Finite Group Databasebeta

← #27429802

Subgroups — #27429802

Order 13568.

906 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810857C211yesmaximal, characteristic
2C2×C4×#11719922C211yesmaximal, characteristic, complemented
3#20278207C211yesmaximal, characteristic, complemented
4#20276129C211yesmaximal, characteristic, complemented
5#20276128C211yesmaximal, characteristic, complemented
6#20275904C211yesmaximal, characteristic, complemented
7#20275901C211yesmaximal, characteristic, complemented
8#20275580C211yesmaximal, characteristic, complemented
9#20275445C211yesmaximal, characteristic, complemented
10#20274976C211yesmaximal, characteristic, complemented
11#20274975C211yesmaximal, characteristic, complemented
12#20274877C211yesmaximal, characteristic, complemented
13#20274742C211yesmaximal, characteristic, complemented
14C22⋊C4×#88322C211yesmaximal, characteristic, complemented
15C53×#2420C211yesmaximal, characteristic, complemented
16C22×#11719922C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20#16810857—24no—
21C4×#11719922—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×#13133137C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810552—21no—
37#16810538—21no—
38#16810525C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810499—21no—
41#16810489C2211yescharacteristic
42#16810485—21no—
43#16810483C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#397—21no—
53C53×#394—21no—
54C53×#390C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#386—21no—
57C53×#382C2211yescharacteristic
58C53×#381—21no—
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C22⋊D53—21no—
63C22×C4×D53—21no—
64C2×#11719922—22no—
65C2×#11719922—22no—
66C2×#11719922D414yes—
67C22×#88322—22no—
68C22×#88322—22no—
69C22×#88322—22no—
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73C22×#88322C2311yescharacteristic
74#13133191—22no—
75#13133191—41no—
76C4×D212—22no—
77C4×D212—41no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87#13133150—22no—
88#13133150—22no—
89#13133150—22no—
90#13133148—41no—
91#13133148—41no—
92#13133138—22no—
93#13133138—22no—
94#13133138—22no—
95#13133137—22no—
96#13133137—22no—
97#13133137—22no—
98C22⋊#88322—22no—
99C4×#88322—22no—
100C4×#88322—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×D4×C53—21no—
106C22×D4×C53C2311yescharacteristic
107C23×C4×C53—21no—
108C23×C4×C53C2311yescharacteristic
109#121×C53—22no—
110#121×C53—41no—
111C4×D4×C53—22no—
112C4×D4×C53—41no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53—21no—
133C24×C53C2×D411yescharacteristic
134C23×D53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—42no—
141C2×D4×C53—42no—
142C2×D4×C53—81no—
143C2×D4×C53C2×D412yes—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—42no—
154C22×C4×C53—42no—
155C22×C4×C53—81no—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2411yescharacteristic
162C2×C22⋊D53—42no—
163C2×C22⋊D53—42no—
164C2×C22⋊D53—81no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C2×D212—21no—
169C2×D212—21no—
170C2×D212—21no—
171C2×D212—21no—
172C2×D212—21no—
173C2×D212C2×D412yes—
174C2×C4×D53—42no—
175C2×C4×D53—42no—
176C2×C4×D53—81no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—81no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—42no—
