Finite Group Databasebeta

← #27289620

Subgroups — #27289620

Order 13568.

1204 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287916C211yesmaximal, characteristic
2C2×#16810641C211yesmaximal, characteristic
3C2×#16810641C211yesmaximal, characteristic
4C2×#16810393C211yesmaximal, characteristic, complemented
5#20274424C211yesmaximal, characteristic, complemented
6C2×#16810379C211yesmaximal, characteristic, complemented
7C53×#1740C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11#16811104—22no—
12#16811103—22no—
13#16811100C412yescomplemented
14#16811097C412yescomplemented
15#16811096—22no—
16C4×#11719945—22no—
17#16811091—22no—
18C2×#13133176C2211yescharacteristic
19#117×D53—22no—
20#16810641—22no—
21#16810641—22no—
22#16810641—22no—
23#16810641C414yescomplemented
24C2×#13133145—22no—
25C2×#13133145C414yescomplemented
26#16810393—22no—
27#16810382—22no—
28#16810379—22no—
29C2×#117×C53C2211yescharacteristic
30C2×#117×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C53×#342—22no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945C2×C412yes—
40C2×#11719945C2×C412yes—
41C2×#11719945D412yes—
42C2×#11719945D412yes—
43C2×#11719945D412yes—
44C2×#11719945Q812yes—
45C22×Dic106C2×C411yescharacteristic
46C22×D212C2×C411yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917C2×C412yescomplemented
60C2×#11719917C2×C412yescomplemented
61C22×#88322—21no—
62C22×#88322—21no—
63C4×Dic106—24no—
64#13133182—24no—
65C4×D212—24no—
66#13133179—24no—
67C4×C22⋊D53—28no—
68#13133177—28no—
69#13133176—21no—
70#13133176—21no—
71#13133176C2×C412yes—
72#13133176D412yes—
73#13133175—24no—
74#13133174—24no—
75#13133173—24no—
76C2×C4⋊Dic53—21no—
77C4⋊C4×D53—24no—
78C2×C22⋊Dic53—21no—
79C22⋊C4×D53—24no—
80C2×C4×Dic53D412yes—
81C42×D53—24no—
82#13133145—28no—
83#13133145—28no—
84C23×C4×C53C2311yescharacteristic
85#117×C53—21no—
86#117×C53—21no—
87#117×C53—21no—
88#117×C53—21no—
89#117×C53—21no—
90#117×C53—21no—
91#117×C53—21no—
92#117×C53—21no—
93#117×C53—21no—
94#117×C53C2×C412yes—
95#117×C53C2×C412yescomplemented
96#117×C53D412yes—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53D412yes—
103C2×C42×C53—21no—
104C2×C42×C53—21no—
105C2×C42×C53C2×C412yescomplemented
106C24×C53C2×D411yescharacteristic
107C23×D53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—41no—
111C22×C4×C53C22×C411yescharacteristic
112C22×C4×C53C22×C411yescharacteristic
113C22×C4×C53C22×C411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×Q811yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53—22no—
129C2×C22⋊D53—22no—
130C2×C22⋊D53—22no—
131C22×Dic53—21no—
132C22×Dic53—22no—
133#11719945—22no—
134#11719945—22no—
135#11719945—22no—
136#11719945—22no—
137#11719945—22no—
138#11719945—22no—
139#11719945—22no—
140#11719945—22no—
141#11719945—22no—
142#11719945—22no—
143#11719945—22no—
144#11719945—22no—
145#11719945C22⋊C414yes—
146#11719945C22⋊C414yes—
147#11719945C4⋊C414yes—
148#11719945C4⋊C414yes—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212C4212yes—
153C2×D212C22⋊C412yes—
154C2×D212C4⋊C412yes—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—28no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106C4212yes—
166C2×Dic106C22⋊C412yes—
167C2×Dic106C4⋊C412yes—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—41no—
172C4⋊C4×C53—41no—
173C4⋊C4×C53—41no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—41no—
177C22⋊C4×C53—41no—
178C22⋊C4×C53—41no—
