Finite Group Databasebeta

← #27541056

Subgroups — #27541056

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287059C211yesmaximal, characteristic, complemented
2#20286394C211yesmaximal, characteristic, complemented
3#20276953C211yesmaximal, characteristic, complemented
4#20276911C211yesmaximal, characteristic, complemented
5#20276901C211yesmaximal, characteristic, complemented
6#20276860C211yesmaximal, characteristic, complemented
7#20276816C211yesmaximal, characteristic, complemented
8#20276808C211yesmaximal, characteristic, complemented
9#20276772C211yesmaximal, characteristic, complemented
10#20276625C211yesmaximal, characteristic, complemented
11#20276591C211yesmaximal, characteristic, complemented
12#20276543C211yesmaximal, characteristic, complemented
13#20276427C211yesmaximal, characteristic, complemented
14#357×D53C211yesmaximal, characteristic, complemented
15C53×#2186C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17#16811273—21no—
18#16811261—21no—
19C2×#13133158C2211yescharacteristic
20C2×#13133157C2211yescharacteristic, complemented
21C2×C16×D53C2211yescharacteristic
22#16811027C2211yescharacteristic, complemented
23#16811008C2211yescharacteristic
24#16810935—21no—
25#16810934—21no—
26#16810901—21no—
27#16810861C2211yescharacteristic
28#16810859C2211yescharacteristic
29#16810856C2211yescharacteristic
30#16810840—21no—
31#16810772—21no—
32#16810768—21no—
33D8×Dic53C2211yescharacteristic
34D8×Dic53C2211yescharacteristic, complemented
35C8×C22⋊D53C2211yescharacteristic
36C2×#13133151C2211yescharacteristic, complemented
37#107×D53C2211yescharacteristic
38#16810451C2211yescharacteristic, complemented
39#16810451C2211yescharacteristic, complemented
40#16810450C2211yescharacteristic, complemented
41#16810441C2211yescharacteristic, complemented
42#16810427C2211yescharacteristic, complemented
43#16810427C2211yescharacteristic, complemented
44#16810425C2211yescharacteristic
45#16810425C2211yescharacteristic
46#16810424C2211yescharacteristic
47#16810424C2211yescharacteristic
48#16810424C2211yescharacteristic, complemented
49#16810423C2211yescharacteristic
50#16810402C2211yescharacteristic
51#16810398C2211yescharacteristic, complemented
52C2×#135×C53C2211yescharacteristic
53C2×SD32×C53C2211yescharacteristic, complemented
54C2×D16×C53C2211yescharacteristic
55C53×#466C2211yescharacteristic, complemented
56C53×#358C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic
58C53×C22⋊C16C2211yescharacteristic, complemented
59C2×#11719947—41no—
60C2×D4×D53—21no—
61C2×#11719924—21no—
62C2×D424C2311yescharacteristic
63C2×#11719922—21no—
64D8×D53—22no—
65D8×D53—41no—
66C2×C8×D53C2311yescharacteristic
67#13133216—21no—
68#13133206—21no—
69#13133200—41no—
70#13133199—41no—
71#13133198—41no—
72#13133194—41no—
73#13133193—21no—
74#13133191—21no—
75#13133189—41no—
76#13133188—21no—
77#13133185—41no—
78D4×Dic53—21no—
79D4×Dic53—21no—
80C4×C22⋊D53D411yescharacteristic
81C2×C4⋊Dic53—21no—
82C4⋊C4×D53—21no—
83C2×#11719921—41no—
84#13133158—41no—
85#13133157—41no—
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88C16×D53—41no—
89#13133151—21no—
90#13133151C2311yescharacteristic
91#13133151C2311yescharacteristic
92#13133151C2311yescharacteristic
93#13133150—21no—
94#13133150—21no—
95#13133146D411yescharacteristic
96#13133141—21no—
97#13133139—21no—
98#13133138—21no—
99#13133138—21no—
100#13133138—21no—
101C22⋊#88322D411yescharacteristic
102#13133129D411yescharacteristic
103C8×Dic53C2311yescharacteristic
104C2×#41×C53—21no—
105#135×C53—22no—
106#135×C53—41no—
107C2×Q16×C53C2311yescharacteristic
108C2×SD16×C53—21no—
109C2×D8×C53C2311yescharacteristic
110C2×D8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#121×C53—21no—
113SD32×C53—41no—
114D16×C53—22no—
115C2×C16×C53C2311yescharacteristic
116C2×C16×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#102×C53—21no—
119C23×D53—41no—
120#41×C53—42no—
121#41×C53—81no—
122C2×Q8×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—41no—
