Finite Group Databasebeta

← D53×#1515 #27351354

Subgroups — D53×#1515 #27351354

Order 13568.

1022 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D53C211yesmaximal, characteristic
2#432×D53C211yesmaximal, characteristic
3#383×D53C211yesmaximal, characteristic
4#20273267C211yesmaximal, characteristic, complemented
5#20273266C211yesmaximal, characteristic, complemented
6#20273265C211yesmaximal, characteristic, complemented
7#20273264C214yesmaximal, complemented
8#328×D53C214yesmaximal
9C53×#1515C211yesmaximal, characteristic, complemented
10#16811180C2211yescharacteristic
11#16811179—21no—
12#16811178C411yescharacteristic, complemented
13#16811178C411yescharacteristic, complemented
14#16811177—21no—
15#16811177C412yescomplemented
16#16811176—21no—
17#16811175—21no—
18#16811175C2212yes—
19C4⋊Q8×D53C411yescharacteristic, complemented
20C4⋊Q8×D53C411yescharacteristic, complemented
21#125×D53—21no—
22C22⋊Q8×D53—21no—
23C22⋊Q8×D53C2212yes—
24C4×Q8×D53—21no—
25C4×Q8×D53C412yescomplemented
26#117×D53—21no—
27C2×C42×D53C2211yescharacteristic
28#16810680C2211yescharacteristic
29#16810679—21no—
30#16810678—21no—
31#16810677C2212yes—
32C4⋊C8×D53—21no—
33C22⋊C8×D53C2212yes—
34#97×D53—21no—
35#16810491C2211yescharacteristic
36#16810490—21no—
37#16810490—21no—
38#16810490C2212yes—
39#95×D53—21no—
40#95×D53—21no—
41#95×D53C2212yes—
42#16810361C2214yes—
43#16810360C2214yes—
44#16810359C2214yes—
45C53×#533C2211yescharacteristic
46C53×#432C2211yescharacteristic
47C53×#383C2211yescharacteristic
48C53×#328C2214yes—
49C2×Q8×D53—21no—
50C2×Q8×D53C2×C412yes—
51C22×C4×D53—22no—
52C22×C4×D53D411yescharacteristic
53C22×C4×D53D411yescharacteristic
54C22×C4×D53C2311yescharacteristic
55C2×C8×D53—22no—
56#13133217C2×C411yescharacteristic, complemented
57#13133217C2×C411yescharacteristic, complemented
58#13133216—21no—
59#13133216—21no—
60#13133216—21no—
61#13133216C2×C412yescomplemented
62#13133215—21no—
63#13133215C2×C412yescomplemented
64#13133209—21no—
65#13133208—21no—
66#13133208—21no—
67#13133207—22no—
68#13133207—22no—
69#13133197—21no—
70#13133197C2×C412yes—
71#13133196—21no—
72#13133196C2312yes—
73#13133195—22no—
74#13133195—22no—
75#13133194—22no—
76#13133194—22no—
77#13133193—21no—
78#13133193C2×C412yes—
79Q8×Dic53—21no—
80Q8×Dic53C2×C412yescomplemented
81C4×Dic106—21no—
82C4×Dic106—22no—
83C4×Dic106—22no—
84C4×Dic106C2×C412yescomplemented
85#13133182—21no—
86#13133182—22no—
87#13133182—22no—
88#13133182C2×C412yescomplemented
89#13133176—21no—
90#13133175—21no—
91#13133175—21no—
92#13133174—22no—
93#13133173—22no—
94C4⋊C4×D53—21no—
95C4⋊C4×D53—21no—
96C4⋊C4×D53—21no—
97C4⋊C4×D53—22no—
98C4⋊C4×D53—22no—
99C4⋊C4×D53C2×C412yes—
100C22⋊C4×D53—22no—
101C2×C4×Dic53D411yescharacteristic
102C2×C4×Dic53D411yescharacteristic
103C2×C4×Dic53C2311yescharacteristic
104C42×D53—22no—
105C42×D53C2×C411yescharacteristic
106C42×D53C2×C411yescharacteristic
107C42×D53C2×C411yescharacteristic
108C42×D53C2×C411yescharacteristic
109C42×D53D411yescharacteristic
110C42×D53D411yescharacteristic
111C42×D53D411yescharacteristic
112C42×D53D411yescharacteristic
113C4⋊#88322—21no—
114#13133146—22no—
115C22⋊#88322C2312yes—
116#13133129—22no—
117#13133128—21no—
118#13133127—22no—
119#13133124—21no—
120#13133124—21no—
121#13133124—22no—
122#13133124—22no—
123#13133124—22no—
124#13133124C2312yes—
125C4⋊Q8×C53C2×C411yescharacteristic, complemented
126C4⋊Q8×C53C2×C411yescharacteristic, complemented
