Finite Group Databasebeta

← #27447997

Subgroups — #27447997

Order 13568.

748 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278428C211yesmaximal, characteristic
2#20276273C211yesmaximal, characteristic, complemented
3#20276211C211yesmaximal, characteristic, complemented
4#20276110C211yesmaximal, characteristic, complemented
5#20275866C211yesmaximal, characteristic, complemented
6#20275672C211yesmaximal, characteristic, complemented
7#20275286C211yesmaximal, characteristic, complemented
8#20275149C211yesmaximal, characteristic
9#20275028C211yesmaximal, characteristic
10#20274969C211yesmaximal, characteristic, complemented
11#20274903C211yesmaximal, characteristic
12#20274900C211yesmaximal, characteristic
13#20274720C211yesmaximal, characteristic, complemented
14#20274434C211yesmaximal, characteristic
15C53×#2360C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810559—21no—
29#16810546—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810520—21no—
33#16810506—21no—
34#16810493C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
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
59C2×#11719917—22no—
60C2×#11719917—41no—
61C2×#11719917—41no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133209—22no—
67#13133209—41no—
68#13133176—22no—
69#13133176—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83#13133150—22no—
84#13133150—41no—
85C4⋊#88322—41no—
86C4⋊#88322—41no—
87#13133141—22no—
88#13133141—41no—
89#13133141—41no—
90#13133141—41no—
91C22⋊#88322—22no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C22×Q8×C53C2311yescharacteristic
96C23×C4×C53—21no—
97C23×C4×C53C2311yescharacteristic
98C22⋊Q8×C53—22no—
99C22⋊Q8×C53Q814yes—
100C4×Q8×C53—22no—
101C4×Q8×C53—41no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53—41no—
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53D411yescharacteristic
123#95×C53D411yescharacteristic
124C24×C53C2×D411yescharacteristic
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53C2×Q812yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53—81no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×Q812yes—
148C22×C4×C53#4112yes—
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53—81no—
162C4⋊C4×C53C2×Q812yes—
163C4⋊C4×C53#4112yes—
164C22⋊C4×C53—24no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—41no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—42no—
172C42×C53—81no—
173C22⋊Dic53—42no—
174C22⋊Dic53—81no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53—81no—
184C4⋊Dic53#4112yes—
185#11719919—42no—
186#11719919—42no—
187#11719919—42no—
188#11719919—81no—
189C4×Dic53—42no—
190#11719917—44no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53C22⋊Q812yes—
210C23×C53C22×D411yescharacteristic
211Q8×C53—22no—
212Q8×C53—22no—
213Q8×C53—41no—
214Q8×C53—41no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
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—84no—
245C2×C4×C53C22⋊Q812yes—
246C2×C4×C53#13712yes—
247C2×C4×C53C22×Q811yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×Dic53—22no—
254C2×Dic53—41no—
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—
265#88322—41no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#26432—531nomaximal, Sylow
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53C2×C22⋊Q811yescharacteristic
278C22×C53#54611yescharacteristic
279C22×C53#54611yescharacteristic
280C22×C53C2×#13711yescharacteristic
281C22×C53#57511yescharacteristic
282C22×C53#57611yescharacteristic
283C22×C53#57811yescharacteristic
284C4×C53—22no—
285C4×C53—22no—
286C4×C53—41no—
287C4×C53—42no—
288C4×C53—42no—
289C4×C53—41no—
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53#52911yescharacteristic
294C4×C53#53811yescharacteristic
295C4×C53#55411yescharacteristic
296C4×C53#55411yescharacteristic
297Dic53—42no—
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301#2441—531no—
302#2360D5311yescharacteristic
303#2099—531no—
304#2087—531no—
305#2068—531no—
306#2026—531no—
307#1989—531no—
308#1918—531no—
309#1894—531no—
310#1873—531no—
311#1863—531no—
312#1852—531no—
313#1851—531no—
314#1803—531no—
315#1736—531no—
316C2×C53—42no—
317C2×C53#245011yescharacteristic
318C2×C53#307811yescharacteristic
319C2×C53#308311yescharacteristic
320C2×C53#331211yescharacteristic
321C2×C53#331511yescharacteristic
322C2×C53#332211yescharacteristic
