Finite Group Databasebeta

← #27350146

Subgroups — #27350146

Order 13568.

940 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811239C211yesmaximal, characteristic
2#20283799C211yesmaximal, characteristic, complemented
3#20283795C211yesmaximal, characteristic, complemented
4#20273084C211yesmaximal, characteristic, complemented
5#20273083C212yesmaximal, complemented
6#20273068C211yesmaximal, characteristic, complemented
7#20273066C211yesmaximal, characteristic, complemented
8#20273065C212yesmaximal, complemented
9#20273034C212yesmaximal, complemented
10#20273028C212yesmaximal, complemented
11C53×#1473C211yesmaximal, characteristic, complemented
12#16811239—22no—
13#16811239—22no—
14#16811239—22no—
15#16811239—22no—
16C2×#13133216C411yescharacteristic
17C2×#13133212C2211yescharacteristic
18C2×#13133205C2211yescharacteristic
19C2×#13133203C411yescharacteristic
20C2×#13133203C2211yescharacteristic
21C2×#13133202C411yescharacteristic
22C2×#13133201C411yescharacteristic
23C2×#13133197C411yescharacteristic
24C2×#13133195C411yescharacteristic
25C2×#13133194C411yescharacteristic
26C2×#13133190C411yescharacteristic
27C2×C4×Dic106C2211yescharacteristic
28C2×#13133177C2211yescharacteristic
29C2×#13133175C2211yescharacteristic
30#16810752—22no—
31#16810751—22no—
32C8×Dic106—22no—
33#16810747—22no—
34#16810680C2211yescharacteristic
35#16810677C2212yes—
36#16810676C2211yescharacteristic
37#16810673C2212yes—
38#16810673C2212yes—
39#16810673C2212yes—
40#16810349C2212yes—
41#16810347C2212yes—
42#16810346C2212yes—
43#16810346C2212yes—
44#16810346C2212yes—
45#16810346C2212yes—
46C2×#126×C53C2211yescharacteristic
47C53×#432C2211yescharacteristic
48C53×#431C2211yescharacteristic
49C53×#324C2212yes—
50C53×#323C2212yes—
51C22×Dic106C2×C411yescharacteristic
52C22×Dic106C2×C411yescharacteristic
53C22×Dic106C2×C411yescharacteristic
54C22×D212C2×C411yescharacteristic
55C22×C22⋊D53C2×C411yescharacteristic
56C22×C22⋊D53C2×C411yescharacteristic
57C23×Dic53C2×C411yescharacteristic
58C22×C4×D53C2×C411yescharacteristic
59C22×C4×D53C2×C411yescharacteristic
60#13133216—22no—
61#13133216—22no—
62#13133212—22no—
63#13133212—22no—
64#13133205—21no—
65#13133205—21no—
66#13133205—21no—
67#13133205—21no—
68#13133203—22no—
69#13133203—22no—
70#13133203—22no—
71#13133203—22no—
72#13133202—22no—
73#13133202—22no—
74#13133201—22no—
75#13133201—22no—
76#13133197—22no—
77#13133197—22no—
78#13133195—22no—
79#13133195—22no—
80#13133194—22no—
81#13133194—22no—
82#13133190—22no—
83#13133190—22no—
84C4×Dic106D412yes—
85C4×Dic106D412yes—
86C4×Dic106D412yes—
87C4×Dic106D412yes—
88#13133177—22no—
89#13133177—22no—
90#13133175—22no—
91#13133175—22no—
92C2×C4⋊Dic53C2×C411yescharacteristic
93C2×C4⋊Dic53C2×C411yescharacteristic
94C2×C4⋊Dic53C2×C411yescharacteristic
95C2×#11719919C2×C411yescharacteristic
96C2×#11719919C2×C411yescharacteristic
97C2×#11719919C2×C411yescharacteristic
98C2×#11719919C2×C411yescharacteristic
99C2×C22⋊Dic53C2×C411yescharacteristic
100C2×#11719921C2×C411yescharacteristic
101C2×#11719921C2×C411yescharacteristic
102C2×#11719921C2×C411yescharacteristic
103C2×#11719921C2×C411yescharacteristic
104C2×#11719921C2×C411yescharacteristic
105C2×#11719921C2×C411yescharacteristic
106C2×#11719921C2311yescharacteristic
107C2×#11719921C2311yescharacteristic
108C2×C4×Dic53C2×C411yescharacteristic
109C2×C4×Dic53C2311yescharacteristic
110C2×C4×Dic53C2311yescharacteristic
111C4⋊#88322—21no—
112C4⋊#88322—21no—
113#13133146—22no—
114#13133146—22no—
115#13133146—22no—
116#13133146—22no—
