Finite Group Databasebeta

← #27306618

Subgroups — #27306618

Order 13568.

1862 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133173C211yesmaximal, characteristic
2C2×#16810490C211yesmaximal, characteristic, complemented
3C2×C4×#11719919C211yesmaximal, characteristic, complemented
4#20272966C212yesmaximal, complemented
5#20272958C212yesmaximal
6#20272952C211yesmaximal, characteristic
7#20272950C211yesmaximal, characteristic, complemented
8#20272948C211yesmaximal, characteristic, complemented
9#20272940C212yesmaximal, complemented
10#20272937C212yesmaximal
11C53×#1437C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C2×#13133173—24no—
14C2×#13133173—24no—
15C2×#13133173C418yescomplemented
16C22×#11719919C2212yes—
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×C22⋊Dic53C2212yes—
20C22×C22⋊Dic53C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24#16810490—24no—
25C4×#11719919—24no—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C22⋊C4×C53C2212yes—
36C22×C22⋊C4×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C23×Dic53—24no—
41C23×Dic53—22no—
42C23×Dic53—22no—
43C23×Dic53—24no—
44C23×Dic53C2×C414yes—
45C23×Dic53C2312yes—
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2312yes—
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51#13133173—232no—
52C2×#11719919—22no—
53C2×#11719919—22no—
54C2×#11719919—24no—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919C2×C414yes—
60C2×#11719919C2×C414yes—
61C2×C22⋊Dic53—24no—
62C2×C22⋊Dic53—22no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—24no—
65C2×C22⋊Dic53—42no—
66C2×C22⋊Dic53C2×C414yes—
67C2×C4×Dic53—22no—
68C2×C4×Dic53—22no—
69C2×C4×Dic53—22no—
70C2×C4×Dic53—42no—
71C2×C4×Dic53C2×C414yes—
72C2×C4×Dic53C2×C414yes—
73C2×C4×Dic53D414yes—
74C2×C4×Dic53D414yes—
75#13133124—22no—
76#13133124—24no—
77#13133124—24no—
78#13133124—24no—
79#13133124—24no—
80#13133124—24no—
81#13133124—24no—
82#13133124—42no—
83#13133124—42no—
84C25×C53C2311yescharacteristic
85C23×C4×C53C2312yes—
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C2×C22⋊C4×C53—24no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53—24no—
93C2×C22⋊C4×C53—42no—
94C2×C22⋊C4×C53C2×C414yes—
95C2×C42×C53—22no—
96C2×C42×C53—42no—
97#95×C53—22no—
98#95×C53—24no—
99C24×C53—21no—
100C24×C53—21no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—22no—
106C24×C53C22×C412yes—
107C24×C53#4114yes—
108C24×C53C2411yescharacteristic
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
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×C53C22×C412yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53#4114yes—
133C22×Dic53—216no—
134C22×Dic53—216no—
135C22×Dic53—28no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—28no—
144C22×Dic53—28no—
145C22×Dic53—22no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
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×Dic53C22×C412yes—
163C22×Dic53C22×C412yes—
164C22×Dic53C22×C412yes—
165C22×Dic53C22×C412yes—
166C22×Dic53C2×D412yes—
167C22×Dic53C2×D412yes—
168C22×Dic53C2×D412yes—
169C22×Dic53C2×D412yes—
170C22×Dic53#4114yes—
171C22×Dic53#4114yes—
172C22⋊C4×C53—28no—
173C22⋊C4×C53—48no—
174C42×C53—44no—
175C22⋊Dic53—28no—
176C22⋊Dic53—48no—
177#11719919—216no—
178#11719919—28no—
179#11719919—28no—
180C4×Dic53—28no—
181C4×Dic53—44no—
182C4×Dic53C22⋊C4116yescomplemented
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—24no—
