Finite Group Databasebeta

← D53×#2330 #27542917

Subgroups — D53×#2330 #27542917

Order 13568.

2398 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D53C211yesmaximal, characteristic
2#20277457C211yesmaximal, characteristic, complemented
3#20277456C211yesmaximal, characteristic, complemented
4#20277455C212yesmaximal, complemented
5#20277454C211yesmaximal, characteristic, complemented
6#20277453C214yesmaximal, complemented
7#20277452C212yesmaximal, complemented
8#20277451C214yesmaximal, complemented
9#389×D53C214yesmaximal
10#389×D53C212yesmaximal
11#387×D53C214yesmaximal
12#384×D53C211yesmaximal, characteristic
13C4×C4⋊C4×D53C212yesmaximal
14#376×D53C211yesmaximal, characteristic
15C53×#2330C211yesmaximal, characteristic, complemented
16C2×#13133209C2211yescharacteristic
17C2×#13133208C2212yes—
18C2×#13133207C2214yes—
19#125×D53C4116yescomplemented
20C2×C4⋊C4×D53C2212yes—
21C2×C4⋊C4×D53C2214yes—
22C2×C4⋊C4×D53C2214yes—
23C2×C4⋊C4×D53C2212yes—
24C2×C4⋊C4×D53C2214yes—
25C2×C42×D53C2211yescharacteristic
26C2×C42×D53C2214yes—
27C2×C42×D53C2211yescharacteristic
28C2×C42×D53C2211yescharacteristic
29#16810522C2214yes—
30#16810522C2212yes—
31#16810521C2214yes—
32#16810521C2212yes—
33#16810520C2214yes—
34#16810520C2214yes—
35#16810520C2214yes—
36#16810519C2212yes—
37#16810519C2214yes—
38#16810518C2214yes—
39#16810518C2214yes—
40#16810518C2214yes—
41#16810510C2214yes—
42#16810509C2214yes—
43#16810508C2214yes—
44#16810507C2214yes—
45#16810506C2214yes—
46#16810505C2214yes—
47#16810504C2214yes—
48#16810493C2211yescharacteristic
49#16810492C2214yes—
50#16810492C2212yes—
51C4⋊C4×Dic53C2212yes—
52C4⋊C4×Dic53C2212yes—
53C4×#11719919C2214yes—
54C4×C4⋊Dic53C2212yes—
55#16810474C2214yes—
56#16810469C2211yescharacteristic
57#16810469C2211yescharacteristic
58#16810469C2211yescharacteristic
59#16810468C2212yes—
60#16810468C2212yes—
61#95×D53C2212yes—
62#95×D53C2212yes—
63#95×D53C2214yes—
64#95×D53C2214yes—
65C2×#125×C53C2211yescharacteristic
66C53×#389C2214yes—
67C53×#389C2212yes—
68C53×#387C2214yes—
69C53×#384C2211yescharacteristic
70C4×C4⋊C4×C53C2212yes—
71C53×#376C2211yescharacteristic
72C22×C4×D53C2311yescharacteristic
73C22×C4×D53C2314yes—
74C22×C4×D53C2311yescharacteristic
75C22×C4×D53C2314yes—
76C22×C4×D53C2312yes—
77C22×C4×D53C2312yes—
78C22×C4×D53C2311yescharacteristic
79#13133209C2×C418yescomplemented
80#13133208C2×C4116yescomplemented
81#13133207C2×C4132yescomplemented
82C2×C4⋊Dic53C2312yes—
83C2×C4⋊Dic53C2314yes—
84C2×C4⋊Dic53C2314yes—
85C2×C4⋊Dic53C2314yes—
86C2×C4⋊Dic53C2312yes—
87C2×#11719919C2314yes—
88C2×#11719919C2314yes—
89C2×#11719919C2314yes—
90C2×#11719919C2314yes—
91C2×#11719919C2314yes—
92C2×#11719919C2314yes—
93C2×#11719919C2314yes—
94C2×#11719919C2314yes—
95C4⋊C4×D53—28no—
96C4⋊C4×D53—48no—
97C4⋊C4×D53—48no—
98C4⋊C4×D53C2×C4132yes—
99C4⋊C4×D53C2×C4116yes—
100C2×C4×Dic53C2311yescharacteristic
101C2×C4×Dic53C2311yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103C2×C4×Dic53C2314yes—
104C2×C4×Dic53C2314yes—
105C2×C4×Dic53C2314yes—
106C2×C4×Dic53C2312yes—
107C2×C4×Dic53C2311yescharacteristic
108C2×C4×Dic53C2312yes—
109C2×C4×Dic53C2311yescharacteristic
110C42×D53—24no—
111C42×D53—48no—
112C42×D53C2×C418yes—
113C42×D53Q818yes—
114#13133124C2314yes—