117C22⋊#88322C2312yes—
118C22⋊#88322C2312yes—
119#13133128—21no—
120#13133127—22no—
121C4×#88322—21no—
122C8×Dic53—22no—
123C8×Dic53—22no—
124C8×Dic53—22no—
125#126×C53—21no—
126#126×C53—21no—
127#126×C53—21no—
128#126×C53—21no—
129C2×C4⋊C4×C53C2×C411yescharacteristic
130C2×C4⋊C4×C53C2×C411yescharacteristic
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C22⋊C4×C53C2×C411yescharacteristic
133C2×C22⋊C4×C53C2×C411yescharacteristic
134C2×C22⋊C4×C53C2311yescharacteristic
135C2×C42×C53C2311yescharacteristic
136C4⋊C8×C53—21no—
137C4⋊C8×C53—21no—
138C22⋊C8×C53C2312yes—
139C22⋊C8×C53C2312yes—
140#97×C53—21no—
141C4×C8×C53—21no—
142C24×C53C22×C411yescharacteristic
143C23×D53C22×C411yescharacteristic
144C23×D53C22×C411yescharacteristic
145C22×C4×C53C22×C411yescharacteristic
146C22×C4×C53C22×C411yescharacteristic
147C22×C4×C53C22×C411yescharacteristic
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C2411yescharacteristic
151C2×C22⋊D53—22no—
152C2×C22⋊D53—22no—
153C2×C22⋊D53—24no—
154C2×C22⋊D53—81no—
155C22×Dic53—22no—
156C22×Dic53—81no—
157C22×Dic53C22×C411yescharacteristic
158C22×Dic53C22×C411yescharacteristic
159C22×Dic53C22×C411yescharacteristic
160C22×Dic53C22×C411yescharacteristic
161C22×Dic53C22×C411yescharacteristic
162C22×Dic53C22×C411yescharacteristic
163C2×D212—21no—
164C2×D212—21no—
165C2×D212—81no—
166C2×C4×D53—22no—
167C2×C4×D53—24no—
168C2×C4×D53—22no—
169C2×C4×D53—81no—
170C2×Dic106—24no—
171C2×Dic106—21no—
172C2×Dic106—21no—
173C2×Dic106—22no—
174C2×Dic106—81no—
175C2×Dic106—81no—
176C2×Dic106C22⋊C412yes—
177C2×Dic106C22⋊C412yes—
178C2×C8×C53—22no—
179C2×C8×C53—22no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C22⋊C4×C53—21no—
187C22⋊C4×C53—21no—
188C22⋊C4×C53—21no—
189C22⋊C4×C53—21no—
190C22⋊C4×C53—21no—
191C22⋊C4×C53—21no—
192C42×C53C2×D411yescharacteristic
193C42×C53C2×D411yescharacteristic
194C42×C53C2×D411yescharacteristic
195C42×C53C2×D411yescharacteristic
196C22⋊Dic53—22no—
197#11719921—21no—
198#11719921—21no—
199#11719921—21no—
200#11719921—21no—
201#11719921—21no—
202#11719921—21no—
203#11719921—21no—
204#11719921—21no—
205#11719921—22no—
206#11719921—22no—
207#11719921—22no—
208#11719921—22no—
209C4⋊Dic53—22no—
210C4⋊Dic53—22no—
211C4⋊Dic53C22⋊C412yes—
212C4⋊Dic53C22⋊C412yes—
213#11719919—22no—
214#11719919—22no—
215#11719919C22⋊C412yes—
216#11719919C22⋊C412yes—
217#11719919C22⋊C412yes—
218#11719919C22⋊C412yes—
219C4×Dic53—22no—
220C4×Dic53C2×D412yes—
221C4×Dic53C2×D412yes—
222C4×Dic53C2×D412yes—
223C4×Dic53C2×D412yes—
224C2×#88322—22no—
225C2×#88322—22no—
226C23×C53—21no—
227C23×C53—22no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53C23×C411yescharacteristic
231C22×D53—21no—
232C22×D53—21no—
233C22×D53—22no—
234C22×D53—22no—
235C22×D53—41no—
236C22×D53—41no—
237C22×D53—41no—
238C22×D53—41no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53C2×C22⋊C411yescharacteristic
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C2×C22⋊C411yescharacteristic
251C2×C4×C53C2×C22⋊C411yescharacteristic
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C22×D411yescharacteristic
254C22⋊D53—44no—
255C22⋊D53—82no—
256C2×Dic53—22no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—82no—
264C2×Dic53C2×C22⋊C412yes—
265C2×Dic53C2×C22⋊C412yes—
266C2×Dic53C2×C22⋊C412yes—
267C2×Dic53C2×C22⋊C412yes—
268C2×Dic53#13114yes—
269D212—81no—
270D212—81no—
271C4×D53—44no—
272C4×D53—82no—
273Dic106—24no—
274Dic106—24no—
275Dic106—82no—
276Dic106—81no—
277Dic106—81no—
278C8×C53—42no—
279C8×C53—42no—
280#88322—42no—
281#88322—42no—
