Finite Group Databasebeta

← #28323631

Subgroups — #28323631

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290888C211yesmaximal, characteristic, complemented
2#20290885C211yesmaximal, characteristic, complemented
3#20290789C211yesmaximal, characteristic
4#20290789C211yesmaximal, characteristic
5#20290784C211yesmaximal, characteristic
6#20290784C211yesmaximal, characteristic
7#20290773C211yesmaximal, characteristic, complemented
8#20290773C211yesmaximal, characteristic, complemented
9#20290772C211yesmaximal, characteristic, complemented
10#20290769C211yesmaximal, characteristic, complemented
11#20290606C211yesmaximal, characteristic
12#20290514C211yesmaximal, characteristic, complemented
13#20290511C211yesmaximal, characteristic, complemented
14#20290441C211yesmaximal, characteristic
15#20277247C211yesmaximal, characteristic
16C2×C4⋊#88324C2211yescharacteristic
17C2×C4⋊#88324C2211yescharacteristic
18C2×C4⋊#88324C2211yescharacteristic
19C2×C4⋊#88324C2211yescharacteristic
20C2×C22⋊#88324C2211yescharacteristic
21C2×C22⋊#88324C2211yescharacteristic
22C2×C22⋊#88324C2211yescharacteristic
23C2×C22⋊#88324C2211yescharacteristic
24C2×#13133272C2211yescharacteristic
25C2×#13133272C2211yescharacteristic
26C2×#13133272C2211yescharacteristic
27C2×#13133272C2211yescharacteristic
28C2×C4×#88324C2211yescharacteristic
29C2×C4×#88324C2211yescharacteristic
30#16811489C2211yescharacteristic
31#16811489C2211yescharacteristic
32#16811489C2211yescharacteristic
33#16811489C2211yescharacteristic
34#16811489C2211yescharacteristic
35#16811489C2211yescharacteristic
36#16811489C2211yescharacteristic
37#16811489C2211yescharacteristic
38#16811489C2211yescharacteristic
39#16811489C2211yescharacteristic
40#16811489C2211yescharacteristic
41#16811489C2211yescharacteristic
42#16811489C2211yescharacteristic
43#16811489C2211yescharacteristic
44C2×#13133194C411yescharacteristic
45C2×#13133182C2211yescharacteristic
46C22×C4×Dic53C2211yescharacteristic
47#16810527C411yescharacteristic
48#16810518C411yescharacteristic
49#16810508C411yescharacteristic
50#16810504C411yescharacteristic
51#16810496C411yescharacteristic
52#16810481C411yescharacteristic
53C4⋊C4×Dic53C2211yescharacteristic
54#16810474C2211yescharacteristic
55#16810474C2211yescharacteristic
56C22⋊C4×Dic53C2211yescharacteristic
57C22⋊C4×Dic53C2211yescharacteristic
58C53×#385C411yescharacteristic
59C22×#88324C2311yescharacteristic
60C22×#88324C2311yescharacteristic
61C22×#88324C2311yescharacteristic
62C22×#88324C2311yescharacteristic
63C22×#88324C2311yescharacteristic
64C22×#88324C2311yescharacteristic
65C22×#88324C2311yescharacteristic
66C22×#88324C2311yescharacteristic
67C4⋊#88324—22no—
68C4⋊#88324—22no—
69C4⋊#88324—41no—
70C4⋊#88324—41no—
71C22⋊#88324—22no—
72C22⋊#88324—22no—
73C22⋊#88324—41no—
74C22⋊#88324—41no—
75#13133272—22no—
76#13133272—41no—
77#13133272—41no—
78#13133272—41no—
79C4×#88324—22no—
80C4×#88324—41no—
81C22×Dic106C2×C411yescharacteristic
82C23×Dic53C2×C411yescharacteristic
83C23×Dic53C2311yescharacteristic
84#13133194—24no—
85#13133182—41no—
86#13133182—41no—
87C2×C4⋊Dic53C2×C411yescharacteristic
88C2×#11719919C2×C411yescharacteristic
89C2×#11719919C2×C411yescharacteristic
90C2×#11719919C2×C411yescharacteristic
91C2×#11719919C2×C411yescharacteristic
92C2×#11719919C2×C411yescharacteristic
93C2×#11719919C2×C411yescharacteristic
94C2×C22⋊Dic53C2×C411yescharacteristic
95C2×C22⋊Dic53C2×C411yescharacteristic
96C2×C4×Dic53—41no—
97C2×C4×Dic53—41no—
98C2×C4×Dic53C2×C411yescharacteristic
99C2×C4×Dic53C2×C411yescharacteristic
100C2×C4×Dic53C2311yescharacteristic
101C2×C4×Dic53C2311yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103C2×C4×Dic53C2311yescharacteristic
104C2×C4×Dic53C2311yescharacteristic
105C2×C4×Dic53C2311yescharacteristic
106#13133124C2×C411yescharacteristic
107#13133124C2×C411yescharacteristic
108#13133124C2×C411yescharacteristic
109#13133124C2×C411yescharacteristic
110#13133124C2×C411yescharacteristic
111#13133124C2×C411yescharacteristic
112#13133124C2×C411yescharacteristic
113#13133124C2×C411yescharacteristic
