Finite Group Databasebeta

← #27591257

Subgroups — #27591257

Order 13568.

1854 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283002C212yesmaximal
2#20281910C214yesmaximal, complemented
3#20281620C214yesmaximal, complemented
4#20281373C214yesmaximal, complemented
5#20281181C214yesmaximal, complemented
6#20280829C211yesmaximal, characteristic, complemented
7#20280685C211yesmaximal, characteristic, complemented
8#20278955C214yesmaximal
9#20278850C214yesmaximal, complemented
10#20277448C212yesmaximal, complemented
11C2×C53×#380C211yesmaximal, characteristic
12C2×#13133210C2214yes—
13C2×#13133207C2214yescomplemented
14C2×#13133196C2212yes—
15C2×#13133196C2212yes—
16C2×#13133196C2212yes—
17#16810565C2214yescomplemented
18#16810561C2214yescomplemented
19#16810557C2214yescomplemented
20#16810556C2214yescomplemented
21#16810555C2214yescomplemented
22#16810554C2214yescomplemented
23#16810553C2214yescomplemented
24#16810551C2214yes—
25#16810551C2214yes—
26#16810548C2214yes—
27#16810546C2214yes—
28#16810544C2214yescomplemented
29#16810527C2214yescomplemented
30#16810520C2214yescomplemented
31#16810518C2214yescomplemented
32#16810517C2214yes—
33#16810516C2214yes—
34#16810511C2214yes—
35#16810510C2214yescomplemented
36#16810509C2214yescomplemented
37#16810506C2214yescomplemented
38#16810504C2214yescomplemented
39#16810496C2214yes—
40#16810496C2214yes—
41#16810496C2214yes—
42#16810496C2214yes—
43#16810496C2214yes—
44#16810483C2212yes—
45#16810483C2212yes—
46#16810483C2214yes—
47#16810483C2212yes—
48#16810479C2214yes—
49C4×C22⋊Dic53C2214yes—
50C24×C4×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2212yes—
52C22×C22⋊C4×C53C2212yes—
53C53×#380C2214yes—
54C53×#380C2214yes—
55C2×#95×C53C2211yescharacteristic
56C2×#95×C53C2211yescharacteristic
57C22×Dic106C2312yes—
58#13133210—48no—
59#13133207—48no—
60#13133196—28no—
61#13133196—44no—
62#13133196—44no—
63C2×C4⋊Dic53C2312yes—
64C2×C4⋊Dic53C2314yes—
65C2×C4⋊Dic53C2312yes—
66C2×C4⋊Dic53C2314yes—
67C2×C4⋊Dic53C2312yes—
68C2×#11719919C2314yes—
69C2×#11719919C2314yes—
70C2×#11719919C2314yes—
71C2×#11719919C2314yes—
72C2×#11719919C2314yes—
73C2×#11719919C2314yes—
74C2×#11719919C2314yes—
75C2×C22⋊Dic53C2314yes—
76C2×C22⋊Dic53C2314yes—
77C2×C22⋊Dic53C2314yes—
78C2×C22⋊Dic53C2314yes—
79C2×C22⋊Dic53C2314yes—
80C2×C22⋊Dic53C2314yes—
81C2×C4×Dic53C2314yes—
82C2×C4×Dic53C2314yes—
83#13133124C2314yes—
84#13133124C2314yes—
85#13133124C2314yes—
86#13133124C2314yes—
87#13133124C2314yes—
88#13133124C2314yes—
89#13133124C2314yes—
90#13133124C2314yes—
91#13133124C2314yes—
92#13133124C2314yes—
93#13133124C2314yes—
94C25×C53C2311yescharacteristic
95C23×C4×C53—24no—
96C23×C4×C53—44no—
97C23×C4×C53C2311yescharacteristic
98C23×C4×C53C2312yes—
99C23×C4×C53C2312yes—
100C23×C4×C53C2312yes—
101C23×C4×C53C2312yes—
