Finite Group Databasebeta

← #27452222

Subgroups — #27452222

Order 13568.

790 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#20279067C211yesmaximal, characteristic
2#20276030C211yesmaximal, characteristic, complemented
3#20275915C211yesmaximal, characteristic, complemented
4#20275866C211yesmaximal, characteristic, complemented
5#20275716C211yesmaximal, characteristic
6#20275672C211yesmaximal, characteristic, complemented
7#20275397C211yesmaximal, characteristic, complemented
8#20275335C211yesmaximal, characteristic, complemented
9#20275323C211yesmaximal, characteristic, complemented
10#20275304C211yesmaximal, characteristic, complemented
11#20275070C211yesmaximal, characteristic, complemented
12#20275032C211yesmaximal, characteristic
13#20274801C211yesmaximal, characteristic, complemented
14C4×#13133141C211yesmaximal, characteristic, complemented
15C53×#2360C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133137C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30#16810552—21no—
31#16810538—21no—
32#16810525C2211yescharacteristic
33#16810525C2211yescharacteristic
34#16810523—21no—
35#16810512—21no—
36#16810483C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C22⋊Q8×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389—21no—
53C53×#389C2211yescharacteristic
54C53×#387—21no—
55C53×#385—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C22⋊C4×C53—21no—
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C2×#11719924—22no—
62C2×#11719924—41no—
63C2×#11719924—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133209—22no—
69#13133209—41no—
70#13133191—22no—
71#13133191—41no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53—21no—
84#13133150—22no—
85C4⋊#88322—41no—
86#13133141—22no—
87#13133141—22no—
88#13133141—41no—
89#13133137—22no—
90#13133137—22no—
91#13133137—41no—
92C22⋊#88322—41no—
93C22⋊#88322—41no—
94C4×#88322—41no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C22⋊Q8×C53—22no—
101C22⋊Q8×C53—22no—
102C4×Q8×C53—22no—
103C4×Q8×C53—41no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—22no—
109C2×C4⋊C4×C53—41no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53—21no—
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
126C24×C53C2×D411yescharacteristic
127C23×D53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53#4112yes—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
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×C53C2411yescharacteristic
153C2×C22⋊D53—42no—
154C2×C22⋊D53—42no—
155C2×C22⋊D53—81no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212—21no—
164C2×D212—21no—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—81no—
173C4⋊C4×C53#4112yes—
174C4⋊C4×C53#4112yes—
175C22⋊C4×C53—24no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—42no—
183C42×C53—81no—
184C22⋊Dic53—81no—
185#11719924—44no—
186#11719921—42no—
187#11719921—42no—
188#11719921—81no—
189#11719921—81no—
190#11719921—81no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53#4112yes—
197#11719919—42no—
198#11719919—42no—
199C4×Dic53—81no—
200C2×#88322—21no—
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—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53#13512yes—
219C23×C53C22×D411yescharacteristic
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227Q8×C53—22no—
228Q8×C53—22no—
229Q8×C53—41no—
230Q8×C53—41no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—22no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
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#13712yes—
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53H(5,𝔽2)11yescharacteristic
268C2×C4×C53H(5,𝔽2)11yescharacteristic
269C22⋊D53—84no—
270C2×Dic53—22no—
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—41no—
284#88322—41no—
285#88322—41no—
286#88322—41no—
287#26727—531nomaximal, Sylow
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53#52511yescharacteristic
296C22×C53#54611yescharacteristic
297C22×C53#54611yescharacteristic
298C22×C53C2×#13511yescharacteristic
299C22×C53C2×#13711yescharacteristic
300C22×C53#57611yescharacteristic
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—41no—
311C4×C53—42no—
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#55511yescharacteristic
321C4×C53#55911yescharacteristic
322Dic53—42no—
323Dic53—81no—
324Dic53—81no—
325#2490—531no—
326#2360D5311yescharacteristic, complemented
327#2055—531no—
328#2036—531no—
329#2026—531no—
330#2000—531no—
331#1989—531no—
332#1934—531no—
333#1925—531no—
334#1923—531no—
335#1921—531no—
336#1880—531no—
337#1874—531no—
338#1823—531no—
339C4×#103—531no—
340C2×C53—42no—
341C2×C53#245911yescharacteristic
342C2×C53#310811yescharacteristic
343C2×C53#311911yescharacteristic
344C2×C53#329611yescharacteristic
345C2×C53#329711yescharacteristic
346C2×C53#332011yescharacteristic
347C2×C53#333311yescharacteristic
348D53—161no—
349C22×SD16—531no—
350C2×#125—531no—
351C2×C22⋊Q8C2×D5311yescharacteristic
