Finite Group Databasebeta

← #27388576

Subgroups — #27388576

Order 13568.

1154 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D53C211yesmaximal, characteristic
2C4×M16×D53C211yesmaximal, characteristic
3#20277119C211yesmaximal, characteristic
4#20275660C211yesmaximal, characteristic, complemented
5#20275659C211yesmaximal, characteristic, complemented
6#20275658C211yesmaximal, characteristic, complemented
7#20275657C214yesmaximal, complemented
8#20274423C214yesmaximal
9C53×#1986C211yesmaximal, characteristic, complemented
10C2×M16×D53C2212yes—
11#16811180C2211yescharacteristic
12#16811179C412yescomplemented
13#16811178C411yescharacteristic, complemented
14#16811178C411yescharacteristic, complemented
15#16811177—21no—
16#16811177—21no—
17#16811176C2212yes—
18#16811175—21no—
19#16811175—21no—
20C4⋊Q8×D53C411yescharacteristic, complemented
21C4⋊Q8×D53C411yescharacteristic, complemented
22#125×D53C412yescomplemented
23C22⋊Q8×D53—21no—
24C22⋊Q8×D53—21no—
25C4×Q8×D53—21no—
26C4×Q8×D53—21no—
27C2×#13133174C2212yes—
28#117×D53C2212yes—
29C2×C42×D53C2211yescharacteristic
30C4×#11719917C2211yescharacteristic
31#16810575—21no—
32M16×Dic53C2212yes—
33C4×#11719912—21no—
34#97×D53—21no—
35C4×C8×D53—21no—
36#16810492—21no—
37#16810487—21no—
38#16810482—21no—
39#16810480—21no—
40C4×#11719919C2212yes—
41C4×C4⋊Dic53C2211yescharacteristic
42#16810384C2214yes—
43#16810383C2214yes—
44#16810382C2214yes—
45C53×#533C2211yescharacteristic
46C4×M16×C53C2211yescharacteristic
47C4×C4⋊C4×C53C2211yescharacteristic
48C53×#339C2214yes—
49C2×Q8×D53—21no—
50C2×Q8×D53—21no—
51C22×C4×D53D411yescharacteristic
52C22×C4×D53D411yescharacteristic
53C22×C4×D53C2311yescharacteristic
54C2×#11719917C2312yes—
55C2×#11719912—22no—
56M16×D53—24no—
57M16×D53Q818yescomplemented
58C2×C8×D53—22no—
59#13133217C2×C411yescharacteristic
60#13133217C2×C411yescharacteristic
61#13133216—21no—
62#13133216—21no—
63#13133216C2×C412yes—
64#13133216C2×C412yes—
65#13133215—21no—
66#13133215—21no—
67#13133209C2×C412yes—
68#13133208C2×C412yes—
69#13133208C2×C412yes—
70#13133207—22no—
71#13133207—22no—
72#13133197—21no—
73#13133197—21no—
74#13133196—21no—
75#13133196—21no—
76#13133195—22no—
77#13133195—22no—
78#13133194—22no—
79#13133194—22no—
80#13133193—21no—
81#13133193—21no—
82Q8×Dic53—21no—
83Q8×Dic53—21no—
84C4×Dic106—21no—
85C4×Dic106—21no—
86C4×Dic106—22no—
87C4×Dic106—22no—
88#13133182—21no—
89#13133182—21no—
90#13133182—22no—
91#13133182—22no—
92#13133176C2312yes—
93#13133175C2×C412yes—
94#13133175C2×C412yes—
95#13133174—22no—
96#13133174—28no—
97#13133173—22no—
98C2×C4⋊Dic53—21no—
99C2×#11719919—22no—
100C2×#11719919—22no—
101C4⋊C4×D53—22no—
102C4⋊C4×D53—22no—
103C4⋊C4×D53—21no—
104C4⋊C4×D53—21no—
105C4⋊C4×D53C2×C412yes—
106C4⋊C4×D53C2×C412yes—
107C2×#11719921—22no—
108C22⋊C4×D53—22no—
109C2×C4×Dic53D411yescharacteristic
110C2×C4×Dic53D411yescharacteristic
