Finite Group Databasebeta

← D53×#1497 #27350713

Subgroups — D53×#1497 #27350713

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 616 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#431×D53C211yesmaximal, characteristic
4#20273189C211yesmaximal, characteristic, complemented
5#20273188C211yesmaximal, characteristic, complemented
6#20273187C211yesmaximal, characteristic, complemented
7#20273186C214yesmaximal, complemented
8Q8⋊C8×D53C214yesmaximal, complemented
9C53×#1497C211yesmaximal, characteristic, complemented
10#16811180C2211yescharacteristic
11#16811179—21no—
12#16811178C411yescharacteristic
13#16811178C411yescharacteristic
14#16811177—21no—
15#16811177C2212yes—
16#16811176—21no—
17#16811175—21no—
18#16811175C412yes—
19C4⋊Q8×D53C411yescharacteristic
20C4⋊Q8×D53C411yescharacteristic
21#125×D53—21no—
22C22⋊Q8×D53—21no—
23C22⋊Q8×D53C412yes—
24C4×Q8×D53—21no—
25C4×Q8×D53C2212yes—
26#117×D53—21no—
27C2×C42×D53C2211yescharacteristic
28#16810680C2211yescharacteristic
29#16810679C2212yes—
30#16810678—21no—
31#16810677—21no—
32#16810676C2211yescharacteristic
33#16810675—21no—
34#16810674C2212yes—
35#16810673—21no—
36C4⋊C8×D53—21no—
37C4⋊C8×D53C2212yes—
38C22⋊C8×D53—21no—
39C22⋊C8×D53—21no—
40#97×D53—21no—
41C4×C8×D53C2212yes—
42#16810355C2214yescomplemented
43Q8⋊#88322C2214yescomplemented
44#16810353C2214yescomplemented
45C53×#533C2211yescharacteristic
46C53×#432C2211yescharacteristic
47C53×#431C2211yescharacteristic
48C53×Q8⋊C8C2214yescomplemented
49C2×Q8×D53—21no—
50C2×Q8×D53C2×C412yes—
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53D411yescharacteristic
53C22×C4×D53D411yescharacteristic
54C2×C8×D53—22no—
55C2×C8×D53—22no—
56#13133217C2×C411yescharacteristic
57#13133217C2×C411yescharacteristic
58#13133216—21no—
59#13133216—21no—
60#13133216—21no—
61#13133216C2×C412yes—
62#13133215—21no—
63#13133215C2×C412yes—
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#13133196C2×C412yes—
73#13133195—22no—
74#13133195—22no—
75#13133194—22no—
76#13133194—22no—
77#13133193—21no—
78#13133193C2×C412yes—
79Q8×Dic53—21no—
80Q8×Dic53C2×C412yes—
81C4×Dic106—21no—
82C4×Dic106—22no—
83C4×Dic106—22no—
84C4×Dic106C2312yes—
85#13133182—21no—
86#13133182—22no—
87#13133182—22no—
88#13133182C2×C412yes—
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×Dic53C2×C411yescharacteristic
102C2×C4×Dic53D411yescharacteristic
103C2×C4×Dic53D411yescharacteristic
104C42×D53—22no—
105C42×D53C2×C411yescharacteristic
106C42×D53C2×C411yescharacteristic
107C42×D53D411yescharacteristic
108C42×D53D411yescharacteristic
109C42×D53D411yescharacteristic
110C42×D53D411yescharacteristic
111C42×D53C2311yescharacteristic
112C42×D53C2311yescharacteristic
113C4⋊#88322—21no—
114C4⋊#88322C2312yes—
115#13133146—22no—
116#13133146—22no—
117C22⋊#88322—21no—
118C22⋊#88322—21no—
119#13133129—22no—
120#13133129—22no—
121#13133128—21no—
122#13133127—22no—
123C4×#88322C2312yes—
124C8×Dic53—22no—
125C4⋊Q8×C53C2×C411yescharacteristic
126C4⋊Q8×C53C2×C411yescharacteristic
127#125×C53—21no—
128C22⋊Q8×C53—21no—
129C22⋊Q8×C53C2×C412yes—
130C4×Q8×C53—21no—
131C4×Q8×C53C2312yes—
