Finite Group Databasebeta

← #27442971

Subgroups — #27442971

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811031C211yesmaximal, characteristic, complemented
2C2×#16810663C211yesmaximal, characteristic
3#20278433C211yesmaximal, characteristic
4#20278407C211yesmaximal, characteristic
5#20276201C211yesmaximal, characteristic, complemented
6#20276199C211yesmaximal, characteristic, complemented
7#20276056C211yesmaximal, characteristic, complemented
8#20276041C211yesmaximal, characteristic, complemented
9#20275452C211yesmaximal, characteristic, complemented
10#20275447C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic
12#20274878C211yesmaximal, characteristic, complemented
13#20274746C211yesmaximal, characteristic
14#20274501C211yesmaximal, characteristic
15C53×#1852C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133208C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20#16811031—24no—
21#16810663—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32#16810562—21no—
33#16810556—21no—
34#16810546—21no—
35#16810535—21no—
36#16810527—21no—
37#16810522C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810519C2211yescharacteristic
41#16810506—21no—
42#16810496—21no—
43#16810488C2211yescharacteristic
44#16810488C2211yescharacteristic
45#16810483—21no—
46#16810483C2211yescharacteristic
47C4×C22⋊Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×C4⋊C4×C53C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×#107×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106—21no—
62C2×#11719917—22no—
63C2×#11719917—22no—
64C2×#11719917Q814yescomplemented
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133216—22no—
68#13133216—41no—
69#13133208—22no—
70#13133208—41no—
71#13133176—22no—
72#13133176—41no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53D411yescharacteristic
87C2×C22⋊Dic53D411yescharacteristic
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133151—41no—
92#13133150—22no—
93#13133150—22no—
94#13133150D414yes—
95#13133149—22no—
96#13133148—22no—
97#13133148—22no—
98#13133148—22no—
99#13133127—22no—
100C8×Dic53—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124D411yescharacteristic
111#13133124D411yescharacteristic
112C23×C4×C53—21no—
113C23×C4×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—22no—
118C2×C4⋊C4×C53—41no—
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C22⋊C4×C53—21no—
124#107×C53—22no—
125#107×C53—41no—
126C22⋊C8×C53Q814yescomplemented
127#95×C53—21no—
128C24×C53C2×D411yescharacteristic
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53—81no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×Q812yes—
142C22×C4×C53C2×Q812yes—
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C2×Dic106—42no—
153C2×Dic106—42no—
154C2×Dic106—81no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53C2×Q812yes—
160C2×C8×C53C2×Q812yes—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53C2×D412yes—
170C22⋊C4×C53—81no—
171C22⋊Dic53—41no—
172C22⋊Dic53—41no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53—42no—
184C4⋊Dic53—81no—
185C4⋊Dic53C2×D412yes—
186#11719919—42no—
187#11719919—42no—
188#11719919—81no—
189#11719919—81no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—42no—
195#11719917—28no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322C2×D412yes—
199C2×#88322C2×Q812yes—
200C2×#88322C2×Q812yes—
201C2×#88322#4112yes—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53C4⋊Q812yes—
209C23×C53C22×D411yescharacteristic
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—84no—
229C2×C4×C53C4⋊Q812yes—
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×Q811yescharacteristic
232C2×C4×C53C22×Q811yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53H(5,𝔽2)11yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×Dic53—21no—
237C2×Dic53—22no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260Dic106—84no—
261C8×C53—22no—
262C8×C53—22no—
263#88322—22no—
264#88322C22⋊Q814yes—
265#26079—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53C2×C4⋊Q811yescharacteristic
274C22×C53D4211yescharacteristic
275C22×C53#54711yescharacteristic
276C22×C53#57511yescharacteristic
277C22×C53#57511yescharacteristic
278C22×C53#57611yescharacteristic
279C22×C53#57811yescharacteristic
280C4×C53—22no—
281C4×C53—42no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53C2×C22⋊Q811yescharacteristic
286C4×C53#54411yescharacteristic
287C4×C53D4×Q811yescharacteristic
288C4×C53#55411yescharacteristic
289Dic53—41no—
290Dic53—41no—
291Dic53—41no—
292Dic53—42no—
293Dic53—41no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297C2×#501—531no—
298C2×#428—531no—
299#2442—531no—
300#2439—531no—
301#2085—531no—
302#2085—531no—
303#2059—531no—
304#2057—531no—
305#1943—531no—
306#1942—531no—
307#1854—531no—
308#1852D5311yescharacteristic
309#1846—531no—
310#1809—531no—
311#1753—531no—
312C2×C53—42no—
313C2×C53#244011yescharacteristic
314C2×C53#316411yescharacteristic
