Finite Group Databasebeta

← #27369343

Subgroups — #27369343

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285851C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic
3#20283739C211yesmaximal, characteristic, complemented
4#20274154C212yesmaximal, complemented
5#20274126C212yesmaximal, complemented
6#20274112C212yesmaximal, complemented
7#20274088C212yesmaximal, complemented
8#20273493C211yesmaximal, characteristic, complemented
9#20273425C211yesmaximal, characteristic, complemented
10#20273375C211yesmaximal, characteristic, complemented
11C53×#3101C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16811026C2211yescharacteristic, complemented
14#16811019C2211yescharacteristic, complemented
15#16810971C2211yescharacteristic, complemented
16#16810971C2211yescharacteristic, complemented
17#16810947—22no—
18#16810941—21no—
19#16810920—22no—
20#16810913C2212yes—
21#16810818C2212yes—
22#16810757—21no—
23#16810733—22no—
24D4×#88322—22no—
25D4×#88322C2212yes—
26D4×#88322C2212yes—
27#16810680C2211yescharacteristic
28#16810676C2211yescharacteristic
29#16810676C2211yescharacteristic
30C2×C22⋊#88322—22no—
31#16810375C2212yescomplemented
32#16810373C2212yescomplemented
33#16810372C2212yescomplemented
34#16810371C2212yescomplemented
35#16810369C2212yescomplemented
36#16810366C2212yescomplemented
37C53×#533C2211yescharacteristic
38C2×C4×D4×C53C2211yescharacteristic
39C53×#483—21no—
40C53×#476C2212yes—
41C53×#461C2212yes—
42C53×#450—21no—
43C4×SD16×C53—22no—
44C53×#431C2211yescharacteristic
45C2×#11719945—21no—
46C2×#11719922—22no—
47C22×#88322—22no—
48C22×#88322—22no—
49#13133217C2311yescharacteristic
50#13133214C2311yescharacteristic
51#13133205—21no—
52#13133191—21no—
53#13133191—21no—
54C4×D212—21no—
55C4×D212—21no—
56#13133150—22no—
57#13133148—22no—
58#13133148—22no—
59C4⋊#88322D412yes—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133138—22no—
63#13133137—22no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68C22⋊#88322—21no—
69C22⋊#88322—21no—
70C22⋊#88322D412yes—
71#13133128—21no—
72C4×#88322—21no—
73C4×#88322C2311yescharacteristic
74C4×#88322C2311yescharacteristic
75C22×D4×C53D411yescharacteristic
76C23×C4×C53—21no—
77C2×SD16×C53—22no—
78C4⋊Q8×C53C2311yescharacteristic
79C4⋊Q8×C53C2311yescharacteristic
80#125×C53—21no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53—21no—
83C4×Q8×C53—21no—
84C4×Q8×C53—21no—
85C4×D4×C53—21no—
86C4×D4×C53—21no—
87C4×D4×C53D412yes—
88C4×D4×C53C2312yes—
89#117×C53—21no—
90C2×C4⋊C4×C53D411yescharacteristic
91C2×C22⋊C4×C53—21no—
92C2×C42×C53C2311yescharacteristic
93#106×C53—22no—
94C4⋊C8×C53C2312yes—
95#103×C53—22no—
96#102×C53—22no—
97C22⋊C8×C53—21no—
98C4×C8×C53—21no—
99C24×C53—21no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×D4×C53—24no—
103C2×D4×C53—21no—
104C2×D4×C53C2×D412yes—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—82no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×C22⋊D53—41no—
113#11719945—82no—
114C2×D212—21no—
115C2×D212—21no—
116C2×D212—21no—
117C2×C4×D53—41no—
118C2×Dic106—21no—
119SD16×C53—44no—
120C2×C8×C53—22no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53C2×D412yes—
129C22⋊C4×C53—22no—
130C22⋊C4×C53—22no—
131C22⋊C4×C53—41no—
132C42×C53—41no—
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53#4111yescharacteristic
136C42×C53C2411yescharacteristic
137#11719922—44no—
138#11719921—41no—
139#11719921—41no—
140C4⋊Dic53—21no—
141C4⋊Dic53—21no—
142C2×#88322—22no—
143C2×#88322—22no—
144C2×#88322—82no—
145C2×#88322—82no—
146C2×#88322C2×D412yes—
147C2×#88322C2×D412yes—
148C23×C53—24no—
149C23×C53—22no—
150C23×C53—81no—
151C23×C53—81no—
152C23×C53#12111yescharacteristic
153C22×D53—41no—
154Q8×C53—42no—
155Q8×C53—81no—
156D4×C53—24no—
157D4×C53—22no—
158D4×C53#13514yes—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—22no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—82no—
168C2×C4×C53—81no—
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53C22×D411yescharacteristic
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C2×#4111yescharacteristic
175C22⋊D53—82no—
176C2×Dic53—41no—
177D212—42no—
178D212—42no—
179D212—81no—
180C4×D53—82no—
181Dic106—81no—
182C8×C53—42no—
183#88322—22no—
184#88322—22no—
185#88322—42no—
186#88322—42no—
187#19394—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—24no—
191C22×C53—21no—
192C22×C53—42no—
193C22×C53—42no—
194C22×C53—84no—
195C22×C53C2×#12111yescharacteristic
196C2×D53—81no—
197C2×D53—81no—
198C2×D53—81no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—42no—
204C4×C53—81no—
205C4×C53—81no—
206C4×C53C2×#13512yes—
207C4×C53#57712yes—
208Dic53—81no—
209#3228—531no—
210#3109—531no—
211#3101D5311yescharacteristic, complemented
212#2972—531no—
213#1681—532no—
