Finite Group Databasebeta

← #27356516

Subgroups — #27356516

Order 13568.

962 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284659C211yesmaximal, characteristic, complemented
2#20284651C211yesmaximal, characteristic
3#20279002C211yesmaximal, characteristic, complemented
4#20273746C214yesmaximal, complemented
5#20273712C214yesmaximal, complemented
6#20273248C211yesmaximal, characteristic, complemented
7#20273244C211yesmaximal, characteristic
8#20273045C211yesmaximal, characteristic, complemented
9C2×C53×#539C211yesmaximal, characteristic
10C2×#13133176C2211yescharacteristic
11#16810961—22no—
12#16810956C2212yescomplemented
13#16810951C2212yescomplemented
14#16810944—22no—
15#16810939C2212yes—
16#16810934C2212yes—
17#16810663—21no—
18#16810663—21no—
19#16810632—21no—
20#16810628—21no—
21#16810587C2211yescharacteristic
22#16810587C2211yescharacteristic
23#16810562C2212yescomplemented
24#16810553C2212yescomplemented
25#16810504—22no—
26#16810496—21no—
27#16810486—21no—
28#16810486C2211yescharacteristic
29#16810360C2214yescomplemented
30#16810357C2214yes—
31#16810349C2214yescomplemented
32C53×#539—22no—
33C53×#539—22no—
34C53×#539—22no—
35C53×#539C2214yes—
36C2×#126×C53—21no—
37C2×#125×C53—21no—
38C2×#123×C53—21no—
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53—21no—
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×#121×C53C2211yescharacteristic
43C2×C4×D4×C53—21no—
44C22×C4⋊C4×C53C2211yescharacteristic
45C2×#11719917—21no—
46C2×#11719917—21no—
47#13133176—21no—
48#13133176—21no—
49#13133176—21no—
50#13133176—21no—
51C2×C4⋊Dic53C2311yescharacteristic
52C2×C4⋊Dic53C2311yescharacteristic
53C2×#11719919—22no—
54C2×C22⋊Dic53—21no—
55C2×C22⋊Dic53—21no—
56C2×C4×Dic53—21no—
57#13133150—22no—
58#13133150—22no—
59#13133148—22no—
60#13133148—22no—
61#13133141—22no—
62#13133141—22no—
63#13133138—22no—
64#13133138—22no—
65C22⋊#88322—21no—
66C22⋊#88322—21no—
67C22⋊#88322C2312yes—
68C22⋊#88322C2312yes—
69#13133124—22no—
70#13133124—21no—
71#13133124—22no—
72#13133124—22no—
73#13133124C2312yes—
74C22×Q8×C53D411yescharacteristic
75C22×D4×C53—21no—
76C22×D4×C53D411yescharacteristic
77C23×C4×C53—21no—
78C23×C4×C53D411yescharacteristic
79C23×C4×C53D411yescharacteristic
80C23×C4×C53C2311yescharacteristic
81#126×C53—24no—
82#126×C53—24no—
83#125×C53—24no—
84#125×C53—24no—
85#123×C53—24no—
86#123×C53—24no—
87#123×C53—24no—
88#123×C53—24no—
89C22⋊Q8×C53—24no—
90C22⋊Q8×C53—24no—
91C22⋊Q8×C53—21no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53D412yes—
94C22⋊Q8×C53C2312yes—
95#121×C53—21no—
96#121×C53—21no—
97#121×C53D412yes—
98#121×C53C2312yes—
99C4×D4×C53—24no—
100C4×D4×C53—24no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D412yes—
112C2×C4⋊C4×C53C2312yes—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53—21no—
119C24×C53—21no—
120C24×C53C2×D411yescharacteristic
121C2×Q8×C53—21no—
122C2×Q8×C53—42no—
123C2×Q8×C53C2×D412yes—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—21no—
127C2×D4×C53—42no—
128C2×D4×C53—44no—
129C2×D4×C53C2×D412yes—
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—24no—
137C22×C4×C53—21no—
