Finite Group Databasebeta

← #27540939

Subgroups — #27540939

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#20286402C211yesmaximal, characteristic
3#20276947C211yesmaximal, characteristic, complemented
4#20276913C211yesmaximal, characteristic, complemented
5#20276907C211yesmaximal, characteristic, complemented
6#20276873C211yesmaximal, characteristic, complemented
7#20276811C211yesmaximal, characteristic, complemented
8#20276806C211yesmaximal, characteristic, complemented
9#20276778C211yesmaximal, characteristic, complemented
10#20276657C211yesmaximal, characteristic, complemented
11#20276646C211yesmaximal, characteristic, complemented
12C4×#13133162C211yesmaximal, characteristic, complemented
13#20276423C211yesmaximal, characteristic, complemented
14C2×#16810425C211yesmaximal, characteristic, complemented
15D4×C16×C53C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×D424C2211yescharacteristic
18#16811313—21no—
19#16811269—21no—
20C2×Dic424C2211yescharacteristic, complemented
21C2×#13133162C2211yescharacteristic
22C2×#13133162C2211yescharacteristic
23#16811025C2211yescharacteristic, complemented
24#16811007C2211yescharacteristic
25#16810965—21no—
26#16810954—21no—
27#16810931—21no—
28#16810864C2211yescharacteristic
29#16810864C2211yescharacteristic
30#16810864C2211yescharacteristic, complemented
31#16810859C2211yescharacteristic
32#16810839—21no—
33#16810776—21no—
34#16810770—21no—
35C4×Dic212C2211yescharacteristic, complemented
36C4×D424C2211yescharacteristic
37#16810661C2211yescharacteristic, complemented
38C2×#13133151C2211yescharacteristic
39#16810451C2211yescharacteristic, complemented
40#16810451C2211yescharacteristic, complemented
41#16810449C2211yescharacteristic, complemented
42#16810428C2211yescharacteristic, complemented
43#16810428C2211yescharacteristic, complemented
44#16810428C2211yescharacteristic, complemented
45#16810428C2211yescharacteristic, complemented
46#16810425C2211yescharacteristic
47#16810425C2211yescharacteristic
48#16810425C2211yescharacteristic
49#16810425C2211yescharacteristic
50#16810425C2211yescharacteristic
51#16810425C2211yescharacteristic
52C22×C16×C53C2211yescharacteristic
53C22×C16×C53C2211yescharacteristic, complemented
54C8×D4×C53C2211yescharacteristic
55C53×C4⋊C16C2211yescharacteristic, complemented
56C53×C22⋊C16C2211yescharacteristic
57C53×C22⋊C16C2211yescharacteristic, complemented
58C4×C16×C53C2211yescharacteristic, complemented
59C2×#11719945—21no—
60C22×D212—21no—
61#13133239—22no—
62#13133239—41no—
63C2×Dic212C2311yescharacteristic
64C2×Dic212C2311yescharacteristic
65C2×#11719913—21no—
66C2×#11719913—21no—
67C2×D424—21no—
68C2×D424—22no—
69C2×D424—41no—
70C2×D424C2311yescharacteristic
71C2×D424C2311yescharacteristic
72#13133210—41no—
73#13133209—21no—
74#13133207—41no—
75#13133205—21no—
76#13133202—41no—
77#13133198—41no—
78#13133196—21no—
79#13133196—21no—
80#13133196—21no—
81#13133191—21no—
82#13133190—41no—
83#13133185—41no—
84C4×Dic106—21no—
85C4×D212—21no—
86#13133176—21no—
87#13133173—41no—
88C2×C4⋊Dic53—21no—
89C22⋊C4×D53—41no—
90Dic424—41no—
91#13133162—22no—
92#13133162—41no—
93#13133151—21no—
94#13133151C2311yescharacteristic
95#13133151C2311yescharacteristic
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133149—21no—
99#13133149—21no—
100#13133142—21no—
101#13133142—21no—
102#13133142—21no—
103#13133142—21no—
104#13133139—21no—
105#13133139—21no—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C4×D4×C53D411yescharacteristic
109C2×C16×C53—41no—
110C2×C16×C53—41no—
111C2×C16×C53C2311yescharacteristic
112C2×C16×C53C2311yescharacteristic
113C2×C16×C53C2311yescharacteristic
114C2×C16×C53C2311yescharacteristic
115C4⋊C8×C53D411yescharacteristic
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C23×D53—41no—
120C2×D4×C53D811yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C2×C22⋊D53—41no—
124C2×C22⋊D53—41no—
125C22×Dic53—41no—
126#11719945—42no—
127#11719945—81no—
128C2×D212—21no—
129C2×D212—21no—
130C2×D212—42no—
131C2×D212—41no—
132C2×D212—81no—