191C23×C53—24no—
192C23×C53—24no—
193C23×C53—42no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53#11714yes—
204C23×C53C23×C411yescharacteristic
205C23×C53C22×D411yescharacteristic
206C23×C53C22×D411yescharacteristic
207C23×C53C2×#4112yes—
208C23×C53C2×#4112yes—
209C23×C53C2×#4112yes—
210C23×C53C2×#4112yes—
211C23×C53H(5,𝔽2)12yes—
212C23×C532−1+412yes—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—22no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53C2×C22⋊C414yes—
237C2×Dic53—232no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—42no—
256C2×Dic53—41no—
257C2×Dic53—42no—
258C2×Dic53—41no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53C2×C22⋊C414yes—
270C2×Dic53C2×C22⋊C414yes—
271C2×Dic53#11714yes—
272#13938—531nomaximal, Sylow
273C22×C53—28no—
274C22×C53—24no—
275C22×C53—42no—
276C22×C53—44no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53—44no—
281C22×C53—42no—
282C22×C53C22×C22⋊C411yescharacteristic
283C22×C53C2×#11711yescharacteristic
284C22×C53C2×#11711yescharacteristic
285C22×C53C4×#4112yes—
286C22×C53#52012yes—
287C22×C53#52512yes—
288C22×C53#52512yes—
289C22×C53#52512yes—
290C22×C53#53211yescharacteristic
291C22×C53#53211yescharacteristic
292C22×C53#53511yescharacteristic
293C22×C53#53611yescharacteristic
294C22×C53#53712yes—
295C22×C53#55112yes—
296C22×C53#55112yes—
297C22×C53#55212yes—
298C22×C53#55212yes—
299C22×C53#55312yes—
300C22×C53#55312yes—
301C22×C53#55612yes—
302C22×C53#55612yes—
303C4×C53—24no—
304C4×C53—44no—
305C4×C53—42no—
306C4×C53—82no—
307Dic53—24no—
308Dic53—24no—
309Dic53—24no—
310Dic53—42no—
311Dic53—44no—
312Dic53—82no—
313C22×#117—531no—
314C2×#383—531no—
315C2×C4×C4⋊C4—531no—
316#1441—532no—
317#1439—532no—
318#1437—531no—
319#1437D5311yescharacteristic
320#1436—531no—
321#1436—531no—
322#1434—532no—
323C22⋊#95—532no—
324C2×C53—44no—
325C2×C53#228011yescharacteristic
326C2×C53#229711yescharacteristic
327C2×C53#231111yescharacteristic
328C2×C53#234312yes—
329C2×C53#234712yes—
330C2×C53#259812yes—
331C2×C53#259912yes—
332C2×C53#260112yes—
333C2×C53#260112yes—
334C24×C4—531no—
335C2×#117—538no—
336C2×#117—1064no—
337C2×#117—1064no—
338C22×C4⋊C4—531no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4—532no—
341C22×C22⋊C4—531no—
342C22×C22⋊C4—532no—
343C22×C22⋊C4C2×D5312yes—
344C22×C22⋊C4C2×D5311yescharacteristic
345C22×C42—531no—
346C22×C42—531no—
347C22×C42—531no—
348C22×C42C2×D5311yescharacteristic
349#383—1064no—
350C4×C4⋊C4—1064no—
351C2×#95—532no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—531no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95C2×D5312yes—
361C2×#95C2×D5311yescharacteristic
362C53#1393811yescharacteristic, complemented, Sylow
363C25C22×D5311yescharacteristic
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—531no—
367C23×C4—534no—
368C23×C4—531no—
369C23×C4—532no—
370C23×C4—532no—
371C23×C4—1062no—
372C23×C4—1062no—
373C23×C4—1064no—
374C23×C4—1064no—
375C23×C4C22×D5312yes—
376C23×C4C22×D5311yescharacteristic
377C23×C4C22×D5312yes—
378C23×C4C22×D5311yescharacteristic
379#117—10632no—
380C2×C4⋊C4—534no—
381C2×C4⋊C4—534no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1064no—
385C2×C4⋊C4—1064no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1064no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—534no—
395C2×C22⋊C4—1062no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4—1064no—
