Finite Group Databasebeta

← #27541057

Subgroups — #27541057

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#20286402C211yesmaximal, characteristic, complemented
2#20276954C211yesmaximal, characteristic, complemented
3#20276912C211yesmaximal, characteristic, complemented
4#20276900C211yesmaximal, characteristic, complemented
5#20276859C211yesmaximal, characteristic, complemented
6#20276816C211yesmaximal, characteristic, complemented
7#20276809C211yesmaximal, characteristic, complemented
8#20276770C211yesmaximal, characteristic, complemented
9#20276636C211yesmaximal, characteristic, complemented
10#20276626C211yesmaximal, characteristic, complemented
11#20276589C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276425C211yesmaximal, characteristic, complemented
14C2×#16810423C211yesmaximal, characteristic, complemented
15C53×#3334C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×D424C2211yescharacteristic
18#16811269—21no—
19C2×#13133161C2211yescharacteristic, complemented
20C2×#13133158C2211yescharacteristic
21C2×#13133158C2211yescharacteristic
22C2×#13133157C2211yescharacteristic, complemented
23C22×#11719908C2211yescharacteristic
24#16811007C2211yescharacteristic
25#16810864C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810839—21no—
28#16810776—21no—
29#16810770—21no—
30C4×D424C2211yescharacteristic
31#16810450C2211yescharacteristic, complemented
32#16810450C2211yescharacteristic, complemented
33#16810450C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810426C2211yescharacteristic, complemented
36#16810426C2211yescharacteristic, complemented
37#16810426C2211yescharacteristic, complemented
38#16810424C2211yescharacteristic, complemented
39#16810423C2211yescharacteristic
40#16810423C2211yescharacteristic
41#16810423C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic, complemented
47#16810399C2211yescharacteristic, complemented
48C53×#547—21no—
49C53×#500C2211yescharacteristic, complemented
50C53×#480—21no—
51C53×#480—21no—
52C53×#474—21no—
53C53×#466C2211yescharacteristic
54C53×#466C2211yescharacteristic, complemented
55C4×D8×C53C2211yescharacteristic, complemented
56C8×D4×C53C2211yescharacteristic
57C2×#107×C53C2211yescharacteristic
58C2×#107×C53C2211yescharacteristic, complemented
59C2×#11719945—21no—
60C22×D212—21no—
61#13133239—22no—
62#13133239—41no—
63C2×Dic212C2311yescharacteristic
64C2×#11719913—21no—
65C2×#11719913—21no—
66C2×D424—21no—
67C2×D424—22no—
68C2×D424—41no—
69C2×D424C2311yescharacteristic
70C2×D424C2311yescharacteristic
71#13133205—21no—
72#13133202—41no—
73#13133196—21no—
74#13133191—21no—
75#13133190—41no—
76#13133185—41no—
77C4×D212—21no—
78C22⋊C4×D53—41no—
79#13133161—41no—
80#13133158—22no—
81#13133158—41no—
82#13133157—41no—
83C2×#11719908—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133142—21no—
90#13133139—21no—
91#13133139—21no—
92C2×#41×C53—41no—
93C2×D8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95C22×C8×C53C2311yescharacteristic
96C4⋊Q8×C53—21no—
97#123×C53—41no—
98#123×C53—41no—
99C22⋊Q8×C53—41no—
100#121×C53—21no—
101#121×C53—21no—
102C4×D4×C53—21no—
103C4×D4×C53D411yescharacteristic
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106#107×C53—21no—
107#107×C53—21no—
108#107×C53—21no—
109#107×C53C2311yescharacteristic
110#107×C53C2311yescharacteristic
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
119C23×D53—41no—
120#41×C53—84no—
121C2×Q8×C53—41no—
122C2×D4×C53—21no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53D811yescharacteristic
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C2×C22⋊D53—41no—
132C2×C22⋊D53—41no—
133#11719945—42no—
134#11719945—81no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—42no—
138C2×D212—41no—
139C2×D212—81no—
140C2×C4×D53—41no—
141C2×C4×D53—41no—
142C2×Dic106—21no—
143D8×C53—21no—
144D8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53C2×D411yescharacteristic
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53C2411yescharacteristic
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—41no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53—41no—
