Finite Group Databasebeta

← #27350075

Subgroups — #27350075

Order 13568.

1020 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811159C211yesmaximal, characteristic
2#20283344C211yesmaximal, characteristic, complemented
3C2×#16810673C211yesmaximal, characteristic, complemented
4#20273083C214yesmaximal, complemented
5#20273079C211yesmaximal, characteristic, complemented
6#20273076C211yesmaximal, characteristic
7#20273074C211yesmaximal, characteristic
8#20273028C214yesmaximal, complemented
9C53×#1472C211yesmaximal, characteristic, complemented
10#16811159—21no—
11#16811159—21no—
12#16811159—21no—
13#16811159—21no—
14#16811159—21no—
15#16811159—21no—
16#16811159—21no—
17#16811159—21no—
18C2×#13133212C2212yes—
19C2×#13133208C411yescharacteristic
20C2×#13133206C411yescharacteristic
21C2×#13133201C412yes—
22C2×#13133191C2211yescharacteristic
23C2×Q8×Dic53C411yescharacteristic
24C2×#13133179C411yescharacteristic
25C2×#13133177C412yes—
26C2×#13133175C2212yes—
27C22×C4×Dic53C2211yescharacteristic
28#16810677—21no—
29#16810677C2212yes—
30#16810673—21no—
31#16810673—21no—
32#16810673—21no—
33#16810673C2212yes—
34#16810673C2212yes—
35#16810673C2212yes—
36#16810646—21no—
37C2×#13133146—21no—
38#16810587C2211yescharacteristic
39C2×C22⋊#88322C2211yescharacteristic
40M16×Dic53—21no—
41C2×C8×Dic53—21no—
42#16810349C2214yes—
43#16810346C2214yes—
44#16810346C2214yes—
45C2×C22⋊Q8×C53C2211yescharacteristic
46C53×#407C2211yescharacteristic
47C2×C22⋊C8×C53C2211yescharacteristic
48C53×#324C2214yes—
49C22×D212C2×C411yescharacteristic
50C22×C22⋊D53C2×C412yes—
51C23×Dic53C2×C411yescharacteristic
52C23×Dic53C2×C411yescharacteristic
53C22×C4×D53C2×C412yes—
54C2×#11719917—21no—
55C22×#88322—21no—
56#13133212—22no—
57#13133212—22no—
58#13133212—22no—
59#13133212—22no—
60#13133208—21no—
61#13133208—21no—
62#13133208—21no—
63#13133208—21no—
64#13133206—21no—
65#13133206—21no—
66#13133206—21no—
67#13133206—21no—
68#13133201—22no—
69#13133201—22no—
70#13133201—22no—
71#13133201—22no—
72#13133191—21no—
73#13133191—21no—
74#13133191—21no—
75#13133191—21no—
76Q8×Dic53—21no—
77Q8×Dic53—21no—
78Q8×Dic53—21no—
79Q8×Dic53—21no—
80#13133179—21no—
81#13133179—21no—
82#13133179—21no—
83#13133179—21no—
84#13133177—22no—
85#13133177—22no—
86#13133177—22no—
87#13133177—22no—
88#13133175—22no—
89#13133175—22no—
90#13133175—22no—
91#13133175—22no—
92C2×C4⋊Dic53C2×C412yes—
93C2×C4⋊Dic53C2×C411yescharacteristic
94C2×C4⋊Dic53C2×C412yes—
95C2×#11719919C2×C412yes—
96C2×#11719921C2×C412yes—
97C2×#11719921C2×C412yes—
98C2×#11719921C2×C412yes—
99C2×#11719921C2312yes—
100C2×C4×Dic53C2×C412yes—
101C2×C4×Dic53C2×C411yescharacteristic
102C2×C4×Dic53C2×C411yescharacteristic
103C2×C4×Dic53D411yescharacteristic
104C2×C4×Dic53D411yescharacteristic
105C2×C4×Dic53D411yescharacteristic
106C2×C4×Dic53D411yescharacteristic
107C2×C4×Dic53D411yescharacteristic
108C2×C4×Dic53D411yescharacteristic
109C2×C4×Dic53D411yescharacteristic
110C2×C4×Dic53D411yescharacteristic
111C2×C4×Dic53C2311yescharacteristic
112C2×C4×Dic53C2311yescharacteristic
113#13133146—22no—
114#13133146—22no—
115#13133146—22no—
116#13133146—22no—
117C22⋊#88322—21no—
118C22⋊#88322—21no—
119C22⋊#88322C2312yes—
120C22⋊#88322C2312yes—
121#13133127—22no—
122C8×Dic53—22no—
123C8×Dic53—22no—
124C8×Dic53—22no—
125C22×Q8×C53C2×C411yescharacteristic
