Finite Group Databasebeta

← D53×#1506 #27351239

Subgroups — D53×#1506 #27351239

Order 13568.

1130 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
1#533×D53C211yesmaximal, characteristic
2#432×D53C211yesmaximal, characteristic
3#376×D53C211yesmaximal, characteristic
4#20273228C211yesmaximal, characteristic, complemented
5#20273227C211yesmaximal, characteristic, complemented
6#20273226C211yesmaximal, characteristic, complemented
7#20273225C214yesmaximal, complemented
8#328×D53C214yesmaximal
9C53×#1506C211yesmaximal, 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×D53—21no—
28C2×C42×D53C2211yescharacteristic
29#16810680C2211yescharacteristic
30#16810679—21no—
31#16810678—21no—
32#16810677C2212yes—
33C4⋊C8×D53—21no—
34C22⋊C8×D53C2212yes—
35#97×D53—21no—
36#16810469—21no—
37#16810469C2211yescharacteristic
38#16810468—21no—
39#16810468C2212yes—
40#95×D53—21no—
41#95×D53C2212yes—
42#16810361C2214yes—
43#16810360C2214yes—
44#16810359C2214yes—
45C53×#533C2211yescharacteristic
46C53×#432C2211yescharacteristic
47C53×#376C2211yescharacteristic
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×Dic53—21no—
102C2×C4×Dic53—22no—
103C2×C4×Dic53D411yescharacteristic
104C2×C4×Dic53D411yescharacteristic
105C2×C4×Dic53C2311yescharacteristic
106C42×D53—28no—
107C42×D53—22no—
108C42×D53C2×C411yescharacteristic
109C42×D53C2×C411yescharacteristic
110C42×D53C2×C411yescharacteristic
111C42×D53C2×C411yescharacteristic
112C42×D53D411yescharacteristic
113C42×D53D411yescharacteristic
114C42×D53D411yescharacteristic
115C42×D53D411yescharacteristic
116C4⋊#88322—21no—
117#13133146—22no—
118C22⋊#88322C2312yes—
119#13133129—22no—
120#13133128—21no—
121#13133127—22no—
122#13133124—21no—
123#13133124—22no—
124#13133124—22no—
125#13133124C2312yes—
126C4⋊Q8×C53C2×C411yescharacteristic, complemented
127C4⋊Q8×C53C2×C411yescharacteristic, complemented
128#125×C53—21no—
129C22⋊Q8×C53—21no—
130C22⋊Q8×C53C2312yes—
131C4×Q8×C53—21no—
132C4×Q8×C53C2×C412yescomplemented
133#117×C53—21no—
134C2×C42×C53—21no—
135C2×C42×C53C2311yescharacteristic
136C4⋊C8×C53—21no—
137C22⋊C8×C53C2312yes—
138#97×C53—21no—
139#95×C53—21no—
140#95×C53C2312yes—
141C23×D53C2×D411yescharacteristic
142C2×Q8×C53—21no—
143C2×Q8×C53C22×C412yes—
144C22×C4×C53—22no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—22no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152Q8×D53—44no—
153Q8×D53—44no—
154C2×C4×D53—24no—
155C2×C4×D53—24no—
156C2×C4×D53—28no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—42no—
160C2×C4×D53—44no—
161C2×C4×D53—44no—
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×D53C22×C411yescharacteristic
172C2×C4×D53C2×D411yescharacteristic
173C2×C4×D53C2×D411yescharacteristic
174C2×Dic106—21no—
175C2×Dic106—22no—
176C2×Dic106—22no—
177C2×Dic106C22×C412yes—
178C2×C8×C53—22no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53C22×C412yes—
185C22⋊C4×C53—22no—
186C42×C53—24no—
187C42×C53—22no—
