Finite Group Databasebeta

← #27310522

Subgroups — #27310522

Order 13568.

1926 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133175C211yesmaximal, characteristic
2C2×#16810520C212yesmaximal, complemented
3C2×#16810482C211yesmaximal, characteristic
4C2×#16810482C211yesmaximal, characteristic
5#20272980C211yesmaximal, characteristic, complemented
6#20272979C211yesmaximal, characteristic, complemented
7#20272974C211yesmaximal, characteristic, complemented
8#20272961C212yesmaximal
9#20272959C212yesmaximal
10#20272943C212yesmaximal, complemented
11C53×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C2×#13133175—24no—
14C2×#13133175—28no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2212yes—
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×#11719919C2212yes—
20C22×#11719919C2212yes—
21C22×#11719921C2212yes—
22C22×#11719921C2211yescharacteristic
23C22×#11719921C2211yescharacteristic
24C22×#11719921C2212yes—
25C22×C4×Dic53C2212yes—
26#16810520—28no—
27#16810482—24no—
28#16810482C418yescomplemented
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C2×#95×C53C2212yes—
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C24×D53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2312yes—
45C23×Dic53C2311yescharacteristic
46C22×C4×D53—24no—
47C22×C4×D53—24no—
48C22×C4×D53C2×C414yes—
49C22×C4×D53D414yes—
50C22×C4×D53Q814yes—
51#13133175—232no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—24no—
54C2×C4⋊Dic53—24no—
55C2×C4⋊Dic53—42no—
56C2×C4⋊Dic53—42no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—42no—
65C2×#11719919—42no—
66C2×#11719919C2×C418yes—
67C2×#11719921—24no—
68C2×#11719921—22no—
69C2×#11719921—22no—
70C2×#11719921—24no—
71C2×#11719921—24no—
72C2×#11719921—42no—
73C2×#11719921—42no—
74C2×#11719921—42no—
75C2×#11719921—42no—
76C2×#11719921C2×C418yes—
77C2×C4×Dic53—24no—
78C2×C4×Dic53—24no—
79C2×C4×Dic53D418yes—
80#13133124—24no—
81#13133124—22no—
82#13133124—42no—
83#13133124—42no—
84#13133124—42no—
85#13133124C2×C414yes—
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2312yes—
89C23×C4×C53C2312yes—
90C23×C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53—24no—
92C2×C4⋊C4×C53—22no—
93#95×C53—24no—
94#95×C53—22no—
95#95×C53—42no—
96#95×C53C2×C414yes—
97C24×C53C2411yescharacteristic
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—21no—
101C23×D53—21no—
102C23×D53—22no—
103C23×D53—21no—
104C23×D53—21no—
105C23×D53—22no—
106C23×D53C22×C412yes—
107C23×D53C2×D412yes—
108C23×D53C2×Q812yes—
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—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—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C22×C414yes—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D414yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×Q812yes—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
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—21no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53C22×C414yes—
161C22×Dic53C22×C412yes—
162C22×Dic53C2×D414yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×D414yes—
165C22×Dic53C2×Q812yes—
166C22×Dic53#4114yes—
167C2×C4×D53—216no—
168C2×C4×D53—28no—
169C2×C4×D53—216no—
170C2×C4×D53—28no—
171C2×C4×D53C4⋊C4116yescomplemented
172C4⋊C4×C53—28no—
173#11719921—216no—
174#11719921—48no—
175#11719921—44no—
176#11719921—44no—
177C4⋊Dic53—28no—
178C4⋊Dic53—48no—
179#11719919—216no—
180#11719919—48no—
181#11719919—44no—
182#11719919—44no—
183C4×Dic53—216no—
184C23×C53C23×C411yescharacteristic
185C23×C53C22×D411yescharacteristic
186C23×C53C22×D412yes—
187C23×C53C22×Q811yescharacteristic
188C23×C53C2×#4112yes—
189C23×C53H(5,𝔽2)11yescharacteristic
190C23×C53H(5,𝔽2)11yescharacteristic
191C23×C53H(5,𝔽2)12yes—
192C23×C532−1+411yescharacteristic
193C23×C532−1+412yes—
194C23×C532−1+411yescharacteristic
195C22×D53—24no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—24no—
199C22×D53—22no—
200C22×D53—24no—