102C23×C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53—44no—
104C2×C4⋊C4×C53C2314yes—
105C2×C4⋊C4×C53C2314yes—
106C2×C22⋊C4×C53—44no—
107C2×C22⋊C4×C53C2314yes—
108C2×C22⋊C4×C53C2314yes—
109#95×C53C2314yes—
110#95×C53C2314yes—
111C24×C53—22no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53—24no—
115C24×C53—42no—
116C24×C53C2411yescharacteristic
117C24×C53C2411yescharacteristic
118C24×C53C2411yescharacteristic
119C22×C4×C53—22no—
120C22×C4×C53—24no—
121C22×C4×C53—24no—
122C22×C4×C53—24no—
123C22×C4×C53—24no—
124C22×C4×C53—24no—
125C22×C4×C53—24no—
126C22×C4×C53—24no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—416no—
135C22×C4×C53#4114yes—
136C22×C4×C53C2414yes—
137C22×C4×C53C2414yes—
138C22×C4×C53C2412yes—
139C22×C4×C53C2412yes—
140C22×Dic53C2414yes—
141C22×Dic53C2414yes—
142C22×Dic53C2414yes—
143C22×Dic53C2414yes—
144C2×Dic106—24no—
145C2×Dic106—42no—
146C2×Dic106—42no—
147C4⋊C4×C53—28no—
148C4⋊C4×C53—44no—
149C22⋊C4×C53—28no—
150C22⋊C4×C53—44no—
151C22⋊Dic53—28no—
152C22⋊Dic53—28no—
153C22⋊Dic53—28no—
154C22⋊Dic53—28no—
155C22⋊Dic53—44no—
156C22⋊Dic53—44no—
157C4⋊Dic53—24no—
158C4⋊Dic53—42no—
159C4⋊Dic53—42no—
160C4⋊Dic53—44no—
161C4⋊Dic53—44no—
162#11719919—28no—
163#11719919—28no—
164#11719919—28no—
165#11719919—44no—
166#11719919—44no—
167#11719919—44no—
168#11719919—44no—
169C4×Dic53—44no—
170C4×Dic53—44no—
171C23×C53—22no—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—28no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—21no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—21no—
183C23×C53—48no—
184C23×C53—48no—
185C23×C53—44no—
186C23×C53—44no—
187C23×C53C2×#4112yes—
188C23×C53H(5,𝔽2)12yes—
189C23×C532−1+414yes—
190C23×C532−1+412yes—
191C23×C53C2511yescharacteristic
192C2×C4×C53—28no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—24no—
198C2×C4×C53—24no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—44no—
204C2×C4×C53—48no—
205C2×C4×C53—48no—
206C2×C4×C53—84no—
207C2×C4×C53—84no—
208C2×C4×C53C2×#4114yes—
209C2×C4×C53H(5,𝔽2)14yes—
210C2×Dic53—24no—
211C2×Dic53—24no—
212C2×Dic53—24no—
213C2×Dic53—24no—
214C2×Dic53—24no—
215C2×Dic53—24no—
216C2×Dic53—24no—
217C2×Dic53—24no—
218C2×Dic53—24no—
219C2×Dic53—24no—
220C2×Dic53—24no—
221C2×Dic53—24no—
222Dic106—84no—
223#36728—531nomaximal, Sylow
224C22×C53—22no—
225C22×C53—22no—
226C22×C53—24no—
227C22×C53—24no—
228C22×C53—22no—
229C22×C53—22no—
230C22×C53—24no—
231C22×C53—24no—
232C22×C53—416no—
233C22×C53—41no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—42no—
238C22×C53—42no—
239C22×C53#54114yes—
240C22×C53C22×#4111yescharacteristic
241C22×C53C2×H(5,𝔽2)12yes—
242C22×C53C2×H(5,𝔽2)11yescharacteristic
243C22×C53C2×2−1+412yes—
244C22×C53C2×2−1+411yescharacteristic
245C4×C53—24no—
246C4×C53—24no—
247C4×C53—44no—
248C4×C53—44no—
