Finite Group Databasebeta

← #27386800

Subgroups — #27386800

Order 13568.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810625C211yesmaximal, characteristic
2C2×#16810625C211yesmaximal, characteristic
3#20277205C211yesmaximal, characteristic
4C2×#16810521C211yesmaximal, characteristic, complemented
5#20275412C212yesmaximal, complemented
6#20275405C212yesmaximal, complemented
7#20275398C212yesmaximal, complemented
8#20275391C212yesmaximal, complemented
9#20274435C211yesmaximal, characteristic, complemented
10#20274435C211yesmaximal, characteristic, complemented
11C2×C4×M16×C53C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919—21no—
17#16810625—24no—
18#16810625C418yescomplemented
19C2×#13133142C2212yes—
20C2×#13133142C2212yes—
21C2×#13133139C2212yes—
22C2×#13133139C2212yes—
23#16810529C2211yescharacteristic
24#16810529C2211yescharacteristic
25#16810525C2211yescharacteristic
26#16810525C2211yescharacteristic
27#16810521—21no—
28#16810521—21no—
29#16810521C2212yes—
30#16810521C2212yes—
31#16810502—21no—
32#16810502—21no—
33C4×#11719921—21no—
34C4×#11719921—21no—
35C2×#13133124—21no—
36#16810386C2212yes—
37#16810386C2212yes—
38#16810386C2212yes—
39#16810386C2212yes—
40C22×M16×C53C2211yescharacteristic
41C22×M16×C53C2211yescharacteristic
42C22×C42×C53C2211yescharacteristic
43C4×M16×C53—24no—
44C2×#97×C53C2212yes—
45C2×C4×C8×C53C2212yes—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945—22no—
51C2×#11719945C2×C414yes—
52C22×Dic106C2311yescharacteristic
53C22×D212C2311yescharacteristic
54C22×C22⋊D53—21no—
55C23×Dic53—21no—
56C23×Dic53—21no—
57C22×C4×D53—21no—
58#13133176—22no—
59#13133176—22no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—22no—
63C2×C4⋊Dic53—22no—
64C2×C4⋊Dic53—22no—
65C2×C4⋊Dic53C2×C414yes—
66C2×C4⋊Dic53C2312yes—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2312yes—
70C2×#11719919—22no—
71C2×#11719919—22no—
72C2×#11719919—44no—
73C2×C22⋊Dic53—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×C4×Dic53—21no—
79#13133142—24no—
80#13133142C2×C418yescomplemented
81#13133139—24no—
82#13133139C2×C418yescomplemented
83#13133124—21no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87#13133124—22no—
88#13133124—22no—
89C23×C4×C53D411yescharacteristic
90C23×C4×C53D411yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C2×M16×C53—22no—
93C2×M16×C53—22no—
94C2×M16×C53—22no—
95C2×M16×C53—22no—
96C2×M16×C53C2×C414yes—
97C2×M16×C53D414yes—
98C22×C8×C53C2312yes—
99C22×C8×C53C2312yes—
100C2×C42×C53—22no—
101C2×C42×C53—22no—
102C2×C42×C53D411yescharacteristic
103C2×C42×C53D411yescharacteristic
104C2×C42×C53C2311yescharacteristic
105C2×C42×C53C2311yescharacteristic
106#97×C53—24no—
107C4×C8×C53—24no—
108C24×C53C2×D411yescharacteristic
109C23×D53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—41no—
115C22×C4×C53—41no—
116C22×C4×C53C22×C412yes—
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D412yes—
123C22×C4×C53#4112yes—
124C22×C4×C53#4112yes—
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
127C22×C4×C53C2411yescharacteristic
128C2×C22⋊D53—24no—
129C2×C22⋊D53—42no—
130C2×C22⋊D53—42no—
131C22×Dic53—24no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53—42no—
141C22×Dic53—42no—
142#11719945—28no—
143#11719945—28no—
144#11719945—44no—
145#11719945—44no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212C22×C412yes—
152C2×C4×D53—24no—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106C22×C412yes—
161M16×C53—28no—
162M16×C53—28no—
163C2×C8×C53—22no—
164C2×C8×C53—22no—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53C22×C414yes—
168C2×C8×C53C2×D414yes—
169C42×C53—22no—
170C42×C53—22no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C22⋊Dic53—42no—
176#11719921—42no—
177#11719921—42no—
178#11719921—42no—
179#11719921—42no—
180C4⋊Dic53—24no—
181C4⋊Dic53—22no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—42no—
189C4⋊Dic53C22×C414yes—
190#11719919—44no—
191#11719919—44no—
192C4×Dic53—42no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53C4×D412yes—