188C42×C53C22×C411yescharacteristic
189C42×C53C22×C411yescharacteristic
190C42×C53C2×D411yescharacteristic
191C42×C53C2×D411yescharacteristic
192C22⋊Dic53—22no—
193#11719921—22no—
194#11719921—22no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4⋊Dic53C22×C412yes—
201#11719919—22no—
202#11719919—22no—
203#11719919—22no—
204#11719919—22no—
205#11719919—22no—
206#11719919—22no—
207#11719919—22no—
208#11719919—22no—
209C4×Dic53—28no—
210C4×Dic53—24no—
211C4×Dic53—22no—
212C4×Dic53—22no—
213C4×Dic53—22no—
214C4×Dic53C22⋊C411yescharacteristic
215C4×Dic53C22⋊C411yescharacteristic
216C4×Dic53C22⋊C411yescharacteristic
217C4×Dic53C22⋊C411yescharacteristic
218C4×Dic53C22⋊C411yescharacteristic
219C4×Dic53C22⋊C411yescharacteristic
220C4×Dic53C22⋊C411yescharacteristic
221C4×Dic53C22⋊C411yescharacteristic
222C4×Dic53C22×C411yescharacteristic
223C4×Dic53C22×C411yescharacteristic
224C4×Dic53C2×D411yescharacteristic
225C4×Dic53C2×D411yescharacteristic
226C2×#88322—22no—
227C8×D53—44no—
228C23×C53C22×D411yescharacteristic
229C22×D53—22no—
230C22×D53—22no—
231C22×D53C2×C22⋊C411yescharacteristic
232C22×D53C2×C22⋊C411yescharacteristic
233C22×D53#13714yes—
234Q8×C53—42no—
235Q8×C53—42no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—24no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53C2×C22⋊C411yescharacteristic
245C2×C4×C53C2×C22⋊C411yescharacteristic
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C2×C22⋊C411yescharacteristic
248C2×C4×C53C23×C411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—24no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—41no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53C2×C22⋊C411yescharacteristic
259C2×Dic53C2×C22⋊C411yescharacteristic
260C2×Dic53C2×C22⋊C411yescharacteristic
261C2×Dic53C2×C22⋊C411yescharacteristic
262C2×Dic53C2×C22⋊C411yescharacteristic
263C2×Dic53C2×C22⋊C411yescharacteristic
264C4×D53—22no—
265C4×D53—22no—
266C4×D53—22no—
267C4×D53—22no—
268C4×D53—22no—
269C4×D53—22no—
270C4×D53—22no—
271C4×D53—22no—
272C4×D53—22no—
273C4×D53—41no—
274C4×D53—41no—
275C4×D53—416no—
276C4×D53—44no—
277C4×D53—44no—
278C4×D53#10414yes—
279Dic106—44no—
280Dic106—42no—
281Dic106—44no—
282Dic106—42no—
283C8×C53—42no—
284#88322—42no—
285C2×#1506—531nomaximal, Sylow
286C22×C53—21no—
287C22×C53—21no—
288C22×C53C22×C22⋊C411yescharacteristic
289C22×C53C2×#13712yes—
290C2×D53—21no—
291C2×D53—21no—
292C2×D53—28no—
293C2×D53#41012yes—
294C2×D53#41912yes—
295C2×D53#41912yes—
296C2×D53C2×#10412yes—
297C4×C53—21no—
298C4×C53—21no—
299C4×C53—21no—
300C4×C53—21no—
301C4×C53—21no—
302C4×C53—44no—
303C4×C53—42no—
304C4×C53—42no—
305C4×C53C2×#10412yes—
306Dic53—21no—
307Dic53—21no—
308Dic53—21no—
309Dic53—21no—
310Dic53—21no—
311Dic53—44no—
312Dic53—42no—
313Dic53—42no—
314Dic53C2×#10412yes—
315C2×#533—531no—
316C2×#432—531no—
317C2×#376—531no—
318#1506—531no—
319#1506—531no—
320#1506—534no—
321#1506—531no—
322#1506D5311yescharacteristic, direct product factor
