Finite Group Databasebeta

← #27540930

Subgroups — #27540930

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#20287078C211yesmaximal, characteristic, complemented
2#20286394C211yesmaximal, characteristic
3#20276948C211yesmaximal, characteristic, complemented
4#20276911C211yesmaximal, characteristic, complemented
5#20276907C211yesmaximal, characteristic, complemented
6#20276870C211yesmaximal, characteristic, complemented
7#20276813C211yesmaximal, characteristic, complemented
8#20276803C211yesmaximal, characteristic, complemented
9#20276779C211yesmaximal, characteristic, complemented
10#20276642C211yesmaximal, characteristic, complemented
11SD32×Dic53C211yesmaximal, characteristic, complemented
12#20276532C211yesmaximal, characteristic, complemented
13#20276422C211yesmaximal, characteristic, complemented
14#357×D53C211yesmaximal, characteristic, complemented
15C53×#2189C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17#16811307—21no—
18#16811261—21no—
19C2×#13133160C2211yescharacteristic
20C22×#11719908C2211yescharacteristic, complemented
21C2×C16×D53C2211yescharacteristic
22#16811024C2211yescharacteristic, complemented
23#16811008C2211yescharacteristic
24#16810961—21no—
25#16810952—21no—
26#16810928—21no—
27#16810861C2211yescharacteristic
28#16810861C2211yescharacteristic
29#16810859C2211yescharacteristic
30#16810840—21no—
31#16810772—21no—
32#16810768—21no—
33Q16×Dic53C2211yescharacteristic, complemented
34D8×Dic53C2211yescharacteristic
35C8×C22⋊D53C2211yescharacteristic
36#16810661C2211yescharacteristic, complemented
37#107×D53C2211yescharacteristic
38#16810451C2211yescharacteristic, complemented
39#16810450C2211yescharacteristic, complemented
40#16810449C2211yescharacteristic, complemented
41#16810441C2211yescharacteristic, complemented
42#16810427C2211yescharacteristic, complemented
43#16810427C2211yescharacteristic, complemented
44#16810427C2211yescharacteristic, complemented
45#16810425C2211yescharacteristic
46#16810425C2211yescharacteristic
47#16810424C2211yescharacteristic
48#16810424C2211yescharacteristic
49#16810423C2211yescharacteristic
50#16810402C2211yescharacteristic
51C16×Dic53C2211yescharacteristic, complemented
52C2×#135×C53C2211yescharacteristic
53C2×Q32×C53C2211yescharacteristic, complemented
54C2×SD32×C53C2211yescharacteristic
55C53×#467C2211yescharacteristic, complemented
56C53×#358C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic
58C53×C22⋊C16C2211yescharacteristic, complemented
59C2×D4×D53—21no—
60C2×#11719924—21no—
61C2×D424C2311yescharacteristic
62C2×#11719922—21no—
63D8×D53—22no—
64D8×D53—41no—
65C2×C8×D53C2311yescharacteristic
66#13133210—41no—
67#13133208—21no—
68#13133207—41no—
69#13133206—21no—
70#13133200—41no—
71#13133199—41no—
72#13133193—21no—
73#13133193—21no—
74#13133191—21no—
75#13133189—41no—
76#13133185—41no—
77Q8×Dic53—21no—
78D4×Dic53—21no—
79C4×C22⋊D53D411yescharacteristic
80#13133176—21no—
81#13133175—41no—
82C4⋊C4×D53—21no—
83C2×#11719921—41no—
84#13133160—41no—
85C2×#11719908—41no—
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88C16×D53—41no—
89#13133151C2311yescharacteristic
90#13133151C2311yescharacteristic
91#13133151C2311yescharacteristic
92#13133150—21no—
93#13133149—21no—
94#13133148—21no—
95#13133146D411yescharacteristic
96#13133141—21no—
97#13133141—21no—
98#13133141—21no—
99#13133139—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×Q16×C53C2311yescharacteristic
109C2×SD16×C53—21no—
110C2×D8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22⋊Q8×C53—21no—
113Q32×C53—41no—
114SD32×C53—22no—
115C2×C16×C53C2311yescharacteristic
116C2×C16×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#103×C53—21no—
119C23×D53—41no—
120#41×C53—42no—
