Finite Group Databasebeta

← #27370010

Subgroups — #27370010

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#20285656C211yesmaximal, characteristic, complemented
2#20284814C211yesmaximal, characteristic, complemented
3#20283750C211yesmaximal, characteristic, complemented
4#20274274C212yesmaximal, complemented
5#20274178C212yesmaximal, complemented
6#20274150C212yesmaximal, complemented
7#20274054C212yesmaximal, complemented
8#20273498C211yesmaximal, characteristic, complemented
9#20273429C211yesmaximal, characteristic, complemented
10#20273413C211yesmaximal, characteristic, complemented
11C2×C53×#459C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16810995C2212yescomplemented
14#16810969C2212yescomplemented
15#16810958—21no—
16#16810903C2211yescharacteristic, complemented
17#16810903C2211yescharacteristic, complemented
18#16810828C2211yescharacteristic, complemented
19#16810828C2211yescharacteristic, complemented
20#16810820—22no—
21#16810774—21no—
22#16810774—21no—
23#16810774—21no—
24#16810723—21no—
25#16810723—21no—
26#16810701C2212yescomplemented
27#16810695—22no—
28D4×#88322C2212yescomplemented
29#16810680C2211yescharacteristic
30#16810680C2211yescharacteristic
31#16810676C2211yescharacteristic
32#16810587—21no—
33C2×C22⋊#88322—21no—
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810370C2212yescomplemented
37#16810368C2212yescomplemented
38#16810367C2212yescomplemented
39#16810367C2212yescomplemented
40C22×D8×C53—21no—
41C2×#127×C53C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#459—21no—
44C53×#459—21no—
45C53×#459C2212yes—
46C53×#459C2212yes—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#102×C53—21no—
49C2×#11719923—22no—
50C2×#11719923—22no—
51C2×#11719923—21no—
52C2×#11719923—21no—
53C2×#11719917—22no—
54C22×#88322—22no—
55#13133217C2311yescharacteristic
56#13133217C2311yescharacteristic
57#13133209—21no—
58#13133196—21no—
59#13133196—21no—
60C4×Dic106—21no—
61C4×Dic106—21no—
62#13133176—21no—
63#13133150—21no—
64#13133150—21no—
65C4⋊#88322D412yes—
66C4⋊#88322C2311yescharacteristic
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2312yes—
69#13133140—21no—
70#13133140—21no—
71#13133138—22no—
72#13133138—22no—
73#13133138—21no—
74#13133138—21no—
75C22⋊#88322—22no—
76C22⋊#88322—22no—
77C22⋊#88322—21no—
78C22⋊#88322—21no—
79C22⋊#88322D411yescharacteristic
80C22⋊#88322D411yescharacteristic
81#13133128—21no—
82#13133128—21no—
83C4×#88322C2312yes—
84C22×D4×C53—21no—
85C22×D4×C53—21no—
86C22×D4×C53—21no—
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89C2×D8×C53—22no—
90C2×D8×C53—22no—
91C2×D8×C53—44no—
92C22×C8×C53—21no—
93#127×C53—21no—
94#127×C53—21no—
95#127×C53—21no—
96#127×C53C2312yes—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53D411yescharacteristic
100C4×D4×C53D411yescharacteristic
101C4×D4×C53C2311yescharacteristic
102C4×D4×C53C2311yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#102×C53—22no—
109#102×C53—22no—
110C24×C53—21no—
111C24×C53—41no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—41no—
118C2×D4×C53—42no—
119C2×D4×C53—42no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—82no—
124C2×D4×C53—82no—
125C2×D4×C53C2×D411yescharacteristic
126C2×D4×C53C2×D411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—82no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53#4111yescharacteristic
136C2×Dic106—21no—
137C2×Dic106—21no—
138D8×C53—44no—
139D8×C53—44no—
140C2×C8×C53—22no—
141C2×C8×C53—42no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53C2×D411yescharacteristic
144C4⋊C4×C53C2×D411yescharacteristic
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C42×C53#4111yescharacteristic
152C42×C53C2411yescharacteristic
153C22⋊Dic53—41no—
