Finite Group Databasebeta

← #27384980

Subgroups — #27384980

Order 13568.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287916C211yesmaximal, characteristic
2C2×#16810663C211yesmaximal, characteristic, complemented
3C2×#16810641C211yesmaximal, characteristic
4C2×#16810622C211yesmaximal, characteristic
5#20274947C214yesmaximal, complemented
6#20274940C214yesmaximal, complemented
7#20274454C211yesmaximal, characteristic, complemented
8#20274425C211yesmaximal, characteristic, complemented
9C53×#2317C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13#16811104C2212yes—
14#16811104C2212yes—
15#16811103—21no—
16#16811103—21no—
17#16811100—21no—
18#16811097—21no—
19#16811096—22no—
20C4×#11719945—22no—
21#16811091—22no—
22C2×#13133176C2211yescharacteristic
23#117×D53—22no—
24#16810663—22no—
25#16810663C2214yescomplemented
26C2×#13133150—21no—
27C2×#13133148—21no—
28#16810641C418yescomplemented
29C2×#13133145C2214yes—
30#16810622—22no—
31#16810622C2214yes—
32C2×#13133140—21no—
33C2×#13133137—21no—
34#16810392C2212yescomplemented
35#16810392C2212yescomplemented
36#16810383C2214yescomplemented
37C2×#117×C53C2211yescharacteristic
38C2×#117×C53C2211yescharacteristic
39C22×C4⋊C4×C53C2211yescharacteristic
40C53×#389—22no—
41C53×#385—22no—
42C53×#384C2212yescomplemented
43C53×#382C2212yescomplemented
44C2×#11719945—21no—
45C2×#11719945—21no—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945C2×C414yes—
49C2×#11719945D412yes—
50C2×#11719945Q814yes—
51C2×#11719945C2312yes—
52C22×Dic106D411yescharacteristic
53C22×D212D411yescharacteristic
54C22×C22⋊D53—21no—
55C22×C4×D53—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—22no—
59C2×#11719917—22no—
60C2×#11719917—22no—
61C2×#11719917C2×C414yescomplemented
62C2×#11719917C2312yes—
63C2×#11719917C2312yes—
64C22×#88322—21no—
65C22×#88322—21no—
66C4×Dic106—22no—
67C4×Dic106—22no—
68#13133182—22no—
69#13133182—22no—
70C4×D212—22no—
71C4×D212—22no—
72#13133179—22no—
73#13133179—22no—
74C4×C22⋊D53—24no—
75C4×C22⋊D53—24no—
76#13133177—24no—
77#13133177—24no—
78#13133176—21no—
79#13133176—21no—
80#13133176D412yes—
81#13133176C2312yes—
82#13133175—22no—
83#13133175—22no—
84#13133174—24no—
85#13133173—24no—
86C2×C4⋊Dic53D411yescharacteristic
87C2×C4⋊Dic53D411yescharacteristic
88C4⋊C4×D53—22no—
89C4⋊C4×D53—22no—
90C2×C22⋊Dic53—21no—
91C22⋊C4×D53—24no—
92C2×C4×Dic53—21no—
93C42×D53—24no—
94#13133150—24no—
95#13133148—24no—
96#13133145C2×C4116yescomplemented
97#13133140—24no—
98#13133137—24no—
99C23×C4×C53—21no—
100C23×C4×C53C2311yescharacteristic
101#117×C53—21no—
102#117×C53—21no—
103#117×C53—21no—
104#117×C53—21no—
105#117×C53—22no—
106#117×C53C2×C414yescomplemented
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—24no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2312yes—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53D412yes—
119C2×C42×C53D412yes—
120C2×C42×C53C2312yes—
121#95×C53—24no—
122C24×C53C2×D411yescharacteristic
123C23×D53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—82no—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×Q812yes—
138C22×C4×C53#4112yes—
139C22×C4×C53#4112yes—
140C22×C4×C53C2411yescharacteristic
141C2×C22⋊D53—24no—
142C2×C22⋊D53—22no—
143C2×C22⋊D53—24no—
144C2×C22⋊D53—22no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148#11719945—24no—
