Finite Group Databasebeta

← #27540937

Subgroups — #27540937

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287084C211yesmaximal, characteristic, complemented
2#20276946C211yesmaximal, characteristic, complemented
3#20276910C211yesmaximal, characteristic, complemented
4#20276903C211yesmaximal, characteristic, complemented
5#20276872C211yesmaximal, characteristic, complemented
6#20276809C211yesmaximal, characteristic, complemented
7#20276804C211yesmaximal, characteristic, complemented
8#20276774C211yesmaximal, characteristic, complemented
9#20276655C211yesmaximal, characteristic, complemented
10#20276644C211yesmaximal, characteristic, complemented
11C4×#13133161C211yesmaximal, characteristic, complemented
12D4×#11719908C211yesmaximal, characteristic, complemented
13#20276422C211yesmaximal, characteristic, complemented
14C2×#16810424C211yesmaximal, characteristic, complemented
15C53×#3280C211yesmaximal, characteristic
16#16811313—21no—
17C2×#13133163C2211yescharacteristic, complemented
18C2×#13133161C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20C22×#11719908C2211yescharacteristic
21C22×#11719908C2211yescharacteristic, complemented
22#16811025C2211yescharacteristic, complemented
23#16810965—21no—
24#16810954—21no—
25#16810931—21no—
26#16810864C2211yescharacteristic
27#16810864C2211yescharacteristic, complemented
28C4×Dic212C2211yescharacteristic, complemented
29#16810661C2211yescharacteristic, complemented
30C2×#13133151C2211yescharacteristic
31#16810450C2211yescharacteristic, complemented
32#16810450C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810427C2211yescharacteristic, complemented
36#16810427C2211yescharacteristic, complemented
37#16810427C2211yescharacteristic, complemented
38#16810426C2211yescharacteristic, complemented
39#16810424C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810424C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic, complemented
47C4×#11719908C2211yescharacteristic, complemented
48C2×#135×C53C2211yescharacteristic
49C22×D8×C53C2211yescharacteristic
50C53×#546—21no—
51C53×#495C2211yescharacteristic
52C53×#467C2211yescharacteristic
53C53×#466C2211yescharacteristic
54C53×#463—21no—
55C53×#450—21no—
56C53×#449—21no—
57C4×D8×C53C2211yescharacteristic
58C8×D4×C53C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60#13133210—41no—
61#13133209—21no—
62#13133207—41no—
63#13133198—41no—
64#13133196—21no—
65#13133196—21no—
66C4×Dic106—21no—
67#13133176—21no—
68#13133173—41no—
69C2×C4⋊Dic53—21no—
70#13133163—41no—
71#13133161—22no—
72#13133161—41no—
73C2×#11719908—41no—
74C2×#11719908—41no—
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79#13133151—21no—
80#13133151C2311yescharacteristic
81#13133151C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133149—21no—
84#13133149—21no—
85#13133142—21no—
86#13133142—21no—
87#13133142—21no—
88C2×#41×C53—21no—
89C22×D4×C53—21no—
90#135×C53—22no—
91#135×C53—41no—
92C2×Q16×C53C2311yescharacteristic
93C2×SD16×C53—21no—
94C2×SD16×C53—21no—
95C2×D8×C53—21no—
96C2×D8×C53—22no—
97C2×D8×C53—41no—
98C2×D8×C53C2311yescharacteristic
99C2×D8×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102#124×C53—21no—
103#123×C53—41no—
104C22⋊Q8×C53—21no—
105#121×C53—21no—
106#121×C53—41no—
107#120×C53—41no—
108C4×D4×C53—21no—
109C4×D4×C53D411yescharacteristic
110C2×C22⋊C4×C53—41no—
111#107×C53C2311yescharacteristic
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#102×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C24×C53—41no—
120#41×C53—42no—
121#41×C53—81no—
122C2×Q8×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—42no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53—81no—
130C2×D4×C53D811yescharacteristic
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×Dic53—41no—
136C2×Dic106—21no—
137Q16×C53—21no—
138Q16×C53—21no—
139SD16×C53—42no—
140SD16×C53—81no—
141D8×C53—21no—
142D8×C53—21no—
143D8×C53—21no—
144D8×C53—42no—