138C22×C4×C53—42no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53—82no—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—22no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53C2×D412yes—
165C4⋊C4×C53C2×D412yes—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C42×C53—24no—
174C22⋊Dic53—42no—
175C22⋊Dic53—41no—
176C22⋊Dic53—41no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181#11719919—44no—
182C4×Dic53—41no—
183C4×Dic53—41no—
184#11719917—42no—
185#11719917—42no—
186C2×#88322—22no—
187C2×#88322—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—41no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53#12011yescharacteristic
196C23×C53#12011yescharacteristic
197C23×C53C22×D411yescharacteristic
198Q8×C53—42no—
199Q8×C53—81no—
200D4×C53—42no—
201D4×C53—44no—
202D4×C53—44no—
203D4×C53—81no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—81no—
225C2×C4×C53—84no—
226C2×C4×C53#12011yescharacteristic
227C2×C4×C53#12011yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237#88322—42no—
238#88322—42no—
239#18139—531nomaximal, Sylow
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—44no—
248C22×C53—42no—
249C22×C53—42no—
250C22×C53—42no—
251C22×C53—84no—
252C22×C53#45812yes—
253C22×C53C2×#12011yescharacteristic
254C22×C53#57512yes—
255C4×C53—21no—
256C4×C53—21no—
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—42no—
260C4×C53—42no—
261C4×C53—42no—
262C4×C53—42no—
263C4×C53—42no—
264Dic53—82no—
265Dic53—81no—
266Dic53—81no—
267C2×#539D5311yescharacteristic
268#3095—531no—
269#3094—531no—
270#2484—531no—
271#1615—534no—
272#1609—534no—
273#1511—531no—
274#1510—531no—
275#1463—531no—
276C2×C53—22no—
277C2×C53—22no—
278C2×C53—41no—
279C2×C53—82no—
280C2×C53#300511yescharacteristic
281C2×C53#302011yescharacteristic
282C2×C53C2×#45811yescharacteristic
283#539—22no—
284#539—22no—
285#539—22no—
286#539C2×D5314yes—
287C2×#126—21no—
288C2×#125—21no—
289C2×#123—21no—
290C2×#123—21no—
291C2×C22⋊Q8—21no—
292C2×C22⋊Q8C2×D5311yescharacteristic
293C2×#121C2×D5311yescharacteristic
294C2×C4×D4—21no—
295C2×#117—531no—
296C22×C4⋊C4C2×D5311yescharacteristic
297#485—1062no—
298#484—532no—
299#483—532no—
300#482—1062no—
301#481—532no—
302#480—532no—
303#428—1061no—
304#428—1061no—
305#419—1061no—
306#418—1061no—
307#407—531no—
308#407—531no—
309#400—532no—
310#398—532no—
311#387—1062no—
312#385—1061no—
313#381—531no—
314#381—1061no—
315#328—534no—
316#327—534no—
317#324—534no—
318C53#1813911yescharacteristic, complemented, Sylow
319C22×Q8C22⋊D5311yescharacteristic
320C22×D4—21no—
321C22×D4C22⋊D5311yescharacteristic
322C23×C4—21no—
323C23×C4C22⋊D5311yescharacteristic
324C23×C4C22⋊D5311yescharacteristic
325C23×C4C22×D5311yescharacteristic
326C2×M16—1061no—
327C2×M16—1061no—
328#126—24no—
329#126—24no—
330#125—24no—
331#125—24no—
332#123—24no—
333#123—24no—
334#123—24no—
335#123—24no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8C22⋊D5312yes—
341C22⋊Q8C22×D5312yes—
342#121—21no—
343#121—21no—
344#121C22⋊D5312yes—
345#121C22×D5312yes—
346C4×D4—24no—
347C4×D4—24no—
348#117—1061no—
349#117—1061no—
350#117—1061no—
351#117—1061no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—1062no—
363C2×C4⋊C4C22⋊D5312yes—
364C2×C4⋊C4C22⋊D5311yescharacteristic
365C2×C4⋊C4C22⋊D5311yescharacteristic