126C2×D4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53C2×D411yescharacteristic
129C2×C22⋊D53—41no—
130C2×C22⋊D53—41no—
131C2×C22⋊D53—41no—
132C2×C22⋊D53D811yescharacteristic
133C22×Dic53—41no—
134C22×Dic53—41no—
135C22×Dic53—41no—
136#11719947—84no—
137D4×D53—42no—
138D4×D53—81no—
139C2×D212—21no—
140C2×C4×D53—41no—
141C2×C4×D53C2×D411yescharacteristic
142C2×Dic106—41no—
143Q16×C53—21no—
144Q16×C53—21no—
145SD16×C53—42no—
146D8×C53—21no—
147D8×C53—21no—
148D8×C53—21no—
149D8×C53#4112yescomplemented
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53C2×D411yescharacteristic
153C2×C8×C53C2411yescharacteristic
154C4⋊C4×C53—21no—
155C22⋊C4×C53—41no—
156C22⋊Dic53—41no—
157C22⋊Dic53—41no—
158C22⋊Dic53—41no—
159C22⋊Dic53D811yescharacteristic
160#11719924—81no—
161#11719922—42no—
162#11719921—41no—
163#11719921—41no—
164#11719921—41no—
165#11719921—41no—
166#11719921D811yescharacteristic
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—41no—
171#11719919—41no—
172#11719919—41no—
173#11719919—41no—
174#11719919D811yescharacteristic
175C4×Dic53C2×D411yescharacteristic
176C2×#88322C2×D411yescharacteristic
177C2×#88322C2×D411yescharacteristic
178D424—21no—
179D424—21no—
180C8×D53—21no—
181C8×D53—21no—
182C16×C53—21no—
183C16×C53—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53C2×D811yescharacteristic
189C22×D53—41no—
190C22×D53—82no—
191C22×D53—82no—
192C22×D53—161no—
193C22×D53C2×D811yescharacteristic
194Q8×C53—41no—
195Q8×C53—41no—
196D4×C53—22no—
197D4×C53—41no—
198D4×C53—41no—
199D4×C53—41no—
200D4×C53—82no—
201D4×C53—82no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—82no—
207C2×C4×C53C2×D811yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C22⋊D53—22no—
210C22⋊D53—82no—
211C22⋊D53—82no—
212C22⋊D53—161no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—82no—
217C2×Dic53—82no—
218C2×Dic53—161no—
219C2×Dic53C2×D811yescharacteristic
220C2×Dic53C2×D811yescharacteristic
221C2×Dic53C2×D811yescharacteristic
222D212—41no—
223D212—41no—
224C4×D53—21no—
225C4×D53—21no—
226C4×D53—82no—
227Dic106—82no—
228C8×C53—21no—
229C8×C53C2×#4111yescharacteristic
230C8×C53H(5,𝔽2)11yescharacteristic
231#88322—21no—
232#88322—21no—
233#33873—531nomaximal, Sylow
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—42no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53—81no—
243C22×C53C22×D811yescharacteristic
244C2×D53—21no—
245C2×D53—21no—
246C2×D53—81no—
247C2×D53—81no—
248C2×D53—81no—
249C2×D53—164no—
250C2×D53#50912yes—
251C4×C53—21no—
252C4×C53—81no—
253C4×C53—81no—
254C4×C53#54611yescharacteristic
255C4×C53#57711yescharacteristic
256Dic53—21no—
257Dic53—21no—
258Dic53—21no—
259Dic53—81no—
260Dic53—81no—
261Dic53—81no—
262#3334—531no—
263#3280—531no—
264#2243—531no—
265#2235—531no—
266#2233—531no—
267#2226—531no—
268#2218—531no—
269#2217—531no—
270#2210—531no—
271#2186D5311yescharacteristic, complemented
272#2184—531no—
273#2177—531no—
274#2168—531no—
275#2139—531no—
276C2×#357—531no—
277C2×C53—41no—
278C2×C53—82no—
279C2×C53—161no—
280C2×C53#329411yescharacteristic
281C2×C53C2×#50911yescharacteristic
282C2×C53#341611yescharacteristic
283D53—22no—
284D53—161no—
285C2×#135C2×D5311yescharacteristic
286C22×D8—531no—
287#547—1061no—
288#546—1061no—
289C2×SD32C2×D5311yescharacteristic, complemented
290C2×D16—531no—
291C2×D16C2×D5311yescharacteristic
292C2×M32—531no—
293C22×C16—531no—
294#500—531no—
295#495—531no—
296#480—1061no—
297#480—1061no—
298#474—1061no—
299#467—531no—
300#466—531no—
301#466—531no—
302#466C2×D5311yescharacteristic, complemented
303#463—1061no—
304#450—1061no—
305#449—1061no—
306C4×D8—531no—
307C4×D8—531no—
308C8×D4—531no—
309C2×#107—531no—
310C2×#107—531no—
311#367—531no—
312#367—531no—
313#367—531no—
314C4⋊C16—531no—
315#358—531no—
316#358—531no—
317#358C2×D5311yescharacteristic, complemented
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322#357—531no—
323#357—531no—
324#357C2×D5311yescharacteristic