323C2×#328—534no—
324C2×C53—22no—
325C2×C53C2×#41011yescharacteristic
326C2×C53C2×#41911yescharacteristic
327C2×C53C22×#10411yescharacteristic
328D53—22no—
329D53#150614yesdirect product factor
330#533—531no—
331#533—531no—
332#533—532no—
333#533—532no—
334#533—531no—
335#533—1061no—
336#533—1061no—
337#533—1061no—
338#533—1061no—
339#533C2×D5311yescharacteristic
340C2×C4⋊Q8—531no—
341C2×C4⋊Q8—531no—
342C2×#125—1061no—
343C2×C22⋊Q8—532no—
344C2×C22⋊Q8—1061no—
345C2×C4×Q8—532no—
346C2×C4×Q8—1061no—
347C2×#117—1061no—
348C22×C42—531no—
349C22×C42—1061no—
350#432—531no—
351#432—532no—
352#432—1061no—
353#432—1061no—
354#432C2×D5311yescharacteristic
355C2×C4⋊C8—1061no—
356C2×C22⋊C8—532no—
357C2×#97—1061no—
358#376—531no—
359#376—532no—
360#376—1061no—
361#376—1061no—
362#376C2×D5311yescharacteristic
363C2×#95—532no—
364C2×#95—1061no—
365#328—534no—
366#328—534no—
367#328—534no—
368#328C2×D5314yes—
369C53C2×#150611yescharacteristic, complemented, Sylow
370C22×Q8—532no—
371C22×Q8—1061no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—1062no—
376C22×C8—1062no—
377C4⋊Q8—532no—
378C4⋊Q8—532no—
379C4⋊Q8—531no—
380C4⋊Q8—531no—
381C4⋊Q8—1061no—
382C4⋊Q8—1061no—
383C4⋊Q8—1061no—
384C4⋊Q8—1061no—
385C4⋊Q8C4×D5311yescharacteristic, complemented
386C4⋊Q8C4×D5311yescharacteristic, complemented
387#125—21no—
388#125—1062no—
389#125—1062no—
390#125—1061no—
391#125—1061no—
392#125—1061no—
393C22⋊Q8—21no—
394C22⋊Q8—532no—
395C22⋊Q8—532no—
396C22⋊Q8—532no—
397C22⋊Q8—1062no—
398C22⋊Q8—1062no—
399C22⋊Q8—1062no—
400C22⋊Q8—1062no—
401C22⋊Q8—1061no—
402C22⋊Q8—1061no—
403C22⋊Q8—1061no—
404C22⋊Q8C22×D5312yes—
405C4×Q8—21no—
406C4×Q8—532no—
407C4×Q8—532no—
408C4×Q8—532no—
409C4×Q8—1062no—
410C4×Q8—1062no—
411C4×Q8—1062no—
412C4×Q8—1061no—
413C4×Q8—1062no—
414C4×Q8—1061no—
415C4×Q8—1061no—
416C4×Q8C4×D5312yescomplemented
417#117—21no—
418#117—1062no—
419#117—1062no—
420#117—1061no—
421#117—1061no—
422#117—1061no—
423C2×C4⋊C4—532no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1062no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4—1061no—
429C2×C22⋊C4—1062no—
430C2×C42—21no—
431C2×C42—531no—
432C2×C42—531no—
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—1062no—
443C2×C42—1068no—
444C2×C42—1062no—
445C2×C42—1061no—
446C2×C42C22×D5311yescharacteristic
447C4⋊C8—21no—
448C4⋊C8—1062no—
449C4⋊C8—1061no—
450C22⋊C8—532no—
451C22⋊C8—1062no—
452C22⋊C8C22×D5312yes—
453#97—21no—
454#97—1062no—
455#97—1061no—
456#95—21no—
457#95—532no—
458#95—1062no—
459#95—1062no—
460#95—1061no—
461#95C22×D5312yes—
462C24—531no—
463C2×Q8—21no—
464C2×Q8—532no—
465C2×Q8—1062no—
466C2×Q8—1062no—
467C2×Q8—1061no—
468C2×Q8—2124no—
469C2×Q8—2124no—
470C2×Q8C2×C4×D5312yes—
471C22×C4—22no—
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—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—531no—
487C22×C4—1064no—
488C22×C4—1064no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1068no—
492C22×C4—1062no—
