Finite Group Databasebeta

← #27556839

Subgroups — #27556839

Order 13568.

2214 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283028C214yesmaximal, complemented
2#20280670C214yesmaximal, complemented
3#20279273C214yesmaximal, complemented
4#20279096C214yesmaximal, complemented
5#20278975C214yesmaximal, complemented
6#20278960C214yesmaximal, complemented
7#20277327C211yesmaximal, characteristic, complemented
8#20277327C211yesmaximal, characteristic, complemented
9#20277293C211yesmaximal, characteristic
10C2×#16810503C211yesmaximal, characteristic
11C2×#16810498C211yesmaximal, characteristic, complemented
12C2×C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
13C53×#2308C211yesmaximal, characteristic
14C24×Dic53C2211yescharacteristic
15C2×#13133210C2214yescomplemented
16C22×C4⋊Dic53C2212yescomplemented
17C22×C22⋊Dic53C2211yescharacteristic
18C22×C22⋊Dic53C2211yescharacteristic
19C22×C22⋊Dic53C2211yescharacteristic
20C22×C22⋊Dic53C2212yes—
21C22×C4×Dic53C2212yescomplemented
22C53⋊#401C2214yescomplemented
23#16810565C2214yescomplemented
24#16810564C2214yescomplemented
25#16810563C2214yescomplemented
26#16810551C2214yescomplemented
27#16810551C2214yescomplemented
28#16810548C2214yescomplemented
29#16810546C2214yescomplemented
30#16810517C2214yescomplemented
31#16810516C2214yescomplemented
32#16810511C2214yescomplemented
33#16810510C2214yescomplemented
34#16810508C2214yescomplemented
35#16810505C2214yescomplemented
36#16810503C2214yes—
37#16810503C2212yes—
38#16810503C2212yes—
39#16810503C2212yes—
40#16810503C2212yes—
41#16810503C2214yes—
42#16810498C2214yes—
43#16810498C2212yescomplemented
44#16810498C2212yescomplemented
45#16810496C2214yescomplemented
46#16810496C2214yescomplemented
47#16810495C2214yes—
48#16810495C2214yescomplemented
49#16810490C2214yescomplemented
50#16810486C2212yes—
51#16810481C2214yes—
52#16810481C2212yescomplemented
53#16810481C2212yescomplemented
54#16810479C2212yes—
55C22⋊C4×Dic53C2214yes—
56C22⋊C4×Dic53C2214yescomplemented
57C2×#13133124C2212yes—
58C2×#13133124C2212yescomplemented
59C2×#13133124C2212yescomplemented
60C23×D4×C53C2211yescharacteristic
61C22×C22⋊C4×C53C2211yescharacteristic
62C22×C22⋊C4×C53C2211yescharacteristic
63C53×#386C2214yes—
64C53×#386C2214yes—
65C53×#381C2212yes—
66C53×#379C2212yes—
67C23×Dic53—24no—
68C23×Dic53—44no—
69C23×Dic53C2311yescharacteristic
70C23×Dic53C2312yes—
71C23×Dic53C2312yes—
72C23×Dic53C2311yescharacteristic
73C23×Dic53C2312yes—
74C23×Dic53C2311yescharacteristic
75C23×Dic53C2311yescharacteristic
76C23×Dic53C2312yes—
77#13133210—48no—
78C2×C4⋊Dic53—44no—
79C2×C4⋊Dic53C2314yes—
80C2×C4⋊Dic53C2314yes—
81C2×#11719919C2314yes—
82C2×#11719919C2314yes—
83C2×#11719919C2314yes—
84C2×#11719919C2314yes—
85C2×C22⋊Dic53—24no—
86C2×C22⋊Dic53—24no—
87C2×C22⋊Dic53—24no—
88C2×C22⋊Dic53—44no—
89C2×C22⋊Dic53C2312yes—
90C2×C22⋊Dic53C2312yes—
91C2×C22⋊Dic53C2312yes—
92C2×C22⋊Dic53C2312yes—
93C2×C22⋊Dic53C2312yes—
94C2×C22⋊Dic53C2314yes—
95C2×C22⋊Dic53C2312yes—
96C2×C22⋊Dic53C2314yes—
97C2×C22⋊Dic53C2314yes—
98C2×C22⋊Dic53C2314yes—
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
115C25×C53C2311yescharacteristic
116C22×D4×C53—24no—
117C22×D4×C53—24no—
118C22×D4×C53—24no—
119C22×D4×C53C2312yes—
120C22×D4×C53C2312yes—