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—
167#11719921—41no—
168#11719921—41no—
169#11719921—41no—
170#11719921—41no—
171C4⋊Dic53—21no—
172#11719919—41no—
173Dic212—21no—
174Dic212—21no—
175D424—21no—
176D424—21no—
177D424—21no—
178D424—42no—
179D424#4112yescomplemented
180#11719913—42no—
181#11719913—81no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53C2×D811yescharacteristic
188C23×C53C2×D811yescharacteristic
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—82no—
192C22×D53—82no—
193C22×D53—161no—
194Q8×C53—82no—
195D4×C53—22no—
196D4×C53—41no—
197D4×C53—41no—
198D4×C53—82no—
199D4×C53—82no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—82no—
208C2×C4×C53—82no—
209C2×C4×C53—82no—
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C2×D811yescharacteristic
212C2×C4×C53C2×D811yescharacteristic
213C2×C4×C53C2×D811yescharacteristic
214C2×C4×C53C22×D411yescharacteristic
215C22⋊D53—82no—
216C22⋊D53—161no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219D212—22no—
220D212—41no—
221D212—41no—
222D212—41no—
223D212—82no—
224C4×D53—82no—
225C4×D53—161no—
226Dic106—41no—
227Dic106—41no—
228C8×C53—21no—
229C8×C53—21no—
230C8×C53—21no—
231C8×C53C2×#4111yescharacteristic
232C8×C53H(5,𝔽2)11yescharacteristic
233#33873—531nomaximal, Sylow
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53#50912yes—
243C22×C53C22×D811yescharacteristic
244C2×D53—42no—
245C2×D53—81no—
246C2×D53—81no—
247C2×D53—81no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—164no—
251C4×C53—21no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—81no—
256C4×C53—81no—
257C4×C53—81no—
258C4×C53#54611yescharacteristic
259C4×C53#57711yescharacteristic
260Dic53—81no—
261Dic53—81no—
262#3334D5311yescharacteristic, complemented
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#2186—531no—
272#2184—531no—
273#2177—531no—
274#2168—531no—
275#2139—531no—
276C2×#357—531no—
277C2×C53—22no—
278C2×C53—41no—
279C2×C53—161no—
280C2×C53#329411yescharacteristic
281C2×C53C2×#50911yescharacteristic
282C2×C53#341611yescharacteristic
283D53—82no—
284D53—161no—
285C2×#135—531no—
286C22×D8—531no—
287#547—21no—
288#546—1061no—
289C2×SD32—531no—
290C2×D16—531no—
291C2×D16—531no—
292C2×M32—531no—
293C22×C16—531no—
294#500C2×D5311yescharacteristic, complemented
295#495—531no—
296#480—21no—
297#480—21no—
298#474—21no—
299#467—531no—
300#466—531no—
301#466C2×D5311yescharacteristic
302#466C2×D5311yescharacteristic, complemented
303#463—1061no—
304#450—1061no—
305#449—1061no—
306C4×D8—531no—
307C4×D8C2×D5311yescharacteristic, complemented
308C8×D4C2×D5311yescharacteristic
309C2×#107C2×D5311yescharacteristic
310C2×#107C2×D5311yescharacteristic, complemented
311#367—531no—
312#367—531no—
313#367—531no—
314C4⋊C16—531no—
315#358—531no—
316#358—531no—
317#358—531no—
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322#357—531no—
323#357—531no—
324#357—531no—
325C22⋊C16—531no—
326C22⋊C16—531no—
327#346—531no—
328C53#3387311yescharacteristic, complemented, Sylow
329C2×#41—41no—
330C2×#41—1061no—
331C22×D4—1061no—
332#135—1062no—
333#135—2121no—
334C2×Q16—531no—
335C2×SD16—1061no—
336C2×SD16—1061no—
337C2×D8—531no—
338C2×D8—531no—
339C2×D8—1062no—
340C2×D8—1061no—
341C2×D8—2121no—
342C2×D8C22×D5311yescharacteristic
343C22×C8C22×D5311yescharacteristic
344C22×C8C22×D5311yescharacteristic
345C4⋊Q8—21no—
346#124—1061no—
347#123—41no—
348#123—41no—
349#123—2121no—
350C22⋊Q8—41no—
351C22⋊Q8—1061no—
352#121—21no—
353#121—21no—
354#121—1061no—
355#121—2121no—
356#120—2121no—
357C4×D4—21no—
358C4×D4—1061no—
359C4×D4C22⋊D5311yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C22⋊C4—2121no—
363SD32—2121no—
364D16—1062no—
365D16—2121no—
366M32—2121no—
367C2×C16—531no—
368C2×C16—531no—
369C2×C16—531no—
370C2×C16—531no—
371C2×C16—2121no—
372#107—21no—
373#107—21no—
374#107—21no—
375#107—531no—
376#107C22×D5311yescharacteristic
377#107C22×D5311yescharacteristic
378#107C22×D5311yescharacteristic
379C4⋊C8C22⋊D5311yescharacteristic
380#103—1061no—
381#102—21no—
382#102—21no—
383#102—21no—
