Finite Group Databasebeta

← #27370016

Subgroups — #27370016

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285653C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3#20283750C211yesmaximal, characteristic, complemented
4#20274278C212yesmaximal, complemented
5#20274182C212yesmaximal, complemented
6#20274154C212yesmaximal, complemented
7#20274058C212yesmaximal, complemented
8#20273501C211yesmaximal, characteristic, complemented
9#20273428C211yesmaximal, characteristic, complemented
10#20273415C211yesmaximal, characteristic, complemented
11C53×#3110C211yesmaximal, characteristic, complemented
12C22×#11719922—21no—
13C2×#13133214C2211yescharacteristic
14#16810997C2212yescomplemented
15#16810971C2212yescomplemented
16#16810823—22no—
17#16810818—21no—
18#16810818—21no—
19#16810818C2212yes—
20#16810818C2212yes—
21#16810773—21no—
22#16810773—21no—
23#16810701C2212yescomplemented
24#16810695—22no—
25D4×#88322C2212yescomplemented
26#16810680C2211yescharacteristic
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133137—21no—
30#16810587—21no—
31C2×C22⋊#88322—21no—
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810369C2212yescomplemented
35#16810369C2212yescomplemented
36#16810366C2212yescomplemented
37#16810366C2212yescomplemented
38C53×#533C2211yescharacteristic
39C2×C4×D4×C53C2211yescharacteristic
40C53×#484—21no—
41C53×#474C2211yescharacteristic, complemented
42C53×#474C2211yescharacteristic, complemented
43C53×#461C2211yescharacteristic, complemented
44C53×#461C2211yescharacteristic, complemented
45C53×#450—21no—
46C53×#440—21no—
47C53×#440—21no—
48C53×#432C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52C2×#11719924—22no—
53C2×#11719924—22no—
54C2×#11719922—22no—
55C2×#11719922—22no—
56C2×#11719922—44no—
57C2×#11719917—22no—
58C22×#88322—22no—
59C22×#88322—21no—
60#13133214—21no—
61#13133214—21no—
62#13133214—21no—
63#13133214C2312yes—
64C4⋊#88322—21no—
65C4⋊#88322D412yes—
66C4⋊#88322C2312yes—
67C4⋊#88322C2312yes—
68#13133137—22no—
69#13133137—22no—
70#13133137—22no—
71#13133137—22no—
72C22⋊#88322—22no—
73C22⋊#88322—22no—
74C22⋊#88322—21no—
75C22⋊#88322D411yescharacteristic
76C22⋊#88322D411yescharacteristic
77#13133128—21no—
78C4×#88322C2312yes—
79C22×D4×C53D411yescharacteristic
80C23×C4×C53—21no—
81C2×SD16×C53—21no—
82C2×SD16×C53—21no—
83C4⋊Q8×C53C2311yescharacteristic
84C4⋊Q8×C53C2311yescharacteristic
85#125×C53—21no—
86C22⋊Q8×C53—21no—
87C22⋊Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53D411yescharacteristic
93C4×D4×C53D411yescharacteristic
94C4×D4×C53C2311yescharacteristic
95C4×D4×C53C2311yescharacteristic
96#117×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C22⋊C4×C53—21no—
99C2×C42×C53C2311yescharacteristic
100#107×C53—21no—
101#107×C53—21no—
102C4⋊C8×C53C2311yescharacteristic
103C4⋊C8×C53C2311yescharacteristic
104#103×C53—21no—
105#103×C53—21no—
106#102×C53—21no—
107#102×C53—21no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C24×C53—21no—
111C23×D53—41no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×D4×C53—22no—
115C2×D4×C53—21no—
116C2×D4×C53—41no—
117C2×D4×C53C2×D411yescharacteristic
118C2×D4×C53C2×D411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—82no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4111yescharacteristic
128C2×D212—22no—
129C2×D212—22no—
130C2×D212—22no—
131C2×D212—42no—
132C2×D212—42no—
133C2×D212—41no—
134C2×D212—41no—
135C2×D212—41no—
136C2×D212—82no—
137C2×D212—82no—
138SD16×C53—42no—
139SD16×C53—42no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53C2×D411yescharacteristic
150C4⋊C4×C53C2×D411yescharacteristic
151C22⋊C4×C53—21no—
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—21no—
155C22⋊C4×C53—41no—
