Finite Group Databasebeta

← #27447757

Subgroups — #27447757

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278439C211yesmaximal, characteristic
2#20276283C211yesmaximal, characteristic, complemented
3#20276216C211yesmaximal, characteristic, complemented
4#20276096C211yesmaximal, characteristic, complemented
5#20275866C211yesmaximal, characteristic, complemented
6#20275672C211yesmaximal, characteristic, complemented
7#20275300C211yesmaximal, characteristic, complemented
8#20275139C211yesmaximal, characteristic
9#20275023C211yesmaximal, characteristic
10#20274983C211yesmaximal, characteristic, complemented
11#20274908C211yesmaximal, characteristic
12#20274895C211yesmaximal, characteristic
13#20274730C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic
15C53×#2360C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810565—21no—
29#16810557—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810495—21no—
33#16810489C2211yescharacteristic
34#16810483C2211yescharacteristic
35#16810481—21no—
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
59C23×Dic53—21no—
60C22×#88322—22no—
61C22×#88322—41no—
62C22×#88322—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133209—22no—
68#13133209—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—22no—
71C2×C4⋊Dic53—41no—
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—
81#13133148—22no—
82#13133148—41no—
83C4⋊#88322—41no—
84C4⋊#88322—41no—
85#13133141—22no—
86#13133141—41no—
87#13133141—41no—
88#13133141—41no—
89C22⋊#88322—22no—
90#13133124—21no—
91#13133124—21no—
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—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—81no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53—81no—
165C4⋊C4×C53C2×Q812yes—
166C4⋊C4×C53#4112yes—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—42no—
175C42×C53—81no—
176C22⋊Dic53—42no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53#4112yes—
186#11719919—42no—
187#11719919—42no—
188#11719919—81no—
189#11719919—81no—
190C2×#88322—21no—
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—44no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53C22⋊Q812yes—
208C23×C53#13512yes—
209C23×C53#13712yes—
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—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
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—21no—
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—84no—
246C2×C4×C53C22⋊Q812yes—
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—
265C2×Dic53—84no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#88322—41no—
270#26416—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53#47914yes—
278C22×C53C2×C22⋊Q811yescharacteristic
279C22×C53#54611yescharacteristic
280C22×C53#54611yescharacteristic
281C22×C53C2×#13511yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57711yescharacteristic
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—42no—
291C4×C53—41no—
292C4×C53—41no—
293C4×C53—81no—
294C4×C53#52911yescharacteristic
295C4×C53#53811yescharacteristic
296C4×C53#55411yescharacteristic
297C4×C53#55411yescharacteristic
298Dic53—42no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302#2442—531no—
303#2360D5311yescharacteristic
304#2101—531no—
305#2088—531no—
306#2066—531no—
307#2026—531no—
308#1989—531no—
309#1920—531no—
310#1892—531no—
311#1872—531no—
312#1865—531no—
313#1853—531no—
314#1850—531no—
315#1805—531no—
316C2×#340—531no—
317C2×C53—24no—
318C2×C53#245011yescharacteristic
319C2×C53C2×#47911yescharacteristic
320C2×C53#308411yescharacteristic
321C2×C53#329611yescharacteristic
322C2×C53#329611yescharacteristic
323C2×C53#331611yescharacteristic
324C2×C53#332011yescharacteristic
325C23×C8—531no—
326C2×#125—531no—
327C2×C22⋊Q8C2×D5311yescharacteristic
328C2×C4×Q8C2×D5311yescharacteristic
329C22×C4⋊C4—531no—
330C22×C4⋊C4C2×D5311yescharacteristic
331C2×#106—531no—
332C2×#106—531no—
333C2×C4⋊C8—531no—
334C2×C4⋊C8—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×C22⋊C8—531no—
340#400—1061no—
341#398—1061no—
342#391C2×D5311yescharacteristic
343#389—21no—
344#389—531no—
345#389—531no—
346#389C2×D5311yescharacteristic
347#387—21no—
348#385—21no—
349#385—1061no—
