Finite Group Databasebeta

← D53×#2608 #27555519

Subgroups — D53×#2608 #27555519

Order 13568.

2270 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280385C211yesmaximal, characteristic, complemented
2#20280384C214yesmaximal, complemented
3#20280383C214yesmaximal, complemented
4#20280382C212yesmaximal, complemented
5#20280381C214yesmaximal, complemented
6#396×D53C214yesmaximal
7#389×D53C214yesmaximal, complemented
8#387×D53C214yesmaximal, complemented
9#381×D53C211yesmaximal, characteristic
10C4×C22⋊C4×D53C212yesmaximal
11C53×#2608C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C2×C4⋊C4×D53C2214yes—
14C2×C4⋊C4×D53C2214yes—
15C2×C4⋊C4×D53C2214yes—
16C2×C22⋊C4×D53C2214yes—
17C2×C22⋊C4×D53C2212yes—
18C2×C42×D53C2212yes—
19C2×C42×D53C2212yes—
20#16810547C2214yes—
21#16810546C2214yes—
22#16810546C2214yes—
23#16810546C2214yes—
24#16810545C2214yes—
25#16810545C2214yes—
26#16810545C2214yes—
27#16810522C2214yescomplemented
28#16810521C2214yescomplemented
29#16810520C2214yescomplemented
30#16810520C2214yescomplemented
31#16810519C2214yescomplemented
32#16810518C2214yescomplemented
33#16810518C2214yescomplemented
34#16810510C2214yescomplemented
35#16810509C2214yescomplemented
36#16810508C2214yescomplemented
37#16810507C2214yescomplemented
38#16810506C2214yescomplemented
39#16810505C2214yescomplemented
40#16810504C2214yescomplemented
41#16810486C2211yescharacteristic
42#16810485C2214yes—
43#16810484C2212yes—
44C22⋊C4×Dic53C2214yes—
45C4×#11719921C2212yes—
46C4×#11719921C2212yes—
47C4×C22⋊Dic53C2212yes—
48#16810470C2214yes—
49#95×D53C2214yes—
50#95×D53C2214yes—
51#95×D53C2214yes—
52C53×#396C2214yes—
53C53×#389C2214yescomplemented
54C53×#387C2214yescomplemented
55C53×#381C2211yescharacteristic
56C4×C22⋊C4×C53C2212yes—
57C24×D53C2311yescharacteristic
58C23×Dic53C2311yescharacteristic
59C22×C4×D53—44no—
60C22×C4×D53Q818yes—
61C22×C4×D53C2314yes—
62C22×C4×D53C2314yes—
63C22×C4×D53C2312yes—
64C22×C4×D53C2314yes—
65C22×C4×D53C2312yes—
66C2×C4⋊Dic53C2314yes—
67C2×C4⋊Dic53C2314yes—
68C2×C4⋊Dic53C2314yes—
69C2×#11719919C2314yes—
70C2×#11719919C2314yes—
71C2×#11719919C2314yes—
72C2×#11719919C2314yes—
73C2×#11719919C2314yes—
74C2×#11719919C2314yes—
75C4⋊C4×D53—216no—
76C4⋊C4×D53—216no—
77C4⋊C4×D53—48no—
78C2×C22⋊Dic53C2314yes—
79C2×C22⋊Dic53C2312yes—
80C2×#11719921C2314yes—
81C2×#11719921C2314yes—
82C2×#11719921C2314yes—
83C22⋊C4×D53—216no—
84C22⋊C4×D53—44no—
85C2×C4×Dic53C2314yes—
86C2×C4×Dic53C2312yes—
87C2×C4×Dic53C2314yes—
88C2×C4×Dic53C2312yes—
89C42×D53—44no—
90C42×D53—44no—
91#13133124C2314yes—
92#13133124C2314yes—
93#13133124C2314yes—
94#13133124C2314yes—
95#13133124C2314yes—
96#13133124C2314yes—
97#13133124C2314yes—
98#13133124C2314yes—
99#13133124C2314yes—
100C23×C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2314yes—