111C2×C4×Dic53C2312yes—
112C2×C4×Dic53C2311yescharacteristic
113C42×D53—22no—
114C42×D53C2×C411yescharacteristic
115C42×D53C2×C411yescharacteristic
116C42×D53C2×C411yescharacteristic
117C42×D53C2×C411yescharacteristic
118C42×D53D411yescharacteristic
119C42×D53D411yescharacteristic
120C42×D53D411yescharacteristic
121C42×D53D411yescharacteristic
122#13133128—21no—
123#13133127—22no—
124C4×#88322—21no—
125C8×Dic53—22no—
126#13133124—21no—
127#13133124—22no—
128C2×M16×C53C2312yes—
129C4⋊Q8×C53C2×C411yescharacteristic
130C4⋊Q8×C53C2×C411yescharacteristic
131#125×C53C2×C412yes—
132C22⋊Q8×C53—21no—
133C22⋊Q8×C53—21no—
134C4×Q8×C53—21no—
135C4×Q8×C53—21no—
136#117×C53C2312yes—
137C2×C4⋊C4×C53—21no—
138C2×C42×C53C2312yes—
139C2×C42×C53C2311yescharacteristic
140#97×C53—21no—
141C4×C8×C53—21no—
142#95×C53—21no—
143C23×D53C2×D411yescharacteristic
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—22no—
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154Q8×D53—44no—
155Q8×D53—44no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53C22⋊C411yescharacteristic
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×D53C2×D411yescharacteristic
171C2×C4×D53C2×D411yescharacteristic
172C2×C4×D53C2×Q814yes—
173C2×C4×D53#4114yes—
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—22no—
177C2×Dic106—22no—
178M16×C53—22no—
179M16×C53C2×Q814yescomplemented
180C2×C8×C53—22no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—42no—
186C4⋊C4×C53C22×C412yes—
187C4⋊C4×C53C22×C412yes—
188C22⋊C4×C53—22no—
189C42×C53—22no—
190C42×C53—24no—
191C42×C53C22×C411yescharacteristic
192C42×C53C22×C411yescharacteristic
193C42×C53C2×D411yescharacteristic
194C42×C53C2×D411yescharacteristic
195C22⋊Dic53—22no—
196#11719921—22no—
197#11719921—22no—
198#11719921—44no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—22no—
202C4⋊Dic53—22no—
203C4⋊Dic53—42no—
204C4⋊Dic53C22×C412yes—
205C4⋊Dic53C22×C412yes—
206#11719919—22no—
207#11719919—22no—
208#11719919—22no—
209#11719919—22no—
210#11719919—22no—
211#11719919—22no—
212#11719919—22no—
213#11719919—22no—
214#11719919—44no—
215#11719919—44no—
216C4×Dic53—22no—
217C4×Dic53—22no—
218C4×Dic53—22no—
219C4×Dic53—24no—
220C4×Dic53C22⋊C411yescharacteristic
221C4×Dic53C22⋊C411yescharacteristic
222C4×Dic53C22⋊C411yescharacteristic
223C4×Dic53C22⋊C411yescharacteristic
224C4×Dic53C22⋊C411yescharacteristic
225C4×Dic53C22⋊C411yescharacteristic
226C4×Dic53C22⋊C411yescharacteristic
227C4×Dic53C22⋊C411yescharacteristic
228C4×Dic53C22×C411yescharacteristic
229C4×Dic53C22×C411yescharacteristic
230C4×Dic53C2×D411yescharacteristic
231C4×Dic53C2×D411yescharacteristic
232#11719917—22no—
233#11719917C2×Q814yescomplemented
234C2×#88322—22no—
235#11719912—28no—
236C8×D53—28no—
237C23×C53C22×D411yescharacteristic
238C22×D53—41no—
239C22×D53—41no—
240C22×D53C2×C22⋊C411yescharacteristic
241C22×D53C2×C22⋊C411yescharacteristic
