Finite Group Databasebeta

← #27384984

Subgroups — #27384984

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×#16810665C211yesmaximal, characteristic, complemented
3C2×#16810639C211yesmaximal, characteristic
4C2×#16810625C211yesmaximal, characteristic
5#20277335C211yesmaximal, characteristic, complemented
6#20274951C214yesmaximal, complemented
7#20274944C214yesmaximal, complemented
8#20274426C211yesmaximal, characteristic, complemented
9C53×#1742C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11#16811104C2212yes—
12#16811104C2212yes—
13#16811103—21no—
14#16811103—21no—
15#16811100—21no—
16#16811097—21no—
17#16811096—22no—
18C4×#11719945—22no—
19#16811091—22no—
20C2×#13133176C2211yescharacteristic
21C2×#13133176C2211yescharacteristic
22#117×D53—22no—
23C22×C4⋊Dic53C2211yescharacteristic
24#16810665—22no—
25#16810665C2214yescomplemented
26C2×#13133151—21no—
27C2×#13133149—21no—
28#16810639C418yescomplemented
29C2×#13133143C2214yes—
30#16810625—22no—
31#16810625C2214yes—
32C2×#13133142—21no—
33C2×#13133139—21no—
34#16810521—22no—
35#16810495—22no—
36#16810492C2212yescomplemented
37#16810488C2212yescomplemented
38#16810384C2214yescomplemented
39C22×M16×C53C2211yescharacteristic
40C22×M16×C53C2211yescharacteristic
41C2×#117×C53C2211yescharacteristic
42C53×#343C2212yescomplemented
43C53×#343C2212yescomplemented
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—
55C23×Dic53—21no—
56C22×C4×D53—21no—
57C4×Dic106—22no—
58C4×Dic106—22no—
59#13133182—22no—
60#13133182—22no—
61C4×D212—22no—
62C4×D212—22no—
63#13133179—22no—
64#13133179—22no—
65C4×C22⋊D53—24no—
66C4×C22⋊D53—24no—
67#13133177—24no—
68#13133177—24no—
69#13133176—21no—
70#13133176—21no—
71#13133176—22no—
72#13133176C2×C414yescomplemented
73#13133176D412yes—
74#13133176C2312yes—
75#13133175—22no—
76#13133175—22no—
77#13133174—24no—
78#13133173—24no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53D411yescharacteristic
86C2×C4⋊Dic53D411yescharacteristic
87C2×C4⋊Dic53C2312yes—
88C2×#11719919—24no—
89C4⋊C4×D53—22no—
90C4⋊C4×D53—22no—
91C2×C22⋊Dic53—21no—
92C2×C22⋊Dic53—21no—
93C22⋊C4×D53—24no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53C2312yes—
96C42×D53—24no—
97#13133151—24no—
98#13133149—24no—
99#13133143C2×C4116yescomplemented
100#13133142—24no—
101#13133139—24no—
102#13133124—24no—
103C23×C4×C53C2311yescharacteristic
104C2×M16×C53—21no—
105C2×M16×C53—21no—
106C2×M16×C53—22no—
107C2×M16×C53—22no—
108C2×M16×C53—22no—
109C2×M16×C53C2×C414yescomplemented
110C2×M16×C53C2312yes—
111C2×M16×C53C2312yes—
112C22×C8×C53—21no—
113C22×C8×C53—21no—
114#117×C53—21no—
115#117×C53—21no—
116#117×C53—21no—
117#117×C53—21no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53D412yes—
121C2×C42×C53D412yes—
122C24×C53C2×D411yescharacteristic
123C23×D53—21no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×Q812yes—
133C22×C4×C53#4112yes—
134C22×C4×C53#4112yes—
135C22×C4×C53C2411yescharacteristic
136C2×C22⋊D53—24no—
137C2×C22⋊D53—22no—
138C2×C22⋊D53—24no—
139C2×C22⋊D53—22no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—22no—
145C22×Dic53—41no—
146C22×Dic53—41no—
147C22×Dic53—82no—
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—
172M16×C53—24no—
173M16×C53—22no—
174M16×C53—22no—
175M16×C53—22no—
176M16×C53—42no—
177M16×C53C22×C414yes—
178C2×C8×C53—24no—
179C2×C8×C53—22no—
180C2×C8×C53—22no—
181C2×C8×C53—42no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—22no—
188C42×C53—22no—
189C42×C53—22no—
190C22⋊Dic53—24no—
191C22⋊Dic53—22no—
192C22⋊Dic53—22no—
193#11719921—24no—
194#11719921—24no—
195C4⋊Dic53—24no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4⋊Dic53—41no—
201C4⋊Dic53—41no—
202C4⋊Dic53C2×D412yes—
203C4⋊Dic53C2×D412yes—
204#11719919—24no—
205#11719919—24no—