127#125×C53—21no—
128C22⋊Q8×C53—21no—
129C22⋊Q8×C53C2312yes—
130C4×Q8×C53—21no—
131C4×Q8×C53C2×C412yescomplemented
132#117×C53—21no—
133C2×C42×C53C2311yescharacteristic
134C4⋊C8×C53—21no—
135C22⋊C8×C53C2312yes—
136#97×C53—21no—
137#95×C53—21no—
138#95×C53—21no—
139#95×C53C2312yes—
140C23×D53C2×D411yescharacteristic
141C2×Q8×C53—21no—
142C2×Q8×C53C22×C412yes—
143C22×C4×C53—22no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—22no—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151Q8×D53—44no—
152Q8×D53—44no—
153C2×C4×D53—24no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53—44no—
159C2×C4×D53—44no—
160C2×C4×D53—44no—
161C2×C4×D53C22⋊C411yescharacteristic
162C2×C4×D53C22⋊C411yescharacteristic
163C2×C4×D53C22⋊C411yescharacteristic
164C2×C4×D53C22⋊C411yescharacteristic
165C2×C4×D53C22⋊C411yescharacteristic
166C2×C4×D53C22⋊C411yescharacteristic
167C2×C4×D53C22⋊C411yescharacteristic
168C2×C4×D53C22⋊C411yescharacteristic
169C2×C4×D53C22×C411yescharacteristic
170C2×C4×D53C22×C411yescharacteristic
171C2×C4×D53C2×D411yescharacteristic
172C2×C4×D53C2×D411yescharacteristic
173C2×Dic106—21no—
174C2×Dic106—22no—
175C2×Dic106—22no—
176C2×Dic106C22×C412yes—
177C2×C8×C53—22no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53C22×C412yes—
184C22⋊C4×C53—22no—
185C42×C53—22no—
186C42×C53C22×C411yescharacteristic
187C42×C53C22×C411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53C2×D411yescharacteristic
190C22⋊Dic53—22no—
191#11719921—22no—
192#11719921—22no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—22no—
197C4⋊Dic53—22no—
198C4⋊Dic53C22×C412yes—
199#11719919—22no—
200#11719919—22no—
201#11719919—22no—
202#11719919—22no—
203#11719919—22no—
204#11719919—22no—
205#11719919—22no—
206#11719919—22no—
207C4×Dic53—22no—
208C4×Dic53—22no—
209C4×Dic53—22no—
210C4×Dic53C22⋊C411yescharacteristic
211C4×Dic53C22⋊C411yescharacteristic
212C4×Dic53C22⋊C411yescharacteristic
213C4×Dic53C22⋊C411yescharacteristic
214C4×Dic53C22⋊C411yescharacteristic
215C4×Dic53C22⋊C411yescharacteristic
216C4×Dic53C22⋊C411yescharacteristic
217C4×Dic53C22⋊C411yescharacteristic
218C4×Dic53C22×C411yescharacteristic
219C4×Dic53C22×C411yescharacteristic
220C4×Dic53C2×D411yescharacteristic
221C4×Dic53C2×D411yescharacteristic
222C2×#88322—22no—
223C8×D53—44no—
224C23×C53C22×D411yescharacteristic
225C22×D53—22no—
226C22×D53—22no—
227C22×D53C2×C22⋊C411yescharacteristic
228C22×D53C2×C22⋊C411yescharacteristic
229C22×D53#13714yes—
230Q8×C53—42no—
231Q8×C53—42no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53C2×C22⋊C411yescharacteristic
241C2×C4×C53C2×C22⋊C411yescharacteristic
242C2×C4×C53C2×C22⋊C411yescharacteristic
243C2×C4×C53C2×C22⋊C411yescharacteristic
244C2×C4×C53C23×C411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53C2×C22⋊C411yescharacteristic
255C2×Dic53C2×C22⋊C411yescharacteristic
256C2×Dic53C2×C22⋊C411yescharacteristic
257C2×Dic53C2×C22⋊C411yescharacteristic
258C2×Dic53C2×C22⋊C411yescharacteristic
259C2×Dic53C2×C22⋊C411yescharacteristic
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—22no—
265C4×D53—22no—
266C4×D53—22no—
267C4×D53—22no—
268C4×D53—41no—
269C4×D53—41no—
270C4×D53—41no—
271C4×D53—41no—
272C4×D53—44no—
273C4×D53—44no—
274C4×D53—88no—
275Dic106—44no—
276Dic106—44no—
277Dic106—42no—
278Dic106—42no—
279C8×C53—42no—
280#88322—42no—
281C2×#1515—531nomaximal, Sylow
282C22×C53—21no—
283C22×C53—21no—