149#11719945—24no—
150#11719945—24no—
151#11719945—24no—
152#11719945—24no—
153#11719945—22no—
154#11719945—22no—
155#11719945C4⋊C418yescomplemented
156#11719945C22×C414yes—
157#11719945C2×Q814yes—
158C2×D212—21no—
159C2×D212—22no—
160C2×D212C4⋊C414yescomplemented
161C2×D212C2×D412yes—
162C2×C4×D53—24no—
163C2×C4×D53—22no—
164C2×C4×D53—24no—
165C2×C4×D53—22no—
166C2×C4×D53—24no—
167C2×C4×D53—24no—
168C2×Dic106—21no—
169C2×Dic106—22no—
170C2×Dic106C4⋊C414yescomplemented
171C2×Dic106C2×D412yes—
172C4⋊C4×C53—24no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53—41no—
180C4⋊C4×C53—41no—
181C4⋊C4×C53—84no—
182C22⋊C4×C53—24no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C42×C53—22no—
186C42×C53—22no—
187C42×C53—22no—
188C42×C53C22×C414yes—
189C22⋊Dic53—22no—
190C22⋊Dic53—22no—
191#11719921—24no—
192#11719921—24no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53C2×D412yes—
196C4⋊Dic53C2×D412yes—
197#11719919—24no—
198#11719919—24no—
199C4×Dic53—24no—
200C4×Dic53—24no—
201C4×Dic53—22no—
202C4×Dic53—22no—
203#11719917—24no—
204#11719917—22no—
205#11719917—22no—
206#11719917—22no—
207#11719917—42no—
208#11719917C22×C414yes—
209C2×#88322—24no—
210C2×#88322—22no—
211C2×#88322—22no—
212C2×#88322—42no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53C4×D412yes—
216C23×C53#12011yescharacteristic
217C23×C53#12011yescharacteristic
218C23×C53C22⋊Q812yes—
219C23×C53C22×D411yescharacteristic
220C22×D53—22no—
221C22×D53—24no—
222C22×D53—81no—
223C22×D53—81no—
224C2×C4×C53—24no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—44no—
232C2×C4×C53—42no—
233C2×C4×C53—82no—
234C2×C4×C53—82no—
235C2×C4×C53C2×C4⋊C412yes—
236C2×C4×C53C2×C4⋊C412yes—
237C2×C4×C53C4×D412yes—
238C2×C4×C53C4×D412yes—
239C2×C4×C53C4×D412yes—
240C2×C4×C53#12011yescharacteristic
241C2×C4×C53#12011yescharacteristic
242C2×C4×C53C22⋊Q812yes—
243C2×C4×C53C22⋊Q812yes—
244C2×C4×C53C22⋊Q812yes—
245C2×C4×C53#12312yes—
246C2×C4×C53#12312yes—
247C2×C4×C53C23×C411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×Q811yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C22⋊D53—24no—
254C22⋊D53—24no—
255C22⋊D53—28no—
256C2×Dic53—22no—
257C2×Dic53—24no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—81no—
261C2×Dic53—81no—
262D212—22no—
263D212—24no—
264D212C2×C4⋊C414yes—
265C4×D53—24no—
266C4×D53—24no—
267C4×D53—28no—
268C4×D53—84no—
269C4×D53—84no—
270Dic106—22no—
271Dic106—24no—
272Dic106C2×C4⋊C414yes—
273#88322—24no—
274#88322—42no—
275#20865—531nomaximal, Sylow
276C22×C53—21no—
277C22×C53—21no—
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53#38512yes—
285C22×C53#38512yes—
286C22×C53C2×C4×D411yescharacteristic
287C22×C53C2×#12011yescharacteristic
288C22×C53C2×C22⋊Q811yescharacteristic
289C2×D53—24no—
290C2×D53—42no—
291C2×D53—42no—
292C2×D53—84no—
293C4×C53—44no—
294C4×C53—42no—
295C4×C53—42no—
296C4×C53—82no—
297C4×C53#38512yes—
298C4×C53#38512yes—
299C4×C53C22×C4⋊C411yescharacteristic
300C4×C53C2×C4×D411yescharacteristic
301C4×C53C2×C22⋊Q811yescharacteristic
302C4×C53C2×#12311yescharacteristic
303Dic53—24no—
304Dic53—42no—
305Dic53—42no—
306#3428—531no—
307C2×#428—531no—
308C2×#421—531no—
309C2×#417—531no—
310#2317D5311yescharacteristic, complemented
311#1860—534no—
312#1859—534no—
313#1742—531no—
314#1733—531no—
