Finite Group Databasebeta

← #27466122

Subgroups — #27466122

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277253C211yesmaximal, characteristic
2#384×D53C211yesmaximal, characteristic
3#20276274C211yesmaximal, characteristic, complemented
4#20276272C211yesmaximal, characteristic, complemented
5#20276258C211yesmaximal, characteristic, complemented
6#20275475C212yesmaximal, complemented
7#20275019C212yesmaximal
8#20274906C212yesmaximal
9#20274718C212yesmaximal, complemented
10#20274694C211yesmaximal, characteristic
11C53×#2096C211yesmaximal, characteristic, complemented
12C2×#13133193C2211yescharacteristic
13C2×Q8×Dic53C2211yescharacteristic
14C2×C4⋊C4×D53—21no—
15C2×C4⋊C4×D53—21no—
16C2×C4⋊C4×D53C2212yes—
17C2×C42×D53C2211yescharacteristic
18C2×#13133148C2212yes—
19C2×#13133146C2212yes—
20C2×#13133141C2212yes—
21C2×#13133129C2212yes—
22#16810521C2211yescharacteristic
23#16810510—21no—
24#16810497—21no—
25#16810493C2211yescharacteristic
26#16810492—21no—
27#16810492—21no—
28#16810492C2212yes—
29#16810480C2211yescharacteristic
30C4⋊C4×Dic53C2211yescharacteristic
31#16810474—21no—
32#16810470—21no—
33#16810386C2212yes—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810378C2211yescharacteristic
38C2×#106×C53C2212yes—
39C2×#103×C53C2212yes—
40C53×#391C2211yescharacteristic
41C53×#384C2211yescharacteristic
42C53×#336C2211yescharacteristic
43C22×C4×D53—22no—
44C22×C4×D53D411yescharacteristic
45C22×C4×D53D411yescharacteristic
46C22×C4×D53C2311yescharacteristic
47C22×#88322C2312yes—
48#13133193—22no—
49#13133193Q814yes—
50Q8×Dic53—24no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2311yescharacteristic
57C2×#11719919—21no—
58C2×#11719919—21no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C4⋊C4×D53—42no—
63C4⋊C4×D53—42no—
64C4⋊C4×D53—44no—
65C4⋊C4×D53—44no—
66C2×#11719921—21no—
67C2×#11719921—21no—
68C2×C4×Dic53—21no—
69C2×C4×Dic53—21no—
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53D411yescharacteristic
72C2×C4×Dic53C2311yescharacteristic
73C42×D53—24no—
74#13133148—42no—
75#13133146—24no—
76#13133141—42no—
77#13133129—24no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81C22×Q8×C53C2311yescharacteristic
82C22×C8×C53C2312yes—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53C2312yes—
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C42×C53C2311yescharacteristic
88#106×C53—42no—
89#103×C53—42no—
90#95×C53—21no—
91#95×C53—21no—
92C23×D53C2×D411yescharacteristic
93C2×Q8×C53—21no—
94C2×Q8×C53—22no—
95C2×Q8×C53—22no—
96C2×Q8×C53C2×Q812yes—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2411yescharacteristic
103C22×Dic53—21no—
104C22×Dic53—21no—
105C22×Dic53—22no—
106C22×Dic53C2×D411yescharacteristic
107C22×Dic53C2×D411yescharacteristic
108C22×Dic53C2×D411yescharacteristic
109C2×C4×D53—22no—
110C2×C4×D53—22no—
111C2×C4×D53—22no—
112C2×C4×D53—22no—
113C2×C4×D53—22no—
114C2×C4×D53—22no—
115C2×C4×D53—22no—
116C2×C4×D53—44no—
117C2×C4×D53—44no—
118C2×C4×D53—44no—
119C2×C4×D53SD1614yes—
120C2×C4×D53C2×Q812yes—
121C2×C4×D53#4112yes—
122C2×C8×C53—22no—
123C2×C8×C53—22no—
124C2×C8×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—42no—
129C4⋊C4×C53—42no—
130C42×C53—22no—
131#11719921—24no—
132#11719921—42no—
133C4⋊Dic53—21no—
134C4⋊Dic53—22no—
135C4⋊Dic53—22no—
136C4⋊Dic53—22no—
137C4⋊Dic53—42no—
138C4⋊Dic53—42no—
139C4⋊Dic53—42no—
140C4⋊Dic53C2×Q812yes—
141#11719919—24no—
142#11719919—44no—
143#11719919—44no—
144#11719919—42no—
145#11719919—44no—
146C4×Dic53—22no—
