Finite Group Databasebeta

← #27414141

Subgroups — #27414141

Order 13568.

752 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#20277113C211yesmaximal, characteristic
2#20276283C211yesmaximal, characteristic, complemented
3#20276263C211yesmaximal, characteristic, complemented
4#20276196C211yesmaximal, characteristic, complemented
5#20276151C211yesmaximal, characteristic
6#20276141C211yesmaximal, characteristic
7#20276133C211yesmaximal, characteristic, complemented
8#20275915C211yesmaximal, characteristic
9#20275908C211yesmaximal, characteristic
10C4×#13133150C211yesmaximal, characteristic, complemented
11C4×#13133148C211yesmaximal, characteristic, complemented
12C4×#13133141C211yesmaximal, characteristic, complemented
13C4×#13133141C211yesmaximal, characteristic, complemented
14C4×C22⋊#88322C211yesmaximal, characteristic
15C4×C22⋊Q8×C53C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C4×#88322C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810491—21no—
29#16810489C2211yescharacteristic
30#16810489C2211yescharacteristic
31#16810486—21no—
32#16810483C2211yescharacteristic
33#16810483C2211yescharacteristic
34C4×C4⋊Dic53C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic
36C4×C22⋊Dic53—21no—
37#16810469—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C22⋊Q8×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C22×C42×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#387—21no—
54C53×#385—21no—
55C53×#380C2211yescharacteristic
56C4×C4⋊C4×C53—21no—
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53—21no—
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C4⋊Dic53C2311yescharacteristic
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C22⋊Dic53—21no—
68C2×C22⋊Dic53—21no—
69C2×C4×Dic53—21no—
70C2×C4×Dic53—21no—
71#13133150—22no—
72#13133150—22no—
73#13133148—22no—
74#13133148—22no—
75#13133141—22no—
76#13133141—22no—
77#13133141—22no—
78#13133141—22no—
79C22⋊#88322—22no—
80C22⋊#88322—22no—
81C4×#88322—22no—
82C4×#88322—22no—
83#13133124—21no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×Q8×C53C2311yescharacteristic
92C23×C4×C53D411yescharacteristic
93C23×C4×C53D411yescharacteristic
94C23×C4×C53C2311yescharacteristic
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97C4×Q8×C53—22no—
98C4×Q8×C53—22no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C42×C53—22no—
108C2×C42×C53—22no—
109C2×C42×C53—21no—
110C2×C42×C53—21no—
111C2×C42×C53D411yescharacteristic
112C2×C42×C53D411yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53#4112yes—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—41no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53#4112yes—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—42no—
164C42×C53—22no—
165C42×C53—22no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53—42no—
169C42×C53—42no—
170C42×C53#4112yes—
171C42×C53#4112yes—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
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—42no—
183C4×Dic53—42no—
184C2×#88322—21no—
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#4112yes—
195C2×#88322#4112yes—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53#13512yes—
203C23×C53C22×D411yescharacteristic
204Q8×C53—22no—
205Q8×C53—22no—
206Q8×C53—41no—
207Q8×C53—41no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—44no—
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—41no—
229C2×C4×C53—41no—
230C2×C4×C53—44no—
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#12312yes—
238C2×C4×C53#12312yes—
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×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×Dic53—41no—
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—22no—
260#88322—22no—
261#88322—41no—
262#88322—41no—
263#23759—531nomaximal, Sylow
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53C2×#12311yescharacteristic
272C22×C53#52511yescharacteristic
273C22×C53#53211yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53C2×#13511yescharacteristic
276C22×C53C2×#13711yescharacteristic
277C22×C53#57511yescharacteristic
278C4×C53—22no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53—42no—
282C4×C53—41no—
283C4×C53—42no—
284C4×C53—81no—
285C4×C53—81no—
286C4×C53#48514yes—
287C4×C53#52911yescharacteristic
288C4×C53#52911yescharacteristic
289C4×C53#52911yescharacteristic
290C4×C53#52911yescharacteristic
291Dic53—81no—
292Dic53—81no—
293Dic53—81no—
294Dic53—81no—
295C4×C22⋊Q8D5311yescharacteristic
296#2283—531no—
297#2101—531no—
298#2097—531no—
299#2085—531no—
300#2077—531no—
301#2075—531no—
302#2074—531no—
303#2036—531no—
304#2035—531no—
305C4×#107—531no—
