Finite Group Databasebeta

← #27369495

Subgroups — #27369495

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285658C211yesmaximal, characteristic, complemented
2#20284503C211yesmaximal, characteristic, complemented
3#20284195C211yesmaximal, characteristic
4#20274131C212yesmaximal, complemented
5#20274124C212yesmaximal, complemented
6#20274063C212yesmaximal, complemented
7#20274056C212yesmaximal, complemented
8#20273486C211yesmaximal, characteristic, complemented
9#20273403C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#2988C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13#16811180C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16810996C2212yes—
16#16810970C2212yes—
17#16810930—21no—
18#16810923—21no—
19#16810918C2211yescharacteristic, complemented
20#16810913C2211yescharacteristic, complemented
21#16810908C2211yescharacteristic, complemented
22#16810903C2211yescharacteristic, complemented
23#16810824—21no—
24#16810824—21no—
25#16810823—21no—
26#16810823C2212yes—
27#16810818—21no—
28#16810818C2212yes—
29#16810776—22no—
30#16810659—21no—
31#16810647C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810627—21no—
34#16810619—21no—
35#16810374C2212yescomplemented
36#16810374C2212yescomplemented
37#16810371C2212yescomplemented
38#16810370C2212yescomplemented
39#16810366C2212yes—
40#16810366C2212yes—
41C4×#41×C53C2211yescharacteristic
42C8×Q8×C53C2212yescomplemented
43C8×D4×C53—22no—
44C8×D4×C53C2212yescomplemented
45C53×#431—21no—
46C53×#431C2211yescharacteristic
47C2×C4⋊C8×C53—21no—
48C2×C4⋊C8×C53C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52#13133242—44no—
53C2×#11719913—22no—
54C2×#11719913—22no—
55C2×#11719924—22no—
56C2×#11719922—22no—
57C2×#11719917—21no—
58C22×#88322—21no—
59#13133217C2311yescharacteristic
60#13133217C2311yescharacteristic
61#13133214—21no—
62#13133214—21no—
63#13133214—21no—
64#13133214C2312yes—
65#13133209—21no—
66#13133196—21no—
67#13133196—21no—
68C4×Dic106—21no—
69C4×Dic106—21no—
70#13133176—21no—
71#13133150—21no—
72#13133150—21no—
73#13133148—21no—
74#13133148—21no—
75C4⋊#88322—21no—
76C4⋊#88322—21no—
77C4⋊#88322C2312yes—
78C4⋊#88322C2311yescharacteristic
79C4⋊#88322C2311yescharacteristic
80#13133141—21no—
81#13133141—21no—
82#13133139—22no—
83#13133139—22no—
84#13133138—21no—
85#13133138—21no—
86#13133137—22no—
87#13133137—22no—
88C2×#41×C53D411yescharacteristic
89C22×C8×C53—22no—
90C22×C8×C53—21no—
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53C2311yescharacteristic
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53D411yescharacteristic
96C4×D4×C53C2311yescharacteristic
97#117×C53—21no—
98#117×C53D411yescharacteristic
99C2×C42×C53—21no—
100C2×C42×C53C2311yescharacteristic
101C4⋊C8×C53—22no—
102C4⋊C8×C53—21no—
103C4⋊C8×C53D411yescharacteristic
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53C2312yes—
106C22⋊C8×C53—22no—
107C22⋊C8×C53D412yes—
108C4×C8×C53—22no—
109C4×C8×C53C2312yes—
110C23×D53—41no—
111#41×C53—22no—
112#41×C53SD1614yescomplemented
113C2×Q8×C53C2×D411yescharacteristic
114C2×D4×C53—21no—
115C2×D4×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C2×D212—22no—
123C2×D212—22no—
124C2×D212—22no—
125C2×D212—42no—
126C2×D212—42no—
127C2×D212—41no—
128C2×D212—41no—
129C2×D212—41no—
130C2×D212—81no—
131C2×D212—81no—
132C2×D212—82no—
133C2×Dic106—21no—
134C2×Dic106—21no—
135C2×C8×C53—22no—
136C2×C8×C53—82no—
137C2×C8×C53—81no—
138C2×C8×C53C2×D412yes—
139C2×C8×C53C2×D412yes—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53C2×D411yescharacteristic
143C4⋊C4×C53C2×D411yescharacteristic
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154C22⋊Dic53—41no—
155#11719924—44no—