114C23×C4×C53C2×C411yescharacteristic
115C2×C4⋊C4×C53C2×C411yescharacteristic
116C2×C4⋊C4×C53C2×C411yescharacteristic
117C2×C22⋊C4×C53C2×C411yescharacteristic
118C2×C22⋊C4×C53C2×C411yescharacteristic
119#95×C53C2×C411yescharacteristic
120#95×C53C2×C411yescharacteristic
121C24×C53C22×C411yescharacteristic
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C22×C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C22×C411yescharacteristic
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—41no—
140C22×Dic53C22×C411yescharacteristic
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C22×C411yescharacteristic
143C22×Dic53C22×C411yescharacteristic
144C22×Dic53C22×C411yescharacteristic
145C22×Dic53C22×C411yescharacteristic
146C22×Dic53C22×C411yescharacteristic
147C22×Dic53C2411yescharacteristic
148C2×Dic106—22no—
149C2×Dic106—22no—
150C2×Dic106—41no—
151C2×#88324—21no—
152C2×#88324—21no—
153C2×#88324—21no—
154C2×#88324—21no—
155C2×#88324—21no—
156C2×#88324—21no—
157C2×#88324—21no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324#4112yes—
174C2×#88324#4112yes—
175C4⋊C4×C53—41no—
176C4⋊C4×C53—41no—
177C22⋊C4×C53—41no—
178C22⋊C4×C53M1614yes—
179C22⋊Dic53—22no—
180C22⋊Dic53—41no—
181C4⋊Dic53—22no—
182#11719919—22no—
183#11719919—22no—
184#11719919—22no—
185#11719919—41no—
186#11719919—41no—
187#11719919—41no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—41no—
201C4×Dic53—41no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53C2×M1612yes—
209C23×C53C23×C411yescharacteristic
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
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—44no—
227C2×C4×C53C2×M1612yes—
228C2×C4×C53C2×M1612yes—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
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—44no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53#13112yes—
255C2×Dic53C2×#4111yescharacteristic
256C2×Dic53C2×#4111yescharacteristic
257C2×Dic53H(5,𝔽2)11yescharacteristic
258C2×Dic53H(5,𝔽2)11yescharacteristic
259C2×Dic532−1+411yescharacteristic
260C2×Dic532−1+411yescharacteristic
261Dic106—44no—
262#88324—22no—
263#88324—22no—
264#88324—41no—
265#88324—41no—
266#88324—41no—
267#88324—41no—
268#88324—41no—
269#88324—41no—
270#22323—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53C4×#4111yescharacteristic
279C22×C53#51811yescharacteristic
280C22×C53#51911yescharacteristic
281C22×C53C22×M1611yescharacteristic
282C22×C53C2×#13111yescharacteristic
283C22×C53#56811yescharacteristic
284C22×C53#56811yescharacteristic
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—41no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—41no—
291Dic53—22no—
292Dic53—41no—
293Dic53—41no—
294Dic53—41no—
295Dic53—41no—
296Dic53—41no—
297Dic53—41no—
298Dic53#53911yescharacteristic
299Dic53#53911yescharacteristic
300Dic53#54211yescharacteristic
301Dic53#54211yescharacteristic
302C4×C22⋊Q8—531no—
303#1985—531no—
304#1984—531no—
305#1924—531no—
306#1924—531no—
307#1923—531no—
308#1923—531no—
309#1917—531no—
310#1917—531no—
311#1917—531no—
312#1916—531no—
313#1817—531no—
314#1771—531no—
315#1770—531no—
316#1749—531no—
317C2×C53—42no—
318C2×C53#233211yescharacteristic
319C2×C53#296911yescharacteristic
320C2×C53#297011yescharacteristic
321C2×C53#297411yescharacteristic
322C2×C53#297811yescharacteristic
323C2×C53#298311yescharacteristic
324C2×C53#298511yescharacteristic
325C2×C22⋊Q8—531no—
326C2×C4×Q8—531no—
327C22×C42—531no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8—531no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—531no—
332C2×C22⋊C8—531no—
333C2×C22⋊C8—531no—
334C2×C22⋊C8—531no—
335C2×C22⋊C8—531no—
336C2×#97—531no—
337C2×#97—531no—
338C2×#97—531no—
339C2×#97—531no—
340C2×C4×C8—531no—
341C2×C4×C8—531no—
342#391—531no—
343#389—531no—
344#387—531no—
345#387—531no—
346#385—531no—
347#385#615711yescharacteristic
348#380—531no—
349C4×C4⋊C4—531no—
350C4×C4⋊C4—531no—
351C4×C4⋊C4—531no—
352C4×C22⋊C4—531no—
353C4×C22⋊C4—531no—
354#336—531no—
355#336—531no—
356#336—531no—
357#336—531no—
358#336—531no—
359#336—531no—
