Finite Group Databasebeta

← #27387169

Subgroups — #27387169

Order 13568.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810724C211yesmaximal, characteristic
2#20277360C211yesmaximal, characteristic, complemented
3#20277076C211yesmaximal, characteristic
4#20275359C211yesmaximal, characteristic, complemented
5#20275346C211yesmaximal, characteristic, complemented
6#20275308C211yesmaximal, characteristic, complemented
7#20275285C211yesmaximal, characteristic, complemented
8#20275051C211yesmaximal, characteristic
9#20274974C211yesmaximal, characteristic
10#20274962C211yesmaximal, characteristic
11#20274611C211yesmaximal, characteristic, complemented
12#103×Dic53C211yesmaximal, characteristic, complemented
13#102×Dic53C211yesmaximal, characteristic
14#20274465C211yesmaximal, characteristic, complemented
15C53×#1877C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19#16810724C418yescomplemented
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810530C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810508—21no—
33#16810506—21no—
34#16810490—21no—
35#16810487—21no—
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53—21no—
43#16810469C2211yescharacteristic
44C2×#13133124—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×SD16×C53C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#391C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64Q8×Dic53—22no—
65Q8×Dic53C2×C414yescomplemented
66D4×Dic53—22no—
67D4×Dic53—22no—
68D4×Dic53—22no—
69D4×Dic53C2×C414yes—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133151C2×C414yescomplemented
87#13133150—22no—
88#13133148—22no—
89#13133141—22no—
90#13133141C2×C414yescomplemented
91#13133138—22no—
92#13133138C2×C414yes—
93#13133127C2×C414yescomplemented
94C8×Dic53—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107C22×Q8×C53C2311yescharacteristic
108C22×D4×C53C2311yescharacteristic
109C23×C4×C53—21no—
110C2×SD16×C53—22no—
111C2×SD16×C53—22no—
112C2×SD16×C53C2×C414yes—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C4×D4×C53—22no—
116C4×D4×C53—22no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#103×C53—22no—
122#102×C53—22no—
123#95×C53—21no—
124#95×C53—21no—
125C24×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C22×C412yes—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53C22×C412yes—
136C2×D4×C53#4112yes—
137C2×D4×C53#4112yes—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—24no—
141C22×C4×C53—21no—
142C22×C4×C53—81no—
143C22×C4×C53—81no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—24no—
151C22×Dic53—24no—
152C22×Dic53—21no—
153C22×Dic53—81no—
154C22×Dic53—81no—
155C22×Dic53—81no—
156C22×Dic53—81no—
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161SD16×C53—28no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53C22×C412yes—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53#4112yes—
172C22⋊C4×C53—24no—
173C42×C53—21no—
174C42×C53—21no—
175C22⋊Dic53—24no—
176C22⋊Dic53—24no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—24no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53C22×C412yes—
184C4⋊Dic53C22×C412yes—
185C4⋊Dic53#4112yes—
186#11719919—81no—
187#11719919—81no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—24no—
191C4×Dic53—81no—
192C4×Dic53—81no—
193C4×Dic53—81no—
194C4×Dic53—81no—
195C4×Dic53C22×C412yes—
196C4×Dic53#4112yes—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322C22×C412yes—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53C22×D411yescharacteristic
211Q8×C53—22no—
212Q8×C53—22no—
213Q8×C53—41no—
214Q8×C53—41no—
215D4×C53—22no—
216D4×C53—22no—
217D4×C53—22no—
218D4×C53#11714yes—
219C2×C4×C53—22no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—84no—
235C2×C4×C53C23×C411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—21no—
248C2×Dic53—22no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—84no—
264C2×Dic53—84no—
265C2×Dic53C4×D412yes—
266C2×Dic53C4×D412yes—
267C8×C53—22no—
268C8×C53—41no—
269#88322—22no—
270#88322—41no—
271#21125—531nomaximal, Sylow
272C22×C53—24no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53C2×C4×D411yescharacteristic
280C22×C53#52511yescharacteristic
281C22×C53#52511yescharacteristic
282C22×C53#57611yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57711yescharacteristic
285C22×C53#57811yescharacteristic
286C4×C53—41no—
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—42no—
290C4×C53—81no—
291C4×C53C2×#11711yescharacteristic
292C4×C53C4×#4111yescharacteristic
293C4×C53#52911yescharacteristic
294C4×C53#52911yescharacteristic
295Dic53—22no—
296Dic53—22no—
297Dic53—41no—
298Dic53—42no—
299Dic53—41no—
300Dic53—42no—
301Dic53—42no—
302Dic53—81no—
303C2×#440—531no—
304#2320—531no—
305#2275—531no—
306#1928—531no—
307#1927—531no—
308#1921—531no—
309#1918—531no—
310#1877—531no—
311#1877D5311yescharacteristic
312#1864—531no—
313#1862—531no—
314#1776—531no—
315C4×#103—531no—
316C4×#102—531no—
317#1746—531no—
318C2×C53—44no—
319C2×C53#229011yescharacteristic
320C2×C53#294611yescharacteristic
321C2×C53#294711yescharacteristic
322C2×C53#311811yescharacteristic
