Finite Group Databasebeta

← #27368508

Subgroups — #27368508

Order 13568.

620 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285108C211yesmaximal, characteristic, complemented
2#20284530C211yesmaximal, characteristic, complemented
3#20283729C211yesmaximal, characteristic, complemented
4#20274295C212yesmaximal, complemented
5#20274253C212yesmaximal, complemented
6#20274164C212yesmaximal, complemented
7#20274122C212yesmaximal, complemented
8#20273483C211yesmaximal, characteristic, complemented
9#20273440C211yesmaximal, characteristic, complemented
10#20273437C211yesmaximal, characteristic, complemented
11C53×#3115C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16810982—21no—
14#16810982—21no—
15#16810973C2212yes—
16#16810969C2212yes—
17#16810918C2212yes—
18#16810913C2212yes—
19#16810908—21no—
20#16810903—21no—
21#16810754C2212yes—
22#16810695—22no—
23D4×#88322C2212yes—
24#16810680—21no—
25#16810680C2211yescharacteristic
26#16810666—21no—
27#16810647C2211yescharacteristic
28#16810624—22no—
29#16810624—21no—
30C4×#11719917C2211yescharacteristic
31C2×#13133128—21no—
32#16810375C2212yes—
33#16810373C2212yes—
34#16810371C2212yes—
35#16810371C2212yes—
36#16810369C2212yes—
37#16810367C2212yes—
38C53×#532C2211yescharacteristic
39C4×#41×C53C2211yescharacteristic
40C53×#482—21no—
41C53×#475C2211yescharacteristic
42C53×#474C2211yescharacteristic
43C53×#460C2211yescharacteristic
44C53×#459C2211yescharacteristic
45C53×#449—21no—
46C53×#442—21no—
47C53×#440—21no—
48C53×#432C2211yescharacteristic
49C22×Dic106—21no—
50C2×#11719917—21no—
51C2×#11719917D412yes—
52C22×#88322—22no—
53#13133217—21no—
54#13133217—21no—
55#13133217—21no—
56#13133217C2312yes—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—21no—
59#13133150—22no—
60#13133148—22no—
61C4⋊#88322—21no—
62C4⋊#88322—22no—
63C4⋊#88322C2312yes—
64C4⋊#88322C2312yes—
65#13133141—22no—
66#13133141—22no—
67#13133141—22no—
68#13133138—22no—
69#13133138—22no—
70#13133138—22no—
71C22⋊#88322—22no—
72C22⋊#88322—21no—
73#13133128—21no—
74#13133128—22no—
75#13133128D411yescharacteristic
76#13133128D411yescharacteristic
77C4×#88322C2312yes—
78C2×#41×C53—21no—
79C2×#41×C53D411yescharacteristic
80C2×SD16×C53—21no—
81C2×D8×C53—21no—
82C4⋊Q8×C53C2311yescharacteristic
83#127×C53C2311yescharacteristic
84#124×C53—21no—
85#121×C53—21no—
86#121×C53—21no—
87C4×Q8×C53D411yescharacteristic
88C4×Q8×C53C2311yescharacteristic
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53D411yescharacteristic
94C4×D4×C53C2311yescharacteristic
95#117×C53—21no—
96#117×C53D411yescharacteristic
97C2×C42×C53—21no—
98C2×C42×C53C2311yescharacteristic
99#107×C53—21no—
100#106×C53—21no—
101C4⋊C8×C53C2311yescharacteristic
102C4⋊C8×C53C2311yescharacteristic
103#103×C53—21no—
104#102×C53—21no—
105#102×C53—21no—
106#102×C53—21no—
107C22⋊C8×C53—21no—
108#97×C53—21no—
109#41×C53—22no—
110#41×C53—41no—
111#41×C53—44no—
112C2×Q8×C53—21no—
113C2×Q8×C53C2×D411yescharacteristic
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C2×D4×C53C2×D411yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—41no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53#4111yescharacteristic
127C22×Dic53—41no—
128C2×Dic106—22no—
129C2×Dic106—41no—
130C2×Dic106—82no—
131SD16×C53—42no—
132D8×C53—42no—