366C2×C4⋊C4C22×D5312yes—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1061no—
373C2×C22⋊C4—1061no—
374C2×C42—21no—
375C2×C42—1061no—
376#107—1062no—
377#107—1062no—
378#106—1062no—
379#106—1062no—
380#103—1062no—
381#103—1062no—
382#102—1062no—
383#102—1062no—
384C22⋊C8—532no—
385C22⋊C8—532no—
386C22⋊C8—1061no—
387C22⋊C8—1061no—
388#95—532no—
389#95—1061no—
390#95—1062no—
391#95—1062no—
392#95—1062no—
393C24—21no—
394C24C2×C22⋊D5311yescharacteristic
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8C2×C22⋊D5312yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4C2×C22⋊D5312yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—82no—
416C22×C4—1062no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4C2×C22⋊D5312yes—
420C22×C4C2×C22⋊D5311yescharacteristic
421C22×C4C2×C22⋊D5311yescharacteristic
422C22×C4C2×C22⋊D5312yes—
423C22×C4C2×C22⋊D5311yescharacteristic
424C22×C4C2×C22⋊D5311yescharacteristic
425C22×C4C2×C22⋊D5311yescharacteristic
426C22×C4C23×D5311yescharacteristic
427M16—2122no—
428M16—2122no—
429C2×C8—1062no—
430C2×C8—1062no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—1061no—
443C4⋊C4—1061no—
444C4⋊C4—1061no—
445C4⋊C4—1061no—
446C4⋊C4—2124no—
447C4⋊C4C2×C22⋊D5312yes—
448C4⋊C4C2×C22⋊D5312yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—2122no—
457C22⋊C4—2121no—
458C22⋊C4—2121no—
459C42—24no—
460C42—2121no—
461C42—2121no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23#1313318611yescharacteristic
470C23#1313318611yescharacteristic
471C23C22×C22⋊D5311yescharacteristic
472Q8—42no—
473Q8—81no—
474D4—42no—
475D4—44no—
476D4—44no—
477D4—81no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—81no—
499C2×C4—84no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2121no—
504C2×C4—2121no—
505C2×C4—2122no—
506C2×C4—2122no—
507C2×C4#1313318611yescharacteristic
508C2×C4#1313318611yescharacteristic
509C2×C4C22×C22⋊D5311yescharacteristic
510C2×C4C22×C22⋊D5311yescharacteristic
511C8—2122no—
512C8—2122no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—84no—
525C22#1681081312yes—
526C22C2×#1313318611yescharacteristic
527C22#1681142112yes—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—4242no—
538C4—4241no—
539C4—4241no—
540C2—22no—
541C2—22no—
542C2—41no—
543C2—82no—
544C2#2028395711yescharacteristic
545C2#2028407911yescharacteristic
546C2C2×#1681081311yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#539C2
socleC22×C53C2×#120
prime radicalC2×#121×C53C22
hypercenterC2×#539D53
hypocenterC53#18139
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#539

Complements

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

NH ≅ G/Ntype
1. #20284659C2semidirect
3. #20279002C2semidirect
4. #20273746C2semidirect
5. #20273712C2semidirect
6. #20273248C2semidirect
8. #20273045C2semidirect
12. #16810956C22semidirect
13. #16810951C22semidirect
23. #16810562C22semidirect
24. #16810553C22semidirect
29. #16810360C22semidirect
31. #16810349C22semidirect
318. C53#18139semidirect

Maximal subgroups

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

subgroupindex
#202846592
#202846512
#202790022
#202737462
#202737122
#202732482
#202732442
#202730452
C2×C53×#5392
#1813953