121C23×C4×C53C2311yescharacteristic
122C23×C4×C53C2312yes—
123C23×C4×C53C2312yes—
124C2×C22⋊C4×C53—24no—
125C2×C22⋊C4×C53—24no—
126C2×C22⋊C4×C53C2312yes—
127C2×C22⋊C4×C53C2312yes—
128C2×C22⋊C4×C53C2312yes—
129C2×C22⋊C4×C53C2312yes—
130C2×C22⋊C4×C53C2314yes—
131C2×C22⋊C4×C53C2314yes—
132#95×C53C2314yes—
133#95×C53C2314yes—
134C24×C53—22no—
135C24×C53—22no—
136C24×C53—22no—
137C24×C53—42no—
138C24×C53—42no—
139C24×C53—42no—
140C24×C53—42no—
141C24×C53#4114yes—
142C24×C53#4114yes—
143C24×C53#4114yes—
144C24×C53C2412yes—
145C24×C53C2411yescharacteristic
146C24×C53C2411yescharacteristic
147C24×C53C2411yescharacteristic
148C2×D4×C53—24no—
149C2×D4×C53—24no—
150C2×D4×C53—24no—
151C2×D4×C53—24no—
152C2×D4×C53—24no—
153C2×D4×C53—24no—
154C2×D4×C53—416no—
155C22×C4×C53—24no—
156C22×C4×C53—24no—
157C22×C4×C53—22no—
158C22×C4×C53—22no—
159C22×C4×C53—22no—
160C22×C4×C53—42no—
161C22×C4×C53—42no—
162C22×C4×C53—42no—
163C22×C4×C53—42no—
164C22×C4×C53C2412yes—
165C22×C4×C53C2414yes—
166C22×C4×C53C2414yes—
167C22×Dic53—22no—
168C22×Dic53—24no—
169C22×Dic53—24no—
170C22×Dic53—24no—
171C22×Dic53—24no—
172C22×Dic53—22no—
173C22×Dic53—24no—
174C22×Dic53—22no—
175C22×Dic53—22no—
176C22×Dic53—24no—
177C22×Dic53—24no—
178C22×Dic53—42no—
179C22×Dic53—42no—
180C22×Dic53—42no—
181C22×Dic53—42no—
182C22×Dic53—42no—
183C22×Dic53—42no—
184C22×Dic53—42no—
185C22×Dic53—42no—
186C22×Dic53—42no—
187C22×Dic53—416no—
188C22×Dic53C2412yes—
189C22×Dic53C2412yes—
190C22×Dic53C2414yes—
191C22×Dic53C2414yes—
192C22×Dic53C2412yes—
193C22×Dic53C2412yes—
194C22⋊C4×C53—24no—
195C22⋊C4×C53—24no—
196C22⋊C4×C53—24no—
197C22⋊C4×C53—24no—
198C22⋊C4×C53—44no—
199C22⋊C4×C53—44no—
200C22⋊Dic53—24no—
201C22⋊Dic53—24no—
202C22⋊Dic53—24no—
203C22⋊Dic53—24no—
204C22⋊Dic53—24no—
205C22⋊Dic53—24no—
206C22⋊Dic53—28no—
207C22⋊Dic53—44no—
208C22⋊Dic53—44no—
209C22⋊Dic53—44no—
210C4⋊Dic53—28no—
211C4⋊Dic53—44no—
212#11719919—44no—
213#11719919—44no—
214#11719919—44no—
215#11719919—44no—
216C4×Dic53—28no—
217C4×Dic53—44no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—22no—
221C23×C53—22no—
222C23×C53—22no—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53—22no—
226C23×C53—22no—
227C23×C53—24no—
228C23×C53—24no—
229C23×C53—44no—
230C23×C53—44no—
231C23×C53—48no—
232C23×C53—48no—
233C23×C53—48no—
234C23×C53—44no—
235C23×C53—82no—
236C23×C53—82no—
237C23×C53—84no—
238C23×C53#126116yescomplemented
239C23×C53C2×#4112yes—
240C23×C53C2×#4112yes—
241C23×C53C2×#4112yes—
242C23×C53C2×#4112yes—
243C23×C53C2×#4112yes—
244C23×C53C2×#4112yes—
245C23×C53C2×#4112yes—
246C23×C53C2×#4112yes—
247C23×C53C2×#4112yes—
248C23×C53C2511yescharacteristic
249D4×C53—48no—
250D4×C53—48no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—24no—
255C2×C4×C53—24no—
256C2×C4×C53—24no—
257C2×C4×C53—24no—
258C2×C4×C53—24no—
259C2×C4×C53—24no—
260C2×C4×C53—48no—
261C2×C4×C53—48no—
262C2×C4×C53—44no—
263C2×Dic53—28no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—24no—
274C2×Dic53—24no—
275C2×Dic53—24no—
276C2×Dic53—24no—
277C2×Dic53—24no—
278C2×Dic53—24no—
279C2×Dic53—48no—
280C2×Dic53—48no—
281C2×Dic53—48no—
282C2×Dic53—44no—
283C2×Dic53—82no—
284C2×Dic53—82no—
285C2×Dic53—84no—
286C2×Dic53H(5,𝔽2)14yes—
287#34995—531nomaximal, Sylow
288C22×C53—24no—
