Finite Group Databasebeta

← #27310530

Subgroups — #27310530

Order 13568.

1782 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
1C2×#16810496C211yesmaximal, characteristic
2C2×#16810495C211yesmaximal, characteristic
3C22×#13133124C211yesmaximal, characteristic
4#20272981C212yesmaximal, complemented
5#20272975C212yesmaximal, complemented
6#20272970C212yesmaximal, complemented
7#20272966C212yesmaximal, complemented
8#20272937C211yesmaximal, characteristic
9#20272937C211yesmaximal, characteristic
10#20272937C211yesmaximal, characteristic
11C53×C22⋊#95C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C22×#11719919C2211yescharacteristic
15C22×#11719919C2211yescharacteristic
16C22×#11719919C2212yes—
17C22×C22⋊Dic53C2211yescharacteristic
18C22×C22⋊Dic53C2211yescharacteristic
19#16810496C418yescomplemented
20#16810495—24no—
21C2×#13133124—24no—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C24×C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2211yescharacteristic
36C2×#95×C53C2212yes—
37C2×#95×C53C2212yes—
38C23×Dic53—22no—
39C23×Dic53—22no—
40C23×Dic53—22no—
41C23×Dic53—22no—
42C23×Dic53—42no—
43C23×Dic53—42no—
44C23×Dic53—42no—
45C23×Dic53—42no—
46C23×Dic53D414yes—
47C23×Dic53Q814yes—
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2312yes—
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53C23×Dic53C2312yes—
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719919—42no—
57C2×#11719919—42no—
58C2×#11719919—42no—
59C2×#11719919—42no—
60C2×#11719919C2×C418yescomplemented
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53—22no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53C2×C414yes—
66C2×C22⋊Dic53C2×C414yes—
67#13133124—24no—
68#13133124—24no—
69#13133124—22no—
70#13133124—22no—
71#13133124—42no—
72#13133124—42no—
73#13133124—42no—
74#13133124—42no—
75#13133124—42no—
76#13133124—42no—
77#13133124C2×C418yescomplemented
78C25×C53C2311yescharacteristic
79C23×C4×C53—22no—
80C23×C4×C53—22no—
81C23×C4×C53—42no—
82C23×C4×C53—42no—
83C23×C4×C53C2×C414yes—
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2312yes—
87C23×C4×C53C2312yes—
88C2×C22⋊C4×C53—22no—
89C2×C22⋊C4×C53—22no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53—22no—
93C2×C22⋊C4×C53—22no—
94#95×C53—42no—
95#95×C53—42no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53C22×C412yes—
101C24×C53C2×D412yes—
102C24×C53C2×Q812yes—
103C24×C53#4112yes—
104C24×C53#4112yes—
105C24×C53#4112yes—
106C24×C53#4112yes—
107C24×C53C2411yescharacteristic
108C22×C4×C53—28no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
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—48no—
133C22×C4×C53—44no—
134C22×C4×C53—44no—
135C22×C4×C53—48no—
136C22×C4×C53C22×C414yes—
137C22×Dic53—28no—
138C22×Dic53—28no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
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×Dic53—44no—
168C22×Dic53—44no—
169C22×Dic53—48no—
170C22×Dic53—44no—
171C22×Dic53—44no—
172C22×Dic53—48no—
173C22×Dic53—48no—
174C22×Dic53—48no—
175C22×Dic53C22×C412yes—
176C22×Dic53C22×C414yes—
177C22×Dic53C22×C412yes—
178C22×Dic53C2×D412yes—
179C22×Dic53C2×D412yes—
180C22×Dic53C2×Q814yes—
181C22⋊C4×C53—44no—
182C22⋊C4×C53—44no—
183C22⋊Dic53—28no—
184C22⋊Dic53—28no—
185#11719919—48no—
186#11719919—44no—
187#11719919—44no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—24no—
195C23×C53—24no—
196C23×C53—24no—
197C23×C53—24no—
198C23×C53—24no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53#11714yes—
205C23×C53C4×D414yes—
206C23×C53C4×Q814yes—
207C23×C53C22⋊Q814yes—
208C23×C53#12314yes—
209C23×C53#12514yes—
210C23×C53#12614yes—
211C23×C53C23×C411yescharacteristic
212C23×C53C22×D411yescharacteristic
