Finite Group Databasebeta

← #27412182

Subgroups — #27412182

Order 13568.

1110 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811253C211yesmaximal, characteristic
2C2×#16810664C211yesmaximal, characteristic
3C2×#16810658C211yesmaximal, characteristic
4#20277330C211yesmaximal, characteristic
5#20276055C211yesmaximal, characteristic, complemented
6#20276052C211yesmaximal, characteristic, complemented
7#20276050C211yesmaximal, characteristic, complemented
8#20276049C211yesmaximal, characteristic, complemented
9#20276047C211yesmaximal, characteristic, complemented
10#20276043C211yesmaximal, characteristic, complemented
11#20276041C211yesmaximal, characteristic, complemented
12#20274433C211yesmaximal, characteristic
13#340×D53C211yesmaximal, characteristic
14#20274410C211yesmaximal, characteristic
15C53×#2059C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18#16811253—24no—
19#16811253—24no—
20C2×#13133217C2211yescharacteristic
21C2×#13133216—21no—
22C2×#13133216C2211yescharacteristic
23C2×#13133208C2211yescharacteristic
24C2×#13133207—21no—
25C2×#13133197—21no—
26C2×#13133195—21no—
27C2×#13133175C2211yescharacteristic
28C2×#13133175C2211yescharacteristic
29C2×#13133174C2211yescharacteristic
30C2×C4⋊C4×D53C2211yescharacteristic
31C2×C4⋊C4×D53C2211yescharacteristic
32#16810664—24no—
33#16810658—24no—
34C2×#13133150C2211yescharacteristic
35C2×#13133150C2211yescharacteristic
36C2×#13133150C2211yescharacteristic
37C2×#13133149C2211yescharacteristic
38C2×#13133149C2211yescharacteristic
39C2×#13133148C2211yescharacteristic
40#16810522C2211yescharacteristic
41#16810521C2211yescharacteristic
42#16810520—21no—
43#16810519C2211yescharacteristic
44#16810518—21no—
45#16810482—21no—
46#16810480—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810378C2211yescharacteristic
54#16810377C2211yescharacteristic
55#16810377C2211yescharacteristic
56C2×#125×C53C2211yescharacteristic
57C2×#107×C53C2211yescharacteristic
58C2×#106×C53C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C53×#336C2211yescharacteristic
63C22×Dic106—21no—
64C22×Dic106—21no—
65C22×C4×D53—21no—
66C22×C4×D53—21no—
67C22×C4×D53C2311yescharacteristic
68C2×#11719912—22no—
69C2×#11719912—22no—
70C2×#11719912Q814yescomplemented
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73C2×C8×D53—22no—
74C2×C8×D53—22no—
75C2×C8×D53Q814yescomplemented
76#13133217—22no—
77#13133217—22no—
78#13133216—28no—
79#13133216—22no—
80#13133216—22no—
81#13133208—22no—
82#13133208—22no—
83#13133207—28no—
84#13133197—28no—
85#13133195—28no—
86#13133175—22no—
87#13133175—22no—
88#13133175—22no—
89#13133175D414yes—
90#13133174—22no—
91#13133174—22no—
92C2×C4⋊Dic53—21no—
93C2×C4⋊Dic53—21no—
94C2×C4⋊Dic53—21no—
95C2×C4⋊Dic53C2311yescharacteristic
96C2×C4⋊Dic53C2311yescharacteristic
97C2×C4⋊Dic53C2311yescharacteristic
98C2×C4⋊Dic53C2311yescharacteristic
99C2×#11719919—21no—
100C2×#11719919—21no—
101C2×#11719919—21no—
102C2×#11719919—21no—
103C2×#11719919—21no—
104C2×#11719919—21no—
105C2×#11719919D411yescharacteristic
106C2×#11719919D411yescharacteristic
107C4⋊C4×D53—22no—
108C4⋊C4×D53—22no—
109C4⋊C4×D53D414yes—
110C4⋊C4×D53Q814yescomplemented
111C2×#11719921—21no—
112C2×#11719921—21no—
113C2×#11719921D411yescharacteristic
114C2×#11719921D411yescharacteristic
115C2×C4×Dic53—21no—
116C2×C4×Dic53—21no—
117C2×C4×Dic53C2311yescharacteristic
