Finite Group Databasebeta

← #27305981

Subgroups — #27305981

Order 13568.

2438 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133174C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic
3C2×#16810485C211yesmaximal, characteristic
4C2×#16810485C212yesmaximal
5#20272967C212yesmaximal, complemented
6#20272966C212yesmaximal, complemented
7#20272950C211yesmaximal, characteristic, complemented
8#20272948C211yesmaximal, characteristic, complemented
9#20272947C211yesmaximal, characteristic, complemented
10#20272936C212yesmaximal
11C53×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C2×#13133174—24no—
16C2×#13133174—28no—
17C22×C4⋊Dic53C2211yescharacteristic
18C22×#11719919C2212yes—
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2211yescharacteristic
21C2×C4⋊C4×D53—24no—
22C2×C4⋊C4×D53—24no—
23C2×C4⋊C4×D53C418yescomplemented
24C22×#11719921C2212yes—
25C22×#11719921C2212yes—
26C22×C4×Dic53C2211yescharacteristic
27#16810485—24no—
28#16810485—28no—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C2×#13133124C2212yes—
37C22×C4⋊C4×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C2×#95×C53C2211yescharacteristic
41C2×#95×C53C2212yes—
42C24×D53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2312yes—
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48C22×C4×D53—22no—
49C22×C4×D53—22no—
50C22×C4×D53—24no—
51C22×C4×D53—24no—
52C22×C4×D53—24no—
53C22×C4×D53—24no—
54C22×C4×D53—22no—
55C22×C4×D53—22no—
56C22×C4×D53—24no—
57C22×C4×D53—24no—
58C22×C4×D53C2×C414yes—
59C22×C4×D53C2×C414yes—
60C22×C4×D53C2×C414yes—
61C22×C4×D53D414yes—
62C22×C4×D53D414yes—
63#13133174—232no—
64C2×C4⋊Dic53—22no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53C2×C414yes—
67C2×#11719919—22no—
68C2×#11719919—22no—
69C2×#11719919—24no—
70C2×#11719919—22no—
71C2×#11719919—22no—
72C2×#11719919—24no—
73C2×#11719919—24no—
74C2×#11719919C2×C414yes—
75C2×#11719919C2×C414yes—
76C4⋊C4×D53—232no—
77C2×#11719921—24no—
78C2×#11719921—24no—
79C2×#11719921—24no—
80C2×#11719921—24no—
81C2×#11719921—24no—
82C2×#11719921D418yes—
83C2×C4×Dic53—22no—
84C2×C4×Dic53—24no—
85#13133124—24no—
86#13133124—22no—
87#13133124—24no—
88#13133124—24no—
89#13133124—24no—
90#13133124—24no—
91#13133124—24no—
92#13133124—24no—
93C23×C4×C53C2312yes—
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2312yes—
96C23×C4×C53C2311yescharacteristic
97C23×C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53C2×C414yes—
101C2×C42×C53—24no—
102C2×C42×C53—22no—
103#95×C53—24no—
104#95×C53—22no—
105#95×C53—24no—
106C24×C53C2411yescharacteristic
107C23×D53—22no—
108C23×D53—22no—
109C23×D53—22no—
110C23×D53—22no—
111C23×D53C22×C412yes—
112C23×D53C2×D412yes—
113C23×D53C2×D412yes—
114C23×D53C2×D414yes—
115C23×D53#4114yes—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53C22×C412yes—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C22×C412yes—
141C22×C4×C53C2×D414yes—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D412yes—
144C22×C4×C53#4114yes—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53C22×C412yes—