147C4×Dic53—22no—
148C4×Dic53—42no—
149C4×Dic53—42no—
150C4×Dic53#4114yes—
151C2×#88322—22no—
152C2×#88322—22no—
153C2×#88322—22no—
154C23×C53C22×D411yescharacteristic
155C22×D53—21no—
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53C22⋊Q812yes—
160C22×D53C2×SD1612yes—
161C22×D53#13712yes—
162Q8×C53—24no—
163Q8×C53—42no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—22no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53C2×SD1612yes—
178C2×C4×C53AGL(1,ℤ/8)12yes—
179C2×C4×C53C22×Q811yescharacteristic
180C2×C4×C53C2×#4111yescharacteristic
181C2×C4×C53C2×#4112yes—
182C2×C4×C53H(5,𝔽2)12yes—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—22no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—42no—
192C2×Dic53—42no—
193C2×Dic53—42no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53C22⋊Q812yes—
197C2×Dic53#12312yes—
198C2×Dic53#12312yes—
199C2×Dic53C2×SD1612yes—
200C2×Dic53#13712yes—
201C4×D53—28no—
202C4×D53—24no—
203C4×D53—24no—
204C4×D53—28no—
205C4×D53—88no—
206C8×C53—42no—
207#88322—42no—
208#27680—531nomaximal, Sylow
209C22×C53C2×C22⋊Q811yescharacteristic
210C22×C53C2×#12311yescharacteristic
211C22×C53#53511yescharacteristic
212C22×C53C22×SD1611yescharacteristic
213C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
214C22×C53C2×#13711yescharacteristic
215C22×C53C2×#13711yescharacteristic
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53#47514yes—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—42no—
226C4×C53—82no—
227C4×C53—81no—
228C4×C53#55312yes—
229C4×C53#55412yes—
230Dic53—22no—
231Dic53—22no—
232Dic53—42no—
233Dic53—81no—
234Dic53—82no—
235Dic53#48314yes—
236C4×C22⋊Q8—531no—
237C2×#384—531no—
238#2099—531no—
239#2099—531no—
240#2096—531no—
241#2096D5311yescharacteristic, complemented
242#1948—532no—
243#1872—532no—
244#1853—532no—
245#1803—532no—
246#1797—531no—
247C2×C53#248711yescharacteristic
248C2×C53C2×#47511yescharacteristic
249C2×C53#307911yescharacteristic
250C2×C53C2×#48311yescharacteristic
251C2×C53#309211yescharacteristic
252C2×C53#320811yescharacteristic
253C2×C53#321811yescharacteristic
254D53—24no—
255C2×C22⋊Q8—531no—
256C2×C4×Q8—531no—
257C22×C4⋊C4—532no—
258C22×C4⋊C4—1061no—
259C22×C4⋊C4—1061no—
260C22×C42—531no—
261C2×#106—532no—
262C2×#106C2×D5312yes—
263C2×C4⋊C8—532no—
264C2×#103—532no—
265C2×#103C2×D5312yes—
266C2×C22⋊C8—532no—
267#391C2×D5311yescharacteristic
268#389—531no—
269#387—1061no—
270#385—1061no—
271#384—532no—
272#384—531no—
273#384—1061no—
274#384—1061no—
275#384C2×D5311yescharacteristic
276#380—531no—
277C4×C4⋊C4—531no—
278C4×C4⋊C4—1061no—
279C4×C22⋊C4—1061no—
280#340—532no—
281#340—532no—
282#340—532no—
283#340—532no—
284#336—531no—
285#336C2×D5311yescharacteristic
286C53#2768011yescharacteristic, complemented, Sylow
287C22×Q8C22×D5311yescharacteristic
288C23×C4—531no—
289C23×C4—531no—
290C23×C4—531no—
291C23×C4—1062no—
292C22×C8—532no—
293C22×C8C22×D5312yes—
294C22⋊Q8—534no—
295C22⋊Q8—1062no—
296C4×Q8—1064no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—532no—
301C2×C4⋊C4—531no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4—1062no—
304C2×C4⋊C4—1062no—
305C2×C4⋊C4—1062no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—1061no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4—2122no—
311C2×C4⋊C4—2122no—
312C2×C4⋊C4—2124no—
313C2×C4⋊C4—2124no—
314C2×C4⋊C4C22×D5312yes—
315C2×C4⋊C4C22×D5311yescharacteristic
316C2×C22⋊C4—1061no—
317C2×C22⋊C4—1061no—
318C2×C42—531no—
319C2×C42—531no—
