Finite Group Databasebeta

← #27541123

Subgroups — #27541123

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287084C211yesmaximal, characteristic, complemented
2#20276956C211yesmaximal, characteristic, complemented
3#20276903C211yesmaximal, characteristic, complemented
4#20276893C211yesmaximal, characteristic, complemented
5#20276872C211yesmaximal, characteristic, complemented
6#20276826C211yesmaximal, characteristic, complemented
7#20276809C211yesmaximal, characteristic, complemented
8#20276752C211yesmaximal, characteristic, complemented
9#20276644C211yesmaximal, characteristic, complemented
10#20276644C211yesmaximal, characteristic, complemented
11#20276579C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276431C211yesmaximal, characteristic, complemented
14C2×#16810424C211yesmaximal, characteristic, complemented
15D4×D8×C53C211yesmaximal, characteristic
16#16811313—21no—
17C2×#13133161C2211yescharacteristic
18C2×#13133161C2211yescharacteristic, complemented
19C2×#13133161C2211yescharacteristic, complemented
20C2×#13133157C2211yescharacteristic, complemented
21C22×#11719908C2211yescharacteristic, complemented
22#16811025C2211yescharacteristic, complemented
23#16810965—21no—
24#16810954—21no—
25#16810931—21no—
26#16810864C2211yescharacteristic, complemented
27#16810864C2211yescharacteristic, complemented
28C4×Dic212C2211yescharacteristic, complemented
29#16810661C2211yescharacteristic, complemented
30C2×#13133151C2211yescharacteristic, complemented
31#16810450C2211yescharacteristic, complemented
32#16810448C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810426C2211yescharacteristic, complemented
36#16810424C2211yescharacteristic
37#16810424C2211yescharacteristic
38#16810424C2211yescharacteristic
39#16810424C2211yescharacteristic, complemented
40#16810424C2211yescharacteristic, complemented
41#16810424C2211yescharacteristic, complemented
42#16810424C2211yescharacteristic, complemented
43#16810424C2211yescharacteristic, complemented
44#16810424C2211yescharacteristic, complemented
45C22⋊#11719908C2211yescharacteristic, complemented
46C22⋊#11719908C2211yescharacteristic, complemented
47#16810399C2211yescharacteristic, complemented
48C22×D8×C53C2211yescharacteristic
49C22×D8×C53C2211yescharacteristic
50D42×C53—21no—
51C53×#493C2211yescharacteristic
52C53×#466C2211yescharacteristic
53C53×#466C2211yescharacteristic
54C53×#459—21no—
55C53×#447—21no—
56C53×#447—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#13133161—22no—
71#13133161—41no—
72#13133161—41no—
73#13133157—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—
88C22×D4×C53—21no—
89C22×D4×C53—21no—
90C22×D4×C53—41no—
91C2×D8×C53—21no—
92C2×D8×C53—21no—
93C2×D8×C53—21no—
94C2×D8×C53—22no—
95C2×D8×C53—22no—
96C2×D8×C53—41no—
97C2×D8×C53—41no—
98C2×D8×C53C2311yescharacteristic
99C2×D8×C53C2311yescharacteristic
100C2×D8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103#127×C53—21no—
104#121×C53—21no—
105#121×C53—21no—
106#121×C53—41no—
107#120×C53—41no—
108#120×C53—41no—
109C4×D4×C53—21no—
110C4×D4×C53D411yescharacteristic
111#107×C53C2311yescharacteristic
112C4⋊C8×C53D411yescharacteristic
113#102×C53—21no—
114#102×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C24×C53—41no—
120C24×C53—41no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53—41no—
130C2×D4×C53—41no—
131C2×D4×C53—41no—
132C2×D4×C53—81no—
133C2×D4×C53—81no—
134C2×D4×C53—84no—
135C2×D4×C53D811yescharacteristic
136C22×C4×C53—41no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×Dic53—41no—
140C2×Dic106—21no—
141D8×C53—21no—
142D8×C53—21no—
143D8×C53—21no—
144D8×C53—21no—
145D8×C53—21no—
146D8×C53—42no—
147D8×C53—42no—
148D8×C53—81no—
149D8×C53#4112yes—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53C2×D411yescharacteristic
155C2×C8×C53C2×D411yescharacteristic
156C2×C8×C53C2×D411yescharacteristic
157C2×C8×C53C2411yescharacteristic
158C4⋊C4×C53—21no—
159C4⋊C4×C53D811yescharacteristic
160C22⋊C4×C53—41no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53D811yescharacteristic
