Finite Group Databasebeta

← #27540983

Subgroups — #27540983

Order 13568.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286458C211yesmaximal, characteristic
2#20276949C211yesmaximal, characteristic, complemented
3#20276923C211yesmaximal, characteristic, complemented
4#20276895C211yesmaximal, characteristic, complemented
5#20276876C211yesmaximal, characteristic, complemented
6#20276813C211yesmaximal, characteristic, complemented
7#20276804C211yesmaximal, characteristic, complemented
8#20276777C211yesmaximal, characteristic, complemented
9#20276653C211yesmaximal, characteristic, complemented
10#20276648C211yesmaximal, characteristic, complemented
11C4×#13133163C211yesmaximal, characteristic, complemented
12D4×#11719908C211yesmaximal, characteristic, complemented
13#20276420C211yesmaximal, characteristic, complemented
14C2×#16810426C211yesmaximal, characteristic, complemented
15C53×#3336C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×Dic212C2211yescharacteristic
18#16811282—21no—
19C2×#13133163C2211yescharacteristic
20C2×#13133163C2211yescharacteristic
21C2×#13133160C2211yescharacteristic
22C22×#11719908C2211yescharacteristic
23C22×#11719908C2211yescharacteristic
24#16811007C2211yescharacteristic
25#16810864C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810846—21no—
28#16810787—21no—
29#16810764—21no—
30C4×Dic212C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810426C2211yescharacteristic
37#16810426C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#552—21no—
49C53×#499C2211yescharacteristic
50C53×#485—21no—
51C53×#483—21no—
52C53×#479—21no—
53C53×#467C2211yescharacteristic
54C53×#467C2211yescharacteristic
55C4×Q16×C53C2211yescharacteristic
56C8×D4×C53C2211yescharacteristic
57C53×#427C2211yescharacteristic
58C2×#107×C53C2211yescharacteristic
59C2×#11719945—21no—
60C22×Dic106—21no—
61#13133239—22no—
62#13133239—41no—
63C2×Dic212—21no—
64C2×Dic212—22no—
65C2×Dic212—41no—
66C2×Dic212C2311yescharacteristic
67C2×Dic212C2311yescharacteristic
68C2×#11719913—21no—
69C2×#11719913—21no—
70C2×D424C2311yescharacteristic
71#13133205—21no—
72#13133203—41no—
73#13133196—21no—
74#13133195—41no—
75#13133191—21no—
76#13133189—41no—
77C4×Dic106—21no—
78#13133177—41no—
79#13133163—22no—
80#13133163—41no—
81#13133160—41no—
82C2×#11719908—41no—
83C2×#11719908—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133142—21no—
90#13133142—21no—
91#13133139—21no—
92C2×Q16×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95#126×C53—41no—
96#125×C53—21no—
97#125×C53—41no—
98#123×C53—41no—
99C22⋊Q8×C53—21no—
100C22⋊Q8×C53—21no—
101C4×Q8×C53—21no—
102C4×D4×C53D411yescharacteristic
103#117×C53—21no—
104#117×C53—41no—
105C2×C4⋊C4×C53—21no—
106#107×C53—21no—
107#107×C53C2311yescharacteristic
108#107×C53C2311yescharacteristic
109#107×C53C2311yescharacteristic
110#106×C53—21no—
111#106×C53—21no—
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#103×C53—21no—
115#103×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C2×Q8×C53—21no—
120C2×D4×C53D811yescharacteristic
121C22×C4×C53—41no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C2×C22⋊D53—41no—
125C2×C22⋊D53—41no—
126C22×Dic53—41no—
127#11719945—42no—
128#11719945—81no—
129C2×D212—21no—
130C2×C4×D53—41no—
131C2×Dic106—21no—
132C2×Dic106—21no—
133C2×Dic106—42no—
134C2×Dic106—41no—
135C2×Dic106—81no—
136Q16×C53—21no—
137Q16×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53C2×D411yescharacteristic
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2×D411yescharacteristic
145C2×C8×C53C2411yescharacteristic
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—41no—
150C4⋊C4×C53—41no—
151C4⋊C4×C53—41no—
152C4⋊C4×C53—41no—