187C22⋊C4×C53—42no—
188C22⋊C4×C53—81no—
189C22⋊C4×C53—81no—
190C22⋊C4×C53—81no—
191C42×C53—21no—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—21no—
195C22⋊Dic53—81no—
196#11719922—28no—
197#11719921—42no—
198#11719921—42no—
199#11719921—81no—
200#11719921—81no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207#11719919—81no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—44no—
217C2×#88322C2×D412yes—
218C2×#88322#4112yes—
219C2×#88322#4112yes—
220C2×#88322#4112yes—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—22no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53—41no—
234C23×C53#13512yes—
235C23×C53C22×D411yescharacteristic
236C22×D53—22no—
237C22×D53—41no—
238C22×D53—41no—
239C22×D53—41no—
240C22×D53—41no—
241C22×D53—41no—
242C22×D53—41no—
243D4×C53—22no—
244D4×C53—22no—
245D4×C53—41no—
246D4×C53—41no—
247D4×C53—84no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—21no—
258C2×C4×C53—21no—
259C2×C4×C53—21no—
260C2×C4×C53—21no—
261C2×C4×C53—21no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53—41no—
271C2×C4×C53—41no—
272C2×C4×C53—41no—
273C2×C4×C53—41no—
274C2×C4×C53—41no—
275C2×C4×C53—84no—
276C2×C4×C53#13512yes—
277C2×C4×C53C22×D411yescharacteristic
278C2×C4×C53C2×#4111yescharacteristic
279C2×C4×C53C2×#4111yescharacteristic
280C2×C4×C53C2×#4111yescharacteristic
281C2×C4×C53H(5,𝔽2)11yescharacteristic
282C2×C4×C532−1+411yescharacteristic
283C22⋊D53—84no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293D212—22no—
294D212—22no—
295D212—41no—
296D212—41no—
297C4×D53—84no—
298#88322—22no—
299#88322—41no—
300#88322—41no—
301#88322#12314yes—
302#25032—531nomaximal, Sylow
303C22×C53—22no—
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—41no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53—41no—
310C22×C53—42no—
311C22×C53—42no—
312C22×C53—42no—
313C22×C53—81no—
314C22×C53—81no—
315C22×C53—81no—
316C22×C53—81no—
317C22×C53#53211yescharacteristic
318C22×C53D4211yescharacteristic
319C22×C53#54711yescharacteristic
320C22×C53C2×#13511yescharacteristic
321C22×C53C2×#13511yescharacteristic
322C22×C53#57611yescharacteristic
323C22×C53#57611yescharacteristic
324C2×D53—42no—
325C2×D53—42no—
326C2×D53—42no—
327C2×D53—81no—
328C2×D53—81no—
329C2×D53—81no—
330C2×D53—81no—
331C4×C53—22no—
332C4×C53—41no—
333C4×C53—41no—
334C4×C53—41no—
335C4×C53—41no—
336C4×C53—81no—
337C4×C53—81no—
338C4×C53—81no—
339C4×C53C2×#12311yescharacteristic
340C4×C53#54211yescharacteristic
341C4×C53#54711yescharacteristic
342C4×C53#55311yescharacteristic
343Dic53—81no—
344Dic53—81no—
345Dic53—81no—
346C2×#466—531no—
347C2×C4×D8—531no—
348#2420—531no—
349#2420D5311yescharacteristic, complemented
350#2073—531no—
351#2073—531no—
352#2034—531no—
353#2034—531no—
354#1970—531no—
355#1942—531no—
356#1864—531no—
357#1864—531no—
358#1846—531no—
359#1808—531no—
360C8×C22⋊C4—531no—
361C2×C53—42no—
362C2×C53—82no—
363C2×C53#242211yescharacteristic
364C2×C53#316811yescharacteristic
365C2×C53#317611yescharacteristic
366C2×C53#328011yescharacteristic
367C2×C53#328011yescharacteristic
368C2×C53#333911yescharacteristic
369C2×C53#334111yescharacteristic
370D53—82no—
371C22×D8—531no—
