Finite Group Databasebeta

← #27370025

Subgroups — #27370025

Order 13568.

660 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285642C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic
3C2×#16810903C211yesmaximal, characteristic, complemented
4#20274290C212yesmaximal, complemented
5#20274180C212yesmaximal, complemented
6#20274166C212yesmaximal, complemented
7#20274056C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273433C211yesmaximal, characteristic, complemented
10#20273411C211yesmaximal, characteristic, complemented
11C53×#2974C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13C2×#13133217C2211yescharacteristic
14#16810996C2212yescomplemented
15#16810974C2212yescomplemented
16#16810941—21no—
17#16810913C2211yescharacteristic
18#16810913C2211yescharacteristic
19#16810903—21no—
20#16810903—21no—
21#16810903C2212yes—
22#16810903C2212yes—
23#16810829—22no—
24#16810818C2211yescharacteristic
25#16810818C2211yescharacteristic
26#16810764—21no—
27#16810764—21no—
28#16810757—21no—
29#16810733—21no—
30#16810733—21no—
31#16810680C2211yescharacteristic
32C2×#13133150—21no—
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133138—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810371C2212yescomplemented
38#16810370C2212yescomplemented
39#16810370C2212yescomplemented
40#16810366C2212yescomplemented
41C2×C4×D4×C53C2211yescharacteristic
42C53×#436C2212yes—
43C53×#435—22no—
44C8×D4×C53C2212yes—
45C53×#432C2211yescharacteristic
46C53×#431C2211yescharacteristic
47C53×#407—21no—
48C2×C22⋊C8×C53—21no—
49C2×#11719945—21no—
50C22×Dic106—21no—
51C2×#11719913—22no—
52C2×#11719913—22no—
53C2×#11719922—21no—
54C2×#11719922—21no—
55C22×#88322—21no—
56#13133217—21no—
57#13133217—21no—
58#13133217—21no—
59#13133217C2312yes—
60#13133217C2311yescharacteristic
61#13133214C2311yescharacteristic
62#13133205—21no—
63#13133191—21no—
64#13133191—21no—
65C4×D212—21no—
66C4×D212—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69#13133150—22no—
70#13133150—22no—
71#13133148—21no—
72#13133148—21no—
73C4⋊#88322—21no—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2312yes—
77#13133142—22no—
78#13133142—22no—
79#13133138—21no—
80#13133138—21no—
81#13133138—22no—
82#13133138—22no—
83#13133137—21no—
84#13133137—21no—
85C22⋊#88322—21no—
86#13133128—21no—
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89C2×M16×C53—22no—
90C22×C8×C53—22no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53D411yescharacteristic
94C4×D4×C53D411yescharacteristic
95C4×D4×C53C2311yescharacteristic
96C4×D4×C53C2311yescharacteristic
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C22⋊C4×C53—21no—
99C2×C42×C53C2311yescharacteristic
100C4⋊C8×C53D412yes—
101C4⋊C8×C53C2312yes—
102C22⋊C8×C53—22no—
103C22⋊C8×C53—22no—
104C22⋊C8×C53—21no—
105C22⋊C8×C53D411yescharacteristic
106C22⋊C8×C53D411yescharacteristic
107#97×C53—21no—
108C4×C8×C53C2312yes—
109C24×C53—21no—
110C2×D4×C53—22no—
111C2×D4×C53—21no—
112C2×D4×C53—41no—
113C2×D4×C53C2×D411yescharacteristic
114C2×D4×C53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—82no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C2×C22⋊D53—41no—
125C22×Dic53—41no—
126#11719945—82no—
127C2×D212—21no—
128C2×D212—21no—
129C2×D212—21no—
130C2×C4×D53—41no—
131C2×Dic106—22no—
132C2×Dic106—21no—
133C2×Dic106—41no—
134C2×Dic106—82no—
135M16×C53—82no—
136C2×C8×C53—22no—
137C2×C8×C53—82no—
138C2×C8×C53C2×D412yes—
139C2×C8×C53C2×D412yes—
140C4⋊C4×C53—21no—
141C4⋊C4×C53C2×D411yescharacteristic
142C4⋊C4×C53C2×D411yescharacteristic
