Finite Group Databasebeta

← #27386498

Subgroups — #27386498

Order 13568.

1218 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×#16810663C211yesmaximal, characteristic, complemented
2C2×#16810622C211yesmaximal, characteristic
3C2×#16810622C211yesmaximal, characteristic
4#20277205C211yesmaximal, characteristic
5#20275356C212yesmaximal, complemented
6#20275349C212yesmaximal, complemented
7#20275342C212yesmaximal, complemented
8#20275335C212yesmaximal, complemented
9#20274434C211yesmaximal, characteristic, complemented
10#20274409C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C53C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C22×#11719917C2211yescharacteristic
15C2×#13133176C2211yescharacteristic
16#16810663—24no—
17C2×#13133150C2212yes—
18C2×#13133148C2212yes—
19#16810622—24no—
20#16810622C418yescomplemented
21C2×#13133140C2212yes—
22C2×#13133140C2212yes—
23C2×#13133137C2212yes—
24C2×#13133137C2212yes—
25#16810529C2211yescharacteristic
26#16810529C2211yescharacteristic
27#16810525C2211yescharacteristic
28#16810525C2211yescharacteristic
29#16810502—21no—
30#16810502—21no—
31C4×#11719921—21no—
32C4×#11719921—21no—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36#16810378C2212yes—
37C22×C4⋊C4×C53C2211yescharacteristic
38C22×C4⋊C4×C53C2211yescharacteristic
39C22×C42×C53—21no—
40C22×C42×C53C2211yescharacteristic
41C4×C4⋊C4×C53—21no—
42C4×C4⋊C4×C53—21no—
43C4×C4⋊C4×C53C2212yes—
44C4×C4⋊C4×C53C2212yes—
45C2×#95×C53—21no—
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—
55C22×C4×D53—21no—
56C2×#11719917—22no—
57C2×#11719917—22no—
58C2×#11719917—22no—
59C2×#11719917—22no—
60C2×#11719917—22no—
61C2×#11719917C2×C414yes—
62C22×#88322C2312yes—
63C22×#88322C2312yes—
64#13133176—22no—
65#13133176—22no—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C22⋊Dic53—21no—
69C2×#11719921—21no—
70C2×#11719921—21no—
71C2×#11719921—21no—
72C2×#11719921—21no—
73C2×C4×Dic53—21no—
74#13133150—24no—
75#13133148—24no—
76#13133140—24no—
77#13133140C2×C418yescomplemented
78#13133137—24no—
79#13133137C2×C418yescomplemented
80#13133124—21no—
81#13133124—21no—
82#13133124—21no—
83#13133124—21no—
84C23×C4×C53—21no—
85C23×C4×C53—21no—
86C23×C4×C53D411yescharacteristic
87C23×C4×C53D411yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53C2×C414yes—
94C2×C4⋊C4×C53D414yes—
95C2×C4⋊C4×C53C2312yes—
96C2×C4⋊C4×C53C2312yes—
97C2×C42×C53—22no—
98C2×C42×C53—22no—
99C2×C42×C53—22no—
100C2×C42×C53—22no—
101C2×C42×C53—44no—
102C2×C42×C53D411yescharacteristic
103C2×C42×C53D411yescharacteristic
104C2×C42×C53C2311yescharacteristic
105C2×C42×C53C2311yescharacteristic
106#95×C53—22no—
107#95×C53—22no—
108C24×C53C2×D411yescharacteristic
109C23×D53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—24no—
114C22×C4×C53—24no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D412yes—
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×C4×C53#4112yes—
134C22×C4×C53#4112yes—
135C22×C4×C53C2411yescharacteristic
136C2×C22⋊D53—24no—
137C2×C22⋊D53—42no—
138C2×C22⋊D53—42no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
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—
161C4⋊C4×C53—24no—
162C4⋊C4×C53—24no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53C22×C414yes—
166C4⋊C4×C53C2×D414yes—
167C42×C53—22no—
168C42×C53—22no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—44no—
174C42×C53—44no—
175C22⋊Dic53—42no—
176#11719921—42no—