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—24no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53C2×C4⋊C414yes—
218C2×C4×C53—24no—
219C2×C4×C53—22no—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
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—41no—
237C2×C4×C53—41no—
238C2×C4×C53—42no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53C2×C4⋊C414yes—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—42no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53C2×C4⋊C414yes—
272C2×Dic53C4×D418yes—
273C2×Dic53#12014yes—
274C2×Dic53C22⋊Q818yes—
275C2×Dic53#12314yes—
276C4×D53—232no—
277#14264—531nomaximal, Sylow
278C22×C53C22×C4⋊C411yescharacteristic
279C22×C53C2×C4×D412yes—
280C22×C53#51811yescharacteristic
281C22×C53#51911yescharacteristic
282C22×C53#52012yes—
283C22×C53C2×#12011yescharacteristic
284C22×C53C2×C22⋊Q812yes—
285C22×C53C2×#12311yescharacteristic
286C22×C53#53511yescharacteristic
287C22×C53#53512yes—
288C22×C53#53611yescharacteristic
289C22×C53#53612yes—
290C22×C53#53712yes—
291C22×C53#53711yescharacteristic
292C22×C53#53711yescharacteristic
293C22×C53#53712yes—
294C22×C53#54012yes—
295C22×C53#54112yes—
296C22×C53#54212yes—
297C22×C53#54212yes—
298C22×C53#54411yescharacteristic
299C22×C53#54411yescharacteristic
300C22×C53#54412yes—
301C2×D53—24no—
302C2×D53—24no—
303C2×D53—24no—
304C2×D53—44no—
305C2×D53—44no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—42no—
310C2×D53—44no—
311C2×D53—44no—
312C4×C53—24no—
313C4×C53—44no—
314C4×C53—44no—
315C4×C53—81no—
316C4×C53—81no—
317Dic53—44no—
318Dic53—44no—
319Dic53—81no—
320Dic53—81no—
321Dic53#38518yes—
322C22×#117—531no—
323C2×#389—532no—
324C2×#380—531no—
325C2×#380—531no—
326#1444—531no—
327#1444—531no—
328#1442—531no—
329#1442D5311yescharacteristic, complemented
330#1439—532no—
331#1439—532no—
332C4⋊#95—532no—
333C2×C53C2×#38511yescharacteristic
334C2×C53#231611yescharacteristic
335C2×C53#231711yescharacteristic
336C2×C53#232312yes—
337C2×C53#232412yes—
338C2×C53#261511yescharacteristic
339C2×C53#261611yescharacteristic
340C2×C53#262411yescharacteristic
341C2×C53#262511yescharacteristic
342C2×C53#262812yes—
343C2×C53#262912yes—
344D53—44no—
345C24×C4—531no—
346C2×#117—1068no—
347C2×#117—1064no—
348C22×C4⋊C4—532no—
349C22×C4⋊C4—531no—
350C22×C4⋊C4—531no—
351C22×C4⋊C4—531no—
352C22×C4⋊C4—532no—
353C22×C4⋊C4—532no—
354C22×C4⋊C4C2×D5311yescharacteristic
355C22×C22⋊C4—532no—
356C22×C22⋊C4—532no—
357C22×C22⋊C4—531no—
358C22×C22⋊C4—531no—
359C22×C42—532no—
360#389—1068no—
361#380—538no—
362#380—1064no—
363C2×#95—531no—
364C2×#95—532no—
365C2×#95—531no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95C2×D5312yes—
370C2×#95C2×D5311yescharacteristic
371C2×#95C2×D5312yes—
372C2×#95C2×D5311yescharacteristic
373C53#1426411yescharacteristic, complemented, Sylow
374C25—531no—
375C23×C4—531no—
376C23×C4—534no—
377C23×C4—534no—
378C23×C4—534no—
379C23×C4—532no—
380C23×C4—532no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4—1064no—
384C23×C4—1064no—
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5312yes—
388C23×C4C22×D5312yes—
389C23×C4C22×D5311yescharacteristic
390#117—10632no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—538no—
394C2×C4⋊C4—1064no—
395C2×C4⋊C4—1064no—
396C2×C4⋊C4—1064no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1064no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1064no—
402C2×C4⋊C4—1064no—
403C2×C4⋊C4—1064no—
404C2×C4⋊C4—2122no—
405C2×C4⋊C4—2122no—
406C2×C4⋊C4—2122no—
407C2×C4⋊C4—2122no—
408C2×C22⋊C4—538no—
409C2×C22⋊C4—1064no—
410C2×C22⋊C4—1064no—
411C2×C22⋊C4—1064no—
412C2×C22⋊C4—1062no—
413C2×C22⋊C4—1062no—
414C2×C22⋊C4—2122no—
415C2×C22⋊C4—2122no—
416C2×C22⋊C4—2122no—
417C2×C22⋊C4—2122no—
418C2×C42—538no—
419C2×C42—1064no—
420C2×C42—1064no—
421#95—22no—
422#95—24no—
423#95—42no—
424#95—534no—
425#95—1062no—
426#95—1064no—
427#95—2122no—
428#95—2122no—
429#95—2122no—
430#95C4×D5314yes—
431C24—532no—
432C24—532no—
433C24—532no—
434C24—1062no—
435C24—1062no—
436C24—1061no—
437C24—1061no—
