Finite Group Databasebeta

← #27556830

Subgroups — #27556830

Order 13568.

2042 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283021C214yesmaximal, complemented
2#20280687C214yesmaximal, complemented
3#20279214C214yesmaximal, complemented
4#20279212C214yesmaximal, complemented
5#20278861C214yesmaximal, complemented
6#20278853C214yesmaximal, complemented
7#20277327C211yesmaximal, characteristic
8#20277327C211yesmaximal, characteristic
9#20277327C211yesmaximal, characteristic
10C2×#16810498C211yesmaximal, characteristic
11C2×#16810495C211yesmaximal, characteristic
12C2×C22⋊C4×Dic53C211yesmaximal, characteristic
13C53×#2315C211yesmaximal, characteristic
14C24×Dic53C2211yescharacteristic
15C2×#13133207C2214yescomplemented
16C22×C4⋊Dic53C2212yes—
17C22×#11719919C2212yes—
18C22×C22⋊Dic53C2211yescharacteristic
19C22×C22⋊Dic53C2211yescharacteristic
20C22×C22⋊Dic53C2211yescharacteristic
21C22×C4×Dic53C2212yes—
22#16810564C2214yescomplemented
23#16810564C2214yescomplemented
24#16810563C2214yescomplemented
25#16810561C2214yescomplemented
26#16810557C2214yescomplemented
27#16810556C2214yescomplemented
28#16810554C2214yescomplemented
29#16810553C2214yescomplemented
30#16810520C2214yescomplemented
31#16810518C2214yescomplemented
32#16810509C2214yescomplemented
33#16810508C2214yescomplemented
34#16810505C2214yescomplemented
35#16810504C2214yescomplemented
36#16810498C2214yes—
37#16810498C2212yes—
38#16810498C2212yes—
39#16810496C2214yes—
40#16810496C2212yes—
41#16810496C2214yes—
42#16810496C2212yes—
43#16810495C2212yes—
44#16810495C2214yes—
45#16810495C2214yes—
46#16810495C2214yes—
47#16810495C2212yes—
48#16810495C2212yes—
49#16810495C2212yes—
50#16810487C2214yescomplemented
51#16810481C2212yes—
52#16810481C2214yes—
53#16810481C2214yes—
54#16810481C2212yes—
55C22⋊C4×Dic53C2214yes—
56C22⋊C4×Dic53C2214yes—
57C2×#13133124C2212yes—
58C2×#13133124C2212yes—
59C2×#13133124C2212yes—
60C22×C22⋊C4×C53C2211yescharacteristic
61C22×C22⋊C4×C53C2211yescharacteristic
62C22×C22⋊C4×C53C2211yescharacteristic
63C53×#385C2214yes—
64C53×#385C2214yes—
65C53×#380C2212yes—
66C53×#380C2212yes—
67C23×Dic53—24no—
68C23×Dic53—44no—
69C23×Dic53C2311yescharacteristic
70C23×Dic53C2312yes—
71C23×Dic53C2312yes—
72C23×Dic53C2311yescharacteristic
73C23×Dic53C2312yes—
74C23×Dic53C2311yescharacteristic
75C23×Dic53C2311yescharacteristic
76C23×Dic53C2312yes—
77#13133207—48no—
78C2×C4⋊Dic53—44no—
79C2×C4⋊Dic53C2314yes—
80C2×C4⋊Dic53C2314yes—
81C2×#11719919—44no—
82C2×#11719919C2314yes—
83C2×#11719919C2314yes—
84C2×#11719919C2314yes—
85C2×#11719919C2314yes—
86C2×#11719919C2314yes—
87C2×#11719919C2314yes—
88C2×#11719919C2314yes—
89C2×#11719919C2314yes—
90C2×C22⋊Dic53—24no—
91C2×C22⋊Dic53—24no—
92C2×C22⋊Dic53Q818yes—
93C2×C22⋊Dic53C2312yes—
94C2×C22⋊Dic53C2312yes—
95C2×C22⋊Dic53C2312yes—
96C2×C22⋊Dic53C2312yes—
97C2×C22⋊Dic53C2312yes—
98C2×C22⋊Dic53C2312yes—
99C2×C4×Dic53—44no—
100C2×C4×Dic53C2314yes—
101C2×C4×Dic53C2314yes—
102#13133124C2314yes—
103#13133124C2314yes—
104#13133124C2314yes—
105#13133124C2314yes—
106#13133124C2314yes—
107#13133124C2314yes—
108#13133124C2314yes—
109#13133124C2314yes—
110#13133124C2314yes—
111#13133124C2314yes—
112#13133124C2314yes—
113#13133124C2314yes—