102C2×C4⋊C4×C53C2314yes—
103C2×C4⋊C4×C53C2314yes—
104C2×C22⋊C4×C53C2314yes—
105C2×C22⋊C4×C53C2312yes—
106C2×C42×C53C2312yes—
107C2×C42×C53C2312yes—
108#95×C53C2314yes—
109#95×C53C2314yes—
110#95×C53C2314yes—
111C24×C53C2411yescharacteristic
112C23×D53—28no—
113C23×D53—24no—
114C23×D53C2×Q814yes—
115C23×D53C2411yescharacteristic
116C23×D53C2411yescharacteristic
117C22×C4×C53—42no—
118C22×C4×C53C2×Q814yes—
119C22×C4×C53C2414yes—
120C22×C4×C53C2414yes—
121C22×C4×C53C2412yes—
122C22×C4×C53C2414yes—
123C22×Dic53—42no—
124C22×Dic53C2×Q814yes—
125C22×Dic53C2414yes—
126C22×Dic53C2414yes—
127C22×Dic53C2412yes—
128C22×Dic53C2414yes—
129C2×C4×D53—28no—
130C2×C4×D53—28no—
131C2×C4×D53—24no—
132C2×C4×D53—24no—
133C2×C4×D53—28no—
134C2×C4×D53—28no—
135C2×C4×D53—28no—
136C2×C4×D53—28no—
137C2×C4×D53—28no—
138C2×C4×D53—28no—
139C2×C4×D53—416no—
140C2×C4×D53—42no—
141C2×C4×D53—42no—
142C2×C4×D53C2×Q818yes—
143C2×C4×D53C2×Q818yes—
144C2×C4×D53#41116yes—
145C4⋊C4×C53—28no—
146C4⋊C4×C53—28no—
147C4⋊C4×C53—44no—
148C22⋊C4×C53—28no—
149C22⋊C4×C53—42no—
150C42×C53—42no—
151C42×C53—42no—
152C22⋊Dic53—28no—
153C22⋊Dic53—42no—
154#11719921—28no—
155#11719921—28no—
156#11719921—44no—
157C4⋊Dic53—28no—
158C4⋊Dic53—28no—
159C4⋊Dic53—44no—
160#11719919—28no—
161#11719919—28no—
162#11719919—28no—
163#11719919—28no—
164#11719919—44no—
165#11719919—44no—
166C4×Dic53—44no—
167C4×Dic53—44no—
168C4×Dic53—42no—
169C4×Dic53—42no—
170C23×C53—24no—
171C23×C53—22no—
172C23×C53C22×Q812yes—
173C23×C53C2511yescharacteristic
174C22×D53—24no—
175C22×D53—22no—
176C22×D53—216no—
177C22×D53—28no—
178C22×D53—416no—
179C22×D53C22×Q812yes—
180C22×D53C22×Q812yes—
181C22×D53C2×#4118yes—
182C22×D53H(5,𝔽2)14yes—
183C2×C4×C53—24no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—24no—
187C2×C4×C53—24no—
188C2×C4×C53—24no—
189C2×C4×C53—24no—
190C2×C4×C53—24no—
191C2×C4×C53—24no—
192C2×C4×C53—24no—
193C2×C4×C53—44no—
194C2×C4×C53C22×Q814yes—
195C2×C4×C53C2×#4118yes—
196C2×Dic53—24no—
197C2×Dic53—22no—
198C2×Dic53—22no—
199C2×Dic53—24no—
200C2×Dic53—24no—
201C2×Dic53—24no—
202C2×Dic53—24no—
203C2×Dic53—24no—
204C2×Dic53—24no—
205C2×Dic53—24no—
206C2×Dic53—44no—
207C2×Dic53C22×Q814yes—
208C2×Dic53C2×#4118yes—
209C4×D53—216no—
210C4×D53—232no—
211C4×D53—48no—
212C4×D53—416no—
213C4×D53—416no—
214C2×#2608—531nomaximal, Sylow
215C22×C53—24no—
216C22×C53—22no—
217C22×C53—44no—
218C22×C53C23×Q811yescharacteristic
219C22×C53C22×#4114yes—
220C22×C53C2×H(5,𝔽2)12yes—
221C2×D53—24no—
222C2×D53—22no—
223C2×D53—44no—
224C2×D53—416no—
225C2×D53#53318yes—
226C2×D53#54314yes—
227C2×D53#553116yes—
228C4×C53—24no—
229C4×C53—28no—
230C4×C53—44no—
231C4×C53—44no—
232Dic53—24no—
233Dic53—28no—
234Dic53—44no—
235Dic53—44no—
236#2608—534no—
237#2608—534no—
238#2608—534no—
239#2608—531no—
240#2608—532no—
