Finite Group Databasebeta

← #27349998

Subgroups — #27349998

Order 13568.

1350 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811095C211yesmaximal, characteristic
2#20283743C212yesmaximal, complemented
3#20273056C211yesmaximal, characteristic, complemented
4#20273050C212yesmaximal, complemented
5#20273047C212yesmaximal, complemented
6#20273040C212yesmaximal, complemented
7#20273030C212yesmaximal, complemented
8#323×D53C212yesmaximal, complemented
9C53×#1466C211yesmaximal, characteristic, complemented
10C22×D4×D53C411yescharacteristic
11#16811095—24no—
12#16811095—24no—
13C2×C4×D212C2211yescharacteristic
14C2×D4×Dic53C411yescharacteristic
15C2×#13133179C411yescharacteristic
16C2×C4×C22⋊D53C411yescharacteristic
17C2×C4×C22⋊D53C411yescharacteristic
18C2×#13133177C2211yescharacteristic
19C2×#13133177C2211yescharacteristic
20C2×#13133175C411yescharacteristic
21C2×#13133174C2211yescharacteristic
22C22×#11719921C411yescharacteristic
23C22×#11719921C411yescharacteristic
24C2×C22⋊C4×D53C2211yescharacteristic
25C2×C22⋊C4×D53C2211yescharacteristic
26#16810696—24no—
27#16810693—24no—
28#16810680C2212yescomplemented
29#16810585C2214yescomplemented
30C22⋊C8×D53C2214yescomplemented
31#16810348C2214yescomplemented
32#16810347C2212yescomplemented
33#16810347C2212yescomplemented
34#16810346C2214yes—
35C2×C4×D4×C53C2211yescharacteristic
36C53×#432C2212yescomplemented
37C53×#323C2212yescomplemented
38C53×#323C2212yescomplemented
39C24×D53C2×C411yescharacteristic
40C24×D53C2×C411yescharacteristic
41C22×D212C2×C411yescharacteristic
42C22×C22⋊D53C2×C411yescharacteristic
43C22×C22⋊D53C2×C411yescharacteristic
44C2×D4×D53—24no—
45C2×D4×D53—24no—
46C2×D4×D53—24no—
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C22×C4×D53C2×C411yescharacteristic
50C22×C4×D53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2311yescharacteristic
53C22×C4×D53C2311yescharacteristic
54C2×#11719912—24no—
55C2×C8×D53—24no—
56C4×D212D414yes—
57C4×D212D414yes—
58D4×Dic53—24no—
59#13133179—24no—
60C4×C22⋊D53—22no—
61C4×C22⋊D53—22no—
62C4×C22⋊D53—22no—
63C4×C22⋊D53—22no—
64#13133177—24no—
65#13133177—24no—
66#13133175—24no—
67#13133174—22no—
68#13133174—22no—
69C2×C4⋊Dic53C2×C411yescharacteristic
70C2×#11719919C2×C411yescharacteristic
71C2×#11719919C2311yescharacteristic
72C2×C22⋊Dic53C2×C411yescharacteristic
73C2×C22⋊Dic53C2×C411yescharacteristic
74C2×#11719921—22no—
75C2×#11719921—22no—
76C2×#11719921—22no—
77C2×#11719921—22no—
78C2×#11719921C2×C411yescharacteristic
79C2×#11719921C2×C411yescharacteristic
80C2×#11719921C2×C411yescharacteristic
81C2×#11719921C2×C411yescharacteristic
82C2×#11719921C2×C411yescharacteristic
83C2×#11719921C2×C411yescharacteristic
84C2×#11719921C2×C411yescharacteristic
85C2×#11719921C2311yescharacteristic
86C22⋊C4×D53—24no—
87C22⋊C4×D53—24no—
88C2×C4×Dic53C2×C411yescharacteristic
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53C2×C411yescharacteristic
91C4⋊#88322—22no—
92C22⋊#88322C2314yes—
93#13133129—24no—
94#13133129—24no—
95#13133128—22no—
96C22×D4×C53C2×C411yescharacteristic
97C23×C4×C53C2×C411yescharacteristic
98C23×C4×C53C2×C411yescharacteristic
99C4×D4×C53—22no—
100C4×D4×C53—22no—
101C2×C4⋊C4×C53C2×C411yescharacteristic
102C2×C22⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53C2311yescharacteristic
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—22no—
106C22⋊C8×C53C2314yes—
107#97×C53—22no—