118#13133150—22no—
119#13133150—22no—
120#13133150—22no—
121#13133149—22no—
122#13133149—22no—
123#13133148—22no—
124#13133124—21no—
125#13133124—21no—
126#13133124—21no—
127C22×C8×C53C2311yescharacteristic
128C22×C8×C53C2311yescharacteristic
129#125×C53—22no—
130#125×C53—22no—
131C2×C4⋊C4×C53—21no—
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53—21no—
134C2×C4⋊C4×C53C2311yescharacteristic
135C2×C4⋊C4×C53C2311yescharacteristic
136C2×C4⋊C4×C53C2311yescharacteristic
137C2×C4⋊C4×C53C2311yescharacteristic
138C2×C42×C53C2311yescharacteristic
139#107×C53—22no—
140#106×C53—22no—
141#95×C53—21no—
142C23×D53C2×D411yescharacteristic
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C2×C4×D53—24no—
158C2×C4×D53—24no—
159C2×C4×D53—81no—
160C2×C4×D53—81no—
161C2×C4×D53—81no—
162C2×C4×D53—81no—
163C2×C4×D53C2×D412yes—
164C2×C4×D53C2×D412yes—
165C2×C4×D53C2×Q812yes—
166C2×C4×D53C2×Q812yes—
167C2×C4×D53C2×Q812yes—
168C2×C4×D53#4112yes—
169C2×Dic106—24no—
170C2×Dic106—24no—
171C2×Dic106—81no—
172C2×Dic106—81no—
173C2×Dic106—81no—
174C2×Dic106—81no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53C2×Q812yes—
180C2×C8×C53C2×Q812yes—
181C4⋊C4×C53—24no—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—81no—
189C4⋊C4×C53C2×D412yes—
190C4⋊C4×C53C2×D412yes—
191C4⋊C4×C53C2×Q812yes—
192C42×C53—21no—
193C42×C53—21no—
194#11719921—24no—
195#11719921—22no—
196#11719921—22no—
197#11719921—42no—
198C4⋊Dic53—24no—
199C4⋊Dic53—24no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—81no—
206C4⋊Dic53C2×D412yes—
207C4⋊Dic53C2×D412yes—
208C4⋊Dic53C2×Q812yes—
209#11719919—24no—
210#11719919—24no—
211#11719919—24no—
212#11719919—22no—
213#11719919—22no—
214#11719919—24no—
215#11719919—81no—
216#11719919—81no—
217C4×Dic53—21no—
218C4×Dic53—21no—
219C4×Dic53—24no—
220C4×Dic53—42no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322C2×Q812yes—
226C2×#88322C2×Q812yes—
227#11719912—28no—
228C8×D53—28no—
229C23×C53C22×D411yescharacteristic
230C22×D53—21no—
231C22×D53—21no—
232C22×D53—21no—
233C22×D53—21no—
234C22×D53#12012yes—
235C22×D53C22⋊Q812yes—
236C22×D53C4⋊Q812yes—
237C2×C4×C53—21no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C22×Q811yescharacteristic
258C2×C4×C53C22×Q811yescharacteristic
259C2×C4×C53C22×Q811yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×Dic53—22no—
262C2×Dic53—21no—
263C2×Dic53—22no—
264C2×Dic53—21no—
265C2×Dic53—22no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53#12012yes—
280C2×Dic53C22⋊Q812yes—
281C2×Dic53C4⋊Q812yes—
282C4×D53—84no—
283C4×D53—84no—
284C4×D53C22⋊Q814yes—
285C4×D53C22⋊Q814yes—
286C4×D53C4⋊Q814yes—
287C4×D53C4⋊Q814yes—
288Dic106—84no—
289Dic106—84no—
290C8×C53—22no—
291C8×C53—22no—
292#88322—22no—
293#88322—22no—
294#23557—531nomaximal, Sylow
295C22×C53C2×#12011yescharacteristic
296C22×C53C2×C22⋊Q811yescharacteristic
297C22×C53C2×C4⋊Q811yescharacteristic
298C22×C53#57611yescharacteristic
299C22×C53#57711yescharacteristic
300C22×C53#57711yescharacteristic
301C22×C53#57811yescharacteristic
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—22no—
305C2×D53—22no—
306C2×D53—22no—
307C2×D53—22no—
308C2×D53#39514yes—
309C4×C53—42no—
310C4×C53—41no—
311C4×C53—42no—
312C4×C53—41no—
313C4×C53—42no—
314C4×C53—81no—
315C4×C53C2×C22⋊Q811yescharacteristic
316C4×C53C2×C22⋊Q811yescharacteristic
317C4×C53C2×C4⋊Q811yescharacteristic
318C4×C53C2×C4⋊Q811yescharacteristic
