Finite Group Databasebeta

← D53×#2646 #27550533

Subgroups — D53×#2646 #27550533

Order 13568.

2374 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280688C212yesmaximal, complemented
2#20280687C214yesmaximal, complemented
3#20280686C214yesmaximal, complemented
4#20280685C211yesmaximal, characteristic, complemented
5#20280684C214yesmaximal, complemented
6#400×D53C214yesmaximal, complemented
7#398×D53C214yesmaximal, complemented
8#385×D53C214yesmaximal
9#380×D53C212yesmaximal
10C2×#95×D53C211yesmaximal, characteristic
11C53×#2646C211yesmaximal, characteristic, complemented
12C23×C4×D53C2212yes—
13C23×C4×D53C2211yescharacteristic
14C2×C4⋊C4×D53C2212yes—
15C2×C4⋊C4×D53C2212yes—
16C2×C4⋊C4×D53C2214yes—
17C2×C4⋊C4×D53C2214yes—
18C2×C22⋊C4×D53C2214yes—
19#16810565C2214yescomplemented
20#16810565C2214yescomplemented
21#16810564C2214yescomplemented
22#16810564C2214yescomplemented
23#16810563C2214yescomplemented
24#16810562C2214yescomplemented
25#16810561C2214yescomplemented
26#16810557C2214yescomplemented
27#16810557C2214yescomplemented
28#16810556C2214yescomplemented
29#16810556C2214yescomplemented
30#16810555C2214yescomplemented
31#16810554C2214yescomplemented
32#16810553C2214yescomplemented
33#16810498C2214yes—
34#16810497C2214yes—
35#16810497C2214yes—
36#16810496C2214yes—
37#16810495C2214yes—
38#16810494C2214yes—
39#16810494C2214yes—
40#16810483C2212yes—
41#16810482C2214yes—
42#16810481C2214yes—
43#16810480C2212yes—
44#16810480C2212yes—
45C2×#13133124C2212yes—
46C2×#13133124C2211yescharacteristic
47#95×D53C2214yes—
48#95×D53C2214yes—
49#95×D53C2214yes—
50#95×D53C2214yes—
51#95×D53C2214yes—
52C53×#400C2214yescomplemented
53C53×#398C2214yescomplemented
54C53×#385C2214yes—
55C53×#380C2212yes—
56C2×#95×C53C2211yescharacteristic
57C24×D53C2311yescharacteristic
58C23×Dic53C2312yes—
59C23×Dic53C2311yescharacteristic
60C22×C4×D53—28no—
61C22×C4×D53—44no—
62C22×C4×D53—44no—
63C22×C4×D53—44no—
64C22×C4×D53Q818yes—
65C22×C4×D53C2314yes—
66C22×C4×D53C2314yes—
67C22×C4×D53C2314yes—
68C22×C4×D53C2312yes—
69C22×C4×D53C2314yes—
70C22×C4×D53C2312yes—
71C2×C4⋊Dic53C2314yes—
72C2×C4⋊Dic53C2314yes—
73C2×C4⋊Dic53C2312yes—
74C2×C4⋊Dic53C2312yes—
75C2×#11719919C2314yes—
76C2×#11719919C2314yes—
77C2×#11719919C2314yes—
78C2×#11719919C2314yes—
79C2×#11719919C2314yes—
80C2×#11719919C2314yes—
81C4⋊C4×D53—44no—
82C4⋊C4×D53—44no—
83C4⋊C4×D53—48no—
84C4⋊C4×D53—48no—
85C2×C22⋊Dic53C2314yes—
86C2×#11719921C2314yes—
87C2×#11719921C2314yes—
88C22⋊C4×D53—216no—
89#13133124C2314yes—
90#13133124C2314yes—
91#13133124C2314yes—
92#13133124C2314yes—
93#13133124C2314yes—
94#13133124C2314yes—
95#13133124C2314yes—
96#13133124C2314yes—
97#13133124C2314yes—
98#13133124C2314yes—
99#13133124C2314yes—
100#13133124C2314yes—
101C23×C4×C53C2312yes—
102C23×C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2314yes—
104C2×C4⋊C4×C53C2312yes—
105C2×C4⋊C4×C53C2314yes—
106C2×C4⋊C4×C53C2312yes—
107C2×C22⋊C4×C53C2314yes—
108#95×C53C2314yes—
109#95×C53C2314yes—
