Finite Group Databasebeta

← #27438506

Subgroups — #27438506

Order 13568.

1096 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811033C211yesmaximal, characteristic, complemented
2C2×#16810969C211yesmaximal, characteristic, complemented
3#20280536C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275681C211yesmaximal, characteristic, complemented
7#20275540C211yesmaximal, characteristic, complemented
8#20274976C212yesmaximal, complemented
9#20274962C212yesmaximal, complemented
10#20274481C211yesmaximal, characteristic, complemented
11C2×C53×#540C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16811033—24no—
14#16810969—24no—
15C2×#13133150C2212yes—
16C2×#13133148C2212yes—
17C2×#13133138C2212yes—
18C2×#13133138C2212yes—
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×C22⋊#88322C2212yes—
22C2×#13133128C2211yescharacteristic
23C2×C4×#88322C2211yescharacteristic
24#16810562—21no—
25#16810559—21no—
26#16810543—21no—
27#16810535—21no—
28#16810521C2212yes—
29#16810489C2211yescharacteristic
30#16810483C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#540—24no—
35C53×#540—24no—
36C2×#127×C53C2211yescharacteristic
37C2×#125×C53C2211yescharacteristic
38C2×#123×C53—21no—
39C2×#123×C53—21no—
40C2×#121×C53—21no—
41C2×#121×C53—21no—
42C2×#121×C53C2211yescharacteristic
43C2×#121×C53C2211yescharacteristic
44C2×C4×D4×C53C2211yescharacteristic
45C2×C4×D4×C53C2211yescharacteristic
46C2×#117×C53C2211yescharacteristic
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2312yes—
55C2×#11719919—22no—
56C2×C22⋊Dic53—22no—
57#13133150—24no—
58#13133148—24no—
59#13133138—24no—
60#13133138—24no—
61#13133138—24no—
62#13133138—24no—
63C22⋊#88322—24no—
64#13133128—22no—
65C4×#88322—22no—
66#13133124—22no—
67#13133124—21no—
68#13133124—22no—
69#13133124—21no—
70C22×D4×C53—21no—
71C22×D4×C53—21no—
72C22×D4×C53—21no—
73C22×D4×C53—21no—
74C22×D4×C53C2311yescharacteristic
75C22×D4×C53C2311yescharacteristic
76C23×C4×C53—21no—
77C23×C4×C53—21no—
78C23×C4×C53C2311yescharacteristic
79#127×C53—22no—
80#127×C53D414yes—
81#125×C53—22no—
82#125×C53D414yes—
83#123×C53—28no—
84#123×C53—28no—
85#121×C53—28no—
86#121×C53—28no—
87#121×C53—22no—
88#121×C53—22no—
89#121×C53—22no—
90#121×C53—22no—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95#117×C53—22no—
96#117×C53—22no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C24×C53—21no—
112C24×C53—21no—
113C24×C53C2×D411yescharacteristic
114C2×D4×C53—24no—
115C2×D4×C53—24no—
116C2×D4×C53—24no—
117C2×D4×C53—24no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—82no—
125C2×D4×C53—82no—
126C2×D4×C53—82no—
127C2×D4×C53—82no—
128C2×D4×C53C2×D412yes—
129C2×D4×C53C2×D412yes—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—24no—
133C22×C4×C53—24no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—82no—
139C22×C4×C53—82no—
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—22no—
147C22×Dic53—22no—
148C2×Dic106—42no—
149C2×Dic106—82no—
150C4⋊C4×C53—24no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C2×D412yes—
157C4⋊C4×C53C2×D412yes—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—24no—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—22no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53C2×D412yes—
167C42×C53C2×D412yes—