398C2×C22⋊C4—1064no—
399C2×C22⋊C4—2122no—
400C2×C22⋊C4C4×D5314yes—
401C2×C42—22no—
402C2×C42—42no—
403C2×C42—534no—
404C2×C42—534no—
405C2×C42—534no—
406C2×C42—534no—
407C2×C42—1062no—
408C2×C42—1062no—
409C2×C42—1062no—
410C2×C42—2122no—
411#95—22no—
412#95—24no—
413#95—1064no—
414#95—1064no—
415#95—1064no—
416#95—1064no—
417#95—1064no—
418#95—1064no—
419#95—1062no—
420#95—2122no—
421#95—2122no—
422C24—21no—
423C24—21no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C2×C4×D5312yes—
430C24#1171994514yes—
431C24C23×D5311yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—532no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—534no—
459C22×C4—532no—
460C22×C4—534no—
461C22×C4—1068no—
462C22×C4—1068no—
463C22×C4—10616no—
464C22×C4—10616no—
465C22×C4—1068no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4C2×C4×D5312yes—
491C22×C4C2×C4×D5312yes—
492C22×C4D4×D5312yes—
493C22×C4D4×D5312yes—
494C22×C4#1171994714yes—
495C4⋊C4—1068no—
496C4⋊C4—10616no—
497C4⋊C4—1068no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—1068no—
501C22⋊C4—2128no—
502C42—44no—
503C42—5316no—
504C42—1068no—
505C42—2124no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23#1313317414yes—
527C23C22×C4×D5311yescharacteristic
528C23C2×D4×D5311yescharacteristic
529C23C2×D4×D5311yescharacteristic
530C23#41×D5312yes—
531C23#41×D5312yes—
532C23C2×#1171994712yes—
533C23C2×#1171994512yes—
534C23#1313326212yes—
535C23#1313326412yes—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—534no—
560C2×C4—534no—
561C2×C4—534no—
562C2×C4—10632no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1062no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2122no—
594C2×C4C22⋊C4×D5314yes—
595C22—28no—
596C22—24no—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22C2×C22⋊C4×D5311yescharacteristic
605C22#117×D5311yescharacteristic
606C22C2×#1313317411yescharacteristic
607C22C4×#1171994712yes—
608C22#1681110212yes—
609C22#1681113212yes—
610C22#1681113212yes—
611C22#1681113312yes—
612C22#1681117111yescharacteristic
613C22#1681117211yescharacteristic
614C22#1681118811yescharacteristic
615C22#1681119511yescharacteristic
616C22#1681120012yes—
617C22#1681129912yes—
618C22#1681130212yes—
619C22#1681130612yes—
620C22#1681131312yes—
621C22#1681131612yes—
622C22#1681131812yes—
623C22#1681134012yes—
624C22#1681134212yes—
625C4—24no—
626C4—42no—
627C4—44no—
628C4—82no—
629C4—1064no—
630C4—1064no—
631C4—1064no—
632C4—4242no—
633C4—2122no—
634C4—2124no—
635C2—44no—
636C2#2027709611yescharacteristic
637C2#2027720211yescharacteristic
638C2#2027730211yescharacteristic
639C2#2027756912yes—
640C2#2027760512yes—
641C2#2028027812yes—
642C2#2028028912yes—
643C2#2028030912yes—
644C2#2028031712yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1437C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1437D53
hypocenterC53#13938
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#16810490C2semidirect
3. C2×C4×#11719919C2semidirect
4. #20272966C2semidirect
7. #20272950C2semidirect
8. #20272948C2semidirect
9. #20272940C2semidirect
15. C2×#13133173C4semidirect
182. C4×Dic53C22⋊C4semidirect
362. C53#13938semidirect

Maximal subgroups

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

subgroupindex
C22×#131331732
C2×#168104902
C2×C4×#117199192
#202729662
#202729582
#202729522
#202729502
#202729482
#202729402
#202729372
C53×#14372
#1393853