Finite Group Databasebeta

← #27306853

Subgroups — #27306853

Order 13568.

1862 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810482C211yesmaximal, characteristic
2C2×C4×#11719921C211yesmaximal, characteristic
3C2×#95×D53C211yesmaximal, characteristic
4#20272973C212yesmaximal, complemented
5#20272970C212yesmaximal, complemented
6#20272942C211yesmaximal, characteristic, complemented
7#20272941C211yesmaximal, characteristic, complemented
8#20272941C211yesmaximal, characteristic, complemented
9#20272936C212yesmaximal
10#20272936C212yesmaximal
11C53×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D53C2212yes—
13C23×C4×D53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810482—24no—
20C4×#11719921C418yescomplemented
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32#95×D53—24no—
33C22×C42×C53C2211yescharacteristic
34C2×#95×C53C2212yes—
35C2×#95×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C24×D53C2311yescharacteristic
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C22×C4×D53—24no—
45C22×C4×D53—24no—
46C22×C4×D53—24no—
47C22×C4×D53—24no—
48C22×C4×D53—24no—
49C22×C4×D53—42no—
50C22×C4×D53—42no—
51C22×C4×D53—42no—
52C22×C4×D53—42no—
53C22×C4×D53C2×C414yes—
54C22×C4×D53D414yes—
55C22×C4×D53Q814yes—
56C2×#11719919—24no—
57C2×#11719919—42no—
58C2×#11719919—42no—
59C2×#11719921—24no—
60C2×#11719921—22no—
61C2×#11719921—22no—
62C2×#11719921—22no—
63C2×#11719921—22no—
64C2×#11719921—24no—
65C2×#11719921—24no—
66C2×#11719921C2×C414yes—
67C2×#11719921C2×C414yes—
68C2×C4×Dic53—42no—
69C2×C4×Dic53C2×C414yescomplemented
70#13133124—22no—
71#13133124—22no—
72#13133124—24no—
73#13133124—24no—
74#13133124—24no—
75#13133124—24no—
76#13133124—42no—
77#13133124—42no—
78#13133124—42no—
79#13133124C2×C414yescomplemented
80#13133124C2×C414yescomplemented
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84C23×C4×C53C2312yes—
85C23×C4×C53C2311yescharacteristic
86C2×C42×C53—42no—
87C2×C42×C53C2×C414yescomplemented
88#95×C53—22no—
89#95×C53—22no—
90#95×C53—24no—
91#95×C53—42no—
92C24×C53C2411yescharacteristic
93C23×D53—22no—
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53C22×C412yes—
98C23×D53C2×D412yes—
99C23×D53C2×Q812yes—
100C23×D53#4112yes—
101C23×D53#4112yes—
102C23×D53#4112yes—
103C23×D53#4112yes—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
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×C53C22×C412yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×Q812yes—
132C22×C4×C53#4114yes—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
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×Dic53C22×C412yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C22×C412yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×Q812yes—
161C22×Dic53#4114yes—
162C2×C4×D53—216no—
163C2×C4×D53—44no—
164C2×C4×D53—44no—
165C2×C4×D53—48no—
166C2×C4×D53—48no—
167C2×C4×D53—48no—
168C2×C4×D53—48no—
169C2×C4×D53—48no—
170C2×C4×D53—48no—
171C2×C4×D53—48no—
172C2×C4×D53—48no—
173C2×C4×D53C4⋊C4116yes—
174C42×C53—44no—
175#11719921—28no—
176#11719921—28no—
177#11719921—48no—
178#11719919—48no—
179C4×Dic53—44no—
180C23×C53C23×C411yescharacteristic
181C23×C53C22×D411yescharacteristic
182C23×C53C22×Q811yescharacteristic
183C23×C53C2×#4111yescharacteristic
184C23×C53C2×#4111yescharacteristic
185C23×C53C2×#4111yescharacteristic
186C23×C53C2×#4111yescharacteristic
187C23×C53C2×#4112yes—
188C23×C53C2×#4112yes—
189C23×C53H(5,𝔽2)12yes—
190C23×C53H(5,𝔽2)12yes—
191C22×D53—22no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—24no—
196C22×D53—24no—
197C22×D53—24no—
198C22×D53—24no—
199C22×D53—24no—
200C22×D53—24no—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53C2×C4⋊C414yes—
208C22×D53#11714yes—
