Finite Group Databasebeta

← #27484282

Subgroups — #27484282

Order 13568.

720 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
1#20280545C211yesmaximal, characteristic
2#20276273C211yesmaximal, characteristic, complemented
3#20276253C211yesmaximal, characteristic, complemented
4#20276053C211yesmaximal, characteristic, complemented
5#20275866C211yesmaximal, characteristic, complemented
6#20275307C211yesmaximal, characteristic, complemented
7#20275286C211yesmaximal, characteristic, complemented
8#20275149C211yesmaximal, characteristic
9#20275139C211yesmaximal, characteristic
10#20274983C211yesmaximal, characteristic, complemented
11#20274969C211yesmaximal, characteristic, complemented
12#20274746C211yesmaximal, characteristic
13#20274742C211yesmaximal, characteristic
14#20274412C211yesmaximal, characteristic
15C53×#2323C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133209C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27#16810556—21no—
28#16810556—21no—
29#16810546—21no—
30#16810546—21no—
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810493C2211yescharacteristic
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C22⋊Q8×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×#117×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389—21no—
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#388—21no—
56C53×#387—21no—
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C2×#11719917—22no—
60C2×#11719917—22no—
61C2×#11719917—41no—
62C22×#88322—22no—
63C22×#88322—22no—
64C22×#88322—41no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133209—41no—
70#13133209—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83#13133150—22no—
84#13133150—22no—
85#13133150—41no—
86#13133148—41no—
87#13133141—22no—
88#13133141—22no—
89#13133141—41no—
90#13133141—41no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C22×Q8×C53C2311yescharacteristic
96C23×C4×C53C2311yescharacteristic
97C22⋊Q8×C53—22no—
98C22⋊Q8×C53Q814yes—
99C4×Q8×C53—41no—
100C4×Q8×C53—41no—
101#117×C53—41no—
102#117×C53—41no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124C24×C53C2×D411yescharacteristic
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53C2×Q812yes—
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—21no—
138C22×C4×C53—21no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×Q812yes—
145C22×C4×C53#4112yes—
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53—41no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—81no—
161C4⋊C4×C53C2×Q812yes—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—41no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53—81no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—42no—
171C42×C53—81no—
172C22⋊Dic53—81no—
173C22⋊Dic53—81no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—81no—
183C4⋊Dic53—81no—
184#11719919—42no—
185#11719919—42no—
186#11719919—81no—
187#11719919—81no—
188#11719917—44no—
189C2×#88322—21no—
190C2×#88322—21no—
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—44no—
200C2×#88322#4112yes—
201C2×#88322#4112yes—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53C22⋊Q812yes—
209C23×C53C22×D411yescharacteristic
210Q8×C53—22no—
211Q8×C53—22no—
212Q8×C53—41no—
213Q8×C53—41no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
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—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
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—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53C22⋊Q812yes—
245C2×C4×C53C22×Q811yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×C4×C53H(5,𝔽2)11yescharacteristic
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—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263#88322—22no—
264#88322—22no—
265#88322—41no—
266#88322—41no—
267#29173—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53C2×C22⋊Q811yescharacteristic
276C22×C53#53211yescharacteristic
277C22×C53#53411yescharacteristic
278C22×C53#57611yescharacteristic
279C22×C53#57711yescharacteristic
280C22×C53#57811yescharacteristic
281C22×C53#57811yescharacteristic
282C4×C53—22no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53—41no—
288C4×C53—41no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53#55111yescharacteristic
292C4×C53#55111yescharacteristic
293C4×C53#55411yescharacteristic
294C4×C53#55411yescharacteristic
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299#2629—531no—
300#2323D5311yescharacteristic
301#2099—531no—
302#2095—531no—
303#2059—531no—
304#2026—531no—
305#1921—531no—
306#1918—531no—
307#1894—531no—
308#1892—531no—
309#1865—531no—
310#1863—531no—
311#1809—531no—
312#1808—531no—