198C23×C53#12112yes—
199C23×C53#12412yes—
200C23×C53C22×D411yescharacteristic
201C22×D53—22no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—44no—
219C2×C4×C53—44no—
220C2×C4×C53#11714yes—
221C2×C4×C53C4×D414yes—
222C2×C4×C53C4×D412yes—
223C2×C4×C53#12112yes—
224C2×C4×C53#12314yes—
225C2×C4×C53#12412yes—
226C2×C4×C53#12614yes—
227C2×C4×C53AGL(1,ℤ/8)12yes—
228C2×C4×C53#13712yes—
229C2×C4×C53C23×C411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C22⋊D53—48no—
236C2×Dic53—22no—
237C2×Dic53—24no—
238C2×Dic53—48no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—42no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—42no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—84no—
253D212—22no—
254D212—22no—
255D212—41no—
256D212—41no—
257C4×D53—48no—
258Dic106—22no—
259Dic106—22no—
260Dic106—41no—
261Dic106—41no—
262C8×C53—24no—
263C8×C53—24no—
264#21093—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53#38814yes—
272C22×C53C2×C4×D411yescharacteristic
273C22×C53C2×#12111yescharacteristic
274C22×C53C2×#12411yescharacteristic
275C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C53C2×#13711yescharacteristic
277C22×C53#57511yescharacteristic
278C22×C53#57511yescharacteristic
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—42no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C2×D53—81no—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53#38814yes—
291C4×C53C2×#11711yescharacteristic
292C4×C53C2×C4×D411yescharacteristic
293C4×C53C2×#12311yescharacteristic
294C4×C53C2×#12611yescharacteristic
295Dic53—44no—
296Dic53—42no—
297Dic53—81no—
298Dic53—81no—
299Dic53—82no—
300C2×#417—531no—
301C2×#417—531no—
302C2×C4×M16D5311yescharacteristic, complemented
303#2297—531no—
304C2×#389—531no—
305#1936—532no—
306#1935—532no—
307#1934—532no—
308#1933—532no—
309#1736—531no—
310#1736—531no—
311C2×C53—24no—
312C2×C53C2×#38811yescharacteristic
313C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C53C4×#13711yescharacteristic
315C2×C53#305511yescharacteristic
316C2×C53#305611yescharacteristic
317C2×C53#314111yescharacteristic
318C2×C53#314211yescharacteristic
319D53—82no—
320C22×#41—531no—
321C22×M16C2×D5311yescharacteristic
322C22×M16C2×D5311yescharacteristic
323C2×#117—531no—
324C22×C4⋊C4—531no—
325C22×C4⋊C4—531no—
326C22×C4⋊C4—1061no—
327C22×C42C2×D5311yescharacteristic
328#417—538no—
329#417—1064no—
330C2×#103—532no—
331C2×#103—532no—
332C2×#102—532no—
333C2×#102—532no—
334C4×M16—24no—
335C2×#97C2×D5312yes—
336C2×C4×C8C2×D5312yes—
337#391—531no—
338#391—531no—
339#390—531no—
340#390—531no—
341#389—532no—
342#389—532no—
343#389—1061no—
344#389—1061no—
345#386—1061no—
346#386—1061no—
347C4×C22⋊C4—1061no—
348C4×C22⋊C4—1061no—
349C2×#95—1061no—
350#340—532no—
351#340—532no—
352#340—532no—
353#340—532no—
354C53#2109311yescharacteristic, complemented, Sylow
355C2×#41—534no—
356C2×#41—1062no—
357C2×#41—1062no—
358C2×#41—1062no—
359C2×#41—1062no—
360C2×#41—1062no—
361C22×Q8—531no—
362C22×D4—531no—
363C22×D4—1061no—
364C23×C4—1061no—
365C23×C4—1061no—
366C23×C4—1061no—
367C23×C4D21211yescharacteristic
368C23×C4D21211yescharacteristic
369C23×C4C22×D5311yescharacteristic
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—22no—
373C2×M16—22no—
374C2×M16C4×D5314yes—
375C2×M16C22⋊D5314yes—
376C22×C8C22×D5312yes—
377C22×C8C22×D5312yes—
378#117—1062no—
379#117—1062no—
380C2×C4⋊C4—534no—
381C2×C4⋊C4—532no—
382C2×C4⋊C4—532no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—2124no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—1061no—
401C2×C42D21211yescharacteristic
402C2×C42D21211yescharacteristic
403C2×C42C22×D5311yescharacteristic
404C2×C42C22×D5311yescharacteristic
405#103—538no—
406#103—1064no—
407#102—538no—
408#102—1064no—
409#97—24no—
410C4×C8—24no—
411#95—1062no—
412#95—1062no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417C24—1061no—
418C24C2×D21211yescharacteristic
419#41—1068no—
420#41—1068no—
421#41—2124no—
422#41—2124no—
423C2×Q8—532no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×D4—532no—