241#2608D5311yescharacteristic, direct product factor
242C2×#396—534no—
243C2×#389—534no—
244C2×#387—534no—
245C2×#381—531no—
246C2×C4×C22⋊C4—532no—
247C2×C53—42no—
248C2×C53C2×#53312yes—
249C2×C53C2×#54311yescharacteristic
250C2×C53C2×#55314yes—
251D53—42no—
252D53#260818yesdirect product factor
253C24×C4—531no—
254C22×C4⋊C4—534no—
255C22×C4⋊C4—534no—
256C22×C4⋊C4—534no—
257C22×C22⋊C4—534no—
258C22×C22⋊C4—532no—
259C22×C42—532no—
260C22×C42—532no—
261#396—534no—
262#396—534no—
263#396—534no—
264#396—534no—
265#396—534no—
266#396—534no—
267#396—534no—
268#396C2×D5314yes—
269#389—534no—
270#389—534no—
271#389—534no—
272#389—534no—
273#389—534no—
274#389—534no—
275#389—534no—
276#389C2×D5314yescomplemented
277#387—534no—
278#387—534no—
279#387—534no—
280#387—534no—
281#387—534no—
282#387—534no—
283#387—534no—
284#387C2×D5314yescomplemented
285#381—531no—
286#381—532no—
287#381—534no—
288#381C2×D5311yescharacteristic
289C4×C22⋊C4—532no—
290C4×C22⋊C4—534no—
291C4×C22⋊C4—532no—
292C4×C22⋊C4—532no—
293C4×C22⋊C4—534no—
294C4×C22⋊C4C2×D5312yes—
295C2×#95—534no—
296C2×#95—534no—
297C2×#95—534no—
298C53C2×#260811yescharacteristic, complemented, Sylow
299C25—531no—
300C23×C4—531no—
301C23×C4—534no—
302C23×C4—532no—
303C23×C4—538no—
304C23×C4—532no—
305C23×C4—534no—
306C23×C4—534no—
307C23×C4—2124no—
308C23×C4C22×D5311yescharacteristic
309C2×C4⋊C4—534no—
310C2×C4⋊C4—534no—
311C2×C4⋊C4—534no—
312C2×C4⋊C4—534no—
313C2×C4⋊C4—534no—
314C2×C4⋊C4—534no—
315C2×C4⋊C4—534no—
316C2×C4⋊C4—534no—
317C2×C4⋊C4—534no—
318C2×C4⋊C4—10616no—
319C2×C4⋊C4—10616no—
320C2×C4⋊C4—2128no—
321C2×C4⋊C4C22×D5314yes—
322C2×C4⋊C4C22×D5314yes—
323C2×C4⋊C4C22×D5314yes—
324C2×C22⋊C4—534no—
325C2×C22⋊C4—534no—
326C2×C22⋊C4—534no—
327C2×C22⋊C4—532no—
328C2×C22⋊C4—534no—
329C2×C22⋊C4—10616no—
330C2×C22⋊C4—2124no—
331C2×C22⋊C4C22×D5314yes—
332C2×C22⋊C4C22×D5312yes—
333C2×C42—534no—
334C2×C42—532no—
335C2×C42—534no—
336C2×C42—532no—
337C2×C42—2124no—
338C2×C42—2124no—
339C2×C42C22×D5312yes—
340C2×C42C22×D5312yes—
341#95—534no—
342#95—534no—
343#95—534no—
344#95—534no—
345#95—534no—
346#95—534no—
347#95—534no—
348#95—534no—
349#95—534no—
350#95C22×D5314yes—
351#95C22×D5314yes—
352#95C22×D5314yes—
353C24—531no—
354C24—534no—
355C24—531no—
356C24—1068no—
357C24—1064no—
358C24C23×D5311yescharacteristic
359C22×C4—42no—
360C22×C4—532no—
361C22×C4—534no—
362C22×C4—534no—
363C22×C4—538no—
364C22×C4—5316no—
365C22×C4—538no—
366C22×C4—534no—
367C22×C4—534no—
368C22×C4—1068no—
369C22×C4—1064no—
370C22×C4—1064no—
371C22×C4—1068no—
372C22×C4—1068no—
373C22×C4—1068no—
374C22×C4—1068no—
375C22×C4—1068no—
376C22×C4—1068no—
377C22×C4—1068no—
378C22×C4—2122no—
379C22×C4—2122no—
380C22×C4—21216no—
381C22×C4—2122no—
382C22×C4Q8×D5314yes—
383C22×C4C23×D5314yes—
384C22×C4C23×D5314yes—
385C22×C4C23×D5312yes—
