Finite Group Databasebeta

← #27436148

Subgroups — #27436148

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810913C211yesmaximal, characteristic
2C2×#16810754C211yesmaximal, characteristic, complemented
3#20279700C211yesmaximal, characteristic, complemented
4#20276217C211yesmaximal, characteristic, complemented
5#20276216C211yesmaximal, characteristic, complemented
6#20276103C211yesmaximal, characteristic
7#20276089C211yesmaximal, characteristic
8#20275687C211yesmaximal, characteristic, complemented
9#20275481C211yesmaximal, characteristic, complemented
10#20275450C211yesmaximal, characteristic, complemented
11#20275440C211yesmaximal, characteristic, complemented
12#20275415C211yesmaximal, characteristic
13#20275401C211yesmaximal, characteristic
14#20274540C211yesmaximal, characteristic
15C53×#2580C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133215—21no—
18#16810913—24no—
19#16810754—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133128C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33C2×C4×#88322C2211yescharacteristic
34#16810553—21no—
35#16810553—21no—
36#16810528—21no—
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810521C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C4×Q8×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#399—21no—
53C53×#393—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#385—21no—
59C53×#385—21no—
60C53×#382C2211yescharacteristic
61C22×Dic106—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133217—22no—
67#13133217—22no—
68#13133215—44no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C4×Dic53—21no—
79#13133150—22no—
80#13133148—22no—
81#13133148—22no—
82#13133148—22no—
83C4⋊#88322—22no—
84C4⋊#88322—22no—
85C4⋊#88322D414yes—
86#13133138—22no—
87#13133138—22no—
88#13133138—22no—
89#13133138Q814yescomplemented
90#13133128—22no—
91#13133128—22no—
92C4×#88322—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98C22×Q8×C53D411yescharacteristic
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53—21no—
101C4×Q8×C53—22no—
102C4×Q8×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125C24×C53—21no—
126C2×Q8×C53—22no—
127C2×Q8×C53—22no—
128C2×Q8×C53—41no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53C2×Q812yes—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—42no—
139C22×C4×C53—81no—
140C22×C4×C53—81no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C2×Dic106—42no—
153C2×Dic106—42no—
154C2×Dic106—42no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—81no—
165C4⋊C4×C53—81no—
166C22⋊C4×C53—42no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—81no—
170C22⋊C4×C53—81no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53C2×D412yes—
176C42×C53#4112yes—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53C2×Q812yes—
185C4⋊Dic53#4112yes—
186#11719919—42no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190C4×Dic53—42no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C2×D412yes—
201C2×#88322C2×D412yes—
202C2×#88322C2×Q812yes—
203C23×C53—22no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53C22×D411yescharacteristic
211Q8×C53—44no—
212D4×C53—22no—
213D4×C53—22no—
214D4×C53—41no—
215D4×C53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
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—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—84no—
241C2×C4×C53#12112yes—
242C2×C4×C53#12112yes—
243C2×C4×C53AGL(1,ℤ/8)12yes—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×Q811yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C532−1+411yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262Dic106—84no—
263#88322—22no—
264#88322—22no—
265#88322—22no—
266#88322—41no—
267#25485—531nomaximal, Sylow
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53C2×#12111yescharacteristic
276C22×C53#54811yescharacteristic
277C22×C53D4×Q811yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53#57511yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57711yescharacteristic
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53—41no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53#53311yescharacteristic
292C4×C53#54111yescharacteristic
293C4×C53#54711yescharacteristic
294C4×C53D4×Q811yescharacteristic
295Dic53—42no—
296Dic53—42no—
297Dic53—81no—
298Dic53—81no—
299C2×#476—531no—
300C2×#446—531no—
301#2580D5311yescharacteristic
302#2546—531no—
303#2088—531no—
304#2088—531no—
305#2067—531no—
306#2065—531no—
307#1993—531no—
308#1949—531no—
309#1943—531no—
310#1941—531no—
311#1936—531no—
312#1934—531no—
313#1762—531no—
314C2×C53—82no—
315C2×C53#242911yescharacteristic