313#1729—531no—
314C2×C53—42no—
315C2×C53#256411yescharacteristic
316C2×C53#308211yescharacteristic
317C2×C53#308311yescharacteristic
318C2×C53#317711yescharacteristic
319C2×C53#317911yescharacteristic
320C2×C53#320111yescharacteristic
321C2×C53#320311yescharacteristic
322C22×M16—531no—
323C23×C8—531no—
324C2×#125—531no—
325C2×C22⋊Q8C2×D5311yescharacteristic
326C2×C4×Q8C2×D5311yescharacteristic
327C2×#117C2×D5311yescharacteristic
328C2×#107—531no—
329C2×#107—531no—
330C2×#107—531no—
331C2×#106—531no—
332C2×#103—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336#398—1061no—
337#398—1061no—
338#396—1061no—
339#396—1061no—
340#391C2×D5311yescharacteristic
341#389—21no—
342#389—531no—
343#389—531no—
344#389C2×D5311yescharacteristic
345#388—21no—
346#388—21no—
347#387—21no—
348#384—531no—
349#380—531no—
350#380—531no—
351#380C2×D5311yescharacteristic
352C4×C4⋊C4C2×D5311yescharacteristic
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#336—531no—
362#336—531no—
363#336—531no—
364#336—531no—
365C53#2917311yescharacteristic, complemented, Sylow
366C22×Q8C22×D5311yescharacteristic
367C23×C4C22×D5311yescharacteristic
368C2×M16—1062no—
369C2×M16—1062no—
370C2×M16—2121no—
371C22×C8—531no—
372C22×C8—531no—
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—1062no—
376C22×C8—1062no—
377C22×C8—2121no—
378#125—2121no—
379#125—2121no—
380C22⋊Q8—22no—
381C22⋊Q8Dic10614yes—
382C4×Q8—41no—
383C4×Q8—41no—
384#117—41no—
385#117—41no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4C22⋊D5311yescharacteristic
400C2×C4⋊C4C22⋊D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4C22⋊D5311yescharacteristic
409C2×C22⋊C4C22⋊D5311yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42C22×D5311yescharacteristic
413C2×C42C22×D5311yescharacteristic
414#107—1062no—
415#107—1062no—
416#107—2121no—
417#106—2121no—
418#103—1062no—
419#103—1062no—
420#103—2121no—
421#103—2121no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431C24C2×C22⋊D5311yescharacteristic
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8C2×Dic10612yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4C2×Dic10612yes—
451C22×C4#1171994512yes—
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C23×D5311yescharacteristic
458M16—2124no—
459C2×C8—532no—
460C2×C8—532no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—2124no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—4241no—
491C4⋊C4—4241no—
492C4⋊C4—4241no—
493C4⋊C4—4241no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4C2×Dic10612yes—
497C22⋊C4—24no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—81no—
501C22⋊C4—4241no—
502C22⋊C4—4241no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—81no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23#1313319612yes—
516C23C22×C22⋊D5311yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
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—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
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—2121no—
562C2×C4—2121no—
563C2×C4#1313319612yes—
564C2×C4C22×Dic10611yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4#1313326211yescharacteristic
569C2×C4#1313326211yescharacteristic
570C8—1062no—
571C8—1062no—
572C8—2121no—
573C8—2121no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1313319611yescharacteristic
582C22#1681116911yescharacteristic
583C22#1681118211yescharacteristic
584C22#1681142611yescharacteristic
585C22#1681143111yescharacteristic
586C22#1681143811yescharacteristic
587C22#1681143811yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—4241no—
598C4—4241no—
599C4—4241no—
600C4—4241no—
601C4#1681130211yescharacteristic
602C4#1681130211yescharacteristic
603C4#1681133211yescharacteristic
604C4#1681133211yescharacteristic
605C2—42no—
606C2#2027989111yescharacteristic
607C2#2028455711yescharacteristic
608C2#2028456611yescharacteristic
609C2#2028536811yescharacteristic
610C2#2028539211yescharacteristic
611C2#2028559711yescharacteristic
612C2#2028561211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2323C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2323D53
hypocenterC53#29173
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. #20276273C2semidirect
3. #20276253C2semidirect
4. #20276053C2semidirect
5. #20275866C2semidirect
6. #20275307C2semidirect
7. #20275286C2semidirect
10. #20274983C2semidirect
11. #20274969C2semidirect
365. C53#29173semidirect

Maximal subgroups

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

subgroupindex
#202805452
#202762732
#202762532
#202760532
#202758662
#202753072
#202752862
#202751492
#202751392
#202749832
#202749692
#202747462
#202747422
#202744122
C53×#23232
#2917353