386C22×C4C23×D5314yes—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—44no—
390C4⋊C4—1068no—
391C4⋊C4—1068no—
392C4⋊C4—1068no—
393C4⋊C4—1068no—
394C4⋊C4—1068no—
395C4⋊C4—1068no—
396C4⋊C4—2124no—
397C4⋊C4—2124no—
398C4⋊C4—2124no—
399C22⋊C4—28no—
400C22⋊C4—42no—
401C22⋊C4—1068no—
402C22⋊C4—1068no—
403C22⋊C4—1068no—
404C22⋊C4—2124no—
405C22⋊C4—2122no—
406C42—42no—
407C42—42no—
408C42—2124no—
409C42—2122no—
410C42—2122no—
411C42—2124no—
412C23—24no—
413C23—22no—
414C23—532no—
415C23—538no—
416C23—534no—
417C23—532no—
418C23—1068no—
419C23—10616no—
420C23—1064no—
421C23—1062no—
422C23—21216no—
423C23C2×Q8×D5312yes—
424C23C24×D5311yescharacteristic
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—44no—
436C2×C4—534no—
437C2×C4—538no—
438C2×C4—10632no—
439C2×C4—10616no—
440C2×C4—1064no—
441C2×C4—1064no—
442C2×C4—1062no—
443C2×C4—1062no—
444C2×C4—1064no—
445C2×C4—1064no—
446C2×C4—1064no—
447C2×C4—1064no—
448C2×C4—1064no—
449C2×C4—1064no—
450C2×C4—2128no—
451C2×C4—21216no—
452C2×C4—21216no—
453C2×C4—2124no—
454C2×C4C2×Q8×D5314yes—
455C2×C4#41×D5318yes—
456C22—24no—
457C22—22no—
458C22—44no—
459C22—538no—
460C22—534no—
461C22—5316no—
462C22—1064no—
463C22—1062no—
464C22—21216no—
465C22—2124no—
466C22C22×Q8×D5311yescharacteristic
467C22C2×#41×D5314yes—
468C22H(5,𝔽2)×D5312yes—
469C4—24no—
470C4—28no—
471C4—44no—
472C4—44no—
473C4—1064no—
474C4—1068no—
475C4—2124no—
476C4—2124no—
477C2—42no—
478C2—538no—
479C2—2122no—
480C2#533×D5312yes—
481C2#543×D5311yescharacteristic
482C2#553×D5314yes—
subgroupG/H
center, FrattiniC23C24×D53
commutator, socle, cosocleC23×C53C25
FittingC53×#2608C2
prime radicalC24×D53C23
hypercenter#2608D53
hypocenterC53C2×#2608
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2608

Complements

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

NH ≅ G/Ntype
1. #20280385C2semidirect
2. #20280384C2semidirect
3. #20280383C2semidirect
4. #20280382C2semidirect
5. #20280381C2semidirect
7. #389×D53C2semidirect
8. #387×D53C2semidirect
11. C53×#2608C2semidirect
27. #16810522C22semidirect
28. #16810521C22semidirect
29. #16810520C22semidirect
30. #16810520C22semidirect
31. #16810519C22semidirect
32. #16810518C22semidirect
33. #16810518C22semidirect
34. #16810510C22semidirect
35. #16810509C22semidirect
36. #16810508C22semidirect
37. #16810507C22semidirect
38. #16810506C22semidirect
39. #16810505C22semidirect
40. #16810504C22semidirect
53. C53×#389C22semidirect
54. C53×#387C22semidirect
241. #2608D53direct (N × H)
252. D53#2608direct (N × H)
276. #389C2×D53semidirect
284. #387C2×D53semidirect
298. C53C2×#2608semidirect

Maximal subgroups

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

subgroupindex
#202803852
#202803842
#202803832
#202803822
#202803812
#396×D532
#389×D532
#387×D532
#381×D532
C4×C22⋊C4×D532
C53×#26082
C2×#260853