430C2×D4—1064no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—2122no—
437C2×D4—2122no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—1064no—
445C22×C4—1061no—
446C22×C4—1062no—
447C22×C4—1064no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1062no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4C2×C4×D5312yes—
459C22×C4C2×D21211yescharacteristic
460C22×C4C2×D21211yescharacteristic
461C22×C4C2×D21211yescharacteristic
462C22×C4C2×D21211yescharacteristic
463C22×C4C2×D21211yescharacteristic
464C22×C4#1171994512yes—
465C22×C4#1171994512yes—
466C22×C4#1171994512yes—
467C22×C4#1171994512yes—
468C22×C4C2×C22⋊D5312yes—
469C22×C4C23×D5311yescharacteristic
470M16—28no—
471M16—28no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8C2×C4×D5314yes—
477C2×C8C2×C22⋊D5314yes—
478C4⋊C4—534no—
479C4⋊C4—1064no—
480C4⋊C4—1062no—
481C4⋊C4—1062no—
482C4⋊C4—1062no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—2124no—
488C4⋊C4—2122no—
489C4⋊C4—2124no—
490C22⋊C4—2122no—
491C22⋊C4—2122no—
492C22⋊C4—2122no—
493C22⋊C4—2122no—
494C22⋊C4—2122no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—2122no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—1062no—
507C23—2121no—
508C23—2121no—
509C23—2121no—
510C23—2121no—
511C23—2121no—
512C23—2121no—
513C23C4×D21212yes—
514C23#1313319112yes—
515C23#1313320512yes—
516C23C22×D21211yescharacteristic
517Q8—1062no—
518Q8—1062no—
519Q8—2121no—
520Q8—2121no—
521D4—1062no—
522D4—1062no—
523D4—2128no—
524D4—2121no—
525D4—2121no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—1064no—
539C2×C4—1062no—
540C2×C4—4244no—
541C2×C4—2122no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2122no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2122no—
550C2×C4—2128no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2128no—
556C2×C4#1313317414yes—
557C2×C4C4×C22⋊D5314yes—
558C2×C4C4×D21212yes—
559C2×C4#1313319112yes—
560C2×C4#1313320014yes—
561C2×C4#1313320512yes—
562C2×C4#1313321114yes—
563C2×C4#1313324512yes—
564C2×C4#1313325012yes—
565C2×C4C22×C4×D5311yescharacteristic
566C2×C4C22×C22⋊D5311yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4C2×#1171994511yescharacteristic
569C2×C4C2×#1171994511yescharacteristic
570C2×C4C2×#1171994511yescharacteristic
571C8—24no—
572C8—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—4241no—
580C22—4241no—
581C22—4241no—
582C22—4241no—
583C22—2122no—
584C22—2122no—
585C22—2122no—
586C22#1681051414yes—
587C22C2×C4×D21211yescharacteristic
588C22C2×#1313319111yescharacteristic
589C22C2×#1313320511yescharacteristic
590C22C2×#1313324511yescharacteristic
591C22C2×#1313325011yescharacteristic
592C22#1681142311yescharacteristic
593C22#1681142311yescharacteristic
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—4242no—
599C4—4241no—
600C4—4241no—
601C4—2124no—
602C4—2122no—
603C4#1681051414yes—
604C4C2×#1313317411yescharacteristic
605C4C2×C4×C22⋊D5311yescharacteristic
606C4C2×#1313320011yescharacteristic
607C4C2×#1313321111yescharacteristic
608C2—24no—
609C2—4242no—
610C2C2×#1681051411yescharacteristic
611C2C4×#1313324511yescharacteristic
612C2C4×#1313325011yescharacteristic
613C2#2028433411yescharacteristic
614C2#2028434511yescharacteristic
615C2#2028510911yescharacteristic
616C2#2028511611yescharacteristic
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C4×M16×C53C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×C4×M16D53
hypocenterC53#21093
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
4. C2×#16810521C2semidirect
5. #20275412C2semidirect
6. #20275405C2semidirect
7. #20275398C2semidirect
8. #20275391C2semidirect
9. #20274435C2semidirect
10. #20274435C2semidirect
11. C2×C4×M16×C53C2semidirect
18. #16810625C4semidirect
80. #13133142C2×C4semidirect
82. #13133139C2×C4semidirect
302. C2×C4×M16D53semidirect
354. C53#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#168106252
#202772052
C2×#168105212
#202754122
#202754052
#202753982
#202753912
#202744352
C2×C4×M16×C532
#2109353