Finite Group Databasebeta

← #27289598

Subgroups — #27289598

Order 13568.

1218 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287916C211yesmaximal, characteristic
2C2×#16810639C211yesmaximal, characteristic
3C2×#16810620C211yesmaximal, characteristic
4#20274448C211yesmaximal, characteristic, complemented
5#20274432C211yesmaximal, characteristic, complemented
6#20274426C211yesmaximal, characteristic, complemented
7C53×#1729C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9#16811104C414yescomplemented
10#16811103—22no—
11#16811100—21no—
12#16811097—21no—
13#16811096—22no—
14C4×#11719945—22no—
15#16811091—22no—
16C2×#13133176C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#117×D53—22no—
20#16810639—22no—
21#16810639C414yescomplemented
22C2×#13133143—22no—
23#16810620—22no—
24#16810620C414yescomplemented
25C2×#13133142—21no—
26C2×#13133139—21no—
27#16810390—22no—
28#16810386—22no—
29#16810384—22no—
30C22×M16×C53C2211yescharacteristic
31C23×C8×C53C2211yescharacteristic
32C2×#117×C53C2211yescharacteristic
33C53×#336—22no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945C2×C412yes—
41C2×#11719945C2×C412yes—
42C2×#11719945C2×C412yes—
43C2×#11719945D412yes—
44C2×#11719945D412yes—
45C2×#11719945Q812yes—
46C22×Dic106D411yescharacteristic
47C22×D212D411yescharacteristic
48C22×C22⋊D53—21no—
49C22×C4×D53—21no—
50C4×Dic106—24no—
51#13133182—24no—
52C4×D212—24no—
53#13133179—24no—
54C4×C22⋊D53—28no—
55#13133177—28no—
56#13133176—21no—
57#13133176—21no—
58#13133176—21no—
59#13133176—21no—
60#13133176—21no—
61#13133176—21no—
62#13133176—21no—
63#13133176—21no—
64#13133176C2×C412yes—
65#13133176C2×C412yescomplemented
66#13133176C2×C412yescomplemented
67#13133176D412yes—
68#13133175—24no—
69#13133174—24no—
70#13133173—24no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53D412yes—
74C4⋊C4×D53—24no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C22⋊C4×D53—24no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C42×D53—24no—
83#13133143—28no—
84#13133142—24no—
85#13133139—24no—
86C23×C4×C53C2311yescharacteristic
87C2×M16×C53—21no—
88C2×M16×C53—21no—
89C2×M16×C53—21no—
90C2×M16×C53—21no—
91C2×M16×C53—21no—
92C2×M16×C53C2×C412yescomplemented
93C22×C8×C53—21no—
94C22×C8×C53—21no—
95C22×C8×C53—21no—
96C22×C8×C53—21no—
97C22×C8×C53—21no—
98C22×C8×C53—21no—
99C22×C8×C53—21no—
100C22×C8×C53C2×C412yescomplemented
101#117×C53—21no—
102#117×C53—21no—
103#117×C53—21no—
104#117×C53—21no—
105C2×C4⋊C4×C53C2×C412yes—
106C2×C22⋊C4×C53D412yes—
107C2×C42×C53D412yes—
108C24×C53C2×D411yescharacteristic
109C23×D53—21no—
110C22×C4×C53C22⋊C414yes—
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—
133C22×Dic53—41no—
134C22×Dic53—41no—
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#11719945—22no—
146#11719945—22no—
147#11719945C4214yes—
148#11719945C22⋊C414yes—
149#11719945C4⋊C414yes—
150#11719945C4⋊C414yes—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212C22⋊C412yes—
155C2×D212C22⋊C412yes—
156C2×D212C4⋊C412yes—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—28no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106C22⋊C412yes—
168C2×Dic106C22⋊C412yes—
169C2×Dic106C4⋊C412yes—
170M16×C53—22no—
171M16×C53—22no—
172M16×C53—41no—
173M16×C53—41no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C2×C8×C53—41no—
179C2×C8×C53—41no—
180C2×C8×C53—41no—
181C2×C8×C53—41no—
