Finite Group Databasebeta

← #27367897

Subgroups — #27367897

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285274C211yesmaximal, characteristic, complemented
2#20284814C211yesmaximal, characteristic, complemented
3#20284806C211yesmaximal, characteristic
4#20274166C212yesmaximal, complemented
5#20274152C212yesmaximal, complemented
6#20274090C212yesmaximal, complemented
7#20274056C212yesmaximal, complemented
8#20273478C211yesmaximal, characteristic, complemented
9#20273413C211yesmaximal, characteristic, complemented
10#20273408C211yesmaximal, characteristic, complemented
11C2×C8×D4×C53C211yesmaximal, characteristic, complemented
12#16811180C2211yescharacteristic
13#16811173C2211yescharacteristic
14#16811021C2211yescharacteristic, complemented
15#16810996C2211yescharacteristic, complemented
16#16810974C2211yescharacteristic, complemented
17#16810970C2211yescharacteristic, complemented
18#16810958—21no—
19#16810941—21no—
20#16810913C2212yes—
21#16810903C2212yes—
22#16810852—21no—
23#16810852—21no—
24#16810828C2212yes—
25#16810818C2212yes—
26#16810774—21no—
27#16810757—21no—
28#16810733—22no—
29#16810723—22no—
30C4×#11719913—21no—
31C4×#11719913—21no—
32#16810680C2211yescharacteristic
33#16810680C2211yescharacteristic
34#16810376C2212yescomplemented
35#16810374C2212yescomplemented
36#16810371C2212yescomplemented
37#16810370C2212yescomplemented
38#16810368C2212yescomplemented
39#16810366C2212yescomplemented
40C23×C8×C53—21no—
41C2×C4×D4×C53C2211yescharacteristic
42C8×D4×C53—21no—
43C8×D4×C53—21no—
44C8×D4×C53C2212yes—
45C8×D4×C53C2212yes—
46C2×C4⋊C8×C53C2211yescharacteristic
47C2×C22⋊C8×C53—21no—
48C2×C4×C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719913—21no—
51C2×#11719913—21no—
52C2×#11719923—22no—
53C2×#11719922—22no—
54#13133217C2311yescharacteristic
55#13133217C2311yescharacteristic
56#13133217C2311yescharacteristic
57#13133214C2311yescharacteristic
58#13133209—21no—
59#13133205—21no—
60#13133196—21no—
61#13133196—21no—
62#13133191—21no—
63#13133191—21no—
64C4×Dic106—21no—
65C4×Dic106—21no—
66C4×D212—21no—
67C4×D212—21no—
68#13133176—21no—
69#13133150—22no—
70#13133149—21no—
71#13133149—21no—
72#13133148—22no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2312yes—
75#13133142—21no—
76#13133142—21no—
77#13133140—22no—
78#13133139—21no—
79#13133139—21no—
80#13133138—22no—
81#13133138—22no—
82#13133137—22no—
83C22⋊#88322—21no—
84C22⋊#88322—21no—
85#13133128—21no—
86#13133128—21no—
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89C22×C8×C53—22no—
90C22×C8×C53—22no—
91C22×C8×C53—21no—
92C22×C8×C53—44no—
93C22×C8×C53D411yescharacteristic
94C22×C8×C53D411yescharacteristic
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53D412yes—
98C4×D4×C53C2312yes—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C22⋊C4×C53—21no—
101C2×C42×C53C2311yescharacteristic
102C4⋊C8×C53—21no—
103C4⋊C8×C53C2312yes—
104C22⋊C8×C53—22no—
105C22⋊C8×C53—22no—
106C4×C8×C53D411yescharacteristic
107C4×C8×C53D411yescharacteristic
108C4×C8×C53C2311yescharacteristic
109C4×C8×C53C2311yescharacteristic
110C24×C53—21no—
111C2×D4×C53—24no—
112C2×D4×C53—21no—
113C2×D4×C53C2×D412yes—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—42no—
118C22×C4×C53—42no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C2×C22⋊D53—41no—
123#11719945—82no—
124C2×D212—21no—
125C2×D212—21no—
126C2×D212—21no—
127C2×C4×D53—41no—
128C2×Dic106—21no—
129C2×Dic106—21no—
130C2×Dic106—21no—
131C2×C8×C53—22no—
132C2×C8×C53—22no—
133C2×C8×C53—41no—
134C2×C8×C53—44no—
135C2×C8×C53—42no—
136C2×C8×C53—44no—
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2×D411yescharacteristic
