Finite Group Databasebeta

← #27466377

Subgroups — #27466377

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278428C211yesmaximal, characteristic
2#20276282C211yesmaximal, characteristic, complemented
3#20276272C211yesmaximal, characteristic, complemented
4#20276123C211yesmaximal, characteristic, complemented
5#20275866C211yesmaximal, characteristic, complemented
6#20275679C211yesmaximal, characteristic, complemented
7#20275394C211yesmaximal, characteristic, complemented
8#20275394C211yesmaximal, characteristic, complemented
9#20275139C211yesmaximal, characteristic
10#20275028C211yesmaximal, characteristic
11#20274842C211yesmaximal, characteristic, complemented
12#20274810C211yesmaximal, characteristic
13C22⋊C4×#88322C211yesmaximal, characteristic
14#20274431C211yesmaximal, characteristic
15C53×#2499C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133141C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C4×#88322C2211yescharacteristic
25C2×C4×#88322C2211yescharacteristic
26#16810559—21no—
27#16810546—21no—
28#16810521C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810520—21no—
31#16810506—21no—
32#16810493C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
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×#117×C53C2211yescharacteristic
50C53×#397—21no—
51C53×#397—21no—
52C53×#391C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#387—21no—
55C53×#387—21no—
56C53×#384C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×#88322—22no—
60C22×#88322—22no—
61C22×#88322—41no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133209—22no—
67#13133209—41no—
68#13133176—22no—
69#13133176—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83#13133141—22no—
84#13133141—22no—
85#13133141—41no—
86#13133141—41no—
87C22⋊#88322—22no—
88C4×#88322—22no—
89C4×#88322—41no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93C22×Q8×C53C2311yescharacteristic
94C23×C4×C53C2311yescharacteristic
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97#117×C53—22no—
98#117×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×C22⋊C4×C53D411yescharacteristic
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C42×C53—21no—
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53D411yescharacteristic
119#95×C53D411yescharacteristic
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—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
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×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53#4112yes—
141C22×C4×C53#4112yes—
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—42no—
154C4⋊C4×C53—42no—
155C4⋊C4×C53—42no—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—42no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—42no—
165C22⋊Dic53—42no—
166C22⋊Dic53—81no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—42no—
175C4⋊Dic53—81no—
176C4⋊Dic53#4112yes—
177#11719919—42no—
178#11719919—42no—
179#11719919—42no—
180#11719919—81no—
181C4×Dic53—42no—
182C2×#88322—21no—
183C2×#88322—21no—
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—44no—
194C2×#88322#4112yes—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53#12412yes—
202C23×C53C22×D411yescharacteristic
203Q8×C53—41no—
204Q8×C53—41no—
205Q8×C53—41no—
206Q8×C53—41no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
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—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—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53#12412yes—
236C2×C4×C53#13512yes—
237C2×C4×C53#13512yes—
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53—41no—
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—
256#88322—22no—
257#88322—41no—
258#88322—41no—
259#88322—41no—
260#27700—531nomaximal, Sylow
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53C2×#12411yescharacteristic
269C22×C53#54611yescharacteristic
270C22×C53#54711yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53C2×#13511yescharacteristic
273C22×C53#57711yescharacteristic
274C22×C53#57811yescharacteristic
275C4×C53—22no—
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53#52911yescharacteristic
285C4×C53#53911yescharacteristic
286C4×C53#55111yescharacteristic
287C4×C53#55211yescharacteristic
288Dic53—42no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2499D5311yescharacteristic
293#2441—531no—
294#2101—531no—
295#2099—531no—
296#2072—531no—
297#2026—531no—
298#1991—531no—
299#1933—531no—
300#1933—531no—
301#1892—531no—
302#1873—531no—
303#1834—531no—
304#1825—531no—
305C8×C22⋊C4—531no—