319Dic53—42no—
320Dic53—41no—
321Dic53—42no—
322Dic53—42no—
323Dic53—41no—
324Dic53—81no—
325Dic53#39514yes—
326C2×#544—531no—
327C2×#428—531no—
328C2×#427—531no—
329#2316—531no—
330#2059—531no—
331#2059—531no—
332#2059—531no—
333#2059D5311yescharacteristic, complemented
334#2058—531no—
335#2058—531no—
336#2057—531no—
337#2057—531no—
338#1736—531no—
339C2×#340—531no—
340#1729—531no—
341C2×C53C2×#39511yescharacteristic
342C2×C53#301211yescharacteristic
343C2×C53#301311yescharacteristic
344C2×C53#308111yescharacteristic
345C2×C53#308211yescharacteristic
346C2×C53#316311yescharacteristic
347C2×C53#316411yescharacteristic
348D53—24no—
349C22×M16—531no—
350C23×C8—531no—
351#544—1064no—
352#544—1064no—
353C2×C4⋊Q8—531no—
354C2×C4⋊Q8—531no—
355C2×C4⋊Q8—1061no—
356C2×#125—531no—
357C2×#125—1061no—
358C2×#125C2×D5311yescharacteristic
359C2×C22⋊Q8—1061no—
360C2×C22⋊Q8—1061no—
361C2×#117—531no—
362C2×#117—531no—
363C2×#117—531no—
364C22×C4⋊C4—531no—
365C22×C4⋊C4—531no—
366#428—1064no—
367#427—1064no—
368C2×#107—531no—
369C2×#107—531no—
370C2×#107—531no—
371C2×#107C2×D5311yescharacteristic
372C2×#106—531no—
373C2×#106—531no—
374C2×#106—531no—
375C2×#106C2×D5311yescharacteristic
376#389—531no—
377#389—531no—
378#389—531no—
379#389—1061no—
380#389—1061no—
381#389C2×D5311yescharacteristic
382#380—1061no—
383#380—1061no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340C2×D5311yescharacteristic
391#340C2×D5311yescharacteristic
392#336—531no—
393#336—531no—
394#336—531no—
395#336C2×D5311yescharacteristic
396C53#2355711yescharacteristic, complemented, Sylow
397C22×Q8—1061no—
398C22×Q8—1061no—
399C23×C4—531no—
400C23×C4—1061no—
401C23×C4—1061no—
402C2×M16—534no—
403C2×M16—1062no—
404C2×M16—1062no—
405C22×C8—531no—
406C22×C8—531no—
407C22×C8—534no—
408C22×C8—1062no—
409C22×C8—1062no—
410C22×C8C22×D5311yescharacteristic
411C22×C8C22×D5311yescharacteristic
412C4⋊Q8—1062no—
413C4⋊Q8—1062no—
414C4⋊Q8—1062no—
415C4⋊Q8—1068no—
416C4⋊Q8—1062no—
417#125—22no—
418#125—22no—
419#125—1062no—
420#125—1068no—
421#125—1062no—
422C22⋊Q8—1068no—
423C22⋊Q8—1068no—
424#117—534no—
425#117—1062no—
426#117—1062no—
427#117—1062no—
428#117—1062no—
429#117—1062no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—534no—
437C2×C4⋊C4—534no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—1062no—
442C2×C4⋊C4—1062no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4—1061no—
449C2×C4⋊C4—1061no—
450C2×C4⋊C4—1061no—
451C2×C4⋊C4—1061no—
452C2×C4⋊C4C22×D5311yescharacteristic
453C2×C4⋊C4C22×D5311yescharacteristic
454C2×C4⋊C4C22×D5311yescharacteristic
455C2×C4⋊C4C22×D5311yescharacteristic
456C2×C22⋊C4—531no—
457C2×C22⋊C4—531no—
458C2×C22⋊C4—1061no—
459C2×C22⋊C4—1061no—
460C2×C42—531no—
461C2×C42—1061no—
462C2×C42—1061no—
463C2×C42C22×D5311yescharacteristic
464#107—22no—
465#107—1062no—
466#107—1062no—
467#107—1062no—
468#106—22no—
469#106—1062no—
470#106—1062no—
471#106—1062no—
472#95—21no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476C24—531no—
477C2×Q8—1064no—
478C2×Q8—1064no—
479C2×Q8—4241no—
480C2×Q8—4241no—
481C2×Q8—4241no—
482C2×Q8—4241no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—531no—
488C22×C4—532no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—532no—
493C22×C4—532no—
494C22×C4—531no—
495C22×C4—531no—
496C22×C4—1064no—
497C22×C4—1064no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—4241no—