156#11719922—44no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161#11719919—41no—
162#11719919—41no—
163C4×Dic53—41no—
164#11719917—42no—
165C2×#88322—22no—
166C2×#88322—21no—
167C2×#88322—21no—
168C2×#88322—42no—
169#11719913—82no—
170#11719913—82no—
171C23×C53—21no—
172C23×C53#12111yescharacteristic
173C22×D53—42no—
174C22×D53—82no—
175C22×D53—82no—
176C22×D53—82no—
177Q8×C53—21no—
178Q8×C53C2×SD1612yes—
179D4×C53—24no—
180D4×C53—21no—
181D4×C53C2×SD1612yes—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—24no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—81no—
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53C2×SD1612yes—
196C2×C4×C53#13512yes—
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C2×#4111yescharacteristic
200C2×Dic53—41no—
201C2×Dic53—41no—
202D212—42no—
203D212—42no—
204D212—42no—
205D212—42no—
206D212—82no—
207D212—82no—
208D212—82no—
209D212—82no—
210Dic106—41no—
211Dic106—41no—
212Dic106—81no—
213C8×C53—22no—
214C8×C53—22no—
215C8×C53—42no—
216#88322—42no—
217#88322—41no—
218#88322—41no—
219#19409—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—22no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53#46512yes—
226C22×C53C2×#12111yescharacteristic
227C2×D53—82no—
228C2×D53—82no—
229C2×D53—82no—
230C2×D53—164no—
231C4×C53—21no—
232C4×C53—22no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—41no—
236C4×C53—41no—
237C4×C53#46512yes—
238C4×C53C22×SD1611yescharacteristic
239C4×C53C2×#13511yescharacteristic
240C4×C53#57611yescharacteristic
241C4×C53#57711yescharacteristic
242Dic53—81no—
243Dic53—81no—
244#3208—531no—
245#3076—531no—
246#3038—531no—
247#2988D5311yescharacteristic, complemented
248#1678—532no—
249#1677—532no—
250#1666—532no—
251#1665—532no—
252#1570—531no—
253#1550—531no—
254C2×#331—531no—
255C2×C53—22no—
256C2×C53—42no—
257C2×C53#304511yescharacteristic
258C2×C53C2×#46511yescharacteristic
259C2×C53#306811yescharacteristic
260D53—162no—
261C2×AGL(1,ℤ/8)—1061no—
262#533—531no—
263C2×#127—531no—
264C4×#41C2×D5311yescharacteristic
265#492—532no—
266#486—532no—
267#479—1061no—
268#478—1061no—
269#477—531no—
270#476—531no—
271#475—531no—
272#474—531no—
273#460—532no—
274#460—1061no—
275#460—1061no—
276#460—1061no—
277#459—532no—
278#459—1061no—
279#450—1062no—
280C8×Q8C2×D5312yescomplemented
281C8×D4—22no—
282C8×D4C2×D5312yescomplemented
283#431—21no—
284#431C2×D5311yescharacteristic
285#427—1061no—
286#423—531no—
287C2×C4⋊C8—21no—
288C2×C4⋊C8—531no—
289C2×C4⋊C8C2×D5311yescharacteristic
290#418—1061no—
291#416—1061no—
292#334—532no—
293#334—532no—
294#333—532no—
295#332—532no—
296#331—532no—
297#331—532no—
298C53#1940911yescharacteristic, complemented, Sylow
299C2×#41D21211yescharacteristic
300C22×D4—1061no—
301C22×D4—1061no—
302C22×D4—1061no—
303AGL(1,ℤ/8)—2124no—
304C2×SD16—1062no—
305C2×SD16—1062no—
306C2×SD16—1062no—
307C2×D8—1062no—
308C2×M16—1061no—
309C22×C8—21no—
310C22×C8—22no—
311C22×C8—1061no—
312C4⋊Q8—531no—
313C4⋊Q8—531no—
314#127—532no—
315#127—1061no—
316#127—1061no—
317#127—1061no—
318#125—1061no—
319C22⋊Q8—1061no—
320C22⋊Q8—1061no—
321C4×Q8—1061no—
322C4×Q8—1061no—
323C4×Q8D21211yescharacteristic
324C4×Q8C22×D5311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4D21211yescharacteristic
328C4×D4C22×D5311yescharacteristic
329#117—21no—
330#117—1061no—
331#117D21211yescharacteristic
332C2×C42—21no—
333C2×C42C22×D5311yescharacteristic
334#107—1061no—
335#107—1061no—
336#106—1061no—
337#106—1061no—
338C4⋊C8—22no—
339C4⋊C8—21no—
340C4⋊C8—531no—
341C4⋊C8—531no—
342C4⋊C8—532no—
343C4⋊C8—1061no—
344C4⋊C8—1061no—
345C4⋊C8C22⋊D5311yescharacteristic
346C4⋊C8C22⋊D5311yescharacteristic
347C4⋊C8C22×D5312yes—
348#103—1061no—
349#103—1061no—
350#102—1061no—