242C22×D53#12412yes—
243C22×D53C4⋊Q812yes—
244Q8×C53—42no—
245Q8×C53—42no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—41no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53C2×C22⋊C411yescharacteristic
254C2×C4×C53C2×C22⋊C411yescharacteristic
255C2×C4×C53C2×C22⋊C411yescharacteristic
256C2×C4×C53C2×C22⋊C411yescharacteristic
257C2×C4×C53C23×C411yescharacteristic
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×Q812yes—
260C2×C4×C53C2×#4112yes—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—41no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53C2×C22⋊C411yescharacteristic
269C2×Dic53C2×C22⋊C411yescharacteristic
270C2×Dic53C2×C22⋊C411yescharacteristic
271C2×Dic53C2×C22⋊C411yescharacteristic
272C2×Dic53C2×C22⋊C411yescharacteristic
273C2×Dic53C2×C22⋊C411yescharacteristic
274C2×Dic53#12412yes—
275C2×Dic53C4⋊Q812yes—
276C4×D53—22no—
277C4×D53—22no—
278C4×D53—22no—
279C4×D53—22no—
280C4×D53—41no—
281C4×D53—41no—
282C4×D53—41no—
283C4×D53—41no—
284C4×D53—44no—
285C4×D53—44no—
286C4×D53C4×Q814yes—
287C4×D53C4×Q814yes—
288C4×D53C22⋊Q814yes—
289C4×D53C22⋊Q814yes—
290Dic106—44no—
291Dic106—42no—
292Dic106—42no—
293Dic106—44no—
294C8×C53—24no—
295#88322—24no—
296#21306—531nomaximal, Sylow
297C22×C53—21no—
298C22×C53—21no—
299C22×C53C22×C22⋊C411yescharacteristic
300C22×C53C2×#12411yescharacteristic
301C22×C53C2×C4⋊Q811yescharacteristic
302C2×D53—21no—
303C2×D53—21no—
304C2×D53—21no—
305C2×D53—21no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53#39112yes—
309C2×D53#39112yes—
310C4×C53—21no—
311C4×C53—21no—
312C4×C53—21no—
313C4×C53—42no—
314C4×C53—42no—
315C4×C53—82no—
316C4×C53#42112yes—
317C4×C53C2×C4×Q812yes—
318C4×C53C2×C22⋊Q812yes—
319Dic53—21no—
320Dic53—21no—
321Dic53—21no—
322Dic53—42no—
323Dic53—42no—
324Dic53—82no—
325Dic53#39112yes—
326Dic53#39112yes—
327Dic53#42112yes—
328C2×#533—531no—
329C2×C4×M16—531no—
330#2285—531no—
331#1986—531no—
332#1986—531no—
333#1986—534no—
334#1986—531no—
335#1986D5311yescharacteristic, complemented
336#1733—534no—
337C2×C53—21no—
338C2×C53—21no—
339C2×C53C2×#39111yescharacteristic
340C2×C53C2×#42111yescharacteristic
341C2×C53C2×#42111yescharacteristic
342D53—22no—
343D53—22no—
344C22×M16—532no—
345#533—531no—
346#533—531no—
347#533—532no—
348#533—532no—
349#533—531no—
350#533—1061no—
351#533—1061no—
352#533—1061no—
353#533—1061no—
354#533C2×D5311yescharacteristic
355C2×C4⋊Q8—531no—
356C2×C4⋊Q8—531no—
357C2×#125—532no—
358C2×C22⋊Q8—1061no—
359C2×C22⋊Q8—1061no—
360C2×C4×Q8—1061no—
361C2×C4×Q8—1061no—
362C2×#117—532no—
363C2×#117—532no—
364C22×C42—531no—
365C4×M16—532no—
366C4×M16—531no—
367C4×M16—1061no—
368C4×M16—1061no—
369C4×M16C2×D5311yescharacteristic
370C2×#97—1061no—
371C2×C4×C8—1061no—
372#384—1061no—
373#382—1061no—
374#380—1061no—
375#380—1061no—
376C4×C4⋊C4—532no—
377C4×C4⋊C4—531no—
378C4×C4⋊C4C2×D5311yescharacteristic
379#339—534no—
380#339—534no—