315C2×C53—21no—
316C2×C53—22no—
317C2×C53—21no—
318C2×C53—22no—
319C2×C53C2×#38511yescharacteristic
320C2×C53#302111yescharacteristic
321C2×C53#302211yescharacteristic
322D53—44no—
323C22×#41—531no—
324C22×M16—531no—
325C22×M16—531no—
326#520—532no—
327#520—532no—
328#520—1061no—
329#520—1061no—
330#519—1061no—
331#518—1061no—
332#518—1062no—
333C4×#41—1062no—
334C4×#41—1062no—
335C2×#117—531no—
336C2×#117—1062no—
337C2×#117C2×D5311yescharacteristic
338C2×#117C2×D5311yescharacteristic
339C22×C4⋊C4C2×D5311yescharacteristic
340#428—534no—
341#428—1062no—
342C2×#107—1061no—
343C2×#106—1061no—
344#421—538no—
345C2×#104—534no—
346#417—534no—
347#417—1062no—
348C2×#103—1061no—
349C2×#102—1061no—
350#389—22no—
351#385—22no—
352#384C2×D5312yescomplemented
353#382C2×D5312yescomplemented
354#343—532no—
355#343—532no—
356#339—534no—
357C53#2086511yescharacteristic, complemented, Sylow
358C2×#41—534no—
359C2×#41—532no—
360C2×#41—534no—
361C2×#41—532no—
362C2×#41—1062no—
363C2×#41—1061no—
364C2×#41—1062no—
365C2×#41—1061no—
366C22×Q8—531no—
367C22×D4—531no—
368C22×D4—1061no—
369C23×C4—21no—
370C23×C4—1061no—
371C23×C4C22×D5311yescharacteristic
372C2×M16—534no—
373C2×M16—532no—
374C2×M16—532no—
375C2×M16—1061no—
376C2×M16—1062no—
377C2×M16—1061no—
378C2×M16—1062no—
379C2×M16—1062no—
380C22×C8—1061no—
381C22×C8—1061no—
382C4×Q8—1062no—
383C4×Q8—1062no—
384C4×Q8—1062no—
385C4×Q8—1062no—
386C4×D4—1062no—
387C4×D4—1062no—
388C4×D4—1064no—
389C4×D4—1064no—
390C4×D4—1062no—
391C4×D4—1062no—
392C4×D4—1064no—
393C4×D4—1064no—
394#117—22no—
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—1062no—
404#117—1062no—
405#117—1061no—
406#117—1061no—
407#117C4×D5314yescomplemented
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—24no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—1062no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4C22×D5312yes—
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1064no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4C22⋊D5312yes—
426C2×C42—1064no—
427C2×C42—1061no—
428C2×C42C22⋊D5312yes—
429C2×C42C22×D5312yes—
430#107—1064no—
431#106—1064no—
432#104—5316no—
433#103—1064no—
434#102—1064no—
435#95—24no—
436C24—1061no—
437C24C2×C22⋊D5311yescharacteristic
438#41—538no—
439#41—534no—
440#41—534no—
441#41—1064no—
442#41—1064no—
443#41—1064no—
444#41—1064no—
445#41—1064no—
446#41—1062no—
447#41—1062no—
448C2×Q8—534no—
449C2×Q8—532no—
450C2×Q8—1061no—
451C2×Q8—1062no—
452C2×D4—534no—
453C2×D4—532no—
454C2×D4—1064no—
455C2×D4—1062no—
456C2×D4—1064no—
457C2×D4—1062no—
458C2×D4—1061no—
459C2×D4—1062no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—82no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—1061no—
468C22×C4—1064no—
469C22×C4—1062no—
470C22×C4—1064no—
471C22×C4—1062no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4C2×C4×D5312yes—
475C22×C4#1171994512yes—
476C22×C4D4×D5311yescharacteristic
477C22×C4D4×D5311yescharacteristic
478C22×C4D4×D5311yescharacteristic
479C22×C4D4×D5311yescharacteristic
480C22×C4Q8×D5312yes—
481C22×C4#1171994912yes—
482C22×C4C2×C22⋊D5312yes—
483C22×C4C2×C22⋊D5312yes—
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C23×D5311yescharacteristic
486M16—534no—
487M16—1064no—
488M16—1062no—
489M16—1062no—
490M16—1062no—
491M16—2122no—
492C2×C8—1064no—