177#11719921—42no—
178#11719921—42no—
179#11719921—42no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4×Dic53—42no—
185#11719917—28no—
186#11719917—44no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322C22×C414yes—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53C4×D412yes—
198C23×C53#12112yes—
199C23×C53#12312yes—
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—28no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—44no—
222C2×C4×C53—48no—
223C2×C4×C53—48no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53#11714yes—
229C2×C4×C53C4×D412yes—
230C2×C4×C53C4×D414yes—
231C2×C4×C53#12112yes—
232C2×C4×C53#12312yes—
233C2×C4×C53#12414yes—
234C2×C4×C53#12614yes—
235C2×C4×C53AGL(1,ℤ/8)12yes—
236C2×C4×C53#13712yes—
237C2×C4×C53C23×C411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C22⋊D53—48no—
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253D212—22no—
254D212—22no—
255D212—41no—
256D212—41no—
257C4×D53—48no—
258Dic106—22no—
259Dic106—22no—
260Dic106—41no—
261Dic106—41no—
262#88322—24no—
263#88322—42no—
264#21059—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×#12311yescharacteristic
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—44no—
291C4×C53—44no—
292C4×C53#38814yes—
293C4×C53C2×#11711yescharacteristic
294C4×C53C2×C4×D411yescharacteristic
295C4×C53C2×#12411yescharacteristic
296C4×C53C2×#12611yescharacteristic
297Dic53—42no—
298Dic53—81no—
299Dic53—81no—
300C2×#428—531no—
301C2×#417—531no—
302C2×#417—531no—
303#2297—531no—
304C2×C4×C4⋊C4D5311yescharacteristic, complemented
305#1928—532no—
306#1927—532no—
307#1926—532no—
308#1925—532no—
309#1736—531no—
310#1728—531no—
311C2×C53—24no—
312C2×C53C2×#38811yescharacteristic
313C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C53C4×#13711yescharacteristic
315C2×C53#303611yescharacteristic
316C2×C53#303711yescharacteristic
317C2×C53#309211yescharacteristic
318C2×C53#309311yescharacteristic
319D53—82no—
320C22×#41—531no—
321C22×M16—531no—
322C22×M16—531no—
323C2×#117—531no—
324C22×C4⋊C4C2×D5311yescharacteristic
325C22×C4⋊C4C2×D5311yescharacteristic
326C22×C42—21no—
327C22×C42C2×D5311yescharacteristic
328#428—1064no—
329C2×#107—532no—
330C2×#106—532no—
331#417—538no—
332#417—1064no—
333C2×#103—532no—
334C2×#103—532no—
335C2×#102—532no—
336C2×#102—532no—
337#391—531no—
338#391—531no—
339#390—531no—
340#390—531no—
341#386—1061no—
342#386—1061no—
343C4×C4⋊C4—21no—
344C4×C4⋊C4—21no—
345C4×C4⋊C4C2×D5312yes—
346C4×C4⋊C4C2×D5312yes—
347C4×C22⋊C4—1061no—
348C4×C22⋊C4—1061no—
349C2×#95—21no—
350#340—532no—
351#340—532no—
352#336—532no—
353#336—532no—
354C53#2105911yescharacteristic, 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—21no—
365C23×C4—21no—
366C23×C4—1061no—
367C23×C4C22⋊D5311yescharacteristic
368C23×C4C22⋊D5311yescharacteristic
369C23×C4C22×D5311yescharacteristic
370C2×M16—534no—
371C2×M16—1062no—
372C2×M16—1062no—
373C2×M16—1062no—
374C2×M16—1062no—
375C2×M16—1062no—
376C22×C8—532no—
377C22×C8—532no—
378#117—1062no—
379#117—1062no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4C4×D5314yes—
387C2×C4⋊C4D21214yes—
388C2×C4⋊C4C22×D5312yes—
389C2×C4⋊C4C22×D5312yes—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—44no—
400C2×C42—1061no—
401C2×C42C22⋊D5311yescharacteristic
402C2×C42C22⋊D5311yescharacteristic
403C2×C42C22×D5311yescharacteristic
404C2×C42C22×D5311yescharacteristic
405#107—1064no—
406#106—1064no—
407#103—538no—