493C22×C4—2122no—
494C22×C4—2124no—
495C22×C4—2124no—
496C22×C4D4×D5311yescharacteristic
497C22×C4D4×D5311yescharacteristic
498C22×C4C23×D5311yescharacteristic
499C2×C8—22no—
500C2×C8—1062no—
501C2×C8—2124no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—532no—
508C4⋊C4—1062no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1062no—
519C4⋊C4—1061no—
520C4⋊C4—1062no—
521C4⋊C4C2×C4×D5312yes—
522C22⋊C4—22no—
523C22⋊C4—1062no—
524C22⋊C4—1062no—
525C22⋊C4—1062no—
526C42—24no—
527C42—22no—
528C42—531no—
529C42—531no—
530C42—531no—
531C42—531no—
532C42—531no—
533C42—531no—
534C42—531no—
535C42—531no—
536C42—531no—
537C42—531no—
538C42—531no—
539C42—531no—
540C42—1062no—
541C42—1062no—
542C42—1064no—
543C42—1068no—
544C42—1062no—
545C42C2×C4×D5311yescharacteristic
546C42C2×C4×D5311yescharacteristic
547C42D4×D5311yescharacteristic
548C42D4×D5311yescharacteristic
549C23—531no—
550C23—531no—
551C23—534no—
552C23—1062no—
553C23—1062no—
554C23C2×D4×D5311yescharacteristic
555Q8—42no—
556Q8—42no—
557Q8—2122no—
558Q8—2124no—
559Q8—2124no—
560Q8—2122no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—531no—
570C2×C4—534no—
571C2×C4—531no—
572C2×C4—531no—
573C2×C4—531no—
574C2×C4—531no—
575C2×C4—531no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1064no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2124no—
593C2×C4—2124no—
594C2×C4—21216no—
595C2×C4—2121no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4C22⋊C4×D5311yescharacteristic
599C2×C4C22⋊C4×D5311yescharacteristic
600C2×C4C22⋊C4×D5311yescharacteristic
601C2×C4C22⋊C4×D5311yescharacteristic
602C2×C4C22×C4×D5311yescharacteristic
603C2×C4C2×D4×D5311yescharacteristic
604C8—42no—
605C8—2122no—
606C22—21no—
607C22—21no—
608C22—532no—
609C22—532no—
610C22—532no—
611C22—532no—
612C22—1068no—
613C22—1061no—
614C22—1061no—
615C22C2×C22⋊C4×D5311yescharacteristic
616C22#137×D5312yes—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—532no—
626C4—1061no—
627C4—1061no—
628C4—1061no—
629C4—1061no—
630C4—1061no—
631C4—2124no—
632C4—2122no—
633C4—2122no—
634C4#104×D5312yes—
635C2—22no—
636C2—534no—
637C2—1062no—
638C2#410×D5311yescharacteristic
639C2#419×D5311yescharacteristic
640C2C2×#104×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1506C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1506D53
hypocenterC53C2×#1506
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
4. #20273228C2semidirect
5. #20273227C2semidirect
6. #20273226C2semidirect
7. #20273225C2semidirect
9. C53×#1506C2semidirect
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
126. C4⋊Q8×C53C2×C4semidirect
127. C4⋊Q8×C53C2×C4semidirect
132. C4×Q8×C53C2×C4semidirect
322. #1506D53direct (N × H)
329. D53#1506direct (N × H)
369. C53C2×#1506semidirect
385. C4⋊Q8C4×D53semidirect
386. C4⋊Q8C4×D53semidirect
416. 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
#376×D532
#202732282
#202732272
#202732262
#202732252
#328×D532
C53×#15062
C2×#150653