289C22×C53—24no—
290C22×C53—24no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—24no—
298C22×C53—24no—
299C22×C53—24no—
300C22×C53—416no—
301C22×C53—42no—
302C22×C53—44no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—44no—
306C22×C53—88no—
307C22×C53C2×#12614yes—
308C22×C53C2×#12614yes—
309C22×C53C2×#12614yes—
310C22×C53C22×#4111yescharacteristic
311C22×C53C22×#4111yescharacteristic
312C22×C53C22×#4111yescharacteristic
313C22×C53C2×H(5,𝔽2)12yes—
314C22×C53#58512yes—
315C4×C53—42no—
316C4×C53—44no—
317C4×C53—44no—
318Dic53—24no—
319Dic53—44no—
320Dic53—42no—
321Dic53—42no—
322Dic53—42no—
323Dic53—44no—
324#2852—534no—
325#2643—534no—
326#2510—534no—
327#2493—534no—
328#2481—534no—
329#2480—534no—
330#2315—531no—
331#2315—531no—
332#2308—531no—
333#2308D5311yescharacteristic
334C2×#386—531no—
335C2×#385—531no—
336C2×C4×C22⋊C4—531no—
337C2×C53—24no—
338C2×C53—24no—
339C2×C53—24no—
340C2×C53—82no—
341C2×C53C22×#12611yescharacteristic
342C2×C53#345912yes—
343C2×C53#349011yescharacteristic
344C2×C53#349111yescharacteristic
345C2×C53#350512yes—
346C23×D4C2×D5311yescharacteristic
347C24×C4—531no—
348C2×#126—534no—
349C22×C4⋊C4—532no—
350C22×C22⋊C4—531no—
351C22×C22⋊C4—531no—
352C22×C22⋊C4—531no—
353C22×C22⋊C4—532no—
354C22×C22⋊C4C2×D5311yescharacteristic
355C22×C22⋊C4C2×D5311yescharacteristic
356C22×C42—532no—
357#401—534no—
358#400—534no—
359#400—534no—
360#400—534no—
361#397—534no—
362#397—534no—
363#397—534no—
364#396—534no—
365#388—534no—
366#388—534no—
367#388—534no—
368#387—534no—
369#387—534no—
370#387—534no—
371#386—534no—
372#386—532no—
373#386—532no—
374#386—532no—
375#386—532no—
376#386—534no—
377#386C2×D5314yes—
378#386C2×D5314yes—
379#385—534no—
380#385—534no—
381#385—534no—
382#385—532no—
383#385—532no—
384#385—534no—
385#385—534no—
386#383—534no—
387#381—532no—
388#381C2×D5312yes—
389#380—534no—
390#380—532no—
391#380—532no—
392#379—532no—
393#379C2×D5312yes—
394C4×C22⋊C4—534no—
395C4×C22⋊C4—534no—
396C2×#95—532no—
397C2×#95—532no—
398C2×#95—532no—
399C53#3499511yescharacteristic, complemented, Sylow
400C25C22×D5311yescharacteristic
401C25C22×D5311yescharacteristic
402C22×D4—24no—
403C22×D4—24no—
404C22×D4—24no—
405C22×D4C22×D5312yes—
406C22×D4C22×D5312yes—
407C23×C4—531no—
408C23×C4—532no—
409C23×C4—532no—
410C23×C4—532no—
411C23×C4—531no—
412C23×C4—531no—
413C23×C4—532no—
414C23×C4—531no—
415C23×C4—1064no—
416C23×C4—2124no—
417C23×C4C22×D5311yescharacteristic
418C23×C4C22×D5312yes—
419C23×C4C22×D5312yes—
420#126—2128no—
421C2×C4⋊C4—534no—
422C2×C4⋊C4—534no—
423C2×C4⋊C4—534no—
424C2×C4⋊C4—534no—
425C2×C4⋊C4—534no—
426C2×C4⋊C4—534no—
427C2×C4⋊C4—2124no—
428C2×C22⋊C4—24no—
429C2×C22⋊C4—24no—
430C2×C22⋊C4—534no—
431C2×C22⋊C4—532no—
432C2×C22⋊C4—532no—
433C2×C22⋊C4—532no—
434C2×C22⋊C4—532no—
435C2×C22⋊C4—532no—
436C2×C22⋊C4—532no—
437C2×C22⋊C4—534no—
438C2×C22⋊C4—534no—
439C2×C22⋊C4—534no—
440C2×C22⋊C4—1064no—
441C2×C22⋊C4—1064no—
442C2×C22⋊C4—1064no—
443C2×C22⋊C4—2124no—
444C2×C22⋊C4C22×D5312yes—
445C2×C22⋊C4C22×D5312yes—
446C2×C22⋊C4C22×D5312yes—
447C2×C22⋊C4C22×D5312yes—
448C2×C22⋊C4C22×D5314yes—
449C2×C22⋊C4C22×D5314yes—
450C2×C42—534no—
451C2×C42—534no—
452C2×C42—2124no—
453#95—534no—
454#95—534no—
455#95—534no—
456#95—534no—
457#95—534no—
458#95—534no—
459#95—534no—