179C22⋊C4×C53C22⋊C414yes—
180C42×C53—22no—
181C42×C53—22no—
182C42×C53—22no—
183C42×C53—22no—
184C42×C53—22no—
185C42×C53—41no—
186C22⋊Dic53—22no—
187C22⋊Dic53—22no—
188#11719921—28no—
189C4⋊Dic53—22no—
190C4⋊Dic53—22no—
191#11719919—28no—
192C4×Dic53—28no—
193C4×Dic53—22no—
194C4×Dic53C22⋊C414yes—
195#11719917—22no—
196#11719917—22no—
197#11719917—22no—
198#11719917—22no—
199#11719917—41no—
200#11719917—41no—
201#11719917—41no—
202#11719917—41no—
203C2×#88322—22no—
204C2×#88322—22no—
205C2×#88322—41no—
206C2×#88322—41no—
207C2×#88322—41no—
208C2×#88322—41no—
209C23×C53—81no—
210C23×C53C2×C22⋊C411yescharacteristic
211C23×C53C4×D411yescharacteristic
212C23×C53C4×D411yescharacteristic
213C23×C53#12011yescharacteristic
214C23×C53#12011yescharacteristic
215C23×C53#12111yescharacteristic
216C23×C53C22⋊Q811yescharacteristic
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—82no—
221C2×C4×C53—24no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—82no—
225C2×C4×C53—82no—
226C2×C4×C53—82no—
227C2×C4×C53C2×C4211yescharacteristic
228C2×C4×C53C2×C22⋊C411yescharacteristic
229C2×C4×C53C2×C22⋊C411yescharacteristic
230C2×C4×C53C2×C22⋊C411yescharacteristic
231C2×C4×C53C2×C22⋊C411yescharacteristic
232C2×C4×C53C2×C4⋊C411yescharacteristic
233C2×C4×C53C2×C4⋊C411yescharacteristic
234C2×C4×C53C2×C4⋊C411yescharacteristic
235C2×C4×C53#11711yescharacteristic
236C2×C4×C53#11711yescharacteristic
237C2×C4×C53C4×D411yescharacteristic
238C2×C4×C53C4×D411yescharacteristic
239C2×C4×C53C4×D411yescharacteristic
240C2×C4×C53C4×D411yescharacteristic
241C2×C4×C53C4×D411yescharacteristic
242C2×C4×C53C4×D411yescharacteristic
243C2×C4×C53#12011yescharacteristic
244C2×C4×C53#12011yescharacteristic
245C2×C4×C53#12111yescharacteristic
246C2×C4×C53#12111yescharacteristic
247C2×C4×C53#12111yescharacteristic
248C2×C4×C53C22⋊Q811yescharacteristic
249C2×C4×C53C22⋊Q811yescharacteristic
250C2×C4×C53C22⋊Q811yescharacteristic
251C2×C4×C53#12311yescharacteristic
252C2×C4×C53#12311yescharacteristic
253C2×C4×C53#12311yescharacteristic
254C2×C4×C53#12311yescharacteristic
255C22⋊D53—24no—
256C22⋊D53—24no—
257C22⋊D53—24no—
258C22⋊D53—24no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—24no—
263C2×Dic53—82no—
264D212—22no—
265D212—22no—
266D212—22no—
267D212#9514yes—
268C4×D53—24no—
269C4×D53—24no—
270C4×D53—24no—
271C4×D53—24no—
272C4×D53—88no—
273Dic106—22no—
274Dic106—22no—
275Dic106—22no—
276Dic106#9514yes—
277#88322—42no—
278#88322—42no—
279#10783—531nomaximal, Sylow
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53C4×C22⋊C411yescharacteristic
288C22×C53#37911yescharacteristic
289C22×C53#38011yescharacteristic
290C22×C53#38511yescharacteristic
291C22×C53#38511yescharacteristic
292C22×C53#38611yescharacteristic
293C22×C53#38611yescharacteristic
294C2×D53—24no—
295C2×D53—44no—
296C2×D53—82no—
297C2×D53—82no—
298C4×C53—44no—
299C4×C53—82no—
300C4×C53—82no—
301C4×C53C2×#9511yescharacteristic
302C4×C53C4×C22⋊C411yescharacteristic
303C4×C53#38011yescharacteristic
304C4×C53#38111yescharacteristic
305C4×C53#38511yescharacteristic
306C4×C53#38511yescharacteristic
307C4×C53#38611yescharacteristic
308C4×C53#38611yescharacteristic
309Dic53—24no—
310Dic53—44no—
311#3428—531no—
312C2×#421—531no—
313C2×#421—531no—
314C2×#344—531no—
315#1740D5311yescharacteristic, complemented
316#1733—531no—
317C2×H(3,ℤ/4)—531no—
318C2×C53—21no—
319C2×C53—21no—
320C2×C53—21no—
321C2×C53—21no—
322C2×C53—21no—
