Finite Group Databasebeta

← #27432266

Subgroups — #27432266

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810961C211yesmaximal, characteristic
2#20279169C211yesmaximal, characteristic
3#20277746C211yesmaximal, characteristic
4#20276189C211yesmaximal, characteristic, complemented
5#20276136C211yesmaximal, characteristic, complemented
6#20276092C211yesmaximal, characteristic, complemented
7#20276035C211yesmaximal, characteristic, complemented
8#20275480C211yesmaximal, characteristic, complemented
9#20275359C211yesmaximal, characteristic, complemented
10#20275058C211yesmaximal, characteristic, complemented
11#20275028C211yesmaximal, characteristic
12#20274893C211yesmaximal, characteristic
13#20274718C211yesmaximal, characteristic, complemented
14#20274432C211yesmaximal, characteristic
15C2×C53×#467C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#16810961—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810551—21no—
30#16810551—21no—
31#16810530C2211yescharacteristic
32#16810530C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810522C2211yescharacteristic
35#16810516—21no—
36#16810516—21no—
37#16810509—21no—
38#16810504—21no—
39#16810504—21no—
40#16810492C2211yescharacteristic
41#16810488C2211yescharacteristic
42C4⋊C4×Dic53—21no—
43C4⋊C4×Dic53C2211yescharacteristic
44C4×C22⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×Q16×C53C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×C22⋊Q8×C53C2211yescharacteristic
56C2×C22⋊Q8×C53C2211yescharacteristic
57C53×#467—24no—
58C2×#107×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#103×C53C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63Q8×Dic53—22no—
64Q8×Dic53—41no—
65#13133176—22no—
66#13133176—22no—
67#13133176—22no—
68#13133176—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D411yescharacteristic
80C2×C22⋊Dic53D411yescharacteristic
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133150—22no—
88#13133148—22no—
89#13133146—22no—
90#13133146—41no—
91#13133141—22no—
92#13133141—22no—
93#13133141—22no—
94#13133141—41no—
95C22⋊#88322—22no—
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
109C22×Q8×C53C2311yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C2×Q16×C53—22no—
112C2×Q16×C53—22no—
113C2×Q16×C53—41no—
114C22×C8×C53—22no—
115C22×C8×C53—22no—
116C22×C8×C53—41no—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22⋊Q8×C53—22no—
120C22⋊Q8×C53—22no—
121C22⋊Q8×C53—22no—
122C22⋊Q8×C53D414yes—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129#107×C53—22no—
130#103×C53—22no—
131#103×C53—22no—
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53C2×D412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D412yes—
155C22×C4×C53#4112yes—
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165Q16×C53—44no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—44no—
172C2×C8×C53#4112yes—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53C2×D412yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—42no—
181C22⋊Dic53—22no—
182C22⋊Dic53—41no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—42no—
194C4⋊Dic53#4112yes—
195#11719919—42no—
196#11719919—42no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C4×Dic53—42no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53#12112yes—
218C23×C53C22×D411yescharacteristic
219Q8×C53—22no—
220Q8×C53—22no—
221Q8×C53—41no—
222Q8×C53—41no—
223Q8×C53—41no—
224Q8×C53—41no—
225Q8×C53—41no—
226Q8×C53—41no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53#12112yes—
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53H(5,𝔽2)11yescharacteristic
251C2×C4×C532−1+411yescharacteristic
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—22no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53#13512yes—
276C8×C53—22no—
277C8×C53—41no—
278#88322—41no—
279#88322—41no—
280#25201—531nomaximal, Sylow
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53C2×#12111yescharacteristic
289C22×C53#54611yescharacteristic
290C22×C53#54711yescharacteristic
291C22×C53C2×#13511yescharacteristic
292C22×C53#57511yescharacteristic
293C22×C53#57811yescharacteristic
294C22×C53#57811yescharacteristic
295C4×C53—22no—
296C4×C53—42no—
297C4×C53—42no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53#52911yescharacteristic
301C4×C53#53911yescharacteristic
302C4×C53#54611yescharacteristic
303C4×C53#54711yescharacteristic
304Dic53—22no—
305Dic53—41no—
306Dic53—41no—
307Dic53—41no—
308Dic53—42no—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312C2×#485—531no—
313C2×#467D5311yescharacteristic
314#2499—531no—
315#2362—531no—
316#2083—531no—
317#2074—531no—
318#2066—531no—
319#2056—531no—
320#1949—531no—
321#1928—531no—
322#1878—531no—
323#1873—531no—
324#1850—531no—
325#1803—531no—
326#1735—531no—
327C2×C53—42no—
328C2×C53#242511yescharacteristic
329C2×C53#303511yescharacteristic
330C2×C53#304411yescharacteristic
331C2×C53#330011yescharacteristic