126C23×C4×C53C2311yescharacteristic
127C2×M16×C53—21no—
128C22×C8×C53—21no—
129C22⋊Q8×C53—21no—
130C22⋊Q8×C53—21no—
131C22⋊Q8×C53—21no—
132C22⋊Q8×C53—21no—
133C2×C4⋊C4×C53C2×C411yescharacteristic
134C2×C4⋊C4×C53C2312yes—
135C2×C22⋊C4×C53C2×C412yes—
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53C2312yes—
139C22⋊C8×C53C2312yes—
140C24×C53C22×C411yescharacteristic
141C23×D53C22×C412yes—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53C22×C412yes—
147C22×C4×C53C22×C411yescharacteristic
148C22×C4×C53C22×C412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2411yescharacteristic
154C2×C22⋊D53—22no—
155C2×C22⋊D53—22no—
156C2×C22⋊D53—44no—
157C22×Dic53—24no—
158C22×Dic53—42no—
159C22×Dic53C22⋊C411yescharacteristic
160C22×Dic53C22⋊C411yescharacteristic
161C22×Dic53C22⋊C411yescharacteristic
162C22×Dic53C22⋊C411yescharacteristic
163C22×Dic53C22⋊C411yescharacteristic
164C22×Dic53C22⋊C411yescharacteristic
165C22×Dic53C22⋊C411yescharacteristic
166C22×Dic53C22⋊C411yescharacteristic
167C22×Dic53C22×C411yescharacteristic
168C22×Dic53C22×C412yes—
169C22×Dic53C22×C411yescharacteristic
170C22×Dic53C22×C411yescharacteristic
171C22×Dic53C22×C411yescharacteristic
172C2×D212—21no—
173C2×D212—21no—
174C2×D212—42no—
175C2×C4×D53—22no—
176C2×C4×D53—22no—
177C2×C4×D53—28no—
178M16×C53—42no—
179C2×C8×C53—22no—
180C2×C8×C53—22no—
181C2×C8×C53—42no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—22no—
188#11719921—22no—
189#11719921—22no—
190#11719921—22no—
191#11719921—22no—
192#11719921—22no—
193#11719921—22no—
194#11719921—22no—
195#11719921—22no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4⋊Dic53—22no—
201C4⋊Dic53—22no—
202#11719919—22no—
203#11719919—22no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53C22⋊C411yescharacteristic
207C4×Dic53C22⋊C411yescharacteristic
208C4×Dic53C22⋊C411yescharacteristic
209C4×Dic53C22⋊C411yescharacteristic
210C4×Dic53C22⋊C411yescharacteristic
211C4×Dic53C22⋊C411yescharacteristic
212C4×Dic53C22⋊C411yescharacteristic
213C4×Dic53C22⋊C411yescharacteristic
214C4×Dic53C2×D411yescharacteristic
215C4×Dic53C2×D411yescharacteristic
216C4×Dic53C2×D411yescharacteristic
217C4×Dic53C2×D411yescharacteristic
218C4×Dic53C2×D411yescharacteristic
219C4×Dic53C2×D411yescharacteristic
220C4×Dic53C2×D411yescharacteristic
221C4×Dic53C2×D411yescharacteristic
222#11719917—42no—
223C2×#88322—22no—
224C2×#88322—22no—
225C2×#88322—42no—
226C23×C53—22no—
227C23×C53—22no—
228C23×C53—41no—
229C23×C53C2×C22⋊C411yescharacteristic
230C23×C53C2×C22⋊C411yescharacteristic
231C23×C53C23×C411yescharacteristic
232C22×D53—22no—
233C22×D53—24no—
234C22×D53—42no—
235C22×D53—42no—
236Q8×C53—42no—
237Q8×C53—42no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—24no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53C2×C22⋊C411yescharacteristic
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C22×D411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C22⋊D53—44no—
253C22⋊D53—44no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—42no—
267C2×Dic53C2×C22⋊C411yescharacteristic
268C2×Dic53C2×C22⋊C411yescharacteristic
269C2×Dic53C2×C22⋊C411yescharacteristic
270C2×Dic53C2×C22⋊C411yescharacteristic
271C2×Dic53C2×C22⋊C411yescharacteristic
272C2×Dic53C2×C22⋊C411yescharacteristic
273C2×Dic53C2×C22⋊C411yescharacteristic
274C2×Dic53C2×C22⋊C411yescharacteristic