108C24×C53C22×C411yescharacteristic
109C24×C53C22×C411yescharacteristic
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—22no—
113C23×D53—22no—
114C23×D53—22no—
115C23×D53—81no—
116C23×D53—81no—
117C23×D53—81no—
118C23×D53—81no—
119C23×D53M1614yescomplemented
120C23×D53C22×C411yescharacteristic
121C23×D53C22×C411yescharacteristic
122C23×D53C22×C411yescharacteristic
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—81no—
131C22×C4×C53—81no—
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C22×C411yescharacteristic
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C2411yescharacteristic
137C2×C22⋊D53—22no—
138C2×C22⋊D53—22no—
139C2×C22⋊D53—22no—
140C2×C22⋊D53—22no—
141C2×C22⋊D53—22no—
142C2×C22⋊D53M1614yescomplemented
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—81no—
146C22×Dic53—81no—
147C22×Dic53C22×C411yescharacteristic
148C22×Dic53C22×C411yescharacteristic
149C22×Dic53C22×C411yescharacteristic
150C22×Dic53C22×C411yescharacteristic
151C22×Dic53C22×C411yescharacteristic
152D4×D53—232no—
153C2×D212—22no—
154C2×D212—22no—
155C2×D212C22⋊C414yes—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—81no—
162C2×C4×D53—81no—
163C2×C4×D53—81no—
164C2×C4×D53—81no—
165C2×C4×D53C2×D414yes—
166C2×C4×D53C2×D414yes—
167C2×C8×C53—24no—
168C4⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C42×C53C2×D412yes—
172C42×C53C2×D412yes—
173C22⋊Dic53—22no—
174C22⋊Dic53—22no—
175#11719921—22no—
176#11719921—22no—
177#11719921—21no—
178#11719921—21no—
179#11719921—22no—
180#11719921—22no—
181#11719921—22no—
182#11719921C22⋊C412yes—
183#11719921C22⋊C412yes—
184#11719921C22⋊C412yes—
185#11719921C22⋊C412yes—
186C4⋊Dic53C22⋊C414yes—
187#11719919—21no—
188#11719919—21no—
189#11719919—22no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192C4×Dic53—22no—
193C2×#88322—24no—
194#11719912—84no—
195C8×D53—84no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53C2×M1612yes—
205C23×C53C2×M1612yes—
206C23×C53C23×C411yescharacteristic
207C22×D53—21no—
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—28no—
212C22×D53—28no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—82no—
220C22×D53—82no—
221C22×D53—82no—
222C22×D53—82no—
223C22×D53—84no—
224C22×D53—84no—
225C22×D53—84no—
226C22×D53—84no—
227C22×D53C2×C22⋊C412yes—
228C22×D53C2×C22⋊C412yes—
229C22×D53C2×M1612yes—
230C22×D53C2×M1612yes—
231C22×D53C2×M1612yes—
232D4×C53—28no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—81no—
242C2×C4×C53—81no—
243C2×C4×C53—81no—
244C2×C4×C53—81no—
245C2×C4×C53C2×C22⋊C412yes—
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C2×C22⋊C411yescharacteristic
248C2×C4×C53C22×D412yes—
249C22⋊D53—28no—
250C22⋊D53—28no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—82no—
260C2×Dic53—82no—
261C2×Dic53C2×C22⋊C412yes—
262C2×Dic53C2×C22⋊C412yes—
263C2×Dic53C2×M1612yes—
264D212—28no—
265C4×D53—28no—
266C4×D53—44no—
267C4×D53—44no—
268C4×D53—82no—
269C4×D53—82no—
270C8×C53—44no—
271#88322—44no—
272#17367—531nomaximal, Sylow
273C22×C53—21no—
274C22×C53—21no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53C22×C22⋊C411yescharacteristic
286C22×C53C22×M1611yescharacteristic
287C22×C53C22×M1611yescharacteristic
288C2×D53—22no—
289C2×D53—22no—
290C2×D53—22no—
291C2×D53—22no—
292C2×D53—22no—
293C2×D53—22no—
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—41no—
300C2×D53—41no—
301C2×D53—42no—
302C2×D53—88no—
303C2×D53—88no—