110#95×C53C2314yes—
111#95×C53C2314yes—
112C24×C53C2411yescharacteristic
113C23×D53—22no—
114C23×D53—24no—
115C23×D53—22no—
116C23×D53C2×Q814yes—
117C23×D53#4118yes—
118C23×D53C2411yescharacteristic
119C23×D53C2411yescharacteristic
120C22×C4×C53—24no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53C2×Q814yes—
125C22×C4×C53C2414yes—
126C22×C4×C53C2414yes—
127C22×C4×C53C2414yes—
128C22×C4×C53C2412yes—
129C22×Dic53—24no—
130C22×Dic53—42no—
131C22×Dic53—42no—
132C22×Dic53—42no—
133C22×Dic53C2×Q814yes—
134C22×Dic53C2414yes—
135C22×Dic53C2414yes—
136C22×Dic53C2414yes—
137C22×Dic53C2412yes—
138C2×C4×D53—24no—
139C2×C4×D53—24no—
140C2×C4×D53—28no—
141C2×C4×D53—28no—
142C2×C4×D53—28no—
143C2×C4×D53—28no—
144C2×C4×D53—28no—
145C2×C4×D53—28no—
146C2×C4×D53—28no—
147C2×C4×D53—28no—
148C2×C4×D53—28no—
149C2×C4×D53—28no—
150C2×C4×D53—416no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—44no—
154C2×C4×D53—44no—
155C2×C4×D53—432no—
156C2×C4×D53C2×Q818yes—
157C2×C4×D53C2×Q818yes—
158C4⋊C4×C53—42no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—44no—
161C4⋊C4×C53—44no—
162C22⋊C4×C53—28no—
163C22⋊Dic53—28no—
164#11719921—28no—
165#11719921—28no—
166C4⋊Dic53—44no—
167C4⋊Dic53—44no—
168C4⋊Dic53—42no—
169C4⋊Dic53—42no—
170#11719919—44no—
171#11719919—44no—
172#11719919—44no—
173#11719919—44no—
174#11719919—44no—
175#11719919—44no—
176C23×C53—21no—
177C23×C53—22no—
178C23×C53—21no—
179C23×C53C22×Q812yes—
180C23×C53C2×#4114yes—
181C23×C53C2511yescharacteristic
182C22×D53—21no—
183C22×D53—22no—
184C22×D53—21no—
185C22×D53—216no—
186C22×D53—216no—
187C22×D53—28no—
188C22×D53—28no—
189C22×D53#125116yes—
190C22×D53C22×Q812yes—
191C22×D53C22×Q812yes—
192C22×D53C2×#4114yes—
193C22×D53C2×#4114yes—
194C22×D53H(5,𝔽2)12yes—
195C22×D53H(5,𝔽2)14yes—
196C22×D532−1+412yes—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—44no—
209C2×C4×C53—48no—
210C2×C4×C53C22×Q814yes—
211C2×Dic53—22no—
212C2×Dic53—22no—
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—
222C2×Dic53—44no—
223C2×Dic53—48no—
224C2×Dic53C22×Q814yes—
225C4×D53—216no—
226C4×D53—48no—
227C4×D53—416no—
228C4×D53—416no—
229C4×D53—416no—
230C4×D53—416no—
231C2×#2646—531nomaximal, Sylow
232C22×C53—24no—
233C22×C53—24no—
234C22×C53—22no—
235C22×C53—22no—
236C22×C53C2×#12514yes—
237C22×C53C23×Q811yescharacteristic
238C22×C53C22×#4112yes—
239C22×C53C2×H(5,𝔽2)11yescharacteristic
240C22×C53C2×H(5,𝔽2)12yes—
241C22×C53C2×2−1+411yescharacteristic
242C2×D53—24no—
243C2×D53—24no—
244C2×D53—22no—
245C2×D53—22no—
246C2×D53—232no—
247C2×D53C2×#12514yes—
248C2×D53C2×#12514yes—
249C2×D53#53818yes—
250C2×D53#53918yes—
251C2×D53#54314yes—
252C2×D53#54414yes—
253C4×C53—24no—
254C4×C53—44no—
255C4×C53—44no—
256C4×C53—44no—
257Dic53—24no—
258Dic53—44no—
259Dic53—44no—
260Dic53—44no—
261#2646—534no—
262#2646—534no—
263#2646—534no—
264#2646—532no—
265#2646—531no—
266#2646D5311yescharacteristic, direct product factor
267C2×#400—534no—
268C2×#398—534no—
269C2×#385—534no—
270C2×#380—532no—
