Finite Group Databasebeta

← #27452762

Subgroups — #27452762

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279242C211yesmaximal, characteristic, complemented
2#20276118C211yesmaximal, characteristic
3#20276103C211yesmaximal, characteristic
4#20275951C211yesmaximal, characteristic
5#20275951C211yesmaximal, characteristic
6#20275394C211yesmaximal, characteristic, complemented
7#20275353C211yesmaximal, characteristic, complemented
8#20275293C211yesmaximal, characteristic, complemented
9#20275286C211yesmaximal, characteristic, complemented
10#20275274C211yesmaximal, characteristic, complemented
11#20274983C211yesmaximal, characteristic, complemented
12#20274976C211yesmaximal, characteristic
13#20274720C211yesmaximal, characteristic, complemented
14C4×#13133138C211yesmaximal, characteristic
15C53×#2367C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133148C2211yescharacteristic
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133141C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810557—21no—
29#16810556—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810520—21no—
35#16810509—21no—
36#16810493C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#124×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#388—21no—
54C53×#388—21no—
55C53×#385—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53—21no—
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×C4×Dic53—21no—
74#13133150—22no—
75#13133148—22no—
76C4⋊#88322—41no—
77#13133141—22no—
78#13133141—22no—
79#13133141—22no—
80#13133141—41no—
81#13133138—22no—
82#13133138—22no—
83#13133138—22no—
84#13133138—41no—
85C4×#88322—41no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×Q8×C53C2311yescharacteristic
92C22×D4×C53C2311yescharacteristic
93C23×C4×C53—21no—
94#124×C53—22no—
95#124×C53—22no—
96C4×Q8×C53—22no—
97C4×Q8×C53—41no—
98C4×D4×C53—22no—
99C4×D4×C53—41no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C42×C53—21no—
111C2×C42×C53—21no—
112C2×C42×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53D411yescharacteristic
119#95×C53D411yescharacteristic
120C24×C53—21no—
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53#4112yes—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53#4112yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53—81no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—42no—
156C4⋊C4×C53#4112yes—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—81no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53—42no—
169C42×C53—81no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—81no—
178C4⋊Dic53—81no—
179C4⋊Dic53#4112yes—
180#11719919—42no—
181#11719919—81no—
182#11719919—81no—
183#11719919—81no—
184C4×Dic53—81no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C23×C53—22no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53C22×D411yescharacteristic
205Q8×C53—22no—
206Q8×C53—22no—
207Q8×C53—41no—
208Q8×C53—41no—
209D4×C53—22no—
210D4×C53—22no—
211D4×C53—41no—
212D4×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
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—84no—
239C2×C4×C53#13512yes—
240C2×C4×C53#13712yes—
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×Dic53—22no—
248C2×Dic53—41no—
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—
259#88322—41no—
260#88322—41no—
261#88322—41no—
262#88322—41no—
263#26763—531nomaximal, Sylow
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53#52511yescharacteristic
272C22×C53#54611yescharacteristic
273C22×C53#54811yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53C2×#13711yescharacteristic
276C22×C53#57611yescharacteristic
277C22×C53#57711yescharacteristic
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—42no—
283C4×C53—41no—
284C4×C53—41no—
285C4×C53—42no—
286C4×C53—81no—
287C4×C53#52911yescharacteristic
288C4×C53#53911yescharacteristic
289C4×C53#55211yescharacteristic
290C4×C53#55311yescharacteristic
291Dic53—42no—
292Dic53—81no—
293Dic53—81no—
294Dic53—81no—
295#2507—531no—
296#2367D5311yescharacteristic
297#2071—531no—
298#2067—531no—
299#2042—531no—
300#2042—531no—
301#1933—531no—
302#1928—531no—
303#1919—531no—
304#1918—531no—
305#1916—531no—
306#1865—531no—
307#1864—531no—
308#1803—531no—
309C4×#102—531no—
310C2×C53—82no—
311C2×C53#246011yescharacteristic
312C2×C53#310711yescharacteristic
313C2×C53#311911yescharacteristic
314C2×C53#330111yescharacteristic