372C23×C8—531no—
373C2×#121—531no—
374C2×#121C2×D5311yescharacteristic
375C2×C4×D4—531no—
376C2×C4×D4C2×D5311yescharacteristic
377#466—1064no—
378C4×D8—1064no—
379C2×#107—531no—
380C2×#107—531no—
381C2×#107—531no—
382C2×#106—531no—
383C2×#106—531no—
384C2×#102—531no—
385C2×#102—531no—
386C2×#102—531no—
387C2×#102—531no—
388C2×#102—531no—
389C2×#102—531no—
390C2×C22⋊C8—531no—
391C2×C4×C8—531no—
392C2×C4×C8—531no—
393#397—21no—
394#397—1061no—
395#394—21no—
396#394—1061no—
397#390—531no—
398#390C2×D5311yescharacteristic
399#389—531no—
400#389C2×D5311yescharacteristic
401#386—21no—
402#386—1061no—
403#382—531no—
404#382C2×D5311yescharacteristic
405#381—21no—
406#381—1061no—
407#380—531no—
408#380C2×D5311yescharacteristic
409C4×C22⋊C4C2×D5311yescharacteristic
410#340—531no—
411#340—531no—
412#340—531no—
413#340—531no—
414#336—531no—
415#336—531no—
416C53#2503211yescharacteristic, complemented, Sylow
417C22×D4—21no—
418C22×D4—531no—
419C22×D4—1061no—
420C22×D4C22×D5311yescharacteristic
421C23×C4—21no—
422C23×C4—1061no—
423C23×C4C22×D5311yescharacteristic
424C2×D8—534no—
425C2×D8—1062no—
426C2×D8—1062no—
427C22×C8—531no—
428C22×C8—531no—
429C22×C8—531no—
430C22×C8—531no—
431C22×C8—1062no—
432C22×C8—1062no—
433C22×C8—1062no—
434#121—22no—
435#121—41no—
436#121—1062no—
437#121—2121no—
438C4×D4—22no—
439C4×D4—41no—
440C4×D4—1062no—
441C4×D4—2121no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—531no—
444C2×C4⋊C4—531no—
445C2×C4⋊C4—531no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4C22×D5311yescharacteristic
448C2×C4⋊C4C22×D5311yescharacteristic
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—1061no—
456C2×C22⋊C4—1061no—
457C2×C22⋊C4—1061no—
458C2×C22⋊C4—1061no—
459C2×C22⋊C4—1061no—
460C2×C22⋊C4C22⋊D5311yescharacteristic
461C2×C22⋊C4C22⋊D5311yescharacteristic
462C2×C42C22×D5311yescharacteristic
463C2×C42C22×D5311yescharacteristic
464#107—1062no—
465#107—1062no—
466#107—1062no—
467#106—2121no—
468#106—2121no—
469#102—1062no—
470#102—1062no—
471#102—1062no—
472#102—1062no—
473#102—1062no—
474#102—1062no—
475C22⋊C8—1062no—
476C4×C8—1062no—
477C4×C8—1062no—
478#95—21no—
479#95—21no—
480#95—21no—
481#95—21no—
482#95—1061no—
483#95—1061no—
484#95—1061no—
485#95—1061no—
486#95C22⋊D5311yescharacteristic
487#95C22⋊D5311yescharacteristic
488C24—21no—
489C24—1061no—
490C24C2×C22⋊D5311yescharacteristic
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—42no—
497C2×D4—42no—
498C2×D4—81no—
499C2×D4—532no—
500C2×D4—1061no—
501C2×D4—1061no—
502C2×D4—1061no—
503C2×D4—1061no—
504C2×D4—1061no—
505C2×D4—4241no—
506C2×D4—2122no—
507C2×D4—2122no—
508C2×D4D4×D5312yes—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—81no—
521C22×C4—1061no—
522C22×C4—1061no—
523C22×C4—1061no—
524C22×C4—4241no—
525C22×C4—2122no—
526C22×C4—2122no—
527C22×C4C2×C22⋊D5311yescharacteristic
528C22×C4C2×C22⋊D5311yescharacteristic
529C22×C4C2×C22⋊D5311yescharacteristic
530C22×C4C2×C22⋊D5311yescharacteristic
531C22×C4C2×C22⋊D5311yescharacteristic
532C22×C4C23×D5311yescharacteristic
533D8—1068no—
534C2×C8—532no—
535C2×C8—532no—
536C2×C8—532no—
537C2×C8—532no—
538C2×C8—1061no—
539C2×C8—1061no—
540C2×C8—1061no—
541C2×C8—1061no—
542C2×C8—1061no—
543C2×C8—1061no—