384#102—1061no—
385#102—1061no—
386C22⋊C8C22⋊D5311yescharacteristic
387C22⋊C8C22⋊D5311yescharacteristic
388C4×C8C22×D5311yescharacteristic
389C24—2121no—
390#41—84no—
391#41—4241no—
392#41—2122no—
393C2×Q8—41no—
394C2×Q8—1061no—
395C2×D4—21no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—1061no—
399C2×D4—1061no—
400C2×D4—4241no—
401C2×D4—2122no—
402C2×D4—2121no—
403C2×D4—2121no—
404C2×D4—2121no—
405C2×D4#1171992211yescharacteristic
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—2121no—
410C22×C4—2121no—
411C22×C4C2×C22⋊D5311yescharacteristic
412C22×C4C2×C22⋊D5311yescharacteristic
413Q16—1061no—
414Q16—1061no—
415SD16—4241no—
416SD16—2122no—
417D8—21no—
418D8—21no—
419D8—532no—
420D8—1061no—
421D8—1061no—
422D8—1061no—
423D8—2122no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×C22⋊D5311yescharacteristic
429C2×C8C2×C22⋊D5311yescharacteristic
430C2×C8C2×C22⋊D5311yescharacteristic
431C2×C8C23×D5311yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—1061no—
439C4⋊C4—2121no—
440C4⋊C4#1171992211yescharacteristic
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—2121no—
445C22⋊C4—2121no—
446C22⋊C4—2121no—
447C22⋊C4—2121no—
448C22⋊C4—2121no—
449C22⋊C4#1171992211yescharacteristic
450C22⋊C4#1171992211yescharacteristic
451C42C2×C22⋊D5311yescharacteristic
452C16—1061no—
453C16—1061no—
454C16—1061no—
455C16—1061no—
456C23—41no—
457C23—8481no—
458C23—4242no—
459C23—4242no—
460C23—2121no—
461C23—2121no—
462C23C2×#1171992211yescharacteristic
463C23C2×#1171992211yescharacteristic
464Q8—82no—
465Q8—2121no—
466Q8—2121no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—82no—
471D4—82no—
472D4—8481no—
473D4—1062no—
474D4—4242no—
475D4—4242no—
476D4—2121no—
477D4—2121no—
478D4—2121no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—8481no—
490C2×C4—4242no—
491C2×C4—2121no—
492C2×C4—2121no—
493C2×C4C2×#1171992211yescharacteristic
494C2×C4C2×#1171992211yescharacteristic
495C2×C4C2×#1171992211yescharacteristic
496C2×C4C2×#1171992211yescharacteristic
497C2×C4C22×C22⋊D5311yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8#41×D5311yescharacteristic
502C8#1313326311yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—8484no—
512C22—4241no—
513C22—4241no—
514C22—4241no—
515C22—4241no—
516C22—4241no—
517C22—2122no—
518C22#1681105912yes—
519C22C22×#1171992211yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—4241no—
528C4—4241no—
529C4#1681126111yescharacteristic
530C4#1681143111yescharacteristic
531C2—22no—
532C2—41no—
533C2—161no—
534C2—8481no—
535C2—4242no—
536C2#2028652211yescharacteristic
537C2C2×#1681105911yescharacteristic
538C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3334C2
socleC22×C53C22×D8
hypercenter#3334D53
hypocenterC53#33873
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #20286402C2semidirect
2. #20276954C2semidirect
3. #20276912C2semidirect
4. #20276900C2semidirect
5. #20276859C2semidirect
6. #20276816C2semidirect
7. #20276809C2semidirect
8. #20276770C2semidirect
9. #20276636C2semidirect
10. #20276626C2semidirect
11. #20276589C2semidirect
12. #20276541C2semidirect
13. #20276425C2semidirect
14. C2×#16810423C2semidirect
15. C53×#3334C2semidirect
19. C2×#13133161C22semidirect
22. C2×#13133157C22semidirect
31. #16810450C22semidirect
32. #16810450C22semidirect
33. #16810450C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810426C22semidirect
36. #16810426C22semidirect
37. #16810426C22semidirect
38. #16810424C22semidirect
46. C22⋊#11719908C22semidirect
47. #16810399C22semidirect
49. C53×#500C22semidirect
54. C53×#466C22semidirect
55. C4×D8×C53C22semidirect
58. C2×#107×C53C22semidirect
179. D424#41semidirect
262. #3334D53semidirect
294. #500C2×D53semidirect
302. #466C2×D53semidirect
307. C4×D8C2×D53semidirect
310. C2×#107C2×D53semidirect
328. C53#33873semidirect

Maximal subgroups

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

subgroupindex
#202864022
#202769542
#202769122
#202769002
#202768592
#202768162
#202768092
#202767702
#202766362
#202766262
#202765892
#202765412
#202764252
C2×#168104232
C53×#33342
#3387353