Finite Group Databasebeta

← #27391304

Subgroups — #27391304

Order 13568.

964 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
1C2×#16810957C211yesmaximal, characteristic
2C4×#13133193C211yesmaximal, characteristic
3C4×C4⋊C4×D53C211yesmaximal, characteristic
4#20275352C211yesmaximal, characteristic, complemented
5#20275351C211yesmaximal, characteristic, complemented
6#20275285C211yesmaximal, characteristic, complemented
7#20275284C211yesmaximal, characteristic, complemented
8#20275019C211yesmaximal, characteristic
9#20274909C211yesmaximal, characteristic
10#20274804C211yesmaximal, characteristic
11#20274613C211yesmaximal, characteristic, complemented
12#20274612C211yesmaximal, characteristic, complemented
13#20274545C211yesmaximal, characteristic, complemented
14#20274478C211yesmaximal, characteristic
15C53×#1763C211yesmaximal, characteristic, complemented
16C2×#13133193C2211yescharacteristic
17C2×C4⋊C4×D53C2211yescharacteristic
18C2×C4⋊C4×D53C2211yescharacteristic
19C2×C42×D53—21no—
20C2×C42×D53C2211yescharacteristic
21#16810957C418yescomplemented
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133129C2211yescharacteristic
29C2×#13133129C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31#16810530C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810504—21no—
34#16810494—21no—
35#16810480C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4×#11719919—21no—
39C4×#11719919—21no—
40C4×C4⋊Dic53C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810474—21no—
43C4×#11719921—21no—
44#95×D53—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#97×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133193—22no—
68#13133193C2×C414yes—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C4⋊C4×D53—22no—
78C4⋊C4×D53—22no—
79C4⋊C4×D53—22no—
80C4⋊C4×D53C2×C414yes—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C42×D53—22no—
91C42×D53—22no—
92C42×D53—44no—
93#13133151C2×C414yescomplemented
94#13133150C2×C414yescomplemented
95#13133149—22no—
96#13133148—22no—
97#13133141—22no—
98#13133141C2×C414yescomplemented
99#13133129—22no—
100#13133129C2×C414yes—
101#13133128—22no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110C22×Q8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C4×Q8×C53—22no—
114C4×Q8×C53—22no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#103×C53—22no—
123#103×C53C2×C414yescomplemented
124#97×C53—22no—
125#95×C53—21no—
126C23×D53C2×D411yescharacteristic
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53C22×C412yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—21no—
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—22no—
154C2×C4×D53—24no—
155C2×C4×D53—24no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—41no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53C22×C412yes—
164C2×C4×D53#4112yes—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53C22×C412yes—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53C22×C412yes—
177C4⋊C4×C53#4112yes—
178C42×C53—42no—
179C42×C53—42no—
180C42×C53#4112yes—
181C42×C53#4112yes—
182#11719921—24no—
183#11719921—42no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53C22×C412yes—
190C4⋊Dic53C22×C412yes—
191C4⋊Dic53#4112yes—
192#11719919—24no—
193#11719919—24no—
194#11719919—24no—
195#11719919—42no—
196C4×Dic53—22no—
197C4×Dic53—22no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—42no—
201C4×Dic53—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322C22×C412yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53—21no—
215C22×D53C4×D412yes—
216C22×D53#13712yes—
217C22×D53#13712yes—
218Q8×C53—22no—
219Q8×C53—22no—
220Q8×C53—41no—
221Q8×C53—41no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
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#12312yes—
239C2×C4×C53#12312yes—
240C2×C4×C53C23×C411yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—21no—
253C2×Dic53—22no—
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×Dic53C4×D412yes—
266C2×Dic53AGL(1,ℤ/8)12yes—
267C4×D53—28no—
268C4×D53—22no—
269C4×D53—22no—
270C4×D53—22no—
271C4×D53—22no—
272C4×D53—44no—
273C4×D53—48no—
274C4×D53—48no—
275C8×C53—22no—
276C8×C53—41no—
277#88322—22no—
278#88322—41no—
279#21581—531nomaximal, Sylow
280C22×C53C2×C4×D411yescharacteristic
281C22×C53C2×#12311yescharacteristic
282C22×C53#52511yescharacteristic
283C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C53C2×#13711yescharacteristic
285C22×C53C2×#13711yescharacteristic
286C22×C53#57511yescharacteristic
287C2×D53—22no—
288C2×D53—22no—
289C2×D53—22no—
290C2×D53—22no—
291C2×D53—22no—
292C2×D53—22no—
293C2×D53#44114yes—
294C4×C53—22no—
295C4×C53—22no—
296C4×C53—42no—
297C4×C53—42no—
298C4×C53—42no—
299C4×C53—81no—
300C4×C53C4×#4111yescharacteristic
301C4×C53C4×#4111yescharacteristic
302C4×C53#52911yescharacteristic
303C4×C53#52911yescharacteristic
304Dic53—22no—
305Dic53—22no—
306Dic53—41no—
307Dic53—42no—
308Dic53—42no—
309Dic53—42no—
310Dic53—81no—
311C2×#484—531no—
312C4×C22⋊Q8—531no—
313C2×C4×C4⋊C4—531no—
314#1927—531no—
315#1927—531no—
316#1918—531no—
317#1918—531no—
318#1872—531no—
319#1853—531no—
320#1824—531no—
321#1776—531no—
322#1776—531no—
323#1763—531no—
324#1763D5311yescharacteristic, complemented
325#1749—531no—
326C2×C53C4×#12311yescharacteristic
327C2×C53C2×#44111yescharacteristic
328C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
329C2×C53#309211yescharacteristic