306C4×#106—531no—
307C4×#103—531no—
308C4×#103—531no—
309C4×C22⋊C8—531no—
310C2×C53—42no—
311C2×C53#239511yescharacteristic
312C2×C53C2×#48511yescharacteristic
313C2×C53#309111yescharacteristic
314C2×C53#310211yescharacteristic
315C2×C53#310811yescharacteristic
316C2×C53#316811yescharacteristic
317C2×C53#317211yescharacteristic
318C2×C22⋊Q8C2×D5311yescharacteristic
319C2×C4×Q8C2×D5311yescharacteristic
320C22×C42C2×D5311yescharacteristic
321C2×#107—531no—
322C2×#107—531no—
323C2×#106—531no—
324C2×#106—531no—
325C2×#103—531no—
326C2×#103—531no—
327C2×#103—531no—
328C2×#103—531no—
329C2×C22⋊C8—531no—
330C2×C22⋊C8—531no—
331C2×C4×C8—531no—
332C2×C4×C8—531no—
333#391C2×D5311yescharacteristic
334#389C2×D5311yescharacteristic
335#387—21no—
336#385—21no—
337#383—1061no—
338#382—531no—
339#382—531no—
340#381—1061no—
341#380—531no—
342#380—531no—
343#380C2×D5311yescharacteristic
344C4×C4⋊C4—21no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4C2×D5311yescharacteristic
348C4×C22⋊C4—21no—
349C4×C22⋊C4—1061no—
350#376—1061no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#336—531no—
360#336—531no—
361C53#2375911yescharacteristic, complemented, Sylow
362C22×Q8C22×D5311yescharacteristic
363C23×C4C22⋊D5311yescharacteristic
364C23×C4C22⋊D5311yescharacteristic
365C23×C4C22×D5311yescharacteristic
366C22×C8—531no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—531no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C4×Q8—22no—
373C4×Q8—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—1061no—
393C2×C42—1061no—
394C2×C42C22⋊D5311yescharacteristic
395C2×C42C22⋊D5311yescharacteristic
396C2×C42C22×D5311yescharacteristic
397C2×C42C22×D5311yescharacteristic
398#107—1062no—
399#107—1062no—
400#106—1062no—
401#106—1062no—
402#103—1062no—
403#103—1062no—
404#103—1062no—
405#103—1062no—
406C22⋊C8—1062no—
407C22⋊C8—1062no—
408C4×C8—1062no—
409C4×C8—1062no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423C24C2×C22⋊D5311yescharacteristic
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1171994512yes—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
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—41no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C23×D5311yescharacteristic
456C2×C8—532no—
457C2×C8—532no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4#1171994512yes—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—42no—
495C42—2122no—
496C42—2122no—
497C42#1171994712yes—
498C42#1171994712yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#1313323712yes—
506C23C22×C22⋊D5311yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
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—44no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
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#1313319812yes—
553C2×C4#1313319812yes—
554C2×C4#1313323712yes—
555C2×C4#1313324812yes—
556C2×C4#41×D5311yescharacteristic
557C2×C4#41×D5311yescharacteristic
558C2×C4C2×#1171994711yescharacteristic
559C2×C4C2×#1171994711yescharacteristic
560C2×C4C2×#1171994511yescharacteristic
561C2×C4C2×#1171994511yescharacteristic
562C8—1062no—
563C8—1062no—
564C8—2121no—
565C8—2121no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#1313319811yescharacteristic
574C22#1681113411yescharacteristic
575C22#1681116911yescharacteristic
576C22C2×#1313323711yescharacteristic
577C22C2×#1313323711yescharacteristic
578C22C2×#1313324811yescharacteristic
579C22#1681142111yescharacteristic
580C4—22no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—4241no—
589C4—4241no—
590C4—4241no—
591C4—4241no—
592C4#1681096114yes—
593C4#1681115311yescharacteristic
594C4#1681115311yescharacteristic
595C4#1681115511yescharacteristic
596C4#1681115511yescharacteristic
597C2—42no—
598C2#2027797411yescharacteristic
599C2C2×#1681096111yescharacteristic
600C2#2028462411yescharacteristic
601C2#2028473311yescharacteristic
602C2#2028479511yescharacteristic
603C2#2028527911yescharacteristic
604C2#2028531511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC4×C22⋊Q8D53
hypocenterC53#23759
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20276283C2semidirect
3. #20276263C2semidirect
4. #20276196C2semidirect
7. #20276133C2semidirect
10. C4×#13133150C2semidirect
11. C4×#13133148C2semidirect
12. C4×#13133141C2semidirect
13. C4×#13133141C2semidirect
361. C53#23759semidirect

Maximal subgroups

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

subgroupindex
#202771132
#202762832
#202762632
#202761962
#202761512
#202761412
#202761332
#202759152
#202759082
C4×#131331502
C4×#131331482
C4×#131331412
C4×C22⋊#883222
C4×C22⋊Q8×C532
#2375953