121#41×C53—81no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53C2×D411yescharacteristic
128C2×C22⋊D53—41no—
129C2×C22⋊D53—41no—
130C2×C22⋊D53D811yescharacteristic
131C22×Dic53—41no—
132D4×D53—42no—
133D4×D53—81no—
134C2×D212—21no—
135C2×C4×D53—41no—
136C2×C4×D53C2×D411yescharacteristic
137Q16×C53—21no—
138Q16×C53—21no—
139Q16×C53—21no—
140Q16×C53—21no—
141SD16×C53—42no—
142D8×C53—21no—
143D8×C53#4112yes—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53C2×D411yescharacteristic
147C2×C8×C53C2411yescharacteristic
148C4⋊C4×C53—21no—
149C4⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C22⋊Dic53—41no—
152C22⋊Dic53—41no—
153C22⋊Dic53D811yescharacteristic
154#11719924—81no—
155#11719922—42no—
156#11719921—41no—
157#11719921—41no—
158#11719921—41no—
159#11719921—41no—
160#11719921—41no—
161#11719921D811yescharacteristic
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—41no—
166C4⋊Dic53—41no—
167#11719919—41no—
168#11719919—41no—
169#11719919—41no—
170#11719919—41no—
171#11719919D811yescharacteristic
172C4×Dic53—41no—
173C4×Dic53—41no—
174C4×Dic53C2×D411yescharacteristic
175C2×#88322C2×D411yescharacteristic
176C2×#88322C2×D411yescharacteristic
177D424—21no—
178D424—21no—
179C8×D53—21no—
180C8×D53—21no—
181C16×C53—21no—
182C16×C53—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53—41no—
186C23×C53C2×D811yescharacteristic
187C22×D53—41no—
188C22×D53—82no—
189C22×D53—82no—
190C22×D53—161no—
191C22×D53C2×D811yescharacteristic
192Q8×C53—41no—
193Q8×C53—41no—
194Q8×C53—41no—
195Q8×C53—41no—
196D4×C53—22no—
197D4×C53—41no—
198D4×C53—82no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—82no—
205C2×C4×C53C2×D811yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C22⋊D53—22no—
208C22⋊D53—82no—
209C22⋊D53—161no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—161no—
214C2×Dic53C2×D811yescharacteristic
215C2×Dic53C2×D811yescharacteristic
216C2×Dic53C2×D811yescharacteristic
217D212—41no—
218D212—41no—
219C4×D53—21no—
220C4×D53—21no—
221C4×D53—82no—
222C8×C53—21no—
223C8×C53C2×#4111yescharacteristic
224C8×C53H(5,𝔽2)11yescharacteristic
225#88322—21no—
226#88322—21no—
227#33863—531nomaximal, Sylow
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—42no—
232C22×C53—81no—
233C22×C53—81no—
234C22×C53C22×D811yescharacteristic
235C2×D53—21no—
236C2×D53—21no—
237C2×D53—81no—
238C2×D53—81no—
239C2×D53—81no—
240C2×D53—164no—
241C2×D53#50812yes—
242C4×C53—21no—
243C4×C53—81no—
244C4×C53—81no—
245C4×C53—81no—
246C4×C53#54611yescharacteristic
247C4×C53#57711yescharacteristic
248Dic53—21no—
249Dic53—21no—
250Dic53—21no—
251Dic53—81no—
252Dic53—81no—
253Dic53—81no—
254#3336—531no—
255#3280—531no—
256#2242—531no—
257#2235—531no—
258#2234—531no—
259#2228—531no—
260#2217—531no—
261#2216—531no—
262#2211—531no—
263#2189D5311yescharacteristic, complemented
264#2187—531no—
265C4×SD32—531no—
266D4×C16—531no—
267#2138—531no—
268C2×#357—531no—
269C2×C53—41no—
270C2×C53—82no—
271C2×C53#329411yescharacteristic
272C2×C53C2×#50811yescharacteristic
273C2×C53#341611yescharacteristic
274D53—22no—
275D53—161no—
276C2×#135C2×D5311yescharacteristic
277C22×D8—531no—
278#552—1061no—
279#546—1061no—
280C2×Q32C2×D5311yescharacteristic, complemented
281C2×SD32—531no—
282C2×SD32C2×D5311yescharacteristic
283C22×C16—531no—
284C22×C16—531no—
285#499—531no—
286#495—531no—
287#485—1061no—
288#483—1061no—
289#479—1061no—
290#467—531no—
291#467—531no—
292#467C2×D5311yescharacteristic, complemented
293#466—531no—
294#463—1061no—
295#450—1061no—
296#449—1061no—
297C4×Q16—531no—
298C4×D8—531no—
299C8×D4—531no—
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#358C2×D5311yescharacteristic, complemented
310#357—531no—
311#357—531no—
312#357—531no—