323C2×C53—21no—
324C2×C53—21no—
325C2×C53C22⋊#9511yescharacteristic
326D53—44no—
327C22×#41—531no—
328C22×M16—531no—
329C22×M16—531no—
330#520—1062no—
331#520—1062no—
332#519—532no—
333#518—532no—
334#518—1062no—
335C4×#41—1062no—
336C4×#41—1062no—
337C2×#117—531no—
338C2×#117—1062no—
339C2×#117C2×D5311yescharacteristic
340C2×#117C2×D5311yescharacteristic
341C2×#117C2×D5311yescharacteristic
342#421—534no—
343#421—1062no—
344#421—1062no—
345#421—1062no—
346C2×#104—534no—
347C2×#104—1062no—
348#344—1062no—
349#342—22no—
350#339—1062no—
351H(3,ℤ/4)—1062no—
352C53#1078311yescharacteristic, complemented, Sylow
353C2×#41—532no—
354C2×#41—532no—
355C2×#41—532no—
356C2×#41—532no—
357C2×#41—532no—
358C2×#41—532no—
359C2×#41—1061no—
360C2×#41—1061no—
361C2×#41—1061no—
362C2×#41—1061no—
363C2×#41—1061no—
364C2×#41—1061no—
365C22×Q8—531no—
366C22×D4—531no—
367C22×D4—1061no—
368C23×C4—1061no—
369C23×C4C22×D5311yescharacteristic
370C2×M16—532no—
371C2×M16—532no—
372C2×M16—1061no—
373C2×M16—1061no—
374C2×M16—1061no—
375C2×M16—1061no—
376C2×M16—1061no—
377C2×M16—1061no—
378C2×M16—1061no—
379C2×M16—1061no—
380C2×M16—1061no—
381C2×M16—1061no—
382C22×C8—1061no—
383C22×C8—1061no—
384C4×Q8—1064no—
385C4×Q8—1064no—
386C4×D4—1064no—
387C4×D4—1064no—
388C4×D4—1068no—
389C4×D4—1068no—
390#117—21no—
391#117—21no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—532no—
400#117—532no—
401#117—1064no—
402#117—1064no—
403#117—1061no—
404#117—1061no—
405#117—1064no—
406#117C4×D5312yes—
407#117C4×D5312yescomplemented
408#117D21212yes—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—1064no—
413C2×C4⋊C4—1061no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1064no—
418C2×C22⋊C4C22⋊D5312yes—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—532no—
422C2×C42—1064no—
423C2×C42C4×D5312yescomplemented
424#104—1068no—
425#104—1068no—
426C24—1061no—
427C24C2×C22⋊D5311yescharacteristic
428#41—534no—
429#41—534no—
430#41—534no—
431#41—534no—
432#41—1062no—
433#41—1062no—
434#41—1062no—
435#41—1062no—
436#41—1062no—
437#41—1062no—
438#41—1062no—
439#41—1062no—
440#41—1062no—
441#41—1062no—
442#41—1062no—
443#41—1062no—
444C2×Q8—532no—
445C2×Q8—532no—
446C2×Q8—532no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×D4—532no—
451C2×D4—532no—
452C2×D4—532no—
453C2×D4—1062no—
454C2×D4—1062no—
455C2×D4—1062no—
456C2×D4—1062no—
457C2×D4—1062no—
458C2×D4—1062no—
459C2×D4—1061no—
460C2×D4—1061no—
461C2×D4—1061no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—1068no—
466C22×C4—1061no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4C2×C4×D5311yescharacteristic
475C22×C4C2×C4×D5311yescharacteristic
476C22×C4C2×C4×D5311yescharacteristic
477C22×C4C2×D21211yescharacteristic
478C22×C4#1171994511yescharacteristic
479C22×C4#1171994511yescharacteristic
480C22×C4D4×D5311yescharacteristic
481C22×C4D4×D5311yescharacteristic
482C22×C4D4×D5311yescharacteristic
483C22×C4D4×D5311yescharacteristic
484C22×C4D4×D5311yescharacteristic
485C22×C4#1171994711yescharacteristic
486C22×C4Q8×D5311yescharacteristic
487C22×C4#1171994911yescharacteristic
488M16—1062no—
489M16—1062no—
490M16—1062no—
491M16—1062no—
492M16—2121no—
493M16—2121no—
494M16—2121no—
495M16—2121no—
496C2×C8—1062no—
497C2×C8—1062no—
498C2×C8—2121no—
499C2×C8—2121no—
500C2×C8—2121no—
501C2×C8—2121no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—1068no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—1068no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C22⋊C4#1171992114yes—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—534no—