168C22⋊Dic53—82no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—42no—
174#11719919—82no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322#4114yes—
181C23×C53—21no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53C22×D411yescharacteristic
197D4×C53—24no—
198D4×C53—24no—
199D4×C53—42no—
200D4×C53—42no—
201D4×C53—84no—
202D4×C53—84no—
203D4×C53—84no—
204D4×C53—84no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—84no—
221C2×C4×C53—84no—
222C2×C4×C53#12012yes—
223C2×C4×C53#12012yes—
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C2×#4112yes—
227C2×C4×C532−1+412yes—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234Dic106—84no—
235#88322—24no—
236#88322—42no—
237#25685—531nomaximal, Sylow
238C22×C53—42no—
239C22×C53—41no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—41no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—41no—
248C22×C53—42no—
249C22×C53—82no—
250C22×C53—82no—
251C22×C53—82no—
252C22×C53—82no—
253C22×C53C2×#12011yescharacteristic
254C22×C53#53211yescharacteristic
255C22×C53#53611yescharacteristic
256C22×C53#57612yes—
257C22×C53#57712yes—
258C4×C53—22no—
259C4×C53—22no—
260C4×C53—41no—
261C4×C53—41no—
262C4×C53—42no—
263C4×C53—41no—
264C4×C53—42no—
265C4×C53#54712yes—
266C4×C53#54812yes—
267Dic53—82no—
268Dic53—82no—
269C2×#540D5311yescharacteristic
270C2×#501—531no—
271C2×#486—531no—
272#2628—531no—
273#2034—532no—
274#2033—532no—
275#1991—531no—
276#1961—531no—
277#1864—532no—
278#1862—532no—
279#1750—531no—
280C2×C53—81no—
281C2×C53—82no—
282C2×C53—82no—
283C2×C53#243211yescharacteristic
284C2×C53#300811yescharacteristic
285C2×C53#300911yescharacteristic
286C2×C53#317912yes—
287C2×C53#324112yes—
288#540—24no—
289#540—24no—
290C2×C4⋊Q8—531no—
291C2×#127C2×D5311yescharacteristic
292C2×#125C2×D5311yescharacteristic
293C2×#123—21no—
294C2×#123—21no—
295C2×#121—21no—
296C2×#121—21no—
297C2×#121C2×D5311yescharacteristic
298C2×#121C2×D5311yescharacteristic
299C2×C4×D4C2×D5311yescharacteristic
300C2×C4×D4C2×D5311yescharacteristic
301C2×#117C2×D5311yescharacteristic
302#501—1064no—
303#486—1064no—
304C2×#107—532no—
305C2×#106—532no—
306C2×#102—532no—
307C2×#102—532no—
308C2×#102—532no—
309C2×#102—532no—
310C2×C22⋊C8—532no—
311C2×#97—531no—
312C2×C4×C8—531no—
313#400—1061no—
314#399—1061no—
315#395—1061no—
316#393—1061no—
317#389—532no—
318#382—531no—
319#380—532no—
320#340—532no—
321#340—532no—
322#336—532no—
323C53#2568511yescharacteristic, complemented, Sylow
324C22×Q8—1061no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—21no—
328C22×D4—21no—
329C22×D4C22×D5311yescharacteristic
330C22×D4C22×D5311yescharacteristic
331C23×C4—21no—
332C23×C4—21no—
333C23×C4C22×D5311yescharacteristic
334C22×C8—532no—
335C22×C8—532no—
336C4⋊Q8—1062no—
337C4⋊Q8—1062no—
338#127—22no—
339#127C22⋊D5314yes—
340#125—22no—
341#125C22⋊D5314yes—
342#123—28no—
343#123—28no—
344#121—28no—
345#121—28no—
346#121—22no—
347#121—22no—
348#121—22no—
349#121—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354#117—22no—
355#117—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—532no—
359C2×C4⋊C4—532no—
360C2×C4⋊C4—1062no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4C22⋊D5311yescharacteristic
363C2×C4⋊C4C22⋊D5311yescharacteristic