164C22⋊C4×C53D811yescharacteristic
165C42×C53C2×D411yescharacteristic
166C22⋊Dic53—41no—
167C22⋊Dic53—41no—
168C22⋊Dic53—41no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—41no—
173#11719919—41no—
174#11719919—41no—
175#11719919—41no—
176#11719919—41no—
177#11719919—41no—
178C4×Dic53—41no—
179C4×Dic53—41no—
180Dic212—21no—
181Dic212—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—82no—
190C23×C53—82no—
191C23×C53—82no—
192C23×C53—82no—
193C23×C53—161no—
194C23×C53—161no—
195C23×C53C2×D811yescharacteristic
196C23×C53C2×D811yescharacteristic
197D4×C53—22no—
198D4×C53—22no—
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—41no—
204D4×C53—82no—
205D4×C53—82no—
206D4×C53—82no—
207D4×C53—82no—
208D4×C53—82no—
209D4×C53—161no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—41no—
215C2×C4×C53—82no—
216C2×C4×C53C2×D811yescharacteristic
217C2×C4×C53C2×D811yescharacteristic
218C2×C4×C53C2×D811yescharacteristic
219C2×C4×C53C2×D811yescharacteristic
220C2×C4×C53C22×D411yescharacteristic
221C2×Dic53—41no—
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—82no—
226Dic106—41no—
227Dic106—41no—
228C8×C53—21no—
229C8×C53—21no—
230C8×C53—21no—
231C8×C53C2×#4111yescharacteristic
232C8×C53H(5,𝔽2)11yescharacteristic
233#33877—531nomaximal, Sylow
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—42no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53—81no—
243C22×C53—81no—
244C22×C53—81no—
245C22×C53—81no—
246C22×C53—81no—
247C22×C53—81no—
248C22×C53—164no—
249C22×C53—164no—
250C22×C53#50912yes—
251C22×C53C22×D811yescharacteristic
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—81no—
257C4×C53#54611yescharacteristic
258C4×C53#57711yescharacteristic
259Dic53—81no—
260Dic53—81no—
261Dic53—81no—
262Dic53—81no—
263#3336—531no—
264D4×D8D5311yescharacteristic
265#2244—531no—
266#2234—531no—
267#2232—531no—
268#2228—531no—
269#2220—531no—
270#2217—531no—
271#2207—531no—
272#2187—531no—
273#2187—531no—
274#2175—531no—
275#2168—531no—
276#2140—531no—
277C2×#357—531no—
278C2×C53—22no—
279C2×C53—41no—
280C2×C53—82no—
281C2×C53—161no—
282C2×C53—161no—
283C2×C53#329411yescharacteristic
284C2×C53C2×#50911yescharacteristic
285C2×C53#341611yescharacteristic
286C22×D8C2×D5311yescharacteristic
287C22×D8C2×D5311yescharacteristic
288#552—1061no—
289D42—21no—
290C2×SD32—531no—
291C2×SD32—531no—
292C2×SD32—531no—
293C2×M32—531no—
294C22×C16—531no—
295#499—531no—
296#493C2×D5311yescharacteristic
297#485—1061no—
298#483—1061no—
299#479—1061no—
300#467—531no—
301#467—531no—
302#466C2×D5311yescharacteristic
303#466C2×D5311yescharacteristic
304#459—21no—
305#447—21no—
306#447—21no—
307C4×Q16—531no—
308C4×D8C2×D5311yescharacteristic
309C8×D4C2×D5311yescharacteristic
310#427—531no—
311C2×#107—531no—
312#367—531no—
313#366—531no—
314#366—531no—
315C4⋊C16—531no—
316#358—531no—
317#357—531no—
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322#357—531no—
323#357—531no—
324#357—531no—
325#357—531no—
326C22⋊C16—531no—
327C22⋊C16—531no—
328#346—531no—
329C53#3387711yescharacteristic, complemented, Sylow
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—41no—
333C2×Q16—531no—
334C2×D8—22no—
335C2×D8—22no—
336C2×D8—21no—
337C2×D8—21no—
338C2×D8—21no—
339C2×D8—41no—
340C2×D8—41no—
341C2×D8C22×D5311yescharacteristic
342C2×D8C22×D5311yescharacteristic
343C2×D8C22×D5311yescharacteristic
344C22×C8C22×D5311yescharacteristic
345C22×C8C22×D5311yescharacteristic
346#127—21no—
347#126—2121no—
348#125—1061no—
349#125—2121no—
350#123—2121no—
351C22⋊Q8—1061no—
352C22⋊Q8—1061no—
353#121—21no—
354#121—21no—
355#121—41no—
356#120—41no—
357#120—41no—
358C4×Q8—1061no—
359C4×D4—21no—
360C4×D4C22⋊D5311yescharacteristic
361#117—1061no—
362#117—2121no—
363C2×C4⋊C4—1061no—
364SD32—1062no—
365SD32—2121no—
366SD32—2121no—
367M32—2121no—
368C2×C16—531no—
369C2×C16—531no—
370C2×C16—531no—
371C2×C16—531no—
372C2×C16—2121no—
373#107—531no—
374#107—531no—
375#107—531no—
376#107—1061no—
377#107C22×D5311yescharacteristic
378#106—1061no—
379#106—1061no—