460#95—534no—
461#95—534no—
462#95—534no—
463#95—534no—
464#95—534no—
465#95C22×D5314yes—
466#95C22×D5314yes—
467C24—22no—
468C24—22no—
469C24—22no—
470C24—42no—
471C24—42no—
472C24—42no—
473C24—42no—
474C24#1171994514yes—
475C24#1171994714yes—
476C24#1171994714yes—
477C24C23×D5312yes—
478C24C23×D5311yescharacteristic
479C24C23×D5311yescharacteristic
480C24C23×D5311yescharacteristic
481C2×D4—24no—
482C2×D4—24no—
483C2×D4—24no—
484C2×D4—24no—
485C2×D4—24no—
486C2×D4—24no—
487C2×D4—416no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—532no—
498C22×C4—532no—
499C22×C4—532no—
500C22×C4—534no—
501C22×C4—534no—
502C22×C4—532no—
503C22×C4—1064no—
504C22×C4—1064no—
505C22×C4—1064no—
506C22×C4—1062no—
507C22×C4—1064no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1064no—
511C22×C4—1064no—
512C22×C4—1064no—
513C22×C4—1062no—
514C22×C4—2122no—
515C22×C4—2122no—
516C22×C4—2122no—
517C22×C4—2122no—
518C22×C4—2122no—
519C22×C4—2122no—
520C22×C4—2122no—
521C22×C4—2122no—
522C22×C4—2122no—
523C22×C4—21216no—
524C22×C4C23×D5312yes—
525C22×C4C23×D5314yes—
526C22×C4C23×D5314yes—
527C4⋊C4—1068no—
528C4⋊C4—2124no—
529C4⋊C4—2124no—
530C4⋊C4—2124no—
531C4⋊C4—2124no—
532C4⋊C4—2124no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—1068no—
540C22⋊C4—1064no—
541C22⋊C4—1064no—
542C22⋊C4—1064no—
543C22⋊C4—1064no—
544C22⋊C4—1064no—
545C22⋊C4—1064no—
546C22⋊C4—2124no—
547C22⋊C4—2124no—
548C22⋊C4—2124no—
549C42—1068no—
550C42—2124no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—44no—
563C23—44no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—44no—
568C23—84no—
569C23—82no—
570C23—82no—
571C23#13133210116yescomplemented
572C23C2×#1171994712yes—
573C23C2×#1171994712yes—
574C23C2×#1171994712yes—
575C23C2×#1171994712yes—
576C23C2×#1171994712yes—
577C23C2×#1171994712yes—
578C23C2×#1171994512yes—
579C23C2×#1171994512yes—
580C23C2×#1171994512yes—
581C23C24×D5311yescharacteristic
582D4—48no—
583D4—48no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—534no—
597C2×C4—1068no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—1062no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—1062no—
608C2×C4—1064no—
609C2×C4—1062no—
610C2×C4—1064no—
611C2×C4—1064no—
612C2×C4—1064no—
613C2×C4—4244no—
614C2×C4—4242no—
615C2×C4—4242no—
616C2×C4—2128no—
617C2×C4—2128no—
618C2×C4—2128no—
619C2×C4—2124no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—416no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—88no—
639C22C2×#1313321014yes—
640C22C2×#1313321014yes—
641C22C2×#1313321014yes—
642C22C22×#1171994711yescharacteristic
643C22C22×#1171994711yescharacteristic
644C22C22×#1171994511yescharacteristic
645C22H(5,𝔽2)×D5312yes—
646C22#1681145812yes—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—1064no—
651C4—2124no—
652C4—2124no—
653C4—2122no—
654C4—2122no—
655C4—2122no—
656C2—24no—
657C2—24no—
658C2—24no—
659C2—82no—
660C2C22×#1313321011yescharacteristic
661C2#2028812112yes—
662C2#2028840011yescharacteristic
663C2#2028840811yescharacteristic
664C2#2028855212yes—
subgroupG/H
center, FrattiniC23C24×D53
commutator, socle, cosocleC23×C53C25
FittingC53×#2308C2
prime radicalC23×D4×C53C22
hypercenter#2308D53
hypocenterC53#34995
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2308

