Finite Group Databasebeta

← #27466321

Subgroups — #27466321

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279192C211yesmaximal, characteristic
2#20278413C211yesmaximal, characteristic
3#20276281C211yesmaximal, characteristic, complemented
4#20276279C211yesmaximal, characteristic, complemented
5#20276126C211yesmaximal, characteristic, complemented
6#20275677C211yesmaximal, characteristic, complemented
7#20275394C211yesmaximal, characteristic, complemented
8#20275393C211yesmaximal, characteristic, complemented
9#20275135C211yesmaximal, characteristic
10#20275019C211yesmaximal, characteristic
11#20274841C211yesmaximal, characteristic, complemented
12#20274804C211yesmaximal, characteristic
13#20274485C211yesmaximal, characteristic
14#340×D53C211yesmaximal, characteristic
15C53×#2024C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C2×#13133193—21no—
18C2×#13133193C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133129C2211yescharacteristic
24C2×C4×#88322C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810557—21no—
27#16810557—21no—
28#16810556—21no—
29#16810542—21no—
30#16810530C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810497—21no—
34#16810497—21no—
35#16810494—21no—
36#16810493C2211yescharacteristic
37#16810492C2211yescharacteristic
38#16810492C2211yescharacteristic
39#16810480C2211yescharacteristic
40C4×#11719921C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4⋊Q8×C53C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C53×#391C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×C4×D53—21no—
60C22×C4×D53—21no—
61C22×C4×D53C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C2×C8×D53—22no—
65C2×C8×D53—22no—
66C2×C8×D53—41no—
67#13133193—22no—
68#13133193—22no—
69#13133193—44no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C4⋊C4×D53—22no—
86C4⋊C4×D53—22no—
87C4⋊C4×D53—22no—
88C4⋊C4×D53—41no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921D411yescharacteristic
92C2×#11719921D411yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133141—22no—
95#13133141—41no—
96#13133129—22no—
97C4×#88322—41no—
98C8×Dic53—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C22×Q8×C53—21no—
107C22×Q8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C4⋊Q8×C53—22no—
111C4⋊Q8×C53—41no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117#103×C53—22no—
118#103×C53—41no—
119#95×C53—21no—
120#95×C53—21no—
121C23×D53C2×D411yescharacteristic
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—42no—
129C2×Q8×C53—42no—
130C2×Q8×C53—81no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2411yescharacteristic
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—81no—
153C2×C4×D53—81no—
154C2×C4×D53—81no—
155C2×C4×D53#4112yes—
156C2×C4×D53#4112yes—
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—42no—
169C4⋊C4×C53#4112yes—
170C42×C53—21no—
171C42×C53—21no—
172#11719921—42no—
173#11719921—42no—
174#11719921SD1614yes—
175#11719921Q1614yes—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53—81no—
186C4⋊Dic53—81no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192#11719919—81no—
193#11719919—81no—
194#11719919—81no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322#4112yes—
203C8×D53—44no—
204C23×C53C22×D411yescharacteristic
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—21no—
209C22×D53#12412yes—
210C22×D53C2×SD1612yes—
211C22×D53C2×Q1612yes—
212Q8×C53—41no—
213Q8×C53—41no—
214Q8×C53—41no—
215Q8×C53—41no—
216Q8×C53—84no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
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×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53C2×SD1612yes—
234C2×C4×C53C2×Q1612yes—
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×C4×C532−1+411yescharacteristic
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53#12412yes—
260C2×Dic53C2×SD1612yes—
261C2×Dic53C2×Q1612yes—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—22no—
265C4×D53—22no—
266C4×D53—84no—
267C4×D53—84no—
268C8×C53—41no—
269C8×C53—41no—
270#88322—22no—
271#88322—41no—
272#27697—531nomaximal, Sylow
273C22×C53C2×#12411yescharacteristic
274C22×C53#54611yescharacteristic
275C22×C53#54711yescharacteristic
276C22×C53C22×SD1611yescharacteristic
277C22×C53C22×Q1611yescharacteristic
278C22×C53#57611yescharacteristic
279C22×C53#57711yescharacteristic
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53—22no—
285C2×D53—22no—
286C2×D53#48714yes—
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—42no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53#52911yescharacteristic
294C4×C53#53911yescharacteristic
295C4×C53#55111yescharacteristic
296C4×C53#55211yescharacteristic
297Dic53—22no—
298Dic53—22no—
299Dic53—22no—
300Dic53—81no—
301Dic53—81no—
302Dic53—81no—
303Dic53—81no—
304#2502—531no—
305#2440—531no—
306#2100—531no—
307#2100—531no—
308#2072—531no—
309#2024D5311yescharacteristic, complemented
310#1990—531no—
311#1933—531no—
312#1933—531no—
313#1892—531no—
314#1872—531no—
315#1834—531no—
316#1824—531no—
317C8×C22⋊C4—531no—
318C2×#340—531no—
319C2×C53#248811yescharacteristic