380C4⋊C8C22⋊D5311yescharacteristic
381#103—1061no—
382#103—1061no—
383#103—1061no—
384#102—21no—
385#102—21no—
386#102—21no—
387C22⋊C8C22⋊D5311yescharacteristic
388C22⋊C8C22⋊D5311yescharacteristic
389C4×C8C22×D5311yescharacteristic
390C24—41no—
391C24—41no—
392C2×Q8—1061no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—84no—
407C2×D4#1171992211yescharacteristic
408C22×C4—41no—
409C22×C4—2121no—
410C22×C4C2×C22⋊D5311yescharacteristic
411C22×C4C2×C22⋊D5311yescharacteristic
412Q16—1061no—
413Q16—1061no—
414D8—21no—
415D8—21no—
416D8—21no—
417D8—21no—
418D8—21no—
419D8—42no—
420D8—42no—
421D8—81no—
422D8#1171994712yes—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×C22⋊D5311yescharacteristic
428C2×C8C2×C22⋊D5311yescharacteristic
429C2×C8C2×C22⋊D5311yescharacteristic
430C2×C8C23×D5311yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—1061no—
433C4⋊C4—1061no—
434C4⋊C4—1061no—
435C4⋊C4—2121no—
436C4⋊C4—2121no—
437C4⋊C4—2121no—
438C4⋊C4—2121no—
439C4⋊C4—2121no—
440C4⋊C4—2121no—
441C4⋊C4#1171992211yescharacteristic
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—2121no—
446C22⋊C4—2121no—
447C22⋊C4—2121no—
448C22⋊C4#1171992211yescharacteristic
449C22⋊C4#1171992211yescharacteristic
450C42—2121no—
451C42—2121no—
452C42C2×C22⋊D5311yescharacteristic
453C16—1061no—
454C16—1061no—
455C16—1061no—
456C16—1061no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—161no—
465C23—161no—
466C23C2×#1171992211yescharacteristic
467C23C2×#1171992211yescharacteristic
468Q8—2121no—
469Q8—2121no—
470D4—22no—
471D4—22no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—82no—
478D4—82no—
479D4—82no—
480D4—82no—
481D4—82no—
482D4—161no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—82no—
489C2×C4—4242no—
490C2×C4—2121no—
491C2×C4—2121no—
492C2×C4—2121no—
493C2×C4—2121no—
494C2×C4C2×#1171992211yescharacteristic
495C2×C4C2×#1171992211yescharacteristic
496C2×C4C2×#1171992211yescharacteristic
497C2×C4C2×#1171992211yescharacteristic
498C2×C4C22×C22⋊D5311yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8C2×#1171994711yescharacteristic
503C8#1313326211yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—164no—
519C22—164no—
520C22#1681105912yes—
521C22C22×#1171992211yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—81no—
527C4—4241no—
528C4—4241no—
529C4—4241no—
530C4—4241no—
531C4#1681126711yescharacteristic
532C4#1681143411yescharacteristic
533C2—22no—
534C2—41no—
535C2—82no—
536C2—161no—
537C2—161no—
538C2#2028652811yescharacteristic
539C2C2×#1681105911yescharacteristic
540C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalD4×D8×C53C2
socleC22×C53C22×D8
hypercenterD4×D8D53
hypocenterC53#33877
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. #20287084C2semidirect
2. #20276956C2semidirect
3. #20276903C2semidirect
4. #20276893C2semidirect
5. #20276872C2semidirect
6. #20276826C2semidirect
7. #20276809C2semidirect
8. #20276752C2semidirect
9. #20276644C2semidirect
10. #20276644C2semidirect
11. #20276579C2semidirect
12. #20276541C2semidirect
13. #20276431C2semidirect
14. C2×#16810424C2semidirect
18. C2×#13133161C22semidirect
19. C2×#13133161C22semidirect
20. C2×#13133157C22semidirect
21. C22×#11719908C22semidirect
22. #16811025C22semidirect
26. #16810864C22semidirect
27. #16810864C22semidirect
28. C4×Dic212C22semidirect
29. #16810661C22semidirect
30. C2×#13133151C22semidirect
31. #16810450C22semidirect
32. #16810448C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810426C22semidirect
39. #16810424C22semidirect
40. #16810424C22semidirect
41. #16810424C22semidirect
42. #16810424C22semidirect
43. #16810424C22semidirect
44. #16810424C22semidirect
45. C22⋊#11719908C22semidirect
46. C22⋊#11719908C22semidirect
47. #16810399C22semidirect
329. C53#33877semidirect

Maximal subgroups

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

subgroupindex
#202870842
#202769562
#202769032
#202768932
#202768722
#202768262
#202768092
#202767522
#202766442
#202765792
#202765412
#202764312
C2×#168104242
D4×D8×C532
#3387753