315C2×C53#331411yescharacteristic
316C2×C53#336111yescharacteristic
317C2×C53#336911yescharacteristic
318C2×#124C2×D5311yescharacteristic
319C2×C4×Q8C2×D5311yescharacteristic
320C2×C4×D4C2×D5311yescharacteristic
321C2×#107—531no—
322C2×#106—531no—
323C2×C4⋊C8—531no—
324C2×#103—531no—
325C2×#103—531no—
326C2×#103—531no—
327C2×#103—531no—
328C2×#102—531no—
329C2×#102—531no—
330C2×#102—531no—
331C2×#102—531no—
332C2×C4×C8—531no—
333#398—1061no—
334#398—1061no—
335#391C2×D5311yescharacteristic
336#390C2×D5311yescharacteristic
337#389—531no—
338#389—531no—
339#389—531no—
340#389—531no—
341#389—1061no—
342#388—21no—
343#388—21no—
344#387—1061no—
345#385—21no—
346#384—531no—
347C4×C4⋊C4—531no—
348C4×C4⋊C4C2×D5311yescharacteristic
349C4×C4⋊C4C2×D5311yescharacteristic
350C4×C22⋊C4—21no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#336—531no—
358#336—531no—
359#336—531no—
360#336—531no—
361C53#2676311yescharacteristic, complemented, Sylow
362C22×Q8C22×D5311yescharacteristic
363C22×D4C22×D5311yescharacteristic
364C23×C4—21no—
365C22×C8—531no—
366C22×C8—531no—
367C22×C8—531no—
368C22×C8—531no—
369#124—22no—
370#124—22no—
371C4×Q8—22no—
372C4×Q8—41no—
373C4×D4—22no—
374C4×D4—41no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4C22⋊D5311yescharacteristic
387C2×C4⋊C4C22⋊D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—1061no—
398C2×C42C22×D5311yescharacteristic
399C2×C42C22×D5311yescharacteristic
400C2×C42C22×D5311yescharacteristic
401#107—1062no—
402#106—1062no—
403C4⋊C8—2121no—
404#103—1062no—
405#103—1062no—
406#103—1062no—
407#103—2121no—
408#102—1062no—
409#102—1062no—
410#102—1062no—
411#102—2121no—
412C4×C8—2121no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95C22⋊D5311yescharacteristic
422#95C22⋊D5311yescharacteristic
423C24—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1171994512yes—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4#1171994512yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C23×D5311yescharacteristic
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—532no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—4241no—
480C4⋊C4—4241no—
481C4⋊C4—4241no—
482C4⋊C4—4241no—
483C4⋊C4—4241no—
484C4⋊C4—2122no—
485C4⋊C4#1171994712yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—81no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—81no—
499C42—4241no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C22×C22⋊D5311yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
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—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—1062no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4#1313323712yes—
555C2×C4#1313324712yes—
556C2×C4#41×D5311yescharacteristic
557C2×C4C2×#1171994711yescharacteristic
558C2×C4C2×#1171994511yescharacteristic
559C2×C4C2×#1171994511yescharacteristic
560C2×C4#1313326211yescharacteristic
561C2×C4#1313326411yescharacteristic
562C8—2121no—
563C8—2121no—
564C8—2121no—
565C8—2121no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22#1681113411yescharacteristic
574C22#1681126711yescharacteristic
575C22#1681128011yescharacteristic
576C22C2×#1313323711yescharacteristic
577C22C2×#1313324711yescharacteristic
578C22#1681142611yescharacteristic
579C22#1681143111yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—4241no—
590C4—4241no—
591C4—4241no—
592C4—2122no—
593C4#1681115311yescharacteristic
594C4#1681121511yescharacteristic
595C4#1681131311yescharacteristic
596C4#1681132211yescharacteristic
597C2—82no—
598C2#2027866111yescharacteristic
599C2#2028478611yescharacteristic
600C2#2028491411yescharacteristic
601C2#2028661711yescharacteristic
602C2#2028679611yescharacteristic
603C2#2028737111yescharacteristic
604C2#2028745111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2367C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2367D53
hypocenterC53#26763
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #20279242C2semidirect
6. #20275394C2semidirect
7. #20275353C2semidirect
8. #20275293C2semidirect
9. #20275286C2semidirect
10. #20275274C2semidirect
11. #20274983C2semidirect
13. #20274720C2semidirect
361. C53#26763semidirect

Maximal subgroups

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

subgroupindex
#202792422
#202761182
#202761032
#202759512
#202753942
#202753532
#202752932
#202752862
#202752742
#202749832
#202749762
#202747202
C4×#131331382
C53×#23672
#2676353