282#17390—531nomaximal, Sylow
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—82no—
289C22×C53C22×C22⋊C411yescharacteristic
290C22×C53C2×#13112yes—
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—41no—
294C2×D53—41no—
295C2×D53—41no—
296C2×D53—41no—
297C2×D53—82no—
298C2×D53—82no—
299C4×C53—21no—
300C4×C53—21no—
301C4×C53—21no—
302C4×C53—21no—
303C4×C53—21no—
304C4×C53—21no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53—41no—
308Dic53—22no—
309Dic53—22no—
310Dic53—22no—
311Dic53—41no—
312Dic53—41no—
313Dic53#40814yes—
314C2×#542—531no—
315#2983—531no—
316#2982—531no—
317#1473—532no—
318#1473—531no—
319#1473D5311yescharacteristic, complemented
320#1469—531no—
321#1469—531no—
322#1469—532no—
323#1460—532no—
324#1458—532no—
325C2×C53—22no—
326C2×C53—81no—
327C2×C53C2×#40811yescharacteristic
328C2×C53#288411yescharacteristic
329C2×C53#288811yescharacteristic
330D53—81no—
331D53—81no—
332#542—1062no—
333#542—1062no—
334#542—1062no—
335#542—1062no—
336C2×C4⋊Q8—531no—
337C2×#126—531no—
338C2×#126C2×D5311yescharacteristic
339C2×#124—531no—
340C2×#124—531no—
341C2×#124—531no—
342C2×#123—531no—
343C2×#123—531no—
344C2×C22⋊Q8—531no—
345C2×C22⋊Q8—531no—
346C2×C22⋊Q8—531no—
347C2×#121—531no—
348C2×C4×Q8—531no—
349C2×C4×D4—531no—
350C2×#117—531no—
351#446—1062no—
352#446—1062no—
353C8×Q8—1062no—
354C8×Q8—1062no—
355#432—531no—
356#432—532no—
357#432C2×D5311yescharacteristic
358#431—531no—
359#431—532no—
360#431—532no—
361#431—532no—
362#431C2×D5311yescharacteristic
363#324—532no—
364#324C2×D5312yes—
365#323—532no—
366#323—532no—
367#323—532no—
368#323—532no—
369#323—532no—
370#323C2×D5312yes—
371C53#1739011yescharacteristic, complemented, Sylow
372C22×Q8—531no—
373C22×Q8—531no—
374C22×Q8—531no—
375C22×D4—531no—
376C22×D4—531no—
377C22×D4—531no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—531no—
381C4⋊Q8—1062no—
382C4⋊Q8—1062no—
383#126—21no—
384#126—21no—
385#126—21no—
386#126—21no—
387#126—1062no—
388#126—1062no—
389#124—1062no—
390#124—1062no—
391#124—1062no—
392#124—1062no—
393#124—1061no—
394#124—1061no—
395#124—1061no—
396#124—1061no—
397#123—1062no—
398#123—1062no—
399#123—1062no—
400#123—1062no—
401C22⋊Q8—1062no—
402C22⋊Q8—1062no—
403C22⋊Q8—1062no—
404C22⋊Q8—1062no—
405C22⋊Q8—1062no—
406C22⋊Q8—1062no—
407#121—1062no—
408#121—1062no—
409C4×Q8—532no—
410C4×Q8—532no—
411C4×Q8—532no—
412C4×Q8—532no—
413C4×D4—1062no—
414C4×D4—1062no—
415#117—1062no—
416#117—1062no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4C4×D5311yescharacteristic
425C2×C4⋊C4C4×D5311yescharacteristic
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C22⋊C4—531no—
428C2×C22⋊C4—531no—
429C2×C22⋊C4—531no—
430C2×C22⋊C4—531no—
431C2×C22⋊C4—531no—
432C2×C22⋊C4—531no—
433C2×C22⋊C4—531no—
434C2×C22⋊C4—531no—
435C2×C22⋊C4—531no—
436C2×C22⋊C4C4×D5311yescharacteristic
437C2×C22⋊C4C4×D5311yescharacteristic
438C2×C22⋊C4C22×D5311yescharacteristic
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42C22×D5311yescharacteristic
443C4⋊C8—21no—
444C4⋊C8—21no—
445C4⋊C8—1062no—
446C4⋊C8—1062no—
447C4⋊C8—1062no—
448C4⋊C8—1062no—
449C4⋊C8—1061no—
450C4⋊C8—1061no—
451C22⋊C8—532no—
452C22⋊C8—532no—
453C22⋊C8C22×D5312yes—
454C22⋊C8C22×D5312yes—
455#97—21no—
456#97—1062no—
457#97—1061no—
458C4×C8—21no—
459C4×C8—1062no—
460C4×C8—1062no—
461C4×C8—1062no—
462C4×C8—1061no—
463C24—531no—
464C24—531no—
465C24C2×C4×D5311yescharacteristic
466C2×Q8—532no—
467C2×Q8—532no—
468C2×Q8—1064no—
469C2×Q8—1062no—
470C2×Q8—1061no—
471C2×Q8—1061no—
472C2×Q8—4241no—
473C2×Q8—4241no—
474C2×D4—1062no—
