Finite Group Databasebeta

← D53×#1894 #27385770

Subgroups — D53×#1894 #27385770

Order 13568.

1966 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D53C211yesmaximal, characteristic
2C2×C22⋊Q8×D53C211yesmaximal, characteristic
3C2×#103×D53C212yesmaximal, complemented
4#380×D53C211yesmaximal, characteristic
5#20275149C211yesmaximal, characteristic, complemented
6#20275148C211yesmaximal, characteristic, complemented
7#20275147C212yesmaximal, complemented
8#20275146C211yesmaximal, characteristic, complemented
9#20275145C212yesmaximal, complemented
10#340×D53C212yesmaximal, complemented
11C53×#1894C211yesmaximal, characteristic, complemented
12C22×Q8×D53C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C22×#11719917C2211yescharacteristic
15C22×#11719912C2212yes—
16C2×M16×D53—24no—
17C2×M16×D53—28no—
18C22×C8×D53C2212yes—
19C2×#13133197C2211yescharacteristic
20C2×#13133196C2211yescharacteristic
21C2×#13133195—21no—
22C2×#13133194—21no—
23C2×#13133193C2211yescharacteristic
24C22⋊Q8×D53—24no—
25C22⋊Q8×D53C418yescomplemented
26C2×C4⋊C4×D53—21no—
27C2×C4⋊C4×D53C2212yes—
28C2×C4⋊C4×D53C2211yescharacteristic
29C2×C22⋊C4×D53—21no—
30C2×C22⋊C4×D53—21no—
31C2×#13133142C2212yescomplemented
32C2×#13133141C2212yescomplemented
33C2×#13133140C2212yescomplemented
34#103×D53—28no—
35#16810483C2211yescharacteristic
36#16810482—21no—
37#16810481—21no—
38#16810480C2212yes—
39#95×D53—21no—
40#16810386C2212yescomplemented
41#16810385C2212yescomplemented
42#16810385C2212yescomplemented
43C22×M16×C53C2211yescharacteristic
44C2×C22⋊Q8×C53C2211yescharacteristic
45C2×#103×C53C2212yescomplemented
46C53×#380C2211yescharacteristic
47C53×#340C2212yescomplemented
48C24×D53D411yescharacteristic
49C22×Dic106—21no—
50C22×Dic106C2311yescharacteristic
51C2×Q8×D53—22no—
52C2×Q8×D53—24no—
53C2×Q8×D53—24no—
54C2×Q8×D53C2×C414yes—
55C23×Dic53D411yescharacteristic
56C22×C4×D53—24no—
57C22×C4×D53—24no—
58C22×C4×D53—21no—
59C22×C4×D53—21no—
60C22×C4×D53—22no—
61C22×C4×D53C2×C414yes—
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53D414yes—
65C22×C4×D53C2311yescharacteristic
66C22×C4×D53C2311yescharacteristic
67C2×#11719917—24no—
68C2×#11719917—22no—
69C2×#11719912—24no—
70C2×#11719912—24no—
71C2×#11719912—24no—
72M16×D53—232no—
73C22×#88322C2312yes—
74C2×C8×D53—24no—
75C2×C8×D53—24no—
76C2×C8×D53D418yes—
77#13133197—22no—
78#13133197C2×C414yescomplemented
79#13133196—22no—
80#13133196C2×C414yescomplemented
81#13133195—28no—
82#13133194—28no—
83#13133193—22no—
84#13133193C2×C414yescomplemented
85C2×C4⋊Dic53—21no—
86C2×C4⋊Dic53C2312yes—
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—22no—
92C4⋊C4×D53—28no—
93C4⋊C4×D53—22no—
94C4⋊C4×D53—24no—
95C4⋊C4×D53—24no—
96C4⋊C4×D53C2×C414yes—
97C2×C22⋊Dic53—21no—
98C2×C22⋊Dic53—21no—
99C2×#11719921—21no—
100C2×#11719921—21no—
101C2×#11719921—22no—
102C22⋊C4×D53—28no—
103C22⋊C4×D53—44no—
104#13133142—24no—
105#13133141—24no—
106#13133140—24no—
107#13133124—21no—
108#13133124—22no—
109C22×Q8×C53C2311yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C2×M16×C53—24no—
112C2×M16×C53—22no—
113C22×C8×C53C2312yes—
114C22⋊Q8×C53—22no—
115C22⋊Q8×C53C2×C414yescomplemented
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2312yes—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121#103×C53—24no—
122#95×C53—21no—
123C24×C53C2×D411yescharacteristic
124C23×D53—24no—
125C23×D53—22no—
126C23×D53—22no—
127C23×D53—24no—
128C23×D53C22⋊C414yes—
129C23×D53C2×D411yescharacteristic
130C23×D53C2×D411yescharacteristic
131C2×Q8×C53—21no—
