Finite Group Databasebeta

← #27416513

Subgroups — #27416513

Order 13568.

1424 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810773C211yesmaximal, characteristic, complemented
2C2×#16810627C211yesmaximal, characteristic, complemented
3#20278071C211yesmaximal, characteristic
4#20275858C212yesmaximal, complemented
5#20275716C212yesmaximal, complemented
6#20275149C211yesmaximal, characteristic, complemented
7#20275070C212yesmaximal, complemented
8#20275032C212yesmaximal, complemented
9#20275013C211yesmaximal, characteristic, complemented
10#20274810C211yesmaximal, characteristic, complemented
11C2×C53×#552C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C22×#11719924C2212yes—
14C22×#11719917C2211yescharacteristic
15C2×#13133191C2212yes—
16C2×#13133185—21no—
17#16810773—24no—
18#16810627—24no—
19C2×#13133141C2212yescomplemented
20C2×#13133137C2212yes—
21C2×#13133137C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2212yescomplemented
25#16810540—21no—
26#16810532—21no—
27#16810525C2212yes—
28#16810483C2211yescharacteristic
29#16810478—21no—
30#16810385C2212yescomplemented
31#16810385C2212yescomplemented
32#16810385C2212yescomplemented
33#16810378C2212yescomplemented
34C53×#552—24no—
35C53×#552—24no—
36C2×#126×C53—21no—
37C2×#125×C53—21no—
38C2×#125×C53C2211yescharacteristic, complemented
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×C22⋊Q8×C53C2211yescharacteristic, complemented
42C2×C4×Q8×C53C2211yescharacteristic, complemented
43C2×C4×D4×C53C2211yescharacteristic
44C2×#117×C53—21no—
45C2×#117×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic, complemented
47C24×D53—21no—
48C22×D212—22no—
49C22×D212—24no—
50C22×D212—22no—
51C22×D212—21no—
52C22×D212—42no—
53C22×D212C2312yes—
54C22×C22⋊D53—22no—
55C2×#11719924—24no—
56C2×#11719924—24no—
57C2×#11719924—42no—
58C2×#11719917—22no—
59C2×#11719917—42no—
60C22×#88322C2312yes—
61C22×#88322C2312yes—
62#13133191—24no—
63#13133191—42no—
64#13133185—44no—
65C2×C4⋊Dic53C2312yes—
66C2×C22⋊Dic53—21no—
67C2×#11719921—22no—
68C2×#11719921—22no—
69C2×#11719921—22no—
70#13133141—42no—
71#13133137—24no—
72#13133137—24no—
73#13133137—24no—
74C22⋊#88322—22no—
75C22⋊#88322—42no—
76#13133124—21no—
77C22×Q8×C53C2311yescharacteristic
78C22×D4×C53D411yescharacteristic
79C23×C4×C53—21no—
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82#126×C53—28no—
83#125×C53—28no—
84#125×C53—22no—
85#125×C53—22no—
86#123×C53—28no—
87C22⋊Q8×C53—22no—
88C22⋊Q8×C53—22no—
89C22⋊Q8×C53—22no—
90C22⋊Q8×C53D414yescomplemented
91C4×Q8×C53—22no—
92C4×Q8×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95#117×C53—28no—
96#117×C53—22no—
97#117×C53D414yescomplemented
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C24×C53C2×D411yescharacteristic
119C24×C53C2×D411yescharacteristic
120C23×D53—22no—
121C23×D53—44no—
122C23×D53—42no—
123C23×D53—42no—
124C23×D53—42no—
125C23×D53—82no—
126C2×Q8×C53—21no—
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C2×Q8×C53C2×D412yes—
130C2×D4×C53—22no—
131C2×D4×C53—22no—
132C2×D4×C53—41no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—24no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2411yescharacteristic
153C2×C22⋊D53—44no—
154C2×C22⋊D53—44no—
155C2×C22⋊D53—82no—
156C22×Dic53—22no—
157C2×D212—28no—
158C2×D212—22no—
159C2×D212—22no—
160C2×D212—22no—
161C2×D212—22no—
162C2×D212—22no—
163C2×D212—48no—
164C2×D212—44no—
165C2×D212—44no—
166C2×D212—42no—
167C2×D212#4114yes—
168C4⋊C4×C53—24no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—24no—
171C4⋊C4×C53—24no—
172C4⋊C4×C53—24no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53C2×D412yes—
179C4⋊C4×C53C2×D412yes—
180C4⋊C4×C53C2×D412yes—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—24no—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C42×C53—24no—
187C42×C53—24no—
188C42×C53—21no—
189C42×C53—21no—
190C22⋊Dic53—42no—
191#11719924—48no—
192#11719921—44no—
193#11719921—44no—