132#117×C53—21no—
133C2×C42×C53C2311yescharacteristic
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2312yes—
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138#97×C53—21no—
139C4×C8×C53C2312yes—
140C23×D53C22⋊C411yescharacteristic
141C2×Q8×C53—21no—
142C2×Q8×C53C22×C412yes—
143C22×C4×C53C22×C411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×Dic53C22⋊C411yescharacteristic
147C22×Dic53C22⋊C411yescharacteristic
148C22×Dic53C22⋊C411yescharacteristic
149Q8×D53—28no—
150Q8×D53—44no—
151C2×C4×D53—24no—
152C2×C4×D53—24no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—42no—
156C2×C4×D53C22⋊C411yescharacteristic
157C2×C4×D53C22⋊C411yescharacteristic
158C2×C4×D53C22⋊C411yescharacteristic
159C2×C4×D53C22⋊C411yescharacteristic
160C2×C4×D53C22⋊C411yescharacteristic
161C2×C4×D53C22⋊C411yescharacteristic
162C2×C4×D53C22×C411yescharacteristic
163C2×C4×D53C22×C411yescharacteristic
164C2×C4×D53C2×D411yescharacteristic
165C2×C4×D53C2×D411yescharacteristic
166C2×C4×D53C2×D411yescharacteristic
167C2×C4×D53C2×D411yescharacteristic
168C2×Dic106—21no—
169C2×Dic106—22no—
170C2×Dic106—22no—
171C2×Dic106C22×C412yes—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53C22×C412yes—
180C22⋊C4×C53—22no—
181C42×C53—22no—
182C42×C53C22×C411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C42×C53C2411yescharacteristic
186C22⋊Dic53—22no—
187#11719921—22no—
188#11719921—22no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53C22×C412yes—
195#11719919—22no—
196#11719919—22no—
197#11719919—22no—
198#11719919—22no—
199#11719919—22no—
200#11719919—22no—
201#11719919—22no—
202#11719919—22no—
203C4×Dic53—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
218C2×#88322—22no—
219C2×#88322—22no—
220C8×D53—44no—
221C8×D53—44no—
222C23×C53C2×C22⋊C411yescharacteristic
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53C2×C22⋊C411yescharacteristic
227C22×D53C2×C22⋊C411yescharacteristic
228Q8×C53—24no—
229Q8×C53—42no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—41no—
235C2×C4×C53C2×C22⋊C411yescharacteristic
236C2×C4×C53C2×C22⋊C411yescharacteristic
237C2×C4×C53C2×C22⋊C411yescharacteristic
238C2×C4×C53C23×C411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53C2×C22⋊C411yescharacteristic
247C2×Dic53C2×C22⋊C411yescharacteristic
248C2×Dic53C2×C22⋊C411yescharacteristic
249C2×Dic53C2×C22⋊C411yescharacteristic
250C2×Dic53C2×C22⋊C411yescharacteristic
251C2×Dic53C2×C22⋊C411yescharacteristic
252C4×D53—22no—
253C4×D53—22no—
254C4×D53—22no—
255C4×D53—22no—
256C4×D53—22no—
257C4×D53—22no—
258C4×D53—22no—
259C4×D53—22no—
260C4×D53—41no—
261C4×D53—41no—
262C4×D53—44no—
263C4×D53—44no—
264C4×D53#13114yes—
265C4×D53#13714yes—
266Dic106—24no—
267Dic106—44no—
268Dic106—42no—
269Dic106—44no—
270C8×C53—42no—
271C8×C53—42no—
272#88322—42no—
273#88322—42no—
274C2×#1497—531nomaximal, Sylow
275C22×C53—21no—
276C22×C53—21no—
277C22×C53—21no—
278C22×C53C22×C22⋊C411yescharacteristic
279C2×D53—21no—
280C2×D53—21no—
281C2×D53—21no—
282C2×D53—44no—
283C2×D53#40812yes—
284C2×D53#41912yes—
285C2×D53#42112yes—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—21no—