323C2×C53#312011yescharacteristic
324C2×C53#312311yescharacteristic
325C2×C53#312411yescharacteristic
326C22×SD16C2×D5311yescharacteristic
327C2×C4×Q8—531no—
328C2×C4×D4—531no—
329C2×C4×D4—531no—
330C2×C4×D4C2×D5311yescharacteristic
331#440—538no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#106—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103C2×D5311yescharacteristic
338C2×#102—531no—
339C2×#102—531no—
340C2×#102C2×D5311yescharacteristic
341C2×#97—531no—
342C2×C4×C8—531no—
343#391—531no—
344#391C2×D5311yescharacteristic
345#389—531no—
346#389—531no—
347#387—1061no—
348#387—1061no—
349#383—1061no—
350#382—1061no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C22⋊C4—1061no—
356C4×C22⋊C4—1061no—
357C4×C22⋊C4—1061no—
358#376—531no—
359C2×#95—1061no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340C2×D5311yescharacteristic
366#336—531no—
367#336—531no—
368#336—531no—
369#336C2×D5311yescharacteristic
370C53#2112511yescharacteristic, complemented, Sylow
371C22×Q8C22×D5311yescharacteristic
372C22×D4C22×D5311yescharacteristic
373C23×C4—21no—
374C23×C4—1061no—
375C23×C4—1061no—
376C2×SD16—22no—
377C2×SD16—22no—
378C2×SD16C4×D5314yes—
379C22×C8—531no—
380C22×C8—531no—
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C4×Q8—534no—
384C4×Q8—1062no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—534no—
388C4×D4—1062no—
389C4×D4—1062no—
390C4×D4—1062no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—1061no—
406C2×C42—1061no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42C22×D5311yescharacteristic
411#107—534no—
412#107—1062no—
413#106—1062no—
414#103—22no—
415#103—534no—
416#103—1062no—
417#102—22no—
418#102—534no—
419#102—1062no—
420#97—534no—
421C4×C8—1062no—
422#95—21no—
423#95—21no—
424#95—531no—
425#95—531no—
426#95—531no—
427#95—531no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D5312yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D5312yes—
447C2×D4#1171994712yes—
448C2×D4#1171994712yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—1064no—
460C22×C4—1064no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—4241no—
466C22×C4—4241no—
467C22×C4—4241no—
468C22×C4—4241no—
469C22×C4D4×D5311yescharacteristic
470C22×C4D4×D5311yescharacteristic
471C22×C4C23×D5311yescharacteristic
472SD16—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—532no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8C2×C4×D5312yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—532no—
489C4⋊C4—532no—
490C4⋊C4—532no—
491C4⋊C4—1064no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4#1171994712yes—
500C22⋊C4—24no—
501C22⋊C4—1064no—
502C22⋊C4—1064no—
503C42—21no—
504C42—21no—
505C42—532no—
506C42—532no—
507C42—1064no—
508C42—1061no—
509C42—1061no—
510C42—4241no—
511C42—4241no—
512C42—4241no—
513C42—4241no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D5311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#1313317314yes—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1062no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—4244no—
562C2×C4—4244no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4C22×C4×D5311yescharacteristic
573C2×C4#41×D5311yescharacteristic
574C2×C4#41×D5311yescharacteristic
575C2×C4C2×#1171994711yescharacteristic
576C2×C4C2×#1171994711yescharacteristic
577C2×C4C2×#1171994711yescharacteristic
578C8—22no—
579C8—41no—
580C8—1062no—
581C8—2121no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C4×D4×D5311yescharacteristic
590C22#1681112911yescharacteristic
591C22#1681113311yescharacteristic
592C22#1681142711yescharacteristic
593C22#1681142911yescharacteristic
594C22#1681143211yescharacteristic
595C22#1681143711yescharacteristic
596C4—41no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—81no—
601C4—1062no—
602C4—1062no—
603C4—4241no—
604C4—2121no—
605C4—2122no—
606C4—2121no—
607C4—2122no—
608C4—2122no—
609C4C2×#1313317311yescharacteristic
610C4C4×#1171994711yescharacteristic
611C4#1681115011yescharacteristic
612C4#1681115411yescharacteristic
613C2—44no—
614C2#2027715511yescharacteristic
615C2#2028354611yescharacteristic
616C2#2028355111yescharacteristic
617C2#2028490011yescharacteristic
618C2#2028493011yescharacteristic
619C2#2028496511yescharacteristic
620C2#2028497411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1877C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1877D53
hypocenterC53#21125
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #20277360C2semidirect
4. #20275359C2semidirect
5. #20275346C2semidirect
6. #20275308C2semidirect
7. #20275285C2semidirect
11. #20274611C2semidirect
12. #103×Dic53C2semidirect
14. #20274465C2semidirect
19. #16810724C4semidirect
65. Q8×Dic53C2×C4semidirect
86. #13133151C2×C4semidirect
90. #13133141C2×C4semidirect
93. #13133127C2×C4semidirect
370. C53#21125semidirect

Maximal subgroups

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

subgroupindex
C2×#168107242
#202773602
#202770762
#202753592
#202753462
#202753082
#202752852
#202750512
#202749742
#202749622
#202746112
#103×Dic532
#102×Dic532
#202744652
C53×#18772
#2112553