133C2×C8×C53—21no—
134C2×C8×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53C2×D411yescharacteristic
140C4⋊C4×C53C2×D411yescharacteristic
141C22⋊C4×C53—21no—
142C22⋊C4×C53—21no—
143C22⋊C4×C53—21no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C42×C53#4111yescharacteristic
152C42×C53C2411yescharacteristic
153C4⋊Dic53—22no—
154C4⋊Dic53—22no—
155C4⋊Dic53—41no—
156C4⋊Dic53—41no—
157#11719917—42no—
158#11719917—42no—
159C2×#88322—22no—
160C2×#88322—22no—
161C2×#88322—44no—
162C2×#88322C2×D412yes—
163C2×#88322C2×D412yes—
164C23×C53—21no—
165C23×C53—41no—
166C23×C53#12111yescharacteristic
167Q8×C53—21no—
168Q8×C53—21no—
169Q8×C53—42no—
170D4×C53—21no—
171D4×C53—42no—
172D4×C53—41no—
173D4×C53—42no—
174D4×C53—41no—
175D4×C53—81no—
176D4×C53—82no—
177D4×C53#13712yes—
178C2×C4×C53—21no—
179C2×C4×C53—22no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—41no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—41no—
188C2×C4×C53—81no—
189C2×C4×C53—82no—
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C2×#4111yescharacteristic
196C2×Dic53—42no—
197C2×Dic53—82no—
198Dic106—82no—
199Dic106—82no—
200C8×C53—41no—
201C8×C53—41no—
202#88322—22no—
203#88322—22no—
204#88322—42no—
205#88322—42no—
206#19318—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—22no—
210C22×C53—21no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53—81no—
216C22×C53C2×#12111yescharacteristic
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—41no—
222C4×C53—41no—
223C4×C53—41no—
224C4×C53—81no—
225C4×C53C2×#13711yescharacteristic
226C4×C53#57512yes—
227C4×C53#57511yescharacteristic
228Dic53—82no—
229#3141—531no—
230#3115D5311yescharacteristic
231#3079—531no—
232#2970—531no—
233#1704—532no—
234#1698—532no—
235#1683—532no—
236#1677—532no—
237#1569—531no—
238#1558—531no—
239#1557—531no—
240C2×C53—41no—
241C2×C53—42no—
242C2×C53—161no—
243C2×C53#304011yescharacteristic
244C2×C53#305611yescharacteristic
245C2×C53#306811yescharacteristic
246#532C2×D5311yescharacteristic
247C2×C4⋊Q8—531no—
248C4×#41C2×D5311yescharacteristic
249#489—1061no—
250#489—1061no—
251#487—532no—
252#486—532no—
253#482—21no—
254#477—532no—
255#476—532no—
256#475—1061no—
257#475C2×D5311yescharacteristic
258#474—1061no—
259#474C2×D5311yescharacteristic
260#460C2×D5311yescharacteristic
261#459C2×D5311yescharacteristic
262#449—21no—
263#446—532no—
264#442—21no—
265#440—21no—
266#435—1062no—
267C8×D4—532no—
268#432—531no—
269#432—1061no—
270#432C2×D5311yescharacteristic
271#428—1061no—
272#423—531no—
273#417—1062no—
274#417—1061no—
275C4×M16—531no—
276C2×#97—1061no—
277#335—532no—
278#334—532no—
279#333—532no—
280#333—532no—
281#332—532no—
282#331—532no—
283C53#1931811yescharacteristic, complemented, Sylow
284C2×#41—21no—
285C2×#41C22⋊D5311yescharacteristic
286C22×Q8—1061no—
287C2×SD16—21no—
288C2×D8—21no—
289C2×M16—532no—
290C2×M16—1061no—
291C22×C8—1062no—
292C4⋊Q8—532no—
293C4⋊Q8—1061no—
294C4⋊Q8—1061no—
295C4⋊Q8—1061no—
296C4⋊Q8C22×D5311yescharacteristic
297#127C22×D5311yescharacteristic
298#124—21no—
299#121—21no—
300#121—21no—
301C4×Q8C22⋊D5311yescharacteristic
302C4×Q8C22×D5311yescharacteristic
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—21no—
307C4×D4C22⋊D5311yescharacteristic
308C4×D4C22×D5311yescharacteristic
309#117—21no—