289C4×C53—21no—
290C4×C53—42no—
291C4×C53—42no—
292C4×C53C2×#13112yes—
293C4×C53C2×#13712yes—
294Dic53—21no—
295Dic53—21no—
296Dic53—21no—
297Dic53—21no—
298Dic53—42no—
299Dic53—42no—
300Dic53#40812yes—
301Dic53#41912yes—
302C2×#533—531no—
303C2×#432—531no—
304C2×#431—531no—
305#1497—531no—
306#1497—531no—
307#1497—534no—
308#1497—531no—
309#1497D5311yescharacteristic, direct product factor
310C2×Q8⋊C8—534no—
311C2×C53—41no—
312C2×C53C2×#40811yescharacteristic
313C2×C53C2×#41911yescharacteristic
314C2×C53C2×#42111yescharacteristic
315D53—41no—
316D53#149714yesdirect product factor
317#533—531no—
318#533—531no—
319#533—532no—
320#533—532no—
321#533—531no—
322#533—1061no—
323#533—1061no—
324#533—1061no—
325#533—1061no—
326#533C2×D5311yescharacteristic
327C2×C4⋊Q8—531no—
328C2×C4⋊Q8—531no—
329C2×#125—1061no—
330C2×C22⋊Q8—532no—
331C2×C22⋊Q8—1061no—
332C2×C4×Q8—532no—
333C2×C4×Q8—1061no—
334C2×#117—1061no—
335C22×C42—531no—
336#432—531no—
337#432—532no—
338#432—1061no—
339#432—1061no—
340#432C2×D5311yescharacteristic
341#431—531no—
342#431—532no—
343#431—1061no—
344#431—1061no—
345#431C2×D5311yescharacteristic
346C2×C4⋊C8—532no—
347C2×C4⋊C8—1061no—
348C2×C22⋊C8—1061no—
349C2×C22⋊C8—1061no—
350C2×#97—1061no—
351C2×C4×C8—532no—
352Q8⋊C8—534no—
353Q8⋊C8—534no—
354Q8⋊C8—534no—
355Q8⋊C8C2×D5314yescomplemented
356C53C2×#149711yescharacteristic, complemented, Sylow
357C22×Q8—532no—
358C22×Q8—1061no—
359C23×C4—531no—
360C23×C4—531no—
361C23×C4—531no—
362C22×C8—1062no—
363C22×C8—1062no—
364C4⋊Q8—532no—
365C4⋊Q8—532no—
366C4⋊Q8—531no—
367C4⋊Q8—531no—
368C4⋊Q8—1061no—
369C4⋊Q8—1061no—
370C4⋊Q8—1061no—
371C4⋊Q8—1061no—
372C4⋊Q8C4×D5311yescharacteristic
373C4⋊Q8C4×D5311yescharacteristic
374#125—21no—
375#125—1062no—
376#125—1062no—
377#125—1061no—
378#125—1061no—
379#125—1061no—
380C22⋊Q8—21no—
381C22⋊Q8—532no—
382C22⋊Q8—532no—
383C22⋊Q8—532no—
384C22⋊Q8—1062no—
385C22⋊Q8—1062no—
386C22⋊Q8—1062no—
387C22⋊Q8—1062no—
388C22⋊Q8—1061no—
389C22⋊Q8—1061no—
390C22⋊Q8—1061no—
391C22⋊Q8C4×D5312yes—
392C4×Q8—21no—
393C4×Q8—532no—
394C4×Q8—532no—
395C4×Q8—532no—
396C4×Q8—1062no—
397C4×Q8—1062no—
398C4×Q8—1062no—
399C4×Q8—1061no—
400C4×Q8—1062no—
401C4×Q8—1061no—
402C4×Q8—1061no—
403C4×Q8C22×D5312yes—
404#117—21no—
405#117—1062no—
406#117—1062no—
407#117—1061no—
408#117—1061no—
409#117—1061no—
410C2×C4⋊C4—532no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1061no—
416C2×C22⋊C4—1062no—
417C2×C42—531no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—531no—
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—1062no—
429C2×C42C22×D5311yescharacteristic
430C4⋊C8—21no—
431C4⋊C8—532no—
432C4⋊C8—1062no—
433C4⋊C8—1062no—
434C4⋊C8—1061no—
435C4⋊C8C22×D5312yes—
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8—1062no—
439C22⋊C8—1062no—
440C22⋊C8—1061no—
441C22⋊C8—1061no—
442#97—21no—
443#97—1062no—
444#97—1061no—
445C4×C8—532no—
446C4×C8—1062no—
447C4×C8C22×D5312yes—
448C24—531no—
449C2×Q8—21no—
450C2×Q8—532no—
451C2×Q8—1062no—
452C2×Q8—1062no—
453C2×Q8—1068no—
454C2×Q8—1061no—
455C2×Q8—2124no—
