Finite Group Databasebeta

← #27403483

Subgroups — #27403483

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277680C211yesmaximal, characteristic, complemented
2D4×C4⋊Dic53C211yesmaximal, characteristic
3#20275407C211yesmaximal, characteristic, complemented
4#20275393C211yesmaximal, characteristic, complemented
5#20275309C211yesmaximal, characteristic, complemented
6#20275288C211yesmaximal, characteristic, complemented
7#20275051C211yesmaximal, characteristic
8#20275051C211yesmaximal, characteristic
9#20274849C211yesmaximal, characteristic, complemented
10#20274725C211yesmaximal, characteristic
11#20274720C211yesmaximal, characteristic, complemented
12#20274587C211yesmaximal, characteristic, complemented
13#20274542C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C2×C53×#440C211yesmaximal, characteristic
16C2×D4×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C22×C4⋊Dic53—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133127C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810530C2211yescharacteristic
27#16810530C2211yescharacteristic
28#16810522C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810509—21no—
31#16810509—21no—
32#16810498—21no—
33#16810498—21no—
34#16810493C2211yescharacteristic
35#16810491—21no—
36#16810489C2211yescharacteristic
37#16810483—21no—
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41#16810469—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×SD16×C53C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C53×#440C418yescomplemented
56C2×#107×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C2×#97×C53C2211yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64D4×Dic53—22no—
65D4×Dic53—22no—
66D4×Dic53—41no—
67D4×Dic53—41no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—44no—
73C2×C4⋊Dic53D411yescharacteristic
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C4⋊#88322—22no—
86#13133141—41no—
87#13133141—41no—
88#13133138—22no—
89#13133138—22no—
90#13133127—41no—
91C8×Dic53—41no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104C22×Q8×C53C2311yescharacteristic
105C22×D4×C53C2311yescharacteristic
106C23×C4×C53—21no—
107C2×SD16×C53—22no—
108C2×SD16×C53—22no—
109C2×SD16×C53C2×C414yes—
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C4×Q8×C53—22no—
113C4×Q8×C53C2×C414yescomplemented
114C4×D4×C53—22no—
115C4×D4×C53C2×C414yes—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#107×C53C2×C414yescomplemented
123#103×C53C2×C414yescomplemented
124#102×C53C2×C414yes—
125#97×C53C2×C414yescomplemented
126C24×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53C22×C412yes—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53C22×C412yes—
137C2×D4×C53#4112yes—
138C2×D4×C53#4112yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—24no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—81no—
157C22×Dic53—81no—
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162SD16×C53—28no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53C22×C412yes—
168C2×C8×C53C22×C412yes—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C22×C412yes—
173C4⋊C4×C53C22×C412yes—
174C22⋊C4×C53—24no—
175C42×C53—24no—
176C42×C53C22×C412yes—
177C42×C53#4112yes—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
180C4⋊Dic53—22no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—41no—
190C4⋊Dic53—42no—
191C4⋊Dic53—42no—
192C4⋊Dic53—42no—
193C4⋊Dic53—81no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53C22×D411yescharacteristic
214Q8×C53—22no—
215Q8×C53—22no—
216Q8×C53—41no—
217Q8×C53—41no—
218D4×C53—22no—
219D4×C53—22no—
220D4×C53—22no—
221D4×C53#11714yes—
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—48no—
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×C53C4×D412yes—
237C2×C4×C53C4×D412yes—
238C2×C4×C53C23×C411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
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—41no—
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—41no—
267C2×Dic53—84no—
268C2×Dic53—84no—
269C2×Dic53AGL(1,ℤ/8)12yes—
270C8×C53—22no—
271C8×C53—22no—
272#88322—41no—
273#88322—41no—
274#22778—531nomaximal, Sylow
275C22×C53—24no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53C2×C4×D411yescharacteristic
283C22×C53#54611yescharacteristic
284C22×C53#54811yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53#57511yescharacteristic
287C22×C53#57711yescharacteristic
288C22×C53#57811yescharacteristic
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53C2×#11711yescharacteristic
295C4×C53#52011yescharacteristic
296C4×C53#55211yescharacteristic
297C4×C53#55311yescharacteristic
298Dic53—22no—
299Dic53—41no—
300Dic53—41no—
301Dic53—41no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305Dic53—81no—
306C2×#440D5311yescharacteristic
307#2356—531no—
308D4×C4⋊C4—531no—
309#1935—531no—
310#1933—531no—
311#1921—531no—
312#1918—531no—
313#1877—531no—
314#1877—531no—
315#1836—531no—
316#1804—531no—
317#1803—531no—
318#1771—531no—
319#1762—531no—
320C4×#102—531no—
321C2×C53—44no—
322C2×C53#235211yescharacteristic
323C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
324C2×C53#294711yescharacteristic
325C2×C53#331011yescharacteristic
326C2×C53#331511yescharacteristic
327C2×C53#336711yescharacteristic
