Finite Group Databasebeta

← #27311275

Subgroups — #27311275

Order 13568.

1926 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133176C211yesmaximal, characteristic
2C2×#16810492C211yesmaximal, characteristic, complemented
3C2×#16810488C211yesmaximal, characteristic, complemented
4C2×#16810483C211yesmaximal, characteristic
5#20272967C214yesmaximal, complemented
6#20272958C214yesmaximal
7#20272944C211yesmaximal, characteristic, complemented
8#20272944C211yesmaximal, characteristic, complemented
9C2×C53×#380C211yesmaximal, characteristic
10C2×#13133176—24no—
11C2×#13133176—24no—
12C2×#13133176C418yescomplemented
13C22×C4⋊Dic53C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4⋊Dic53C2212yes—
16C22×#11719919C2214yes—
17C22×C22⋊Dic53C2212yes—
18C22×C22⋊Dic53C2212yes—
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21#16810492—24no—
22#16810488—24no—
23#16810483—24no—
24C2×#13133124C2214yes—
25C2×#13133124C2212yes—
26C2×#13133124C2214yes—
27C2×#13133124C2214yes—
28C24×C4×C53C2211yescharacteristic
29C22×C4⋊C4×C53C2211yescharacteristic
30C22×C4⋊C4×C53C2211yescharacteristic
31C22×C22⋊C4×C53C2212yes—
32C53×#380—24no—
33C2×#95×C53C2212yes—
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C23×Dic53C2312yes—
38#13133176—232no—
39C2×C4⋊Dic53—22no—
40C2×C4⋊Dic53—22no—
41C2×C4⋊Dic53—22no—
42C2×C4⋊Dic53—22no—
43C2×C4⋊Dic53—22no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—24no—
46C2×C4⋊Dic53—24no—
47C2×C4⋊Dic53C2×C418yes—
48C2×#11719919—28no—
49C2×#11719919—44no—
50C2×#11719919—44no—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—24no—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—44no—
55C2×C22⋊Dic53C2×C418yes—
56C2×C4×Dic53—22no—
57C2×C4×Dic53—22no—
58C2×C4×Dic53C2×C414yes—
59C2×C4×Dic53C2×C414yes—
60C2×C4×Dic53D414yes—
61C2×C4×Dic53D414yes—
62#13133124—24no—
63#13133124—28no—
64#13133124—44no—
65#13133124—44no—
66C25×C53C2311yescharacteristic
67C23×C4×C53—22no—
68C23×C4×C53—22no—
69C23×C4×C53—22no—
70C23×C4×C53—22no—
71C23×C4×C53C2×C414yes—
72C23×C4×C53D414yes—
73C23×C4×C53Q814yes—
74C23×C4×C53C2312yes—
75C23×C4×C53C2312yes—
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C2×C4⋊C4×C53—22no—
79C2×C4⋊C4×C53—22no—
80C2×C4⋊C4×C53—22no—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53—22no—
84C2×C22⋊C4×C53—24no—
85C2×C22⋊C4×C53—44no—
86#95×C53—24no—
87C24×C53—21no—
88C24×C53—21no—
89C24×C53—21no—
90C24×C53—21no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—22no—
95C24×C53C22×C412yes—
96C24×C53C2×D412yes—
97C24×C53C2×Q812yes—
98C24×C53C2411yescharacteristic
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—28no—
104C22×C4×C53—28no—
105C22×C4×C53—28no—
106C22×C4×C53—28no—
107C22×C4×C53—28no—
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—24no—
116C22×C4×C53—24no—
117C22×C4×C53—24no—
118C22×C4×C53—24no—
119C22×C4×C53C4⋊C4116yescomplemented
120C22×C4×C53C22×C412yes—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×Q812yes—
127C22×C4×C53C2×Q812yes—
128C22×C4×C53#4112yes—
129C22×C4×C53#4112yes—
130C22×Dic53—22no—
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—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—24no—
147C22×Dic53—24no—
148C22×Dic53—24no—
149C22×Dic53—24no—
150C22×Dic53C22×C414yes—
151C22×Dic53C22×C414yes—
152C22×Dic53C2×D414yes—
153C22×Dic53C2×D414yes—
154C4⋊C4×C53—44no—
155C4⋊C4×C53—44no—
156C22⋊C4×C53—48no—
157C22⋊Dic53—216no—
158C22⋊Dic53—48no—
159C4⋊Dic53—216no—
160C4⋊Dic53—44no—
161C4⋊Dic53—44no—
162#11719919—416no—
163C4×Dic53—28no—
164C4×Dic53C22⋊C4116yescomplemented
165C23×C53—22no—
166C23×C53—22no—
167C23×C53—24no—
168C23×C53—24no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—24no—
178C23×C53—24no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53C2×C4⋊C414yes—
188C23×C53C23×C411yescharacteristic
189C23×C53C22×D411yescharacteristic
190C23×C53C22×D411yescharacteristic