115#13133124C2312yes—
116#13133124C2314yes—
117#13133124C2312yes—
118#13133124C2314yes—
119#13133124C2312yes—
120#13133124C2314yes—
121#13133124C2314yes—
122#13133124C2314yes—
123#13133124C2312yes—
124#13133124C2312yes—
125#13133124C2312yes—
126#125×C53C2×C418yescomplemented
127C2×C4⋊C4×C53C2312yes—
128C2×C4⋊C4×C53C2314yes—
129C2×C4⋊C4×C53C2314yes—
130C2×C4⋊C4×C53C2314yes—
131C2×C4⋊C4×C53C2312yes—
132C2×C42×C53C2311yescharacteristic
133C2×C42×C53C2314yes—
134C2×C42×C53C2311yescharacteristic
135C2×C42×C53C2311yescharacteristic
136#95×C53C2314yes—
137#95×C53C2314yes—
138#95×C53C2312yes—
139#95×C53C2312yes—
140C23×D53C2411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×C4×C53C2414yes—
144C22×C4×C53C2414yes—
145C22×C4×C53C2412yes—
146C22×C4×C53C2412yes—
147C22×C4×C53C2411yescharacteristic
148C22×Dic53C2411yescharacteristic
149C22×Dic53C2411yescharacteristic
150C22×Dic53C2414yes—
151C22×Dic53C2414yes—
152C22×Dic53C2412yes—
153C22×Dic53C2412yes—
154C22×Dic53C2411yescharacteristic
155C2×C4×D53—24no—
156C2×C4×D53—28no—
157C2×C4×D53—24no—
158C2×C4×D53—28no—
159C2×C4×D53—24no—
160C2×C4×D53—24no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—28no—
164C2×C4×D53—28no—
165C2×C4×D53—24no—
166C2×C4×D53—28no—
167C2×C4×D53—24no—
168C2×C4×D53C22×C418yes—
169C2×C4×D53C22×C4116yes—
170C2×C4×D53C22×C414yes—
171C2×C4×D53C2×Q818yes—
172C2×C4×D53C2×Q814yes—
173C2×C4×D53#4114yes—
174C4⋊C4×C53—24no—
175C4⋊C4×C53—44no—
176C4⋊C4×C53—44no—
177C4⋊C4×C53C22×C4116yes—
178C4⋊C4×C53C22×C418yes—
179C42×C53—22no—
180C42×C53—44no—
181C42×C53C22×C414yes—
182C42×C53C2×Q814yes—
183C4⋊Dic53—24no—
184C4⋊Dic53—44no—
185C4⋊Dic53—44no—
186C4⋊Dic53C22×C4116yes—
187C4⋊Dic53C22×C418yes—
188#11719919—28no—
189#11719919—44no—
190#11719919—44no—
191#11719919—44no—
192#11719919—44no—
193#11719919C22×C4116yes—
194#11719919C22×C4116yes—
195#11719919C22×C4116yes—
196C4×Dic53—22no—
197C4×Dic53—22no—
198C4×Dic53—22no—
199C4×Dic53—44no—
200C4×Dic53—44no—
201C4×Dic53—44no—
202C4×Dic53C22×C418yes—
203C4×Dic53C22×C414yes—
204C4×Dic53C2×Q814yes—
205C4×Dic53C2×Q818yes—
206C23×C53C2511yescharacteristic
207C22×D53C23×C412yes—
208C22×D53C22×Q812yes—
209C22×D53C2×#4112yes—
210C22×D53H(5,𝔽2)14yes—
211C22×D532−1+414yes—
212C2×C4×C53—22no—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53C23×C414yes—
226C2×C4×C53C23×C418yes—
227C2×C4×C53C23×C412yes—
228C2×C4×C53C22×Q814yes—
229C2×C4×C53C22×Q812yes—
230C2×C4×C53C2×#4112yes—
231C2×Dic53—22no—
232C2×Dic53—24no—
233C2×Dic53—22no—
234C2×Dic53—24no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—22no—
242C2×Dic53—24no—
243C2×Dic53—22no—
244C2×Dic53C23×C414yes—
245C2×Dic53C23×C418yes—
246C2×Dic53C23×C412yes—
247C2×Dic53C22×Q814yes—
248C2×Dic53C22×Q812yes—
249C2×Dic53C2×#4112yes—
250C4×D53—232no—
251C4×D53—216no—
252C4×D53—216no—
253C4×D53—44no—
254C4×D53—416no—
255C4×D53—44no—
256C4×D53C4×Q8116yes—
257C2×#2330—531nomaximal, Sylow
258C22×C53C24×C411yescharacteristic
259C22×C53C23×Q811yescharacteristic
260C22×C53C22×#4111yescharacteristic
261C22×C53C2×H(5,𝔽2)12yes—
262C22×C53C2×2−1+412yes—
263C2×D53C2×C4×Q814yes—
264C2×D53#52018yes—
265C2×D53#54014yes—
266C2×D53#54114yes—
267C2×D53#54418yes—
268C4×C53—28no—
269C4×C53—24no—
270C4×C53—24no—
271C4×C53—41no—