206#11719919—84no—
207C4×Dic53—24no—
208C4×Dic53—24no—
209C4×Dic53—22no—
210C4×Dic53—22no—
211C4×Dic53—22no—
212C4×Dic53C22×C414yes—
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—22no—
225C2×C4×C53—22no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53C2×C4⋊C412yes—
229C2×C4×C53C2×C4⋊C412yes—
230C2×C4×C53C4×D412yes—
231C2×C4×C53C4×D412yes—
232C2×C4×C53C4×D412yes—
233C2×C4×C53#12011yescharacteristic
234C2×C4×C53#12011yescharacteristic
235C2×C4×C53C22⋊Q812yes—
236C2×C4×C53C22⋊Q812yes—
237C2×C4×C53C22⋊Q812yes—
238C2×C4×C53#12312yes—
239C2×C4×C53#12312yes—
240C2×C4×C53C23×C411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×Q811yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C22⋊D53—24no—
247C22⋊D53—24no—
248C22⋊D53—28no—
249C2×Dic53—22no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—44no—
257C2×Dic53—42no—
258C2×Dic53—81no—
259C2×Dic53—81no—
260C2×Dic53—82no—
261C2×Dic53—82no—
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—
273C8×C53—24no—
274C8×C53—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—42no—
294C4×C53—42no—
295C4×C53#38512yes—
296C4×C53#38512yes—
297C4×C53C22×C4⋊C411yescharacteristic
298C4×C53C2×C4×D411yescharacteristic
299C4×C53C2×C22⋊Q811yescharacteristic
300C4×C53C2×#12311yescharacteristic
301Dic53—24no—
302Dic53—44no—
303Dic53—42no—
304Dic53—42no—
305Dic53—82no—
306#3428—531no—
307C2×#428—531no—
308C2×#421—531no—
309C2×#417—531no—
310#2317—531no—
311#1860—534no—
312#1859—534no—
313#1742D5311yescharacteristic, complemented
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×M16C2×D5311yescharacteristic
325C22×M16C2×D5311yescharacteristic
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—531no—
337C2×#117—1062no—
338C2×#117C2×D5311yescharacteristic
339C22×C4⋊C4—531no—
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—1062no—
351#385—1062no—
352#384—532no—
353#382—532no—
354#343C2×D5312yescomplemented
355#343C2×D5312yescomplemented
356#339—534no—
357C53#2086511yescharacteristic, complemented, Sylow
358C2×#41—534no—
359C2×#41—532no—
360C2×#41—534no—
361C2×#41—532no—
362C2×#41—1061no—
363C2×#41—1062no—
364C2×#41—1061no—
365C2×#41—1062no—
366C22×Q8—531no—
367C22×D4—531no—
368C22×D4—1061no—
369C23×C4—1061no—
370C23×C4—1061no—
371C23×C4C22×D5311yescharacteristic
372C2×M16—21no—
373C2×M16—22no—
374C2×M16—21no—
375C2×M16—22no—
376C2×M16—22no—
377C2×M16C4×D5314yescomplemented
378C2×M16C22×D5312yes—
379C2×M16C22×D5312yes—
380C22×C8—21no—
381C22×C8—21no—
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—21no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—534no—
399#117—532no—
400#117—532no—
401#117—1064no—
402#117—1064no—
403#117—1062no—
404#117—1062no—
405#117—1061no—
406#117—1062no—
407#117—1061no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—532no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1064no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C22⋊C4—1064no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4D21212yes—
426C2×C42—532no—
427C2×C42—1064no—
428C2×C42—1061no—
429C2×C42D21212yes—
430#107—1064no—
431#106—1064no—
432#104—5316no—
433#103—1064no—
434#102—1064no—
435#95—1064no—
436C24—1061no—
437C24C2×D21211yescharacteristic
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—1064no—
461C22×C4—1064no—
462C22×C4—1061no—
463C22×C4—1064no—
464C22×C4—1062no—
465C22×C4—1064no—
466C22×C4—1062no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—4242no—
472C22×C4—2121no—
473C22×C4—2121no—
474C22×C4C2×C4×D5312yes—
475C22×C4C2×D21212yes—
476C22×C4C2×D21212yes—
477C22×C4C2×D21211yescharacteristic
478C22×C4#1171994512yes—
479C22×C4D4×D5311yescharacteristic
480C22×C4D4×D5311yescharacteristic
481C22×C4D4×D5311yescharacteristic
482C22×C4D4×D5311yescharacteristic