381#339—534no—
382#339C2×D5314yes—
383C53#2130611yescharacteristic, complemented, Sylow
384C22×Q8—1061no—
385C22×Q8—1061no—
386C23×C4—531no—
387C23×C4—531no—
388C23×C4—531no—
389C2×M16—532no—
390C2×M16—538no—
391C2×M16—1062no—
392C2×M16—1064no—
393C2×M16C22×D5312yes—
394C22×C8—1062no—
395C4⋊Q8—532no—
396C4⋊Q8—532no—
397C4⋊Q8—531no—
398C4⋊Q8—531no—
399C4⋊Q8—1061no—
400C4⋊Q8—1061no—
401C4⋊Q8—1061no—
402C4⋊Q8—1061no—
403C4⋊Q8C4×D5311yescharacteristic
404C4⋊Q8C4×D5311yescharacteristic
405#125—532no—
406#125—532no—
407#125—532no—
408#125—1062no—
409#125—1062no—
410#125C4×D5312yes—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—1062no—
414C22⋊Q8—1062no—
415C22⋊Q8—1062no—
416C22⋊Q8—1062no—
417C22⋊Q8—1061no—
418C22⋊Q8—1061no—
419C22⋊Q8—1061no—
420C22⋊Q8—1061no—
421C22⋊Q8—1061no—
422C22⋊Q8—1061no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—1062no—
426C4×Q8—1062no—
427C4×Q8—1062no—
428C4×Q8—1062no—
429C4×Q8—1061no—
430C4×Q8—1061no—
431C4×Q8—1061no—
432C4×Q8—1061no—
433C4×Q8—1061no—
434C4×Q8—1061no—
435#117—532no—
436#117—532no—
437#117—532no—
438#117—1062no—
439#117—1068no—
440#117—1062no—
441#117C22×D5312yes—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—532no—
444C2×C4⋊C4—532no—
445C2×C4⋊C4—1062no—
446C2×C4⋊C4—1062no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4—1061no—
449C2×C4⋊C4—1062no—
450C2×C4⋊C4—1062no—
451C2×C4⋊C4—1061no—
452C2×C22⋊C4—1062no—
453C2×C22⋊C4—1062no—
454C2×C42—531no—
455C2×C42—531no—
456C2×C42—531no—
457C2×C42—531no—
458C2×C42—531no—
459C2×C42—531no—
460C2×C42—531no—
461C2×C42—531no—
462C2×C42—531no—
463C2×C42—531no—
464C2×C42—531no—
465C2×C42—532no—
466C2×C42—1062no—
467C2×C42C22×D5312yes—
468C2×C42C22×D5311yescharacteristic
469#97—21no—
470#97—1062no—
471#97—1061no—
472C4×C8—21no—
473C4×C8—1062no—
474C4×C8—1061no—
475#95—21no—
476#95—1062no—
477#95—1061no—
478C24—531no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—1062no—
482C2×Q8—1062no—
483C2×Q8—1061no—
484C2×Q8—1061no—
485C2×Q8—2124no—
486C2×Q8—2124no—
487C22×C4—22no—
488C22×C4—531no—
489C22×C4—531no—
490C22×C4—531no—
491C22×C4—531no—
492C22×C4—531no—
493C22×C4—534no—
494C22×C4—534no—
495C22×C4—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—531no—
500C22×C4—531no—
501C22×C4—531no—
502C22×C4—531no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—2122no—
509C22×C4—2122no—
510C22×C4D4×D5311yescharacteristic
511C22×C4D4×D5311yescharacteristic
512C22×C4C23×D5311yescharacteristic
513M16—22no—
514M16—534no—
515M16—1068no—
516M16—1062no—
517M16Q8×D5314yescomplemented
518C2×C8—22no—
519C2×C8—1068no—
520C2×C8—1062no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—532no—
527C4⋊C4—532no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—1062no—
533C4⋊C4—1062no—
534C4⋊C4—1062no—
535C4⋊C4—1062no—
536C4⋊C4—1062no—
537C4⋊C4—1062no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—2122no—
541C4⋊C4—2124no—
542C4⋊C4—2124no—
