Finite Group Databasebeta

← #27384839

Subgroups — #27384839

Order 13568.

1630 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
1C22×#13133175C211yesmaximal, characteristic
2C2×#16810662C212yesmaximal, complemented
3C2×#16810585C211yesmaximal, characteristic
4#380×D53C211yesmaximal, characteristic
5#20274917C212yesmaximal, complemented
6#20274913C212yesmaximal, complemented
7#20274908C211yesmaximal, characteristic, complemented
8#20274905C211yesmaximal, characteristic, complemented
9#20274902C211yesmaximal, characteristic, complemented
10#20274433C212yesmaximal, complemented
11C53×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719912C2212yes—
14C2×#13133175—21no—
15C2×#13133175—21no—
16C2×#13133175—28no—
17C2×#13133175C2212yes—
18C2×#13133175C2212yes—
19C22×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C22×#11719921—21no—
23C2×C22⋊C4×D53—21no—
24C22×C4×Dic53—21no—
25#16810662—28no—
26C2×#13133151C2212yescomplemented
27C2×#13133150C2212yescomplemented
28C2×#13133148C2212yescomplemented
29#16810585C418yescomplemented
30C2×C22⋊#88322C2211yescharacteristic
31C2×#13133129C2212yes—
32#16810483C2211yescharacteristic
33#16810482—21no—
34#16810481—21no—
35#16810480C2211yescharacteristic
36#16810480C2211yescharacteristic
37#95×D53—21no—
38#16810386C2212yescomplemented
39#16810385C2212yescomplemented
40#16810385C2212yescomplemented
41C22×C4⋊C4×C53C2211yescharacteristic
42C2×#106×C53C2212yescomplemented
43C2×C22⋊C8×C53C2211yescharacteristic
44C53×#380C2211yescharacteristic
45C53×#340C2212yescomplemented
46C24×D53D411yescharacteristic
47C23×Dic53—21no—
48C23×Dic53D411yescharacteristic
49C22×C4×D53—24no—
50C22×C4×D53—24no—
51C22×C4×D53—21no—
52C22×C4×D53—21no—
53C22×C4×D53C2×C414yes—
54C22×C4×D53D411yescharacteristic
55C22×C4×D53D411yescharacteristic
56C22×C4×D53Q814yes—
57C22×C4×D53C2311yescharacteristic
58C22×C4×D53C2311yescharacteristic
59C2×#11719912—24no—
60C2×#11719912—24no—
61C2×#11719912C2×C418yescomplemented
62C22×#88322C2312yes—
63#13133175—28no—
64#13133175—28no—
65#13133175—24no—
66#13133175—24no—
67#13133175—24no—
68#13133175D418yes—
69C2×C4⋊Dic53—24no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2312yes—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C4⋊C4×D53—24no—
77C4⋊C4×D53—24no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—28no—
80C2×#11719921—22no—
81C2×#11719921—22no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C22⋊C4×D53—44no—
85C2×C4×Dic53—28no—
86C2×C4×Dic53—22no—
87C2×C4×Dic53—22no—
88#13133151—24no—
89#13133150—24no—
90#13133148—24no—
91C22⋊#88322C2×C414yescomplemented
92#13133129C2×C418yescomplemented
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96C23×C4×C53—21no—
97C23×C4×C53C2311yescharacteristic
98C22×C8×C53C2312yes—
99C2×C4⋊C4×C53—24no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2312yes—
104C2×C22⋊C4×C53—21no—
105#106×C53—24no—
106C22⋊C8×C53C2×C414yescomplemented
107#95×C53—21no—
108C24×C53C2×D411yescharacteristic
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—22no—
113C23×D53—41no—
114C23×D53—41no—
115C23×D53C4⋊C414yes—
116C23×D53C2×D411yescharacteristic
117C23×D53C2×D411yescharacteristic
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—24no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×Q812yes—
129C22×C4×C53C2411yescharacteristic
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—24no—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—42no—
137C22×Dic53—42no—
138C22×Dic53C4⋊C414yes—
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—216no—
148C2×C4×D53—216no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×C4×D53C4⋊C414yes—
156C2×C4×D53C4⋊C414yes—
157C2×C4×D53C22×C412yes—
158C2×C4×D53C22×C412yes—
159C2×C4×D53C2×D414yes—
160C2×C4×D53C2×Q812yes—
161C2×C4×D53C2×Q812yes—
162C2×C4×D53#4114yes—
163C2×C8×C53—22no—
164C2×C8×C53—22no—
165C2×C8×C53C22×C414yes—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53C2×D414yes—
171C22⋊C4×C53—42no—
172C22⋊Dic53—42no—
173#11719921—28no—
174#11719921—28no—
175#11719921—42no—
176#11719921—42no—
177C4⋊Dic53—24no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181C4⋊Dic53C2×D414yes—
182#11719919—42no—
183#11719919—42no—
184C4×Dic53—28no—
185C4×Dic53—28no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322C22×C414yes—