114C25×C53C2311yescharacteristic
115C23×C4×C53C2312yes—
116C23×C4×C53C2311yescharacteristic
117C23×C4×C53C2312yes—
118C23×C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2314yes—
120C2×C4⋊C4×C53C2314yes—
121C2×C22⋊C4×C53—24no—
122C2×C22⋊C4×C53—24no—
123C2×C22⋊C4×C53—24no—
124C2×C22⋊C4×C53C2312yes—
125C2×C22⋊C4×C53C2312yes—
126C2×C22⋊C4×C53C2312yes—
127C2×C22⋊C4×C53C2312yes—
128C2×C22⋊C4×C53C2312yes—
129C2×C22⋊C4×C53C2312yes—
130#95×C53C2314yes—
131#95×C53C2314yes—
132C24×C53—42no—
133C24×C53—42no—
134C24×C53C2×Q814yes—
135C24×C53#4114yes—
136C24×C53#4114yes—
137C24×C53C2411yescharacteristic
138C24×C53C2411yescharacteristic
139C24×C53C2411yescharacteristic
140C22×C4×C53—24no—
141C22×C4×C53—24no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53—42no—
150C22×C4×C53C2412yes—
151C22×C4×C53C2414yes—
152C22×C4×C53C2414yes—
153C22×C4×C53C2412yes—
154C22×Dic53—22no—
155C22×Dic53—24no—
156C22×Dic53—24no—
157C22×Dic53—24no—
158C22×Dic53—24no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—24no—
163C22×Dic53—24no—
164C22×Dic53—42no—
165C22×Dic53—42no—
166C22×Dic53—42no—
167C22×Dic53—42no—
168C22×Dic53—42no—
169C22×Dic53—42no—
170C22×Dic53—42no—
171C22×Dic53—42no—
172C22×Dic53—42no—
173C22×Dic53—416no—
174C22×Dic53C2×Q818yes—
175C22×Dic53C2412yes—
176C22×Dic53C2412yes—
177C22×Dic53C2414yes—
178C22×Dic53C2412yes—
179C22×Dic53C2414yes—
180C22×Dic53C2412yes—
181C4⋊C4×C53—44no—
182C4⋊C4×C53—44no—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—24no—
185C22⋊C4×C53—24no—
186C22⋊C4×C53—24no—
187C22⋊C4×C53—24no—
188C22⋊C4×C53—24no—
189C22⋊Dic53—24no—
190C22⋊Dic53—24no—
191C22⋊Dic53—24no—
192C22⋊Dic53—24no—
193C22⋊Dic53C2×Q818yes—
194C22⋊Dic53C2×Q818yes—
195C4⋊Dic53—28no—
196C4⋊Dic53—44no—
197#11719919—28no—
198#11719919—44no—
199#11719919—44no—
200#11719919—44no—
201#11719919—44no—
202#11719919—44no—
203#11719919—44no—
204#11719919—44no—
205C4×Dic53—28no—
206C4×Dic53—44no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—44no—
214C23×C53—48no—
215C23×C53—44no—
216C23×C53—48no—
217C23×C53#125116yescomplemented
218C23×C53C22×Q812yes—
219C23×C53C22×Q812yes—
220C23×C53C22×Q812yes—
221C23×C53C2×#4112yes—
222C23×C53C2×#4112yes—
223C23×C53C2×#4112yes—
224C23×C53C2×#4112yes—
225C23×C53C2×#4112yes—
226C23×C53C2×#4112yes—
227C23×C53C2511yescharacteristic
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—48no—
239C2×C4×C53—48no—
240C2×C4×C53—82no—
241C2×C4×C53—82no—
242C2×Dic53—28no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—22no—
248C2×Dic53—24no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—48no—
258C2×Dic53—48no—
259C2×Dic53—48no—
260C2×Dic53—48no—
261C2×Dic53—44no—
262C2×Dic53—82no—
263C2×Dic53—82no—
264C2×Dic53C22×Q818yes—
265C2×Dic53H(5,𝔽2)14yes—
266#34994—531nomaximal, Sylow
267C22×C53—24no—
268C22×C53—24no—
269C22×C53—24no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—24no—
277C22×C53—24no—
278C22×C53—24no—
279C22×C53—416no—
280C22×C53C2×#12514yes—
281C22×C53C2×#12514yes—
282C22×C53C2×#12514yes—
283C22×C53C23×Q811yescharacteristic
284C22×C53C22×#4111yescharacteristic
285C22×C53C22×#4111yescharacteristic
286C22×C53C2×H(5,𝔽2)12yes—
287C22×C53#58512yes—
288C4×C53—44no—
289C4×C53—44no—
290C4×C53—42no—
291C4×C53—42no—