209C22×D53#11714yes—
210C22×D53#12414yes—
211C22×D53#12414yes—
212C22×D53#12514yes—
213C22×D53#12514yes—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
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×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53C2×C4⋊C414yes—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
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×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53C2×C4⋊C414yes—
272C4×D53—416no—
273C4×D53—432no—
274#13966—531nomaximal, Sylow
275C22×C53C22×C4⋊C411yescharacteristic
276C22×C53C2×#11711yescharacteristic
277C22×C53C2×#11711yescharacteristic
278C22×C53C4×#4111yescharacteristic
279C22×C53C4×#4111yescharacteristic
280C22×C53#51811yescharacteristic
281C22×C53#51811yescharacteristic
282C22×C53#52512yes—
283C22×C53C2×#12411yescharacteristic
284C22×C53C2×#12411yescharacteristic
285C22×C53C2×#12511yescharacteristic
286C22×C53C2×#12511yescharacteristic
287C22×C53#53312yes—
288C22×C53#53412yes—
289C22×C53#54312yes—
290C22×C53#55112yes—
291C22×C53#55112yes—
292C22×C53#55112yes—
293C22×C53#55112yes—
294C22×C53#55312yes—
295C22×C53#55312yes—
296C22×C53#55312yes—
297C22×C53#55312yes—
298C2×D53—28no—
299C2×D53—28no—
300C2×D53—28no—
301C2×D53—24no—
302C2×D53—28no—
303C2×D53—24no—
304C2×D53—24no—
305C2×D53—24no—
306C2×D53—24no—
307C2×D53—24no—
308C2×D53#38218yes—
309C4×C53—24no—
310C4×C53—44no—
311C4×C53—42no—
312C4×C53—42no—
313C4×C53—82no—
314Dic53—24no—
315Dic53—44no—
316Dic53—42no—
317Dic53—42no—
318Dic53—82no—
319C2×#380—531no—
320C2×C4×C22⋊C4—531no—
321C22×#95—531no—
322#1442—532no—
323#1442—532no—
324#1434—531no—
325#1434—531no—
326#1434—531no—
327#1434D5311yescharacteristic, complemented
328C22⋊#95—532no—
329C22⋊#95—532no—
330C2×C53C2×#38211yescharacteristic
331C2×C53#229611yescharacteristic
332C2×C53#231511yescharacteristic
333C2×C53#235311yescharacteristic
334C2×C53#235311yescharacteristic
335C2×C53#235311yescharacteristic
336C2×C53#235311yescharacteristic
337C2×C53#256112yes—
338C2×C53#256912yes—
339C2×C53#256912yes—
340C2×C53#259112yes—
341D53—28no—
342C24×C4—531no—
343C24×C4—532no—
344C22×C4⋊C4—532no—
345C22×C22⋊C4—531no—
346C22×C22⋊C4—531no—
347C22×C22⋊C4—532no—
348C22×C42—531no—
349C22×C42C2×D5311yescharacteristic
350#380—1064no—
351C4×C22⋊C4—538no—
352C2×#95—531no—
353C2×#95—531no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—531no—
358C2×#95—531no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—1064no—
364C2×#95C2×D5312yes—
365C2×#95C2×D5311yescharacteristic
366C2×#95C2×D5311yescharacteristic
367C2×#95C2×D5312yes—
368C53#1396611yescharacteristic, complemented, Sylow
369C25—531no—
370C23×C4—531no—
371C23×C4—534no—
372C23×C4—534no—
373C23×C4—534no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—532no—
377C23×C4—532no—
378C23×C4—1064no—
379C23×C4—1064no—
380C23×C4—1064no—
381C23×C4—1064no—
382C23×C4—1064no—
383C23×C4—2122no—
384C23×C4—2122no—
385C23×C4—2122no—
386C23×C4—2122no—
387C23×C4C22×D5312yes—
388C23×C4C22×D5311yescharacteristic
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5312yes—
391C23×C4C22×D5311yescharacteristic
392C2×C4⋊C4—1064no—
393C2×C4⋊C4—2122no—
394C2×C4⋊C4—2122no—
395C2×C22⋊C4—534no—
396C2×C22⋊C4—534no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1062no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1064no—
404C2×C42—42no—
405C2×C42—534no—
406C2×C42—2122no—
407C2×C42C4×D5314yescomplemented
408#95—24no—
409#95—22no—
410#95—22no—
411#95—42no—
412#95—534no—
413#95—534no—
414#95—1064no—
415#95—1064no—
416#95—1064no—
417#95—1064no—
418#95—1062no—
419#95—1062no—
420#95—2122no—
421#95—2122no—
422#95—2122no—
423C24—532no—
424C24—532no—
425C24—532no—
426C24—532no—
427C24—532no—
428C24—532no—