310#117C22⋊D5311yescharacteristic
311C2×C4⋊C4—1061no—
312C2×C4⋊C4—1061no—
313C2×C42—21no—
314C2×C42C22×D5311yescharacteristic
315#107—21no—
316#107—1062no—
317#106—21no—
318#106—1062no—
319C4⋊C8—532no—
320C4⋊C8—532no—
321C4⋊C8—1062no—
322C4⋊C8—1061no—
323C4⋊C8C22×D5311yescharacteristic
324C4⋊C8C22×D5311yescharacteristic
325#103—21no—
326#103—1062no—
327#103—1062no—
328#103—1062no—
329#102—21no—
330#102—21no—
331#102—21no—
332#102—1062no—
333#102—1062no—
334#102—1062no—
335C22⋊C8—21no—
336C22⋊C8—1062no—
337C22⋊C8—1061no—
338#97—21no—
339#97—531no—
340#97—531no—
341#97—1062no—
342#97—1061no—
343C4×C8—532no—
344#41—22no—
345#41—41no—
346#41—44no—
347C2×Q8—21no—
348C2×Q8—1062no—
349C2×Q8—4242no—
350C2×Q8—2121no—
351C2×Q8C2×C22⋊D5311yescharacteristic
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—41no—
357C2×D4C2×C22⋊D5311yescharacteristic
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—2121no—
363C22×C4D4×D5311yescharacteristic
364C22×C4#1171994711yescharacteristic
365C22×C4C2×C22⋊D5311yescharacteristic
366SD16—42no—
367D8—42no—
368M16—2122no—
369M16—2122no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—532no—
373C2×C8—532no—
374C2×C8—1062no—
375C2×C8—1062no—
376C2×C8—2124no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—1062no—
382C4⋊C4—1062no—
383C4⋊C4—2121no—
384C4⋊C4—2121no—
385C4⋊C4C2×C22⋊D5311yescharacteristic
386C4⋊C4C2×C22⋊D5311yescharacteristic
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—21no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42D4×D5311yescharacteristic
396C42#1171994711yescharacteristic
397C42C2×C22⋊D5311yescharacteristic
398C42C23×D5311yescharacteristic
399C23—21no—
400C23—41no—
401C23#1313318911yescharacteristic
402Q8—21no—
403Q8—21no—
404Q8—42no—
405Q8—4242no—
406Q8—4242no—
407D4—21no—
408D4—42no—
409D4—41no—
410D4—42no—
411D4—41no—
412D4—81no—
413D4—82no—
414D4#1313324712yes—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—4242no—
428C2×C4—2122no—
429C2×C4#1313318911yescharacteristic
430C2×C4#1313318911yescharacteristic
431C2×C4#1313318911yescharacteristic
432C2×C4C2×D4×D5311yescharacteristic
433C2×C4C22×C22⋊D5311yescharacteristic
434C2×C4C2×#1171994711yescharacteristic
435C8—41no—
436C8—41no—
437C8—1062no—
438C8—1062no—
439C8—2122no—
440C8—2122no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22C2×#1313318911yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—81no—
459C4—4242no—
460C4C2×#1313324711yescharacteristic
461C4#1681141912yes—
462C4#1681142111yescharacteristic
463C2—41no—
464C2—42no—
465C2—161no—
466C2#2028420711yescharacteristic
467C2#2028434711yescharacteristic
468C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3115C2
socleC22×C53C2×#121
hypercenter#3115D53
hypocenterC53#19318
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #20285108C2semidirect
2. #20284530C2semidirect
3. #20283729C2semidirect
4. #20274295C2semidirect
5. #20274253C2semidirect
6. #20274164C2semidirect
7. #20274122C2semidirect
8. #20273483C2semidirect
9. #20273440C2semidirect
10. #20273437C2semidirect
283. C53#19318semidirect

Maximal subgroups

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

subgroupindex
#202851082
#202845302
#202837292
#202742952
#202742532
#202741642
#202741222
#202734832
#202734402
#202734372
C53×#31152
#1931853