140C2×C8×C53C2×D411yescharacteristic
141C4⋊C4×C53—21no—
142C4⋊C4×C53C2×D412yes—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—22no—
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C42×C53#4111yescharacteristic
148C42×C53C2411yescharacteristic
149C22⋊Dic53—41no—
150#11719923—44no—
151#11719922—44no—
152#11719921—41no—
153#11719921—41no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160#11719919—41no—
161#11719919—41no—
162C4×Dic53—41no—
163C2×#88322—22no—
164C2×#88322—22no—
165#11719913—42no—
166#11719913—42no—
167C23×C53—24no—
168C23×C53—22no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53#12111yescharacteristic
172C22×D53—41no—
173D4×C53—24no—
174D4×C53—22no—
175D4×C53#13514yes—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—22no—
181C2×C4×C53—44no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C2×#4111yescharacteristic
191C22⋊D53—82no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195D212—42no—
196D212—41no—
197D212—41no—
198D212—81no—
199C4×D53—82no—
200Dic106—42no—
201Dic106—41no—
202Dic106—41no—
203Dic106—81no—
204C8×C53—21no—
205C8×C53—21no—
206C8×C53—21no—
207C8×C53—42no—
208C8×C53AGL(1,ℤ/8)12yes—
209#88322—42no—
210#88322—42no—
211#19257—531nomaximal, Sylow
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—24no—
215C22×C53—48no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53#47112yes—
219C22×C53C2×#12111yescharacteristic
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—42no—
228C4×C53C2×#13512yes—
229C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C53#57511yescharacteristic
231Dic53—81no—
232Dic53—81no—
233Dic53—81no—
234#3167—531no—
235#3110—531no—
236#3109—531no—
237C2×C8×D4D5311yescharacteristic, complemented
238#1683—532no—
239#1681—532no—
240#1671—532no—
241#1665—532no—
242#1568—531no—
243#1552—531no—
244#1551—531no—
245C2×C53—22no—
246C2×C53—44no—
247C2×C53#303611yescharacteristic
248C2×C53#305011yescharacteristic
249C2×C53C2×#47111yescharacteristic
250D53—161no—
251C23×C8—21no—
252#533—531no—
253#532—531no—
254C2×C4×D4C2×D5311yescharacteristic
255#498—531no—
256#492—531no—
257#487—531no—
258#486—531no—
259#484—1061no—
260#481—1061no—
261#476—532no—
262#474—532no—
263#465—1061no—
264#465—1061no—
265#461—532no—
266#459—532no—
267#450—1061no—
268#447—1061no—
269#442—1062no—
270#440—1062no—
271C4×SD16—1061no—
272C4×SD16—1061no—
273C8×D4—21no—
274C8×D4—21no—
275C8×D4C2×D5312yes—
276C8×D4C2×D5312yes—
277#432—531no—
278#432—531no—
279C2×C4⋊C8C2×D5311yescharacteristic
280C2×C22⋊C8—21no—
281C2×C4×C8C2×D5311yescharacteristic
282#335—532no—
283#334—532no—
284#333—532no—
285#332—532no—
286#332—532no—
287#331—532no—
288C53#1925711yescharacteristic, complemented, Sylow
289C2×#41—1061no—
290C22×D4D21211yescharacteristic
291C23×C4—21no—
292C2×SD16—1062no—
293C2×SD16—1061no—
294C2×SD16—1061no—
295C2×D8—1062no—
296C22×C8—21no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—44no—
300C22×C8C22⋊D5311yescharacteristic
301C22×C8C22⋊D5311yescharacteristic
302C4⋊Q8—531no—
303C4⋊Q8—531no—
304C4⋊Q8—531no—
305#127—531no—
306#125—1061no—
307#124—1061no—
308C22⋊Q8—1061no—
309C22⋊Q8—1061no—
310#121—1061no—
311#121—1061no—
312C4×Q8—1061no—
313C4×Q8—1061no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—1061no—
317C4×D4—1061no—
318C4×D4D21212yes—
319C4×D4C22×D5312yes—
320#117—1061no—
321C2×C4⋊C4D21211yescharacteristic
322C2×C22⋊C4—21no—
323C2×C42C22×D5311yescharacteristic
324#107—1062no—
325#106—1061no—
326#106—1061no—
327#106—1062no—
328C4⋊C8—21no—
329C4⋊C8—532no—
330C4⋊C8—532no—
331C4⋊C8C22×D5312yes—
332#103—1061no—
333#103—1061no—
334#103—1062no—
335#102—1062no—
336#102—1062no—