191C23×C53C22×D411yescharacteristic
192C23×C53C22×Q811yescharacteristic
193C23×C53C2×#4111yescharacteristic
194C23×C53C2×#4111yescharacteristic
195C23×C53H(5,𝔽2)12yes—
196C23×C53H(5,𝔽2)12yes—
197C23×C532−1+412yes—
198C23×C532−1+412yes—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—232no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—44no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—44no—
214C2×C4×C53—44no—
215C2×C4×C53—44no—
216C2×C4×C53C2×C22⋊C414yes—
217C2×C4×C53C2×C4⋊C414yes—
218C2×C4×C53C2×C4⋊C414yes—
219C2×C4×C53#11714yes—
220C2×C4×C53#12114yes—
221C2×C4×C53#12114yes—
222C2×C4×C53C22⋊Q814yes—
223C2×C4×C53C22⋊Q814yes—
224C2×Dic53—24no—
225C2×Dic53—24no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—44no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—44no—
245C2×Dic53—44no—
246C2×Dic53—44no—
247C2×Dic53C2×C22⋊C418yes—
248#14340—531nomaximal, Sylow
249C22×C53—24no—
250C22×C53—24no—
251C22×C53—24no—
252C22×C53—44no—
253C22×C53—44no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—44no—
259C22×C53—44no—
260C22×C53C22×C22⋊C411yescharacteristic
261C22×C53C22×C4⋊C411yescharacteristic
262C22×C53C2×#11711yescharacteristic
263C22×C53#52012yes—
264C22×C53#52012yes—
265C22×C53C2×#12111yescharacteristic
266C22×C53C2×#12111yescharacteristic
267C22×C53C2×C22⋊Q811yescharacteristic
268C22×C53C2×C22⋊Q811yescharacteristic
269C22×C53#53511yescharacteristic
270C22×C53#53512yes—
271C22×C53#53511yescharacteristic
272C22×C53#53611yescharacteristic
273C22×C53#53612yes—
274C22×C53#53611yescharacteristic
275C22×C53#53712yes—
276C22×C53#53712yes—
277C22×C53#54012yes—
278C22×C53#54112yes—
279C22×C53#54212yes—
280C22×C53#54212yes—
281C22×C53#54412yes—
282C22×C53#54412yes—
283C4×C53—24no—
284C4×C53—82no—
285C4×C53—82no—
286C4×C53#38018yes—
287Dic53—28no—
288Dic53—44no—
289Dic53—82no—
290Dic53—82no—
291C22×#117—531no—
292C2×#384—531no—
293C2×#382—531no—
294C2×#380—531no—
295C2×#380D5311yescharacteristic
296#1441—534no—
297#1439—534no—
298C4⋊#95—531no—
299C4⋊#95—531no—
300C2×C53—44no—
301C2×C53C2×#38011yescharacteristic
302C2×C53#231112yes—
303C2×C53#231412yes—
304C2×C53#231711yescharacteristic
305C2×C53#231711yescharacteristic
306C2×C53#261911yescharacteristic
307C2×C53#262011yescharacteristic
308C2×C53#262011yescharacteristic
309C2×C53#262111yescharacteristic
310C2×C53#262812yes—
311C2×C53#262912yes—
312C24×C4C2×D5311yescharacteristic
313C2×#117—538no—
314C2×#117—1064no—
315C2×#117—1064no—
316C22×C4⋊C4—532no—
317C22×C4⋊C4—531no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4—534no—
320C22×C4⋊C4C2×D5311yescharacteristic
321C22×C4⋊C4C2×D5311yescharacteristic
322C22×C22⋊C4—532no—
323C22×C22⋊C4—532no—
324C22×C22⋊C4C2×D5312yes—
325C22×C42—531no—
326C22×C42—531no—
327#384—1064no—
328#382—1064no—
329#380—24no—
330#380—1064no—
331C2×#95—534no—
332C2×#95—532no—
333C2×#95—534no—
334C2×#95—534no—
335C2×#95C2×D5312yes—
336C53#1434011yescharacteristic, complemented, Sylow
337C25C22×D5311yescharacteristic
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—532no—
343C23×C4—532no—
344C23×C4—532no—
345C23×C4—532no—
346C23×C4Dic10614yes—
347C23×C4C4×D5314yes—
348C23×C4C22⋊D5314yes—
349C23×C4C22×D5312yes—
350C23×C4C22×D5312yes—
351C23×C4C22×D5311yescharacteristic
352C23×C4C22×D5311yescharacteristic
353#117—10632no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—538no—
361C2×C4⋊C4—1064no—
362C2×C4⋊C4—1064no—
363C2×C4⋊C4—1068no—
364C2×C4⋊C4—1062no—
365C2×C4⋊C4—1062no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—2124no—
371C2×C4⋊C4—2124no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4—538no—
375C2×C22⋊C4—1064no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—1064no—
378C2×C22⋊C4—2124no—
379C2×C42—534no—
380C2×C42—534no—
381C2×C42—534no—
382C2×C42—534no—
383C2×C42—1062no—
384C2×C42—1062no—
385#95—24no—
386#95—1068no—
387#95—1064no—
388#95—2124no—
389#95—2124no—