503C22×C4—4241no—
504C22×C4—4241no—
505C22×C4—4241no—
506C22×C4D4×D5311yescharacteristic
507C22×C4D4×D5311yescharacteristic
508C22×C4C23×D5311yescharacteristic
509M16—1068no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—532no—
515C2×C8—532no—
516C2×C8—1068no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8Q8×D5312yes—
522C2×C8Q8×D5312yes—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—81no—
531C4⋊C4—532no—
532C4⋊C4—532no—
533C4⋊C4—532no—
534C4⋊C4—1064no—
535C4⋊C4—1064no—
536C4⋊C4—1062no—
537C4⋊C4—1062no—
538C4⋊C4—1064no—
539C4⋊C4—1064no—
540C4⋊C4—1064no—
541C4⋊C4—1064no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—4241no—
548C4⋊C4—4241no—
549C4⋊C4—4241no—
550C4⋊C4D4×D5312yes—
551C4⋊C4D4×D5312yes—
552C4⋊C4Q8×D5312yes—
553C22⋊C4—1062no—
554C22⋊C4—1062no—
555C22⋊C4—1064no—
556C22⋊C4—2122no—
557C42—21no—
558C42—21no—
559C42—1064no—
560C42—1061no—
561C42—1061no—
562C42—2122no—
563C23—532no—
564C23—532no—
565C23—532no—
566C23—1061no—
567C23—1061no—
568C23—1061no—
569C23—1061no—
570C23C2×D4×D5311yescharacteristic
571Q8—4244no—
572Q8—4244no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—532no—
592C2×C4—532no—
593C2×C4—532no—
594C2×C4—534no—
595C2×C4—534no—
596C2×C4—534no—
597C2×C4—534no—
598C2×C4—1062no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—4244no—
608C2×C4—4244no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4C2×D4×D5311yescharacteristic
619C2×C4C2×D4×D5311yescharacteristic
620C2×C4C2×Q8×D5311yescharacteristic
621C2×C4C2×Q8×D5311yescharacteristic
622C2×C4C2×Q8×D5311yescharacteristic
623C2×C4#41×D5311yescharacteristic
624C8—22no—
625C8—22no—
626C8—1062no—
627C8—1062no—
628C22—534no—
629C22—1062no—
630C22—1062no—
631C22—1062no—
632C22—1062no—
633C22—1062no—
634C22—1062no—
635C22#120×D5311yescharacteristic
636C22C22⋊Q8×D5311yescharacteristic
637C22C4⋊Q8×D5311yescharacteristic
638C22#1681142711yescharacteristic
639C22#1681143011yescharacteristic
640C22#1681143011yescharacteristic
641C22#1681143911yescharacteristic
642C4—41no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—81no—
648C4—534no—
649C4—4241no—
650C4—2121no—
651C4—2122no—
652C4—2122no—
653C4—2122no—
654C4—2121no—
655C4C22⋊Q8×D5311yescharacteristic
656C4C22⋊Q8×D5311yescharacteristic
657C4C4⋊Q8×D5311yescharacteristic
658C4C4⋊Q8×D5311yescharacteristic
659C2—1064no—
660C2#395×D5311yescharacteristic
661C2#2028401711yescharacteristic
662C2#2028403411yescharacteristic
663C2#2028454511yescharacteristic
664C2#2028455411yescharacteristic
665C2#2028524411yescharacteristic
666C2#2028525011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2059C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2059D53
hypocenterC53#23557
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
5. #20276055C2semidirect
6. #20276052C2semidirect
7. #20276050C2semidirect
8. #20276049C2semidirect
9. #20276047C2semidirect
10. #20276043C2semidirect
11. #20276041C2semidirect
15. C53×#2059C2semidirect
70. C2×#11719912Q8semidirect
75. C2×C8×D53Q8semidirect
110. C4⋊C4×D53Q8semidirect
333. #2059D53semidirect
396. C53#23557semidirect

Maximal subgroups

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

subgroupindex
C2×#168112532
C2×#168106642
C2×#168106582
#202773302
#202760552
#202760522
#202760502
#202760492
#202760472
#202760432
#202760412
#202744332
#340×D532
#202744102
C53×#20592
#2355753