271C22×#95—531no—
272C2×C53—24no—
273C2×C53C22×#12511yescharacteristic
274C2×C53C2×#53812yes—
275C2×C53C2×#53912yes—
276C2×C53C2×#54311yescharacteristic
277C2×C53C2×#54411yescharacteristic
278D53—24no—
279D53#264618yesdirect product factor
280C24×C4—531no—
281C24×C4—532no—
282C22×C4⋊C4—532no—
283C22×C4⋊C4—534no—
284C22×C4⋊C4—534no—
285C22×C4⋊C4—532no—
286C22×C22⋊C4—534no—
287#400—534no—
288#400—534no—
289#400—534no—
290#400—534no—
291#400—534no—
292#400—534no—
293#400—534no—
294#400C2×D5314yescomplemented
295#398—534no—
296#398—534no—
297#398—534no—
298#398—534no—
299#398—534no—
300#398—534no—
301#398—534no—
302#398C2×D5314yescomplemented
303#385—534no—
304#385—534no—
305#385—534no—
306#385—534no—
307#385—534no—
308#385—534no—
309#385—534no—
310#385C2×D5314yes—
311#380—532no—
312#380—532no—
313#380—532no—
314#380—534no—
315#380—534no—
316#380C2×D5312yes—
317C2×#95—532no—
318C2×#95—531no—
319C2×#95—534no—
320C2×#95—534no—
321C2×#95—534no—
322C2×#95—534no—
323C2×#95—534no—
324C2×#95C2×D5311yescharacteristic
325C53C2×#264611yescharacteristic, complemented, Sylow
326C25—531no—
327C23×C4—531no—
328C23×C4—532no—
329C23×C4—538no—
330C23×C4—532no—
331C23×C4—532no—
332C23×C4—534no—
333C23×C4—534no—
334C23×C4—534no—
335C23×C4—534no—
336C23×C4—1068no—
337C23×C4—2124no—
338C23×C4—2124no—
339C23×C4—2124no—
340C23×C4C22×D5312yes—
341C23×C4C22×D5311yescharacteristic
342C2×C4⋊C4—534no—
343C2×C4⋊C4—534no—
344C2×C4⋊C4—534no—
345C2×C4⋊C4—534no—
346C2×C4⋊C4—534no—
347C2×C4⋊C4—534no—
348C2×C4⋊C4—532no—
349C2×C4⋊C4—532no—
350C2×C4⋊C4—534no—
351C2×C4⋊C4—534no—
352C2×C4⋊C4—2124no—
353C2×C4⋊C4—2124no—
354C2×C4⋊C4—2128no—
355C2×C4⋊C4—2128no—
356C2×C4⋊C4C22×D5312yes—
357C2×C4⋊C4C22×D5312yes—
358C2×C4⋊C4C22×D5314yes—
359C2×C4⋊C4C22×D5314yes—
360C2×C22⋊C4—534no—
361C2×C22⋊C4—534no—
362C2×C22⋊C4—534no—
363C2×C22⋊C4—10616no—
364C2×C22⋊C4C22×D5314yes—
365#95—534no—
366#95—534no—
367#95—534no—
368#95—534no—
369#95—534no—
370#95—534no—
371#95—534no—
372#95—534no—
373#95—534no—
374#95—534no—
375#95—534no—
376#95—534no—
377#95C22×D5314yes—
378#95C22×D5314yes—
379#95C22×D5314yes—
380#95C22×D5314yes—
381C24—531no—
382C24—534no—
383C24—538no—
384C24—531no—
385C24—1062no—
386C24—1064no—
387C24—1062no—
388C24C23×D5311yescharacteristic
389C22×C4—24no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—532no—
394C22×C4—534no—
395C22×C4—538no—
396C22×C4—538no—
397C22×C4—534no—
398C22×C4—534no—
399C22×C4—534no—
400C22×C4—1064no—
401C22×C4—1064no—
402C22×C4—1068no—
403C22×C4—1068no—
404C22×C4—1068no—
405C22×C4—1068no—
406C22×C4—1068no—
407C22×C4—1068no—
408C22×C4—1068no—
409C22×C4—1068no—
410C22×C4—1068no—
411C22×C4—1068no—
412C22×C4—1064no—
413C22×C4—2122no—
414C22×C4—2122no—
415C22×C4—2124no—
416C22×C4—2124no—
417C22×C4—21216no—
418C22×C4—21232no—
419C22×C4—2122no—
420C22×C4—2122no—
421C22×C4—2122no—
422C22×C4Q8×D5314yes—
423C22×C4C23×D5314yes—
424C22×C4C23×D5314yes—
425C22×C4C23×D5314yes—
426C22×C4C23×D5312yes—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—2124no—