332C2×C53#331511yescharacteristic
333C2×C53#334211yescharacteristic
334C2×C53#334911yescharacteristic
335C22×Q16C2×D5311yescharacteristic
336C23×C8C2×D5311yescharacteristic
337C2×C22⋊Q8C2×D5311yescharacteristic
338C2×C22⋊Q8C2×D5311yescharacteristic
339C2×C4×Q8—531no—
340C2×#117—531no—
341C2×#117—531no—
342#485—1064no—
343#467—24no—
344C2×#107—531no—
345C2×#107C2×D5311yescharacteristic
346C2×#106—531no—
347C2×C4⋊C8—531no—
348C2×C4⋊C8—531no—
349C2×#103—531no—
350C2×#103—531no—
351C2×#103—531no—
352C2×#103—531no—
353C2×#103C2×D5311yescharacteristic
354C2×#103C2×D5311yescharacteristic
355C2×C22⋊C8—531no—
356#397—1061no—
357#397—1061no—
358#391—531no—
359#391—531no—
360#391—531no—
361#389—531no—
362#388—1061no—
363#388—1061no—
364#387—1061no—
365#387—1061no—
366#387—1061no—
367#384—531no—
368#382—531no—
369C4×C4⋊C4—531no—
370C4×C4⋊C4—1061no—
371C4×C22⋊C4—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#336—531no—
379#336—531no—
380C53#2520111yescharacteristic, complemented, Sylow
381C22×Q8C22×D5311yescharacteristic
382C22×Q8C22×D5311yescharacteristic
383C23×C4C22×D5311yescharacteristic
384C2×Q16—22no—
385C2×Q16—22no—
386C2×Q16—41no—
387C22×C8—22no—
388C22×C8—22no—
389C22×C8—41no—
390C22×C8—531no—
391C22×C8—531no—
392C22×C8C22×D5311yescharacteristic
393C22×C8C22×D5311yescharacteristic
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8C22⋊D5314yes—
398C4×Q8—1062no—
399C4×Q8—2121no—
400#117—1062no—
401#117—1062no—
402#117—1062no—
403#117—2121no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—531no—
419C2×C22⋊C4—531no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—1061no—
425C2×C42—1061no—
426C2×C42—1061no—
427C2×C42—1061no—
428#107—22no—
429#107—1062no—
430#106—1062no—
431C4⋊C8—1062no—
432C4⋊C8—2121no—
433#103—22no—
434#103—22no—
435#103—1062no—
436#103—1062no—
437#103—1062no—
438#103—2121no—
439C22⋊C8—1062no—
440#95—531no—
441#95—531no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452C24D4×D5311yescharacteristic
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×C22⋊D5312yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4D4×D5311yescharacteristic
482C22×C4#1171994712yes—
483C22×C4C2×C22⋊D5312yes—
484C22×C4C23×D5311yescharacteristic
485Q16—44no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—44no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8#1171994712yes—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—532no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4—2122no—
516C4⋊C4C2×C22⋊D5312yes—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—1062no—
520C22⋊C4—2121no—
521C22⋊C4—2122no—
522C22⋊C4—2122no—
523C42—1061no—
524C42—1061no—
525C42—1061no—
526C42—1061no—
527C42—2122no—
528C42—2122no—
529C42—2122no—
530C42—2122no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23#1313318912yes—
538C23C2×D4×D5311yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
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—41no—
564C2×C4—41no—
565C2×C4—532no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1062no—
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—2121no—
589C2×C4#1313318912yes—
590C2×C4C22×C22⋊D5311yescharacteristic
591C2×C4#41×D5311yescharacteristic
592C2×C4C2×#1171994711yescharacteristic
593C2×C4C2×#1171994711yescharacteristic
594C2×C4#1313326211yescharacteristic
595C2×C4#1313326411yescharacteristic
596C8—22no—
597C8—41no—
598C8—2121no—
599C8—2121no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22C2×#1313318911yescharacteristic
608C22#1681126211yescharacteristic
609C22#1681127311yescharacteristic
610C22#135×D5311yescharacteristic
611C22#1681142011yescharacteristic
612C22#1681143611yescharacteristic
613C22#1681143611yescharacteristic
614C4—22no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—1062no—
620C4—4241no—
621C4—4241no—
622C4—4241no—
623C4—2121no—
624C4—2121no—
625C4—2122no—
626C4—2121no—
627C4#1681115411yescharacteristic
628C4#1681121811yescharacteristic
629C4#1681126711yescharacteristic
630C4#1681127211yescharacteristic
631C2—42no—
632C2#2027827211yescharacteristic
633C2#2028416611yescharacteristic
634C2#2028424211yescharacteristic
635C2#2028659711yescharacteristic
636C2#2028681711yescharacteristic
637C2#2028714211yescharacteristic
638C2#2028725111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#467C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#467D53
hypocenterC53#25201
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
4. #20276189C2semidirect
5. #20276136C2semidirect
6. #20276092C2semidirect
7. #20276035C2semidirect
8. #20275480C2semidirect
9. #20275359C2semidirect
10. #20275058C2semidirect
13. #20274718C2semidirect
380. C53#25201semidirect

Maximal subgroups

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

subgroupindex
C2×#168109612
#202791692
#202777462
#202761892
#202761362
#202760922
#202760352
#202754802
#202753592
#202750582
#202750282
#202748932
#202747182
#202744322
C2×C53×#4672
#2520153