320C2×C42—531no—
321C2×C42—1064no—
322C2×C42—1061no—
323C2×C42—1061no—
324C2×C42C22×D5311yescharacteristic
325#106—42no—
326#106—2122no—
327C4⋊C8—1064no—
328#103—42no—
329#103—2122no—
330C22⋊C8—1064no—
331#95—21no—
332#95—21no—
333#95—1061no—
334#95—1061no—
335#95—1061no—
336C24—531no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8Q8×D5312yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—531no—
345C22×C4—531no—
346C22×C4—531no—
347C22×C4—534no—
348C22×C4—532no—
349C22×C4—532no—
350C22×C4—1062no—
351C22×C4—1062no—
352C22×C4—1062no—
353C22×C4—1062no—
354C22×C4—1062no—
355C22×C4—1062no—
356C22×C4—1062no—
357C22×C4—1061no—
358C22×C4—1061no—
359C22×C4—1062no—
360C22×C4—2124no—
361C22×C4—2124no—
362C22×C4—2124no—
363C22×C4D4×D5311yescharacteristic
364C22×C4D4×D5311yescharacteristic
365C22×C4C23×D5311yescharacteristic
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—1062no—
370C2×C8—1062no—
371C2×C8—1062no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—532no—
378C4⋊C4—1064no—
379C4⋊C4—1061no—
380C4⋊C4—1062no—
381C4⋊C4—1062no—
382C4⋊C4—1062no—
383C4⋊C4—2124no—
384C4⋊C4—2122no—
385C4⋊C4—2124no—
386C4⋊C4—2124no—
387C4⋊C4—2122no—
388C4⋊C4—2122no—
389C4⋊C4—2122no—
390C22⋊C4—1064no—
391C22⋊C4—2122no—
392C42—22no—
393C42—534no—
394C42—1062no—
395C42—1062no—
396C42—2122no—
397C42—2122no—
398C23—532no—
399C23—532no—
400C23—532no—
401C23—1061no—
402C23—1061no—
403C23—1061no—
404C23—1061no—
405C23C2×D4×D5311yescharacteristic
406Q8—24no—
407Q8—42no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—532no—
422C2×C4—532no—
423C2×C4—532no—
424C2×C4—532no—
425C2×C4—532no—
426C2×C4—1068no—
427C2×C4—1068no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1064no—
433C2×C4—1064no—
434C2×C4—1062no—
435C2×C4—4248no—
436C2×C4—2121no—
437C2×C4—2122no—
438C2×C4—2122no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2122no—
442C2×C4—2121no—
443C2×C4—2121no—
444C2×C4SD16×D5312yes—
445C2×C4#1313324412yes—
446C2×C4C2×Q8×D5311yescharacteristic
447C2×C4#41×D5311yescharacteristic
448C2×C4#41×D5312yes—
449C2×C4#1313326212yes—
450C8—42no—
451C8—2122no—
452C22—534no—
453C22—1062no—
454C22—1062no—
455C22—1062no—
456C22—1062no—
457C22—1062no—
458C22—1062no—
459C22C22⋊Q8×D5311yescharacteristic
460C22#123×D5311yescharacteristic
461C22#1681118611yescharacteristic
462C22C2×SD16×D5311yescharacteristic
463C22C2×#1313324411yescharacteristic
464C22#137×D5311yescharacteristic
465C22#137×D5311yescharacteristic
466C4—22no—
467C4—22no—
468C4—42no—
469C4—81no—
470C4—82no—
471C4—534no—
472C4—1062no—
473C4—1062no—
474C4—4242no—
475C4—4241no—
476C4—2122no—
477C4#1681131812yes—
478C4#1681132712yes—
479C2—1064no—
480C2#2027902511yescharacteristic
481C2#475×D5311yescharacteristic
482C2#2028452511yescharacteristic
483C2#483×D5311yescharacteristic
484C2#2028463911yescharacteristic
485C2#2028565211yescharacteristic
486C2#2028573811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2096C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2096D53
hypocenterC53#27680
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
3. #20276274C2semidirect
4. #20276272C2semidirect
5. #20276258C2semidirect
6. #20275475C2semidirect
9. #20274718C2semidirect
11. C53×#2096C2semidirect
241. #2096D53semidirect
286. C53#27680semidirect

Maximal subgroups

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

subgroupindex
#202772532
#384×D532
#202762742
#202762722
#202762582
#202754752
#202750192
#202749062
#202747182
#202746942
C53×#20962
#2768053