249Dic53—44no—
250Dic53—44no—
251Dic53—44no—
252Dic53—44no—
253#2848—532no—
254#2747—534no—
255#2722—534no—
256#2704—534no—
257#2690—534no—
258#2659—531no—
259#2646—531no—
260#2480—534no—
261#2473—534no—
262#2329—532no—
263C2×#380D5311yescharacteristic
264C2×C53—24no—
265C2×C53—42no—
266C2×C53—42no—
267C2×C53C2×#54111yescharacteristic
268C2×C53#349011yescharacteristic
269C2×C53#349211yescharacteristic
270C2×C53#353612yes—
271C2×C53#353812yes—
272C24×C4C2×D5311yescharacteristic
273C2×#126—534no—
274C2×#125—534no—
275C2×C22⋊Q8—532no—
276C2×C22⋊Q8—532no—
277C2×C22⋊Q8—532no—
278C22×C4⋊C4C2×D5312yes—
279C22×C22⋊C4C2×D5312yes—
280#400—534no—
281#400—534no—
282#398—534no—
283#398—534no—
284#398—534no—
285#398—534no—
286#398—534no—
287#397—534no—
288#397—534no—
289#397—534no—
290#396—534no—
291#395—534no—
292#391—534no—
293#389—534no—
294#389—534no—
295#388—534no—
296#388—534no—
297#388—534no—
298#387—534no—
299#387—534no—
300#387—534no—
301#387—534no—
302#385—534no—
303#385—534no—
304#385—534no—
305#385—534no—
306#385—534no—
307#380—532no—
308#380—532no—
309#380—532no—
310#380—534no—
311#380C2×D5314yes—
312#380C2×D5314yes—
313#379—534no—
314C4×C22⋊C4—534no—
315C2×#95C2×D5311yescharacteristic
316C2×#95C2×D5311yescharacteristic
317C53#3672811yescharacteristic, complemented, Sylow
318C25C22×D5311yescharacteristic
319C22×Q8—532no—
320C23×C4—24no—
321C23×C4—44no—
322C23×C4C22×D5311yescharacteristic
323C23×C4C22×D5312yes—
324C23×C4C22×D5312yes—
325C23×C4C22×D5312yes—
326C23×C4C22×D5312yes—
327C23×C4C22×D5311yescharacteristic
328#126—2128no—
329#125—2128no—
330C22⋊Q8—1068no—
331C22⋊Q8—2124no—
332C22⋊Q8—2124no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—534no—
335C2×C4⋊C4—534no—
336C2×C4⋊C4—534no—
337C2×C4⋊C4—532no—
338C2×C4⋊C4—532no—
339C2×C4⋊C4—532no—
340C2×C4⋊C4—534no—
341C2×C4⋊C4—534no—
342C2×C4⋊C4—534no—
343C2×C4⋊C4—534no—
344C2×C4⋊C4—534no—
345C2×C4⋊C4—534no—
346C2×C4⋊C4C22×D5314yes—
347C2×C4⋊C4C22×D5314yes—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—534no—
350C2×C22⋊C4—534no—
351C2×C22⋊C4—534no—
352C2×C22⋊C4—534no—
353C2×C22⋊C4—534no—
354C2×C22⋊C4—534no—
355C2×C22⋊C4C22×D5314yes—
356C2×C22⋊C4C22×D5314yes—
357C2×C42—534no—
358C2×C42—534no—
359#95—534no—
360#95—534no—
361#95—534no—
362#95—534no—
363#95—534no—
364#95—534no—
365#95—534no—
366#95—534no—
367#95—534no—
368#95—534no—
369#95—534no—
370#95C22×D5314yes—
371#95C22×D5314yes—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—42no—
377C24C23×D5311yescharacteristic
378C24C23×D5311yescharacteristic
379C24C23×D5311yescharacteristic
380C2×Q8—1064no—
381C2×Q8—2122no—
382C2×Q8—2122no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4—534no—
400C22×C4—534no—
401C22×C4—534no—
402C22×C4—534no—
403C22×C4#1171994514yes—
404C22×C4C23×D5314yes—
405C22×C4C23×D5314yes—
