Finite Group Databasebeta

← #27387108

Subgroups — #27387108

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810723C211yesmaximal, characteristic
2#20277192C211yesmaximal, characteristic, complemented
3#20275356C211yesmaximal, characteristic, complemented
4#20275349C211yesmaximal, characteristic, complemented
5#20275307C211yesmaximal, characteristic, complemented
6#20275286C211yesmaximal, characteristic, complemented
7#20275049C211yesmaximal, characteristic
8#20275049C211yesmaximal, characteristic
9#20274976C211yesmaximal, characteristic
10#20274962C211yesmaximal, characteristic
11#20274613C211yesmaximal, characteristic, complemented
12#20274548C211yesmaximal, characteristic, complemented
13#20274540C211yesmaximal, characteristic
14#20274464C211yesmaximal, characteristic, complemented
15C42×D4×C53C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18#16810723C418yescomplemented
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29C2×#13133128C2211yescharacteristic
30#16810529C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810507—21no—
35#16810507—21no—
36C4×#11719919—21no—
37C4×C4⋊Dic53C2211yescharacteristic
38#16810469—21no—
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49C2×C4×D4×C53C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C22×C42×C53—21no—
53C4×C4⋊C4×C53C2211yescharacteristic
54C4×C4⋊C4×C53C2211yescharacteristic
55C4×C4⋊C4×C53C2211yescharacteristic
56C4×C22⋊C4×C53—21no—
57C4×C22⋊C4×C53—21no—
58C4×C22⋊C4×C53—21no—
59C43×C53C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C2×#11719923—22no—
62C2×#11719923—22no—
63C2×#11719923C2×C414yes—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C4×Dic106—22no—
69C4×Dic106C2×C414yescomplemented
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78#13133150—22no—
79#13133150C2×C414yescomplemented
80#13133148—22no—
81#13133140—22no—
82#13133140—22no—
83#13133140C2×C414yescomplemented
84#13133138—22no—
85#13133138—22no—
86#13133138C2×C414yes—
87#13133128—22no—
88#13133128C2×C414yescomplemented
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96C22×D4×C53C2311yescharacteristic
97C23×C4×C53—21no—
98C23×C4×C53—21no—
99C23×C4×C53—21no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C4×D4×C53C2×C414yes—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C42×C53—21no—
113C2×C42×C53—21no—
114C2×C42×C53—21no—
115C2×C42×C53—44no—
116C2×C42×C53D411yescharacteristic
117C2×C42×C53D411yescharacteristic
118C2×C42×C53D411yescharacteristic
119C2×C42×C53D411yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53C22×C412yes—
131C2×D4×C53#4112yes—
132C2×D4×C53#4112yes—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—24no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C22×C412yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53C22×C412yes—
166C4⋊C4×C53#4112yes—
167C4⋊C4×C53#4112yes—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—24no—
171C42×C53—22no—
172C42×C53—22no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—41no—
178C42×C53—41no—
179C42×C53—42no—
180C42×C53—42no—
181C42×C53—42no—
182C42×C53C22×C412yes—
183C42×C53#4112yes—
184#11719923—28no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53C22×C412yes—
191#11719919—24no—
192#11719919—42no—
193#11719919—42no—
194C4×Dic53—24no—
195C4×Dic53—42no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322C22×C412yes—
207C2×#88322C22×C412yes—
208C23×C53—22no—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53C22×D411yescharacteristic
216D4×C53—22no—
217D4×C53—22no—
218D4×C53—22no—
219D4×C53#11714yes—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—48no—
238C2×C4×C53—48no—
239C2×C4×C53—48no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53C4×D412yes—
247C2×C4×C53C4×D412yes—
248C2×C4×C53AGL(1,ℤ/8)12yes—
249C2×C4×C53#13712yes—
250C2×C4×C53C23×C411yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272#88322—22no—
273#88322—22no—
274#88322—41no—
275#88322—41no—
276#21121—531nomaximal, Sylow
277C22×C53—24no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53C2×C4×D411yescharacteristic
285C22×C53#52511yescharacteristic
286C22×C53#52511yescharacteristic
287C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57511yescharacteristic
290C22×C53#57511yescharacteristic
291C4×C53—22no—
292C4×C53—22no—
293C4×C53—22no—
294C4×C53—22no—
295C4×C53—42no—
296C4×C53—41no—
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—42no—
300C4×C53C2×#11711yescharacteristic
301C4×C53C4×#4111yescharacteristic
302C4×C53#52911yescharacteristic
303C4×C53#52911yescharacteristic
304Dic53—42no—
305Dic53—42no—
306Dic53—81no—
307Dic53—81no—
308C2×#440—531no—
309#2295—531no—
310C42×D4D5311yescharacteristic
311#1928—531no—
312#1927—531no—
313#1921—531no—
314#1918—531no—
315#1877—531no—
316#1877—531no—
317#1864—531no—
318#1862—531no—
319#1776—531no—
320#1763—531no—
321#1762—531no—
322#1745—531no—
323C2×C53—44no—
