Finite Group Databasebeta

← #27452212

Subgroups — #27452212

Order 13568.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279091C211yesmaximal, characteristic
2#20277720C211yesmaximal, characteristic
3#20276036C211yesmaximal, characteristic, complemented
4#20275919C211yesmaximal, characteristic
5#20275855C211yesmaximal, characteristic, complemented
6#20275723C211yesmaximal, characteristic
7#20275661C211yesmaximal, characteristic, complemented
8#20275388C211yesmaximal, characteristic, complemented
9#20275358C211yesmaximal, characteristic, complemented
10#20275324C211yesmaximal, characteristic
11#20275306C211yesmaximal, characteristic, complemented
12#20275053C211yesmaximal, characteristic, complemented
13#20274805C211yesmaximal, characteristic
14#20274534C211yesmaximal, characteristic, complemented
15C53×#1872C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133142C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810561—21no—
29#16810535—21no—
30#16810530C2211yescharacteristic
31#16810530C2211yescharacteristic
32#16810528—21no—
33#16810528C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810518—21no—
36#16810517—21no—
37#16810504—21no—
38#16810496—21no—
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C22⋊Dic53—21no—
43C4×C22⋊Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106C2311yescharacteristic
61C2×#11719925—22no—
62C2×#11719925—41no—
63C2×#11719925—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133216—22no—
67#13133216—41no—
68#13133196—22no—
69#13133196—22no—
70#13133182—22no—
71#13133182—41no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53D411yescharacteristic
82C2×C22⋊Dic53D411yescharacteristic
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133151—22no—
89#13133146—41no—
90#13133142—22no—
91#13133142—41no—
92#13133141—22no—
93#13133140—22no—
94C22⋊#88322—41no—
95C8×Dic53—41no—
96#13133124—21no—
97#13133124—21no—
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#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C22×Q8×C53C2311yescharacteristic
109C23×C4×C53—21no—
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22⋊Q8×C53—22no—
114C22⋊Q8×C53—41no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—22no—
118C2×C4⋊C4×C53—41no—
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C22⋊C4×C53—21no—
123#103×C53—22no—
124#103×C53—41no—
125C22⋊C8×C53—41no—
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—21no—
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×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—81no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
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—42no—
162C2×Dic106—42no—
163C2×Dic106—81no—
164C2×Dic106#4112yes—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53#4112yes—
178C4⋊C4×C53#4112yes—
179C22⋊C4×C53—42no—
180C22⋊Dic53—24no—
181C22⋊Dic53—41no—
182C22⋊Dic53—81no—
183C22⋊Dic53Q1614yescomplemented
184#11719925—44no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53#4112yes—
191#11719919—24no—
192#11719919—42no—
193#11719919—42no—
194#11719919—81no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—42no—
200C4×Dic53—81no—
201C4×Dic53—81no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53C2×Q1612yes—
215C23×C53C22×D411yescharacteristic
216Q8×C53—41no—
217Q8×C53—41no—
218Q8×C53—41no—
219Q8×C53—41no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
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×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—84no—
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
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—
267C2×Dic53—41no—
268C2×Dic53C2×Q1612yes—
269C2×Dic53C2×Q1612yes—
270Dic106—22no—
271Dic106—22no—
272Dic106—41no—
273Dic106—41no—
274Dic106—84no—
275C8×C53—41no—
276C8×C53—41no—
277#88322—41no—
278#88322—41no—
279#26726—531nomaximal, Sylow
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53#52511yescharacteristic
288C22×C53#54611yescharacteristic
289C22×C53#54611yescharacteristic
290C22×C53C22×Q1611yescharacteristic
291C22×C53#57511yescharacteristic
292C22×C53#57711yescharacteristic
293C22×C53#57711yescharacteristic
294C4×C53—41no—
295C4×C53—42no—
296C4×C53—42no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53#52911yescharacteristic
300C4×C53#53811yescharacteristic
301C4×C53#55511yescharacteristic
302C4×C53#55911yescharacteristic
303Dic53—22no—
304Dic53—22no—
305Dic53—42no—
306Dic53—42no—
307Dic53—41no—
308Dic53—41no—
309Dic53—81no—
310Dic53—81no—
311#2492—531no—
312#2360—531no—
313#2056—531no—
314#2036—531no—
315#2024—531no—
316#2001—531no—
317#1987—531no—
318#1933—531no—
319#1928—531no—
320#1923—531no—
321#1921—531no—
322#1878—531no—
323#1872D5311yescharacteristic
324#1824—531no—
325C4×#103—531no—
326C2×C53—42no—
327C2×C53#245911yescharacteristic
328C2×C53#310411yescharacteristic
329C2×C53#312511yescharacteristic
330C2×C53#329911yescharacteristic
331C2×C53#329911yescharacteristic