352C2×#121—531no—
353C2×C4×Q8C2×D5311yescharacteristic
354C22×C4⋊C4C2×D5311yescharacteristic
355C2×#107—531no—
356C2×C4⋊C8—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×#102—531no—
363C2×C22⋊C8—531no—
364C2×C22⋊C8—531no—
365C2×C4×C8—531no—
366#397—1061no—
367#394—1061no—
368#391C2×D5311yescharacteristic
369#390—531no—
370#390—531no—
371#390—1061no—
372#389—21no—
373#389C2×D5311yescharacteristic
374#388—1061no—
375#387—21no—
376#385—21no—
377#380—531no—
378C4×C4⋊C4—531no—
379C4×C4⋊C4C2×D5311yescharacteristic
380C4×C22⋊C4—21no—
381C4×C22⋊C4C2×D5311yescharacteristic
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#336—531no—
391#336—531no—
392C53#2672711yescharacteristic, complemented, Sylow
393C22×Q8C22×D5311yescharacteristic
394C22×D4—531no—
395C22×D4—1061no—
396C23×C4—21no—
397C23×C4C22×D5311yescharacteristic
398C2×SD16—1062no—
399C2×SD16—2121no—
400C2×SD16—2121no—
401C22×C8—531no—
402C22×C8—531no—
403C22×C8—531no—
404C22×C8—531no—
405#125—1062no—
406#125—2121no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409#121—1062no—
410#121—2121no—
411C4×Q8—22no—
412C4×Q8—41no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—41no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4C22⋊D5311yescharacteristic
435C2×C22⋊C4C22⋊D5311yescharacteristic
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—1061no—
439C2×C42C22×D5311yescharacteristic
440C2×C42C22×D5311yescharacteristic
441#107—1062no—
442C4⋊C8—2121no—
443#103—1062no—
444#103—1062no—
445#103—2121no—
446#102—1062no—
447#102—1062no—
448#102—2121no—
449C22⋊C8—2121no—
450C22⋊C8—2121no—
451C4×C8—2121no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—1061no—
457#95—1061no—
458#95C22⋊D5311yescharacteristic
459#95C22⋊D5311yescharacteristic
460C24—1061no—
461C24C2×C22⋊D5311yescharacteristic
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8#1171994512yes—
468C2×D4—1061no—
469C2×D4—1061no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C2×D4—1061no—
474C2×D4—4241no—
475C2×D4—2122no—
476C2×D4—2122no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C23×D5311yescharacteristic
499SD16—2124no—
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—
511C2×C8—1061no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—532no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4#1171994512yes—
529C4⋊C4#1171994712yes—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—4241no—
534C22⋊C4—4241no—
535C22⋊C4—4241no—
536C22⋊C4—4241no—
537C22⋊C4—2122no—
538C22⋊C4—2122no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—42no—
544C42—81no—
545C42—4241no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23—2121no—
559C23#1313323712yes—
560C23C22×C22⋊D5311yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565D4—4244no—
566D4—2121no—
567D4—2121no—
568D4—2121no—
569D4—2121no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—1062no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4#1313324812yes—
611C2×C4#41×D5311yescharacteristic
612C2×C4C2×#1171994711yescharacteristic
613C2×C4C2×#1171994511yescharacteristic
614C2×C4C2×#1171994511yescharacteristic
615C2×C4#1313326211yescharacteristic
616C2×C4#1313326311yescharacteristic
617C8—2121no—
618C8—2121no—
619C8—2121no—
620C8—2121no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—4241no—
629C22—4241no—
630C22—4241no—
631C22—4241no—
632C22—4241no—
633C22—4241no—
634C22—4241no—
635C22#1681113411yescharacteristic
636C22#1681126111yescharacteristic
637C22#1681126711yescharacteristic
638C22C2×#1313323711yescharacteristic
639C22C2×#1313324811yescharacteristic
640C22#1681142611yescharacteristic
641C22#1681143111yescharacteristic
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—4241no—
652C4—4241no—
653C4—2122no—
654C4#1681115311yescharacteristic
655C4#1681120611yescharacteristic
656C4#1681133811yescharacteristic
657C4#1681135611yescharacteristic
658C2—42no—
659C2—8481no—
660C2#2027864611yescharacteristic
661C2#2028479511yescharacteristic
662C2#2028491411yescharacteristic
663C2#2028654411yescharacteristic
664C2#2028656511yescharacteristic
665C2#2028688611yescharacteristic
666C2#2028704711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2360C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2360D53
hypocenterC53#26727
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #20276030C2semidirect
3. #20275915C2semidirect
4. #20275866C2semidirect
6. #20275672C2semidirect
7. #20275397C2semidirect
8. #20275335C2semidirect
9. #20275323C2semidirect
10. #20275304C2semidirect
11. #20275070C2semidirect
13. #20274801C2semidirect
14. C4×#13133141C2semidirect
15. C53×#2360C2semidirect
326. #2360D53semidirect
392. C53#26727semidirect

Maximal subgroups

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

subgroupindex
#202790672
#202760302
#202759152
#202758662
#202757162
#202756722
#202753972
#202753352
#202753232
#202753042
#202750702
#202750322
#202748012
C4×#131331412
C53×#23602
#2672753