Finite Group Databasebeta

← #27466146

Subgroups — #27466146

Order 13568.

770 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277243C211yesmaximal, characteristic
2#20277118C211yesmaximal, characteristic
3#20276287C211yesmaximal, characteristic, complemented
4#20276285C211yesmaximal, characteristic, complemented
5#20276264C211yesmaximal, characteristic, complemented
6#20276260C211yesmaximal, characteristic, complemented
7#20275475C211yesmaximal, characteristic, complemented
8#20275470C211yesmaximal, characteristic, complemented
9#20275027C211yesmaximal, characteristic
10#20275025C211yesmaximal, characteristic
11#20274912C211yesmaximal, characteristic
12#20274719C211yesmaximal, characteristic, complemented
13#20274719C211yesmaximal, characteristic, complemented
14#20274693C211yesmaximal, characteristic
15C53×#1854C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810528C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810505—21no—
32#16810495—21no—
33#16810492C2211yescharacteristic
34#16810492C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810488C2211yescharacteristic
37#16810481—21no—
38#16810481—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4×#11719919—21no—
41C4×#11719919—21no—
42#16810474—21no—
43C22⋊C4×Dic53—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53D411yescharacteristic
61C23×Dic53D411yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133196—22no—
65#13133196Q814yes—
66#13133182—22no—
67#13133182—22no—
68#13133176—41no—
69#13133176—41no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133150—41no—
92#13133148—41no—
93#13133146—22no—
94#13133146—22no—
95#13133142—41no—
96#13133142—41no—
97#13133140—41no—
98#13133140—41no—
99C22⋊#88322—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C23×C4×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111#117×C53—41no—
112#117×C53—41no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—21no—
120#107×C53—41no—
121#106×C53—41no—
122C22⋊C8×C53—22no—
123#95×C53—21no—
124C24×C53C2×D411yescharacteristic
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×Q812yes—
135C22×C4×C53#4112yes—
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—41no—
146C22×Dic53—41no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106C2×Q812yes—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C22⋊C4×C53—42no—
172C22⋊C4×C53—42no—
173C42×C53—42no—
174C22⋊Dic53—24no—
175C22⋊Dic53—42no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53C2×Q812yes—
182#11719919—24no—
183#11719919—42no—
184#11719919—42no—
185#11719919—42no—
186#11719919—42no—
187#11719919—42no—
188#11719919—42no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—22no—
192C4×Dic53—22no—
193C4×Dic53—42no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53#4112yes—
197C4×Dic53#4112yes—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53C22⋊Q812yes—
211C23×C53C22×D411yescharacteristic
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
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×C53C22⋊Q812yes—
231C2×C4×C53C22×Q811yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53H(5,𝔽2)11yescharacteristic
236C2×C4×C53H(5,𝔽2)11yescharacteristic
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—44no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—44no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53#12312yes—
260C2×Dic53#12312yes—
261C2×Dic53#13512yes—
262C2×Dic53#13712yes—
263Dic106—22no—
264Dic106—22no—
265Dic106—41no—
266Dic106—41no—
267C8×C53—41no—
268C8×C53—41no—
269#88322—41no—
270#88322—41no—
271#27682—531nomaximal, Sylow
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53C2×C22⋊Q811yescharacteristic
280C22×C53C2×#12311yescharacteristic
281C22×C53#53511yescharacteristic
282C22×C53C2×#13511yescharacteristic
283C22×C53C2×#13711yescharacteristic
284C22×C53#57511yescharacteristic
285C22×C53#57511yescharacteristic
286C4×C53—22no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53#55311yescharacteristic
292C4×C53#55311yescharacteristic
293C4×C53#55411yescharacteristic
294C4×C53#55411yescharacteristic
295Dic53—22no—
296Dic53—42no—
297Dic53—41no—
298Dic53—41no—
299Dic53—42no—
300Dic53—81no—
301Dic53—81no—
302Dic53#48514yes—
303C4×C22⋊Q8—531no—
304#2285—531no—
305#2101—531no—
306#2101—531no—
307#2097—531no—
308#2097—531no—
309#1948—531no—
310#1947—531no—
311#1873—531no—
312#1873—531no—
313#1854—531no—
314#1854D5311yescharacteristic
315#1803—531no—
316#1803—531no—
317#1797—531no—
318C2×C53—42no—
319C2×C53#248711yescharacteristic
320C2×C53#307611yescharacteristic
321C2×C53#307811yescharacteristic