313#357—531no—
314#357—531no—
315#357C2×D5311yescharacteristic
316C22⋊C16—531no—
317C22⋊C16C2×D5311yescharacteristic, complemented
318C4×C16—531no—
319C53#3386311yescharacteristic, complemented, Sylow
320C2×#41—21no—
321C22×D4—1061no—
322#135—22no—
323#135—41no—
324C2×Q16C22×D5311yescharacteristic
325C2×Q16C22×D5311yescharacteristic
326C2×SD16—21no—
327C2×SD16—1061no—
328C2×D8—531no—
329C2×D8—1061no—
330C2×D8—1062no—
331C2×D8—2121no—
332C2×D8C22×D5311yescharacteristic
333C22×C8—531no—
334C22×C8C22×D5311yescharacteristic
335#126—2121no—
336#125—1061no—
337#125—2121no—
338#124—1061no—
339#123—2121no—
340#123—2121no—
341C22⋊Q8—21no—
342C22⋊Q8—1061no—
343C22⋊Q8—1061no—
344#121—1061no—
345#121—2121no—
346#120—2121no—
347C4×Q8—1061no—
348C4×D4—531no—
349C4×D4—1061no—
350#117—1061no—
351#117—2121no—
352C2×C4⋊C4—1061no—
353C2×C22⋊C4—2121no—
354Q32—41no—
355SD32—22no—
356SD32—2121no—
357C2×C16—531no—
358C2×C16—531no—
359C2×C16—2121no—
360C2×C16—2121no—
361C2×C16C22×D5311yescharacteristic
362C2×C16C22×D5311yescharacteristic
363#107—531no—
364#107—531no—
365#107—531no—
366#107—1061no—
367#107C22×D5311yescharacteristic
368#106—1061no—
369#106—1061no—
370C4⋊C8—531no—
371#103—21no—
372#103—1061no—
373#103—1061no—
374#103—1061no—
375#102—1061no—
376#102—1061no—
377C22⋊C8—531no—
378C22⋊C8—531no—
379C4×C8—531no—
380C24—2121no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—531no—
388C2×D4—1061no—
389C2×D4—4241no—
390C2×D4—2121no—
391C2×D4—2122no—
392C2×D4—2121no—
393C22×C4—41no—
394C22×C4—531no—
395C22×C4—2121no—
396C22×C4—2121no—
397C22×C4D4×D5311yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—21no—
401Q16—21no—
402SD16—42no—
403SD16—4241no—
404D8—21no—
405D8—1061no—
406D8—1061no—
407D8—2122no—
408D8#1171994712yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—531no—
412C2×C8—531no—
413C2×C8—1061no—
414C2×C8—1061no—
415C2×C8D4×D5311yescharacteristic
416C2×C8C23×D5311yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—531no—
420C4⋊C4—1061no—
421C4⋊C4—1061no—
422C4⋊C4—1061no—
423C4⋊C4—2121no—
424C4⋊C4—2121no—
425C4⋊C4—2121no—
426C4⋊C4—2121no—
427C4⋊C4—2121no—
428C4⋊C4—2121no—
429C22⋊C4—41no—
430C22⋊C4—531no—
431C22⋊C4—531no—
432C22⋊C4—2121no—
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C42—531no—
440C42—2121no—
441C42—2121no—
442C16—21no—
443C16—21no—
444C16—1061no—
445C16—1061no—
446C23—41no—
447C23—531no—
448C23—8481no—
449C23—4242no—
450C23—4242no—
451C23—2121no—
452C23D8×D5311yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457D4—22no—
458D4—41no—
459D4—82no—
460D4—8481no—
461D4—1062no—
462D4—4242no—
463D4—2121no—
464D4—2121no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—531no—
472C2×C4—531no—
473C2×C4—531no—
474C2×C4—8481no—
475C2×C4—1061no—
476C2×C4—1061no—
477C2×C4—4242no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4D8×D5311yescharacteristic
482C2×C4C2×D4×D5311yescharacteristic
483C8—21no—
484C8—1061no—
485C8—1061no—
486C8C2×#1171994711yescharacteristic
487C8#1313326211yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—532no—
495C22—8484no—
496C22—1061no—
497C22—1061no—
498C22—4241no—
499C22—4241no—
500C22—4241no—
501C22C2×D8×D5311yescharacteristic
502C4—21no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—1061no—
507C4—1061no—
508C4—1061no—
509C4—4241no—
510C4—4241no—
511C4—4241no—
512C4#1681126011yescharacteristic
513C4#1681143311yescharacteristic
514C2—41no—
515C2—82no—
516C2—8481no—
517C2—1062no—
518C2#2028652111yescharacteristic
519C2#508×D5311yescharacteristic
520C2#2028786011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2189C2
socleC22×C53C22×D8
prime radical#20286394C2
hypercenter#2189D53
hypocenterC53#33863
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2189