360#336—531no—
361#336—531no—
362#336—531no—
363#336—531no—
364#336—531no—
365#336—531no—
366#336—531no—
367#336—531no—
368C53#2232311yescharacteristic, complemented, Sylow
369C22×Q8—531no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4C2×#615711yescharacteristic
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—531no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8—531no—
381C22⋊Q8—1064no—
382C4×Q8—2121no—
383C4×Q8—2121no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4C2×#615711yescharacteristic
392C2×C4⋊C4C2×#615711yescharacteristic
393C2×C22⋊C4—531no—
394C2×C22⋊C4—531no—
395C2×C22⋊C4C2×#615711yescharacteristic
396C2×C22⋊C4C2×#615711yescharacteristic
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—2121no—
406C2×C42—2121no—
407C4⋊C8—1062no—
408C4⋊C8—1062no—
409C4⋊C8—2121no—
410C4⋊C8—2121no—
411C22⋊C8—1062no—
412C22⋊C8—1062no—
413C22⋊C8—2121no—
414C22⋊C8—2121no—
415#97—1062no—
416#97—2121no—
417#97—2121no—
418#97—2121no—
419C4×C8—1062no—
420C4×C8—2121no—
421#95—531no—
422#95—531no—
423#95—531no—
424#95—531no—
425#95—531no—
426#95—531no—
427#95—531no—
428#95—531no—
429#95C2×#615711yescharacteristic
430#95C2×#615711yescharacteristic
431C24C22×#615711yescharacteristic
432C2×Q8—1062no—
433C2×Q8—1062no—
434C2×Q8—2121no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—2121no—
455C22×C4C22×#615711yescharacteristic
456C22×C4C22×#615711yescharacteristic
457C22×C4C22×#615711yescharacteristic
458C22×C4C22×#615711yescharacteristic
459C22×C4C22×#615711yescharacteristic
460C22×C4C22×#615711yescharacteristic
461C2×C8—532no—
462C2×C8—532no—
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—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—1062no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—2121no—
492C4⋊C4—2121no—
493C4⋊C4—2121no—
494C22⋊C4—41no—
495C22⋊C4—1062no—
496C22⋊C4—2121no—
497C22⋊C4#1171993614yes—
498C42—1061no—
499C42—1061no—
500C42—1061no—
501C42—1061no—
502C42—1061no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C42—1061no—
507C42—1061no—
508C42—1061no—
509C42—1061no—
510C42—2121no—
511C42—2121no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C2×#1171993612yes—
519C23C23×#615711yescharacteristic
520Q8—2124no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
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—44no—
538C2×C4—531no—
539C2×C4—531no—
540C2×C4—531no—
541C2×C4—531no—
542C2×C4—531no—
543C2×C4—531no—
544C2×C4—532no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2124no—
570C2×C4C2×#1171993612yes—
571C2×C4C2×#1171993612yes—
572C8—1062no—
573C8—1062no—
574C8—2121no—
575C8—2121no—
576C8—2121no—
577C8—2121no—
578C8—2121no—
579C8—2121no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22#41×#615711yescharacteristic
588C22#1681169011yescharacteristic
589C22#1681169111yescharacteristic
590C22C22×#1171993611yescharacteristic
591C22D53⋊#13111yescharacteristic
592C22#1681170111yescharacteristic
593C22#1681170111yescharacteristic
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—531no—
601C4—531no—
602C4—531no—
603C4—531no—
604C4—1062no—
605C4—2121no—
606C4—2121no—
607C4—2121no—
608C4—2121no—
609C4—2121no—
610C4—2121no—
611C2—42no—
612C2#2029121111yescharacteristic
613C2#2029154911yescharacteristic
614C2#2029155311yescharacteristic
615C2#2029157511yescharacteristic
616C2#2029159111yescharacteristic
617C2#2029161311yescharacteristic
618C2#2029162111yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC53×#385C4
prime radicalC24×C53C22×C4
cosocleC22×Dic53C24
hypercenter#385#6157
hypocenterC53#22323
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #20290888C2semidirect
2. #20290885C2semidirect
7. #20290773C2semidirect
8. #20290773C2semidirect
9. #20290772C2semidirect
10. #20290769C2semidirect
12. #20290514C2semidirect
13. #20290511C2semidirect
368. C53#22323semidirect

Maximal subgroups

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

subgroupindex
#202908882
#202908852
#202907892
#202907842
#202907732
#202907722
#202907692
#202906062
#202905142
#202905112
#202904412
#202772472
#2232353