Complements

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

NH ≅ G/Ntype
1. #20283028C2semidirect
2. #20280670C2semidirect
3. #20279273C2semidirect
4. #20279096C2semidirect
5. #20278975C2semidirect
6. #20278960C2semidirect
7. #20277327C2semidirect
8. #20277327C2semidirect
11. C2×#16810498C2semidirect
12. C2×C22⋊C4×Dic53C2semidirect
15. C2×#13133210C22semidirect
16. C22×C4⋊Dic53C22semidirect
21. C22×C4×Dic53C22semidirect
22. C53⋊#401C22semidirect
23. #16810565C22semidirect
24. #16810564C22semidirect
25. #16810563C22semidirect
26. #16810551C22semidirect
27. #16810551C22semidirect
28. #16810548C22semidirect
29. #16810546C22semidirect
30. #16810517C22semidirect
31. #16810516C22semidirect
32. #16810511C22semidirect
33. #16810510C22semidirect
34. #16810508C22semidirect
35. #16810505C22semidirect
43. #16810498C22semidirect
44. #16810498C22semidirect
45. #16810496C22semidirect
46. #16810496C22semidirect
48. #16810495C22semidirect
49. #16810490C22semidirect
52. #16810481C22semidirect
53. #16810481C22semidirect
56. C22⋊C4×Dic53C22semidirect
58. C2×#13133124C22semidirect
59. C2×#13133124C22semidirect
238. C23×C53#126semidirect
399. C53#34995semidirect
571. C23#13133210semidirect

Maximal subgroups

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

subgroupindex
#202830282
#202806702
#202792732
#202790962
#202789752
#202789602
#202773272
#202772932
C2×#168105032
C2×#168104982
C2×C22⋊C4×Dic532
C53×#23082
#3499553