304C2×D53#40714yes—
305C4×C53—22no—
306C4×C53—22no—
307C4×C53—21no—
308C4×C53—21no—
309C4×C53—41no—
310C4×C53—41no—
311Dic53—22no—
312Dic53—22no—
313Dic53—22no—
314Dic53—41no—
315Dic53—41no—
316C2×#518—531no—
317#2973—532no—
318#1466—531no—
319#1466D5311yescharacteristic, complemented
320#1465—532no—
321#1464—532no—
322#1462—532no—
323#1459—532no—
324C2×#323—532no—
325C2×C53—21no—
326C2×C53—21no—
327C2×C53—42no—
328C2×C53—42no—
329C2×C53C2×#40711yescharacteristic
330C2×C53#288311yescharacteristic
331C2×C53#288811yescharacteristic
332D53—24no—
333D53—42no—
334D53—42no—
335C23×D4—531no—
336#518—1064no—
337#518—1064no—
338C2×C4×D4—531no—
339C2×C4×D4—531no—
340C2×C4×D4—531no—
341C2×C4×D4—531no—
342C2×C4×D4—531no—
343C2×C4×D4—531no—
344C2×C4×D4—531no—
345C2×C4×D4C2×D5311yescharacteristic
346C2×#117—531no—
347C2×#117—531no—
348C22×C22⋊C4—531no—
349C22×C22⋊C4—531no—
350C22×C22⋊C4—531no—
351C22×C22⋊C4—531no—
352#435—1064no—
353#435—1064no—
354#432—532no—
355#432C2×D5312yescomplemented
356#407—534no—
357C2×C22⋊C8—534no—
358#324—534no—
359#323—532no—
360#323—532no—
361#323—534no—
362#323C2×D5312yescomplemented
363#323C2×D5312yescomplemented
364C53#1736711yescharacteristic, complemented, Sylow
365C25—531no—
366C25—531no—
367C22×D4—531no—
368C22×D4—531no—
369C22×D4—531no—
370C22×D4—1064no—
371C22×D4—1064no—
372C22×D4—1064no—
373C22×D4C4×D5311yescharacteristic
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—531no—
377C23×C4—531no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4C4×D5311yescharacteristic
382C23×C4C4×D5311yescharacteristic
383C2×M16—1064no—
384C22×C8—1064no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—534no—
388C4×D4—534no—
389C4×D4—1064no—
390C4×D4—1064no—
391C4×D4—1062no—
392C4×D4—1062no—
393C4×D4—1062no—
394C4×D4—1062no—
395C4×D4—1064no—
396C4×D4—1064no—
397#117—1064no—
398#117—1062no—
399#117—1062no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4C4×D5311yescharacteristic
404C2×C22⋊C4—531no—
405C2×C22⋊C4—531no—
406C2×C22⋊C4—531no—
407C2×C22⋊C4—531no—
408C2×C22⋊C4—531no—
409C2×C22⋊C4—531no—
410C2×C22⋊C4—531no—
411C2×C22⋊C4—531no—
412C2×C22⋊C4—531no—
413C2×C22⋊C4—531no—
414C2×C22⋊C4—1064no—
415C2×C22⋊C4—1064no—
416C2×C22⋊C4—1062no—
417C2×C22⋊C4—1062no—
418C2×C22⋊C4—1062no—
419C2×C22⋊C4—1062no—
420C2×C22⋊C4C22×D5311yescharacteristic
421C2×C22⋊C4C22×D5311yescharacteristic
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—531no—
425C2×C42C22×D5311yescharacteristic
426C4⋊C8—22no—
427C4⋊C8—1062no—
428C22⋊C8—534no—
429C22⋊C8—1064no—
430C22⋊C8—1064no—
431C22⋊C8C22×D5314yes—
432#97—22no—
433#97—1062no—
434C24—531no—
435C24—534no—
436C24—531no—
437C24—531no—
438C24—1062no—
439C24—1062no—
440C24—1062no—
441C24—1062no—
442C24—1062no—
443C24—4241no—
444C24—4241no—
445C24—4241no—
446C24—4241no—
447C24C2×C4×D5311yescharacteristic
448C24C2×C4×D5311yescharacteristic
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—534no—
453C2×D4—534no—
454C2×D4—1062no—
455C2×D4—1062no—
456C2×D4—1062no—
457C2×D4—10632no—
458C2×D4—1062no—
459C2×D4—1062no—
460C2×D4—1062no—
461C2×D4—1062no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—534no—
472C22×C4—534no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—4241no—
483C22×C4—4241no—
484C22×C4—4241no—
485C22×C4—4241no—
486C22×C4—4241no—
487C22×C4—4241no—
488C22×C4C2×C4×D5311yescharacteristic
489C22×C4C2×C4×D5311yescharacteristic
490C22×C4C2×C4×D5311yescharacteristic