153C4⋊C4×C53—41no—
154C4⋊C4×C53—41no—
155C4⋊C4×C53D811yescharacteristic
156C22⋊C4×C53—41no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53D811yescharacteristic
160C22⋊C4×C53D811yescharacteristic
161C42×C53—41no—
162C42×C53—41no—
163C42×C53C2×D411yescharacteristic
164C22⋊Dic53—41no—
165#11719921—41no—
166#11719921—41no—
167#11719921—41no—
168C4⋊Dic53—21no—
169#11719919—41no—
170#11719919—41no—
171C4×Dic53—41no—
172Dic212—21no—
173Dic212—21no—
174Dic212—21no—
175Dic212—42no—
176Dic212#4112yes—
177D424—21no—
178D424—21no—
179#11719913—42no—
180#11719913—81no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53C2×D811yescharacteristic
186C23×C53C2×D811yescharacteristic
187C22×D53—41no—
188Q8×C53—41no—
189Q8×C53—41no—
190D4×C53—22no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—82no—
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C2×D811yescharacteristic
203C2×C4×C53C2×D811yescharacteristic
204C2×C4×C53C22×D411yescharacteristic
205C22⋊D53—82no—
206C22⋊D53—82no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—82no—
211D212—41no—
212D212—41no—
213C4×D53—82no—
214Dic106—22no—
215Dic106—41no—
216Dic106—41no—
217Dic106—41no—
218Dic106—82no—
219C8×C53—21no—
220C8×C53—21no—
221C8×C53—21no—
222C8×C53C2×#4111yescharacteristic
223C8×C53H(5,𝔽2)11yescharacteristic
224#33867—531nomaximal, Sylow
225C22×C53—21no—
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—21no—
230C22×C53#50812yes—
231C22×C53C22×D811yescharacteristic
232C2×D53—81no—
233C2×D53—81no—
234C2×D53—81no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—81no—
240C4×C53—81no—
241C4×C53—81no—
242C4×C53—81no—
243C4×C53#54611yescharacteristic
244C4×C53#57711yescharacteristic
245Dic53—42no—
246Dic53—81no—
247Dic53—81no—
248#3336D5311yescharacteristic, complemented
249#3285—531no—
250#2242—531no—
251#2237—531no—
252#2232—531no—
253#2229—531no—
254#2217—531no—
255#2216—531no—
256#2211—531no—
257#2189—531no—
258#2188—531no—
259C4×Q32—531no—
260D4×C16—531no—
261#2138—531no—
262C2×#358—531no—
263C2×C53—22no—
264C2×C53—41no—
265C2×C53#329411yescharacteristic
266C2×C53C2×#50811yescharacteristic
267C2×C53#341711yescharacteristic
268D53—161no—
269C2×#135—531no—
270C22×Q16—531no—
271#552—21no—
272#548—1061no—
273C2×Q32—531no—
274C2×Q32—531no—
275C2×SD32—531no—
276C22×C16—531no—
277C22×C16—531no—
278#499C2×D5311yescharacteristic
279#495—531no—
280#485—21no—
281#483—21no—
282#479—21no—
283#467—531no—
284#467C2×D5311yescharacteristic
285#467C2×D5311yescharacteristic
286#466—531no—
287#464—1061no—
288#452—1061no—
289#448—1061no—
290C4×Q16—531no—
291C4×Q16C2×D5311yescharacteristic
292C8×D4C2×D5311yescharacteristic
293#427C2×D5311yescharacteristic
294C2×#107C2×D5311yescharacteristic
295#367—531no—
296#366—531no—
297#366—531no—
298C4⋊C16—531no—
299#358—531no—
300#358—531no—
301#358—531no—
302#358—531no—
303#358—531no—
304#358—531no—
305#358—531no—
306#357—531no—
307#357—531no—
308#357—531no—
309C22⋊C16—531no—
310C22⋊C16—531no—
311C4×C16—531no—
312C53#3386711yescharacteristic, complemented, Sylow
313C2×#41—1061no—
314C22×Q8—1061no—
315#135—1062no—
316#135—2121no—
317C2×Q16—531no—
318C2×Q16—531no—
319C2×Q16—1062no—
320C2×Q16—1061no—
321C2×Q16—2121no—
322C2×Q16C22×D5311yescharacteristic
323C2×SD16—1061no—
324C2×SD16—1061no—
325C2×D8—531no—
326C22×C8C22×D5311yescharacteristic
327C22×C8C22×D5311yescharacteristic
328#126—41no—
329#125—21no—
330#125—41no—
331#124—1061no—
332#124—2121no—
333#123—41no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—1061no—
337C22⋊Q8—2121no—
338#121—1061no—
339#121—2121no—
340C4×Q8—21no—
341C4×Q8—1061no—
342C4×D4—2121no—
343C4×D4C22⋊D5311yescharacteristic
344#117—21no—
345#117—41no—