145D8×C53#4112yes—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53C2×D411yescharacteristic
153C2×C8×C53C2411yescharacteristic
154C4⋊C4×C53—21no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53D811yescharacteristic
157C22⋊C4×C53—41no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—41no—
160C22⋊C4×C53—41no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53D811yescharacteristic
163C22⋊C4×C53D811yescharacteristic
164C42×C53C2×D411yescharacteristic
165C22⋊Dic53—41no—
166C22⋊Dic53—41no—
167C22⋊Dic53—41no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—41no—
172#11719919—41no—
173#11719919—41no—
174#11719919—41no—
175#11719919—41no—
176#11719919—41no—
177C4×Dic53—41no—
178C4×Dic53—41no—
179Dic212—21no—
180Dic212—21no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—82no—
188C23×C53—82no—
189C23×C53—161no—
190C23×C53C2×D811yescharacteristic
191C23×C53C2×D811yescharacteristic
192Q8×C53—41no—
193Q8×C53—41no—
194D4×C53—22no—
195D4×C53—22no—
196D4×C53—41no—
197D4×C53—41no—
198D4×C53—41no—
199D4×C53—82no—
200D4×C53—82no—
201D4×C53—161no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—82no—
209C2×C4×C53—161no—
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C2×D811yescharacteristic
212C2×C4×C53C2×D811yescharacteristic
213C2×C4×C53C2×D811yescharacteristic
214C2×C4×C53C22×D411yescharacteristic
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—82no—
220Dic106—41no—
221Dic106—41no—
222C8×C53—21no—
223C8×C53—21no—
224C8×C53—21no—
225C8×C53C2×#4111yescharacteristic
226C8×C53H(5,𝔽2)11yescharacteristic
227#33863—531nomaximal, Sylow
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—42no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53—164no—
240C22×C53#50812yes—
241C22×C53C22×D811yescharacteristic
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—81no—
247C4×C53—81no—
248C4×C53#54611yescharacteristic
249C4×C53#57711yescharacteristic
250Dic53—81no—
251Dic53—81no—
252Dic53—81no—
253Dic53—81no—
254#3336—531no—
255#3280D5311yescharacteristic
256#2242—531no—
257#2235—531no—
258#2234—531no—
259#2228—531no—
260#2217—531no—
261#2216—531no—
262#2211—531no—
263#2189—531no—
264#2187—531no—
265C4×SD32—531no—
266D4×C16—531no—
267#2138—531no—
268C2×#357—531no—
269C2×C53—22no—
270C2×C53—41no—
271C2×C53—82no—
272C2×C53—161no—
273C2×C53#329411yescharacteristic
274C2×C53C2×#50811yescharacteristic
275C2×C53#341611yescharacteristic
276C2×#135C2×D5311yescharacteristic
277C22×D8C2×D5311yescharacteristic
278#552—1061no—
279#546—21no—
280C2×Q32—531no—
281C2×SD32—531no—
282C2×SD32—531no—
283C22×C16—531no—
284C22×C16—531no—
285#499—531no—
286#495C2×D5311yescharacteristic
287#485—1061no—
288#483—1061no—
289#479—1061no—
290#467—531no—
291#467—531no—
292#467C2×D5311yescharacteristic
293#466C2×D5311yescharacteristic
294#463—21no—
295#450—21no—
296#449—21no—
297C4×Q16—531no—
298C4×D8C2×D5311yescharacteristic
299C8×D4C2×D5311yescharacteristic
300#427—531no—
301C2×#107—531no—
302#367—531no—
303#367—531no—
304#366—531no—
305C4⋊C16—531no—
306#358—531no—
307#358—531no—
308#358—531no—
309#358—531no—
310#357—531no—
311#357—531no—
312#357—531no—
313#357—531no—
314#357—531no—
315#357—531no—
316C22⋊C16—531no—
317C22⋊C16—531no—
318C4×C16—531no—
319C53#3386311yescharacteristic, complemented, Sylow
320C2×#41—21no—
321C22×D4—21no—
322#135—22no—
323#135—41no—
324C2×Q16—531no—
325C2×Q16C22×D5311yescharacteristic
326C2×SD16—21no—
327C2×SD16—21no—
328C2×D8—22no—
329C2×D8—21no—
330C2×D8—41no—
331C2×D8C22×D5311yescharacteristic
332C2×D8C22×D5311yescharacteristic
333C22×C8C22×D5311yescharacteristic
334C22×C8C22×D5311yescharacteristic
335#126—2121no—
336#125—1061no—
337#125—2121no—
338#124—21no—
339#123—41no—
340#123—2121no—
341C22⋊Q8—21no—
342C22⋊Q8—1061no—
343C22⋊Q8—1061no—
344#121—21no—
345#121—41no—
346#120—41no—
347C4×Q8—1061no—
348C4×D4—21no—
349C4×D4C22⋊D5311yescharacteristic
350#117—1061no—
351#117—2121no—
352C2×C4⋊C4—1061no—
353C2×C22⋊C4—41no—
354Q32—2121no—
355SD32—1062no—
356SD32—2121no—