332C2×C53#331111yescharacteristic
333C2×C53#332311yescharacteristic
334C22×Q16—531no—
335C2×C4⋊Q8—531no—
336C2×C22⋊Q8—531no—
337C2×C22⋊Q8C2×D5311yescharacteristic
338C2×C4×Q8—531no—
339C22×C4⋊C4C2×D5311yescharacteristic
340C2×#107—531no—
341C2×C4⋊C8—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×#103C2×D5311yescharacteristic
347C2×#103C2×D5311yescharacteristic
348C2×C22⋊C8—531no—
349C2×C22⋊C8C2×D5311yescharacteristic
350C2×C4×C8—531no—
351#400—1061no—
352#393—1061no—
353#391—531no—
354#391—531no—
355#391—531no—
356#391—1061no—
357#389—531no—
358#389—1061no—
359#388—1061no—
360#387—1061no—
361#385—1061no—
362#380—531no—
363C4×C4⋊C4—531no—
364C4×C4⋊C4—531no—
365C4×C22⋊C4—531no—
366C4×C22⋊C4—1061no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340C2×D5311yescharacteristic
374#340C2×D5311yescharacteristic
375#336—531no—
376#336—531no—
377C53#2672611yescharacteristic, complemented, Sylow
378C22×Q8—531no—
379C22×Q8—1061no—
380C22×Q8C22×D5311yescharacteristic
381C23×C4—21no—
382C23×C4C22×D5311yescharacteristic
383C2×Q16—1062no—
384C2×Q16—2121no—
385C2×Q16—2121no—
386C22×C8—531no—
387C22×C8—531no—
388C22×C8C22×D5311yescharacteristic
389C22×C8C22×D5311yescharacteristic
390C4⋊Q8—1062no—
391C4⋊Q8—2121no—
392C22⋊Q8—22no—
393C22⋊Q8—41no—
394C22⋊Q8—1062no—
395C22⋊Q8—1062no—
396C4×Q8—1062no—
397C4×Q8—2121no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—41no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—531no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C42—531no—
418C2×C42—531no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42—1061no—
422#107—1062no—
423C4⋊C8—2121no—
424#103—22no—
425#103—41no—
426#103—1062no—
427#103—1062no—
428#103—1062no—
429#103—2121no—
430C22⋊C8—41no—
431C22⋊C8—2121no—
432C4×C8—2121no—
433#95—531no—
434#95—531no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445C24D4×D5311yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—532no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C2×Q8—1061no—
457C2×Q8—1061no—
458C2×Q8—4241no—
459C2×Q8—2122no—
460C2×Q8—2122no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4D4×D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484Q16—2124no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—532no—
504C4⋊C4—1064no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—4241no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4#1171994512yes—
514C4⋊C4#1171994712yes—
515C22⋊C4—42no—
516C22⋊C4—534no—
517C22⋊C4—1064no—
518C22⋊C4—4241no—
519C22⋊C4—2121no—
520C42—1061no—
521C42—1061no—
522C42—1061no—
523C42—1061no—
524C42—4241no—
525C42—4241no—
526C42—2122no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23Q16×D5312yes—
534C23C2×D4×D5311yescharacteristic
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—1062no—
540Q8—1062no—
541Q8—4244no—
542Q8—2121no—
543Q8—2121no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
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—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—84no—
564C2×C4—532no—
565C2×C4—532no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—2121no—
579C2×C4—2121no—
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#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4C2×#1171994711yescharacteristic
591C2×C4C2×#1171994511yescharacteristic
592C2×C4#1313326211yescharacteristic
593C2×C4#1313326211yescharacteristic
594C8—41no—
595C8—41no—
596C8—2121no—
597C8—2121no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1681113111yescharacteristic
606C22#1681126011yescharacteristic
607C22#1681126311yescharacteristic
608C22C2×Q16×D5311yescharacteristic
609C22#1681141911yescharacteristic
610C22#1681142911yescharacteristic
611C22#1681143211yescharacteristic
612C4—42no—
613C4—41no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—1062no—
618C4—1062no—
619C4—4241no—
620C4—4241no—
621C4—2121no—
622C4—2122no—
623C4—2122no—
624C4—2121no—
625C4#1681115511yescharacteristic
626C4#1681120511yescharacteristic
627C4#1681133711yescharacteristic
628C4#1681135611yescharacteristic
629C2—42no—
630C2#2027865111yescharacteristic
631C2#2028475111yescharacteristic
632C2#2028498711yescharacteristic
633C2#2028658411yescharacteristic
634C2#2028658711yescharacteristic
635C2#2028674411yescharacteristic
636C2#2028693211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1872D53
hypocenterC53#26726
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
3. #20276036C2semidirect
5. #20275855C2semidirect
7. #20275661C2semidirect
8. #20275388C2semidirect
9. #20275358C2semidirect
11. #20275306C2semidirect
12. #20275053C2semidirect
14. #20274534C2semidirect
183. C22⋊Dic53Q16semidirect
377. C53#26726semidirect

Maximal subgroups

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

subgroupindex
#202790912
#202777202
#202760362
#202759192
#202758552
#202757232
#202756612
#202753882
#202753582
#202753242
#202753062
#202750532
#202748052
#202745342
C53×#18722
#2672653