408#103—1064no—
409#102—538no—
410#102—1064no—
411#95—22no—
412#95—22no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417C24—1061no—
418C24C2×C22⋊D5311yescharacteristic
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—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—1061no—
453C22×C4—1064no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4C2×C4×D5312yes—
459C22×C4C2×D21212yes—
460C22×C4#1171994512yes—
461C22×C4#1171994512yes—
462C22×C4#1171994512yes—
463C22×C4#1171994512yes—
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470M16—1068no—
471M16—2124no—
472C2×C8—534no—
473C2×C8—1062no—
474C2×C8—1062no—
475C2×C8—1062no—
476C2×C8—1062no—
477C2×C8—1062no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4C2×C4×D5314yes—
487C4⋊C4C2×D21214yes—
488C22⋊C4—2122no—
489C22⋊C4—2122no—
490C22⋊C4—2122no—
491C22⋊C4—2122no—
492C22⋊C4—2122no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—44no—
500C42—44no—
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×C22⋊D5312yes—
514C23#1313319112yes—
515C23#1313320012yes—
516C23C22×C22⋊D5311yescharacteristic
517Q8—1062no—
518Q8—1062no—
519Q8—2121no—
520Q8—2121no—
521D4—1062no—
522D4—1062no—
523D4—2128no—
524D4—2121no—
525D4—2121no—
526C2×C4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—1062no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2128no—
556C2×C4#1313317414yes—
557C2×C4C4×C22⋊D5312yes—
558C2×C4C4×D21214yes—
559C2×C4#1313319112yes—
560C2×C4#1313320012yes—
561C2×C4#1313320514yes—
562C2×C4#1313321114yes—
563C2×C4#1313324312yes—
564C2×C4#1313324812yes—
565C2×C4C22×C4×D5311yescharacteristic
566C2×C4C22×D21211yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4C2×#1171994511yescharacteristic
569C2×C4C2×#1171994511yescharacteristic
570C2×C4C2×#1171994511yescharacteristic
571C8—1064no—
572C8—2122no—
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×C22⋊D5311yescharacteristic
588C22C2×#1313319111yescharacteristic
589C22C2×#1313320011yescharacteristic
590C22C2×#1313324311yescharacteristic
591C22C2×#1313324811yescharacteristic
592C22#1681142111yescharacteristic
593C22#1681142111yescharacteristic
594C4—22no—
595C4—22no—
596C4—44no—
597C4—44no—
598C4—41no—
599C4—41no—
600C4—4241no—
601C4—4241no—
602C4—2122no—
603C4#1681051414yes—
604C4C2×#1313317411yescharacteristic
605C4C2×C4×D21211yescharacteristic
606C4C2×#1313320511yescharacteristic
607C4C2×#1313321111yescharacteristic
608C2—24no—
609C2—4242no—
610C2C2×#1681051411yescharacteristic
611C2C4×#1313324311yescharacteristic
612C2C4×#1313324811yescharacteristic
613C2#2028417911yescharacteristic
614C2#2028418711yescharacteristic
615C2#2028463811yescharacteristic
616C2#2028464611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C4×C4⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×C4×C4⋊C4D53
hypocenterC53#21059
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16810663C2semidirect
5. #20275356C2semidirect
6. #20275349C2semidirect
7. #20275342C2semidirect
8. #20275335C2semidirect
9. #20274434C2semidirect
10. #20274409C2semidirect
11. C2×C4×C4⋊C4×C53C2semidirect
20. #16810622C4semidirect
77. #13133140C2×C4semidirect
79. #13133137C2×C4semidirect
304. C2×C4×C4⋊C4D53semidirect
354. C53#21059semidirect

Maximal subgroups

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

subgroupindex
C2×#168106632
C2×#168106222
#202772052
#202753562
#202753492
#202753422
#202753352
#202744342
#202744092
C2×C4×C4⋊C4×C532
#2105953