Finite Group Databasebeta

← #27349997

Subgroups — #27349997

Order 13568.

1350 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811095C211yesmaximal, characteristic
2#20283741C212yesmaximal, complemented
3#20273054C212yesmaximal, complemented
4#20273052C212yesmaximal, complemented
5#20273048C212yesmaximal, complemented
6#20273040C212yesmaximal, complemented
7#20273033C211yesmaximal, characteristic, complemented
8C2×#16810346C212yesmaximal, complemented
9C53×#1459C211yesmaximal, characteristic, complemented
10C22×D4×D53C411yescharacteristic
11#16811095—24no—
12#16811095—24no—
13C2×C4×D212C411yescharacteristic
14C2×D4×Dic53C411yescharacteristic
15C2×#13133179C2211yescharacteristic
16C2×C4×C22⋊D53C411yescharacteristic
17C2×C4×C22⋊D53C2211yescharacteristic
18C2×#13133177C411yescharacteristic
19C2×#13133177C2211yescharacteristic
20C2×#13133175C2211yescharacteristic
21C2×#13133174C411yescharacteristic
22C22×#11719921C411yescharacteristic
23C22×#11719921C2211yescharacteristic
24C2×C22⋊C4×D53C411yescharacteristic
25C2×C22⋊C4×D53C2211yescharacteristic
26#16810694—24no—
27#16810691—24no—
28#16810677C2214yescomplemented
29#16810585C2214yescomplemented
30C2×C22⋊#88322C2212yescomplemented
31#16810349C2212yescomplemented
32#16810347C2212yescomplemented
33#16810346C2214yes—
34#16810346C2214yescomplemented
35C2×C4×D4×C53C2211yescharacteristic
36C2×C22⋊C8×C53C2212yescomplemented
37C53×#324C2212yescomplemented
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×D53C2×C411yescharacteristic
53C22×C4×D53C2311yescharacteristic
54C2×#11719912—24no—
55C22×#88322—22no—
56C4×D212—22no—
57C4×D212—22no—
58D4×Dic53—24no—
59#13133179—24no—
60C4×C22⋊D53—22no—
61C4×C22⋊D53—22no—
62C4×C22⋊D53D414yes—
63C4×C22⋊D53D414yes—
64#13133177—24no—
65#13133177—24no—
66#13133175—24no—
67#13133174—22no—
68#13133174—22no—
69C2×C4⋊Dic53C2×C411yescharacteristic
70C2×#11719919C2×C411yescharacteristic
71C2×#11719919C2×C411yescharacteristic
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×#11719921C2311yescharacteristic
85C2×#11719921C2311yescharacteristic
86C22⋊C4×D53—24no—
87C22⋊C4×D53—24no—
88C2×C4×Dic53C2×C411yescharacteristic
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133146—24no—
92C22⋊#88322—22no—
93C22⋊#88322C2314yes—
94#13133129—24no—
95#13133127—24no—
96C22×D4×C53C2×C411yescharacteristic
97C23×C4×C53C2×C411yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53—22no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53C2×C411yescharacteristic
104C2×C22⋊C4×C53C2311yescharacteristic
105C2×C42×C53C2×C411yescharacteristic
106C22⋊C8×C53—22no—
107C22⋊C8×C53C2314yes—
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×C53M1614yescomplemented
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—81no—
129C22×C4×C53—81no—
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C22×C411yescharacteristic
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
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⋊D53C22⋊C414yes—
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×D212—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—81no—
163C2×C4×D53—81no—
164C2×C4×D53—81no—
165C2×C4×D53—81no—
166C2×C4×D53C2×D414yes—
167C2×C8×C53—24no—
168C2×C8×C53—82no—
169C4⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C42×C53—21no—
173C42×C53—21no—
174C22⋊Dic53—22no—
175C22⋊Dic53C22⋊C414yes—
176#11719921—22no—
177#11719921—21no—
178#11719921—21no—
179#11719921—21no—
180#11719921—21no—
181#11719921—22no—
182#11719921—22no—
183#11719921—22no—
184#11719921—22no—
185#11719921C22⋊C412yes—
186#11719921C22⋊C412yes—
187C4⋊Dic53—22no—
188#11719919—22no—
189#11719919C22⋊C412yes—
190#11719919C22⋊C412yes—
191C4×Dic53—22no—
192C4×Dic53—22no—
193C4×Dic53C2×D414yes—
194C2×#88322—24no—
195C2×#88322—82no—
196#11719912—84no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53C2×C22⋊C412yes—
204C23×C53C2×M1612yes—
205C23×C53C2×M1612yes—
206C23×C53C2×M1612yes—
207C23×C53C23×C411yescharacteristic