337#102—1061no—
338#102—1061no—
339#102—1062no—
340C22⋊C8—22no—
341C22⋊C8—22no—
342C22⋊C8—1061no—
343C22⋊C8—1061no—
344#97—1061no—
345#97—1061no—
346C4×C8C22⋊D5311yescharacteristic
347C4×C8C22⋊D5311yescharacteristic
348C4×C8C22×D5311yescharacteristic
349C4×C8C22×D5311yescharacteristic
350C24—21no—
351#41—4242no—
352C2×Q8—1061no—
353C2×Q8—1061no—
354C2×Q8—1061no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—1061no—
358C2×D4—1061no—
359C2×D4—1061no—
360C2×D4—2121no—
361C2×D4C2×D21212yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—2121no—
368C22×C4C2×D21211yescharacteristic
369C22×C4#1171994511yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371SD16—2122no—
372SD16—2124no—
373SD16—2122no—
374D8—2124no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—41no—
378C2×C8—44no—
379C2×C8—42no—
380C2×C8—44no—
381C2×C8—1062no—
382C2×C8—1062no—
383C2×C8C2×C22⋊D5311yescharacteristic
384C2×C8C2×C22⋊D5311yescharacteristic
385C2×C8C2×C22⋊D5311yescharacteristic
386C2×C8C2×C22⋊D5311yescharacteristic
387C4⋊C4—21no—
388C4⋊C4—1061no—
389C4⋊C4—1061no—
390C4⋊C4—1061no—
391C4⋊C4—1061no—
392C4⋊C4—1061no—
393C4⋊C4—1061no—
394C4⋊C4—2121no—
395C4⋊C4—2121no—
396C4⋊C4C2×D21212yes—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—2121no—
400C22⋊C4—2121no—
401C22⋊C4—2121no—
402C42—2121no—
403C42C2×D21211yescharacteristic
404C42#1171994511yescharacteristic
405C42C2×C22⋊D5311yescharacteristic
406C42C23×D5311yescharacteristic
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—2121no—
412C23#1313319111yescharacteristic
413Q8—4241no—
414Q8—2122no—
415Q8—2121no—
416Q8—2121no—
417D4—24no—
418D4—22no—
419D4—4241no—
420D4—4242no—
421D4—2122no—
422D4—2121no—
423D4—2121no—
424D4#1313323914yes—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—4242no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4#1313319111yescharacteristic
439C2×C4#1313319111yescharacteristic
440C2×C4#1313319111yescharacteristic
441C2×C4C22×C22⋊D5311yescharacteristic
442C2×C4C22×D21211yescharacteristic
443C2×C4C2×#1171994511yescharacteristic
444C8—21no—
445C8—21no—
446C8—21no—
447C8—42no—
448C8—2122no—
449C8—2122no—
450C8#1313324312yes—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—48no—
455C22—42no—
456C22—42no—
457C22—4241no—
458C22—4241no—
459C22—4241no—
460C22#1681088612yes—
461C22C2×#1313319111yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—4241no—
468C4—4241no—
469C4—4241no—
470C4C2×#1313323912yes—
471C4C2×#1313324311yescharacteristic
472C4#1681142111yescharacteristic
473C2—22no—
474C2—44no—
475C2—8481no—
476C2#2028417911yescharacteristic
477C2#2028429611yescharacteristic
478C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C8×D4×C53C2
socleC22×C53C2×#121
prime radical#20284806C2
hypercenterC2×C8×D4D53
hypocenterC53#19257
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. #20285274C2semidirect
2. #20284814C2semidirect
4. #20274166C2semidirect
5. #20274152C2semidirect
6. #20274090C2semidirect
7. #20274056C2semidirect
8. #20273478C2semidirect
9. #20273413C2semidirect
10. #20273408C2semidirect
11. C2×C8×D4×C53C2semidirect
14. #16811021C22semidirect
15. #16810996C22semidirect
16. #16810974C22semidirect
17. #16810970C22semidirect
34. #16810376C22semidirect
35. #16810374C22semidirect
36. #16810371C22semidirect
37. #16810370C22semidirect
38. #16810368C22semidirect
39. #16810366C22semidirect
237. C2×C8×D4D53semidirect
288. C53#19257semidirect

Maximal subgroups

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

subgroupindex
#202852742
#202848142
#202848062
#202741662
#202741522
#202740902
#202740562
#202734782
#202734132
#202734082
C2×C8×D4×C532
#1925753