316C2×C53#305511yescharacteristic
317C2×C53#305711yescharacteristic
318C2×C53#337111yescharacteristic
319C2×C53#337311yescharacteristic
320C2×C53#338411yescharacteristic
321C2×C53#338811yescharacteristic
322C2×C4⋊Q8—531no—
323C2×C4⋊Q8—1061no—
324C2×C4×Q8C2×D5311yescharacteristic
325C2×C4×D4C2×D5311yescharacteristic
326#476—1064no—
327#446—1064no—
328C2×#107—531no—
329C2×#106—531no—
330C2×#106—531no—
331C2×#106—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×C4⋊C8—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×#102—531no—
338C2×#102—531no—
339C2×#97—531no—
340C2×#97—531no—
341C2×C4×C8—531no—
342#399—21no—
343#398—1061no—
344#398—1061no—
345#393—21no—
346#391—1061no—
347#390C2×D5311yescharacteristic
348#390C2×D5311yescharacteristic
349#389—531no—
350#389—531no—
351#389—531no—
352#389—531no—
353#389C2×D5311yescharacteristic
354#389C2×D5311yescharacteristic
355#385—21no—
356#385—21no—
357#382C2×D5311yescharacteristic
358C4×C4⋊C4—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367C53#2548511yescharacteristic, complemented, Sylow
368C22×Q8—1061no—
369C22×Q8C22⋊D5311yescharacteristic
370C22×D4C22×D5311yescharacteristic
371C23×C4—21no—
372C22×C8—531no—
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—531no—
376C4⋊Q8—1062no—
377C4⋊Q8—1062no—
378C4⋊Q8—2124no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—22no—
382C4×D4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4C22⋊D5311yescharacteristic
395C2×C4⋊C4C22⋊D5311yescharacteristic
396C2×C4⋊C4C22⋊D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C42—1061no—
406C2×C42C22×D5311yescharacteristic
407C2×C42C22×D5311yescharacteristic
408C2×C42C22×D5311yescharacteristic
409#107—1062no—
410#106—1062no—
411#106—1062no—
412#106—1062no—
413C4⋊C8—534no—
414C4⋊C8—1062no—
415C4⋊C8—1062no—
416#102—534no—
417#102—1062no—
418#102—1062no—
419#102—1062no—
420#97—1062no—
421#97—1062no—
422C4×C8—1062no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432C24—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—41no—
436C2×Q8—2122no—
437C2×Q8—2122no—
438C2×Q8—2122no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4Q8×D5312yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C23×D5311yescharacteristic
462C2×C8—532no—
463C2×C8—532no—
464C2×C8—532no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—532no—
486C4⋊C4—532no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—2122no—
508C42D4×D5312yes—
509C42#1171994712yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C22×C22⋊D5311yescharacteristic
518Q8—44no—
519Q8—4244no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
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—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—84no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4#1313318912yes—
562C2×C4#1313318912yes—
563C2×C4#1313324212yes—
564C2×C4C2×D4×D5311yescharacteristic
565C2×C4C2×Q8×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4C2×#1171994711yescharacteristic
568C2×C4#1313326411yescharacteristic
569C2×C4#1313326511yescharacteristic
570C8—1062no—
571C8—1062no—
572C8—1062no—
573C8—2121no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#1313318911yescharacteristic
582C22#1681128011yescharacteristic
583C22Q8×C22⋊D5311yescharacteristic
584C22C2×#1313324211yescharacteristic
585C22#1681142111yescharacteristic
586C22#1681143411yescharacteristic
587C22#1681143411yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—4241no—
598C4—4241no—
599C4—2122no—
600C4—2122no—
601C4#1681117511yescharacteristic
602C4#1681123311yescharacteristic
603C4#1681127511yescharacteristic
604C4#1681128611yescharacteristic
605C2—82no—
606C2#2027831611yescharacteristic
607C2#2028433611yescharacteristic
608C2#2028435111yescharacteristic
609C2#2028747711yescharacteristic
610C2#2028750311yescharacteristic
611C2#2028761411yescharacteristic
612C2#2028766011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2580C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2580D53
hypocenterC53#25485
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. C2×#16810754C2semidirect
3. #20279700C2semidirect
4. #20276217C2semidirect
5. #20276216C2semidirect
8. #20275687C2semidirect
9. #20275481C2semidirect
10. #20275450C2semidirect
11. #20275440C2semidirect
89. #13133138Q8semidirect
367. C53#25485semidirect

Maximal subgroups

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

subgroupindex
C2×#168109132
C2×#168107542
#202797002
#202762172
#202762162
#202761032
#202760892
#202756872
#202754812
#202754502
#202754402
#202754152
#202754012
#202745402
C53×#25802
#2548553