132C2×Q8×C53—22no—
133C2×Q8×C53—22no—
134C2×Q8×C53C22×C412yes—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53C22×C412yes—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—24no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—22no—
148C22×Dic53C22⋊C414yes—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151Q8×D53—216no—
152Q8×D53—44no—
153Q8×D53—44no—
154C2×C4×D53—24no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—24no—
158C2×C4×D53—216no—
159C2×C4×D53—28no—
160C2×C4×D53—28no—
161C2×C4×D53—24no—
162C2×C4×D53—24no—
163C2×C4×D53—44no—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
166C2×C4×D53—44no—
167C2×C4×D53—44no—
168C2×C4×D53—44no—
169C2×C4×D53C22⋊C414yes—
170C2×C4×D53C22⋊C414yes—
171C2×C4×D53C22×C412yes—
172C2×C4×D53C22×C412yes—
173C2×C4×D53C2×D412yes—
174C2×C4×D53C2×D414yes—
175C2×C4×D53C2×D412yes—
176C2×C4×D53#4114yes—
177C2×Dic106—22no—
178C2×Dic106—22no—
179C2×Dic106—21no—
180C2×Dic106—24no—
181C2×Dic106—44no—
182C2×Dic106C22×C412yes—
183M16×C53—28no—
184C2×C8×C53—22no—
185C2×C8×C53—22no—
186C2×C8×C53C2×D414yes—
187C4⋊C4×C53—24no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—22no—
190C4⋊C4×C53—22no—
191C4⋊C4×C53C22×C412yes—
192C22⋊C4×C53—24no—
193C22⋊C4×C53—42no—
194C22⋊Dic53—24no—
195C22⋊Dic53—42no—
196#11719921—24no—
197#11719921—24no—
198#11719921—44no—
199C4⋊Dic53—21no—
200C4⋊Dic53—24no—
201C4⋊Dic53—22no—
202C4⋊Dic53—22no—
203C4⋊Dic53C22×C412yes—
204#11719919—24no—
205#11719919—24no—
206#11719919—24no—
207#11719919—44no—
208#11719917—28no—
209C2×#88322—22no—
210C2×#88322—22no—
211C2×#88322C2×D414yes—
212#11719912—216no—
213C8×D53—216no—
214C23×C53—22no—
215C23×C53—21no—
216C23×C53—22no—
217C23×C53—21no—
218C23×C53C2×C22⋊C412yes—
219C23×C53C22×D411yescharacteristic
220C22×D53—22no—
221C22×D53—21no—
222C22×D53—22no—
223C22×D53—21no—
224C22×D53—28no—
225C22×D53—28no—
226C22×D53—28no—
227C22×D53—48no—
228C22×D53—48no—
229C22×D53C2×C22⋊C412yes—
230C22×D53C2×C22⋊C412yes—
231C22×D53#12114yes—
232C22×D53AGL(1,ℤ/8)12yes—
233C22×D53#13712yes—
234C22×D53#13714yes—
235Q8×C53—24no—
236Q8×C53—41no—
237Q8×C53—41no—
238C2×C4×C53—22no—
239C2×C4×C53—24no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—42no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53C2×C22⋊C412yes—
251C2×C4×C53C23×C411yescharacteristic
252C2×C4×C53C22×D412yes—
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C2×#4112yes—
255C2×Dic53—22no—
256C2×Dic53—28no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—42no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53C2×C22⋊C412yes—
266C2×Dic53C2×C22⋊C412yes—
267C2×Dic53#12114yes—
268C4×D53—28no—
269C4×D53—24no—
270C4×D53—22no—
271C4×D53—22no—
272C4×D53—28no—
273C4×D53—48no—
274C4×D53—48no—
275C4×D53—88no—
276C4×D53C4×D418yes—
277C4×D53#12014yes—
278C4×D53#12314yes—
279Dic106—24no—
280Dic106—41no—
281Dic106—41no—
282Dic106—48no—
283C8×C53—24no—
284#88322—24no—
285C2×#1894—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53C22×C22⋊C411yescharacteristic
292C22×C53C2×#12112yes—
293C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C53C2×#13712yes—
295C22×C53C2×#13711yescharacteristic
296C2×D53—22no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—416no—
302C2×D53#38614yes—
303C2×D53#41714yes—
304C2×D53#41914yes—
305C2×D53#46818yes—
306C2×D53#47018yes—
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—82no—
311C4×C53C2×C4×D412yes—
312C4×C53C2×#12011yescharacteristic
313C4×C53C2×#12311yescharacteristic
314Dic53—22no—
315Dic53—42no—
316Dic53—42no—
317Dic53—82no—