491C22×C4C2×C4×D5311yescharacteristic
492C22×C4C23×D5311yescharacteristic
493M16—4244no—
494C2×C8—24no—
495C2×C8—1064no—
496C2×C8—4244no—
497C4⋊C4—22no—
498C4⋊C4—534no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1062no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—532no—
505C22⋊C4—532no—
506C22⋊C4—532no—
507C22⋊C4—532no—
508C22⋊C4—1062no—
509C22⋊C4—1062no—
510C22⋊C4—1062no—
511C22⋊C4—1062no—
512C22⋊C4—1062no—
513C22⋊C4—1062no—
514C22⋊C4—1061no—
515C22⋊C4—1061no—
516C22⋊C4—1062no—
517C42—1062no—
518C42—1062no—
519C42—1062no—
520C42D4×D5312yes—
521C42D4×D5312yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—532no—
531C23—532no—
532C23—532no—
533C23—532no—
534C23—532no—
535C23—1068no—
536C23—1068no—
537C23—1061no—
538C23—1061no—
539C23—1061no—
540C23—1061no—
541C23—4242no—
542C23—4242no—
543C23—4242no—
544C23—4242no—
545C23—4244no—
546C23—4244no—
547C23—4244no—
548C23—4244no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23M16×D5312yes—
556C23M16×D5312yes—
557C23C22×C4×D5311yescharacteristic
558D4—28no—
559D4—1068no—
560D4—1068no—
561D4—1068no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—532no—
575C2×C4—532no—
576C2×C4—532no—
577C2×C4—1068no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—4242no—
583C2×C4—4242no—
584C2×C4—4242no—
585C2×C4—4242no—
586C2×C4—2124no—
587C2×C4—2124no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4C22⋊C4×D5312yes—
593C2×C4C22⋊C4×D5311yescharacteristic
594C2×C4C22⋊C4×D5311yescharacteristic
595C2×C4C2×D4×D5312yes—
596C8—44no—
597C8—2124no—
598C22—21no—
599C22—21no—
600C22—22no—
601C22—22no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—534no—
611C22—1062no—
612C22—1062no—
613C22—1062no—
614C22—1062no—
615C22—1062no—
616C22—1062no—
617C22—4248no—
618C22—4248no—
619C22—2122no—
620C22—2122no—
621C22—2122no—
622C22—2122no—
623C22—2122no—
624C22—2122no—
625C22—2121no—
626C22—2121no—
627C22C2×C22⋊C4×D5311yescharacteristic
628C22C2×M16×D5311yescharacteristic
629C22C2×M16×D5311yescharacteristic
630C4—22no—
631C4—22no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—41no—
636C4—1062no—
637C4—1062no—
638C4—1062no—
639C4—2121no—
640C4—2121no—
641C2—21no—
642C2—21no—
643C2—42no—
644C2—42no—
645C2—1064no—
646C2—2122no—
647C2—2122no—
648C2#407×D5311yescharacteristic
649C2#2028317411yescharacteristic
650C2#2028319111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1466C2
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×D53C4
cosocleC22×C4×C53C24
hypercenter#1466D53
hypocenterC53#17367
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #20283743C2semidirect
3. #20273056C2semidirect
4. #20273050C2semidirect
5. #20273047C2semidirect
6. #20273040C2semidirect
7. #20273030C2semidirect
8. #323×D53C2semidirect
9. C53×#1466C2semidirect
28. #16810680C22semidirect
29. #16810585C22semidirect
30. C22⋊C8×D53C22semidirect
31. #16810348C22semidirect
32. #16810347C22semidirect
33. #16810347C22semidirect
36. C53×#432C22semidirect
37. C53×#323C22semidirect
38. C53×#323C22semidirect
119. C23×D53M16semidirect
142. C2×C22⋊D53M16semidirect
319. #1466D53semidirect
355. #432C2×D53semidirect
362. #323C2×D53semidirect
363. #323C2×D53semidirect
364. C53#17367semidirect

Maximal subgroups

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

subgroupindex
C2×#168110952
#202837432
#202730562
#202730502
#202730472
#202730402
#202730302
#323×D532
C53×#14662
#1736753