189#11719912—216no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53C2×C4⋊C412yes—
195C23×C53AGL(1,ℤ/8)12yes—
196C23×C53#13712yes—
197C23×C53C22×D411yescharacteristic
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53—21no—
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—28no—
209C22×D53—28no—
210C22×D53—44no—
211C22×D53—44no—
212C22×D53—44no—
213C22×D53—44no—
214C22×D53—44no—
215C22×D53C2×C4⋊C412yes—
216C22×D53C2×C4⋊C412yes—
217C22×D53#12012yes—
218C22×D53#12312yes—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—48no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
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×C53C2×C4⋊C412yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C22×D412yes—
236C2×C4×C53C22×Q811yescharacteristic
237C2×C4×C53C2×#4112yes—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—28no—
241C2×Dic53—24no—
242C2×Dic53—48no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53C2×C4⋊C412yes—
252C2×Dic53C2×C4⋊C412yes—
253C2×Dic53#12012yes—
254C2×Dic53#12312yes—
255C4×D53—28no—
256C4×D53—24no—
257C4×D53—24no—
258C4×D53—28no—
259C4×D53—84no—
260C4×D53—84no—
261C4×D53C4×D418yes—
262C4×D53C22⋊Q818yes—
263C8×C53—24no—
264#88322—24no—
265#20846—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53#42814yescomplemented
273C22×C53C22×C4⋊C411yescharacteristic
274C22×C53C2×#12011yescharacteristic
275C22×C53C2×#12311yescharacteristic
276C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C53C2×#13711yescharacteristic
278C22×C53#57511yescharacteristic
279C22×C53#57511yescharacteristic
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53—22no—
285C2×D53—22no—
286C2×D53—22no—
287C2×D53—22no—
288C2×D53—22no—
289C2×D53—416no—
290C2×D53—41no—
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—41no—
294C2×D53#38514yes—
295C4×C53—22no—
296C4×C53—44no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53C2×C4×D412yes—
300C4×C53C2×C22⋊Q812yes—
301Dic53—22no—
302Dic53—44no—
303Dic53—81no—
304Dic53—81no—
305Dic53#38514yes—
306C22×#117—531no—
307C2×#428—532no—
308C2×#407—531no—
309C2×#380—531no—
310#1854—532no—
311#1854—532no—
312#1853—531no—
313#1853D5311yescharacteristic, complemented
314#1852—531no—
315#1852—531no—
316#1736—532no—
317C2×C53—24no—
318C2×C53C2×#38511yescharacteristic
319C2×C53C2×#42811yescharacteristic
320C2×C53#291111yescharacteristic
321C2×C53#300611yescharacteristic
322C2×C53#300711yescharacteristic
323C2×C53#309411yescharacteristic
324C2×C53#309511yescharacteristic
325D53—24no—
326D53—42no—
327C24×C4—531no—
328C22×M16—532no—
329C2×#117—532no—
330C2×#117—532no—
331C2×#117—1068no—
332C2×#117—1061no—
333C2×#117—1061no—
334C22×C4⋊C4—531no—
335C22×C4⋊C4—531no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4C2×D5311yescharacteristic
338C22×C22⋊C4—1061no—
339C22×C22⋊C4—1061no—
340C22×C42—1061no—
341#428—1068no—
342C2×#107—532no—
343C2×#107—532no—
344C2×#106—532no—
345C2×#106C2×D5312yescomplemented
346#407—538no—
347C2×C22⋊C8—531no—
348C2×C22⋊C8—532no—
349C2×C22⋊C8C2×D5311yescharacteristic
350#380—531no—
351#380—531no—
352#380—531no—
353#380—1061no—
354#380—1061no—
355#380C2×D5311yescharacteristic
356C2×#95—1061no—
357#340—532no—
358#340—532no—
359#340—532no—
360#340C2×D5312yescomplemented
361C53#2084611yescharacteristic, complemented, Sylow
362C25—531no—
363C23×C4—21no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—534no—
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—534no—
370C23×C4—531no—
371C23×C4—1064no—
372C23×C4—1064no—
373C23×C4—1061no—
374C23×C4—1061no—
375C23×C4—1061no—
376C23×C4C22×D5311yescharacteristic
377C2×M16—538no—
378C2×M16—1064no—
379C2×M16—1064no—
380C22×C8—532no—
381C22×C8C22×D5312yes—
382#117—538no—
383#117—1064no—
384#117—1064no—
385#117—1068no—
386#117—1068no—
387#117—1064no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—532no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1064no—
394C2×C4⋊C4—1064no—
395C2×C4⋊C4—1064no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5312yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—1062no—
405C2×C22⋊C4—1068no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—2124no—
410C2×C42—1062no—
411C2×C42—1062no—
412C2×C42—1068no—
413#107—1064no—
414#107—1064no—
415#106—24no—
416#106—1064no—
417C22⋊C8—534no—
418C22⋊C8—538no—
419C22⋊C8C4×D5314yescomplemented