325C22⋊C16—531no—
326C22⋊C16C2×D5311yescharacteristic, complemented
327#346—531no—
328C53#3387311yescharacteristic, complemented, Sylow
329C2×#41—21no—
330C2×#41—2121no—
331C22×D4—1061no—
332#135—22no—
333#135—41no—
334C2×Q16C22×D5311yescharacteristic
335C2×SD16—21no—
336C2×SD16—1061no—
337C2×D8—531no—
338C2×D8—1061no—
339C2×D8—1062no—
340C2×D8—2121no—
341C2×D8C22×D5311yescharacteristic
342C2×D8C22×D5311yescharacteristic
343C22×C8—531no—
344C22×C8C22×D5311yescharacteristic
345C4⋊Q8—1061no—
346#124—1061no—
347#123—2121no—
348#123—2121no—
349#123—2121no—
350C22⋊Q8—1061no—
351C22⋊Q8—2121no—
352#121—21no—
353#121—1061no—
354#121—1061no—
355#121—2121no—
356#120—2121no—
357C4×D4—531no—
358C4×D4—1061no—
359C4×D4—1061no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C22⋊C4—2121no—
363SD32—41no—
364D16—22no—
365D16—2121no—
366M32—2121no—
367C2×C16—531no—
368C2×C16—531no—
369C2×C16—2121no—
370C2×C16C22×D5311yescharacteristic
371C2×C16C22×D5311yescharacteristic
372#107—531no—
373#107—531no—
374#107—531no—
375#107—1061no—
376#107—1061no—
377#107—1061no—
378#107C22×D5311yescharacteristic
379C4⋊C8—531no—
380#103—1061no—
381#102—21no—
382#102—1061no—
383#102—1061no—
384#102—1061no—
385#102—1061no—
386C22⋊C8—531no—
387C22⋊C8—531no—
388C4×C8—531no—
389C24—2121no—
390#41—42no—
391#41—81no—
392#41—4244no—
393C2×Q8—21no—
394C2×Q8—2121no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—531no—
400C2×D4—1061no—
401C2×D4—4241no—
402C2×D4—2121no—
403C2×D4—2122no—
404C2×D4—2121no—
405C2×D4—2121no—
406C22×C4—41no—
407C22×C4—531no—
408C22×C4—2121no—
409C22×C4—2121no—
410C22×C4—2121no—
411C22×C4—2121no—
412C22×C4D4×D5311yescharacteristic
413Q16—21no—
414Q16—21no—
415SD16—42no—
416SD16—4241no—
417D8—21no—
418D8—21no—
419D8—21no—
420D8—1061no—
421D8—1061no—
422D8—2122no—
423D8#1171994712yescomplemented
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—531no—
427C2×C8—531no—
428C2×C8—1061no—
429C2×C8—1061no—
430C2×C8D4×D5311yescharacteristic
431C2×C8C23×D5311yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—531no—
434C4⋊C4—1061no—
435C4⋊C4—1061no—
436C4⋊C4—1061no—
437C4⋊C4—2121no—
438C4⋊C4—2121no—
439C4⋊C4—2121no—
440C4⋊C4—2121no—
441C22⋊C4—41no—
442C22⋊C4—531no—
443C22⋊C4—531no—
444C22⋊C4—2121no—
445C22⋊C4—2121no—
446C22⋊C4—2121no—
447C22⋊C4—2121no—
448C22⋊C4—2121no—
449C22⋊C4—2121no—
450C22⋊C4—2121no—
451C42—531no—
452C16—21no—
453C16—21no—
454C16—1061no—
455C16—1061no—
456C23—41no—
457C23—41no—
458C23—531no—
459C23—8481no—
460C23—4242no—
461C23—4242no—
462C23—2121no—
463C23D8×D5311yescharacteristic
464Q8—41no—
465Q8—41no—
466Q8—4242no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—82no—
472D4—82no—
473D4—8481no—
474D4—1062no—
475D4—4242no—
476D4—4242no—
477D4—2121no—
478D4—2121no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—82no—
484C2×C4—531no—
485C2×C4—531no—
486C2×C4—531no—
487C2×C4—8481no—
488C2×C4—1061no—
489C2×C4—1061no—
490C2×C4—4242no—
491C2×C4—4242no—
492C2×C4—4242no—
493C2×C4—2121no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4D8×D5311yescharacteristic
497C2×C4C2×D4×D5311yescharacteristic
498C8—21no—
499C8—1061no—
500C8—1061no—
501C8C2×#1171994711yescharacteristic
502C8#1313326211yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—42no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—532no—
513C22—8484no—
514C22—1061no—
515C22—1061no—
516C22—4241no—
517C22—4241no—
518C22—4241no—
519C22C2×D8×D5311yescharacteristic
520C4—21no—
521C4—81no—
522C4—81no—
523C4—1061no—
524C4—1061no—
525C4—1061no—
526C4—4241no—
527C4—4241no—
528C4—4241no—
529C4#1681126011yescharacteristic
530C4#1681143311yescharacteristic
531C2—41no—
532C2—82no—
533C2—161no—
534C2—8481no—
535C2—1062no—
536C2#2028652111yescharacteristic
537C2#509×D5311yescharacteristic
538C2#2028785811yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2186C2
socleC22×C53C22×D8
hypercenter#2186D53
hypocenterC53#33873
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2186