328C2×C53#337611yescharacteristic
329C22×SD16C2×D5311yescharacteristic
330C2×C4×Q8C2×D5311yescharacteristic
331C2×C4×D4—531no—
332C2×C4×D4—531no—
333C2×C4×D4C2×D5311yescharacteristic
334C22×C4⋊C4—1061no—
335#440Dic5318yescomplemented
336C2×#107C2×D5311yescharacteristic
337C2×C4⋊C8—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103C2×D5311yescharacteristic
341C2×#102—531no—
342C2×#102—531no—
343C2×#102C2×D5311yescharacteristic
344C2×#97—531no—
345C2×#97C2×D5311yescharacteristic
346C2×C4×C8—531no—
347#391—531no—
348#391—531no—
349#389—531no—
350#389—531no—
351#387—1061no—
352#387—1061no—
353#385—1061no—
354#385—1061no—
355#384—531no—
356#383—1061no—
357#382—531no—
358#380—1061no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
362#376—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#336—531no—
372#336—531no—
373C53#2277811yescharacteristic, complemented, Sylow
374C22×Q8C22×D5311yescharacteristic
375C22×D4C22×D5311yescharacteristic
376C23×C4—21no—
377C23×C4—1061no—
378C23×C4—1061no—
379C2×SD16—22no—
380C2×SD16—22no—
381C2×SD16C2×Dic5314yes—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386C4×Q8—22no—
387C4×Q8C2×Dic5314yescomplemented
388C4×D4—22no—
389C4×D4—1062no—
390C4×D4—1062no—
391C4×D4—2121no—
392C4×D4—2121no—
393C4×D4C2×Dic5314yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—2124no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C42—21no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42C22×D5311yescharacteristic
417#107C2×Dic5314yescomplemented
418C4⋊C8—1062no—
419#103—2121no—
420#103—2121no—
421#103C2×Dic5314yescomplemented
422#102—1062no—
423#102—1062no—
424#102C2×Dic5314yes—
425#97—2121no—
426#97C2×Dic5314yescomplemented
427C4×C8—2121no—
428#95—531no—
429#95—531no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8C22×Dic5312yes—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4#1171994512yes—
451C2×D4#1171994512yes—
452C2×D4C22×Dic5312yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—4241no—
468C22×C4—4241no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4D4×D5311yescharacteristic
474C22×C4D4×D5311yescharacteristic
475C22×C4C23×D5311yescharacteristic
476SD16—28no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8C22×Dic5312yes—
488C2×C8C22×Dic5312yes—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—1062no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—4241no—
502C4⋊C4—2121no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4C22×Dic5312yes—
507C4⋊C4C22×Dic5312yes—
508C22⋊C4—24no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C42—24no—
512C42—1061no—
513C42—1061no—
514C42—1061no—
515C42—1061no—
516C42—2122no—
517C42—2122no—
518C42#1171994712yes—
519C42C22×Dic5312yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23C2×D4×D5311yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532D4—22no—
533D4—22no—
534D4—22no—
535D4#1313317614yes—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—48no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—532no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—4244no—
563C2×C4—4244no—
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×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4D4×Dic5312yes—
577C2×C4D4×Dic5312yes—
578C2×C4C23×Dic5311yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4C2×#1171994511yescharacteristic
581C2×C4C2×#1171994511yescharacteristic
582C2×C4#1313326311yescharacteristic
583C2×C4#1313326411yescharacteristic
584C8—22no—
585C8—22no—
586C8—2121no—
587C8—2121no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22C2×D4×Dic5311yescharacteristic
596C22#1681126911yescharacteristic
597C22#1681128211yescharacteristic
598C22AGL(1,ℤ/8)×D5311yescharacteristic
599C22#1681141911yescharacteristic
600C22#1681143211yescharacteristic
601C22#1681143911yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—1062no—
608C4—4241no—
609C4—4241no—
610C4—4241no—
611C4—4241no—
612C4—2121no—
613C4—2121no—
614C4—2121no—
615C4C2×#1313317611yescharacteristic
616C4#1681110611yescharacteristic
617C4#1681131311yescharacteristic
618C4#1681131911yescharacteristic
619C2—44no—
620C2#2027764411yescharacteristic
621C2AGL(1,ℤ/8)×Dic5311yescharacteristic
622C2#2028355911yescharacteristic
623C2#2028674111yescharacteristic
624C2#2028681611yescharacteristic
625C2#2028743411yescharacteristic
626C2#2028754711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#440C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterC2×#440D53
hypocenterC53#22778
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #20277680C2semidirect
3. #20275407C2semidirect
4. #20275393C2semidirect
5. #20275309C2semidirect
6. #20275288C2semidirect
9. #20274849C2semidirect
11. #20274720C2semidirect
12. #20274587C2semidirect
55. C53×#440C4semidirect
113. C4×Q8×C53C2×C4semidirect
122. #107×C53C2×C4semidirect
123. #103×C53C2×C4semidirect
125. #97×C53C2×C4semidirect
335. #440Dic53semidirect
373. C53#22778semidirect
387. C4×Q8C2×Dic53semidirect
417. #107C2×Dic53semidirect
421. #103C2×Dic53semidirect
426. #97C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202776802
D4×C4⋊Dic532
#202754072
#202753932
#202753092
#202752882
#202750512
#202748492
#202747252
#202747202
#202745872
#202745422
#102×Dic532
C2×C53×#4402
#2277853