Complements

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

NH ≅ G/Ntype
1. #20287078C2semidirect
3. #20276948C2semidirect
4. #20276911C2semidirect
5. #20276907C2semidirect
6. #20276870C2semidirect
7. #20276813C2semidirect
8. #20276803C2semidirect
9. #20276779C2semidirect
10. #20276642C2semidirect
11. SD32×Dic53C2semidirect
12. #20276532C2semidirect
13. #20276422C2semidirect
14. #357×D53C2semidirect
15. C53×#2189C2semidirect
20. C22×#11719908C22semidirect
22. #16811024C22semidirect
33. Q16×Dic53C22semidirect
36. #16810661C22semidirect
38. #16810451C22semidirect
39. #16810450C22semidirect
40. #16810449C22semidirect
41. #16810441C22semidirect
42. #16810427C22semidirect
43. #16810427C22semidirect
44. #16810427C22semidirect
51. C16×Dic53C22semidirect
53. C2×Q32×C53C22semidirect
55. C53×#467C22semidirect
56. C53×#358C22semidirect
58. C53×C22⋊C16C22semidirect
263. #2189D53semidirect
280. C2×Q32C2×D53semidirect
292. #467C2×D53semidirect
309. #358C2×D53semidirect
317. C22⋊C16C2×D53semidirect
319. C53#33863semidirect

Maximal subgroups

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

subgroupindex
#202870782
#202863942
#202769482
#202769112
#202769072
#202768702
#202768132
#202768032
#202767792
#202766422
SD32×Dic532
#202765322
#202764222
#357×D532
C53×#21892
#3386353