346C2×C4⋊C4—21no—
347Q32—1062no—
348Q32—2121no—
349SD32—2121no—
350C2×C16—531no—
351C2×C16—531no—
352C2×C16—531no—
353C2×C16—531no—
354C2×C16—2121no—
355C2×C16—2121no—
356#107—21no—
357#107—531no—
358#107C22×D5311yescharacteristic
359#107C22×D5311yescharacteristic
360#107C22×D5311yescharacteristic
361#106—21no—
362#106—21no—
363C4⋊C8C22⋊D5311yescharacteristic
364#103—21no—
365#103—21no—
366#103—21no—
367#103—1061no—
368#103—1061no—
369#102—1061no—
370C22⋊C8C22⋊D5311yescharacteristic
371C22⋊C8C22⋊D5311yescharacteristic
372C4×C8C22×D5311yescharacteristic
373#41—4241no—
374#41—2122no—
375C2×Q8—21no—
376C2×Q8—1061no—
377C2×Q8—1061no—
378C2×Q8—4241no—
379C2×Q8—2122no—
380C2×Q8—2121no—
381C2×D4—1061no—
382C2×D4—2121no—
383C2×D4—2121no—
384C2×D4#1171992211yescharacteristic
385C22×C4—41no—
386C22×C4—2121no—
387C22×C4—2121no—
388C22×C4C2×C22⋊D5311yescharacteristic
389C22×C4C2×C22⋊D5311yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—532no—
393Q16—1061no—
394Q16—1061no—
395Q16—1061no—
396Q16—2122no—
397SD16—4241no—
398SD16—2122no—
399D8—1061no—
400D8—1061no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D5311yescharacteristic
406C2×C8C2×C22⋊D5311yescharacteristic
407C2×C8C2×C22⋊D5311yescharacteristic
408C2×C8C23×D5311yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—1061no—
419C4⋊C4—2121no—
420C4⋊C4—2121no—
421C4⋊C4#1171992211yescharacteristic
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—2121no—
426C22⋊C4—2121no—
427C22⋊C4—2121no—
428C22⋊C4—2121no—
429C22⋊C4#1171992211yescharacteristic
430C22⋊C4#1171992211yescharacteristic
431C42—41no—
432C42—41no—
433C42—2121no—
434C42C2×C22⋊D5311yescharacteristic
435C16—1061no—
436C16—1061no—
437C16—1061no—
438C16—1061no—
439C23—2121no—
440C23C2×#1171992211yescharacteristic
441C23C2×#1171992211yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—1062no—
445Q8—4242no—
446Q8—2121no—
447Q8—2121no—
448Q8—2121no—
449D4—22no—
450D4—4242no—
451D4—4242no—
452D4—2121no—
453D4—2121no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—4242no—
464C2×C4—4242no—
465C2×C4—2121no—
466C2×C4—2121no—
467C2×C4—2121no—
468C2×C4C2×#1171992211yescharacteristic
469C2×C4C2×#1171992211yescharacteristic
470C2×C4C2×#1171992211yescharacteristic
471C2×C4C2×#1171992211yescharacteristic
472C2×C4C22×C22⋊D5311yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8#41×D5311yescharacteristic
477C8#1313326311yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—4241no—
484C22—4241no—
485C22—4241no—
486C22#1681105312yes—
487C22C22×#1171992211yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—4241no—
497C4—4241no—
498C4—2122no—
499C4#1681126111yescharacteristic
500C4#1681143111yescharacteristic
501C2—22no—
502C2—41no—
503C2—8481no—
504C2#2028652211yescharacteristic
505C2C2×#1681105311yescharacteristic
506C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3336C2
socleC22×C53C22×D8
prime radical#20286458C2
hypercenter#3336D53
hypocenterC53#33867
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #20276949C2semidirect
3. #20276923C2semidirect
4. #20276895C2semidirect
5. #20276876C2semidirect
6. #20276813C2semidirect
7. #20276804C2semidirect
8. #20276777C2semidirect
9. #20276653C2semidirect
10. #20276648C2semidirect
11. C4×#13133163C2semidirect
12. D4×#11719908C2semidirect
13. #20276420C2semidirect
14. C2×#16810426C2semidirect
15. C53×#3336C2semidirect
248. #3336D53semidirect
312. C53#33867semidirect

Maximal subgroups

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

subgroupindex
#202864582
#202769492
#202769232
#202768952
#202768762
#202768132
#202768042
#202767772
#202766532
#202766482
C4×#131331632
D4×#117199082
#202764202
C2×#168104262
C53×#33362
#3386753