527C42—1068no—
528C42—1062no—
529C23—81no—
530C23—1062no—
531C23—1062no—
532C23—1062no—
533C23—4242no—
534C23C2×#1171992111yescharacteristic
535C23C4×C22⋊D5311yescharacteristic
536C23C4×C22⋊D5311yescharacteristic
537C23#1313318511yescharacteristic
538C23#1313318511yescharacteristic
539C23#1313318811yescharacteristic
540C23#1313319311yescharacteristic
541Q8—534no—
542Q8—1062no—
543Q8—1062no—
544Q8—1062no—
545D4—534no—
546D4—1064no—
547D4—1064no—
548D4—1064no—
549D4—1064no—
550D4—1062no—
551D4—1062no—
552D4—1062no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—1064no—
566C2×C4—1064no—
567C2×C4—4248no—
568C2×C4—4242no—
569C2×C4C42×D5311yescharacteristic
570C2×C4C22⋊C4×D5311yescharacteristic
571C2×C4C22⋊C4×D5311yescharacteristic
572C2×C4C22⋊C4×D5311yescharacteristic
573C2×C4C22⋊C4×D5311yescharacteristic
574C2×C4C4⋊C4×D5311yescharacteristic
575C2×C4C4⋊C4×D5311yescharacteristic
576C2×C4C4⋊C4×D5311yescharacteristic
577C2×C4#1313317411yescharacteristic
578C2×C4#1313317511yescharacteristic
579C2×C4#1313317711yescharacteristic
580C2×C4#1313317711yescharacteristic
581C2×C4#1313317911yescharacteristic
582C2×C4#1313317911yescharacteristic
583C2×C4C4×D21211yescharacteristic
584C2×C4C4×D21211yescharacteristic
585C2×C4#1313318711yescharacteristic
586C2×C4#1313318711yescharacteristic
587C2×C4#1313319011yescharacteristic
588C2×C4#1313319011yescharacteristic
589C2×C4#1313319211yescharacteristic
590C2×C4#1313319511yescharacteristic
591C2×C4#1313319511yescharacteristic
592C2×C4#1313319711yescharacteristic
593C2×C4#1313319911yescharacteristic
594C2×C4#1313319911yescharacteristic
595C2×C4#1313320211yescharacteristic
596C2×C4#1313320211yescharacteristic
597C8—2122no—
598C8—2122no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—1064no—
607C22—4242no—
608C22—4242no—
609C22—2124no—
610C22C4×#1171992111yescharacteristic
611C22#1681047811yescharacteristic
612C22#1681048011yescharacteristic
613C22#1681049411yescharacteristic
614C22#1681049411yescharacteristic
615C22#1681049911yescharacteristic
616C22#1681049911yescharacteristic
617C4—44no—
618C4—82no—
619C4—82no—
620C4—1064no—
621C4—2124no—
622C4#95×D5311yescharacteristic
623C4#1681047011yescharacteristic
624C4#1681048211yescharacteristic
625C4#1681048511yescharacteristic
626C4#1681049711yescharacteristic
627C4#1681049711yescharacteristic
628C4#1681050111yescharacteristic
629C4#1681050111yescharacteristic
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—2124no—
638C2#2027293611yescharacteristic
subgroupG/H
centerC4#16810501
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1740C2
socleC2×C53C22⋊#95
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenter#1740D53
hypocenterC53#10783
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. C2×#16810393C2semidirect
5. #20274424C2semidirect
6. C2×#16810379C2semidirect
7. C53×#1740C2semidirect
13. #16811100C4semidirect
14. #16811097C4semidirect
23. #16810641C4semidirect
25. C2×#13133145C4semidirect
59. C2×#11719917C2×C4semidirect
60. C2×#11719917C2×C4semidirect
95. #117×C53C2×C4semidirect
105. C2×C42×C53C2×C4semidirect
315. #1740D53semidirect
352. C53#10783semidirect
407. #117C4×D53semidirect
423. C2×C42C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202879162
C2×#168106412
C2×#168103932
#202744242
C2×#168103792
C53×#17402
#1078353