133C2×C4×D53—41no—
134C2×C4×D53—41no—
135C2×Dic106—21no—
136C2×Dic106—21no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53C2×D411yescharacteristic
142C2×C8×C53C2×D411yescharacteristic
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2411yescharacteristic
145C4⋊C4×C53D811yescharacteristic
146C22⋊C4×C53D811yescharacteristic
147C22⋊C4×C53D811yescharacteristic
148C42×C53C2×D411yescharacteristic
149C22⋊Dic53—41no—
150C22⋊Dic53—41no—
151C22⋊Dic53—41no—
152C22⋊Dic53—41no—
153#11719921—41no—
154#11719921—41no—
155#11719921—41no—
156#11719921—41no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—41no—
162#11719919—41no—
163#11719919—41no—
164#11719919—41no—
165#11719919—41no—
166#11719919—41no—
167#11719919—41no—
168C4×Dic53—41no—
169C4×Dic53—41no—
170Dic212—21no—
171Dic212—21no—
172Dic212—21no—
173Dic212—21no—
174D424—21no—
175D424—21no—
176D424—21no—
177D424—42no—
178D424#4112yes—
179#11719913—42no—
180#11719913—81no—
181C16×C53—21no—
182C16×C53—21no—
183C16×C53—21no—
184C16×C53—21no—
185C23×C53C2×D811yescharacteristic
186C23×C53C2×D811yescharacteristic
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—82no—
190C22×D53—82no—
191C22×D53—161no—
192D4×C53—22no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C22⋊D53—82no—
203C22⋊D53—161no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—82no—
211D212—22no—
212D212—41no—
213D212—41no—
214D212—41no—
215D212—82no—
216C4×D53—82no—
217C4×D53—161no—
218Dic106—41no—
219Dic106—41no—
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#50812yes—
234C22×C53C22×D811yescharacteristic
235C2×D53—42no—
236C2×D53—81no—
237C2×D53—81no—
238C2×D53—81no—
239C2×D53—81no—
240C2×D53—81no—
241C2×D53—164no—
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53#54611yescharacteristic
247C4×C53#57711yescharacteristic
248Dic53—81no—
249Dic53—81no—
250Dic53—81no—
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#2189—531no—
264#2187—531no—
265C4×SD32—531no—
266D4×C16D5311yescharacteristic, complemented
267#2138—531no—
268C2×#357—531no—
269C2×C53—22no—
270C2×C53—41no—
271C2×C53#329411yescharacteristic
272C2×C53C2×#50811yescharacteristic
273C2×C53#341611yescharacteristic
274D53—82no—
275D53—161no—
276C2×#135—531no—
277C22×D8—531no—
278#552—1061no—
279#546—1061no—
280C2×Q32—531no—
281C2×SD32—531no—
282C2×SD32—531no—
283C22×C16C2×D5311yescharacteristic
284C22×C16C2×D5311yescharacteristic, complemented
285#499—531no—
286#495—531no—
287#485—1061no—
288#483—1061no—
289#479—1061no—
290#467—531no—
291#467—531no—
292#467—531no—
293#466—531no—
294#463—1061no—
295#450—1061no—
296#449—1061no—
297C4×Q16—531no—
298C4×D8—531no—
299C8×D4C2×D5311yescharacteristic
300#427—531no—
301C2×#107—531no—
302#367—531no—
303#367—531no—
304#366—531no—
305C4⋊C16C2×D5311yescharacteristic, complemented
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⋊C16C2×D5311yescharacteristic
317C22⋊C16C2×D5311yescharacteristic, complemented
318C4×C16C2×D5311yescharacteristic, complemented
319C53#3386311yescharacteristic, complemented, Sylow
320C2×#41—1061no—
321C22×D4—1061no—
322#135—1062no—
323#135—2121no—
324C2×Q16—531no—
325C2×Q16—531no—
326C2×SD16—1061no—
327C2×SD16—1061no—
328C2×D8—531no—
329C2×D8—531no—
330C2×D8—1062no—
331C2×D8—1061no—
332C2×D8—2121no—
333C22×C8C22×D5311yescharacteristic
334C22×C8C22×D5311yescharacteristic
335#126—2121no—
336#125—1061no—
337#125—2121no—
338#124—1061no—
339#123—2121no—
340#123—2121no—
341C22⋊Q8—1061no—
342C22⋊Q8—1061no—
343C22⋊Q8—1061no—
344#121—1061no—
345#121—2121no—
346#120—2121no—
347C4×Q8—1061no—
348C4×D4—1061no—
349C4×D4D21211yescharacteristic
350#117—1061no—
351#117—2121no—
352C2×C4⋊C4—1061no—
353C2×C22⋊C4—2121no—
354Q32—2121no—
355SD32—1062no—
356SD32—2121no—
357C2×C16—41no—
358C2×C16—41no—
359C2×C16C22×D5311yescharacteristic
360C2×C16C22×D5311yescharacteristic
361C2×C16C22×D5311yescharacteristic
362C2×C16C22×D5311yescharacteristic
363#107—531no—
364#107—531no—
365#107—531no—
366#107—531no—
367#107—1061no—
368#106—1061no—
369#106—1061no—
370C4⋊C8D21211yescharacteristic