323C2×C53#333311yescharacteristic
324C22×M16—531no—
325C2×#125—531no—
326C2×C22⋊Q8C2×D5311yescharacteristic
327C2×C4×Q8C2×D5311yescharacteristic
328C2×#117—531no—
329C22×C4⋊C4C2×D5311yescharacteristic
330C2×#107—531no—
331C2×#107—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×C22⋊C8—531no—
339#399—1061no—
340#396—1061no—
341#391C2×D5311yescharacteristic
342#389—21no—
343#389—531no—
344#389—531no—
345#389—1061no—
346#389C2×D5311yescharacteristic
347#387—21no—
348#387—1061no—
349#385—21no—
350#384—531no—
351#380—531no—
352C4×C4⋊C4C2×D5311yescharacteristic
353C4×C22⋊C4—21no—
354C4×C22⋊C4C2×D5311yescharacteristic
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#336—531no—
366#336—531no—
367C53#2643211yescharacteristic, complemented, Sylow
368C22×Q8C22×D5311yescharacteristic
369C23×C4—21no—
370C23×C4C22×D5311yescharacteristic
371C2×M16—1062no—
372C2×M16—2121no—
373C2×M16—2121no—
374C22×C8—531no—
375C22×C8—531no—
376C22×C8—531no—
377C22×C8—531no—
378#125—1062no—
379#125—2121no—
380C22⋊Q8—22no—
381C22⋊Q8Dic10614yes—
382C4×Q8—22no—
383C4×Q8—41no—
384#117—1062no—
385#117—2121no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—41no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4C22⋊D5311yescharacteristic
409C2×C22⋊C4C22⋊D5311yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—1061no—
413C2×C42C22×D5311yescharacteristic
414C2×C42C22×D5311yescharacteristic
415#107—1062no—
416#107—2121no—
417C4⋊C8—2121no—
418C4⋊C8—2121no—
419#103—1062no—
420#103—2121no—
421#103—2121no—
422#103—2121no—
423C22⋊C8—1062no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95C22⋊D5311yescharacteristic
432#95C22⋊D5311yescharacteristic
433C24C2×C22⋊D5311yescharacteristic
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C2×Dic10612yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4C2×Dic10612yes—
456C22×C4#1171994512yes—
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C23×D5311yescharacteristic
463M16—2124no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—532no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—4241no—
493C4⋊C4—4241no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4C2×Dic10612yes—
499C4⋊C4#1171994712yes—
500C22⋊C4—24no—
501C22⋊C4—22no—
502C22⋊C4—41no—
503C22⋊C4—4241no—
504C22⋊C4—2122no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—81no—
511C42—2122no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23#1313319612yes—
519C23C22×C22⋊D5311yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—84no—
554C2×C4—1062no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4#1313319612yes—
567C2×C4#1313324812yes—
568C2×C4C22×Dic10611yescharacteristic
569C2×C4#41×D5311yescharacteristic
570C2×C4C2×#1171994711yescharacteristic
571C2×C4C2×#1171994511yescharacteristic
572C2×C4#1313326211yescharacteristic
573C2×C4#1313326311yescharacteristic
574C8—2121no—
575C8—2121no—
576C8—2121no—
577C8—2121no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×#1313319611yescharacteristic
586C22#1681126111yescharacteristic
587C22#1681126711yescharacteristic
588C22C2×#1313324811yescharacteristic
589C22#1681142111yescharacteristic
590C22#1681142611yescharacteristic
591C22#1681143811yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—81no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—2122no—
605C4#1681115311yescharacteristic
606C4#1681120611yescharacteristic
607C4#1681133011yescharacteristic
608C4#1681133211yescharacteristic
609C2—42no—
610C2#2027853311yescharacteristic
611C2#2028452011yescharacteristic
612C2#2028456611yescharacteristic
613C2#2028676611yescharacteristic
614C2#2028680511yescharacteristic
615C2#2028691611yescharacteristic
616C2#2028704711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2360C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2360D53
hypocenterC53#26432
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. #20276273C2semidirect
3. #20276211C2semidirect
4. #20276110C2semidirect
5. #20275866C2semidirect
6. #20275672C2semidirect
7. #20275286C2semidirect
10. #20274969C2semidirect
13. #20274720C2semidirect
367. C53#26432semidirect

Maximal subgroups

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

subgroupindex
#202784282
#202762732
#202762112
#202761102
#202758662
#202756722
#202752862
#202751492
#202750282
#202749692
#202749032
#202749002
#202747202
#202744342
C53×#23602
#2643253