182C2×C8×C53—41no—
183C2×C8×C53—41no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—22no—
188C42×C53—22no—
189C42×C53—22no—
190C22⋊Dic53—22no—
191C22⋊Dic53—22no—
192C22⋊Dic53—22no—
193C22⋊Dic53—22no—
194C22⋊Dic53—41no—
195C22⋊Dic53—41no—
196#11719921—28no—
197C4⋊Dic53—22no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4⋊Dic53—41no—
201C4⋊Dic53—41no—
202C4⋊Dic53C22⋊C414yes—
203#11719919—28no—
204C4×Dic53—28no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—22no—
208C4×Dic53—22no—
209C4×Dic53—41no—
210C4×Dic53—41no—
211C23×C53—42no—
212C23×C53C2×C22⋊C411yescharacteristic
213C23×C53C4×D411yescharacteristic
214C23×C53C4×D411yescharacteristic
215C23×C53#12011yescharacteristic
216C23×C53#12011yescharacteristic
217C23×C53#12111yescharacteristic
218C23×C53C22⋊Q811yescharacteristic
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—22no—
222C22×D53—82no—
223C2×C4×C53—24no—
224C2×C4×C53—44no—
225C2×C4×C53C2×C4211yescharacteristic
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×C4×C53C2×C22⋊C411yescharacteristic
229C2×C4×C53C2×C22⋊C411yescharacteristic
230C2×C4×C53C2×C4⋊C411yescharacteristic
231C2×C4×C53C2×C4⋊C411yescharacteristic
232C2×C4×C53C2×C4⋊C411yescharacteristic
233C2×C4×C53#11711yescharacteristic
234C2×C4×C53#11711yescharacteristic
235C2×C4×C53C4×D411yescharacteristic
236C2×C4×C53C4×D411yescharacteristic
237C2×C4×C53C4×D411yescharacteristic
238C2×C4×C53C4×D411yescharacteristic
239C2×C4×C53C4×D411yescharacteristic
240C2×C4×C53C4×D411yescharacteristic
241C2×C4×C53#12011yescharacteristic
242C2×C4×C53#12011yescharacteristic
243C2×C4×C53#12111yescharacteristic
244C2×C4×C53#12111yescharacteristic
245C2×C4×C53#12111yescharacteristic
246C2×C4×C53C22⋊Q811yescharacteristic
247C2×C4×C53C22⋊Q811yescharacteristic
248C2×C4×C53C22⋊Q811yescharacteristic
249C2×C4×C53#12311yescharacteristic
250C2×C4×C53#12311yescharacteristic
251C2×C4×C53#12311yescharacteristic
252C2×C4×C53#12311yescharacteristic
253C22⋊D53—24no—
254C22⋊D53—24no—
255C22⋊D53—24no—
256C22⋊D53—24no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—24no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—82no—
264C2×Dic53—82no—
265C2×Dic53—82no—
266D212—22no—
267D212—22no—
268D212—22no—
269D212#9514yes—
270C4×D53—24no—
271C4×D53—24no—
272C4×D53—24no—
273C4×D53—24no—
274C4×D53—88no—
275Dic106—22no—
276Dic106—22no—
277Dic106—22no—
278Dic106#9514yes—
279C8×C53—42no—
280C8×C53—42no—
281#10779—531nomaximal, Sylow
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53C4×C22⋊C411yescharacteristic
290C22×C53#37911yescharacteristic
291C22×C53#38011yescharacteristic
292C22×C53#38511yescharacteristic
293C22×C53#38511yescharacteristic
294C22×C53#38611yescharacteristic
295C22×C53#38611yescharacteristic
296C2×D53—24no—
297C2×D53—44no—
298C2×D53—82no—
299C2×D53—82no—
300C4×C53—44no—
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—
311Dic53—82no—
312Dic53—82no—
313#3428—531no—
314C2×#421—531no—
315C2×#416—531no—
316#1740—531no—
317#1735—531no—
318#1733—531no—
319#1729D5311yescharacteristic, complemented
320C2×C53—21no—
321C2×C53—21no—
322C2×C53—21no—
323C2×C53—21no—
324C2×C53—21no—
325C2×C53—21no—
326C2×C53C22⋊#9511yescharacteristic
327C2×C53#179412yes—
328D53—44no—
329C22×#41—531no—
330C22×M16C2×D5311yescharacteristic
331C23×C8C2×D5311yescharacteristic
332#520—534no—
333#520—1062no—
334#519—1061no—
335#518—1061no—