168C22×Dic53C22×C412yes—
169C22×Dic53C22×C412yes—
170C22×Dic53C2×D414yes—
171C22×Dic53C2×D412yes—
172C22×Dic53C2×D412yes—
173C22×Dic53#4114yes—
174C2×C4×D53—216no—
175C2×C4×D53—216no—
176C2×C4×D53—28no—
177C2×C4×D53—216no—
178C2×C4×D53—28no—
179C2×C4×D53—216no—
180C2×C4×D53—28no—
181C2×C4×D53—28no—
182C2×C4×D53—28no—
183C2×C4×D53—28no—
184C2×C4×D53—216no—
185C2×C4×D53—44no—
186C2×C4×D53—48no—
187C2×C4×D53—44no—
188C2×C4×D53C22⋊C4116yes—
189C4⋊C4×C53—28no—
190C42×C53—28no—
191#11719921—216no—
192#11719921—48no—
193C4⋊Dic53—28no—
194#11719919—216no—
195#11719919—28no—
196#11719919—28no—
197C4×Dic53—28no—
198C23×C53C23×C411yescharacteristic
199C23×C53C22×D411yescharacteristic
200C23×C53C22×D411yescharacteristic
201C23×C53C22×D412yes—
202C23×C53C2×#4112yes—
203C23×C53C2×#4112yes—
204C23×C53C2×#4112yes—
205C23×C532−1+412yes—
206C23×C532−1+412yes—
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—22no—
213C22×D53—24no—
214C22×D53—22no—
215C22×D53—24no—
216C22×D53—24no—
217C22×D53—24no—
218C22×D53—24no—
219C22×D53—24no—
220C22×D53—24no—
221C22×D53—24no—
222C22×D53—24no—
223C22×D53C2×C22⋊C414yes—
224C22×D53C4×D418yes—
225C22×D53#12014yes—
226C22×D53#12118yes—
227C22×D53#12314yes—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—22no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—41no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53C2×C22⋊C414yes—
256C2×C4×C53#11714yescomplemented
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—24no—
260C2×Dic53—22no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—22no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—41no—
271C2×Dic53—42no—
272C2×Dic53—41no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53C2×C22⋊C414yes—
285C2×Dic53#11714yescomplemented
286C4×D53—232no—
287C4×D53—232no—
288C4×D53—416no—
289#13870—531nomaximal, Sylow
290C22×C53C22×C22⋊C411yescharacteristic
291C22×C53C2×#11711yescharacteristic
292C22×C53C2×#11711yescharacteristic
293C22×C53C2×C4×D412yes—
294C22×C53#51912yes—
295C22×C53C2×#12011yescharacteristic
296C22×C53C2×#12112yes—
297C22×C53C2×#12311yescharacteristic
298C22×C53#52512yes—
299C22×C53#52512yes—
300C22×C53#53611yescharacteristic
301C22×C53#53612yes—
302C22×C53#53611yescharacteristic
303C22×C53#54712yes—
304C22×C53#54712yes—
305C22×C53#54712yes—
306C22×C53#54712yes—
307C22×C53#55212yes—
308C22×C53#55212yes—
309C22×C53#55212yes—
310C22×C53#55212yes—
311C2×D53—24no—
312C2×D53—28no—
313C2×D53—28no—
314C2×D53—28no—
315C2×D53—28no—
316C2×D53—28no—
317C2×D53—28no—
318C2×D53—24no—
319C2×D53#38618yes—
320C4×C53—24no—
321C4×C53—24no—
322C4×C53—44no—
323C4×C53—42no—
324C4×C53—82no—
325Dic53—24no—
326Dic53—24no—
327Dic53—44no—
328Dic53—42no—
329Dic53—82no—
330C22×#117—531no—
331C23×C4⋊C4—531no—
332C2×#381—532no—
333C2×#381—531no—
334#1441—532no—
335#1441—532no—
336#1436—531no—
337#1436—531no—
338#1436—531no—
339#1436D5311yescharacteristic, complemented
340C22⋊#95—532no—
341C2×C53#228011yescharacteristic
342C2×C53C2×#38611yescharacteristic