Complements

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

NH ≅ G/Ntype
1. #20287059C2semidirect
2. #20286394C2semidirect
3. #20276953C2semidirect
4. #20276911C2semidirect
5. #20276901C2semidirect
6. #20276860C2semidirect
7. #20276816C2semidirect
8. #20276808C2semidirect
9. #20276772C2semidirect
10. #20276625C2semidirect
11. #20276591C2semidirect
12. #20276543C2semidirect
13. #20276427C2semidirect
14. #357×D53C2semidirect
15. C53×#2186C2semidirect
20. C2×#13133157C22semidirect
22. #16811027C22semidirect
34. D8×Dic53C22semidirect
36. C2×#13133151C22semidirect
38. #16810451C22semidirect
39. #16810451C22semidirect
40. #16810450C22semidirect
41. #16810441C22semidirect
42. #16810427C22semidirect
43. #16810427C22semidirect
48. #16810424C22semidirect
51. #16810398C22semidirect
53. C2×SD32×C53C22semidirect
55. C53×#466C22semidirect
56. C53×#358C22semidirect
58. C53×C22⋊C16C22semidirect
149. D8×C53#41semidirect
271. #2186D53semidirect
289. C2×SD32C2×D53semidirect
302. #466C2×D53semidirect
317. #358C2×D53semidirect
326. C22⋊C16C2×D53semidirect
328. C53#33873semidirect
423. D8#11719947semidirect

Maximal subgroups

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

subgroupindex
#202870592
#202863942
#202769532
#202769112
#202769012
#202768602
#202768162
#202768082
#202767722
#202766252
#202765912
#202765432
#202764272
#357×D532
C53×#21862
#3387353