Finite Group Databasebeta

← #27311313

Subgroups — #27311313

Order 13568.

1782 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810482C211yesmaximal, characteristic
2C2×#16810480C211yesmaximal, characteristic
3C2×#95×D53C211yesmaximal, characteristic
4#20272974C211yesmaximal, characteristic, complemented
5#20272973C212yesmaximal, complemented
6#20272972C211yesmaximal, characteristic, complemented
7#20272971C211yesmaximal, characteristic, complemented
8#20272970C212yesmaximal, complemented
9#20272936C212yesmaximal
10#20272936C212yesmaximal
11C53×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D53C2212yes—
13C23×C4×D53C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919C2211yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×#11719921C2212yes—
18C22×#11719921C2211yescharacteristic
19C22×#11719921C2211yescharacteristic
20#16810482—24no—
21#16810480—24no—
22C2×#13133124C2212yes—
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32#95×D53C418yescomplemented
33C22×C4⋊C4×C53C2211yescharacteristic
34C2×#95×C53C2211yescharacteristic
35C2×#95×C53C2212yes—
36C2×#95×C53C2212yes—
37C2×#95×C53C2211yescharacteristic
38C24×D53C2311yescharacteristic
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C22×C4×D53—24no—
45C22×C4×D53—24no—
46C22×C4×D53—42no—
47C22×C4×D53—42no—
48C22×C4×D53—42no—
49C22×C4×D53—42no—
50C22×C4×D53—42no—
51C22×C4×D53—42no—
52C22×C4×D53C2×C418yes—
53C22×C4×D53C2×C414yes—
54C22×C4×D53D414yes—
55C22×C4×D53Q814yes—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—42no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—42no—
61C2×#11719919—42no—
62C2×#11719921—24no—
63C2×#11719921—22no—
64C2×#11719921—22no—
65C2×#11719921—24no—
66C2×#11719921—24no—
67C2×#11719921—24no—
68C2×#11719921—24no—
69#13133124—22no—
70#13133124—24no—
71#13133124—42no—
72#13133124—42no—
73#13133124—42no—
74#13133124—42no—
75#13133124—42no—
76#13133124—42no—
77#13133124C2×C414yes—
78#13133124C2×C418yes—
79C23×C4×C53C2312yes—
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—42no—
86#95×C53—22no—
87#95×C53—42no—
88#95×C53—42no—
89#95×C53C2×C414yes—
90C24×C53C2411yescharacteristic
91C23×D53—22no—
92C23×D53—22no—
93C23×D53—21no—
94C23×D53—21no—
95C23×D53—21no—
96C23×D53—21no—
97C23×D53C22×C412yes—
98C23×D53C2×D412yes—
99C23×D53C2×Q812yes—
100C23×D53#4114yes—
101C23×D53#4114yes—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53C22×C414yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×Q812yes—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
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—21no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53C22×C414yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×Q812yes—
160C2×C4×D53—216no—
161C2×C4×D53—48no—
162C2×C4×D53—48no—
163C2×C4×D53—48no—
164C2×C4×D53—44no—
165C2×C4×D53—44no—
166C2×C4×D53—48no—
167C2×C4×D53—48no—
168C2×C4×D53—48no—
169C2×C4×D53—48no—
170C2×C4×D53—48no—
171C2×C4×D53C4⋊C4116yes—
172C4⋊C4×C53—44no—
173#11719921—48no—
174#11719921—44no—
175#11719921—44no—
176C4⋊Dic53—44no—
177#11719919—44no—
178#11719919—44no—
179C23×C53C23×C411yescharacteristic
180C23×C53C22×D411yescharacteristic
181C23×C53C22×Q811yescharacteristic
182C23×C53C2×#4112yes—
183C23×C53C2×#4112yes—
184C23×C53H(5,𝔽2)12yes—
185C23×C53H(5,𝔽2)11yescharacteristic
186C23×C53H(5,𝔽2)12yes—
187C23×C53H(5,𝔽2)11yescharacteristic
188C23×C532−1+411yescharacteristic
189C23×C532−1+411yescharacteristic
190C22×D53—22no—
191C22×D53—24no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—22no—
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—22no—
207C22×D53—22no—
208C22×D53C2×C4⋊C414yes—