336#518—1062no—
337C4×#41—1062no—
338C4×#41—1062no—
339C2×#117—531no—
340C2×#117—531no—
341C2×#117—531no—
342C2×#117—1062no—
343C2×#117C2×D5311yescharacteristic
344#421—534no—
345#421—1062no—
346C2×#104—1062no—
347#416—534no—
348#416—1062no—
349C2×#103—1061no—
350C2×#102—1061no—
351#342—1062no—
352#340—1062no—
353#339—1062no—
354#336—22no—
355C53#1077911yescharacteristic, complemented, Sylow
356C2×#41—532no—
357C2×#41—532no—
358C2×#41—532no—
359C2×#41—532no—
360C2×#41—532no—
361C2×#41—532no—
362C2×#41—1061no—
363C2×#41—1061no—
364C2×#41—1061no—
365C2×#41—1061no—
366C2×#41—1061no—
367C2×#41—1061no—
368C22×Q8—531no—
369C22×D4—531no—
370C22×D4—1061no—
371C23×C4—1061no—
372C23×C4C22×D5311yescharacteristic
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16C4×D5312yescomplemented
379C22×C8—21no—
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—21no—
383C22×C8—21no—
384C22×C8—21no—
385C22×C8—21no—
386C22×C8C4×D5312yescomplemented
387C4×Q8—1064no—
388C4×Q8—1064no—
389C4×D4—1064no—
390C4×D4—1068no—
391C4×D4—1064no—
392C4×D4—1068no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—532no—
398#117—532no—
399#117—532no—
400#117—532no—
401#117—1064no—
402#117—1064no—
403#117—1064no—
404#117—1061no—
405#117—1061no—
406#117—1061no—
407#117—1061no—
408#117—1061no—
409#117—1061no—
410#117—1061no—
411#117—1061no—
412C2×C4⋊C4—532no—
413C2×C4⋊C4—1064no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4C4×D5312yes—
417C2×C22⋊C4—1064no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4D21212yes—
422C2×C42—1064no—
423C2×C42—1061no—
424C2×C42—1061no—
425C2×C42—1061no—
426C2×C42C22⋊D5312yes—
427#104—1068no—
428#103—1064no—
429#102—1064no—
430C24—1061no—
431C24C2×D21211yescharacteristic
432#41—534no—
433#41—534no—
434#41—534no—
435#41—534no—
436#41—1062no—
437#41—1062no—
438#41—1062no—
439#41—1062no—
440#41—1062no—
441#41—1062no—
442#41—1062no—
443#41—1062no—
444#41—1062no—
445#41—1062no—
446#41—1062no—
447#41—1062no—
448C2×Q8—532no—
449C2×Q8—532no—
450C2×Q8—532no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×D4—532no—
455C2×D4—532no—
456C2×D4—532no—
457C2×D4—1062no—
458C2×D4—1062no—
459C2×D4—1062no—
460C2×D4—1062no—
461C2×D4—1062no—
462C2×D4—1062no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1061no—
466C22×C4—1068no—
467C22×C4—1061no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—2121no—
476C22×C4—2121no—
477C22×C4#1171992114yes—
478C22×C4C2×C4×D5311yescharacteristic
479C22×C4C2×C4×D5311yescharacteristic
480C22×C4C2×C4×D5311yescharacteristic
481C22×C4#1171994511yescharacteristic
482C22×C4#1171994511yescharacteristic
483C22×C4D4×D5311yescharacteristic
484C22×C4D4×D5311yescharacteristic
485C22×C4D4×D5311yescharacteristic
486C22×C4D4×D5311yescharacteristic
487C22×C4D4×D5311yescharacteristic
488C22×C4#1171994711yescharacteristic
489C22×C4Q8×D5311yescharacteristic
490C22×C4#1171994911yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492M16—22no—
493M16—22no—
494M16—41no—
495M16—41no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—534no—
509C4⋊C4—1068no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—2121no—
514C4⋊C4—2121no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—1068no—
518C22⋊C4—1062no—
519C22⋊C4—1062no—
520C22⋊C4—1062no—
521C22⋊C4—1062no—
522C22⋊C4—2121no—
523C22⋊C4—2121no—
524C42—22no—
525C42—22no—
526C42—1068no—
527C42—1062no—
528C42—1062no—