284C22×C53C22×C22⋊C411yescharacteristic
285C22×C53C2×#13712yes—
286C2×D53—21no—
287C2×D53—21no—
288C2×D53—28no—
289C2×D53#41012yes—
290C2×D53#41912yes—
291C2×D53#41912yes—
292C2×D53#42112yes—
293C4×C53—21no—
294C4×C53—21no—
295C4×C53—21no—
296C4×C53—21no—
297C4×C53—21no—
298C4×C53—21no—
299C4×C53—42no—
300C4×C53—42no—
301C4×C53—82no—
302Dic53—21no—
303Dic53—21no—
304Dic53—21no—
305Dic53—21no—
306Dic53—21no—
307Dic53—21no—
308Dic53—42no—
309Dic53—42no—
310Dic53—82no—
311C2×#533—531no—
312C2×#432—531no—
313C2×#383—531no—
314#1515—531no—
315#1515—531no—
316#1515—534no—
317#1515—531no—
318#1515D5311yescharacteristic, direct product factor
319C2×#328—534no—
320C2×C53—22no—
321C2×C53C2×#41011yescharacteristic
322C2×C53C2×#41911yescharacteristic
323C2×C53C2×#42111yescharacteristic
324D53—22no—
325D53#151514yesdirect product factor
326#533—531no—
327#533—531no—
328#533—532no—
329#533—532no—
330#533—531no—
331#533—1061no—
332#533—1061no—
333#533—1061no—
334#533—1061no—
335#533C2×D5311yescharacteristic
336C2×C4⋊Q8—531no—
337C2×C4⋊Q8—531no—
338C2×#125—1061no—
339C2×C22⋊Q8—532no—
340C2×C22⋊Q8—1061no—
341C2×C4×Q8—532no—
342C2×C4×Q8—1061no—
343C2×#117—1061no—
344C22×C42—531no—
345#432—531no—
346#432—532no—
347#432—1061no—
348#432—1061no—
349#432C2×D5311yescharacteristic
350C2×C4⋊C8—1061no—
351C2×C22⋊C8—532no—
352C2×#97—1061no—
353#383—531no—
354#383—532no—
355#383—1061no—
356#383—1061no—
357#383C2×D5311yescharacteristic
358C2×#95—532no—
359C2×#95—1061no—
360C2×#95—1061no—
361#328—534no—
362#328—534no—
363#328—534no—
364#328C2×D5314yes—
365C53C2×#151511yescharacteristic, complemented, Sylow
366C22×Q8—532no—
367C22×Q8—1061no—
368C23×C4—531no—
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—1062no—
372C22×C8—1062no—
373C4⋊Q8—532no—
374C4⋊Q8—532no—
375C4⋊Q8—531no—
376C4⋊Q8—531no—
377C4⋊Q8—1061no—
378C4⋊Q8—1061no—
379C4⋊Q8—1061no—
380C4⋊Q8—1061no—
381C4⋊Q8C4×D5311yescharacteristic, complemented
382C4⋊Q8C4×D5311yescharacteristic, complemented
383#125—21no—
384#125—1062no—
385#125—1062no—
386#125—1061no—
387#125—1061no—
388#125—1061no—
389C22⋊Q8—21no—
390C22⋊Q8—532no—
391C22⋊Q8—532no—
392C22⋊Q8—532no—
393C22⋊Q8—1062no—
394C22⋊Q8—1062no—
395C22⋊Q8—1062no—
396C22⋊Q8—1062no—
397C22⋊Q8—1061no—
398C22⋊Q8—1061no—
399C22⋊Q8—1061no—
400C22⋊Q8C22×D5312yes—
401C4×Q8—21no—
402C4×Q8—532no—
403C4×Q8—532no—
404C4×Q8—532no—
405C4×Q8—1062no—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C4×Q8—1061no—
409C4×Q8—1062no—
410C4×Q8—1061no—
411C4×Q8—1061no—
412C4×Q8C4×D5312yescomplemented
413#117—21no—
414#117—1062no—
415#117—1062no—
416#117—1061no—
417#117—1061no—
418#117—1061no—
419C2×C4⋊C4—532no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1062no—
423C2×C4⋊C4—1062no—
424C2×C4⋊C4—1061no—
425C2×C22⋊C4—1062no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—531no—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—1062no—
438C2×C42C22×D5311yescharacteristic
439C4⋊C8—21no—
440C4⋊C8—1062no—
441C4⋊C8—1061no—
442C22⋊C8—532no—
443C22⋊C8—1062no—
444C22⋊C8C22×D5312yes—
445#97—21no—
446#97—1062no—
447#97—1061no—
448#95—21no—
449#95—21no—
450#95—532no—
451#95—1062no—
452#95—1062no—
453#95—1062no—
454#95—1061no—
455#95—1061no—
456#95C22×D5312yes—
457C24—531no—
458C2×Q8—21no—
459C2×Q8—532no—
460C2×Q8—1062no—
461C2×Q8—1062no—
462C2×Q8—1061no—
463C2×Q8—2124no—
464C2×Q8—2124no—
465C2×Q8C2×C4×D5312yes—
466C22×C4—22no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—531no—