275C2×Dic53#13114yes—
276D212—42no—
277D212—42no—
278C4×D53—44no—
279C4×D53—44no—
280C8×C53—42no—
281C8×C53—42no—
282#88322—42no—
283#88322—42no—
284#17380—531nomaximal, Sylow
285C22×C53—21no—
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—21no—
289C22×C53—21no—
290C22×C53—21no—
291C22×C53—21no—
292C22×C53—44no—
293C22×C53#41312yes—
294C22×C53C22×C22⋊C411yescharacteristic
295C22×C53C2×#13112yes—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—84no—
300C4×C53—21no—
301C4×C53—21no—
302C4×C53—21no—
303C4×C53—42no—
304C4×C53—42no—
305C4×C53#41012yes—
306Dic53—21no—
307Dic53—21no—
308Dic53—21no—
309Dic53—21no—
310Dic53—21no—
311Dic53—21no—
312Dic53—42no—
313Dic53#40812yes—
314Dic53#40812yes—
315C2×#529—531no—
316#2920—531no—
317C2×#431—531no—
318#1473—534no—
319#1472—531no—
320#1472D5311yescharacteristic, complemented
321#1471—531no—
322#1471—531no—
323#1458—534no—
324C2×C53—22no—
325C2×C53—22no—
326C2×C53—41no—
327C2×C53C2×#40811yescharacteristic
328C2×C53C2×#41011yescharacteristic
329C2×C53C2×#41311yescharacteristic
330D53—82no—
331#529—1061no—
332#529—1061no—
333#529—1061no—
334#529—1061no—
335#529—1061no—
336#529—1061no—
337#529—1061no—
338#529—1061no—
339C2×#126—532no—
340C2×#125—531no—
341C2×#124—531no—
342C2×#123—532no—
343C2×C22⋊Q8C2×D5311yescharacteristic
344C2×#121—531no—
345C2×C4×Q8—531no—
346C2×C4×D4—531no—
347C2×C4×D4—532no—
348C2×#117—532no—
349C22×C42—531no—
350#432—532no—
351#432—1061no—
352#431—532no—
353#431—532no—
354#431—532no—
355#431—1061no—
356#431—1061no—
357#431—1061no—
358#423—1061no—
359C2×C4⋊C8—1061no—
360#407—531no—
361#407C2×D5311yescharacteristic
362C2×C22⋊C8—531no—
363C2×C22⋊C8C2×D5311yescharacteristic
364C4×M16—1061no—
365C2×C4×C8—1061no—
366#324—534no—
367#324C2×D5314yes—
368#323—534no—
369#323—534no—
370C53#1738011yescharacteristic, complemented, Sylow
371C22×Q8C4×D5311yescharacteristic
372C22×D4—532no—
373C22×D4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—532no—
377C23×C4C22×D5311yescharacteristic
378C2×M16—21no—
379C2×M16—1061no—
380C22×C8—21no—
381C22×C8—1061no—
382#126—1062no—
383#126—1062no—
384#126—1062no—
385#126—1062no—
386#125—1061no—
387#125—1061no—
388#125—1061no—
389#125—1061no—
390#124—1061no—
391#124—1061no—
392#124—1061no—
393#124—1061no—
394#123—1062no—
395#123—1062no—
396#123—1062no—
397#123—1062no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402#121—1061no—
403#121—1061no—
404#121—1061no—
405#121—1061no—
406C4×Q8—1061no—
407C4×Q8—1061no—
408C4×Q8—1061no—
409C4×Q8—1061no—
410C4×D4—1061no—
411C4×D4—1061no—
412C4×D4—1061no—
413C4×D4—1061no—
414C4×D4—1062no—
415C4×D4—1062no—
416C4×D4—1062no—
417C4×D4—1062no—
418#117—1062no—
419#117—1062no—
420#117—1062no—
421#117—1062no—
422C2×C4⋊C4—532no—
423C2×C4⋊C4—532no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—532no—
426C2×C4⋊C4C4×D5311yescharacteristic
427C2×C4⋊C4C22×D5312yes—
428C2×C22⋊C4—532no—
429C2×C22⋊C4—532no—
430C2×C22⋊C4—532no—
431C2×C22⋊C4—532no—
432C2×C22⋊C4C4×D5312yes—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—531no—
444C2×C42—531no—
445C2×C42—532no—
446C4⋊C8—1062no—
447C4⋊C8—1062no—
448C4⋊C8—1062no—
449C4⋊C8—1062no—
450C22⋊C8—21no—
451C22⋊C8—21no—
452C22⋊C8—532no—
453C22⋊C8—532no—
454C22⋊C8—1061no—
455C22⋊C8—1061no—
456C22⋊C8C22×D5312yes—
457C22⋊C8C22×D5312yes—
458#97—1062no—