390C24—21no—
391C24—21no—
392C24—21no—
393C24—21no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24C2×Dic10612yes—
399C24C2×C4×D5312yes—
400C24C2×C22⋊D5312yes—
401C24C23×D5311yescharacteristic
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—534no—
423C22×C4—534no—
424C22×C4—534no—
425C22×C4—534no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1064no—
443C22×C4—1064no—
444C22×C4—1064no—
445C22×C4—1064no—
446C22×C4#11719919116yescomplemented
447C22×C4C2×Dic10612yes—
448C22×C4C2×Dic10612yes—
449C22×C4C2×C4×D5312yes—
450C22×C4C2×C4×D5312yes—
451C22×C4D4×D5312yes—
452C22×C4D4×D5312yes—
453C22×C4#1171994712yes—
454C22×C4#1171994712yes—
455C22×C4C2×C22⋊D5312yes—
456C22×C4C2×C22⋊D5312yes—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—10616no—
460C4⋊C4—21216no—
461C4⋊C4—2124no—
462C4⋊C4—2124no—
463C22⋊C4—48no—
464C22⋊C4—10616no—
465C22⋊C4—2128no—
466C42—5316no—
467C42—1068no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23C2×#1171991914yes—
491C23C22×C4×D5311yescharacteristic
492C23C2×D4×D5311yescharacteristic
493C23C2×D4×D5311yescharacteristic
494C23C22×C22⋊D5311yescharacteristic
495C23C22×Dic10611yescharacteristic
496C23C2×#1171994711yescharacteristic
497C23C2×#1171994711yescharacteristic
498C23#1313326212yes—
499C23#1313326312yes—
500C23#1313326412yes—
501C23#1313326512yes—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—232no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—538no—
520C2×C4—1064no—
521C2×C4—1064no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—1064no—
527C2×C4—1064no—
528C2×C4—1064no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2124no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2124no—
537C2×C4—2124no—
538C2×C4—2124no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4C22⋊C4×D5314yes—
544C2×C4C2×#1171991914yes—
545C2×C4C2×#1171991914yes—
546C2×C4#1313317314yes—
547C2×C4#1313318914yes—
548C2×C4#1313318914yes—
549C2×C4#1313319414yes—
550C2×C4#1313319414yes—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—44no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22C2×C22⋊C4×D5311yescharacteristic
563C22C22×#1171991911yescharacteristic
564C22C2×#1313317311yescharacteristic
565C22#1681110312yes—
566C22#1681110412yes—
567C22C2×#1313318911yescharacteristic
568C22C2×#1313318911yescharacteristic
569C22C2×#1313319411yescharacteristic
570C22C2×#1313319411yescharacteristic
571C22#1681118711yescharacteristic
572C22#1681118912yes—
573C22#1681119011yescharacteristic
574C22#1681119411yescharacteristic
575C22#1681119612yes—
576C22#1681119711yescharacteristic
577C22#1681120212yes—
578C22#1681120312yes—
579C22#1681122612yes—
580C22#1681123312yes—
581C22#1681124012yes—
582C22#1681124612yes—
583C22#1681125112yes—
584C22#1681125412yes—
585C4—24no—
586C4—82no—
587C4—82no—
588C4—1068no—
589C4—4242no—
590C4—4242no—
591C4—2124no—
592C4#1681048118yes—
593C2—44no—
594C2C2×#1681048111yescharacteristic
595C2#2027730412yes—
596C2#2027732212yes—
597C2#2027733511yescharacteristic
598C2#2027733911yescharacteristic
599C2#2028046911yescharacteristic
600C2#2028047611yescharacteristic
601C2#2028047711yescharacteristic
602C2#2028048511yescharacteristic
603C2#2028054012yes—
604C2#2028054912yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#380C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×#380D53
hypocenterC53#14340
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#16810492C2semidirect
3. C2×#16810488C2semidirect
5. #20272967C2semidirect
7. #20272944C2semidirect
8. #20272944C2semidirect
12. C2×#13133176C4semidirect
119. C22×C4×C53C4⋊C4semidirect
164. C4×Dic53C22⋊C4semidirect
336. C53#14340semidirect
446. C22×C4#11719919semidirect

Maximal subgroups

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

subgroupindex
C22×#131331762
C2×#168104922
C2×#168104882
C2×#168104832
#202729672
#202729582
#202729442
C2×C53×#3802
#1434053