Finite Group Databasebeta

← #27457358

Subgroups — #27457358

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279301C211yesmaximal, characteristic, complemented
2#20276248C211yesmaximal, characteristic, complemented
3#20276025C211yesmaximal, characteristic
4#20275951C211yesmaximal, characteristic, complemented
5#20275792C211yesmaximal, characteristic
6#20275572C211yesmaximal, characteristic, complemented
7#20275346C211yesmaximal, characteristic, complemented
8#20275342C211yesmaximal, characteristic, complemented
9#20275300C211yesmaximal, characteristic, complemented
10#20275274C211yesmaximal, characteristic, complemented
11#20275070C211yesmaximal, characteristic, complemented
12#20275063C211yesmaximal, characteristic
13#20274720C211yesmaximal, characteristic, complemented
14C4×#13133138C211yesmaximal, characteristic, complemented
15C53×#2367C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C22×#11719922C2211yescharacteristic
18C2×#13133212—21no—
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810550—21no—
32#16810545—21no—
33#16810525C2211yescharacteristic
34#16810525C2211yescharacteristic
35#16810525C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810515—21no—
38#16810493C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#124×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#388—21no—
54C53×#388—21no—
55C53×#385—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53—21no—
59C22×D212C2311yescharacteristic
60C2×#11719924—22no—
61C2×#11719924—22no—
62C2×#11719924—41no—
63C2×#11719922—22no—
64C2×#11719922—41no—
65C2×#11719922—41no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133212—44no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53—21no—
84#13133148—22no—
85#13133148—41no—
86C4⋊#88322—41no—
87#13133141—22no—
88#13133141—41no—
89#13133141—41no—
90#13133138—22no—
91#13133138—22no—
92#13133138—41no—
93#13133137—22no—
94#13133137—41no—
95C4×#88322—41no—
96#13133124—21no—
97#13133124—21no—
98C22×Q8×C53C2311yescharacteristic
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53—21no—
101#124×C53—22no—
102#124×C53—22no—
103C4×Q8×C53—22no—
104C4×Q8×C53—41no—
105C4×D4×C53—22no—
106C4×D4×C53—41no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C42×C53—21no—
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53D411yescharacteristic
126#95×C53D411yescharacteristic
127C24×C53—21no—
128C23×D53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53#4112yes—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53#4112yes—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53—81no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53#4112yes—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—24no—
172C22⋊C4×C53—42no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—81no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—42no—
182C42×C53—81no—
183#11719924—44no—
184#11719922—44no—
185#11719921—42no—
186#11719921—42no—
187#11719921—42no—
188#11719921—42no—
189#11719921—81no—
190#11719921—81no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—42no—
198C4⋊Dic53—81no—
199#11719919—42no—
200C4×Dic53—42no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322#4112yes—
213C23×C53—22no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53C22×D411yescharacteristic
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228Q8×C53—22no—
229Q8×C53—22no—
230Q8×C53—41no—
231Q8×C53—41no—
232D4×C53—22no—
233D4×C53—22no—
234D4×C53—41no—
235D4×C53—41no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—84no—
262C2×C4×C53#13512yes—
263C2×C4×C53AGL(1,ℤ/8)12yes—
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53H(5,𝔽2)11yescharacteristic
269C2×C4×C53H(5,𝔽2)11yescharacteristic
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279D212—41no—
280D212—41no—
281D212—41no—
282D212—41no—
283#88322—22no—
284#88322—41no—
285#88322—41no—
286#88322—41no—
287#27071—531nomaximal, Sylow
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53#52511yescharacteristic
296C22×C53#54611yescharacteristic
297C22×C53#55011yescharacteristic
298C22×C53C2×#13511yescharacteristic
299C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C53#57711yescharacteristic
301C22×C53#57711yescharacteristic
302C2×D53—81no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C2×D53—81no—
308C2×D53—81no—
309C4×C53—22no—
310C4×C53—22no—
311C4×C53—41no—
312C4×C53—41no—
313C4×C53—42no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53—42no—
317C4×C53—81no—
318C4×C53#52911yescharacteristic
319C4×C53#53811yescharacteristic
320C4×C53#55111yescharacteristic
321C4×C53#55311yescharacteristic
322Dic53—81no—
323Dic53—81no—
324Dic53—81no—
325#2512—531no—
326#2367D5311yescharacteristic, complemented
327#2094—531no—
328#2054—531no—
329#2042—531no—
330#2012—531no—
331#1968—531no—
332#1927—531no—
333#1926—531no—
334#1920—531no—
335#1916—531no—
336#1880—531no—
337#1879—531no—
338#1803—531no—
339C4×#102—531no—
340C2×C53—82no—
341C2×C53#246711yescharacteristic
342C2×C53#310611yescharacteristic
343C2×C53#312511yescharacteristic
344C2×C53#330111yescharacteristic
345C2×C53#330611yescharacteristic
