Finite Group Databasebeta

← #27447768

Subgroups — #27447768

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#20278437C211yesmaximal, characteristic
2#20277716C211yesmaximal, characteristic
3#20276284C211yesmaximal, characteristic, complemented
4#20276217C211yesmaximal, characteristic, complemented
5#20276092C211yesmaximal, characteristic, complemented
6#20275864C211yesmaximal, characteristic, complemented
7#20275671C211yesmaximal, characteristic, complemented
8#20275302C211yesmaximal, characteristic, complemented
9#20275135C211yesmaximal, characteristic
10#20275019C211yesmaximal, characteristic
11#20274906C211yesmaximal, characteristic
12#20274894C211yesmaximal, characteristic
13#20274730C211yesmaximal, characteristic, complemented
14#340×D53C211yesmaximal, characteristic
15C53×#1865C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×#13133193C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133129C2211yescharacteristic
27#16810563—21no—
28#16810553—21no—
29#16810530C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810518—21no—
33#16810507—21no—
34#16810497—21no—
35#16810494—21no—
36#16810489C2211yescharacteristic
37#16810480—21no—
38#16810480C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×#11719921—21no—
41C4×#11719921C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4×Q8×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×C4×D53—21no—
60C22×C4×D53—21no—
61C22×C4×D53C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C2×C8×D53—22no—
65C2×C8×D53—41no—
66C2×C8×D53—41no—
67#13133208—22no—
68#13133208—41no—
69#13133193—22no—
70#13133193Q814yes—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C4⋊C4×D53—22no—
83C4⋊C4×D53—22no—
84C4⋊C4×D53—41no—
85C4⋊C4×D53—41no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921D411yescharacteristic
89C2×#11719921D411yescharacteristic
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92#13133148—41no—
93C4⋊#88322—41no—
94#13133146—41no—
95#13133141—41no—
96#13133141—41no—
97#13133129—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C22×Q8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C4×Q8×C53—22no—
112C4×Q8×C53—41no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120#106×C53—22no—
121#103×C53—22no—
122#103×C53—41no—
123#95×C53—21no—
124C23×D53C2×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×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2411yescharacteristic
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—81no—
154C2×C4×D53—81no—
155C2×C4×D53C2×Q812yes—
156C2×C4×D53#4112yes—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—81no—
169C4⋊C4×C53#4112yes—
170C4⋊C4×C53#4112yes—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—42no—
174#11719921—24no—
175#11719921—22no—
176#11719921—22no—
177#11719921—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53—81no—
187C4⋊Dic53C2×Q812yes—
188#11719919—24no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192#11719919—81no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—81no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C8×D53—44no—
203C23×C53C22×D411yescharacteristic
204C22×D53—21no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53C22⋊Q812yes—
209C22×D53#13512yes—
210C22×D53#13712yes—
211Q8×C53—22no—
212Q8×C53—22no—
213Q8×C53—41no—
214Q8×C53—41no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53C22×Q811yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53H(5,𝔽2)11yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—22no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53C22⋊Q812yes—
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—84no—
265C4×D53—84no—
266C8×C53—41no—
267C8×C53—41no—
268#88322—41no—
269#88322—41no—
270#26416—531nomaximal, Sylow
271C22×C53C2×C22⋊Q811yescharacteristic
272C22×C53#54611yescharacteristic
273C22×C53#54611yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53C2×#13711yescharacteristic
276C22×C53#57711yescharacteristic
277C22×C53#57711yescharacteristic
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53#47914yes—
285C4×C53—42no—
286C4×C53—42no—
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—81no—
291C4×C53#52911yescharacteristic
292C4×C53#53811yescharacteristic
293C4×C53#55411yescharacteristic
294C4×C53#55411yescharacteristic
295Dic53—22no—
296Dic53—42no—
297Dic53—41no—
298Dic53—41no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302#2442—531no—
303#2360—531no—
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#1865D5311yescharacteristic, complemented
313#1853—531no—
314#1850—531no—
315#1805—531no—
316C2×#340—531no—
317C2×C53#245011yescharacteristic
318C2×C53C2×#47911yescharacteristic
319C2×C53#308411yescharacteristic
320C2×C53#329611yescharacteristic
321C2×C53#329611yescharacteristic
322C2×C53#331611yescharacteristic