318Dic53#38614yes—
319C23×M16—531no—
320C22×C22⋊Q8—531no—
321C22×#103—532no—
322C2×#380—531no—
323#1894—532no—
324#1894—531no—
325#1894—532no—
326#1894—531no—
327#1894—531no—
328#1894D5311yescharacteristic, direct product factor
329C2×#340—532no—
330C2×C53—42no—
331C2×C53C2×#38611yescharacteristic
332C2×C53C2×#41911yescharacteristic
333C2×C53C2×#41711yescharacteristic
334C2×C53C2×#46812yes—
335C2×C53C2×#47012yes—
336D53—42no—
337D53#189418yesdirect product factor
338C23×Q8—531no—
339C24×C4—531no—
340C22×M16—531no—
341C22×M16—532no—
342C22×M16—1068no—
343C22×M16—1064no—
344C22×M16C2×D5311yescharacteristic
345C23×C8—532no—
346C2×C22⋊Q8—531no—
347C2×C22⋊Q8—531no—
348C2×C22⋊Q8—538no—
349C2×C22⋊Q8—531no—
350C2×C22⋊Q8—1061no—
351C2×C22⋊Q8—1061no—
352C2×C22⋊Q8—1064no—
353C2×C22⋊Q8C2×D5311yescharacteristic
354C22×C4⋊C4—531no—
355C22×C4⋊C4—532no—
356C22×C4⋊C4—1061no—
357C22×C22⋊C4—1061no—
358C22×C22⋊C4—1061no—
359C2×#103—532no—
360C2×#103—532no—
361C2×#103—532no—
362C2×#103—1068no—
363C2×#103C2×D5312yescomplemented
364#380—531no—
365#380—532no—
366#380—1061no—
367#380—1061no—
368#380C2×D5311yescharacteristic
369C2×#95—1061no—
370#340—532no—
371#340—532no—
372#340—532no—
373#340C2×D5312yescomplemented
374C53C2×#189411yescharacteristic, complemented, Sylow
375C25—531no—
376C22×Q8—534no—
377C22×Q8—531no—
378C22×Q8—1061no—
379C22×Q8—1062no—
380C22×Q8—1064no—
381C22×Q8—1064no—
382C22×Q8C22×D5311yescharacteristic
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—534no—
386C23×C4—531no—
387C23×C4—534no—
388C23×C4—531no—
389C23×C4—531no—
390C23×C4—1064no—
391C23×C4—1064no—
392C23×C4—1061no—
393C23×C4—1061no—
394C23×C4—1062no—
395C23×C4C22×D5311yescharacteristic
396C2×M16—22no—
397C2×M16—24no—
398C2×M16—1064no—
399C2×M16—1064no—
400C2×M16—1064no—
401C2×M16—10632no—
402C2×M16—1062no—
403C2×M16—1064no—
404C22×C8—532no—
405C22×C8—538no—
406C22×C8—1064no—
407C22×C8—1064no—
408C22×C8C22×D5312yes—
409C22⋊Q8—22no—
410C22⋊Q8—534no—
411C22⋊Q8—534no—
412C22⋊Q8—534no—
413C22⋊Q8—1068no—
414C22⋊Q8—1068no—
415C22⋊Q8—1062no—
416C22⋊Q8—1062no—
417C22⋊Q8—1062no—
418C22⋊Q8C4×D5314yescomplemented
419C2×C4⋊C4—21no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—534no—
422C2×C4⋊C4—532no—
423C2×C4⋊C4—1062no—
424C2×C4⋊C4—1068no—
425C2×C4⋊C4—1064no—
426C2×C4⋊C4—1064no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4C22×D5312yes—
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1068no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1062no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—2124no—
443#103—24no—
444#103—1064no—
445#103—1064no—
446#103—1064no—
447#95—21no—
448#95—1062no—
449#95—1061no—
450C24—531no—
451C24—534no—
452C24—531no—
453C24—1062no—
454C24—1064no—
455C24—1064no—
456C24—1062no—
457C24D4×D5311yescharacteristic
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—532no—
462C2×Q8—1064no—
463C2×Q8—10616no—
464C2×Q8—1061no—
465C2×Q8—1062no—
466C2×Q8—1062no—
467C2×Q8—2124no—
468C2×Q8—2124no—
469C2×Q8—2124no—
470C2×Q8C2×C4×D5312yes—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—531no—
477C22×C4—532no—
478C22×C4—534no—
479C22×C4—532no—
480C22×C4—534no—
481C22×C4—534no—
482C22×C4—531no—
483C22×C4—534no—
484C22×C4—534no—
485C22×C4—532no—
486C22×C4—532no—
487C22×C4—1068no—
488C22×C4—1064no—
489C22×C4—10616no—
490C22×C4—1068no—
491C22×C4—1064no—
492C22×C4—1064no—
493C22×C4—1064no—
494C22×C4—1064no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1062no—
500C22×C4—2124no—
501C22×C4—2124no—
502C22×C4—2124no—
503C22×C4—2122no—
504C22×C4—2122no—