364C2×C4⋊C4C22×D5311yescharacteristic
365C2×C4⋊C4C22×D5311yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—1062no—
371C2×C22⋊C4C22⋊D5311yescharacteristic
372C2×C22⋊C4C22⋊D5311yescharacteristic
373C2×C42C22×D5311yescharacteristic
374C2×C42C22×D5311yescharacteristic
375#107—1064no—
376#106—1064no—
377#102—1064no—
378#102—1064no—
379#102—1064no—
380#102—1064no—
381C22⋊C8—1064no—
382#97—1062no—
383C4×C8—1062no—
384#95—1062no—
385#95—1061no—
386#95—1061no—
387#95—1062no—
388C24—21no—
389C24—21no—
390C24C2×C22⋊D5311yescharacteristic
391C2×Q8—4242no—
392C2×Q8—2122no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4C2×C22⋊D5312yes—
408C2×D4C2×C22⋊D5312yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—1062no—
420C22×C4—1062no—
421C22×C4C2×C22⋊D5311yescharacteristic
422C22×C4C2×C22⋊D5311yescharacteristic
423C22×C4C2×C22⋊D5311yescharacteristic
424C22×C4C2×C22⋊D5311yescharacteristic
425C22×C4C2×C22⋊D5311yescharacteristic
426C22×C4C23×D5311yescharacteristic
427C2×C8—534no—
428C2×C8—1062no—
429C2×C8—1062no—
430C2×C8—1062no—
431C2×C8—1062no—
432C2×C8—1062no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—1062no—
440C4⋊C4—1062no—
441C4⋊C4—1062no—
442C4⋊C4—1062no—
443C4⋊C4—4242no—
444C4⋊C4—2122no—
445C4⋊C4C2×C22⋊D5312yes—
446C4⋊C4C2×C22⋊D5312yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—4242no—
454C42—21no—
455C42—21no—
456C42C2×C22⋊D5312yes—
457C42C2×C22⋊D5312yes—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—22no—
463C23—22no—
464C23—21no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23C22×C22⋊D5311yescharacteristic
474Q8—4244no—
475D4—24no—
476D4—24no—
477D4—42no—
478D4—42no—
479D4—84no—
480D4—84no—
481D4—84no—
482D4—84no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4#1313318612yes—
507C2×C4#1313318612yes—
508C2×C4C22×C22⋊D5311yescharacteristic
509C2×C4C22×C22⋊D5311yescharacteristic
510C2×C4#41×D5312yes—
511C2×C4#1313326412yes—
512C8—1064no—
513C8—2122no—
514C22—42no—
515C22—41no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22—42no—
521C22—42no—
522C22—41no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22C2×#1313318611yescharacteristic
530C22#1681116911yescharacteristic
531C22#1681119411yescharacteristic
532C22#1681142612yes—
533C22#1681143412yes—
534C4—22no—
535C4—22no—
536C4—41no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—42no—
541C4—4242no—
542C4—4242no—
543C4#1681127212yes—
544C4#1681128012yes—
545C2—81no—
546C2—82no—
547C2—82no—
548C2#2027833911yescharacteristic
549C2#2028398311yescharacteristic
550C2#2028399211yescharacteristic
551C2#2028540012yes—
552C2#2028596212yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#540C2
socleC23×C53C22×D4
hypercenterC2×#540D53
hypocenterC53#25685
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×#16811033C2semidirect
2. C2×#16810969C2semidirect
3. #20280536C2semidirect
4. #20275901C2semidirect
5. #20275894C2semidirect
6. #20275681C2semidirect
7. #20275540C2semidirect
8. #20274976C2semidirect
9. #20274962C2semidirect
10. #20274481C2semidirect
323. C53#25685semidirect

Maximal subgroups

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

subgroupindex
C2×#168110332
C2×#168109692
#202805362
#202759012
#202758942
#202756812
#202755402
#202749762
#202749622
#202744812
C2×C53×#5402
#2568553