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53—28no—
215C22×D53—28no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—82no—
223C22×D53—82no—
224C22×D53—82no—
225C22×D53—82no—
226C22×D53—84no—
227C22×D53—84no—
228C22×D53—84no—
229C22×D53—84no—
230C22×D53C2×C22⋊C412yes—
231C22×D53C2×M1612yes—
232C22×D53C2×M1612yes—
233D4×C53—28no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—81no—
245C2×C4×C53—81no—
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C2×C22⋊C411yescharacteristic
248C2×C4×C53C2×M1612yes—
249C2×C4×C53C22×D412yes—
250C22⋊D53—28no—
251C22⋊D53—28no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—82no—
261C2×Dic53—82no—
262C2×Dic53C2×C22⋊C412yes—
263C2×Dic53C2×C22⋊C412yes—
264C2×Dic53C2×C22⋊C412yes—
265D212—28no—
266C4×D53—28no—
267C4×D53—44no—
268C4×D53—82no—
269C4×D53—82no—
270C4×D53—82no—
271C8×C53—44no—
272#88322—44no—
273#17367—531nomaximal, Sylow
274C22×C53—21no—
275C22×C53—22no—
276C22×C53—21no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53#40714yes—
286C22×C53C22×C22⋊C411yescharacteristic
287C22×C53C22×M1611yescharacteristic
288C22×C53C22×M1611yescharacteristic
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—41no—
299C2×D53—41no—
300C2×D53—41no—
301C2×D53—41no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—88no—
305C2×D53—88no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—21no—
309C4×C53—21no—
310C4×C53—41no—
311C4×C53—41no—
312Dic53—22no—
313Dic53—22no—
314Dic53—22no—
315Dic53—41no—
316Dic53—41no—
317C2×#518—531no—
318#2973—532no—
319#1466—532no—
320#1465—532no—
321#1464—532no—
322#1462—532no—
323#1459—531no—
324#1459D5311yescharacteristic, complemented
325C2×#323—532no—
326C2×C53—21no—
327C2×C53—21no—
328C2×C53—24no—
329C2×C53—42no—
330C2×C53C2×#40711yescharacteristic
331C2×C53#288311yescharacteristic
332C2×C53#288811yescharacteristic
333D53—42no—
334D53—42no—
335D53—42no—
336C23×D4—531no—
337#518—1064no—
338#518—1064no—
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×D4—531no—
346C2×C4×D4C2×D5311yescharacteristic
347C2×#117—531no—
348C2×#117—531no—
349C22×C22⋊C4—531no—
350C22×C22⋊C4—531no—
351C22×C22⋊C4—531no—
352C22×C22⋊C4—531no—
353#435—1064no—
354#435—1064no—
355#432—534no—
356#407—534no—
357C2×C22⋊C8—532no—
358C2×C22⋊C8C2×D5312yescomplemented
359#324—532no—
360#324C2×D5312yescomplemented
361#323—532no—
362#323—534no—
363#323—534no—
364#323C2×D5312yescomplemented
365C53#1736711yescharacteristic, complemented, Sylow
366C25—531no—
367C25—531no—
368C22×D4—531no—
369C22×D4—531no—
370C22×D4—531no—
371C22×D4—1064no—
372C22×D4—1064no—
373C22×D4—1064no—
374C22×D4C4×D5311yescharacteristic
375C23×C4—531no—
376C23×C4—531no—
377C23×C4—531no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4—531no—
382C23×C4C4×D5311yescharacteristic
383C23×C4C22×D5311yescharacteristic
384C2×M16—1064no—
385C22×C8—22no—
386C22×C8—1062no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—534no—
390C4×D4—534no—
391C4×D4—1062no—
392C4×D4—1062no—
393C4×D4—1064no—
394C4×D4—1064no—
395C4×D4—1064no—
396C4×D4—1064no—
397C4×D4—1062no—
398C4×D4—1062no—
399#117—1062no—
400#117—1062no—
401#117—1064no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4C22×D5311yescharacteristic
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—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—1064no—
417C2×C22⋊C4—1064no—
418C2×C22⋊C4—1062no—
419C2×C22⋊C4—1062no—
420C2×C22⋊C4—1062no—
421C2×C22⋊C4—1062no—
422C2×C22⋊C4C4×D5311yescharacteristic
423C2×C22⋊C4C22×D5311yescharacteristic
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42C4×D5311yescharacteristic
428C4⋊C8—1064no—
429C22⋊C8—22no—
430C22⋊C8—534no—
431C22⋊C8—1064no—
432C22⋊C8—1062no—
433C22⋊C8C22×D5314yes—
434#97—1064no—
435C24—534no—
436C24—531no—
437C24—531no—
438C24—531no—
439C24—1062no—
440C24—1062no—
441C24—1062no—
442C24—1062no—
443C24—1062no—
444C24—4241no—
445C24—4241no—
446C24—4241no—
447C24—4241no—
448C24C2×C4×D5311yescharacteristic
449C24C2×C4×D5311yescharacteristic