292Dic53—24no—
293Dic53—28no—
294Dic53—44no—
295Dic53—42no—
296Dic53—42no—
297Dic53—42no—
298#2851—534no—
299#2646—534no—
300#2505—534no—
301#2505—534no—
302#2473—534no—
303#2473—534no—
304#2315—531no—
305#2315—531no—
306#2315—531no—
307#2315D5311yescharacteristic
308C2×#385—531no—
309C2×#385—531no—
310C2×C4×C22⋊C4—531no—
311C2×C53—24no—
312C2×C53—24no—
313C2×C53—24no—
314C2×C53C22×#12511yescharacteristic
315C2×C53#345912yes—
316C2×C53#349011yescharacteristic
317C2×C53#349111yescharacteristic
318C2×C53#350112yes—
319C24×C4—531no—
320C2×#125—534no—
321C22×C4⋊C4—532no—
322C22×C4⋊C4—532no—
323C22×C22⋊C4—531no—
324C22×C22⋊C4—531no—
325C22×C22⋊C4—531no—
326C22×C22⋊C4C2×D5311yescharacteristic
327C22×C22⋊C4C2×D5311yescharacteristic
328C22×C22⋊C4C2×D5311yescharacteristic
329C22×C42—532no—
330#400—534no—
331#400—534no—
332#400—534no—
333#400—534no—
334#398—534no—
335#398—534no—
336#398—534no—
337#398—534no—
338#389—534no—
339#389—534no—
340#387—534no—
341#387—534no—
342#387—534no—
343#387—534no—
344#385—534no—
345#385—532no—
346#385—534no—
347#385—532no—
348#385—532no—
349#385—532no—
350#385—534no—
351#385—532no—
352#385—534no—
353#385—534no—
354#385—532no—
355#385—532no—
356#385—532no—
357#385—534no—
358#385C2×D5314yes—
359#385C2×D5314yes—
360#382—534no—
361#380—534no—
362#380—534no—
363#380—532no—
364#380—532no—
365#380C2×D5312yes—
366#380C2×D5312yes—
367C4×C22⋊C4—534no—
368C4×C22⋊C4—534no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—532no—
372C53#3499411yescharacteristic, complemented, Sylow
373C25C22×D5311yescharacteristic
374C23×C4—532no—
375C23×C4—531no—
376C23×C4—532no—
377C23×C4—532no—
378C23×C4—531no—
379C23×C4—532no—
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—1064no—
383C23×C4—2124no—
384C23×C4C22×D5311yescharacteristic
385C23×C4C22×D5312yes—
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5312yes—
388#125—2128no—
389C2×C4⋊C4—534no—
390C2×C4⋊C4—534no—
391C2×C4⋊C4—534no—
392C2×C4⋊C4—534no—
393C2×C4⋊C4—534no—
394C2×C4⋊C4—534no—
395C2×C4⋊C4—534no—
396C2×C4⋊C4—534no—
397C2×C4⋊C4—534no—
398C2×C4⋊C4—534no—
399C2×C4⋊C4—2124no—
400C2×C4⋊C4—2124no—
401C2×C4⋊C4C22×D5314yes—
402C2×C4⋊C4C22×D5314yes—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—538no—
407C2×C22⋊C4—532no—
408C2×C22⋊C4—532no—
409C2×C22⋊C4—532no—
410C2×C22⋊C4—532no—
411C2×C22⋊C4—532no—
412C2×C22⋊C4—532no—
413C2×C22⋊C4—1064no—
414C2×C22⋊C4—1064no—
415C2×C22⋊C4C22×D5312yes—
416C2×C22⋊C4C22×D5312yes—
417C2×C22⋊C4C22×D5312yes—
418C2×C22⋊C4C22×D5312yes—
419C2×C22⋊C4C22×D5312yes—
420C2×C22⋊C4C22×D5312yes—
421C2×C42—534no—
422C2×C42—534no—
423C2×C42—2124no—
424#95—534no—
425#95—534no—
426#95—534no—
427#95—534no—
428#95—534no—
429#95—534no—
430#95—534no—
431#95—534no—
432#95—534no—
433#95—534no—
434#95—534no—
435#95—534no—
436#95C22×D5314yes—
437#95C22×D5314yes—
438C24—42no—
439C24—42no—
440C24#1171994514yes—
441C24#1171994714yes—
442C24Q8×D5314yes—
443C24C23×D5311yescharacteristic
444C24C23×D5311yescharacteristic
445C24C23×D5311yescharacteristic
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—534no—
457C22×C4—532no—
458C22×C4—538no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—534no—
462C22×C4—532no—
463C22×C4—1064no—
464C22×C4—1064no—