323C2×C53#332011yescharacteristic
324D53—24no—
325C23×C8—531no—
326C2×#125—531no—
327C2×C22⋊Q8—531no—
328C2×C4×Q8C2×D5311yescharacteristic
329C22×C4⋊C4—531no—
330C22×C4⋊C4—531no—
331C2×#106—531no—
332C2×#106C2×D5311yescharacteristic
333C2×C4⋊C8—531no—
334C2×C4⋊C8—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103C2×D5311yescharacteristic
338C2×#103C2×D5311yescharacteristic
339C2×C22⋊C8—531no—
340#400—1061no—
341#398—1061no—
342#391—531no—
343#389—531no—
344#389—531no—
345#389—1061no—
346#389C2×D5311yescharacteristic
347#387—1061no—
348#385—1061no—
349#385—1061no—
350#382—531no—
351#380—531no—
352#380—1061no—
353C4×C4⋊C4—531no—
354C4×C22⋊C4—531no—
355C4×C22⋊C4—1061no—
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#340C2×D5311yescharacteristic
366#336—531no—
367#336C2×D5311yescharacteristic
368C53#2641611yescharacteristic, complemented, Sylow
369C22×Q8C22×D5311yescharacteristic
370C23×C4—531no—
371C23×C4—1061no—
372C23×C4—1061no—
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—1062no—
376C22×C8—2121no—
377C22×C8—2121no—
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380#125—1062no—
381#125—2121no—
382C22⋊Q8—534no—
383C22⋊Q8—1062no—
384C4×Q8—22no—
385C4×Q8—41no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—2121no—
402C2×C4⋊C4—2121no—
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C22⋊C4—531no—
407C2×C22⋊C4—531no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C42—21no—
411C2×C42—531no—
412C2×C42—1061no—
413C2×C42C22×D5311yescharacteristic
414#106—22no—
415#106—2121no—
416C4⋊C8—2121no—
417C4⋊C8—2121no—
418#103—22no—
419#103—41no—
420#103—2121no—
421#103—2121no—
422C22⋊C8—1062no—
423#95—21no—
424#95—531no—
425#95—531no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434C24—531no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8Q8×D5312yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—531no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—4241no—
459C22×C4—4241no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4D4×D5311yescharacteristic
465C22×C4D4×D5311yescharacteristic
466C22×C4C23×D5311yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—2124no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—532no—
487C4⋊C4—1064no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4#1171994512yes—
502C4⋊C4#1171994712yes—
503C22⋊C4—1062no—
504C22⋊C4—1062no—
505C22⋊C4—1064no—
506C22⋊C4—2122no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—1061no—
511C42—1061no—
512C42—4241no—
513C23—532no—
514C23—532no—
515C23—532no—
516C23—1061no—
517C23—1061no—
518C23—1061no—
519C23—1061no—
520C23C2×D4×D5311yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—532no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1062no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—4244no—
558C2×C4—4244no—
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—2121no—
570C2×C4C2×Q8×D5311yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C2×C4C2×#1171994711yescharacteristic
573C2×C4C2×#1171994511yescharacteristic
574C2×C4#1313326211yescharacteristic
575C2×C4#1313326311yescharacteristic
576C8—41no—
577C8—41no—
578C8—2121no—
579C8—2121no—
580C22—534no—
581C22—1062no—
582C22—1062no—
583C22—1062no—
584C22—1062no—
585C22—1062no—
586C22—1062no—
587C22C22⋊Q8×D5311yescharacteristic
588C22#1681126011yescharacteristic
589C22#1681126911yescharacteristic
590C22#135×D5311yescharacteristic
591C22#137×D5311yescharacteristic
592C22#1681142911yescharacteristic
593C22#1681143211yescharacteristic
594C4—42no—
595C4—41no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—81no—
600C4—1062no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—2121no—
605C4—2121no—
606C4—2122no—
607C4#1681115511yescharacteristic
608C4#1681120911yescharacteristic
609C4#1681132811yescharacteristic
610C4#1681133111yescharacteristic
611C2—1064no—
612C2#2027854411yescharacteristic
613C2#479×D5311yescharacteristic
614C2#2028457411yescharacteristic
615C2#2028654311yescharacteristic
616C2#2028655511yescharacteristic
617C2#2028683111yescharacteristic
618C2#2028687911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1865C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1865D53
hypocenterC53#26416
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
3. #20276284C2semidirect
4. #20276217C2semidirect
5. #20276092C2semidirect
6. #20275864C2semidirect
7. #20275671C2semidirect
8. #20275302C2semidirect
13. #20274730C2semidirect
15. C53×#1865C2semidirect
312. #1865D53semidirect
368. C53#26416semidirect

Maximal subgroups

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

subgroupindex
#202784372
#202777162
#202762842
#202762172
#202760922
#202758642
#202756712
#202753022
#202751352
#202750192
#202749062
#202748942
#202747302
#340×D532
C53×#18652
#2641653