346C2×C53#339411yescharacteristic
347C2×C53#339711yescharacteristic
348D53—161no—
349C22×SD16—531no—
350C22×D8—531no—
351C2×#126—1061no—
352C2×#124C2×D5311yescharacteristic
353C2×C4×Q8C2×D5311yescharacteristic
354C2×C4×D4C2×D5311yescharacteristic
355C2×#106—531no—
356C2×#106—531no—
357C2×C4⋊C8—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×#103—531no—
361C2×#102—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#102—531no—
366C2×C4×C8—531no—
367#397—1061no—
368#396—1061no—
369#391C2×D5311yescharacteristic
370#390—531no—
371#390—531no—
372#390—531no—
373#390C2×D5311yescharacteristic
374#389—531no—
375#388—21no—
376#388—21no—
377#388—1061no—
378#385—21no—
379#384—531no—
380C4×C4⋊C4—531no—
381C4×C4⋊C4C2×D5311yescharacteristic
382C4×C4⋊C4C2×D5311yescharacteristic
383C4×C22⋊C4—21no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#336—531no—
389#336—531no—
390#336—531no—
391#336—531no—
392C53#2707111yescharacteristic, complemented, Sylow
393C22×Q8C22×D5311yescharacteristic
394C22×D4—531no—
395C22×D4C22×D5311yescharacteristic
396C23×C4—21no—
397C2×SD16—1062no—
398C2×SD16—1062no—
399C2×SD16—2121no—
400C2×D8—1062no—
401C2×D8—2121no—
402C2×D8—2121no—
403C22×C8—531no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8—531no—
407#126—2124no—
408#124—22no—
409#124—22no—
410C4×Q8—22no—
411C4×Q8—41no—
412C4×D4—22no—
413C4×D4—41no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4C22⋊D5311yescharacteristic
422C2×C4⋊C4C22⋊D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—1061no—
439C2×C42C22×D5311yescharacteristic
440C2×C42C22×D5311yescharacteristic
441C2×C42C22×D5311yescharacteristic
442#106—1062no—
443#106—2121no—
444C4⋊C8—2121no—
445#103—1062no—
446#103—2121no—
447#103—2121no—
448#102—1062no—
449#102—1062no—
450#102—1062no—
451#102—2121no—
452#102—2121no—
453C4×C8—2121no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—1061no—
458#95—1061no—
459#95C22⋊D5311yescharacteristic
460#95C22⋊D5311yescharacteristic
461C24—21no—
462C24—1061no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8#1171994512yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—1061no—
475C2×D4—1061no—
476C2×D4—1061no—
477C2×D4—1061no—
478C2×D4—1061no—
479C2×D4—1061no—
480C2×D4#1171994512yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C23×D5311yescharacteristic
497SD16—2124no—
498D8—2124no—
499C2×C8—532no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—4241no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4#1171994712yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—81no—
532C22⋊C4—4241no—
533C22⋊C4—4241no—
534C22⋊C4—2122no—
535C22⋊C4—2122no—
536C22⋊C4—2122no—
537C22⋊C4—2122no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—42no—
545C42—81no—
546C42—2122no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23—2121no—
559C23—2121no—
560C23—2121no—
561C23C22×C22⋊D5311yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—41no—
565Q8—41no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—2121no—
571D4—2121no—
572D4—2121no—
573D4—2121no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4#1313323712yes—
610C2×C4#1313324212yes—
611C2×C4#41×D5311yescharacteristic
612C2×C4C2×#1171994711yescharacteristic
613C2×C4C2×#1171994511yescharacteristic
614C2×C4C2×#1171994511yescharacteristic
615C2×C4#1313326211yescharacteristic
616C2×C4#1313326311yescharacteristic
617C8—1062no—
618C8—2121no—
619C8—2121no—
620C8—2121no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—4241no—
629C22—4241no—
630C22—4241no—
631C22—4241no—
632C22—4241no—
633C22—4241no—
634C22—4241no—
635C22#1681113411yescharacteristic
636C22#1681126711yescharacteristic
637C22#1681129211yescharacteristic
638C22C2×#1313323711yescharacteristic
639C22C2×#1313324211yescharacteristic
640C22#1681143111yescharacteristic
641C22#1681143411yescharacteristic
642C4—22no—
643C4—22no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—4241no—
652C4—4241no—
653C4—4241no—
654C4#1681115311yescharacteristic
655C4#1681120611yescharacteristic
656C4#1681130211yescharacteristic
657C4#1681131911yescharacteristic
658C2—82no—
659C2—8481no—
660C2#2027876011yescharacteristic
661C2#2028477511yescharacteristic
662C2#2028499211yescharacteristic
663C2#2028661711yescharacteristic
664C2#2028667611yescharacteristic
665C2#2028770711yescharacteristic
666C2#2028773511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2367C2
socleC23×C53C22×D4
prime radicalC22×#11719922C22
hypercenter#2367D53
hypocenterC53#27071
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #20279301C2semidirect
2. #20276248C2semidirect
4. #20275951C2semidirect
6. #20275572C2semidirect
7. #20275346C2semidirect
8. #20275342C2semidirect
9. #20275300C2semidirect
10. #20275274C2semidirect
11. #20275070C2semidirect
13. #20274720C2semidirect
14. C4×#13133138C2semidirect
15. C53×#2367C2semidirect
326. #2367D53semidirect
392. C53#27071semidirect

Maximal subgroups

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

subgroupindex
#202793012
#202762482
#202760252
#202759512
#202757922
#202755722
#202753462
#202753422
#202753002
#202752742
#202750702
#202750632
#202747202
C4×#131331382
C53×#23672
#2707153