Finite Group Databasebeta

← #27452191

Subgroups — #27452191

Order 13568.

748 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279052C211yesmaximal, characteristic
2#20277724C211yesmaximal, characteristic
3#20276036C211yesmaximal, characteristic, complemented
4#20275864C211yesmaximal, characteristic, complemented
5#20275723C211yesmaximal, characteristic
6#20275671C211yesmaximal, characteristic, complemented
7#20275392C211yesmaximal, characteristic, complemented
8#20275348C211yesmaximal, characteristic, complemented
9#20275324C211yesmaximal, characteristic
10#20275302C211yesmaximal, characteristic, complemented
11#20275054C211yesmaximal, characteristic, complemented
12#20275020C211yesmaximal, characteristic
13#20274799C211yesmaximal, characteristic
14#103×Dic53C211yesmaximal, characteristic, complemented
15C53×#2035C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133142C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810565—21no—
29#16810551—21no—
30#16810530C2211yescharacteristic
31#16810528C2211yescharacteristic
32#16810528C2211yescharacteristic
33#16810522—21no—
34#16810522C2211yescharacteristic
35#16810516—21no—
36#16810509—21no—
37#16810498—21no—
38#16810487—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53—21no—
61C2×#11719925—22no—
62C2×#11719925—41no—
63C2×#11719925—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—22no—
67#13133208—41no—
68#13133196—22no—
69#13133196—41no—
70Q8×Dic53—22no—
71Q8×Dic53—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—41no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53D411yescharacteristic
85C2×C22⋊Dic53D411yescharacteristic
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133146—41no—
92#13133142—22no—
93#13133142—41no—
94#13133141—22no—
95#13133140—22no—
96C22⋊#88322—41no—
97C8×Dic53—41no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×Q8×C53C2311yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22⋊Q8×C53—22no—
114C22⋊Q8×C53—22no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121#106×C53—22no—
122#103×C53—22no—
123#103×C53—41no—
124C22⋊C8×C53—41no—
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×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—81no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C4⋊C4×C53—24no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53#4112yes—
175C22⋊C4×C53—24no—
176C22⋊C4×C53—42no—
177C22⋊Dic53—22no—
178C22⋊Dic53—22no—
179C22⋊Dic53—42no—
180#11719925—44no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53—42no—
188C4⋊Dic53—81no—
189C4⋊Dic53#4112yes—
190C4⋊Dic53#4112yes—
191#11719919—42no—
192#11719919—42no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53—81no—
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#13512yes—
213C23×C53C22×D411yescharacteristic
214Q8×C53—22no—
215Q8×C53—22no—
216Q8×C53—41no—
217Q8×C53—41no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
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—
266C2×Dic53—84no—
267C2×Dic53#13712yes—
268Dic106—41no—
269Dic106—41no—
270Dic106—41no—
271Dic106—41no—
272C8×C53—41no—
273C8×C53—41no—
274#88322—41no—
275#88322—41no—
276#26725—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53#52511yescharacteristic
285C22×C53#54611yescharacteristic
286C22×C53#54611yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57711yescharacteristic
290C22×C53#57811yescharacteristic
291C4×C53—42no—
292C4×C53—41no—
293C4×C53—42no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53#52911yescharacteristic
297C4×C53#53811yescharacteristic
298C4×C53#55511yescharacteristic
299C4×C53#55911yescharacteristic
300Dic53—22no—
301Dic53—41no—
302Dic53—42no—
303Dic53—41no—
304Dic53—42no—
305Dic53—41no—
306Dic53—81no—
307Dic53—81no—
308#2489—531no—
309#2360—531no—
310#2056—531no—
311#2035D5311yescharacteristic
312#2026—531no—
313#2001—531no—
314#1989—531no—
315#1933—531no—
316#1927—531no—
317#1923—531no—
318#1920—531no—
319#1878—531no—
320#1872—531no—
321#1823—531no—
322C4×#103—531no—
323C2×C53—42no—
324C2×C53#245911yescharacteristic
325C2×C53#310811yescharacteristic
326C2×C53#312011yescharacteristic
327C2×C53#329611yescharacteristic
328C2×C53#330011yescharacteristic
329C2×C53#331211yescharacteristic
330C2×C53#333111yescharacteristic
331C22×Q16—531no—
332C2×#125—531no—
333C2×C22⋊Q8—531no—
334C2×C22⋊Q8C2×D5311yescharacteristic
335C2×C4×Q8—531no—
336C22×C4⋊C4—531no—
337C2×#106C2×D5311yescharacteristic