351#102—1061no—
352#102—1062no—
353#102—1062no—
354#102—1062no—
355#102—1062no—
356C22⋊C8—22no—
357C22⋊C8C22⋊D5312yes—
358C4×C8—22no—
359C4×C8C22×D5312yes—
360C24—2121no—
361#41—22no—
362#41#1171991314yescomplemented
363C2×Q8—1061no—
364C2×Q8—1061no—
365C2×Q8C2×D21211yescharacteristic
366C2×D4—21no—
367C2×D4—1062no—
368C2×D4—1062no—
369C2×D4—1062no—
370C2×D4—4241no—
371C2×D4—4241no—
372C2×D4—4242no—
373C2×D4—2121no—
374C2×D4—2121no—
375C2×D4—2121no—
376C2×D4—2122no—
377C2×D4—2122no—
378C2×D4C2×D21211yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4C2×D21211yescharacteristic
383C22×C4#1171994511yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385SD16—4242no—
386SD16—4242no—
387SD16—2124no—
388D8—2124no—
389M16—2122no—
390C2×C8—22no—
391C2×C8—82no—
392C2×C8—81no—
393C2×C8—1062no—
394C2×C8—1061no—
395C2×C8—1061no—
396C2×C8—2122no—
397C2×C8C2×C22⋊D5312yes—
398C2×C8C2×C22⋊D5312yes—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—1061no—
402C4⋊C4—1061no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C4⋊C4C2×D21211yescharacteristic
408C4⋊C4C2×D21211yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—2121no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—2121no—
417C42C2×D21211yescharacteristic
418C42#1171994511yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—21no—
422C23—4242no—
423C23—4242no—
424C23—4242no—
425C23—2122no—
426C23#1313319111yescharacteristic
427Q8—21no—
428Q8—4241no—
429Q8—2121no—
430Q8—2121no—
431Q8C2×#1171991312yes—
432D4—24no—
433D4—21no—
434D4—4242no—
435D4—4242no—
436D4—4242no—
437D4—4242no—
438D4—2122no—
439D4—2122no—
440D4—2122no—
441D4—2122no—
442D4C2×#1171991312yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—2121no—
454C2×C4—2121no—
455C2×C4#1313319111yescharacteristic
456C2×C4#1313319111yescharacteristic
457C2×C4#1313319111yescharacteristic
458C2×C4C2×#1171991312yes—
459C2×C4#1313323912yes—
460C2×C4C22×C22⋊D5311yescharacteristic
461C2×C4C22×D21211yescharacteristic
462C2×C4C2×#1171994511yescharacteristic
463C8—22no—
464C8—22no—
465C8—42no—
466C8—2122no—
467C8—2121no—
468C8—2121no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—8484no—
475C22—4242no—
476C22—4242no—
477C22—4242no—
478C22#1681085212yes—
479C22C2×#1313319111yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—4241no—
487C4—4241no—
488C4#1681085212yes—
489C4C22×#1171991311yescharacteristic
490C4C2×#1313323911yescharacteristic
491C4#1681142611yescharacteristic
492C4#1681143111yescharacteristic
493C2—22no—
494C2—42no—
495C2—8482no—
496C2#2028425411yescharacteristic
497C2C2×#1681085211yescharacteristic
498C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2988C2
socleC22×C53C2×#121
prime radical#20284195C2
hypercenter#2988D53
hypocenterC53#19409
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #20285658C2semidirect
2. #20284503C2semidirect
4. #20274131C2semidirect
5. #20274124C2semidirect
6. #20274063C2semidirect
7. #20274056C2semidirect
8. #20273486C2semidirect
9. #20273403C2semidirect
10. C2×#16810366C2semidirect
11. C53×#2988C2semidirect
19. #16810918C22semidirect
20. #16810913C22semidirect
21. #16810908C22semidirect
22. #16810903C22semidirect
35. #16810374C22semidirect
36. #16810374C22semidirect
37. #16810371C22semidirect
38. #16810370C22semidirect
42. C8×Q8×C53C22semidirect
44. C8×D4×C53C22semidirect
112. #41×C53SD16semidirect
247. #2988D53semidirect
280. C8×Q8C2×D53semidirect
282. C8×D4C2×D53semidirect
298. C53#19409semidirect
362. #41#11719913semidirect

Maximal subgroups

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

subgroupindex
#202856582
#202845032
#202841952
#202741312
#202741242
#202740632
#202740562
#202734862
#202734032
C2×#168103662
C53×#29882
#1940953