214#1677—532no—
215#1675—532no—
216#1671—532no—
217#1572—531no—
218#1554—531no—
219#1543—531no—
220C2×C53—41no—
221C2×C53—44no—
222C2×C53#304211yescharacteristic
223C2×C53#305011yescharacteristic
224C2×C53#306811yescharacteristic
225D53—161no—
226#533C2×D5311yescharacteristic
227#532—531no—
228C2×C4×D4C2×D5311yescharacteristic
229#500—531no—
230#498—531no—
231#486—531no—
232#486—531no—
233#483—21no—
234#482—1062no—
235#481—1061no—
236#478—1062no—
237#476—532no—
238#476C2×D5312yes—
239#461C2×D5312yes—
240#459—532no—
241#450—21no—
242#447—1061no—
243#442—1062no—
244C4×SD16—22no—
245C8×D4—532no—
246C8×D4—532no—
247C8×D4—1062no—
248#432—531no—
249#431—531no—
250#431—531no—
251#431C2×D5311yescharacteristic
252C2×C22⋊C8—1062no—
253#335—532no—
254#334—532no—
255#333—532no—
256#333—532no—
257#332—532no—
258#331—532no—
259C53#1939411yescharacteristic, complemented, Sylow
260C2×#41—1061no—
261C22×D4C22⋊D5311yescharacteristic
262C23×C4—21no—
263C2×SD16—22no—
264C2×D8—1062no—
265C22×C8—1062no—
266C22×C8—1062no—
267C4⋊Q8—531no—
268C4⋊Q8C22×D5311yescharacteristic
269C4⋊Q8C22×D5311yescharacteristic
270#127—531no—
271#125—21no—
272#124—1061no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275#121—1061no—
276#121—1061no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—1061no—
282C4×D4—1061no—
283C4×D4C22⋊D5312yes—
284C4×D4C22×D5312yes—
285#117—21no—
286C2×C4⋊C4C22⋊D5311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C42C22×D5311yescharacteristic
289#107—1062no—
290#106—22no—
291#106—1062no—
292#106—1062no—
293C4⋊C8—532no—
294C4⋊C8—532no—
295C4⋊C8—532no—
296C4⋊C8C22×D5312yes—
297#103—22no—
298#102—22no—
299#102—1062no—
300#102—1062no—
301#102—1062no—
302#102—1062no—
303C22⋊C8—21no—
304C22⋊C8—532no—
305C22⋊C8—1062no—
306C22⋊C8—1062no—
307C22⋊C8—1061no—
308C22⋊C8—1061no—
309#97—1061no—
310C4×C8—21no—
311C4×C8—531no—
312C4×C8—531no—
313C4×C8—1061no—
314C24—21no—
315#41—4242no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—1061no—
319C2×D4—24no—
320C2×D4—21no—
321C2×D4—1061no—
322C2×D4—1061no—
323C2×D4—1061no—
324C2×D4—2121no—
325C2×D4C2×C22⋊D5312yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—82no—
330C22×C4—2121no—
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334SD16—44no—
335D8—2124no—
336C2×C8—22no—
337C2×C8—532no—
338C2×C8—532no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8—4242no—
342C2×C8—4242no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—1061no—
351C4⋊C4—1061no—
352C4⋊C4C2×C22⋊D5312yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—41no—
356C22⋊C4—2121no—
357C22⋊C4—2121no—
358C42—41no—
359C42D4×D5311yescharacteristic
360C42#1171994711yescharacteristic
361C42C2×C22⋊D5311yescharacteristic
362C42C23×D5311yescharacteristic
363C23—24no—
364C23—22no—
365C23—81no—
366C23—81no—
367C23—2121no—
368C23#1313318911yescharacteristic
369Q8—42no—
370Q8—81no—
371Q8—4241no—
372D4—24no—
373D4—22no—
374D4—4241no—
375D4—4242no—
376D4—2122no—
377D4—2122no—
378D4#1313323714yes—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—4242no—
390C2×C4—2121no—
391C2×C4#1313318911yescharacteristic
392C2×C4#1313318911yescharacteristic
393C2×C4#1313318911yescharacteristic
394C2×C4C2×D4×D5311yescharacteristic
395C2×C4C22×C22⋊D5311yescharacteristic
396C2×C4C2×#1171994711yescharacteristic
397C8—42no—
398C8—1062no—
399C8—1062no—
400C8—2122no—
401C8—2122no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—24no—
406C22—42no—
407C22—42no—
408C22—84no—
409C22—4241no—
410C22—4241no—
411C22—4241no—
412C22C2×#1313318911yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—81no—
419C4—81no—
420C4—4241no—
421C4C2×#1313323712yes—
422C4#1681142912yes—
423C2—41no—
424C2—44no—
425C2—8481no—
426C2#2028422411yescharacteristic
427C2#2028429311yescharacteristic
428C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3101C2
socleC22×C53C2×#121
prime radical#20284806C2
hypercenter#3101D53
hypocenterC53#19394
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #20285851C2semidirect
3. #20283739C2semidirect
4. #20274154C2semidirect
5. #20274126C2semidirect
6. #20274112C2semidirect
7. #20274088C2semidirect
8. #20273493C2semidirect
9. #20273425C2semidirect
10. #20273375C2semidirect
11. C53×#3101C2semidirect
13. #16811026C22semidirect
14. #16811019C22semidirect
15. #16810971C22semidirect
16. #16810971C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810372C22semidirect
34. #16810371C22semidirect
35. #16810369C22semidirect
36. #16810366C22semidirect
211. #3101D53semidirect
259. C53#19394semidirect

Maximal subgroups

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

subgroupindex
#202858512
#202848062
#202837392
#202741542
#202741262
#202741122
#202740882
#202734932
#202734252
#202733752
C53×#31012
#1939453