343C2×C53#231011yescharacteristic
344C2×C53#234412yes—
345C2×C53#234412yes—
346C2×C53#240612yes—
347C2×C53#241312yes—
348C2×C53#241312yes—
349C2×C53#248412yes—
350D53—28no—
351C24×C4—531no—
352C24×C4—531no—
353C24×C4—531no—
354C2×#117—1068no—
355C2×#117—1064no—
356C22×C4⋊C4—531no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4—531no—
359C22×C4⋊C4—538no—
360C22×C4⋊C4—532no—
361C22×C4⋊C4—1064no—
362C22×C4⋊C4—1064no—
363C22×C4⋊C4C2×D5311yescharacteristic
364C22×C22⋊C4—532no—
365C22×C22⋊C4—532no—
366C22×C42—531no—
367C22×C42C2×D5311yescharacteristic
368#381—1068no—
369#381—1064no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—531no—
374C2×#95—532no—
375C2×#95—532no—
376C2×#95—532no—
377C2×#95—532no—
378C2×#95C2×D5312yes—
379C2×#95C2×D5311yescharacteristic
380C2×#95C2×D5312yes—
381C53#1387011yescharacteristic, complemented, Sylow
382C25—531no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—534no—
386C23×C4—534no—
387C23×C4—534no—
388C23×C4—532no—
389C23×C4—531no—
390C23×C4—534no—
391C23×C4—534no—
392C23×C4—532no—
393C23×C4—1064no—
394C23×C4—1062no—
395C23×C4—1062no—
396C23×C4—1064no—
397C23×C4—1064no—
398C23×C4—1062no—
399C23×C4—1062no—
400C23×C4—1064no—
401C23×C4—1064no—
402C23×C4—1064no—
403C23×C4C22×D5312yes—
404C23×C4C22×D5311yescharacteristic
405C23×C4C22×D5312yes—
406C23×C4C22×D5311yescharacteristic
407C23×C4C22×D5311yescharacteristic
408#117—10632no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—534no—
412C2×C4⋊C4—534no—
413C2×C4⋊C4—534no—
414C2×C4⋊C4—10632no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1062no—
417C2×C4⋊C4—1062no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4—1064no—
420C2×C4⋊C4—1064no—
421C2×C4⋊C4—1062no—
422C2×C4⋊C4—1062no—
423C2×C4⋊C4—1064no—
424C2×C4⋊C4C4×D5314yes—
425C2×C22⋊C4—538no—
426C2×C22⋊C4—1064no—
427C2×C22⋊C4—1064no—
428C2×C22⋊C4—1064no—
429C2×C22⋊C4—1064no—
430C2×C22⋊C4—1064no—
431C2×C42—24no—
432C2×C42—22no—
433C2×C42—1064no—
434C2×C42—1062no—
435#95—24no—
436#95—24no—
437#95—22no—
438#95—1064no—
439#95—1064no—
440#95—1064no—
441#95—1064no—
442#95—1064no—
443#95—1064no—
444#95—1064no—
445#95—1062no—
446C24—532no—
447C24—534no—
448C24—532no—
449C24—532no—
450C24—534no—
451C24—1062no—
452C24—1062no—
453C24—1062no—
454C24—1062no—
455C24C23×D5311yescharacteristic
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—532no—
479C22×C4—532no—
480C22×C4—532no—
481C22×C4—532no—
482C22×C4—534no—
483C22×C4—5316no—
484C22×C4—532no—
485C22×C4—534no—
486C22×C4—1068no—
487C22×C4—1068no—
488C22×C4—1068no—
489C22×C4—10616no—
490C22×C4—10616no—
491C22×C4—1068no—
492C22×C4—10616no—
493C22×C4—10616no—
494C22×C4—1068no—
495C22×C4—10616no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1068no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—2124no—
520C22×C4—2128no—
521C22×C4—2124no—
522C22×C4C2×C4×D5312yes—
523C22×C4C2×C4×D5312yes—
524C22×C4C2×C4×D5312yes—
525C22×C4#1171994514yes—
526C22×C4D4×D5314yes—
527C22×C4D4×D5312yes—
528C22×C4D4×D5312yes—
529C4⋊C4—28no—
530C4⋊C4—1068no—
531C4⋊C4—1068no—