194#11719921—82no—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197#11719917—44no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C23×C53—21no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53#12012yes—
213C23×C53AGL(1,ℤ/8)12yes—
214C23×C53C22×D411yescharacteristic
215C22×D53—24no—
216C22×D53—48no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—42no—
223C22×D53—88no—
224C22×D53—88no—
225C22×D53—84no—
226C22×D53—84no—
227Q8×C53—24no—
228Q8×C53—41no—
229Q8×C53—41no—
230D4×C53—44no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—21no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—48no—
247C2×C4×C53—42no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—42no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—42no—
256C2×C4×C53#12012yes—
257C2×C4×C53C22×D411yescharacteristic
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C2×#4112yes—
260C2×C4×C53H(5,𝔽2)12yes—
261C22⋊D53—88no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265D212—24no—
266D212—24no—
267D212—48no—
268D212—42no—
269D212—42no—
270#88322—42no—
271#88322—42no—
272#23982—531nomaximal, Sylow
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—24no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53C2×#12011yescharacteristic
284C22×C53#54612yes—
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53#57511yescharacteristic
287C22×C53#57712yes—
288C2×D53—44no—
289C2×D53—44no—
290C2×D53—44no—
291C2×D53—816no—
292C2×D53—82no—
293C2×D53—82no—
294C2×D53—82no—
295C2×D53—82no—
296C4×C53—22no—
297C4×C53—41no—
298C4×C53—42no—
299C4×C53—41no—
300C4×C53—42no—
301C4×C53—42no—
302C4×C53—41no—
303C4×C53—42no—
304C4×C53#52511yescharacteristic
305C4×C53#53511yescharacteristic
306C4×C53#54612yes—
307Dic53—82no—
308C2×#552D5311yescharacteristic, complemented
309C2×#450—531no—
310C2×#418—531no—
311#2408—531no—
312#2025—532no—
313#2000—532no—
314#1894—531no—
315#1880—532no—
316#1874—532no—
317#1870—531no—
318#1825—531no—
319C2×C53—42no—
320C2×C53—42no—
321C2×C53#240411yescharacteristic
322C2×C53#300611yescharacteristic
323C2×C53#301011yescharacteristic
324C2×C53#331012yes—
325C2×C53#332312yes—
326D53—84no—
327C23×D4—531no—
328C22×SD16—532no—
329C22×M16—531no—
330#552—24no—
331#552—24no—
332C2×#126—21no—
333C2×#125—21no—
334C2×#125C2×D5311yescharacteristic, complemented
335C2×#123—21no—
336C2×C22⋊Q8C2×D5311yescharacteristic
337C2×C22⋊Q8C2×D5311yescharacteristic, complemented
338C2×#121—532no—
339C2×#120—1061no—
340C2×C4×Q8C2×D5311yescharacteristic, complemented
341C2×C4×D4C2×D5311yescharacteristic
342C2×#117—21no—
343C2×#117C2×D5311yescharacteristic
344C22×C4⋊C4C2×D5311yescharacteristic, complemented
345#450—1064no—
346#418—1064no—
347C2×#103—532no—
348C2×#102—532no—
349C2×#102—532no—
350C2×#102—532no—
351C2×C22⋊C8—531no—
352C2×C22⋊C8—532no—
353#394—1061no—
354#392—1061no—
355#390—532no—
356#380—531no—
357#379—1061no—
358#340—532no—
359#340—532no—
360#340—532no—
361#336—532no—
362C53#2398211yescharacteristic, complemented, Sylow
363C25—1061no—
364C22×Q8C22×D5311yescharacteristic
365C22×D4—532no—
366C22×D4—1064no—
367C22×D4—1062no—
368C22×D4—1062no—
369C22×D4—1062no—
370C22×D4—1061no—
371C22×D4—2122no—
372C22×D4C22⋊D5311yescharacteristic
373C23×C4—21no—
374C23×C4C22×D5311yescharacteristic
375C23×C4C22×D5311yescharacteristic
376C2×SD16—1064no—
377C2×SD16—1064no—
378C2×SD16—2122no—
379C2×M16—1062no—
380C2×M16—2122no—
381C22×C8—532no—
382C22×C8—532no—
383#126—28no—
384#125—28no—
385#125—22no—
386#125—22no—
387#123—28no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8C22⋊D5314yescomplemented
392#121—1064no—
393#121—2122no—
394#120—2124no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×D4—22no—
398C4×D4—22no—
399#117—28no—
400#117—22no—
401#117C22⋊D5314yescomplemented
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—532no—
411C2×C4⋊C4C22⋊D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1062no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1062no—
421C2×C22⋊C4—1062no—
422C2×C22⋊C4C22⋊D5311yescharacteristic