272C4×C53—44no—
273C4×C53—41no—
274C4×C53C2×C4×Q814yes—
275Dic53—28no—
276Dic53—24no—
277Dic53—24no—
278Dic53—41no—
279Dic53—44no—
280Dic53—41no—
281Dic53C2×C4×Q814yes—
282C22×#125—531no—
283#2330—534no—
284#2330—532no—
285#2330—534no—
286#2330—531no—
287#2330—532no—
288#2330—531no—
289#2330—531no—
290#2330D5311yescharacteristic, direct product factor
291C2×#389—532no—
292C2×#389—534no—
293C2×#387—534no—
294C2×#384—531no—
295C2×C4×C4⋊C4—532no—
296C2×#376—531no—
297C2×C53C22×C4×Q811yescharacteristic
298C2×C53C2×#52012yes—
299C2×C53C2×#54011yescharacteristic
300C2×C53C2×#54111yescharacteristic
301C2×C53C2×#54412yes—
302D53#233018yesdirect product factor
303C2×#125—5316no—
304C2×#125—534no—
305C2×#125—532no—
306C2×#125—531no—
307C2×#125C2×D5311yescharacteristic
308C22×C4⋊C4—532no—
309C22×C4⋊C4—534no—
310C22×C4⋊C4—532no—
311C22×C4⋊C4—534no—
312C22×C4⋊C4—534no—
313C22×C42—531no—
314C22×C42—531no—
315C22×C42—531no—
316C22×C42—534no—
317#389—534no—
318#389—534no—
319#389—534no—
320#389—534no—
321#389—532no—
322#389—534no—
323#389—532no—
324#389—534no—
325#389—532no—
326#389—534no—
327#389—534no—
328#389—534no—
329#389C2×D5314yes—
330#389C2×D5312yes—
331#387—534no—
332#387—534no—
333#387—534no—
334#387—534no—
335#387—534no—
336#387—534no—
337#387—534no—
338#387C2×D5314yes—
339#384—534no—
340#384—532no—
341#384—531no—
342#384C2×D5311yescharacteristic
343C4×C4⋊C4—532no—
344C4×C4⋊C4—532no—
345C4×C4⋊C4—532no—
346C4×C4⋊C4—534no—
347C4×C4⋊C4—534no—
348C4×C4⋊C4C2×D5312yes—
349#376—531no—
350#376—531no—
351#376—531no—
352#376—532no—
353#376—532no—
354#376C2×D5311yescharacteristic
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—534no—
358C2×#95—534no—
359C53C2×#233011yescharacteristic, complemented, Sylow
360C23×C4—531no—
361C23×C4—532no—
362C23×C4—534no—
363C23×C4—532no—
364C23×C4—531no—
365C23×C4—534no—
366C23×C4—531no—
367#125—5332no—
368#125—5316no—
369#125—538no—
370#125C4×D5318yescomplemented
371C2×C4⋊C4—534no—
372C2×C4⋊C4—534no—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—534no—
376C2×C4⋊C4—5316no—
377C2×C4⋊C4—5332no—
378C2×C4⋊C4—534no—
379C2×C4⋊C4—532no—
380C2×C4⋊C4—534no—
381C2×C4⋊C4—534no—
382C2×C4⋊C4—534no—
383C2×C4⋊C4—534no—
384C2×C4⋊C4—534no—
385C2×C4⋊C4—534no—
386C2×C4⋊C4—1068no—
387C2×C4⋊C4—2128no—
388C2×C4⋊C4—2128no—
389C2×C4⋊C4C22×D5312yes—
390C2×C4⋊C4C22×D5314yes—
391C2×C4⋊C4C22×D5314yes—
392C2×C4⋊C4C22×D5312yes—
393C2×C4⋊C4C22×D5314yes—
394C2×C42—532no—
395C2×C42—531no—
396C2×C42—532no—
397C2×C42—531no—
398C2×C42—538no—
399C2×C42—538no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—534no—
404C2×C42—534no—
405C2×C42—534no—
406C2×C42—1064no—
407C2×C42—2128no—
408C2×C42C22×D5311yescharacteristic
409C2×C42C22×D5314yes—
410C2×C42C22×D5311yescharacteristic
411C2×C42C22×D5311yescharacteristic
412#95—532no—
413#95—532no—
414#95—532no—
415#95—532no—
416#95—532no—
417#95—532no—
418#95—534no—
419#95—534no—
420#95—534no—
421#95—534no—
422#95—534no—
423#95—534no—
424#95C22×D5312yes—
425#95C22×D5312yes—
426#95C22×D5314yes—
427#95C22×D5314yes—
428C24—531no—
429C22×C4—531no—
430C22×C4—534no—
431C22×C4—534no—
432C22×C4—534no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—534no—
436C22×C4—538no—
437C22×C4—5316no—
