Finite Group Databasebeta

← #27367900

Subgroups — #27367900

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#20285275C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic
3C2×D4×#88322C211yesmaximal, characteristic, complemented
4#20274164C212yesmaximal, complemented
5#20274154C212yesmaximal, complemented
6#20274088C212yesmaximal, complemented
7#20274058C212yesmaximal, complemented
8#20273476C211yesmaximal, characteristic, complemented
9#20273415C211yesmaximal, characteristic, complemented
10#20273411C211yesmaximal, characteristic, complemented
11C53×#3110C211yesmaximal, characteristic, complemented
12C23×#88322—21no—
13#16811173C2211yescharacteristic
14#16811019C2211yescharacteristic, complemented
15#16810997C2211yescharacteristic, complemented
16#16810973C2211yescharacteristic, complemented
17#16810971C2211yescharacteristic, complemented
18#16810941—21no—
19#16810913C2212yes—
20#16810854—21no—
21#16810854—21no—
22#16810818C2212yes—
23#16810757—21no—
24#16810733—22no—
25C4×#11719924—21no—
26C4×#11719924—21no—
27D4×#88322—21no—
28D4×#88322—21no—
29D4×#88322C2212yes—
30D4×#88322C2212yes—
31#16810680C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33C2×C22⋊#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yescomplemented
36#16810373C2212yescomplemented
37#16810371C2212yescomplemented
38#16810369C2212yescomplemented
39#16810369C2212yescomplemented
40#16810366C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#484—21no—
44C53×#474C2212yes—
45C53×#461C2212yes—
46C53×#450—21no—
47C53×#440—22no—
48C53×#432C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719924—21no—
51C2×#11719924—21no—
52C2×#11719922—22no—
53C22×#88322—22no—
54C22×#88322—22no—
55C22×#88322—21no—
56C22×#88322—44no—
57C22×#88322D411yescharacteristic
58C22×#88322D411yescharacteristic
59#13133217C2311yescharacteristic
60#13133214C2311yescharacteristic
61#13133205—21no—
62#13133191—21no—
63#13133191—21no—
64C4×D212—21no—
65C4×D212—21no—
66#13133148—22no—
67#13133148—21no—
68#13133148—21no—
69C4⋊#88322—21no—
70C4⋊#88322C2312yes—
71C4⋊#88322C2312yes—
72#13133141—21no—
73#13133141—21no—
74#13133138—22no—
75#13133137—22no—
76#13133137—21no—
77#13133137—21no—
78C22⋊#88322—22no—
79C22⋊#88322—22no—
80C22⋊#88322—21no—
81#13133128—21no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C22×D4×C53D411yescharacteristic
87C23×C4×C53—21no—
88C2×SD16×C53—22no—
89C4⋊Q8×C53C2311yescharacteristic
90C4⋊Q8×C53C2311yescharacteristic
91#125×C53—21no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D412yes—
99C4×D4×C53C2312yes—
100#117×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#107×C53—22no—
105C4⋊C8×C53C2312yes—
106#103×C53—22no—
107#102×C53—22no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C24×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×D4×C53—24no—
114C2×D4×C53—21no—
115C2×D4×C53C2×D412yes—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C2×C22⋊D53—41no—
125#11719945—82no—
126C2×D212—21no—
127C2×D212—21no—
128C2×D212—21no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131SD16×C53—44no—
132C2×C8×C53—22no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53C2×D412yes—
141C22⋊C4×C53—22no—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—41no—
144C42×C53—41no—
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C42×C53#4111yescharacteristic
148C42×C53C2411yescharacteristic
149#11719924—42no—
150#11719924—42no—
151#11719922—44no—
152#11719921—41no—
153#11719921—41no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156C2×#88322—22no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—41no—
160C2×#88322—44no—
161C2×#88322—42no—
162C2×#88322—44no—
163C2×#88322C2×D411yescharacteristic
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166C2×#88322C2×D411yescharacteristic
167C23×C53—24no—
168C23×C53—22no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53#12111yescharacteristic
172C22×D53—41no—
173Q8×C53—42no—
174Q8×C53—41no—
175Q8×C53—41no—
176D4×C53—24no—
177D4×C53—22no—
178D4×C53#13514yes—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—44no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C2×#4111yescharacteristic
196C22⋊D53—82no—
197C2×Dic53—41no—
198D212—42no—
199D212—41no—
200D212—41no—
201D212—81no—
202C4×D53—82no—
203Dic106—81no—
204C8×C53—42no—
205#88322—21no—
206#88322—21no—
207#88322—21no—
208#88322—42no—
209#88322—42no—
210#88322AGL(1,ℤ/8)12yes—
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×C53—81no—
229C4×C53—81no—
230C4×C53C2×#13512yes—
231C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C53#57511yescharacteristic
233Dic53—81no—
234#3167—531no—
235#3110D5311yescharacteristic, complemented
236#3109—531no—
237C2×C8×D4—531no—
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—1061no—
252#533C2×D5311yescharacteristic