529C42—1062no—
530C42—1062no—
531C42—2121no—
532C42—2121no—
533C23—42no—
534C23—1062no—
535C23—1062no—
536C23—1062no—
537C23—4242no—
538C23C2×#1171992111yescharacteristic
539C23C4×D21211yescharacteristic
540C23C4×D21211yescharacteristic
541C23#1313318711yescharacteristic
542C23#1313318711yescharacteristic
543C23#1313319211yescharacteristic
544C23#1313319711yescharacteristic
545Q8—534no—
546Q8—1062no—
547Q8—1062no—
548Q8—1062no—
549D4—534no—
550D4—1064no—
551D4—1064no—
552D4—1064no—
553D4—1064no—
554D4—1062no—
555D4—1062no—
556D4—1062no—
557C2×C4—24no—
558C2×C4—44no—
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×C4—4242no—
570C2×C4—4242no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4C42×D5311yescharacteristic
574C2×C4C22⋊C4×D5311yescharacteristic
575C2×C4C22⋊C4×D5311yescharacteristic
576C2×C4C22⋊C4×D5311yescharacteristic
577C2×C4C22⋊C4×D5311yescharacteristic
578C2×C4C4⋊C4×D5311yescharacteristic
579C2×C4C4⋊C4×D5311yescharacteristic
580C2×C4C4⋊C4×D5311yescharacteristic
581C2×C4#1313317411yescharacteristic
582C2×C4#1313317511yescharacteristic
583C2×C4#1313317711yescharacteristic
584C2×C4#1313317711yescharacteristic
585C2×C4C4×C22⋊D5311yescharacteristic
586C2×C4C4×C22⋊D5311yescharacteristic
587C2×C4#1313317911yescharacteristic
588C2×C4#1313317911yescharacteristic
589C2×C4#1313318511yescharacteristic
590C2×C4#1313318511yescharacteristic
591C2×C4#1313318811yescharacteristic
592C2×C4#1313319011yescharacteristic
593C2×C4#1313319011yescharacteristic
594C2×C4#1313319311yescharacteristic
595C2×C4#1313319511yescharacteristic
596C2×C4#1313319511yescharacteristic
597C2×C4#1313319911yescharacteristic
598C2×C4#1313319911yescharacteristic
599C2×C4#1313320211yescharacteristic
600C2×C4#1313320211yescharacteristic
601C8—42no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—1064no—
611C22—4242no—
612C22—4242no—
613C22—2124no—
614C22C4×#1171992111yescharacteristic
615C22#1681047811yescharacteristic
616C22#1681048011yescharacteristic
617C22#1681049711yescharacteristic
618C22#1681049711yescharacteristic
619C22#1681050111yescharacteristic
620C22#1681050111yescharacteristic
621C4—44no—
622C4—1064no—
623C4—4242no—
624C4—4242no—
625C4—2124no—
626C4#95×D5311yescharacteristic
627C4#1681047011yescharacteristic
628C4#1681048211yescharacteristic
629C4#1681048511yescharacteristic
630C4#1681049411yescharacteristic
631C4#1681049411yescharacteristic
632C4#1681049911yescharacteristic
633C4#1681049911yescharacteristic
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—21no—
640C2—2124no—
641C2#2027293611yescharacteristic
642C2#2027467812yes—
subgroupG/H
centerC2×C4#13133185
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1729C2
socleC22×C53#379
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenter#1729D53
hypocenterC53#10779
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
4. #20274448C2semidirect
5. #20274432C2semidirect
6. #20274426C2semidirect
7. C53×#1729C2semidirect
9. #16811104C4semidirect
21. #16810639C4semidirect
24. #16810620C4semidirect
65. #13133176C2×C4semidirect
66. #13133176C2×C4semidirect
92. C2×M16×C53C2×C4semidirect
100. C22×C8×C53C2×C4semidirect
319. #1729D53semidirect
355. C53#10779semidirect
378. C2×M16C4×D53semidirect
386. C22×C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202879162
C2×#168106392
C2×#168106202
#202744482
#202744322
#202744262
C53×#17292
#1077953