20275059C211yesmaximal, characteristic, complemented
11#20275056C211yesmaximal, characteristic, complemented
12#20274969C211yesmaximal, characteristic, complemented
13#20274968C211yesmaximal, characteristic, complemented
14#20274786C211yesmaximal, characteristic
15C53×#2323C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810529C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810518—21no—
33#16810517—21no—
34#16810511—21no—
35#16810504—21no—
36#16810483C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C22⋊Q8×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×#117×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389—21no—
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#388—21no—
56C53×#387—21no—
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C2×#11719925—22no—
61C2×#11719925—22no—
62C2×#11719925—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133196—22no—
68#13133196—22no—
69C4×Dic106—41no—
70C4×Dic106—41no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81#13133150—22no—
82#13133150—22no—
83#13133150—41no—
84#13133141—22no—
85#13133141—22no—
86#13133141—41no—
87#13133140—22no—
88#13133140—22no—
89#13133140—41no—
90C22⋊#88322—41no—
91C22⋊#88322—41no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22⋊Q8×C53—22no—
100C22⋊Q8×C53—22no—
101C4×Q8×C53—41no—
102C4×Q8×C53—41no—
103#117×C53—41no—
104#117×C53—41no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
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×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C42×C53—21no—
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
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
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53#4112yes—
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×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106#4112yes—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—41no—
166C4⋊C4×C53—41no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—81no—
169C4⋊C4×C53#4112yes—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—41no—
172C22⋊C4×C53—41no—
173C22⋊C4×C53—81no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—42no—
179C42×C53—81no—
180C22⋊Dic53—24no—
181C22⋊Dic53—81no—
182#11719925—44no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53#4112yes—
189#11719919—24no—
190#11719919—42no—
191#11719919—81no—
192C4×Dic53—42no—
193C4×Dic53—81no—
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—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C22×D411yescharacteristic
214Q8×C53—22no—
215Q8×C53—22no—
216Q8×C53—41no—
217Q8×C53—41no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
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—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—41no—
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×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×Dic53—22no—
256C2×Dic53—22no—
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—
266C2×Dic53—41no—
267Dic106—22no—
268Dic106—22no—
269Dic106—41no—
270Dic106—41no—
271#88322—41no—
272#88322—41no—
273#88322—41no—
274#88322—41no—
275#29427—531nomaximal, Sylow
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53#52511yescharacteristic
284C22×C53#52511yescharacteristic
285C22×C53#53411yescharacteristic
286C22×C53#57611yescharacteristic
287C22×C53#57811yescharacteristic
288C22×C53#57811yescharacteristic
289C22×C53#57811yescharacteristic
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53#55111yescharacteristic
300C4×C53#55311yescharacteristic
301C4×C53#55311yescharacteristic
302C4×C53#55911yescharacteristic
303Dic53—42no—
304Dic53—42no—
305Dic53—81no—
306Dic53—81no—
307#2323—531no—
308#2323D5311yescharacteristic
309#2075—531no—
310#2036—531no—
311#2036—531no—
312#2001—531no—
313#1928—531no—
314#1928—531no—
315#1918—531no—
316#1918—531no—
317#1878—531no—
318#1878—531no—
319#1863—531no—
320#1863—531no—
321#1818—531no—
322C2×C53—81no—
323C2×C53#260011yescharacteristic
324C2×C53#312411yescharacteristic
325C2×C53#312411yescharacteristic
326C2×C53#312711yescharacteristic
327C2×C53#312711yescharacteristic
328C2×C53#320211yescharacteristic
329C2×C53#320311yescharacteristic
330C22×Q16—531no—
331C2×C22⋊Q8—531no—
332C2×C22⋊Q8C2×D5311yescharacteristic
333C2×C4×Q8—531no—
334C2×C4×Q8C2×D5311yescharacteristic
335C2×#117C2×D5311yescharacteristic
336C2×#107—531no—
337C2×#107—531no—