483C22×C4#1171994712yes—
484C22×C4Q8×D5312yes—
485C22×C4C23×D5311yescharacteristic
486M16—24no—
487M16—22no—
488M16—22no—
489M16—22no—
490M16—42no—
491M16C2×C4×D5314yes—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—42no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—532no—
501C4⋊C4—532no—
502C4⋊C4—1064no—
503C4⋊C4—1064no—
504C4⋊C4—1064no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1062no—
508C4⋊C4—1062no—
509C4⋊C4—4244no—
510C4⋊C4—2121no—
511C4⋊C4—2121no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—1064no—
515C22⋊C4—1064no—
516C22⋊C4—1064no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C42—22no—
520C42—22no—
521C42—534no—
522C42—1064no—
523C42—1064no—
524C42—1062no—
525C42—1062no—
526C42—1062no—
527C23—41no—
528C23—41no—
529C23—1064no—
530C23—1062no—
531C23—4241no—
532C23—4241no—
533C23C4×D21212yes—
534C23#1313318711yescharacteristic
535C23#1313318711yescharacteristic
536C23#1313319712yes—
537C23C22×D21211yescharacteristic
538Q8—534no—
539Q8—1064no—
540Q8—1062no—
541D4—534no—
542D4—1068no—
543D4—1064no—
544D4—1064no—
545D4—1064no—
546D4—1062no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—1064no—
552C2×C4—1062no—
553C2×C4—1068no—
554C2×C4—1064no—
555C2×C4—1064no—
556C2×C4—1064no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—4244no—
560C2×C4—4244no—
561C2×C4—4242no—
562C2×C4—4242no—
563C2×C4—4241no—
564C2×C4—4241no—
565C2×C4—2124no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4C4⋊C4×D5312yes—
570C2×C4C4⋊C4×D5312yes—
571C2×C4#1313317712yes—
572C2×C4#1313317712yes—
573C2×C4C4×D21212yes—
574C2×C4#1313318711yescharacteristic
575C2×C4#1313318711yescharacteristic
576C2×C4#1313319512yes—
577C2×C4#1313319512yes—
578C2×C4#1313319712yes—
579C2×C4#1313319912yes—
580C2×C4#1313319912yes—
581C2×C4C22×C4×D5311yescharacteristic
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C2×D4×D5311yescharacteristic
584C2×C4C2×Q8×D5311yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4C2×#1171994511yescharacteristic
587C8—24no—
588C8—42no—
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#1681049712yes—
602C22#1681049712yes—
603C22C2×C4×D21211yescharacteristic
604C22C2×#1313318711yescharacteristic
605C22C2×#1313319711yescharacteristic
606C4—42no—
607C4—42no—
608C4—1064no—
609C4—4242no—
610C4—2124no—
611C4—2122no—
612C4—2122no—
613C4#1681049712yes—
614C4#1681049712yes—
615C4C2×C4⋊C4×D5311yescharacteristic
616C4C2×#1313317711yescharacteristic
617C4C2×#1313319511yescharacteristic
618C4C2×#1313319911yescharacteristic
619C2—21no—
620C2—22no—
621C2—22no—
622C2—21no—
623C2—2124no—
624C2C2×#1681049711yescharacteristic
625C2#2028408911yescharacteristic
626C2#2028409611yescharacteristic
subgroupG/H
centerC22C2×#13133187
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1742C2
socleC22×C53C2×#120
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#1742D53
hypocenterC53#20865
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. C2×#16810665C2semidirect
5. #20277335C2semidirect
6. #20274951C2semidirect
7. #20274944C2semidirect
8. #20274426C2semidirect
9. C53×#1742C2semidirect
25. #16810665C22semidirect
28. #16810639C4semidirect
36. #16810492C22semidirect
37. #16810488C22semidirect
38. #16810384C22semidirect
42. C53×#343C22semidirect
43. C53×#343C22semidirect
72. #13133176C2×C4semidirect
99. #13133143C2×C4semidirect
109. C2×M16×C53C2×C4semidirect
155. #11719945C4⋊C4semidirect
160. C2×D212C4⋊C4semidirect
170. C2×Dic106C4⋊C4semidirect
313. #1742D53semidirect
354. #343C2×D53semidirect
355. #343C2×D53semidirect
357. C53#20865semidirect
377. C2×M16C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202879162
C2×#168106652
C2×#168106392
C2×#168106252
#202773352
#202749512
#202749442
#202744262
C53×#17422
#2086553