213C23×C53C22×Q811yescharacteristic
214C23×C53C2×#4111yescharacteristic
215C23×C53C2×#4111yescharacteristic
216C23×C53C2×#4111yescharacteristic
217C23×C53C2×#4111yescharacteristic
218C23×C53H(5,𝔽2)12yes—
219C23×C53H(5,𝔽2)12yes—
220C23×C53H(5,𝔽2)12yes—
221C23×C532−1+412yes—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—416no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—42no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×Dic53—24no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—416no—
264C2×Dic53—416no—
265C2×Dic53—42no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—42no—
270C2×Dic53—41no—
271C2×Dic53—42no—
272C2×Dic53—42no—
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—
283#14265—531nomaximal, Sylow
284C22×C53—28no—
285C22×C53—24no—
286C22×C53—24no—
287C22×C53—28no—
288C22×C53—24no—
289C22×C53—24no—
290C22×C53—24no—
291C22×C53—24no—
292C22×C53—28no—
293C22×C53—28no—
294C22×C53#38718yes—
295C22×C53C2×#11711yescharacteristic
296C22×C53C2×C4×D411yescharacteristic
297C22×C53C2×C4×Q811yescharacteristic
298C22×C53#51812yes—
299C22×C53#52012yes—
300C22×C53C2×C22⋊Q811yescharacteristic
301C22×C53C2×#12311yescharacteristic
302C22×C53C2×#12511yescharacteristic
303C22×C53C2×#12611yescharacteristic
304C22×C53#53411yescharacteristic
305C22×C53#53411yescharacteristic
306C22×C53#53511yescharacteristic
307C22×C53#53611yescharacteristic
308C22×C53#53812yes—
309C22×C53#53812yes—
310C22×C53#53812yes—
311C22×C53#53812yes—
312C22×C53#53912yes—
313C22×C53#53912yes—
314C22×C53#53912yes—
315C22×C53#53912yes—
316C22×C53#54312yes—
317C22×C53#54412yes—
318C4×C53—44no—
319C4×C53—81no—
320C4×C53—81no—
321C4×C53—82no—
322Dic53—44no—
323Dic53—42no—
324Dic53—42no—
325Dic53—42no—
326Dic53—42no—
327Dic53—44no—
328C2×#385—531no—
329C2×#385—531no—
330C22×#95—531no—
331#1445—532no—
332#1443—532no—
333#1442—532no—
334#1441—532no—
335C22⋊#95—531no—
336C22⋊#95—531no—
337C22⋊#95—531no—
338C22⋊#95D5311yescharacteristic
339C2×C53—28no—
340C2×C53C2×#38711yescharacteristic
341C2×C53#231812yes—
342C2×C53#232111yescharacteristic
343C2×C53#232211yescharacteristic
344C2×C53#232912yes—
345C2×C53#262211yescharacteristic
346C2×C53#262311yescharacteristic
347C2×C53#262611yescharacteristic
348C2×C53#262711yescharacteristic
349C2×C53#264312yes—
350C2×C53#264612yes—
351C24×C4—531no—
352C24×C4—531no—
353C24×C4C2×D5311yescharacteristic
354C22×C4⋊C4—532no—
355C22×C4⋊C4—531no—
356C22×C4⋊C4—531no—
357C22×C22⋊C4—531no—
358C22×C22⋊C4—531no—
359C22×C22⋊C4C2×D5311yescharacteristic
360C22×C22⋊C4C2×D5311yescharacteristic
361#385—538no—
362#385—1064no—
363C2×#95—532no—
364C2×#95—532no—
365C2×#95—532no—
366C2×#95—531no—
367C2×#95—531no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—532no—
374C2×#95—1064no—
375C2×#95C2×D5312yes—
376C2×#95C2×D5312yes—
377C53#1426511yescharacteristic, complemented, Sylow
378C25C22×D5311yescharacteristic
379C23×C4—22no—
380C23×C4—22no—
381C23×C4—42no—
382C23×C4—42no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—532no—
386C23×C4—531no—
387C23×C4—531no—
388C23×C4—532no—
389C23×C4—534no—
390C23×C4—534no—
391C23×C4—1062no—
392C23×C4—1062no—
393C23×C4—1062no—
394C23×C4—1062no—
395C23×C4—2122no—
396C23×C4—2122no—
397C23×C4—2122no—
398C23×C4—2122no—
399C23×C4C4×D5314yes—
400C23×C4C22×D5311yescharacteristic
401C23×C4C22×D5311yescharacteristic
402C23×C4C22×D5312yes—
403C23×C4C22×D5312yes—
404C2×C4⋊C4—538no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1062no—
407C2×C4⋊C4—2122no—
408C2×C4⋊C4—2122no—
409C2×C4⋊C4—2122no—
410C2×C4⋊C4—2122no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—534no—
418C2×C22⋊C4—534no—
419C2×C22⋊C4—1062no—
420C2×C22⋊C4—1062no—
421C2×C22⋊C4—1062no—
422C2×C22⋊C4—1062no—
423#95—42no—
424#95—42no—