338C2×#107—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×C22⋊C8—531no—
346C2×C22⋊C8—531no—
347#391—531no—
348#391C2×D5311yescharacteristic
349#389—21no—
350#389—531no—
351#389—1061no—
352#389C2×D5311yescharacteristic
353#388—21no—
354#388—21no—
355#388—1061no—
356#388—1061no—
357#387—21no—
358#387—1061no—
359#380—531no—
360#380C2×D5311yescharacteristic
361C4×C4⋊C4—531no—
362C4×C4⋊C4C2×D5311yescharacteristic
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#336—531no—
370#336—531no—
371#336—531no—
372#336—531no—
373C53#2942711yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C22×Q8C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C2×Q16—1062no—
378C2×Q16—1062no—
379C2×Q16—2121no—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—531no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—1062no—
387C22⋊Q8—1062no—
388C4×Q8—41no—
389C4×Q8—41no—
390C4×Q8—2121no—
391C4×Q8—2121no—
392#117—41no—
393#117—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C22⋊D5311yescharacteristic
404C2×C4⋊C4C22⋊D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4C22⋊D5311yescharacteristic
413C2×C22⋊C4C22⋊D5311yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42C22×D5311yescharacteristic
419C2×C42C22×D5311yescharacteristic
420#107—1062no—
421#107—1062no—
422#107—2121no—
423#103—1062no—
424#103—1062no—
425#103—1062no—
426#103—1062no—
427#103—2121no—
428#103—2121no—
429C22⋊C8—2121no—
430C22⋊C8—2121no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441C24C2×C22⋊D5311yescharacteristic
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—532no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8#1171994512yes—
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—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C23×D5311yescharacteristic
474Q16—2124no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—532no—
498C4⋊C4—1064no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—4241no—
505C4⋊C4—2122no—
506C4⋊C4#1171994512yes—
507C22⋊C4—24no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—81no—
511C22⋊C4—1064no—
512C22⋊C4—4241no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—81no—
519C42—4241no—
520C42—2122no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23C22×C22⋊D5311yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—41no—
532Q8—41no—
533Q8—1062no—
534Q8—1062no—
535Q8—2121no—
536Q8—2121no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
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—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
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—1062no—
569C2×C4—1062no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4#41×D5311yescharacteristic
581C2×C4#41×D5311yescharacteristic
582C2×C4C2×#1171994511yescharacteristic
583C2×C4C2×#1171994511yescharacteristic
584C2×C4#1313326211yescharacteristic
585C2×C4#1313326211yescharacteristic
586C8—2121no—
587C8—2121no—
588C8—2121no—
589C8—2121no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22#1681113011yescharacteristic
598C22#1681113411yescharacteristic
599C22#1681118211yescharacteristic
600C22#1681142611yescharacteristic
601C22#1681143811yescharacteristic
602C22#1681143811yescharacteristic
603C22#1681143811yescharacteristic
604C4—42no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—81no—
612C4—81no—
613C4—4241no—
614C4—4241no—
615C4—2122no—
616C4—2122no—
617C4#1681130211yescharacteristic
618C4#1681132211yescharacteristic
619C4#1681132311yescharacteristic
620C4#1681135611yescharacteristic
621C2—81no—
622C2#2028029711yescharacteristic
623C2#2028497111yescharacteristic
624C2#2028497711yescharacteristic
625C2#2028501011yescharacteristic
626C2#2028501511yescharacteristic
627C2#2028560611yescharacteristic
628C2#2028561211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2323C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2323D53
hypocenterC53#29427
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
6. #20275356C2semidirect
7. #20275353C2semidirect
8. #20275286C2semidirect
9. #20275283C2semidirect
10. #20275059C2semidirect
11. #20275056C2semidirect
12. #20274969C2semidirect
13. #20274968C2semidirect
373. C53#29427semidirect

Maximal subgroups

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

subgroupindex
#202773972
#202761412
#202759182
#202759152
#202757232
#202753562
#202753532
#202752862
#202752832
#202750592
#202750562
#202749692
#202749682
#202747862
C53×#23232
#2942753