320C2×C53C2×#48711yescharacteristic
321C2×C53#313611yescharacteristic
322C2×C53#329511yescharacteristic
323C2×C53#329911yescharacteristic
324C2×C53#333511yescharacteristic
325C2×C53#333811yescharacteristic
326D53—24no—
327C23×C8—531no—
328C2×C4⋊Q8C2×D5311yescharacteristic
329C2×C22⋊Q8—531no—
330C2×C22⋊Q8—1061no—
331C22×C4⋊C4—531no—
332C22×C4⋊C4—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×#103C2×D5311yescharacteristic
336C2×#103C2×D5311yescharacteristic
337C2×C22⋊C8—531no—
338C2×C4×C8—531no—
339C2×C4×C8—531no—
340#398—1061no—
341#398—1061no—
342#398—1061no—
343#395—1061no—
344#391—531no—
345#391C2×D5311yescharacteristic
346#389—531no—
347#389—531no—
348#385—1061no—
349#385—1061no—
350#385—1061no—
351#384—531no—
352#384—531no—
353#384—531no—
354#380—531no—
355C4×C22⋊C4—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340C2×D5311yescharacteristic
367#340C2×D5311yescharacteristic
368#336—531no—
369#336C2×D5311yescharacteristic
370C53#2769711yescharacteristic, complemented, Sylow
371C22×Q8—21no—
372C22×Q8C22×D5311yescharacteristic
373C23×C4—531no—
374C23×C4—1061no—
375C23×C4—1061no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8—1062no—
379C22×C8—1062no—
380C22×C8—2121no—
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C4⋊Q8—22no—
384C4⋊Q8—41no—
385C22⋊Q8—1062no—
386C22⋊Q8—1062no—
387C22⋊Q8—2124no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
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—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—2121no—
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C22⋊C4—531no—
412C2×C22⋊C4—531no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C42—531no—
416C2×C42C22×D5311yescharacteristic
417#103—22no—
418#103—41no—
419#103—1062no—
420#103—2121no—
421C22⋊C8—1062no—
422C4×C8—1062no—
423C4×C8—2121no—
424#95—21no—
425#95—21no—
426#95—531no—
427#95—531no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433C24—531no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—81no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—532no—
451C22×C4—532no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—4241no—
461C22×C4—4241no—
462C22×C4—4241no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4D4×D5311yescharacteristic
467C22×C4D4×D5311yescharacteristic
468C22×C4C23×D5311yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—532no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—2124no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4—4241no—
500C4⋊C4—4241no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4#1171994712yes—
508C22⋊C4—534no—
509C22⋊C4—534no—
510C22⋊C4—2122no—
511C22⋊C4—2122no—
512C42—21no—
513C42—21no—
514C42—1061no—
515C42—1061no—
516C23—532no—
517C23—532no—
518C23—532no—
519C23—1061no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23C2×D4×D5311yescharacteristic
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—84no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
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—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—4244no—
559C2×C4—4244no—
560C2×C4—2121no—
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—2121no—
572C2×C4SD16×D5312yes—
573C2×C4Q16×D5312yes—
574C2×C4#41×D5311yescharacteristic
575C2×C4#41×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4C2×#1171994711yescharacteristic
578C2×C4#1313326211yescharacteristic
579C2×C4#1313326511yescharacteristic
580C8—41no—
581C8—41no—
582C8—1062no—
583C8—2121no—
584C22—534no—
585C22—1062no—
586C22—1062no—
587C22—1062no—
588C22—1062no—
589C22—1062no—
590C22—1062no—
591C22#124×D5311yescharacteristic
592C22#1681126011yescharacteristic
593C22#1681127511yescharacteristic
594C22C2×SD16×D5311yescharacteristic
595C22C2×Q16×D5311yescharacteristic
596C22#1681142711yescharacteristic
597C22#1681143211yescharacteristic
598C4—22no—
599C4—22no—
600C4—42no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—1062no—
605C4—1062no—
606C4—1062no—
607C4—4241no—
608C4—4241no—
609C4—4241no—
610C4—4241no—
611C4#1681115211yescharacteristic
612C4#1681121911yescharacteristic
613C4#1681130311yescharacteristic
614C4#1681130711yescharacteristic
615C2—1064no—
616C2#2027903411yescharacteristic
617C2#487×D5311yescharacteristic
618C2#2028506611yescharacteristic
619C2#2028653211yescharacteristic
620C2#2028658411yescharacteristic
621C2#2028707511yescharacteristic
622C2#2028710611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2024C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2024D53
hypocenterC53#27697
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
3. #20276281C2semidirect
4. #20276279C2semidirect
5. #20276126C2semidirect
6. #20275677C2semidirect
7. #20275394C2semidirect
8. #20275393C2semidirect
11. #20274841C2semidirect
15. C53×#2024C2semidirect
309. #2024D53semidirect
370. C53#27697semidirect

Maximal subgroups

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

subgroupindex
#202791922
#202784132
#202762812
#202762792
#202761262
#202756772
#202753942
#202753932
#202751352
#202750192
#202748412
#202748042
#202744852
#340×D532
C53×#20242
#2769753