315C2×C53#316611yescharacteristic
316C2×C53#327311yescharacteristic
317C2×C53#327711yescharacteristic
318C2×C53#334711yescharacteristic
319C2×C53#334911yescharacteristic
320C22×M16—531no—
321C2×C4⋊Q8—531no—
322C2×#125—531no—
323C2×#117—531no—
324C22×C4⋊C4C2×D5311yescharacteristic
325#501—1064no—
326#428—1064no—
327C2×#107—531no—
328C2×#107—531no—
329C2×#107—531no—
330C2×#107—531no—
331C2×#107C2×D5311yescharacteristic
332C2×#107C2×D5311yescharacteristic
333C2×#106—531no—
334C2×#106—531no—
335C2×#106—531no—
336C2×#106—531no—
337C2×C22⋊C8C2×D5311yescharacteristic
338C2×#97—531no—
339C2×C4×C8—531no—
340#400—1061no—
341#398—1061no—
342#396—1061no—
343#393—1061no—
344#391—1061no—
345#389—531no—
346#389—531no—
347#389—531no—
348#389—531no—
349#387—1061no—
350#385—1061no—
351#382—531no—
352#382—531no—
353#380—531no—
354#380—1061no—
355#380C2×D5311yescharacteristic
356C4×C22⋊C4—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340C2×D5311yescharacteristic
362#340C2×D5311yescharacteristic
363#336—531no—
364#336—531no—
365C53#2607911yescharacteristic, complemented, Sylow
366C22×Q8—1061no—
367C23×C4—21no—
368C23×C4C22×D5311yescharacteristic
369C2×M16—534no—
370C2×M16—1062no—
371C2×M16—1062no—
372C22×C8—531no—
373C22×C8—531no—
374C22×C8C22×D5311yescharacteristic
375C22×C8C22×D5311yescharacteristic
376C4⋊Q8—1062no—
377C4⋊Q8—2121no—
378#125—1062no—
379#125—2121no—
380#117—1062no—
381#117—2121no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—41no—
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⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—531no—
402C2×C22⋊C4—531no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—1061no—
408#107—22no—
409#107—41no—
410#107—534no—
411#107—1062no—
412#107—1062no—
413#107—2121no—
414#106—1062no—
415#106—1062no—
416#106—1062no—
417#106—1062no—
418C22⋊C8Dic10614yescomplemented
419#97—1062no—
420C4×C8—1062no—
421#95—21no—
422#95—531no—
423#95—531no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433C24D4×D5311yescharacteristic
434C2×Q8—4241no—
435C2×Q8—2122no—
436C2×Q8—2122no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4C2×Dic10612yes—
457C22×C4D4×D5311yescharacteristic
458C22×C4Q8×D5312yes—
459C22×C4C23×D5311yescharacteristic
460M16—1068no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—532no—
466C2×C8—532no—
467C2×C8—532no—
468C2×C8—532no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8C2×Dic10612yes—
472C2×C8C2×Dic10612yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—532no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—4241no—
490C4⋊C4—4241no—
491C4⋊C4—4241no—
492C4⋊C4—2122no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4D4×D5312yes—
497C22⋊C4—81no—
498C22⋊C4—2121no—
499C22⋊C4—2121no—
500C22⋊C4—2122no—
501C22⋊C4—2122no—
502C42—1061no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C42—2122no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1313321612yes—
514C23C2×D4×D5311yescharacteristic
515Q8—4244no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1062no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
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#1313321612yes—
560C2×C4C2×D4×D5311yescharacteristic
561C2×C4C2×Q8×D5311yescharacteristic
562C2×C4C22×Dic10611yescharacteristic
563C2×C4#41×D5311yescharacteristic
564C2×C4#1313326211yescharacteristic
565C2×C4#1313326411yescharacteristic
566C8—22no—
567C8—22no—
568C8—534no—
569C8—1062no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1313321611yescharacteristic
578C22#1681125511yescharacteristic
579C22#1681127111yescharacteristic
580C22#1681141911yescharacteristic
581C22#1681142211yescharacteristic
582C22#1681142411yescharacteristic
583C22#1681143611yescharacteristic
584C4—22no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—4241no—
590C4—4241no—
591C4—4241no—
592C4—2121no—
593C4—2121no—
594C4—2122no—
595C4—2121no—
596C4—2121no—
597C4C22⋊Q8×D5311yescharacteristic
598C4#1681125211yescharacteristic
599C4D4×Dic10611yescharacteristic
600C4#1681133211yescharacteristic
601C2—42no—
602C2#2027841611yescharacteristic
603C2#2028525111yescharacteristic
604C2#2028526511yescharacteristic
605C2#2028632611yescharacteristic
606C2#2028637211yescharacteristic
607C2#2028721811yescharacteristic
608C2#2028723811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1852D53
hypocenterC53#26079
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C2×#16811031C2semidirect
5. #20276201C2semidirect
6. #20276199C2semidirect
7. #20276056C2semidirect
8. #20276041C2semidirect
9. #20275452C2semidirect
10. #20275447C2semidirect
12. #20274878C2semidirect
64. C2×#11719917Q8semidirect
126. C22⋊C8×C53Q8semidirect
365. C53#26079semidirect
418. C22⋊C8Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168110312
C2×#168106632
#202784332
#202784072
#202762012
#202761992
#202760562
#202760412
#202754522
#202754472
#202749162
#202748782
#202747462
#202745012
C53×#18522
#2607953