493C2×C8—1062no—
494C2×C8—1062no—
495C2×C8—2122no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—84no—
506C4⋊C4—532no—
507C4⋊C4—532no—
508C4⋊C4—1064no—
509C4⋊C4—1064no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—1064no—
516C22⋊C4—1064no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—1064no—
523C42—1064no—
524C42—1062no—
525C42—1062no—
526C42C2×C4×D5314yes—
527C23—41no—
528C23—41no—
529C23—1064no—
530C23—1062no—
531C23—4241no—
532C23—4241no—
533C23C4×C22⋊D5312yes—
534C23#1313318511yescharacteristic
535C23#1313318511yescharacteristic
536C23#1313319312yes—
537C23C22×C22⋊D5311yescharacteristic
538Q8—534no—
539Q8—1062no—
540Q8—1064no—
541D4—534no—
542D4—1064no—
543D4—1064no—
544D4—1068no—
545D4—1062no—
546D4—1064no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—1062no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1068no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—4244no—
566C2×C4—4244no—
567C2×C4—4241no—
568C2×C4—4241no—
569C2×C4C4⋊C4×D5312yes—
570C2×C4C4⋊C4×D5312yes—
571C2×C4C4×C22⋊D5312yes—
572C2×C4#1313317912yes—
573C2×C4#1313317912yes—
574C2×C4#1313318511yescharacteristic
575C2×C4#1313318511yescharacteristic
576C2×C4#1313319312yes—
577C2×C4#1313319512yes—
578C2×C4#1313319512yes—
579C2×C4#1313320212yes—
580C2×C4#1313320212yes—
581C2×C4C22×C4×D5311yescharacteristic
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C2×D4×D5311yescharacteristic
584C2×C4C2×Q8×D5311yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4C2×#1171994911yescharacteristic
587C8—1064no—
588C8—2122no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—42no—
596C22—42no—
597C22—1064no—
598C22—4244no—
599C22—2122no—
600C22—2122no—
601C22#1681049412yes—
602C22#1681049412yes—
603C22C2×C4×C22⋊D5311yescharacteristic
604C22C2×#1313318511yescharacteristic
605C22C2×#1313319311yescharacteristic
606C4—44no—
607C4—42no—
608C4—42no—
609C4—82no—
610C4—1064no—
611C4—2122no—
612C4—2122no—
613C4#1681049412yes—
614C4#1681049412yes—
615C4C2×C4⋊C4×D5311yescharacteristic
616C4C2×#1313317911yescharacteristic
617C4C2×#1313319511yescharacteristic
618C4C2×#1313320211yescharacteristic
619C2—21no—
620C2—22no—
621C2—21no—
622C2—22no—
623C2—2124no—
624C2C2×#1681049411yescharacteristic
625C2#2028408611yescharacteristic
626C2#2028409311yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2317C2
socleC22×C53C2×#120
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#2317D53
hypocenterC53#20865
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C2×#16810663C2semidirect
5. #20274947C2semidirect
6. #20274940C2semidirect
7. #20274454C2semidirect
8. #20274425C2semidirect
9. C53×#2317C2semidirect
25. #16810663C22semidirect
28. #16810641C4semidirect
34. #16810392C22semidirect
35. #16810392C22semidirect
36. #16810383C22semidirect
42. C53×#384C22semidirect
43. C53×#382C22semidirect
61. C2×#11719917C2×C4semidirect
96. #13133145C2×C4semidirect
106. #117×C53C2×C4semidirect
155. #11719945C4⋊C4semidirect
160. C2×D212C4⋊C4semidirect
170. C2×Dic106C4⋊C4semidirect
310. #2317D53semidirect
352. #384C2×D53semidirect
353. #382C2×D53semidirect
357. C53#20865semidirect
407. #117C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202879162
C2×#168106632
C2×#168106412
C2×#168106222
#202749472
#202749402
#202744542
#202744252
C53×#23172
#2086553