322C2×C53C2×#48511yescharacteristic
323C2×C53#309311yescharacteristic
324C2×C53#321211yescharacteristic
325C2×C53#321611yescharacteristic
326C2×C22⋊Q8—531no—
327C2×C4×Q8—531no—
328C2×#117—531no—
329C2×#117C2×D5311yescharacteristic
330C22×C42—531no—
331C2×#107—531no—
332C2×#107C2×D5311yescharacteristic
333C2×#106—531no—
334C2×#106C2×D5311yescharacteristic
335C2×C4⋊C8—531no—
336C2×C4⋊C8—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×C22⋊C8—531no—
342C2×C22⋊C8C2×D5311yescharacteristic
343#391—531no—
344#389—531no—
345#387—1061no—
346#385—1061no—
347#384—531no—
348#384—531no—
349#382—531no—
350#382—531no—
351#380—1061no—
352#380—1061no—
353#380C2×D5311yescharacteristic
354C4×C4⋊C4—531no—
355C4×C4⋊C4—1061no—
356C4×C4⋊C4—1061no—
357C4×C4⋊C4—1061no—
358C4×C22⋊C4—1061no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340C2×D5311yescharacteristic
366#340C2×D5311yescharacteristic
367#336—531no—
368#336—531no—
369C53#2768211yescharacteristic, complemented, Sylow
370C22×Q8—531no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4C22×D5311yescharacteristic
374C22×C8—531no—
375C22×C8—531no—
376C22×C8C22×D5311yescharacteristic
377C22×C8C22×D5311yescharacteristic
378C22⋊Q8—534no—
379C22⋊Q8—1062no—
380C4×Q8—1062no—
381C4×Q8—1062no—
382#117—41no—
383#117—41no—
384#117—2121no—
385#117—2121no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
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—1061no—
403C2×C22⋊C4—1061no—
404C2×C42—21no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—531no—
408C2×C42—531no—
409C2×C42—1062no—
410C2×C42—1062no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414#107—41no—
415#107—2121no—
416#106—41no—
417#106—2121no—
418C4⋊C8—1062no—
419C4⋊C8—1062no—
420#103—2121no—
421#103—2121no—
422#103—2121no—
423#103—2121no—
424C22⋊C8—22no—
425C22⋊C8—1062no—
426#95—21no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435C24D4×D5311yescharacteristic
436C2×Q8—532no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×Q8—1061no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—2121no—
463C22×C4—2121no—
464C22×C4#1171994512yes—
465C22×C4D4×D5311yescharacteristic
466C22×C4Q8×D5312yes—
467C22×C4C23×D5311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—532no—
489C4⋊C4—1064no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—1064no—
504C22⋊C4—2122no—
505C42—42no—
506C42—532no—
507C42—532no—
508C42—1062no—
509C42—1062no—
510C42—1061no—
511C42—1061no—
512C42—2122no—
513C42—2122no—
514C42—2122no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#1313319512yes—
522C23C2×D4×D5311yescharacteristic
523Q8—1062no—
524Q8—1062no—
525Q8—2121no—
526Q8—2121no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—2124no—
560C2×C4—2124no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4#1313319512yes—
572C2×C4C2×Q8×D5311yescharacteristic
573C2×C4#41×D5311yescharacteristic
574C2×C4#41×D5311yescharacteristic
575C2×C4C2×#1171994511yescharacteristic
576C2×C4#1313326211yescharacteristic
577C2×C4#1313326211yescharacteristic
578C8—41no—
579C8—41no—
580C8—2121no—
581C8—2121no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1313319511yescharacteristic
590C22#123×D5311yescharacteristic
591C22#1681118811yescharacteristic
592C22#135×D5311yescharacteristic
593C22#137×D5311yescharacteristic
594C22#1681141911yescharacteristic
595C22#1681142011yescharacteristic
596C4—22no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—534no—
602C4—1062no—
603C4—4241no—
604C4—4241no—
605C4—2122no—
606C4—2121no—
607C4—2121no—
608C4—2122no—
609C4#1681132211yescharacteristic
610C4#1681132211yescharacteristic
611C4#1681132711yescharacteristic
612C4#1681132711yescharacteristic
613C2—42no—
614C2#2027903111yescharacteristic
615C2#2028450111yescharacteristic
616C2#2028451911yescharacteristic
617C2#485×D5311yescharacteristic
618C2#2028464311yescharacteristic
619C2#2028568811yescharacteristic
620C2#2028572311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1854C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1854D53
hypocenterC53#27682
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
3. #20276287C2semidirect
4. #20276285C2semidirect
5. #20276264C2semidirect
6. #20276260C2semidirect
7. #20275475C2semidirect
8. #20275470C2semidirect
12. #20274719C2semidirect
13. #20274719C2semidirect
369. C53#27682semidirect

Maximal subgroups

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

subgroupindex
#202772432
#202771182
#202762872
#202762852
#202762642
#202762602
#202754752
#202754702
#202750272
#202750252
#202749122
#202747192
#202746932
C53×#18542
#2768253