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—534no—
453C2×D4—1062no—
454C2×D4—1062no—
455C2×D4—1062no—
456C2×D4—10632no—
457C2×D4—1062no—
458C2×D4—1062no—
459C2×D4—1062no—
460C2×D4—1062no—
461C2×D4—1062no—
462C2×D4#1171991214yescomplemented
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—534no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—4241no—
482C22×C4—4241no—
483C22×C4—4241no—
484C22×C4—4241no—
485C22×C4—4241no—
486C22×C4—4241no—
487C22×C4C2×C4×D5311yescharacteristic
488C22×C4C2×C4×D5311yescharacteristic
489C22×C4C2×C4×D5311yescharacteristic
490C22×C4C2×C4×D5311yescharacteristic
491C22×C4D4×D5312yes—
492C22×C4D4×D5312yes—
493C22×C4C23×D5311yescharacteristic
494M16—4244no—
495C2×C8—24no—
496C2×C8—82no—
497C2×C8—1064no—
498C2×C8—4242no—
499C4⋊C4—22no—
500C4⋊C4—532no—
501C4⋊C4—532no—
502C4⋊C4—1062no—
503C4⋊C4—1062no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—534no—
507C22⋊C4—532no—
508C22⋊C4—532no—
509C22⋊C4—1062no—
510C22⋊C4—1062no—
511C22⋊C4—1062no—
512C22⋊C4—1062no—
513C22⋊C4—1061no—
514C22⋊C4—1061no—
515C22⋊C4—1061no—
516C22⋊C4—1061no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C42—21no—
520C42—21no—
521C42—534no—
522C42—1062no—
523C42—1062no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—532no—
531C23—532no—
532C23—532no—
533C23—1068no—
534C23—1068no—
535C23—1061no—
536C23—1061no—
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—
555C23C22⋊C4×D5312yes—
556C23M16×D5312yes—
557C23C2×#1171991212yes—
558C23C2×#1171991212yes—
559C23C22×C4×D5311yescharacteristic
560D4—28no—
561D4—1068no—
562D4—1068no—
563D4—1068no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—532no—
577C2×C4—532no—
578C2×C4—532no—
579C2×C4—1068no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—4242no—
585C2×C4—4242no—
586C2×C4—4242no—
587C2×C4—4242no—
588C2×C4—4242no—
589C2×C4—2124no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4C22⋊C4×D5311yescharacteristic
595C2×C4C22⋊C4×D5311yescharacteristic
596C2×C4C2×#1171991212yes—
597C2×C4C2×D4×D5312yes—
598C8—44no—
599C8—2124no—
600C22—21no—
601C22—22no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—41no—
610C22—41no—
611C22—1062no—
612C22—1062no—
613C22—1062no—
614C22—1062no—
615C22—1062no—
616C22—4248no—
617C22—4248no—
618C22—2122no—
619C22—2122no—
620C22—2122no—
621C22—2122no—
622C22—2122no—
623C22—2122no—
624C22—2121no—
625C22—2121no—
626C22—2121no—
627C22—2121no—
628C22#1681058514yes—
629C22C2×C22⋊C4×D5311yescharacteristic
630C22C2×M16×D5311yescharacteristic
631C22C22×#1171991211yescharacteristic
632C4—22no—
633C4—22no—
634C4—21no—
635C4—21no—
636C4—41no—
637C4—41no—
638C4—1062no—
639C4—1062no—
640C4—1062no—
641C4—2121no—
642C4—2121no—
643C2—21no—
644C2—21no—
645C2—24no—
646C2—42no—
647C2—2122no—
648C2—2122no—
649C2—2122no—
650C2C2×#1681058511yescharacteristic
651C2#2028317311yescharacteristic
652C2#2028319211yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1459C2
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×D53C4
cosocleC22×C4×C53C24
hypercenter#1459D53
hypocenterC53#17367
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #20283741C2semidirect
3. #20273054C2semidirect
4. #20273052C2semidirect
5. #20273048C2semidirect
6. #20273040C2semidirect
7. #20273033C2semidirect
8. C2×#16810346C2semidirect
9. C53×#1459C2semidirect
28. #16810677C22semidirect
29. #16810585C22semidirect
30. C2×C22⋊#88322C22semidirect
31. #16810349C22semidirect
32. #16810347C22semidirect
34. #16810346C22semidirect
36. C2×C22⋊C8×C53C22semidirect
37. C53×#324C22semidirect
38. C53×#323C22semidirect
119. C23×D53M16semidirect
125. C2×D4×C53M16semidirect
324. #1459D53semidirect
358. C2×C22⋊C8C2×D53semidirect
360. #324C2×D53semidirect
364. #323C2×D53semidirect
365. C53#17367semidirect
462. C2×D4#11719912semidirect

Maximal subgroups

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

subgroupindex
C2×#168110952
#202837412
#202730542
#202730522
#202730482
#202730402
#202730332
C2×#168103462
C53×#14592
#1736753