532C4⋊C4—10616no—
533C4⋊C4—1068no—
534C22⋊C4—10616no—
535C22⋊C4—2128no—
536C42—28no—
537C42—1068no—
538C23—534no—
539C23—538no—
540C23—534no—
541C23—538no—
542C23—534no—
543C23—1064no—
544C23—1064no—
545C23—1064no—
546C23—1064no—
547C23—1064no—
548C23—1064no—
549C23—1064no—
550C23—1064no—
551C23—1062no—
552C23—1062no—
553C23—1064no—
554C23—1064no—
555C23—1064no—
556C23—1062no—
557C23—1064no—
558C23—1062no—
559C23C22×C4×D5311yescharacteristic
560C23C2×D4×D5312yes—
561C23C2×D4×D5311yescharacteristic
562C23C2×D4×D5311yescharacteristic
563C23#41×D5312yes—
564C23#41×D5312yes—
565C23C2×#1171994512yes—
566C23#1313326412yes—
567C23#1313326512yes—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—534no—
596C2×C4—534no—
597C2×C4—10632no—
598C2×C4—10632no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1064no—
602C2×C4—1062no—
603C2×C4—1064no—
604C2×C4—1062no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—1064no—
608C2×C4—1064no—
609C2×C4—1062no—
610C2×C4—1064no—
611C2×C4—1064no—
612C2×C4—21216no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2121no—
623C2×C4—2122no—
624C2×C4—2121no—
625C2×C4—2122no—
626C2×C4—2122no—
627C2×C4C22⋊C4×D5314yes—
628C2×C4#1313317414yescomplemented
629C22—538no—
630C22—1068no—
631C22—1064no—
632C22—1068no—
633C22—1068no—
634C22—1068no—
635C22—1064no—
636C22—1068no—
637C22—1068no—
638C22C2×C22⋊C4×D5311yescharacteristic
639C22#117×D5311yescharacteristic
640C22C2×#1313317411yescharacteristic
641C22C4×D4×D5312yes—
642C22#1681109912yes—
643C22#120×D5311yescharacteristic
644C22#121×D5312yes—
645C22#123×D5311yescharacteristic
646C22#1681113212yes—
647C22#1681113212yes—
648C22#1681119311yescharacteristic
649C22#1681119612yes—
650C22#1681119811yescharacteristic
651C22#1681127112yes—
652C22#1681127412yes—
653C22#1681127512yes—
654C22#1681127812yes—
655C22#1681130512yes—
656C22#1681130912yes—
657C22#1681131212yes—
658C22#1681131412yes—
659C4—24no—
660C4—24no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—1064no—
665C4—1064no—
666C4—4242no—
667C4—2124no—
668C4—2122no—
669C2—1068no—
670C2#2027709611yescharacteristic
671C2#386×D5311yescharacteristic
672C2#2027729811yescharacteristic
673C2#2027758012yes—
674C2#2027758112yes—
675C2#2027804812yes—
676C2#2027811912yes—
677C2#2027812112yes—
678C2#2027900012yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1436C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1436D53
hypocenterC53#13870
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
5. #20272967C2semidirect
6. #20272966C2semidirect
7. #20272950C2semidirect
8. #20272948C2semidirect
9. #20272947C2semidirect
11. C53×#1436C2semidirect
23. C2×C4⋊C4×D53C4semidirect
256. C2×C4×C53#117semidirect
285. C2×Dic53#117semidirect
339. #1436D53semidirect
381. C53#13870semidirect
628. C2×C4#13133174semidirect

Maximal subgroups

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

subgroupindex
C22×#131331742
C22×C4⋊C4×D532
C2×#168104852
#202729672
#202729662
#202729502
#202729482
#202729472
#202729362
C53×#14362
#1387053