456C2×Q8C2×C4×D5312yes—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—1064no—
473C22×C4—1064no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—2122no—
477C22×C4C2×C4×D5311yescharacteristic
478C22×C4D4×D5311yescharacteristic
479C22×C4D4×D5311yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—2124no—
485C2×C8—2124no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—532no—
492C4⋊C4—1062no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—1062no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1062no—
503C4⋊C4—1061no—
504C4⋊C4—1062no—
505C4⋊C4C2×C4×D5312yes—
506C22⋊C4—22no—
507C22⋊C4—1062no—
508C22⋊C4—1062no—
509C22⋊C4—1062no—
510C42—22no—
511C42—531no—
512C42—531no—
513C42—531no—
514C42—531no—
515C42—531no—
516C42—531no—
517C42—531no—
518C42—531no—
519C42—531no—
520C42—531no—
521C42—531no—
522C42—531no—
523C42—1062no—
524C42—1062no—
525C42—1062no—
526C42C2×C4×D5311yescharacteristic
527C42D4×D5311yescharacteristic
528C42D4×D5311yescharacteristic
529C42C23×D5311yescharacteristic
530C23—531no—
531C23—531no—
532C23—1062no—
533C23—1062no—
534C23—1062no—
535C23C22⋊C4×D5311yescharacteristic
536Q8—24no—
537Q8—42no—
538Q8—1064no—
539Q8—2122no—
540Q8—2124no—
541Q8—2124no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—531no—
548C2×C4—531no—
549C2×C4—534no—
550C2×C4—534no—
551C2×C4—531no—
552C2×C4—531no—
553C2×C4—531no—
554C2×C4—531no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2124no—
570C2×C4—2124no—
571C2×C4—2121no—
572C2×C4C22⋊C4×D5311yescharacteristic
573C2×C4C22⋊C4×D5311yescharacteristic
574C2×C4C22⋊C4×D5311yescharacteristic
575C2×C4C22×C4×D5311yescharacteristic
576C2×C4C2×D4×D5311yescharacteristic
577C2×C4C2×D4×D5311yescharacteristic
578C8—42no—
579C8—42no—
580C8—2122no—
581C8—2122no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—532no—
586C22—532no—
587C22—532no—
588C22—1061no—
589C22—1061no—
590C22—1061no—
591C22—2124no—
592C22C2×C22⋊C4×D5311yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—42no—
598C4—42no—
599C4—532no—
600C4—532no—
601C4—1061no—
602C4—1061no—
603C4—1061no—
604C4—1061no—
605C4—2122no—
606C4—2122no—
607C4#131×D5312yes—
608C4#137×D5312yes—
609C2—41no—
610C2—534no—
611C2—2121no—
612C2#408×D5311yescharacteristic
613C2#419×D5311yescharacteristic
614C2#421×D5311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1497C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC42×C53C24
hypercenter#1497D53
hypocenterC53C2×#1497
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1497

Complements

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

NH ≅ G/Ntype
4. #20273189C2semidirect
5. #20273188C2semidirect
6. #20273187C2semidirect
7. #20273186C2semidirect
8. Q8⋊C8×D53C2semidirect
9. C53×#1497C2semidirect
42. #16810355C22semidirect
43. Q8⋊#88322C22semidirect
44. #16810353C22semidirect
48. C53×Q8⋊C8C22semidirect
309. #1497D53direct (N × H)
316. D53#1497direct (N × H)
355. Q8⋊C8C2×D53semidirect
356. C53C2×#1497semidirect

Maximal subgroups

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

subgroupindex
#533×D532
#432×D532
#431×D532
#202731892
#202731882
#202731872
#202731862
Q8⋊C8×D532
C53×#14972
C2×#149753