543C4⋊C4C2×C4×D5312yes—
544C4⋊C4C2×C4×D5312yes—
545C22⋊C4—22no—
546C22⋊C4—1062no—
547C22⋊C4—1062no—
548C22⋊C4—1062no—
549C22⋊C4—2124no—
550C42—22no—
551C42—24no—
552C42—531no—
553C42—531no—
554C42—531no—
555C42—531no—
556C42—531no—
557C42—531no—
558C42—531no—
559C42—531no—
560C42—531no—
561C42—531no—
562C42—531no—
563C42—531no—
564C42—1062no—
565C42—1062no—
566C42—1062no—
567C42—1064no—
568C42C2×C4×D5311yescharacteristic
569C42C2×C4×D5311yescharacteristic
570C42D4×D5311yescharacteristic
571C42D4×D5311yescharacteristic
572C23—531no—
573C23—531no—
574C23—532no—
575C23—532no—
576C23—2121no—
577C23—2121no—
578C23C2×D4×D5311yescharacteristic
579Q8—42no—
580Q8—42no—
581Q8—2122no—
582Q8—2124no—
583Q8—2124no—
584Q8—2122no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—531no—
593C2×C4—531no—
594C2×C4—531no—
595C2×C4—532no—
596C2×C4—532no—
597C2×C4—534no—
598C2×C4—534no—
599C2×C4—534no—
600C2×C4—534no—
601C2×C4—531no—
602C2×C4—531no—
603C2×C4—531no—
604C2×C4—1062no—
605C2×C4—1062no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2124no—
616C2×C4—2124no—
617C2×C4—2121no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4C22⋊C4×D5311yescharacteristic
622C2×C4C22⋊C4×D5311yescharacteristic
623C2×C4C22⋊C4×D5311yescharacteristic
624C2×C4C22⋊C4×D5311yescharacteristic
625C2×C4C22×C4×D5311yescharacteristic
626C2×C4C2×D4×D5311yescharacteristic
627C2×C4C2×Q8×D5312yes—
628C2×C4#41×D5312yes—
629C8—24no—
630C8—1064no—
631C22—21no—
632C22—21no—
633C22—532no—
634C22—532no—
635C22—1061no—
636C22—1061no—
637C22—1061no—
638C22—1061no—
639C22—2122no—
640C22—2122no—
641C22C2×C22⋊C4×D5311yescharacteristic
642C22#124×D5311yescharacteristic
643C22C4⋊Q8×D5311yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—532no—
651C4—532no—
652C4—532no—
653C4—1061no—
654C4—1061no—
655C4—1061no—
656C4—4242no—
657C4—2122no—
658C4—2122no—
659C4#1681063812yes—
660C4C4×Q8×D5312yes—
661C4C22⋊Q8×D5312yes—
662C2—21no—
663C2—21no—
664C2—1062no—
665C2—1062no—
666C2#391×D5311yescharacteristic
667C2#421×D5311yescharacteristic
668C2C2×#1681063811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1986C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1986D53
hypocenterC53#21306
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
4. #20275660C2semidirect
5. #20275659C2semidirect
6. #20275658C2semidirect
7. #20275657C2semidirect
9. C53×#1986C2semidirect
12. #16811179C4semidirect
13. #16811178C4semidirect
14. #16811178C4semidirect
20. C4⋊Q8×D53C4semidirect
21. C4⋊Q8×D53C4semidirect
22. #125×D53C4semidirect
57. M16×D53Q8semidirect
179. M16×C53C2×Q8semidirect
233. #11719917C2×Q8semidirect
335. #1986D53semidirect
383. C53#21306semidirect
517. M16Q8×D53semidirect

Maximal subgroups

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

subgroupindex
#533×D532
C4×M16×D532
#202771192
#202756602
#202756592
#202756582
#202756572
#202744232
C53×#19862
#2130653