330C2×C53#309511yescharacteristic
331C2×C53#311011yescharacteristic
332C2×C53#311711yescharacteristic
333D53—24no—
334C2×C22⋊Q8—531no—
335C2×C4×Q8C2×D5311yescharacteristic
336C22×C4⋊C4—531no—
337C22×C4⋊C4—531no—
338C22×C42—531no—
339C22×C42—1061no—
340#484—538no—
341C2×#107—531no—
342C2×#107—531no—
343C2×#106—531no—
344C2×#106—531no—
345C2×#103—531no—
346C2×#103—531no—
347C2×#103C2×D5311yescharacteristic
348C2×#103C2×D5311yescharacteristic
349C2×C22⋊C8—531no—
350C2×C22⋊C8—531no—
351C2×#97—531no—
352C2×#97C2×D5311yescharacteristic
353#391—531no—
354#389—531no—
355#387—1061no—
356#385—1061no—
357#380—531no—
358C4×C4⋊C4—531no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
362C4×C4⋊C4—1061no—
363C4×C4⋊C4—1061no—
364C4×C4⋊C4—1061no—
365C4×C4⋊C4C2×D5311yescharacteristic
366C4×C22⋊C4—1061no—
367C2×#95—1061no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340C2×D5311yescharacteristic
375#340C2×D5311yescharacteristic
376#336—531no—
377#336—531no—
378C53#2158111yescharacteristic, complemented, Sylow
379C22×Q8C22×D5311yescharacteristic
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4—1061no—
384C23×C4—1061no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8C22×D5311yescharacteristic
388C22×C8C22×D5311yescharacteristic
389C22⋊Q8—534no—
390C22⋊Q8—1062no—
391C4×Q8—22no—
392C4×Q8—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—534no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—1062no—
417C2×C42—1062no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42—2124no—
423C2×C42C22×D5311yescharacteristic
424#107—534no—
425#107—534no—
426#106—1062no—
427#106—1062no—
428#103—22no—
429#103—534no—
430#103—1062no—
431#103C4×D5314yescomplemented
432C22⋊C8—534no—
433C22⋊C8—1062no—
434#97—22no—
435#97—1062no—
436#95—21no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445C24—531no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×C4×D5312yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—531no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1064no—
464C22×C4—1062no—
465C22×C4—1064no—
466C22×C4—1062no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1062no—
475C22×C4—1061no—
476C22×C4—2121no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4D4×D5311yescharacteristic
482C22×C4D4×D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—532no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8C2×C4×D5312yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—532no—
502C4⋊C4—532no—
503C4⋊C4—532no—
504C4⋊C4—1064no—
505C4⋊C4—1064no—
506C4⋊C4—1064no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—2122no—
513C4⋊C4C2×C4×D5312yes—
514C4⋊C4#1171994512yes—
515C22⋊C4—1064no—
516C22⋊C4—2122no—
517C42—42no—
518C42—42no—
519C42—1062no—
520C42—1062no—
521C42—1061no—
522C42—1061no—
523C42—2122no—
524C42—2122no—
525C42—2122no—
526C42—2122no—
527C42#1171994512yes—
528C42#1171994712yes—
529C23—532no—
530C23—532no—
531C23—532no—
532C23—1061no—
533C23—1061no—
534C23—1061no—
535C23—1061no—
536C23C2×D4×D5311yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
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—532no—
559C2×C4—1068no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1061no—
571C2×C4—1062no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—2124no—
575C2×C4—2128no—
576C2×C4—2128no—
577C2×C4—2121no—
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#1313319912yes—
587C2×C4#1313319912yes—
588C2×C4C22×C4×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4#41×D5311yescharacteristic
591C2×C4C2×#1171994711yescharacteristic
592C2×C4C2×#1171994511yescharacteristic
593C2×C4C2×#1171994511yescharacteristic
594C8—22no—
595C8—41no—
596C8—1062no—
597C8—2121no—
598C22—534no—
599C22—1062no—
600C22—1062no—
601C22—1062no—
602C22—1062no—
603C22—1062no—
604C22—1062no—
605C22C4×D4×D5311yescharacteristic
606C22C2×#1313319911yescharacteristic
607C22#1681113211yescharacteristic
608C22AGL(1,ℤ/8)×D5311yescharacteristic
609C22#137×D5311yescharacteristic
610C22#137×D5311yescharacteristic
611C22#1681142211yescharacteristic
612C4—22no—
613C4—22no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—1062no—
619C4—1062no—
620C4—4241no—
621C4—2122no—
622C4—2122no—
623C4—2122no—
624C4—2121no—
625C4C4×#1171994711yescharacteristic
626C4C4×#1171994511yescharacteristic
627C4#1681115111yescharacteristic
628C4#1681115511yescharacteristic
629C2—1064no—
630C2C4×#1313319911yescharacteristic
631C2#441×D5311yescharacteristic
632C2#2028352011yescharacteristic
633C2#2028463611yescharacteristic
634C2#2028466111yescharacteristic
635C2#2028481211yescharacteristic
636C2#2028489011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1763C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1763D53
hypocenterC53#21581
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
4. #20275352C2semidirect
5. #20275351C2semidirect
6. #20275285C2semidirect
7. #20275284C2semidirect
11. #20274613C2semidirect
12. #20274612C2semidirect
13. #20274545C2semidirect
15. C53×#1763C2semidirect
21. #16810957C4semidirect
93. #13133151C2×C4semidirect
94. #13133150C2×C4semidirect
98. #13133141C2×C4semidirect
123. #103×C53C2×C4semidirect
324. #1763D53semidirect
378. C53#21581semidirect
431. #103C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109572
C4×#131331932
C4×C4⋊C4×D532
#202753522
#202753512
#202752852
#202752842
#202750192
#202749092
#202748042
#202746132
#202746122
#202745452
#202744782
C53×#17632
#2158153