438C24—1062no—
439C24—1061no—
440C24—1062no—
441C24—1061no—
442C24C23×D5311yescharacteristic
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—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—532no—
468C22×C4—534no—
469C22×C4—532no—
470C22×C4—532no—
471C22×C4—534no—
472C22×C4—5316no—
473C22×C4—534no—
474C22×C4—534no—
475C22×C4—1068no—
476C22×C4—10616no—
477C22×C4—10616no—
478C22×C4—1068no—
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×C4—1062no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1061no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4C2×C4×D5314yes—
502C22×C4C2×C4×D5312yes—
503C22×C4D4×D5314yes—
504C22×C4D4×D5312yes—
505C22×C4Q8×D5312yes—
506C4⋊C4—28no—
507C4⋊C4—10616no—
508C4⋊C4—1068no—
509C4⋊C4—2124no—
510C4⋊C4—2124no—
511C4⋊C4—2128no—
512C4⋊C4—2128no—
513C22⋊C4—10616no—
514C22⋊C4—2128no—
515C22⋊C4—2124no—
516C22⋊C4—2124no—
517C42—10616no—
518C23—534no—
519C23—1064no—
520C23—1062no—
521C23—1062no—
522C23—1064no—
523C23—1062no—
524C23—1064no—
525C23—1062no—
526C23—1062no—
527C23—1062no—
528C23—1062no—
529C23—1062no—
530C23—1062no—
531C23—1062no—
532C23—1064no—
533C23—2122no—
534C23—2122no—
535C23—2122no—
536C23—2122no—
537C23—2122no—
538C23—2122no—
539C23—2122no—
540C23—2122no—
541C23C22×C4×D5311yescharacteristic
542C23C2×D4×D5311yescharacteristic
543C23C2×D4×D5312yes—
544C23C2×Q8×D5311yescharacteristic
545C23#41×D5312yes—
546C23#1313326211yescharacteristic
547C23#1313326311yescharacteristic
548C23#1313326312yes—
549C23#1313326411yescharacteristic
550C23#1313326412yes—
551C23#1313326511yescharacteristic
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
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—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—534no—
581C2×C4—538no—
582C2×C4—534no—
583C2×C4—538no—
584C2×C4—534no—
585C2×C4—10632no—
586C2×C4—1064no—
587C2×C4—1064no—
588C2×C4—1064no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2122no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4C4⋊C4×D5314yes—
611C22—1064no—
612C22—1064no—
613C22—1064no—
614C22—2122no—
615C22—2122no—
616C22—2124no—
617C22—2124no—
618C22—2122no—
619C22—2122no—
620C22—2124no—
621C22—2124no—
622C22C2×C4⋊C4×D5311yescharacteristic
623C22C4×D4×D5312yes—
624C22#1681109511yescharacteristic
625C22#1681109911yescharacteristic
626C22#1681110312yes—
627C22#120×D5311yescharacteristic
628C22C22⋊Q8×D5312yes—
629C22#123×D5311yescharacteristic
630C22#1681119111yescharacteristic
631C22#1681119112yes—
632C22#1681119311yescharacteristic
633C22#1681119512yes—
634C22#1681120012yes—
635C22#1681120211yescharacteristic
636C22#1681120311yescharacteristic
637C22#1681120312yes—
638C22#1681122512yes—
639C22#1681123212yes—
640C22#1681123912yes—
641C22#1681123912yes—
642C22#1681125011yescharacteristic
643C22#1681125311yescharacteristic
644C22#1681125312yes—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—81no—
649C4—81no—
650C4—538no—
651C4—4241no—
652C4—4241no—
653C4—2124no—
654C4—2124no—
655C2—2124no—
656C2#385×D5311yescharacteristic
657C2#2027733011yescharacteristic
658C2#2027734011yescharacteristic
659C2#2027739612yes—
660C2#2027740312yes—
661C2#2028044211yescharacteristic
662C2#2028044311yescharacteristic
663C2#2028051011yescharacteristic
664C2#2028051611yescharacteristic
665C2#2028054312yes—
666C2#2028054712yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1442C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1442D53
hypocenterC53#14264
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
2. C2×#16810520C2semidirect
5. #20272980C2semidirect
6. #20272979C2semidirect
7. #20272974C2semidirect
10. #20272943C2semidirect
11. C53×#1442C2semidirect
28. #16810482C4semidirect
171. C2×C4×D53C4⋊C4semidirect
329. #1442D53semidirect
373. C53#14264semidirect

Maximal subgroups

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

subgroupindex
C22×#131331752
C2×#168105202
C2×#168104822
#202729802
#202729792
#202729742
#202729612
#202729592
#202729432
C53×#14422
#1426453