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—1064no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—2122no—
487C22×C4—2122no—
488C22×C4—2124no—
489C22×C4—2124no—
490C22×C4—2124no—
491C22×C4D4×D5311yescharacteristic
492C22×C4D4×D5311yescharacteristic
493C22×C4C23×D5311yescharacteristic
494C2×C8—22no—
495C2×C8—1062no—
496C2×C8—2124no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—532no—
503C4⋊C4—1062no—
504C4⋊C4—1062no—
505C4⋊C4—1062no—
506C4⋊C4—1062no—
507C4⋊C4—1062no—
508C4⋊C4—1062no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1062no—
514C4⋊C4—1061no—
515C4⋊C4—1062no—
516C4⋊C4C2×C4×D5312yes—
517C22⋊C4—22no—
518C22⋊C4—1062no—
519C22⋊C4—1062no—
520C22⋊C4—1062no—
521C42—22no—
522C42—531no—
523C42—531no—
524C42—531no—
525C42—531no—
526C42—531no—
527C42—531no—
528C42—531no—
529C42—531no—
530C42—531no—
531C42—531no—
532C42—531no—
533C42—531no—
534C42—1062no—
535C42—1062no—
536C42—1062no—
537C42C2×C4×D5311yescharacteristic
538C42C2×C4×D5311yescharacteristic
539C42D4×D5311yescharacteristic
540C42D4×D5311yescharacteristic
541C23—531no—
542C23—531no—
543C23—534no—
544C23—1062no—
545C23—1062no—
546C23C2×D4×D5311yescharacteristic
547Q8—42no—
548Q8—42no—
549Q8—2122no—
550Q8—2122no—
551Q8—2124no—
552Q8—2124no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—531no—
562C2×C4—531no—
563C2×C4—531no—
564C2×C4—531no—
565C2×C4—531no—
566C2×C4—531no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—4248no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2124no—
584C2×C4—2124no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4C22⋊C4×D5311yescharacteristic
591C2×C4C22⋊C4×D5311yescharacteristic
592C2×C4C22⋊C4×D5311yescharacteristic
593C2×C4C22⋊C4×D5311yescharacteristic
594C2×C4C22×C4×D5311yescharacteristic
595C2×C4C2×D4×D5311yescharacteristic
596C8—42no—
597C8—2122no—
598C22—21no—
599C22—21no—
600C22—532no—
601C22—532no—
602C22—532no—
603C22—532no—
604C22—1068no—
605C22—1061no—
606C22—1061no—
607C22C2×C22⋊C4×D5311yescharacteristic
608C22#137×D5312yes—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—42no—
616C4—42no—
617C4—82no—
618C4—1061no—
619C4—1061no—
620C4—1061no—
621C4—1061no—
622C4—1061no—
623C4—1061no—
624C4—4242no—
625C4—2122no—
626C4—2122no—
627C2—22no—
628C2—534no—
629C2—1062no—
630C2#410×D5311yescharacteristic
631C2#419×D5311yescharacteristic
632C2#421×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1515C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1515D53
hypocenterC53C2×#1515
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
4. #20273267C2semidirect
5. #20273266C2semidirect
6. #20273265C2semidirect
7. #20273264C2semidirect
9. C53×#1515C2semidirect
12. #16811178C4semidirect
13. #16811178C4semidirect
15. #16811177C4semidirect
19. C4⋊Q8×D53C4semidirect
20. C4⋊Q8×D53C4semidirect
25. C4×Q8×D53C4semidirect
56. #13133217C2×C4semidirect
57. #13133217C2×C4semidirect
61. #13133216C2×C4semidirect
63. #13133215C2×C4semidirect
80. Q8×Dic53C2×C4semidirect
84. C4×Dic106C2×C4semidirect
88. #13133182C2×C4semidirect
125. C4⋊Q8×C53C2×C4semidirect
126. C4⋊Q8×C53C2×C4semidirect
131. C4×Q8×C53C2×C4semidirect
318. #1515D53direct (N × H)
325. D53#1515direct (N × H)
365. C53C2×#1515semidirect
381. C4⋊Q8C4×D53semidirect
382. C4⋊Q8C4×D53semidirect
412. C4×Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#533×D532
#432×D532
#383×D532
#202732672
#202732662
#202732652
#202732642
#328×D532
C53×#15152
C2×#151553