143C22⋊C4×C53—21no—
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149C42×C53#4111yescharacteristic
150C42×C53C2411yescharacteristic
151#11719922—42no—
152#11719922—42no—
153#11719921—41no—
154#11719921—41no—
155C4⋊Dic53—21no—
156C4⋊Dic53—22no—
157C4⋊Dic53—21no—
158C4⋊Dic53—22no—
159C4⋊Dic53—41no—
160C4⋊Dic53—41no—
161C2×#88322—21no—
162C2×#88322—21no—
163C2×#88322—22no—
164C2×#88322—42no—
165#11719913—82no—
166#11719913—82no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53—81no—
172C23×C53#12111yescharacteristic
173C22×D53—41no—
174D4×C53—21no—
175D4×C53—21no—
176D4×C53—21no—
177D4×C53—42no—
178D4×C53#13712yes—
179C2×C4×C53—22no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—22no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—81no—
188C2×C4×C53—81no—
189C2×C4×C53—81no—
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—42no—
198C2×Dic53—41no—
199C2×Dic53—82no—
200D212—41no—
201D212—41no—
202D212—42no—
203D212—81no—
204C4×D53—82no—
205Dic106—42no—
206Dic106—42no—
207Dic106—82no—
208Dic106—81no—
209C8×C53—22no—
210C8×C53—22no—
211C8×C53—42no—
212#88322—41no—
213#88322—42no—
214#88322—41no—
215#19457—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—22no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—84no—
225C22×C53#47112yes—
226C22×C53C2×#12111yescharacteristic
227C2×D53—81no—
228C2×D53—81no—
229C2×D53—81no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—41no—
235C4×C53—41no—
236C4×C53C2×#13711yescharacteristic
237C4×C53#57511yescharacteristic
238C4×C53#57612yes—
239Dic53—82no—
240Dic53—81no—
241#3206—531no—
242#3109—531no—
243C2×#474—531no—
244#2974D5311yescharacteristic, complemented
245#1703—532no—
246#1685—532no—
247#1683—532no—
248#1665—532no—
249#1573—531no—
250#1556—531no—
251#1551—531no—
252C2×C53—22no—
253C2×C53—42no—
254C2×C53—81no—
255C2×C53#304111yescharacteristic
256C2×C53#305611yescharacteristic
257C2×C53C2×#47111yescharacteristic
258D53—161no—
259#532—531no—
260C2×C4⋊Q8—531no—
261C2×C4×D4C2×D5311yescharacteristic
262#492—532no—
263#487—532no—
264#481—1061no—
265#476—531no—
266#476—531no—
267#474—532no—
268#474—532no—
269#474—1061no—
270#474—1061no—
271#461—1062no—
272#459—531no—
273#459—531no—
274#448—1061no—
275#448—1061no—
276#447—1061no—
277#442—1061no—
278#442—1061no—
279#436C2×D5312yes—
280#435—22no—
281C8×D4C2×D5312yes—
282#432—531no—
283#432C2×D5311yescharacteristic
284#431C2×D5311yescharacteristic
285C2×#107—1061no—
286C2×C4⋊C8—531no—
287C2×#102—1061no—
288#407—21no—
289C2×C22⋊C8—21no—
290#335—532no—
291#334—532no—
292#333—532no—
293#332—532no—
294#332—532no—
295#331—532no—
296C53#1945711yescharacteristic, complemented, Sylow
297C2×#41—1061no—
298C22×Q8—1061no—
299C22×D4D21211yescharacteristic
300C23×C4—21no—
301C2×SD16—1062no—
302C2×SD16—1062no—
303C2×D8—1061no—
304C2×D8—1061no—
305C2×M16—22no—
306C22×C8—22no—
307C22×C8—1061no—
308C4⋊Q8—532no—
309C4⋊Q8—531no—
310C4⋊Q8—1061no—
311C4⋊Q8—1061no—
312C4⋊Q8—1061no—
313#127—531no—
314#124—1061no—
315#121—1061no—
316#121—1061no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—1061no—
320C4×D4—1061no—
321C4×D4D21211yescharacteristic
322C4×D4D21211yescharacteristic
323C4×D4C22×D5311yescharacteristic
324C4×D4C22×D5311yescharacteristic
325C2×C4⋊C4—1061no—
326C2×C4⋊C4—1061no—
327C2×C4⋊C4D21211yescharacteristic
328C2×C22⋊C4—21no—
329C2×C42C22×D5311yescharacteristic
330#107—1062no—
331#107—1062no—
332#106—1061no—
333#106—1061no—
334C4⋊C8—532no—
335C4⋊C8—531no—
336C4⋊C8—531no—
337C4⋊C8—1061no—
338C4⋊C8C22⋊D5312yes—
339C4⋊C8C22×D5312yes—
