Finite Group Databasebeta

← #27457193

Subgroups — #27457193

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279148C211yesmaximal, characteristic
2#20276123C211yesmaximal, characteristic, complemented
3#20276110C211yesmaximal, characteristic, complemented
4#20275908C211yesmaximal, characteristic
5#20275908C211yesmaximal, characteristic
6#20275478C211yesmaximal, characteristic, complemented
7#20275455C211yesmaximal, characteristic, complemented
8#20275330C211yesmaximal, characteristic
9#20275300C211yesmaximal, characteristic, complemented
10#20275297C211yesmaximal, characteristic, complemented
11#20274983C211yesmaximal, characteristic, complemented
12#20274914C211yesmaximal, characteristic
13#20274810C211yesmaximal, characteristic
14#20274545C211yesmaximal, characteristic, complemented
15C53×#2362C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133148C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×#13133128C2211yescharacteristic
28#16810565—21no—
29#16810546—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810516—21no—
33#16810510—21no—
34#16810493C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
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
50C2×#117×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#388—21no—
54C53×#388—21no—
55C53×#387—21no—
56C4×C4⋊C4×C53—21no—
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
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×C22⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C4×Dic53—21no—
74#13133150—41no—
75#13133148—22no—
76#13133148—22no—
77#13133148—22no—
78C4⋊#88322—22no—
79#13133141—22no—
80#13133141—22no—
81#13133141—22no—
82#13133141—41no—
83C22⋊#88322—22no—
84C22⋊#88322—41no—
85#13133128—41no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×Q8×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C22⋊Q8×C53—22no—
94C22⋊Q8×C53—22no—
95C4×Q8×C53—22no—
96C4×Q8×C53—41no—
97#117×C53—22no—
98#117×C53—41no—
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×C53—21no—
111C2×C42×C53—21no—
112C2×C42×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
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#4112yes—
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×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C4⋊C4×C53—24no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—42no—
153C4⋊C4×C53—42no—
154C4⋊C4×C53#4112yes—
155C4⋊C4×C53#4112yes—
156C22⋊C4×C53—24no—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—42no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—42no—
165C42×C53—42no—
166C42×C53—42no—
167C22⋊Dic53—81no—
168C22⋊Dic53—81no—
169C4⋊Dic53—21no—
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—
179#11719919—42no—
180#11719919—81no—
181C4×Dic53—81no—
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#4112yes—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53C22×D411yescharacteristic
202Q8×C53—22no—
203Q8×C53—22no—
204Q8×C53—41no—
205Q8×C53—41no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
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—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—41no—
236C2×C4×C53#13712yes—
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C2×Dic53—41no—
244C2×Dic53—41no—
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—
255#88322—22no—
256#88322—41no—
257#88322—41no—
258#88322—41no—
259#27060—531nomaximal, Sylow
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53#52511yescharacteristic
268C22×C53#54611yescharacteristic
269C22×C53#54711yescharacteristic
270C22×C53C2×#13711yescharacteristic
271C22×C53#57511yescharacteristic
272C22×C53#57711yescharacteristic
273C22×C53#57711yescharacteristic
274C4×C53—22no—
275C4×C53—41no—
276C4×C53—42no—
277C4×C53—41no—
278C4×C53—42no—
279C4×C53—41no—
280C4×C53—42no—
281C4×C53—41no—
282C4×C53—81no—
283C4×C53#52911yescharacteristic
284C4×C53#53911yescharacteristic
285C4×C53#55311yescharacteristic
286C4×C53#55611yescharacteristic
287Dic53—81no—
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291#2497—531no—
292#2362D5311yescharacteristic
293#2072—531no—
294#2068—531no—
295#2035—531no—
296#2035—531no—
297#1948—531no—
298#1944—531no—
299#1924—531no—
300#1920—531no—
301#1920—531no—
302#1865—531no—
303#1854—531no—
304#1825—531no—
305#1763—531no—
306C2×C53—81no—
307C2×C53#246611yescharacteristic
308C2×C53#311611yescharacteristic
309C2×C53#312611yescharacteristic
310C2×C53#332111yescharacteristic
311C2×C53#333111yescharacteristic