475C2×D4—1062no—
476C2×D4—1064no—
477C2×D4—1061no—
478C2×D4—1061no—
479C2×D4—4241no—
480C2×D4—4241no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—531no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1064no—
490C22×C4—1062no—
491C22×C4—4241no—
492C22×C4—4241no—
493C22×C4C2×C4×D5311yescharacteristic
494C22×C4C2×C4×D5311yescharacteristic
495C22×C4C2×C4×D5311yescharacteristic
496C22×C4C2×C4×D5311yescharacteristic
497C22×C4C2×C4×D5311yescharacteristic
498C22×C4C23×D5311yescharacteristic
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—1062no—
502C2×C8—1062no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—532no—
510C4⋊C4—532no—
511C4⋊C4—532no—
512C4⋊C4—532no—
513C4⋊C4—532no—
514C4⋊C4—532no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C4⋊C4—1062no—
518C4⋊C4—1062no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—1062no—
526C22⋊C4—1062no—
527C22⋊C4—1062no—
528C22⋊C4—1062no—
529C22⋊C4—1062no—
530C22⋊C4—1061no—
531C22⋊C4—1061no—
532C22⋊C4—1061no—
533C22⋊C4—1061no—
534C22⋊C4—1061no—
535C22⋊C4—1061no—
536C22⋊C4—1061no—
537C22⋊C4—1061no—
538C42—532no—
539C42—532no—
540C42—532no—
541C42—532no—
542C42—1062no—
543C42D4×D5311yescharacteristic
544C42D4×D5311yescharacteristic
545C42D4×D5311yescharacteristic
546C42D4×D5311yescharacteristic
547C23—21no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—1062no—
552C23—1062no—
553C23—1061no—
554C23—1061no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23—2121no—
559C23C22×C4×D5311yescharacteristic
560Q8—1064no—
561Q8—1064no—
562Q8—4241no—
563Q8—4241no—
564Q8—4242no—
565D4—4241no—
566D4—4241no—
567D4—4242no—
568D4—2124no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—532no—
579C2×C4—534no—
580C2×C4—532no—
581C2×C4—532no—
582C2×C4—532no—
583C2×C4—1062no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—4242no—
587C2×C4—4242no—
588C2×C4—2124no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4C22⋊C4×D5311yescharacteristic
594C2×C4C22⋊C4×D5311yescharacteristic
595C2×C4C22⋊C4×D5311yescharacteristic
596C2×C4C22⋊C4×D5311yescharacteristic
597C2×C4C2×D4×D5311yescharacteristic
598C2×C4C2×D4×D5311yescharacteristic
599C8—42no—
600C8—42no—
601C8—2122no—
602C8—2122no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—82no—
609C22—4242no—
610C22—4242no—
611C22—2121no—
612C22—2121no—
613C22—2121no—
614C22—2121no—
615C22—2121no—
616C22—2121no—
617C22C2×C22⋊C4×D5311yescharacteristic
618C22#131×D5312yes—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—534no—
629C4—1062no—
630C4—1062no—
631C4—1062no—
632C4—2121no—
633C4—2121no—
634C2—22no—
635C2—81no—
636C2—4241no—
637C2—4241no—
638C2#408×D5311yescharacteristic
639C2#2028317711yescharacteristic
640C2#2028319211yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1473C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133190C4
cosocleC22×C4×C53C24
hypercenter#1473D53
hypocenterC53#17390
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1473

Complements

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

NH ≅ G/Ntype
2. #20283799C2semidirect
3. #20283795C2semidirect
4. #20273084C2semidirect
5. #20273083C2semidirect
6. #20273068C2semidirect
7. #20273066C2semidirect
8. #20273065C2semidirect
9. #20273034C2semidirect
10. #20273028C2semidirect
11. C53×#1473C2semidirect
319. #1473D53semidirect
371. C53#17390semidirect

Maximal subgroups

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

subgroupindex
C2×#168112392
#202837992
#202837952
#202730842
#202730832
#202730682
#202730662
#202730652
#202730342
#202730282
C53×#14732
#1739053