340#103—1062no—
341#103—1062no—
342#102—1062no—
343#102—1062no—
344#102—1061no—
345#102—1061no—
346#102—1061no—
347#102—1061no—
348C22⋊C8—21no—
349C22⋊C8—22no—
350C22⋊C8—22no—
351C22⋊C8—1061no—
352C22⋊C8C22⋊D5311yescharacteristic
353C22⋊C8C22⋊D5311yescharacteristic
354#97—21no—
355#97—1061no—
356C4×C8C22×D5312yes—
357C24—21no—
358#41—4242no—
359C2×Q8—1062no—
360C2×Q8—1061no—
361C2×Q8—4242no—
362C2×Q8—2121no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—1061no—
367C2×D4—1061no—
368C2×D4—1061no—
369C2×D4—2121no—
370C2×D4C2×D21211yescharacteristic
371C2×D4C2×D21211yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—82no—
378C22×C4—2121no—
379C22×C4—2121no—
380C22×C4C2×D21211yescharacteristic
381C22×C4#1171994511yescharacteristic
382C22×C4C2×C22⋊D5311yescharacteristic
383SD16—4242no—
384SD16—4242no—
385D8—2122no—
386D8—2122no—
387M16—82no—
388C2×C8—22no—
389C2×C8—82no—
390C2×C8—1061no—
391C2×C8—1061no—
392C2×C8—1062no—
393C2×C8—2122no—
394C2×C8C2×C22⋊D5312yes—
395C2×C8C2×C22⋊D5312yes—
396C4⋊C4—21no—
397C4⋊C4—1061no—
398C4⋊C4—1061no—
399C4⋊C4—1062no—
400C4⋊C4—1062no—
401C4⋊C4—2121no—
402C4⋊C4—2121no—
403C4⋊C4C2×D21211yescharacteristic
404C4⋊C4C2×D21211yescharacteristic
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—2121no—
410C22⋊C4—2121no—
411C42C2×D21211yescharacteristic
412C42#1171994511yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C42C23×D5311yescharacteristic
415C23—21no—
416C23—21no—
417C23—42no—
418C23—42no—
419C23—81no—
420C23—2121no—
421C23#1313319111yescharacteristic
422Q8—4241no—
423Q8—4242no—
424Q8—2122no—
425Q8—2122no—
426D4—21no—
427D4—21no—
428D4—21no—
429D4—42no—
430D4—4241no—
431D4—4242no—
432D4—2121no—
433D4—2121no—
434D4—2122no—
435D4#1313325012yes—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—4242no—
448C2×C4—4242no—
449C2×C4—2121no—
450C2×C4—2122no—
451C2×C4#1313319111yescharacteristic
452C2×C4#1313319111yescharacteristic
453C2×C4#1313319111yescharacteristic
454C2×C4C22×C22⋊D5311yescharacteristic
455C2×C4C22×D21211yescharacteristic
456C2×C4C2×#1171994511yescharacteristic
457C8—22no—
458C8—22no—
459C8—42no—
460C8—2121no—
461C8—2122no—
462C8—2121no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—84no—
472C22—4241no—
473C22—4241no—
474C22—4241no—
475C22#1681088612yes—
476C22C2×#1313319111yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—41no—
482C4—41no—
483C4—4242no—
484C4—4241no—
485C4C2×#1313325011yescharacteristic
486C4#1681142311yescharacteristic
487C4#1681142612yes—
488C2—22no—
489C2—42no—
490C2—81no—
491C2—8481no—
492C2#2028421811yescharacteristic
493C2#2028434511yescharacteristic
494C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2974C2
socleC22×C53C2×#121
prime radical#20284806C2
hypercenter#2974D53
hypocenterC53#19457
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. #20285642C2semidirect
3. C2×#16810903C2semidirect
4. #20274290C2semidirect
5. #20274180C2semidirect
6. #20274166C2semidirect
7. #20274056C2semidirect
8. #20273500C2semidirect
9. #20273433C2semidirect
10. #20273411C2semidirect
11. C53×#2974C2semidirect
14. #16810996C22semidirect
15. #16810974C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810371C22semidirect
38. #16810370C22semidirect
39. #16810370C22semidirect
40. #16810366C22semidirect
244. #2974D53semidirect
296. C53#19457semidirect

Maximal subgroups

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

subgroupindex
#202856422
#202848062
C2×#168109032
#202742902
#202741802
#202741662
#202740562
#202735002
#202734332
#202734112
C53×#29742
#1945753