324C2×C53#229011yescharacteristic
325C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
326C2×C53C4×#13711yescharacteristic
327C2×C53#311011yescharacteristic
328C2×C53#311111yescharacteristic
329C2×C53#311411yescharacteristic
330C2×C53#311611yescharacteristic
331C22×SD16—531no—
332C2×C4×Q8—531no—
333C2×C4×D4C2×D5311yescharacteristic
334C2×C4×D4C2×D5311yescharacteristic
335C2×C4×D4C2×D5311yescharacteristic
336C22×C42—21no—
337#440—538no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#106—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×#97—531no—
348C2×#97—531no—
349#391—531no—
350#391—531no—
351#389—531no—
352#389—531no—
353#387—1061no—
354#387—1061no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—1061no—
357C4×C4⋊C4C2×D5311yescharacteristic
358C4×C4⋊C4C2×D5311yescharacteristic
359C4×C4⋊C4C2×D5311yescharacteristic
360C4×C22⋊C4—21no—
361C4×C22⋊C4—21no—
362C4×C22⋊C4—21no—
363#376—1061no—
364C43C2×D5311yescharacteristic
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#336—531no—
372#336—531no—
373#336—531no—
374#336—531no—
375C53#2112111yescharacteristic, complemented, Sylow
376C22×Q8—531no—
377C22×D4C22×D5311yescharacteristic
378C23×C4—21no—
379C23×C4—21no—
380C23×C4—21no—
381C2×SD16—534no—
382C2×SD16—1062no—
383C2×SD16—1062no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8—531no—
388C4×Q8—534no—
389C4×Q8—1062no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4C4×D5314yes—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—44no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42C22⋊D5311yescharacteristic
415C2×C42C22⋊D5311yescharacteristic
416C2×C42C22⋊D5311yescharacteristic
417C2×C42C22⋊D5311yescharacteristic
418C2×C42C22×D5311yescharacteristic
419C2×C42C22×D5311yescharacteristic
420C2×C42C22×D5311yescharacteristic
421#107—534no—
422#107—1062no—
423#106—1062no—
424#103—534no—
425#103—1062no—
426#103—1062no—
427#102—534no—
428#102—1062no—
429#102—1062no—
430#97—534no—
431#97—1062no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442C24—21no—
443C2×Q8—532no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D5312yes—
453C2×D4#1171994512yes—
454C2×D4#1171994512yes—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—1068no—
479C2×C8—532no—
480C2×C8—532no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—532no—
495C4⋊C4—1064no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4C2×C4×D5312yes—
504C4⋊C4#1171994512yes—
505C4⋊C4#1171994512yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—41no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—1064no—
521C42—2122no—
522C42C2×C4×D5312yes—
523C42#1171994512yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C22×C22⋊D5311yescharacteristic
532Q8—1062no—
533Q8—1062no—
534Q8—2121no—
535Q8—2121no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1313317414yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4C4×C22⋊D5312yes—
579C2×C4C4×C22⋊D5312yes—
580C2×C4#1313324312yes—
581C2×C4#1313324712yes—
582C2×C4C22×C4×D5311yescharacteristic
583C2×C4C2×#1171994511yescharacteristic
584C2×C4C2×#1171994511yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4C2×#1171994511yescharacteristic
587C2×C4C2×#1171994511yescharacteristic
588C8—1062no—
589C8—1062no—
590C8—2121no—
591C8—2121no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C2×C4×C22⋊D5311yescharacteristic
600C22#1681113411yescharacteristic
601C22#1681113411yescharacteristic
602C22C2×#1313324311yescharacteristic
603C22C2×#1313324711yescharacteristic
604C22#1681142111yescharacteristic
605C22#1681142111yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—4241no—
616C4—4241no—
617C4—2122no—
618C4—2122no—
619C4C2×#1313317411yescharacteristic
620C4C4×#1171994511yescharacteristic
621C4#1681115711yescharacteristic
622C4#1681115711yescharacteristic
623C2—44no—
624C2#2027715811yescharacteristic
625C2C4×#1313324311yescharacteristic
626C2C4×#1313324711yescharacteristic
627C2#2028481411yescharacteristic
628C2#2028482411yescharacteristic
629C2#2028485311yescharacteristic
630C2#2028487711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC42×D4×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterC42×D4D53
hypocenterC53#21121
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #20277192C2semidirect
3. #20275356C2semidirect
4. #20275349C2semidirect
5. #20275307C2semidirect
6. #20275286C2semidirect
11. #20274613C2semidirect
12. #20274548C2semidirect
14. #20274464C2semidirect
18. #16810723C4semidirect
69. C4×Dic106C2×C4semidirect
79. #13133150C2×C4semidirect
83. #13133140C2×C4semidirect
88. #13133128C2×C4semidirect
375. C53#21121semidirect

Maximal subgroups

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

subgroupindex
C2×#168107232
#202771922
#202753562
#202753492
#202753072
#202752862
#202750492
#202749762
#202749622
#202746132
#202745482
#202745402
#202744642
C42×D4×C532
#2112153