312C2×C53#334411yescharacteristic
313C2×C53#335011yescharacteristic
314C2×C22⋊Q8C2×D5311yescharacteristic
315C2×C4×Q8C2×D5311yescharacteristic
316C2×#117C2×D5311yescharacteristic
317C2×#107—531no—
318C2×#106—531no—
319C2×#106—531no—
320C2×#106—531no—
321C2×C4⋊C8—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×#103—531no—
325C2×#103—531no—
326C2×C22⋊C8—531no—
327C2×C22⋊C8—531no—
328C2×#97—531no—
329#400—1061no—
330#396—1061no—
331#391C2×D5311yescharacteristic
332#389—531no—
333#389—531no—
334#389C2×D5311yescharacteristic
335#388—21no—
336#388—21no—
337#388—1061no—
338#387—21no—
339#387—1061no—
340#384—531no—
341#380—531no—
342#380—531no—
343C4×C4⋊C4—21no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4C2×D5311yescharacteristic
346C4×C22⋊C4C2×D5311yescharacteristic
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#336—531no—
356#336—531no—
357C53#2706011yescharacteristic, complemented, Sylow
358C22×Q8C22×D5311yescharacteristic
359C23×C4C22×D5311yescharacteristic
360C22×C8—531no—
361C22×C8—531no—
362C22×C8—531no—
363C22×C8—531no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C4×Q8—22no—
367C4×Q8—41no—
368#117—22no—
369#117—41no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
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×C22⋊C4C22⋊D5311yescharacteristic
389C2×C22⋊C4C22⋊D5311yescharacteristic
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—1061no—
394C2×C42C22×D5311yescharacteristic
395C2×C42C22×D5311yescharacteristic
396#107—2121no—
397#106—1062no—
398#106—1062no—
399#106—1062no—
400C4⋊C8—1062no—
401#103—1062no—
402#103—1062no—
403#103—1062no—
404#103—2121no—
405C22⋊C8—1062no—
406C22⋊C8—2121no—
407#97—2121no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95C22⋊D5311yescharacteristic
418#95C22⋊D5311yescharacteristic
419C24C2×C22⋊D5311yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#1171994512yes—
426C22×C4—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—21no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C23×D5311yescharacteristic
446C2×C8—532no—
447C2×C8—1061no—
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—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—4241no—
474C4⋊C4—4241no—
475C4⋊C4—4241no—
476C4⋊C4—2122no—
477C4⋊C4#1171994512yes—
478C4⋊C4#1171994712yes—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—41no—
482C22⋊C4—42no—
483C22⋊C4—4241no—
484C22⋊C4—4241no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—4241no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×C22⋊D5311yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
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—2121no—
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#1313324712yes—
548C2×C4#41×D5311yescharacteristic
549C2×C4C2×#1171994711yescharacteristic
550C2×C4C2×#1171994511yescharacteristic
551C2×C4C2×#1171994511yescharacteristic
552C2×C4#1313326211yescharacteristic
553C2×C4#1313326411yescharacteristic
554C8—1062no—
555C8—2121no—
556C8—2121no—
557C8—2121no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#1681113411yescharacteristic
566C22#1681126711yescharacteristic
567C22#1681127211yescharacteristic
568C22C2×#1313324711yescharacteristic
569C22#1681142111yescharacteristic
570C22#1681143111yescharacteristic
571C22#1681143111yescharacteristic
572C4—22no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—4241no—
582C4—4241no—
583C4—4241no—
584C4—4241no—
585C4#1681115311yescharacteristic
586C4#1681121511yescharacteristic
587C4#1681132311yescharacteristic
588C4#1681134311yescharacteristic
589C2—81no—
590C2#2027874511yescharacteristic
591C2#2028487711yescharacteristic
592C2#2028500611yescharacteristic
593C2#2028690111yescharacteristic
594C2#2028703111yescharacteristic
595C2#2028717011yescharacteristic
596C2#2028726011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2362C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2362D53
hypocenterC53#27060
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #20276123C2semidirect
3. #20276110C2semidirect
6. #20275478C2semidirect
7. #20275455C2semidirect
9. #20275300C2semidirect
10. #20275297C2semidirect
11. #20274983C2semidirect
14. #20274545C2semidirect
357. C53#27060semidirect

Maximal subgroups

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

subgroupindex
#202791482
#202761232
#202761102
#202759082
#202754782
#202754552
#202753302
#202753002
#202752972
#202749832
#202749142
#202748102
#202745452
C53×#23622
#2706053