438C22×C4—538no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—534no—
442C22×C4—1064no—
443C22×C4—1068no—
444C22×C4—1064no—
445C22×C4—1064no—
446C22×C4—1068no—
447C22×C4—1064no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1068no—
451C22×C4—1068no—
452C22×C4—1064no—
453C22×C4—1068no—
454C22×C4—1064no—
455C22×C4C23×D5311yescharacteristic
456C22×C4C23×D5311yescharacteristic
457C22×C4C23×D5314yes—
458C22×C4C23×D5314yes—
459C22×C4C23×D5312yes—
460C22×C4C23×D5312yes—
461C22×C4C23×D5311yescharacteristic
462C4⋊C4—24no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—5316no—
466C4⋊C4—5316no—
467C4⋊C4—5316no—
468C4⋊C4—5316no—
469C4⋊C4—538no—
470C4⋊C4—1068no—
471C4⋊C4—1064no—
472C4⋊C4—2124no—
473C4⋊C4—2124no—
474C4⋊C4—2124no—
475C4⋊C4—2124no—
476C4⋊C4—2124no—
477C4⋊C4—2124no—
478C4⋊C4C2×C4×D53116yes—
479C4⋊C4C2×C4×D5318yes—
480C42—22no—
481C42—44no—
482C42—538no—
483C42—534no—
484C42—538no—
485C42—534no—
486C42—1062no—
487C42—1062no—
488C42—1062no—
489C42—2124no—
490C42—2124no—
491C42—2124no—
492C42C2×C4×D5314yes—
493C42Q8×D5314yes—
494C23—532no—
495C23—532no—
496C23—532no—
497C23—534no—
498C23—534no—
499C23C24×D5311yescharacteristic
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—532no—
514C2×C4—532no—
515C2×C4—532no—
516C2×C4—5316no—
517C2×C4—534no—
518C2×C4—538no—
519C2×C4—534no—
520C2×C4—10616no—
521C2×C4—10616no—
522C2×C4—10632no—
523C2×C4—1062no—
524C2×C4—1064no—
525C2×C4—1062no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—1064no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1064no—
532C2×C4—1064no—
533C2×C4—1062no—
534C2×C4—1064no—
535C2×C4—1062no—
536C2×C4—2124no—
537C2×C4—21216no—
538C2×C4—2124no—
539C2×C4C22×C4×D5314yes—
540C2×C4C22×C4×D5318yes—
541C2×C4C22×C4×D5312yes—
542C2×C4C2×Q8×D5314yes—
543C2×C4C2×Q8×D5312yes—
544C2×C4#41×D5312yes—
545C22—534no—
546C22—534no—
547C22—538no—
548C22—534no—
549C22—538no—
550C22C23×C4×D5311yescharacteristic
551C22C22×Q8×D5311yescharacteristic
552C22C2×#41×D5311yescharacteristic
553C22H(5,𝔽2)×D5312yes—
554C222−1+4×D5312yes—
555C4—24no—
556C4—28no—
557C4—24no—
558C4—41no—
559C4—44no—
560C4—41no—
561C4—534no—
562C4—1064no—
563C4—1064no—
564C4—1068no—
565C4—2121no—
566C4—2124no—
567C4—2121no—
568C4C4×Q8×D5314yes—
569C2—538no—
570C2C2×C4×Q8×D5311yescharacteristic
571C2#520×D5312yes—
572C2#540×D5311yescharacteristic
573C2#541×D5311yescharacteristic
574C2#544×D5312yes—
subgroupG/H
center, FrattiniC23C24×D53
commutatorC22×C53C24×C4
FittingC53×#2330C2
socle, cosocleC23×C53C25
prime radicalC23×D53C24
hypercenter#2330D53
hypocenterC53C2×#2330
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #20277457C2semidirect
3. #20277456C2semidirect
4. #20277455C2semidirect
5. #20277454C2semidirect
6. #20277453C2semidirect
7. #20277452C2semidirect
8. #20277451C2semidirect
15. C53×#2330C2semidirect
19. #125×D53C4semidirect
79. #13133209C2×C4semidirect
80. #13133208C2×C4semidirect
81. #13133207C2×C4semidirect
126. #125×C53C2×C4semidirect
290. #2330D53direct (N × H)
302. D53#2330direct (N × H)
359. C53C2×#2330semidirect
370. #125C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D532
#202774572
#202774562
#202774552
#202774542
#202774532
#202774522
#202774512
#389×D532
#387×D532
#384×D532
C4×C4⋊C4×D532
#376×D532
C53×#23302
C2×#233053