420#95—21no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424C24—531no—
425C24—531no—
426C24—534no—
427C24—1062no—
428C24—1062no—
429C24—1062no—
430C24—1062no—
431C24—2121no—
432C24—2121no—
433C24D4×D5311yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—531no—
443C22×C4—532no—
444C22×C4—534no—
445C22×C4—532no—
446C22×C4—534no—
447C22×C4—534no—
448C22×C4—531no—
449C22×C4—534no—
450C22×C4—534no—
451C22×C4—532no—
452C22×C4—532no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—10616no—
456C22×C4—10616no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1064no—
464C22×C4—1062no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4C2×C4×D5312yes—
476C22×C4D4×D5311yescharacteristic
477C22×C4Q8×D5312yes—
478C22×C4C23×D5311yescharacteristic
479M16—10616no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—534no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8C2×C4×D5314yes—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—534no—
491C4⋊C4—1064no—
492C4⋊C4—1062no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4D4×D5314yes—
498C22⋊C4—42no—
499C22⋊C4—1068no—
500C22⋊C4—1068no—
501C22⋊C4—2122no—
502C22⋊C4—2122no—
503C22⋊C4—2122no—
504C42—1068no—
505C42—1068no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—532no—
511C23—532no—
512C23—532no—
513C23—532no—
514C23—1068no—
515C23—1061no—
516C23—1061no—
517C23—1061no—
518C23—1061no—
519C23—1061no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23—1061no—
525C23—1068no—
526C23—2124no—
527C23—2124no—
528C23—2124no—
529C23—2124no—
530C23—2124no—
531C23C4⋊C4×D5312yes—
532C23#1313324412yes—
533C23#1313324612yes—
534C23C2×D4×D5311yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—538no—
553C2×C4—538no—
554C2×C4—532no—
555C2×C4—1068no—
556C2×C4—1068no—
557C2×C4—1068no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1064no—
563C2×C4—4244no—
564C2×C4—4244no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2128no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4C4⋊C4×D5312yes—
575C2×C4C22×C4×D5311yescharacteristic
576C2×C4C2×D4×D5312yes—
577C2×C4C2×Q8×D5311yescharacteristic
578C2×C4#41×D5312yes—
579C8—24no—
580C8—1064no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—534no—
588C22—1062no—
589C22—1062no—
590C22—1062no—
591C22—1062no—
592C22—1062no—
593C22—1062no—
594C22—1062no—
595C22—1062no—
596C22—1062no—
597C22—21216no—
598C22—2121no—
599C22—2121no—
600C22—2121no—
601C22—2121no—
602C22#1681066214yescomplemented
603C22C2×C4⋊C4×D5311yescharacteristic
604C22#120×D5311yescharacteristic
605C22#123×D5311yescharacteristic
606C22C2×#1313324411yescharacteristic
607C22C2×#1313324611yescharacteristic
608C22#1681141911yescharacteristic
609C22#1681142211yescharacteristic
610C4—22no—
611C4—44no—
612C4—81no—
613C4—81no—
614C4—534no—
615C4—1062no—
616C4—4241no—
617C4—4241no—
618C4—2124no—
619C4C4×D4×D5312yes—
620C4C22⋊Q8×D5312yes—
621C2—24no—
622C2—1064no—
623C2—2122no—
624C2#385×D5311yescharacteristic
625C2C2×#1681066211yescharacteristic
626C2#2028330211yescharacteristic
627C2#2028396611yescharacteristic
628C2#2028397011yescharacteristic
629C2#2028465211yescharacteristic
630C2#2028465811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1853C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1853D53
hypocenterC53#20846
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#16810662C2semidirect
5. #20274917C2semidirect
6. #20274913C2semidirect
7. #20274908C2semidirect
8. #20274905C2semidirect
9. #20274902C2semidirect
10. #20274433C2semidirect
11. C53×#1853C2semidirect
26. C2×#13133151C22semidirect
27. C2×#13133150C22semidirect
28. C2×#13133148C22semidirect
29. #16810585C4semidirect
38. #16810386C22semidirect
39. #16810385C22semidirect
40. #16810385C22semidirect
42. C2×#106×C53C22semidirect
45. C53×#340C22semidirect
61. C2×#11719912C2×C4semidirect
91. C22⋊#88322C2×C4semidirect
92. #13133129C2×C4semidirect
106. C22⋊C8×C53C2×C4semidirect
272. C22×C53#428semidirect
313. #1853D53semidirect
345. C2×#106C2×D53semidirect
360. #340C2×D53semidirect
361. C53#20846semidirect
419. C22⋊C8C4×D53semidirect
602. C22#16810662semidirect

Maximal subgroups

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

subgroupindex
C22×#131331752
C2×#168106622
C2×#168105852
#380×D532
#202749172
#202749132
#202749082
#202749052
#202749022
#202744332
C53×#18532
#2084653