544C2×C8—1061no—
545C2×C8—1061no—
546C2×C8—2124no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—81no—
554C4⋊C4—1061no—
555C4⋊C4—1061no—
556C4⋊C4—1061no—
557C4⋊C4—1061no—
558C4⋊C4—1061no—
559C4⋊C4—1061no—
560C4⋊C4—4241no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—81no—
566C22⋊C4—81no—
567C22⋊C4—81no—
568C22⋊C4—4241no—
569C22⋊C4—4241no—
570C22⋊C4—4241no—
571C22⋊C4—2122no—
572C22⋊C4—2122no—
573C42—21no—
574C42—21no—
575C42—21no—
576C42—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—1062no—
591C23—2121no—
592C23—2121no—
593C23—2121no—
594C23—2121no—
595C23—2121no—
596C23—2121no—
597C23#1313323712yes—
598C23C22×C22⋊D5311yescharacteristic
599D4—22no—
600D4—22no—
601D4—41no—
602D4—41no—
603D4—84no—
604D4—1062no—
605D4—1062no—
606D4—4244no—
607D4—2121no—
608D4—2121no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—84no—
637C2×C4—4244no—
638C2×C4—2121no—
639C2×C4—2121no—
640C2×C4—2121no—
641C2×C4—2121no—
642C2×C4—2121no—
643C2×C4—2121no—
644C2×C4—2121no—
645C2×C4—2121no—
646C2×C4—2121no—
647C2×C4#1313323712yes—
648C2×C4C2×D4×D5311yescharacteristic
649C2×C4#41×D5311yescharacteristic
650C2×C4#41×D5311yescharacteristic
651C2×C4#41×D5311yescharacteristic
652C2×C4#1313326211yescharacteristic
653C2×C4#1313326411yescharacteristic
654C8—534no—
655C8—1062no—
656C8—2121no—
657C8—2121no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—4241no—
673C22—4241no—
674C22—4241no—
675C22—4241no—
676C22—2122no—
677C22—2122no—
678C22—2122no—
679C22#1681116911yescharacteristic
680C22D4×C22⋊D5311yescharacteristic
681C22#1681127211yescharacteristic
682C22C2×#1313323711yescharacteristic
683C22C2×#1313323711yescharacteristic
684C22#1681142611yescharacteristic
685C22#1681142611yescharacteristic
686C4—22no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—81no—
692C4—81no—
693C4—81no—
694C4—4241no—
695C4—4241no—
696C4—4241no—
697C4#123×D5311yescharacteristic
698C4#1681124111yescharacteristic
699C4#1681127111yescharacteristic
700C4#1681131811yescharacteristic
701C2—42no—
702C2—82no—
703C2—4242no—
704C2#2027822611yescharacteristic
705C2#2028527911yescharacteristic
706C2#2028535911yescharacteristic
707C2#2028639611yescharacteristic
708C2#2028639611yescharacteristic
709C2#2028711111yescharacteristic
710C2#2028713311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2420C2
socleC23×C53C22×D4
prime radicalC2×#16810857C2
hypercenter#2420D53
hypocenterC53#25032
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2420

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719922C2semidirect
3. #20278207C2semidirect
4. #20276129C2semidirect
5. #20276128C2semidirect
6. #20275904C2semidirect
7. #20275901C2semidirect
8. #20275580C2semidirect
9. #20275445C2semidirect
10. #20274976C2semidirect
11. #20274975C2semidirect
12. #20274877C2semidirect
13. #20274742C2semidirect
14. C22⋊C4×#88322C2semidirect
15. C53×#2420C2semidirect
349. #2420D53semidirect
416. C53#25032semidirect

Maximal subgroups

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

subgroupindex
C2×#168108572
C2×C4×#117199222
#202782072
#202761292
#202761282
#202759042
#202759012
#202755802
#202754452
#202749762
#202749752
#202748772
#202747422
C22⋊C4×#883222
C53×#24202
#2503253