505C22×C4—2124no—
506C22×C4C2×C4×D5312yes—
507C22×C4D4×D5311yescharacteristic
508C22×C4D4×D5312yes—
509C22×C4C23×D5311yescharacteristic
510M16—28no—
511M16—10616no—
512M16—1068no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—534no—
516C2×C8—10616no—
517C2×C8—1062no—
518C2×C8—1062no—
519C2×C8D4×D5314yes—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—532no—
525C4⋊C4—1064no—
526C4⋊C4—1064no—
527C4⋊C4—1064no—
528C4⋊C4—1064no—
529C4⋊C4—1061no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—2124no—
533C4⋊C4C2×C4×D5312yes—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—1064no—
537C22⋊C4—1064no—
538C22⋊C4—1064no—
539C22⋊C4—2124no—
540C22⋊C4—2122no—
541C23—22no—
542C23—21no—
543C23—22no—
544C23—21no—
545C23—532no—
546C23—534no—
547C23—532no—
548C23—534no—
549C23—532no—
550C23—532no—
551C23—1068no—
552C23—1068no—
553C23—1068no—
554C23—1062no—
555C23—1061no—
556C23—1062no—
557C23—1061no—
558C23—2128no—
559C23—2128no—
560C23C22⋊C4×D5312yes—
561C23C2×D4×D5311yescharacteristic
562Q8—24no—
563Q8—41no—
564Q8—41no—
565Q8—1064no—
566Q8—2128no—
567Q8—2121no—
568Q8—2121no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—532no—
582C2×C4—534no—
583C2×C4—538no—
584C2×C4—534no—
585C2×C4—534no—
586C2×C4—532no—
587C2×C4—1068no—
588C2×C4—1068no—
589C2×C4—1068no—
590C2×C4—1062no—
591C2×C4—1064no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—4248no—
597C2×C4—2128no—
598C2×C4—2128no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4C22⋊C4×D5312yes—
606C2×C4C22×C4×D5311yescharacteristic
607C2×C4C2×D4×D5312yes—
608C2×C4C2×D4×D5311yescharacteristic
609C2×C4#41×D5312yes—
610C8—24no—
611C8—1064no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—42no—
616C22—42no—
617C22—534no—
618C22—534no—
619C22—534no—
620C22—538no—
621C22—538no—
622C22—1062no—
623C22—1062no—
624C22—1062no—
625C22—21216no—
626C22—2122no—
627C22—2122no—
628C22C2×C22⋊C4×D5311yescharacteristic
629C22#121×D5312yes—
630C22AGL(1,ℤ/8)×D5311yescharacteristic
631C22#137×D5312yes—
632C22#137×D5311yescharacteristic
633C4—22no—
634C4—42no—
635C4—42no—
636C4—82no—
637C4—534no—
638C4—1062no—
639C4—4242no—
640C4—2122no—
641C4—2122no—
642C4C4×D4×D5312yes—
643C4#120×D5311yescharacteristic
644C4#123×D5311yescharacteristic
645C2—42no—
646C2—538no—
647C2—2122no—
648C2#386×D5311yescharacteristic
649C2#419×D5311yescharacteristic
650C2#417×D5311yescharacteristic
651C2#468×D5312yes—
652C2#470×D5312yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1894C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1894D53
hypocenterC53C2×#1894
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
3. C2×#103×D53C2semidirect
5. #20275149C2semidirect
6. #20275148C2semidirect
7. #20275147C2semidirect
8. #20275146C2semidirect
9. #20275145C2semidirect
10. #340×D53C2semidirect
11. C53×#1894C2semidirect
25. C22⋊Q8×D53C4semidirect
31. C2×#13133142C22semidirect
32. C2×#13133141C22semidirect
33. C2×#13133140C22semidirect
40. #16810386C22semidirect
41. #16810385C22semidirect
42. #16810385C22semidirect
45. C2×#103×C53C22semidirect
47. C53×#340C22semidirect
78. #13133197C2×C4semidirect
80. #13133196C2×C4semidirect
84. #13133193C2×C4semidirect
115. C22⋊Q8×C53C2×C4semidirect
328. #1894D53direct (N × H)
337. D53#1894direct (N × H)
363. C2×#103C2×D53semidirect
373. #340C2×D53semidirect
374. C53C2×#1894semidirect
418. C22⋊Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D532
C2×C22⋊Q8×D532
C2×#103×D532
#380×D532
#202751492
#202751482
#202751472
#202751462
#202751452
#340×D532
C53×#18942
C2×#189453