253#532—531no—
254C2×C4×D4C2×D5311yescharacteristic
255#498—531no—
256#492—531no—
257#487—531no—
258#486—531no—
259#484—21no—
260#481—1061no—
261#476—532no—
262#474C2×D5312yes—
263#465—1061no—
264#465—1061no—
265#461C2×D5312yes—
266#459—532no—
267#450—21no—
268#447—1061no—
269#442—1062no—
270#440—22no—
271C4×SD16—1061no—
272C4×SD16—1061no—
273C8×D4—532no—
274C8×D4—532no—
275C8×D4—1061no—
276C8×D4—1061no—
277#432—531no—
278#432C2×D5311yescharacteristic
279C2×C4⋊C8—531no—
280C2×C22⋊C8—1061no—
281C2×C4×C8—531no—
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×D4C22⋊D5311yescharacteristic
291C23×C4—21no—
292C2×SD16—22no—
293C2×SD16—1061no—
294C2×SD16—1061no—
295C2×D8—1062no—
296C22×C8—531no—
297C22×C8—531no—
298C22×C8—1062no—
299C22×C8—1062no—
300C22×C8—1061no—
301C22×C8—2124no—
302C4⋊Q8—531no—
303C4⋊Q8C22×D5311yescharacteristic
304C4⋊Q8C22×D5311yescharacteristic
305#127—531no—
306#125—21no—
307#124—1061no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310#121—1061no—
311#121—1061no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—1061no—
317C4×D4—1061no—
318C4×D4C22⋊D5312yes—
319C4×D4C22×D5312yes—
320#117—21no—
321C2×C4⋊C4C22⋊D5311yescharacteristic
322C2×C22⋊C4—21no—
323C2×C42C22×D5311yescharacteristic
324#107—22no—
325#106—1061no—
326#106—1061no—
327#106—1062no—
328C4⋊C8—532no—
329C4⋊C8—532no—
330C4⋊C8—1061no—
331C4⋊C8C22×D5312yes—
332#103—22no—
333#103—1061no—
334#103—1061no—
335#102—22no—
336#102—1061no—
337#102—1061no—
338#102—1062no—
339#102—1062no—
340C22⋊C8—21no—
341C22⋊C8—1062no—
342C22⋊C8—1062no—
343C22⋊C8—1061no—
344#97—21no—
345#97—1061no—
346C4×C8—531no—
347C4×C8—531no—
348C4×C8—531no—
349C4×C8—531no—
350C24—21no—
351#41—4242no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—1061no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—1061no—
358C2×D4—1061no—
359C2×D4—1061no—
360C2×D4—2121no—
361C2×D4C2×C22⋊D5312yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—2121no—
368C22×C4D4×D5311yescharacteristic
369C22×C4#1171994711yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371SD16—44no—
372SD16—2122no—
373SD16—2122no—
374D8—2124no—
375C2×C8—22no—
376C2×C8—531no—
377C2×C8—531no—
378C2×C8—531no—
379C2×C8—531no—
380C2×C8—1062no—
381C2×C8—1062no—
382C2×C8—1062no—
383C2×C8—2124no—
384C2×C8—2121no—
385C2×C8—2124no—
386C2×C8—2122no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—1061no—
395C4⋊C4—1061no—
396C4⋊C4C2×C22⋊D5312yes—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—2121no—
401C22⋊C4—2121no—
402C42—41no—
403C42D4×D5311yescharacteristic
404C42#1171994711yescharacteristic
405C42C2×C22⋊D5311yescharacteristic
406C42C23×D5311yescharacteristic
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—2121no—
412C23#1313318911yescharacteristic
413Q8—42no—
414Q8—41no—
415Q8—41no—
416Q8—4241no—
417D4—24no—
418D4—22no—
419D4—4241no—
420D4—4242no—
421D4—2122no—
422D4—2121no—
423D4—2121no—
424D4#1313323714yes—
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—41no—
435C2×C4—41no—
436C2×C4—4242no—
437C2×C4—2121no—
438C2×C4#1313318911yescharacteristic
439C2×C4#1313318911yescharacteristic
440C2×C4#1313318911yescharacteristic
441C2×C4C2×D4×D5311yescharacteristic
442C2×C4C22×C22⋊D5311yescharacteristic
443C2×C4C2×#1171994711yescharacteristic
444C8—42no—
445C8—532no—
446C8—1061no—
447C8—1061no—
448C8—1061no—
449C8—2122no—
450C8—2122no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—48no—
455C22—42no—
456C22—42no—
457C22—4241no—
458C22—4241no—
459C22—4241no—
460C22#1681088812yes—
461C22C2×#1313318911yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—4241no—
470C4C2×#1313323712yes—
471C4AGL(1,ℤ/8)×D5311yescharacteristic
472C4#1681141911yescharacteristic
473C2—22no—
474C2—44no—
475C2—8481no—
476C2#2028417611yescharacteristic
477C2#2028429311yescharacteristic
478C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3110C2
socleC22×C53C2×#121
prime radical#20284806C2
hypercenter#3110D53
hypocenterC53#19257
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #20285275C2semidirect
3. C2×D4×#88322C2semidirect
4. #20274164C2semidirect
5. #20274154C2semidirect
6. #20274088C2semidirect
7. #20274058C2semidirect
8. #20273476C2semidirect
9. #20273415C2semidirect
10. #20273411C2semidirect
11. C53×#3110C2semidirect
14. #16811019C22semidirect
15. #16810997C22semidirect
16. #16810973C22semidirect
17. #16810971C22semidirect
35. #16810375C22semidirect
36. #16810373C22semidirect
37. #16810371C22semidirect
38. #16810369C22semidirect
39. #16810369C22semidirect
40. #16810366C22semidirect
235. #3110D53semidirect
288. C53#19257semidirect

Maximal subgroups

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

subgroupindex
#202852752
#202848062
C2×D4×#883222
#202741642
#202741542
#202740882
#202740582
#202734762
#202734152
#202734112
C53×#31102
#1925753