350#382—531no—
351#380—531no—
352#380—1061no—
353C4×C4⋊C4C2×D5311yescharacteristic
354C4×C22⋊C4—21no—
355C4×C22⋊C4C2×D5311yescharacteristic
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#340—531no—
366#336—531no—
367#336—531no—
368C53#2641611yescharacteristic, complemented, Sylow
369C22×Q8C22×D5311yescharacteristic
370C23×C4—21no—
371C23×C4—1061no—
372C23×C4C22×D5311yescharacteristic
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—531no—
376C22×C8—531no—
377C22×C8—1062no—
378C22×C8—2121no—
379C22×C8—2121no—
380#125—1062no—
381#125—2121no—
382C22⋊Q8—22no—
383C22⋊Q8Dic10614yes—
384C4×Q8—22no—
385C4×Q8—41no—
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—1062no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—2121no—
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4C22⋊D5311yescharacteristic
409C2×C22⋊C4C22⋊D5311yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42C22×D5311yescharacteristic
413C2×C42C22×D5311yescharacteristic
414#106—1062no—
415#106—2121no—
416C4⋊C8—2121no—
417C4⋊C8—2121no—
418#103—1062no—
419#103—2121no—
420#103—2121no—
421#103—2121no—
422C22⋊C8—1062no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95C22⋊D5311yescharacteristic
433#95C22⋊D5311yescharacteristic
434C24C2×C22⋊D5311yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×Dic10612yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—81no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—4241no—
457C22×C4—2122no—
458C22×C4—2122no—
459C22×C4C2×Dic10612yes—
460C22×C4#1171994512yes—
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C23×D5311yescharacteristic
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—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—2124no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—532no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4C2×Dic10612yes—
502C4⋊C4#1171994712yes—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—2122no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—81no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23#1313319612yes—
518C23#1313323712yes—
519C23#1313324812yes—
520C23C22×C22⋊D5311yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
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—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4—1062no—
557C2×C4—4244no—
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—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4#1313319612yes—
570C2×C4C22×Dic10611yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C2×C4C2×#1171994711yescharacteristic
573C2×C4C2×#1171994511yescharacteristic
574C2×C4#1313326211yescharacteristic
575C2×C4#1313326311yescharacteristic
576C8—2121no—
577C8—2121no—
578C8—2121no—
579C8—2121no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22#1681093014yes—
587C22C2×#1313319611yescharacteristic
588C22#1681126111yescharacteristic
589C22#1681126711yescharacteristic
590C22C2×#1313323711yescharacteristic
591C22C2×#1313324811yescharacteristic
592C22#1681143111yescharacteristic
593C22#1681143411yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—4241no—
604C4—4241no—
605C4—4241no—
606C4—2122no—
607C4#1681115311yescharacteristic
608C4#1681120611yescharacteristic
609C4#1681133011yescharacteristic
610C4#1681133211yescharacteristic
611C2—24no—
612C2#2027853311yescharacteristic
613C2C2×#1681093011yescharacteristic
614C2#2028457711yescharacteristic
615C2#2028654411yescharacteristic
616C2#2028655011yescharacteristic
617C2#2028682011yescharacteristic
618C2#2028688611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2360C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2360D53
hypocenterC53#26416
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. #20276283C2semidirect
3. #20276216C2semidirect
4. #20276096C2semidirect
5. #20275866C2semidirect
6. #20275672C2semidirect
7. #20275300C2semidirect
10. #20274983C2semidirect
13. #20274730C2semidirect
368. C53#26416semidirect

Maximal subgroups

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

subgroupindex
#202784392
#202762832
#202762162
#202760962
#202758662
#202756722
#202753002
#202751392
#202750232
#202749832
#202749082
#202748952
#202747302
C2×#168103852
C53×#23602
#2641653