425#95—538no—
426#95—1064no—
427#95—1064no—
428#95—1062no—
429#95—1062no—
430#95—2122no—
431#95—2122no—
432#95—2122no—
433#95—2122no—
434#95—2122no—
435#95—2122no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24C2×C4×D5312yes—
441C24#1171994512yes—
442C24#1171994512yes—
443C24D4×D5312yes—
444C24#1171994712yes—
445C24#1171994712yes—
446C24Q8×D5312yes—
447C24C23×D5311yescharacteristic
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—532no—
477C22×C4—534no—
478C22×C4—532no—
479C22×C4—532no—
480C22×C4—532no—
481C22×C4—534no—
482C22×C4—1068no—
483C22×C4—1068no—
484C22×C4—1062no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—2124no—
513C22×C4—2124no—
514C22×C4—2128no—
515C22×C4—2124no—
516C22×C4—2124no—
517C22×C4—2128no—
518C22×C4—2128no—
519C22×C4—2128no—
520C22×C4C2×C4×D5314yes—
521C4⋊C4—2128no—
522C4⋊C4—2124no—
523C4⋊C4—2124no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—1068no—
527C22⋊C4—1068no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23#1313317414yes—
545C23#1313317714yes—
546C23#1313318214yes—
547C23#1313319514yes—
548C23#1313319914yes—
549C23#1313320714yes—
550C23#1313321014yes—
551C23C22×C4×D5311yescharacteristic
552C23C2×D4×D5311yescharacteristic
553C23C2×Q8×D5311yescharacteristic
554C23C2×#1171994711yescharacteristic
555C23C2×#1171994711yescharacteristic
556C23C2×#1171994511yescharacteristic
557C23C2×#1171994511yescharacteristic
558C23#1313326212yes—
559C23#1313326212yes—
560C23#1313326312yes—
561C23#1313326412yes—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—416no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—1064no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—21216no—
604C2×C4—21216no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2122no—
608C2×C4—2121no—
609C2×C4—2122no—
610C2×C4—2121no—
611C2×C4—2122no—
612C2×C4—2122no—
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—2122no—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—28no—
631C22—24no—
632C22—24no—
633C22#1681050518yes—
634C22C2×#1313317411yescharacteristic
635C22C2×#1313317711yescharacteristic
636C22C2×#1313318211yescharacteristic
637C22#1681109612yes—
638C22#1681110312yes—
639C22C2×#1313319511yescharacteristic
640C22C2×#1313319911yescharacteristic
641C22C2×#1313320711yescharacteristic
642C22C2×#1313321011yescharacteristic
643C22#1681118311yescharacteristic
644C22#1681118411yescharacteristic
645C22#1681118811yescharacteristic
646C22#1681119511yescharacteristic
647C22#1681120612yes—
648C22#1681120612yes—
649C22#1681120812yes—
650C22#1681121012yes—
651C22#1681121512yes—
652C22#1681121512yes—
653C22#1681121712yes—
654C22#1681121812yes—
655C22#1681124812yes—
656C22#1681125212yes—
657C4—44no—
658C4—82no—
659C4—81no—
660C4—81no—
661C4—2124no—
662C4—2122no—
663C4—2122no—
664C4—2122no—
665C4—2122no—
666C4—2124no—
667C2—28no—
668C2C2×#1681050511yescharacteristic
669C2#2027734212yes—
670C2#2027737011yescharacteristic
671C2#2027738111yescharacteristic
672C2#2027744512yes—
673C2#2028049211yescharacteristic
674C2#2028049911yescharacteristic
675C2#2028052611yescharacteristic
676C2#2028053411yescharacteristic
677C2#2028067012yes—
678C2#2028068712yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22⋊#95D53
hypocenterC53#14265
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. #20272981C2semidirect
5. #20272975C2semidirect
6. #20272970C2semidirect
7. #20272966C2semidirect
19. #16810496C4semidirect
60. C2×#11719919C2×C4semidirect
77. #13133124C2×C4semidirect
377. C53#14265semidirect

Maximal subgroups

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

subgroupindex
C2×#168104962
C2×#168104952
C22×#131331242
#202729812
#202729752
#202729702
#202729662
#202729372
C53×C22⋊#952
#1426553