209C22×D53#11718yes—
210C22×D53#12414yes—
211C22×D53#12414yes—
212C22×D53#12518yes—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—42no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—41no—
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×C53C2×C4⋊C414yes—
242C2×Dic53—24no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—42no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—42no—
259C2×Dic53—41no—
260C2×Dic53—41no—
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×Dic53C2×C4⋊C414yes—
271C4×D53—416no—
272C4×D53—432no—
273#14347—531nomaximal, Sylow
274C22×C53C22×C4⋊C411yescharacteristic
275C22×C53C2×#11712yes—
276C22×C53#51811yescharacteristic
277C22×C53#51812yes—
278C22×C53#51911yescharacteristic
279C22×C53C2×#12411yescharacteristic
280C22×C53C2×#12411yescharacteristic
281C22×C53C2×#12512yes—
282C22×C53#53411yescharacteristic
283C22×C53#53411yescharacteristic
284C22×C53#53711yescharacteristic
285C22×C53#53711yescharacteristic
286C22×C53#53812yes—
287C22×C53#53812yes—
288C22×C53#53812yes—
289C22×C53#53812yes—
290C22×C53#53912yes—
291C22×C53#53912yes—
292C22×C53#53912yes—
293C22×C53#53912yes—
294C22×C53#54312yes—
295C22×C53#54411yescharacteristic
296C22×C53#54411yescharacteristic
297C2×D53—28no—
298C2×D53—28no—
299C2×D53—24no—
300C2×D53—24no—
301C2×D53—24no—
302C2×D53—24no—
303C2×D53—24no—
304C2×D53—24no—
305C2×D53—28no—
306C2×D53—28no—
307C2×D53#38218yes—
308C4×C53—24no—
309C4×C53—44no—
310C4×C53—81no—
311C4×C53—81no—
312C4×C53—82no—
313Dic53—24no—
314Dic53—44no—
315Dic53—81no—
316Dic53—81no—
317Dic53—82no—
318C2×#380—531no—
319C2×#380—531no—
320C22×#95—531no—
321#1442—532no—
322#1442—531no—
323#1442—531no—
324#1442—532no—
325#1442—531no—
326#1442D5311yescharacteristic, complemented
327C22⋊#95—532no—
328C22⋊#95—532no—
329C2×C53C2×#38211yescharacteristic
330C2×C53#231212yes—
331C2×C53#231312yes—
332C2×C53#231511yescharacteristic
333C2×C53#231611yescharacteristic
334C2×C53#263911yescharacteristic
335C2×C53#263911yescharacteristic
336C2×C53#263911yescharacteristic
337C2×C53#263911yescharacteristic
338C2×C53#264612yes—
339C2×C53#264612yes—
340D53—28no—
341C24×C4—531no—
342C24×C4—532no—
343C22×C4⋊C4—531no—
344C22×C4⋊C4—531no—
345C22×C4⋊C4—531no—
346C22×C4⋊C4C2×D5311yescharacteristic
347C22×C22⋊C4—531no—
348C22×C22⋊C4—531no—
349C22×C22⋊C4—532no—
350#380—1064no—
351#380—1064no—
352C2×#95—532no—
353C2×#95—531no—
354C2×#95—538no—
355C2×#95—532no—
356C2×#95—531no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95C2×D5311yescharacteristic
364C2×#95C2×D5312yes—
365C2×#95C2×D5312yes—
366C2×#95C2×D5311yescharacteristic
367C53#1434711yescharacteristic, complemented, Sylow
368C25—531no—
369C23×C4—531no—
370C23×C4—534no—
371C23×C4—534no—
372C23×C4—534no—
373C23×C4—532no—
374C23×C4—538no—
375C23×C4—531no—
376C23×C4—531no—
377C23×C4—532no—
378C23×C4—1064no—
379C23×C4—1064no—
380C23×C4—2122no—
381C23×C4—2122no—
382C23×C4—2122no—
383C23×C4—2122no—
384C23×C4—2122no—
385C23×C4—2122no—
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5312yes—
389C23×C4C22×D5312yes—
390C23×C4C22×D5311yescharacteristic
391C2×C4⋊C4—22no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—2122no—
397C2×C4⋊C4—2122no—
398C2×C4⋊C4—2122no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—1062no—
405C2×C22⋊C4—1064no—
406#95—22no—
407#95—42no—
408#95—42no—
409#95—538no—
410#95—534no—
411#95—1064no—
412#95—1062no—
413#95—2122no—
414#95—2122no—
415#95—2122no—
416#95—2122no—
417#95—2122no—
418#95—2122no—
419#95C4×D5314yes—
420C24—532no—
421C24—532no—
422C24—532no—
423C24—534no—
424C24—534no—
425C24—1062no—
426C24—1061no—
427C24—1062no—
428C24—1061no—
429C24—1061no—