423C2×C22⋊C4C22⋊D5311yescharacteristic
424C2×C42—21no—
425C2×C42—21no—
426C2×C42C22×D5311yescharacteristic
427#103—2122no—
428#102—1064no—
429#102—1064no—
430#102—1064no—
431C22⋊C8—1062no—
432C22⋊C8—2122no—
433#95—1061no—
434C24—1062no—
435C24—4242no—
436C24—2124no—
437C24—2122no—
438C24—2122no—
439C24—2122no—
440C24C2×C22⋊D5311yescharacteristic
441C24C2×C22⋊D5311yescharacteristic
442C2×Q8—21no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8C2×C22⋊D5312yes—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—41no—
449C2×D4—534no—
450C2×D4—1068no—
451C2×D4—1062no—
452C2×D4—1062no—
453C2×D4—1062no—
454C2×D4—1062no—
455C2×D4—1062no—
456C2×D4—4242no—
457C2×D4—2128no—
458C2×D4—2124no—
459C2×D4—2124no—
460C2×D4—2124no—
461C2×D4—2124no—
462C2×D4—2122no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—1062no—
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5312yes—
482C22×C4C2×C22⋊D5312yes—
483C22×C4C23×D5311yescharacteristic
484SD16—2128no—
485M16—2124no—
486C2×C8—1062no—
487C2×C8—1062no—
488C2×C8—1062no—
489C2×C8—1062no—
490C2×C8—1062no—
491C2×C8—1062no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—1062no—
503C4⋊C4—1062no—
504C4⋊C4C2×C22⋊D5312yes—
505C4⋊C4C2×C22⋊D5312yes—
506C4⋊C4C2×C22⋊D5312yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—4242no—
513C22⋊C4—2124no—
514C22⋊C4—2124no—
515C22⋊C4—2122no—
516C42—24no—
517C42—24no—
518C42—21no—
519C42—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—1064no—
529C23—4244no—
530C23—4244no—
531C23—4248no—
532C23—4248no—
533C23—2128no—
534C23—2122no—
535C23—2122no—
536C23—2122no—
537C23—2122no—
538C23—2122no—
539C23—2122no—
540C23#1313318612yes—
541C23#1313324212yes—
542C23C22×C22⋊D5311yescharacteristic
543Q8—24no—
544Q8—41no—
545Q8—41no—
546D4—44no—
547D4—1064no—
548D4—1064no—
549D4—4248no—
550D4—2128no—
551D4—2122no—
552D4—2122no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4#1313318612yes—
582C2×C4C22×C22⋊D5311yescharacteristic
583C2×C4C22×C22⋊D5311yescharacteristic
584C2×C4#41×D5312yes—
585C2×C4#1313326312yes—
586C8—2122no—
587C8—2122no—
588C22—24no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—42no—
598C22—42416no—
599C22—4242no—
600C22—4242no—
601C22—4242no—
602C22—4242no—
603C22—2124no—
604C22—2124no—
605C22—2124no—
606C22C2×#1313318611yescharacteristic
607C22#1681126112yes—
608C22C2×#1313324211yescharacteristic
609C22#1681142111yescharacteristic
610C22#1681143112yes—
611C4—22no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—4242no—
620C4#1681113011yescharacteristic
621C4#1681118711yescharacteristic
622C4#1681126112yes—
623C2—42no—
624C2—42no—
625C2—4244no—
626C2#2027803011yescharacteristic
627C2#2028396711yescharacteristic
628C2#2028400111yescharacteristic
629C2#2028673012yes—
630C2#2028692512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#552C2
socleC23×C53C22×D4
prime radical#20278071C2
hypercenterC2×#552D53
hypocenterC53#23982
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#16810773C2semidirect
2. C2×#16810627C2semidirect
4. #20275858C2semidirect
5. #20275716C2semidirect
6. #20275149C2semidirect
7. #20275070C2semidirect
8. #20275032C2semidirect
9. #20275013C2semidirect
10. #20274810C2semidirect
11. C2×C53×#552C2semidirect
19. C2×#13133141C22semidirect
24. C2×C22⋊#88322C22semidirect
30. #16810385C22semidirect
31. #16810385C22semidirect
32. #16810385C22semidirect
33. #16810378C22semidirect
38. C2×#125×C53C22semidirect
41. C2×C22⋊Q8×C53C22semidirect
42. C2×C4×Q8×C53C22semidirect
46. C22×C4⋊C4×C53C22semidirect
90. C22⋊Q8×C53D4semidirect
97. #117×C53D4semidirect
308. C2×#552D53semidirect
334. C2×#125C2×D53semidirect
337. C2×C22⋊Q8C2×D53semidirect
340. C2×C4×Q8C2×D53semidirect
344. C22×C4⋊C4C2×D53semidirect
362. C53#23982semidirect
391. C22⋊Q8C22⋊D53semidirect
401. #117C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107732
C2×#168106272
#202780712
#202758582
#202757162
#202751492
#202750702
#202750322
#202750132
#202748102
C2×C53×#5522
#2398253