Finite Group Databasebeta

← #27447190

Subgroups — #27447190

Order 13568.

848 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810920C211yesmaximal, characteristic
2C2×C4×#11719924C211yesmaximal, characteristic
3#20277626C211yesmaximal, characteristic
4#20276263C211yesmaximal, characteristic, complemented
5#20276196C211yesmaximal, characteristic, complemented
6#20276096C211yesmaximal, characteristic, complemented
7#20276050C211yesmaximal, characteristic, complemented
8#20275440C211yesmaximal, characteristic, complemented
9#20275274C211yesmaximal, characteristic, complemented
10#20275052C211yesmaximal, characteristic
11#20275052C211yesmaximal, characteristic
12#20274975C211yesmaximal, characteristic
13#20274969C211yesmaximal, characteristic, complemented
14#20274961C211yesmaximal, characteristic
15C53×#2453C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20#16810920—24no—
21C4×#11719924—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133137C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810494—21no—
38#16810494—21no—
39#16810489C2211yescharacteristic
40#16810482—21no—
41#95×D53—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#400—21no—
53C53×#398—21no—
54C53×#391C2211yescharacteristic
55C53×#391C2211yescharacteristic
56C53×#389—21no—
57C53×#389C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#387—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C2×#11719924—22no—
66C2×#11719924—22no—
67C2×#11719924Q814yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C4×D212—22no—
73C4×D212—22no—
74C4×D212—22no—
75C4×D212—22no—
76C4×D212—41no—
77C4×D212—41no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86#13133150—22no—
87#13133150—41no—
88#13133148—22no—
89#13133148—22no—
90#13133148—22no—
91C4⋊#88322—22no—
92#13133141—22no—
93#13133141—22no—
94#13133141—41no—
95#13133137—22no—
96#13133137—22no—
97#13133137D414yes—
98C4×#88322—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×Q8×C53C2311yescharacteristic
104#125×C53—22no—
105#125×C53—41no—
106C4×Q8×C53—22no—
107C4×Q8×C53—41no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
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×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53D411yescharacteristic
129#95×C53D411yescharacteristic
130C23×D53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53C2×Q812yes—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212C2×D412yes—
155C2×D212C2×Q812yes—
156C2×D212#4112yes—
157C2×C4×D53—24no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53—42no—
162C2×C4×D53—81no—
163C2×C4×D53—81no—
164C2×C4×D53—81no—
165C2×C4×D53—81no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—41no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53C2×D412yes—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—42no—
186#11719924—28no—
187#11719921—24no—
188#11719921—42no—
189#11719921—42no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53#4112yes—
196#11719919—81no—
197#11719919—81no—
198C2×#88322—21no—
199C2×#88322—21no—
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×#88322C2×D412yes—
209C2×#88322C2×Q812yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—22no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218Q8×C53—22no—
219Q8×C53—22no—
220Q8×C53—41no—
221Q8×C53—41no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53#13512yes—
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C22×Q811yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C532−1+411yescharacteristic
258C2×Dic53—22no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267D212—22no—
268D212—22no—
269D212—22no—
270D212C22⋊Q814yes—
271C4×D53—84no—
272C4×D53—84no—
273C4×D53—84no—
274#88322—22no—
275#88322—22no—
276#88322—41no—
277#88322—41no—
278#26378—531nomaximal, Sylow
279C22×C53#52511yescharacteristic
280C22×C53D4211yescharacteristic
281C22×C53D4×Q811yescharacteristic
282C22×C53C2×#13511yescharacteristic
283C22×C53#57511yescharacteristic
284C22×C53#57711yescharacteristic
285C22×C53#57711yescharacteristic
286C2×D53—24no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—42no—
291C2×D53—42no—
292C2×D53—42no—
293C4×C53—22no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—42no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53—42no—
300C4×C53—41no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53C2×C22⋊Q811yescharacteristic
304C4×C53#53911yescharacteristic
305C4×C53#54711yescharacteristic
306C4×C53#55411yescharacteristic
307Dic53—42no—
308Dic53—81no—
309Dic53—81no—
310C2×#478—531no—
311C2×C4×SD16—531no—
312#2453D5311yescharacteristic, complemented
313#2349—531no—
314#2097—531no—
315#2085—531no—
316#2066—531no—
317#2059—531no—
318#1941—531no—
319#1916—531no—
320#1877—531no—
321#1877—531no—
322#1864—531no—
323#1863—531no—
324#1862—531no—
325C2×C53#244911yescharacteristic
326C2×C53#310211yescharacteristic
327C2×C53#312311yescharacteristic
328C2×C53#326711yescharacteristic
329C2×C53#328211yescharacteristic