156C42×C53—41no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53#4111yescharacteristic
160C42×C53C2411yescharacteristic
161#11719924—82no—
162#11719924—82no—
163#11719922—44no—
164#11719922—44no—
165#11719917—82no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—42no—
169C2×#88322—82no—
170C2×#88322C2×D412yes—
171C2×#88322C2×D412yes—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—81no—
177C23×C53#12111yescharacteristic
178C22×D53—42no—
179C22×D53—82no—
180C22×D53—82no—
181C22×D53—82no—
182Q8×C53—41no—
183Q8×C53—41no—
184Q8×C53—42no—
185D4×C53—21no—
186D4×C53—21no—
187D4×C53—21no—
188D4×C53—42no—
189D4×C53AGL(1,ℤ/8)12yes—
190C2×C4×C53—22no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—22no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—81no—
201C2×C4×C53—81no—
202C2×C4×C53—81no—
203C2×C4×C53#12111yescharacteristic
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C2×#4111yescharacteristic
209D212—42no—
210D212—42no—
211D212—42no—
212D212—42no—
213D212—82no—
214D212—82no—
215D212—82no—
216D212—82no—
217C8×C53—41no—
218C8×C53—41no—
219#88322—22no—
220#88322—22no—
221#88322—42no—
222#88322—42no—
223#19456—531nomaximal, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—22no—
227C22×C53—41no—
228C22×C53—41no—
229C22×C53—41no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—84no—
233C22×C53#47112yes—
234C22×C53C2×#12111yescharacteristic
235C2×D53—82no—
236C2×D53—82no—
237C2×D53—82no—
238C2×D53—164no—
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—21no—
242C4×C53—21no—
243C4×C53—41no—
244C4×C53—41no—
245C4×C53—81no—
246C4×C53—81no—
247C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
248C4×C53#57511yescharacteristic
249C4×C53#57612yes—
250#3208—531no—
251#3110D5311yescharacteristic, complemented
252C2×#459—531no—
253#2974—531no—
254#1701—532no—
255#1685—532no—
256#1681—532no—
257#1665—532no—
258#1573—531no—
259#1555—531no—
260#1552—531no—
261C2×C53—22no—
262C2×C53—42no—
263C2×C53—81no—
264C2×C53#304111yescharacteristic
265C2×C53#305511yescharacteristic
266C2×C53C2×#47111yescharacteristic
267D53—162no—
268C22×D8—1061no—
269#533C2×D5311yescharacteristic
270C2×#127—531no—
271C2×C4×D4C2×D5311yescharacteristic
272#492—532no—
273#486—532no—
274#484—21no—
275#474C2×D5311yescharacteristic, complemented
276#474C2×D5311yescharacteristic, complemented
277#461C2×D5311yescharacteristic, complemented
278#461C2×D5311yescharacteristic, complemented
279#460—1062no—
280#459—532no—
281#459—532no—
282#459—1061no—
283#459—1061no—
284#450—21no—
285#450—1061no—
286#450—1061no—
287#440—21no—
288#440—21no—
289#436—532no—
290#435—1062no—
291C8×D4—532no—
292#432—531no—
293#432C2×D5311yescharacteristic
294#431—531no—
295C2×C4⋊C8—531no—
296C2×#102—1061no—
297#407—1061no—
298C2×C22⋊C8—1061no—
299#335—532no—
300#334—532no—
301#332—532no—
302#332—532no—
303#331—532no—
304#331—532no—
305C53#1945611yescharacteristic, complemented, Sylow
306C22×D4—1061no—
307C22×D4—1061no—
308C22×D4—1061no—
309C22×D4C22⋊D5311yescharacteristic
310C23×C4—21no—
311C2×SD16—21no—
312C2×SD16—21no—
313C2×SD16—1062no—
314C2×SD16—1062no—
315C2×D8—1062no—
316C2×D8—1062no—
317C2×D8—2124no—
318C2×M16—1062no—
319C22×C8—1062no—
320C22×C8—1061no—
321C4⋊Q8C22×D5311yescharacteristic
322C4⋊Q8C22×D5311yescharacteristic
323#127—532no—
324#127—1061no—
325#127—1061no—
326#127—1061no—
327#125—21no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4C22⋊D5311yescharacteristic
335C4×D4C22⋊D5311yescharacteristic
336C4×D4C22×D5311yescharacteristic
337C4×D4C22×D5311yescharacteristic
338#117—21no—
339C2×C4⋊C4C22⋊D5311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C42C22×D5311yescharacteristic
342#107—21no—
343#107—21no—
344C4⋊C8—532no—
345C4⋊C8—532no—
346C4⋊C8—532no—
347C4⋊C8—1061no—
348C4⋊C8C22×D5311yescharacteristic