465C22×C4—1064no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1064no—
470C22×C4—1064no—
471C22×C4—1064no—
472C22×C4—1062no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4—2122no—
482C22×C4—21216no—
483C22×C4C23×D5312yes—
484C22×C4C23×D5314yes—
485C22×C4C23×D5314yes—
486C22×C4C23×D5312yes—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—1068no—
490C4⋊C4—1068no—
491C4⋊C4—2124no—
492C4⋊C4—2124no—
493C4⋊C4—2124no—
494C4⋊C4—2124no—
495C4⋊C4—2124no—
496C4⋊C4—2124no—
497C4⋊C4—2124no—
498C4⋊C4—2124no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—538no—
506C22⋊C4—538no—
507C22⋊C4—1064no—
508C22⋊C4—1064no—
509C22⋊C4—1064no—
510C22⋊C4—1064no—
511C42—1068no—
512C42—2124no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—44no—
520C23—44no—
521C23—48no—
522C23—48no—
523C23#13133207116yescomplemented
524C23C2×Q8×D5312yes—
525C23C2×Q8×D5312yes—
526C23C2×Q8×D5312yes—
527C23C2×#1171994712yes—
528C23C2×#1171994712yes—
529C23C2×#1171994712yes—
530C23C2×#1171994512yes—
531C23C2×#1171994512yes—
532C23C2×#1171994512yes—
533C23C24×D5311yescharacteristic
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—538no—
549C2×C4—534no—
550C2×C4—1068no—
551C2×C4—1062no—
552C2×C4—1064no—
553C2×C4—1062no—
554C2×C4—1064no—
555C2×C4—1064no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—4242no—
566C2×C4—4242no—
567C2×C4—2128no—
568C2×C4—2128no—
569C2×C4—2128no—
570C2×C4—2128no—
571C2×C4—2124no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—416no—
585C22C2×#1313320714yes—
586C22C2×#1313320714yes—
587C22C2×#1313320714yes—
588C22C22×Q8×D5311yescharacteristic
589C22C22×#1171994711yescharacteristic
590C22C22×#1171994511yescharacteristic
591C22H(5,𝔽2)×D5312yes—
592C22#1681145812yes—
593C4—44no—
594C4—44no—
595C4—42no—
596C4—42no—
597C4—1068no—
598C4—1064no—
599C4—2124no—
600C4—2122no—
601C4—2122no—
602C4—2122no—
603C2—24no—
604C2—24no—
605C2—24no—
606C2C22×#1313320711yescharacteristic
607C2#2028812112yes—
608C2#2028840011yescharacteristic
609C2#2028840811yescharacteristic
610C2#2028851512yes—
subgroupG/H
center, FrattiniC23C24×D53
commutator, socle, cosocleC23×C53C25
FittingC53×#2315C2
prime radicalC25×C53C23
hypercenter#2315D53
hypocenterC53#34994
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2315

Complements

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

NH ≅ G/Ntype
1. #20283021C2semidirect
2. #20280687C2semidirect
3. #20279214C2semidirect
4. #20279212C2semidirect
5. #20278861C2semidirect
6. #20278853C2semidirect
15. C2×#13133207C22semidirect
22. #16810564C22semidirect
23. #16810564C22semidirect
24. #16810563C22semidirect
25. #16810561C22semidirect
26. #16810557C22semidirect
27. #16810556C22semidirect
28. #16810554C22semidirect
29. #16810553C22semidirect
30. #16810520C22semidirect
31. #16810518C22semidirect
32. #16810509C22semidirect
33. #16810508C22semidirect
34. #16810505C22semidirect
35. #16810504C22semidirect
50. #16810487C22semidirect
217. C23×C53#125semidirect
372. C53#34994semidirect
523. C23#13133207semidirect

Maximal subgroups

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

subgroupindex
#202830212
#202806872
#202792142
#202792122
#202788612
#202788532
#202773272
C2×#168104982
C2×#168104952
C2×C22⋊C4×Dic532
C53×#23152
#3499453