406C22×C4C23×D5312yes—
407C22×C4C23×D5312yes—
408C4⋊C4—28no—
409C4⋊C4—44no—
410C4⋊C4—1068no—
411C4⋊C4—1064no—
412C4⋊C4—1068no—
413C4⋊C4—1068no—
414C4⋊C4—2122no—
415C4⋊C4—2122no—
416C4⋊C4—2124no—
417C4⋊C4—2124no—
418C4⋊C4—2124no—
419C4⋊C4—2124no—
420C4⋊C4—2124no—
421C4⋊C4—2124no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—1068no—
425C22⋊C4—1068no—
426C22⋊C4—1068no—
427C22⋊C4—1068no—
428C22⋊C4—2124no—
429C22⋊C4—2124no—
430C42—2124no—
431C42—2124no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—28no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—48no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23C2×#1171994512yes—
449C23#1313326212yes—
450C23#1313326414yes—
451C23#1313326512yes—
452C23C24×D5311yescharacteristic
453Q8—4244no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—1064no—
471C2×C4—1064no—
472C2×C4—1064no—
473C2×C4—1064no—
474C2×C4—1064no—
475C2×C4—1064no—
476C2×C4—1064no—
477C2×C4—1064no—
478C2×C4—1064no—
479C2×C4—1064no—
480C2×C4—1064no—
481C2×C4—1064no—
482C2×C4C2×#1171994514yes—
483C2×C4#1313326214yes—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—24no—
491C22—24no—
492C22—416no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—42no—
498C22—42no—
499C22#1681123514yes—
500C22C22×#1171994511yescharacteristic
501C22C2×#1313326212yes—
502C22C2×#1313326211yescharacteristic
503C22C2×#1313326412yes—
504C22C2×#1313326511yescharacteristic
505C4—24no—
506C4—24no—
507C4—44no—
508C4—44no—
509C4—2124no—
510C4—2124no—
511C4—2124no—
512C4—2124no—
513C2—24no—
514C2—42no—
515C2—42no—
516C2C2×#1681123511yescharacteristic
517C2#2028840411yescharacteristic
518C2#2028842411yescharacteristic
519C2#2028893712yes—
520C2#2028896012yes—
subgroupG/H
center, FrattiniC23C24×D53
commutator, socle, cosocleC23×C53C25
FittingC2×C53×#380C2
prime radicalC25×C53C23
hypercenterC2×#380D53
hypocenterC53#36728
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. #20281910C2semidirect
3. #20281620C2semidirect
4. #20281373C2semidirect
5. #20281181C2semidirect
6. #20280829C2semidirect
7. #20280685C2semidirect
9. #20278850C2semidirect
10. #20277448C2semidirect
13. C2×#13133207C22semidirect
17. #16810565C22semidirect
18. #16810561C22semidirect
19. #16810557C22semidirect
20. #16810556C22semidirect
21. #16810555C22semidirect
22. #16810554C22semidirect
23. #16810553C22semidirect
28. #16810544C22semidirect
29. #16810527C22semidirect
30. #16810520C22semidirect
31. #16810518C22semidirect
35. #16810510C22semidirect
36. #16810509C22semidirect
37. #16810506C22semidirect
38. #16810504C22semidirect
317. C53#36728semidirect

Maximal subgroups

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

subgroupindex
#202830022
#202819102
#202816202
#202813732
#202811812
#202808292
#202806852
#202789552
#202788502
#202774482
C2×C53×#3802
#3672853