349C4⋊C8C22×D5311yescharacteristic
350#103—21no—
351#103—21no—
352#102—21no—
353#102—21no—
354#102—1062no—
355#102—1062no—
356#102—1062no—
357#102—1062no—
358C22⋊C8—21no—
359C22⋊C8—531no—
360C22⋊C8—531no—
361C22⋊C8—1062no—
362C22⋊C8—1062no—
363C22⋊C8—1061no—
364#97—21no—
365#97—1061no—
366C4×C8—532no—
367C24—21no—
368C24—2121no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—1062no—
375C2×D4—1062no—
376C2×D4—1062no—
377C2×D4—4242no—
378C2×D4—4242no—
379C2×D4—2122no—
380C2×D4—2122no—
381C2×D4—2121no—
382C2×D4—2121no—
383C2×D4—2121no—
384C2×D4C2×C22⋊D5311yescharacteristic
385C2×D4C2×C22⋊D5311yescharacteristic
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—82no—
392C22×C4D4×D5311yescharacteristic
393C22×C4#1171994711yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395SD16—42no—
396SD16—42no—
397SD16—4242no—
398SD16—4242no—
399D8—2124no—
400D8—2124no—
401M16—4242no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—532no—
405C2×C8—532no—
406C2×C8—1062no—
407C2×C8—1062no—
408C2×C8—4242no—
409C2×C8—2122no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4C2×C22⋊D5311yescharacteristic
418C4⋊C4C2×C22⋊D5311yescharacteristic
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—41no—
424C42—41no—
425C42D4×D5311yescharacteristic
426C42#1171994711yescharacteristic
427C42C2×C22⋊D5311yescharacteristic
428C42C23×D5311yescharacteristic
429C23—21no—
430C23—21no—
431C23—42no—
432C23—42no—
433C23—81no—
434C23—4242no—
435C23—4242no—
436C23—4242no—
437C23—2122no—
438C23#1313318911yescharacteristic
439Q8—41no—
440Q8—41no—
441Q8—42no—
442D4—21no—
443D4—21no—
444D4—21no—
445D4—42no—
446D4—4242no—
447D4—4242no—
448D4—4242no—
449D4—4242no—
450D4—2122no—
451D4—2122no—
452D4—2122no—
453D4—2122no—
454D4#1313324212yes—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#1313318911yescharacteristic
469C2×C4#1313318911yescharacteristic
470C2×C4#1313318911yescharacteristic
471C2×C4C2×D4×D5311yescharacteristic
472C2×C4C22×C22⋊D5311yescharacteristic
473C2×C4C2×#1171994711yescharacteristic
474C8—41no—
475C8—41no—
476C8—1062no—
477C8—1062no—
478C8—2122no—
479C8—2122no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—8484no—
490C22—4242no—
491C22—4242no—
492C22—4242no—
493C22#1681088812yes—
494C22C2×#1313318911yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—41no—
500C4—41no—
501C4—81no—
502C4—81no—
503C4C2×#1313324211yescharacteristic
504C4#1681142111yescharacteristic
505C4#1681142712yes—
506C2—22no—
507C2—42no—
508C2—81no—
509C2—8482no—
510C2#2028422011yescharacteristic
511C2#2028433611yescharacteristic
512C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3110C2
socleC22×C53C2×#121
prime radicalC2×#16810818C2
hypercenter#3110D53
hypocenterC53#19456
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #20285653C2semidirect
3. #20283750C2semidirect
4. #20274278C2semidirect
5. #20274182C2semidirect
6. #20274154C2semidirect
7. #20274058C2semidirect
8. #20273501C2semidirect
9. #20273428C2semidirect
10. #20273415C2semidirect
11. C53×#3110C2semidirect
14. #16810997C22semidirect
15. #16810971C22semidirect
23. #16810701C22semidirect
25. D4×#88322C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810369C22semidirect
35. #16810369C22semidirect
36. #16810366C22semidirect
37. #16810366C22semidirect
41. C53×#474C22semidirect
42. C53×#474C22semidirect
43. C53×#461C22semidirect
44. C53×#461C22semidirect
251. #3110D53semidirect
275. #474C2×D53semidirect
276. #474C2×D53semidirect
277. #461C2×D53semidirect
278. #461C2×D53semidirect
305. C53#19456semidirect

Maximal subgroups

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

subgroupindex
#202856532
C2×#168108182
#202837502
#202742782
#202741822
#202741542
#202740582
#202735012
#202734282
#202734152
C53×#31102
#1945653