330C2×C53#338111yescharacteristic
331C2×C53#338811yescharacteristic
332D53—44no—
333C22×SD16—531no—
334C2×#125C2×D5311yescharacteristic
335C2×C4×Q8C2×D5311yescharacteristic
336C2×C4×D4—531no—
337C2×C4×D4—531no—
338C2×C4×D4—531no—
339#478—1064no—
340C4×SD16—1064no—
341C2×#107—531no—
342C2×#107—531no—
343C2×#106—531no—
344C2×#106—531no—
345C2×#106—531no—
346C2×C4⋊C8—531no—
347C2×#103—531no—
348C2×#103—531no—
349C2×#103—531no—
350C2×#102—531no—
351C2×#102—531no—
352C2×#102—531no—
353C2×C4×C8—531no—
354#400—21no—
355#398—21no—
356#391C2×D5311yescharacteristic
357#391C2×D5311yescharacteristic
358#389—21no—
359#389—531no—
360#389—531no—
361#389C2×D5311yescharacteristic
362#389C2×D5311yescharacteristic
363#387—21no—
364#385—1061no—
365#385—1061no—
366#382—531no—
367#380—1061no—
368C4×C4⋊C4C2×D5311yescharacteristic
369C2×#95—1061no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#336—531no—
375#336—531no—
376#336—531no—
377#336—531no—
378C53#2637811yescharacteristic, complemented, Sylow
379C22×Q8C22×D5311yescharacteristic
380C22×D4—531no—
381C23×C4—1061no—
382C23×C4—1061no—
383C23×C4—1061no—
384C2×SD16—534no—
385C2×SD16—1062no—
386C2×SD16—1062no—
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—531no—
390C22×C8—531no—
391#125—22no—
392#125—41no—
393C4×Q8—22no—
394C4×Q8—41no—
395C4×D4—1062no—
396C4×D4—1062no—
397C4×D4—1062no—
398C4×D4—1062no—
399C4×D4—2121no—
400C4×D4—2121no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4C22⋊D5311yescharacteristic
412C2×C4⋊C4C22⋊D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C42—21no—
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422C2×C42C22×D5311yescharacteristic
423#107—1062no—
424#107—2121no—
425#106—1062no—
426#106—1062no—
427#106—1062no—
428C4⋊C8—1062no—
429#103—1062no—
430#103—1062no—
431#103—2121no—
432#102—534no—
433#102—1062no—
434#102—1062no—
435C4×C8—1062no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95C22⋊D5311yescharacteristic
447#95C22⋊D5311yescharacteristic
448C24—1061no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8Q8×D5312yes—
455C2×D4—532no—
456C2×D4—532no—
457C2×D4—532no—
458C2×D4—1061no—
459C2×D4—1061no—
460C2×D4—1061no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—1064no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—4241no—
470C22×C4—4241no—
471C22×C4—4241no—
472C22×C4—4241no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484SD16—1068no—
485C2×C8—532no—
486C2×C8—532no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—532no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—4241no—
516C4⋊C4—4241no—
517C4⋊C4D4×D5312yes—
518C22⋊C4—1064no—
519C22⋊C4—2122no—
520C22⋊C4—2122no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—42no—
528C23—1062no—
529C23—1062no—
530C23—1062no—
531C23—2121no—
532C23—2121no—
533C23—2121no—
534C23—2121no—
535C23C22×C22⋊D5311yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—534no—
541D4—1062no—
542D4—1062no—
543D4—1062no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1062no—
574C2×C4—4244no—
575C2×C4—4244no—
576C2×C4—4244no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4#1313323712yes—
586C2×C4C2×D4×D5311yescharacteristic
587C2×C4C2×Q8×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4#1313326211yescharacteristic
591C2×C4#1313326511yescharacteristic
592C8—1062no—
593C8—1062no—
594C8—2121no—
595C8—2121no—
596C22—1064no—
597C22—2122no—
598C22—2122no—
599C22—2122no—
600C22—2122no—
601C22—2122no—
602C22—2122no—
603C22#1681113011yescharacteristic
604C22D4×C22⋊D5311yescharacteristic
605C22Q8×C22⋊D5311yescharacteristic
606C22C2×#1313323711yescharacteristic
607C22#1681142111yescharacteristic
608C22#1681143111yescharacteristic
609C22#1681143111yescharacteristic
610C4—22no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—4241no—
621C4—4241no—
622C4—2122no—
623C4C22⋊Q8×D5311yescharacteristic
624C4#1681121911yescharacteristic
625C4#1681127511yescharacteristic
626C4#1681132711yescharacteristic
627C2—2124no—
628C2#2027852111yescharacteristic
629C2#2028472711yescharacteristic
630C2#2028496211yescharacteristic
631C2#2028624111yescharacteristic
632C2#2028642311yescharacteristic
633C2#2028758611yescharacteristic
634C2#2028765411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2453C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2453D53
hypocenterC53#26378
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2453

Complements

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

NH ≅ G/Ntype
4. #20276263C2semidirect
5. #20276196C2semidirect
6. #20276096C2semidirect
7. #20276050C2semidirect
8. #20275440C2semidirect
9. #20275274C2semidirect
13. #20274969C2semidirect
15. C53×#2453C2semidirect
312. #2453D53semidirect
378. C53#26378semidirect

Maximal subgroups

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

subgroupindex
C2×#168109202
C2×C4×#117199242
#202776262
#202762632
#202761962
#202760962
#202760502
#202754402
#202752742
#202750522
#202749752
#202749692
#202749612
C53×#24532
#2637853