371#103—1061no—
372#103—1061no—
373#103—1061no—
374#103—1061no—
375#102—1061no—
376#102—1061no—
377C22⋊C8D21211yescharacteristic
378C22⋊C8D21211yescharacteristic
379C4×C8C22×D5311yescharacteristic
380C24—2121no—
381#41—4241no—
382#41—2122no—
383C2×Q8—1061no—
384C2×Q8—1061no—
385C2×D4—1061no—
386C2×D4—1061no—
387C2×D4—4241no—
388C2×D4—2122no—
389C2×D4—2121no—
390C2×D4—2121no—
391C2×D4—2121no—
392C2×D4D42411yescharacteristic
393C22×C4—2121no—
394C22×C4—2121no—
395C22×C4—2121no—
396C22×C4C2×D21211yescharacteristic
397C22×C4C2×D21211yescharacteristic
398Q16—1061no—
399Q16—1061no—
400Q16—1061no—
401Q16—1061no—
402SD16—4241no—
403SD16—2122no—
404D8—532no—
405D8—1061no—
406D8—1061no—
407D8—1061no—
408D8—2122no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×D21211yescharacteristic
414C2×C8C2×D21211yescharacteristic
415C2×C8C2×D21211yescharacteristic
416C2×C8C23×D5311yescharacteristic
417C4⋊C4—1061no—
418C4⋊C4—1061no—
419C4⋊C4—1061no—
420C4⋊C4—1061no—
421C4⋊C4—2121no—
422C4⋊C4—2121no—
423C4⋊C4—2121no—
424C4⋊C4—2121no—
425C4⋊C4—2121no—
426C4⋊C4—2121no—
427C4⋊C4—2121no—
428C4⋊C4D42411yescharacteristic
429C22⋊C4—2121no—
430C22⋊C4—2121no—
431C22⋊C4—2121no—
432C22⋊C4—2121no—
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4D42411yescharacteristic
438C22⋊C4D42411yescharacteristic
439C42—2121no—
440C42—2121no—
441C42C2×D21211yescharacteristic
442C16—21no—
443C16—21no—
444C16—21no—
445C16—21no—
446C23—8481no—
447C23—4242no—
448C23—4242no—
449C23—2121no—
450C23—2121no—
451C23C2×D42411yescharacteristic
452C23C2×D42411yescharacteristic
453Q8—2121no—
454Q8—2121no—
455Q8—2121no—
456Q8—2121no—
457D4—22no—
458D4—8481no—
459D4—1062no—
460D4—4242no—
461D4—4242no—
462D4—2121no—
463D4—2121no—
464D4—2121no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—8481no—
470C2×C4—4242no—
471C2×C4—4242no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4C2×D42411yescharacteristic
479C2×C4C2×D42411yescharacteristic
480C2×C4C2×D42411yescharacteristic
481C2×C4C2×D42411yescharacteristic
482C2×C4C22×D21211yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D5311yescharacteristic
487C8#1313326211yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—8484no—
494C22—4241no—
495C22—4241no—
496C22—4241no—
497C22—4241no—
498C22—4241no—
499C22—2122no—
500C22#1681105512yes—
501C22C22×D42411yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—4241no—
507C4—4241no—
508C4—4241no—
509C4—4241no—
510C4—4241no—
511C4—4241no—
512C4#1681126811yescharacteristic
513C4#1681143511yescharacteristic
514C2—22no—
515C2—41no—
516C2—8481no—
517C2—4242no—
518C2#2028652911yescharacteristic
519C2C2×#1681105511yescharacteristic
520C2#2028786311yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingD4×C16×C53C2
socleC22×C53C22×D8
prime radical#20286402C2
hypercenterD4×C16D53
hypocenterC53#33863
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #20287084C2semidirect
3. #20276947C2semidirect
4. #20276913C2semidirect
5. #20276907C2semidirect
6. #20276873C2semidirect
7. #20276811C2semidirect
8. #20276806C2semidirect
9. #20276778C2semidirect
10. #20276657C2semidirect
11. #20276646C2semidirect
12. C4×#13133162C2semidirect
13. #20276423C2semidirect
14. C2×#16810425C2semidirect
15. D4×C16×C53C2semidirect
20. C2×Dic424C22semidirect
23. #16811025C22semidirect
30. #16810864C22semidirect
35. C4×Dic212C22semidirect
37. #16810661C22semidirect
39. #16810451C22semidirect
40. #16810451C22semidirect
41. #16810449C22semidirect
42. #16810428C22semidirect
43. #16810428C22semidirect
44. #16810428C22semidirect
45. #16810428C22semidirect
53. C22×C16×C53C22semidirect
55. C53×C4⋊C16C22semidirect
57. C53×C22⋊C16C22semidirect
58. C4×C16×C53C22semidirect
266. D4×C16D53semidirect
284. C22×C16C2×D53semidirect
305. C4⋊C16C2×D53semidirect
317. C22⋊C16C2×D53semidirect
318. C4×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
#202870842
#202864022
#202769472
#202769132
#202769072
#202768732
#202768112
#202768062
#202767782
#202766572
#202766462
C4×#131331622
#202764232
C2×#168104252
D4×C16×C532
#3386353