306#1735—531no—
307C2×C53—42no—
308C2×C53#248811yescharacteristic
309C2×C53#313111yescharacteristic
310C2×C53#313711yescharacteristic
311C2×C53#330211yescharacteristic
312C2×C53#330411yescharacteristic
313C2×C53#333611yescharacteristic
314C2×C53#333711yescharacteristic
315C23×C8—531no—
316C2×#125—531no—
317C2×C22⋊Q8C2×D5311yescharacteristic
318C2×#117—531no—
319C2×#117C2×D5311yescharacteristic
320C2×#103—531no—
321C2×#103—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×C22⋊C8—531no—
325C2×C4×C8—531no—
326C2×C4×C8—531no—
327#399—1061no—
328#397—21no—
329#397—21no—
330#396—1061no—
331#391C2×D5311yescharacteristic
332#391C2×D5311yescharacteristic
333#389—531no—
334#389—531no—
335#389—1061no—
336#387—21no—
337#387—21no—
338#387—1061no—
339#384—531no—
340#384C2×D5311yescharacteristic
341#382C2×D5311yescharacteristic
342#380—531no—
343C4×C22⋊C4C2×D5311yescharacteristic
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#336—531no—
357#336—531no—
358C53#2770011yescharacteristic, complemented, Sylow
359C22×Q8C22×D5311yescharacteristic
360C23×C4C22×D5311yescharacteristic
361C22×C8—531no—
362C22×C8—531no—
363C22×C8—531no—
364C22×C8—531no—
365C22×C8—1062no—
366C22×C8—1062no—
367C22×C8—2121no—
368#125—1062no—
369#125—2121no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372#117—22no—
373#117—22no—
374#117—1062no—
375#117—2121no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4C22⋊D5311yescharacteristic
397C2×C22⋊C4C22⋊D5311yescharacteristic
398C2×C42—21no—
399C2×C42—1061no—
400C2×C42C22×D5311yescharacteristic
401C2×C42C22×D5311yescharacteristic
402#103—1062no—
403#103—1062no—
404#103—2121no—
405#103—2121no—
406C22⋊C8—1062no—
407C4×C8—1062no—
408C4×C8—2121no—
409#95—21no—
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#95C22⋊D5311yescharacteristic
419#95C22⋊D5311yescharacteristic
420C24C2×C22⋊D5311yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4#1171994712yes—
440C22×C4#1171994712yes—
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C23×D5311yescharacteristic
447C2×C8—532no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—2124no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—532no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—4241no—
478C4⋊C4—4241no—
479C4⋊C4—2122no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—4241no—
488C22⋊C4—2122no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—2122no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23#1313320412yes—
502C23C22×C22⋊D5311yescharacteristic
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
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—41no—
535C2×C4—1062no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4#1313320412yes—
548C2×C4#1313323712yes—
549C2×C4#1313323712yes—
550C2×C4#41×D5311yescharacteristic
551C2×C4#41×D5311yescharacteristic
552C2×C4C2×#1171994711yescharacteristic
553C2×C4C2×#1171994711yescharacteristic
554C2×C4#1313326311yescharacteristic
555C2×C4#1313326411yescharacteristic
556C8—1062no—
557C8—2121no—
558C8—2121no—
559C8—2121no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22C2×#1313320411yescharacteristic
568C22#1681126111yescharacteristic
569C22#1681127211yescharacteristic
570C22C2×#1313323711yescharacteristic
571C22C2×#1313323711yescharacteristic
572C22#1681143111yescharacteristic
573C22#1681143811yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—4241no—
584C4—4241no—
585C4—4241no—
586C4—2122no—
587C4#1681115011yescharacteristic
588C4#1681121711yescharacteristic
589C4#1681129611yescharacteristic
590C4#1681131011yescharacteristic
591C2—42no—
592C2#2027904511yescharacteristic
593C2#2028503711yescharacteristic
594C2#2028508011yescharacteristic
595C2#2028662611yescharacteristic
596C2#2028664811yescharacteristic
597C2#2028707711yescharacteristic
598C2#2028709411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2499C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2499D53
hypocenterC53#27700
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2499

Complements

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

NH ≅ G/Ntype
2. #20276282C2semidirect
3. #20276272C2semidirect
4. #20276123C2semidirect
5. #20275866C2semidirect
6. #20275679C2semidirect
7. #20275394C2semidirect
8. #20275394C2semidirect
11. #20274842C2semidirect
358. C53#27700semidirect

Maximal subgroups

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

subgroupindex
#202784282
#202762822
#202762722
#202761232
#202758662
#202756792
#202753942
#202751392
#202750282
#202748422
#202748102
C22⋊C4×#883222
#202744312
C53×#24992
#2770053