459C4×C8—1062no—
460C4×C8—1062no—
461C4×C8—1062no—
462C24—532no—
463C24C2×C4×D5311yescharacteristic
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—42no—
467C2×D4—1062no—
468C2×D4—1062no—
469C2×D4—1061no—
470C2×D4—1061no—
471C2×D4—2122no—
472C2×D4—2124no—
473C22×C4—42no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—532no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1068no—
490C22×C4—1064no—
491C22×C4—2122no—
492C22×C4C2×C4×D5312yes—
493C22×C4C2×C4×D5311yescharacteristic
494C22×C4C2×C4×D5312yes—
495C22×C4D4×D5311yescharacteristic
496C22×C4D4×D5311yescharacteristic
497C22×C4D4×D5311yescharacteristic
498C22×C4D4×D5311yescharacteristic
499C22×C4C23×D5311yescharacteristic
500M16—42no—
501M16—2122no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—42no—
505C2×C8—1062no—
506C2×C8—1062no—
507C2×C8—2122no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C4⋊C4—1062no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—1062no—
523C22⋊C4—1062no—
524C22⋊C4—1062no—
525C22⋊C4—1062no—
526C22⋊C4—1062no—
527C22⋊C4—1062no—
528C22⋊C4—1062no—
529C22⋊C4—1062no—
530C42—531no—
531C42—531no—
532C42—531no—
533C42—531no—
534C42—531no—
535C42—531no—
536C42—531no—
537C42—531no—
538C42—531no—
539C42—531no—
540C42—531no—
541C42—531no—
542C42—531no—
543C42—531no—
544C42—531no—
545C42—531no—
546C42—1062no—
547C42—1062no—
548C23—22no—
549C23—22no—
550C23—41no—
551C23—1064no—
552C23—1062no—
553C23—2122no—
554C23—2122no—
555C23C22⋊C4×D5311yescharacteristic
556C23C22⋊C4×D5311yescharacteristic
557C23C22×C4×D5311yescharacteristic
558Q8—42no—
559Q8—42no—
560D4—2122no—
561D4—2124no—
562D4—2124no—
563D4—2122no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—531no—
575C2×C4—531no—
576C2×C4—534no—
577C2×C4—531no—
578C2×C4—531no—
579C2×C4—531no—
580C2×C4—531no—
581C2×C4—531no—
582C2×C4—531no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—2124no—
592C2×C4—2124no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2122no—
598C2×C4C22⋊C4×D5311yescharacteristic
599C2×C4C22⋊C4×D5311yescharacteristic
600C2×C4C2×D4×D5311yescharacteristic
601C2×C4C2×D4×D5311yescharacteristic
602C8—42no—
603C8—42no—
604C8—2122no—
605C8—2122no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—44no—
614C22—4244no—
615C22—2122no—
616C22—2122no—
617C22—2122no—
618C22#1681060612yes—
619C22C2×C22⋊C4×D5311yescharacteristic
620C22#131×D5312yes—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—42no—
625C4—42no—
626C4—532no—
627C4—532no—
628C4—1061no—
629C4—1061no—
630C4—1061no—
631C4—1061no—
632C4—1061no—
633C4—1061no—
634C4—2122no—
635C4#1681059512yes—
636C2—22no—
637C2—22no—
638C2—41no—
639C2—4242no—
640C2#408×D5311yescharacteristic
641C2C2×#1681059511yescharacteristic
642C2C2×#1681060611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1472C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenter#1472D53
hypocenterC53#17380
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
2. #20283344C2semidirect
3. C2×#16810673C2semidirect
4. #20273083C2semidirect
5. #20273079C2semidirect
8. #20273028C2semidirect
9. C53×#1472C2semidirect
320. #1472D53semidirect
370. C53#17380semidirect

Maximal subgroups

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

subgroupindex
C2×#168111592
#202833442
C2×#168106732
#202730832
#202730792
#202730762
#202730742
#202730282
C53×#14722
#1738053