432C4⋊C4—2124no—
433C4⋊C4—2124no—
434C4⋊C4—2124no—
435C4⋊C4—2124no—
436C4⋊C4—2124no—
437C4⋊C4—2122no—
438C4⋊C4—2122no—
439C4⋊C4—2124no—
440C4⋊C4—2124no—
441C22⋊C4—28no—
442C22⋊C4—1068no—
443C22⋊C4—1068no—
444C22⋊C4—1068no—
445C23—21no—
446C23—22no—
447C23—21no—
448C23—532no—
449C23—534no—
450C23—532no—
451C23—534no—
452C23—532no—
453C23—5316no—
454C23—532no—
455C23—534no—
456C23—10616no—
457C23—1068no—
458C23—10616no—
459C23—1068no—
460C23—1061no—
461C23—1062no—
462C23—1061no—
463C23C2×Q8×D5312yes—
464C23#41×D5314yes—
465C23C24×D5311yescharacteristic
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—534no—
480C2×C4—10616no—
481C2×C4—1062no—
482C2×C4—1062no—
483C2×C4—1064no—
484C2×C4—1064no—
485C2×C4—1064no—
486C2×C4—1064no—
487C2×C4—1064no—
488C2×C4—1064no—
489C2×C4—1064no—
490C2×C4—1064no—
491C2×C4—1064no—
492C2×C4—2128no—
493C2×C4—21216no—
494C2×C4—21216no—
495C2×C4—21216no—
496C2×C4—21216no—
497C2×C4—2124no—
498C2×C4—2128no—
499C2×C4C2×Q8×D5314yes—
500C22—24no—
501C22—24no—
502C22—22no—
503C22—22no—
504C22—538no—
505C22—538no—
506C22—534no—
507C22—534no—
508C22—534no—
509C22—534no—
510C22—10632no—
511C22—1064no—
512C22—1064no—
513C22—1062no—
514C22—1062no—
515C22#125×D5314yes—
516C22C22×Q8×D5311yescharacteristic
517C22C2×#41×D5312yes—
518C22H(5,𝔽2)×D5311yescharacteristic
519C22H(5,𝔽2)×D5312yes—
520C222−1+4×D5311yescharacteristic
521C4—24no—
522C4—44no—
523C4—44no—
524C4—44no—
525C4—1064no—
526C4—2124no—
527C4—2124no—
528C4—2124no—
529C2—24no—
530C2—538no—
531C2—1064no—
532C2C2×#125×D5311yescharacteristic
533C2#538×D5312yes—
534C2#539×D5312yes—
535C2#543×D5311yescharacteristic
536C2#544×D5311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D53
commutator, socle, cosocleC23×C53C25
FittingC53×#2646C2
prime radicalC24×D53C23
hypercenter#2646D53
hypocenterC53C2×#2646
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2646

Complements

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

NH ≅ G/Ntype
1. #20280688C2semidirect
2. #20280687C2semidirect
3. #20280686C2semidirect
4. #20280685C2semidirect
5. #20280684C2semidirect
6. #400×D53C2semidirect
7. #398×D53C2semidirect
11. C53×#2646C2semidirect
19. #16810565C22semidirect
20. #16810565C22semidirect
21. #16810564C22semidirect
22. #16810564C22semidirect
23. #16810563C22semidirect
24. #16810562C22semidirect
25. #16810561C22semidirect
26. #16810557C22semidirect
27. #16810557C22semidirect
28. #16810556C22semidirect
29. #16810556C22semidirect
30. #16810555C22semidirect
31. #16810554C22semidirect
32. #16810553C22semidirect
52. C53×#400C22semidirect
53. C53×#398C22semidirect
266. #2646D53direct (N × H)
279. D53#2646direct (N × H)
294. #400C2×D53semidirect
302. #398C2×D53semidirect
325. C53C2×#2646semidirect

Maximal subgroups

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

subgroupindex
#202806882
#202806872
#202806862
#202806852
#202806842
#400×D532
#398×D532
#385×D532
#380×D532
C2×#95×D532
C53×#26462
C2×#264653