430C24—1061no—
431C24C23×D5311yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—532no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—5316no—
461C22×C4—534no—
462C22×C4—10616no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1061no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1062no—
488C22×C4—2128no—
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×C4C2×C4×D5314yes—
499C22×C4C2×C4×D5312yes—
500C22×C4D4×D5312yes—
501C22×C4Q8×D5312yes—
502C4⋊C4—44no—
503C4⋊C4—2124no—
504C4⋊C4—2124no—
505C4⋊C4—2124no—
506C22⋊C4—2124no—
507C22⋊C4—2124no—
508C22⋊C4—2128no—
509C23—534no—
510C23—538no—
511C23—534no—
512C23—534no—
513C23—538no—
514C23—1062no—
515C23—1064no—
516C23—1062no—
517C23—1062no—
518C23—1062no—
519C23—1062no—
520C23—1062no—
521C23—1064no—
522C23—1064no—
523C23—1064no—
524C23—1064no—
525C23—1064no—
526C23—1064no—
527C23—1064no—
528C23—1064no—
529C23—1064no—
530C23—1062no—
531C23—1062no—
532C23C22×C4×D5311yescharacteristic
533C23C2×D4×D5311yescharacteristic
534C23C2×Q8×D5311yescharacteristic
535C23#41×D5312yes—
536C23#41×D5312yes—
537C23#1313326212yes—
538C23#1313326211yescharacteristic
539C23#1313326312yes—
540C23#1313326311yescharacteristic
541C23#1313326411yescharacteristic
542C23#1313326511yescharacteristic
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
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—534no—
572C2×C4—1064no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—21216no—
580C2×C4—21232no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2122no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2122no—
587C2×C4—2121no—
588C2×C4—2121no—
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×C4C4⋊C4×D5314yes—
603C22—538no—
604C22—1068no—
605C22—1068no—
606C22—1064no—
607C22—1064no—
608C22—1064no—
609C22—1064no—
610C22—1064no—
611C22—1064no—
612C22—1068no—
613C22—1068no—
614C22C2×C4⋊C4×D5311yescharacteristic
615C22#117×D5312yes—
616C22#1681109511yescharacteristic
617C22#1681109512yes—
618C22#1681109911yescharacteristic
619C22#124×D5311yescharacteristic
620C22#124×D5311yescharacteristic
621C22#125×D5312yes—
622C22#1681118311yescharacteristic
623C22#1681118411yescharacteristic
624C22#1681120211yescharacteristic
625C22#1681120311yescharacteristic
626C22#1681120512yes—
627C22#1681120712yes—
628C22#1681120912yes—
629C22#1681121212yes—
630C22#1681121412yes—
631C22#1681121612yes—
632C22#1681121912yes—
633C22#1681122112yes—
634C22#1681124812yes—
635C22#1681125011yescharacteristic
636C22#1681125311yescharacteristic
637C4—24no—
638C4—44no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4—1064no—
643C4—4241no—
644C4—4241no—
645C4—4242no—
646C4—2124no—
647C2—1068no—
648C2#382×D5311yescharacteristic
649C2#2027730912yes—
650C2#2027731412yes—
651C2#2027732611yescharacteristic
652C2#2027733011yescharacteristic
653C2#2028064211yescharacteristic
654C2#2028064411yescharacteristic
655C2#2028064611yescharacteristic
656C2#2028064711yescharacteristic
657C2#2028068412yes—
658C2#2028068612yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1442C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1442D53
hypocenterC53#14347
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #20272974C2semidirect
5. #20272973C2semidirect
6. #20272972C2semidirect
7. #20272971C2semidirect
8. #20272970C2semidirect
11. C53×#1442C2semidirect
32. #95×D53C4semidirect
326. #1442D53semidirect
367. C53#14347semidirect

Maximal subgroups

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

subgroupindex
C2×#168104822
C2×#168104802
C2×#95×D532
#202729742
#202729732
#202729722
#202729712
#202729702
#202729362
C53×#14422
#1434753