357C2×C16—531no—
358C2×C16—531no—
359C2×C16—531no—
360C2×C16—531no—
361C2×C16—2121no—
362C2×C16—2121no—
363#107—531no—
364#107—531no—
365#107—531no—
366#107—1061no—
367#107C22×D5311yescharacteristic
368#106—1061no—
369#106—1061no—
370C4⋊C8C22⋊D5311yescharacteristic
371#103—21no—
372#103—1061no—
373#103—1061no—
374#103—1061no—
375#102—21no—
376#102—21no—
377C22⋊C8C22⋊D5311yescharacteristic
378C22⋊C8C22⋊D5311yescharacteristic
379C4×C8C22×D5311yescharacteristic
380C24—41no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—1061no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—81no—
392C2×D4#1171992211yescharacteristic
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—2121no—
396C22×C4C2×C22⋊D5311yescharacteristic
397C22×C4C2×C22⋊D5311yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1061no—
401Q16—1061no—
402SD16—42no—
403SD16—81no—
404D8—21no—
405D8—21no—
406D8—21no—
407D8—42no—
408D8#1171994712yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D5311yescharacteristic
414C2×C8C2×C22⋊D5311yescharacteristic
415C2×C8C2×C22⋊D5311yescharacteristic
416C2×C8C23×D5311yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—1061no—
420C4⋊C4—1061no—
421C4⋊C4—1061no—
422C4⋊C4—2121no—
423C4⋊C4—2121no—
424C4⋊C4—2121no—
425C4⋊C4—2121no—
426C4⋊C4—2121no—
427C4⋊C4—2121no—
428C4⋊C4#1171992211yescharacteristic
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4#1171992211yescharacteristic
438C22⋊C4#1171992211yescharacteristic
439C42—2121no—
440C42—2121no—
441C42C2×C22⋊D5311yescharacteristic
442C16—1061no—
443C16—1061no—
444C16—1061no—
445C16—1061no—
446C23—41no—
447C23—41no—
448C23—82no—
449C23—82no—
450C23—161no—
451C23C2×#1171992211yescharacteristic
452C23C2×#1171992211yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—2121no—
456Q8—2121no—
457D4—22no—
458D4—22no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—161no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—161no—
473C2×C4—4242no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4C2×#1171992211yescharacteristic
479C2×C4C2×#1171992211yescharacteristic
480C2×C4C2×#1171992211yescharacteristic
481C2×C4C2×#1171992211yescharacteristic
482C2×C4C22×C22⋊D5311yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8C2×#1171994711yescharacteristic
487C8#1313326211yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—164no—
500C22#1681105312yes—
501C22C22×#1171992211yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—4241no—
509C4—4241no—
510C4—4241no—
511C4—4241no—
512C4#1681126711yescharacteristic
513C4#1681143411yescharacteristic
514C2—22no—
515C2—41no—
516C2—82no—
517C2—161no—
518C2#2028652811yescharacteristic
519C2C2×#1681105311yescharacteristic
520C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3280C2
socleC22×C53C22×D8
hypercenter#3280D53
hypocenterC53#33863
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #20287084C2semidirect
2. #20276946C2semidirect
3. #20276910C2semidirect
4. #20276903C2semidirect
5. #20276872C2semidirect
6. #20276809C2semidirect
7. #20276804C2semidirect
8. #20276774C2semidirect
9. #20276655C2semidirect
10. #20276644C2semidirect
11. C4×#13133161C2semidirect
12. D4×#11719908C2semidirect
13. #20276422C2semidirect
14. C2×#16810424C2semidirect
17. C2×#13133163C22semidirect
21. C22×#11719908C22semidirect
22. #16811025C22semidirect
27. #16810864C22semidirect
28. C4×Dic212C22semidirect
29. #16810661C22semidirect
31. #16810450C22semidirect
32. #16810450C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810427C22semidirect
36. #16810427C22semidirect
37. #16810427C22semidirect
38. #16810426C22semidirect
46. C22⋊#11719908C22semidirect
47. C4×#11719908C22semidirect
319. C53#33863semidirect

Maximal subgroups

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

subgroupindex
#202870842
#202769462
#202769102
#202769032
#202768722
#202768092
#202768042
#202767742
#202766552
#202766442
C4×#131331612
D4×#117199082
#202764222
C2×#168104242
C53×#32802
#3386353