429C24—532no—
430C24—1062no—
431C24—1062no—
432C24—1062no—
433C24—1062no—
434C24C23×D5311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
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—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—5316no—
462C22×C4—532no—
463C22×C4—532no—
464C22×C4—534no—
465C22×C4—10616no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1062no—
475C22×C4—1062no—
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—2124no—
490C22×C4—2124no—
491C22×C4—2128no—
492C22×C4—2128no—
493C22×C4—2128no—
494C22×C4—2128no—
495C22×C4—2128no—
496C22×C4—2128no—
497C22×C4—2128no—
498C22×C4—2128no—
499C22×C4C2×C4×D5312yes—
500C22×C4C2×C4×D5312yes—
501C22×C4C2×C4×D5312yes—
502C22×C4#1171994514yes—
503C22×C4D4×D5312yes—
504C22×C4Q8×D5312yes—
505C4⋊C4—2128no—
506C22⋊C4—1068no—
507C22⋊C4—1068no—
508C22⋊C4—2128no—
509C42—44no—
510C42—2124no—
511C23—534no—
512C23—534no—
513C23—534no—
514C23—534no—
515C23—534no—
516C23—534no—
517C23—534no—
518C23—1064no—
519C23—1064no—
520C23—1064no—
521C23—1064no—
522C23—1064no—
523C23—1064no—
524C23—1064no—
525C23—1064no—
526C23—1062no—
527C23—1062no—
528C23—1062no—
529C23—1062no—
530C23—1064no—
531C23—1064no—
532C23—1064no—
533C23—1064no—
534C23C22×C4×D5311yescharacteristic
535C23C2×D4×D5311yescharacteristic
536C23C2×Q8×D5311yescharacteristic
537C23#41×D5311yescharacteristic
538C23#41×D5311yescharacteristic
539C23#41×D5312yes—
540C23#41×D5311yescharacteristic
541C23#41×D5311yescharacteristic
542C23C2×#1171994512yes—
543C23#1313326212yes—
544C23#1313326312yes—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—534no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1064no—
587C2×C4—21216no—
588C2×C4—21232no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4C4⋊C4×D5314yes—
605C22—538no—
606C22—1068no—
607C22—1064no—
608C22—1068no—
609C22—1068no—
610C22—1068no—
611C22—1064no—
612C22—1064no—
613C22—1064no—
614C22—1064no—
615C22—1064no—
616C22C2×C4⋊C4×D5311yescharacteristic
617C22#117×D5311yescharacteristic
618C22#117×D5311yescharacteristic
619C22#1681109111yescharacteristic
620C22C4×#1171994511yescharacteristic
621C22#1681109611yescharacteristic
622C22#1681109711yescharacteristic
623C22#1681113212yes—
624C22#124×D5311yescharacteristic
625C22#124×D5311yescharacteristic
626C22#125×D5311yescharacteristic
627C22#125×D5311yescharacteristic
628C22#1681117712yes—
629C22#1681118112yes—
630C22#1681124812yes—
631C22#1681129712yes—
632C22#1681129812yes—
633C22#1681130112yes—
634C22#1681130212yes—
635C22#1681131712yes—
636C22#1681131812yes—
637C22#1681132312yes—
638C22#1681132412yes—
639C4—24no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—1064no—
645C4—4242no—
646C4—2124no—
647C4—2122no—
648C4—2122no—
649C2—1068no—
650C2#382×D5311yescharacteristic
651C2#2027719511yescharacteristic
652C2#2027732811yescharacteristic
653C2#2027764811yescharacteristic
654C2#2027765211yescharacteristic
655C2#2027765611yescharacteristic
656C2#2027765811yescharacteristic
657C2#2027987112yes—
658C2#2027993712yes—
659C2#2027993812yes—
660C2#2028019312yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1434D53
hypocenterC53#13966
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #20272973C2semidirect
5. #20272970C2semidirect
6. #20272942C2semidirect
7. #20272941C2semidirect
8. #20272941C2semidirect
11. C53×#1434C2semidirect
20. C4×#11719921C4semidirect
69. C2×C4×Dic53C2×C4semidirect
79. #13133124C2×C4semidirect
80. #13133124C2×C4semidirect
87. C2×C42×C53C2×C4semidirect
327. #1434D53semidirect
368. C53#13966semidirect
407. C2×C42C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104822
C2×C4×#117199212
C2×#95×D532
#202729732
#202729702
#202729422
#202729412
#202729362
C53×#14342
#1396653