338C2×C4⋊C8—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103C2×D5311yescharacteristic
344C2×#103C2×D5311yescharacteristic
345C2×C22⋊C8—531no—
346C2×C22⋊C8C2×D5311yescharacteristic
347C2×C4×C8—531no—
348#400—1061no—
349#397—1061no—
350#391—531no—
351#391—531no—
352#391—531no—
353#389—531no—
354#389—1061no—
355#388—1061no—
356#387—1061no—
357#385—1061no—
358#382—1061no—
359#380C2×D5311yescharacteristic
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
362C4×C22⋊C4—531no—
363C4×C22⋊C4—1061no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#336—531no—
373#336—531no—
374C53#2672511yescharacteristic, complemented, Sylow
375C22×Q8—531no—
376C22×Q8C22×D5311yescharacteristic
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C2×Q16—1062no—
380C2×Q16—2121no—
381C2×Q16—2121no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386#125—1062no—
387#125—2121no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—1062no—
391C22⋊Q8—2121no—
392C4×Q8—1062no—
393C4×Q8—2121no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—2121no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—531no—
412C2×C22⋊C4—531no—
413C2×C22⋊C4—1061no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42—1061no—
419#106—22no—
420C4⋊C8—2121no—
421#103—22no—
422#103—41no—
423#103—1062no—
424#103—1062no—
425#103—1062no—
426#103—2121no—
427C22⋊C8—41no—
428C22⋊C8—2121no—
429C4×C8—2121no—
430#95—21no—
431#95—531no—
432#95—531no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442C24D4×D5311yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8#1171994712yes—
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—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—4241no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4D4×D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478Q16—2124no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—532no—
498C4⋊C4—532no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—4241no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4#1171994712yes—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—1062no—
513C22⋊C4—1062no—
514C22⋊C4—2122no—
515C42—1061no—
516C42—1061no—
517C42—1061no—
518C42—1061no—
519C42—4241no—
520C42—2122no—
521C42—2122no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1313323812yes—
529C23C2×D4×D5311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—2121no—
535Q8—2121no—
536Q8—2121no—
537Q8—2121no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—532no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1062no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1062no—
571C2×C4—4244no—
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—2121no—
581C2×C4—2121no—
582C2×C4#41×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4C2×#1171994711yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326311yescharacteristic
588C8—41no—
589C8—41no—
590C8—2121no—
591C8—2121no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1681112911yescharacteristic
600C22#1681126211yescharacteristic
601C22#1681126511yescharacteristic
602C22C2×#1313323811yescharacteristic
603C22#137×D5311yescharacteristic
604C22#1681143211yescharacteristic
605C22#1681143711yescharacteristic
606C4—42no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—1062no—
612C4—4241no—
613C4—4241no—
614C4—2121no—
615C4—2122no—
616C4—2121no—
617C4—2122no—
618C4—2121no—
619C4#1681115011yescharacteristic
620C4#1681121011yescharacteristic
621C4#1681133711yescharacteristic
622C4#1681135511yescharacteristic
623C2—42no—
624C2#2027864511yescharacteristic
625C2#2028479011yescharacteristic
626C2#2028493011yescharacteristic
627C2#2028654811yescharacteristic
628C2#2028660811yescharacteristic
629C2#2028676911yescharacteristic
630C2#2028702611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2035C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2035D53
hypocenterC53#26725
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #20276036C2semidirect
4. #20275864C2semidirect
6. #20275671C2semidirect
7. #20275392C2semidirect
8. #20275348C2semidirect
10. #20275302C2semidirect
11. #20275054C2semidirect
14. #103×Dic53C2semidirect
374. C53#26725semidirect

Maximal subgroups

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

subgroupindex
#202790522
#202777242
#202760362
#202758642
#202757232
#202756712
#202753922
#202753482
#202753242
#202753022
#202750542
#202750202
#202747992
#103×Dic532
C53×#20352
#2672553