154#11719923—42no—
155#11719923—42no—
156#11719923—82no—
157#11719923—82no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162#11719919—41no—
163#11719919—41no—
164C4×Dic53—41no—
165#11719917—82no—
166C2×#88322—22no—
167C2×#88322—21no—
168C2×#88322—21no—
169C2×#88322—82no—
170C2×#88322C2×D412yes—
171C2×#88322C2×D412yes—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—81no—
178C23×C53—82no—
179C23×C53—82no—
180C23×C53—82no—
181C23×C53#12111yescharacteristic
182D4×C53—21no—
183D4×C53—21no—
184D4×C53—21no—
185D4×C53—42no—
186D4×C53—42no—
187D4×C53—42no—
188D4×C53—42no—
189D4×C53—42no—
190D4×C53—82no—
191D4×C53—82no—
192D4×C53—82no—
193D4×C53—82no—
194D4×C53AGL(1,ℤ/8)12yes—
195C2×C4×C53—22no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—22no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—81no—
204C2×C4×C53—81no—
205C2×C4×C53—81no—
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53#12111yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C2×Dic53—41no—
213C2×Dic53—41no—
214Dic106—41no—
215Dic106—41no—
216Dic106—42no—
217C8×C53—42no—
218#88322—22no—
219#88322—22no—
220#88322—41no—
221#88322—41no—
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—82no—
234C22×C53—82no—
235C22×C53—82no—
236C22×C53—164no—
237C22×C53#47112yes—
238C22×C53C2×#12111yescharacteristic
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—21no—
242C4×C53—21no—
243C4×C53—41no—
244C4×C53—41no—
245C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C53#57511yescharacteristic
247C4×C53#57612yes—
248Dic53—81no—
249Dic53—81no—
250#3208—531no—
251#3110—531no—
252C2×#459D5311yescharacteristic
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—162no—
265C2×C53#304111yescharacteristic
266C2×C53#305511yescharacteristic
267C2×C53C2×#47111yescharacteristic
268C22×D8—21no—
269#533—531no—
270C2×#127C2×D5311yescharacteristic
271C2×C4×D4C2×D5311yescharacteristic
272#492—532no—
273#486—532no—
274#484—1061no—
275#474—531no—
276#474—531no—
277#461—531no—
278#461—531no—
279#460—1062no—
280#459—21no—
281#459—21no—
282#459C2×D5312yes—
283#459C2×D5312yes—
284#450—1061no—
285#450—1061no—
286#450—1061no—
287#440—1061no—
288#440—1061no—
289#436—532no—
290#435—1062no—
291C8×D4—532no—
292#432—531no—
293#432—531no—
294#431—531no—
295C2×C4⋊C8C2×D5311yescharacteristic
296C2×#102—21no—
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—21no—
307C22×D4—21no—
308C22×D4—21no—
309C22×D4C22⋊D5311yescharacteristic
310C23×C4—21no—
311C2×SD16—1062no—
312C2×SD16—1062no—
313C2×SD16—1061no—
314C2×SD16—1061no—
315C2×D8—22no—
316C2×D8—22no—
317C2×D8—44no—
318C2×M16—1062no—
319C22×C8—21no—
320C22×C8—1062no—
321C4⋊Q8—531no—
322C4⋊Q8—531no—
323#127—21no—
324#127—21no—
325#127—21no—
326#127C22×D5312yes—
327#125—1061no—
328C22⋊Q8—1061no—
329C22⋊Q8—1061no—
330C4×Q8—1061no—
331C4×Q8—1061no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4C22⋊D5311yescharacteristic
335C4×D4C22⋊D5311yescharacteristic
336C4×D4C22×D5311yescharacteristic
337C4×D4C22×D5311yescharacteristic
338#117—1061no—
339C2×C4⋊C4C22⋊D5311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C42C22×D5311yescharacteristic
342#107—1061no—
343#107—1061no—
344C4⋊C8—21no—
345C4⋊C8—532no—
346C4⋊C8—532no—
347C4⋊C8—531no—
348C4⋊C8—531no—
349C4⋊C8C22×D5312yes—
350#103—1061no—
351#103—1061no—
352#102—22no—
353#102—22no—
354#102—1062no—
355#102—1062no—
356#102—1061no—
357#102—1061no—
358C22⋊C8—531no—
359C22⋊C8—531no—
360C22⋊C8—1062no—
361C22⋊C8—1062no—
362C22⋊C8—1061no—
363C22⋊C8—1061no—
364#97—1061no—
365#97—1061no—
366C4×C8—532no—
367C24—21no—
368C24—41no—
369C2×Q8—1061no—
370C2×Q8—1061no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—41no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—82no—
383C2×D4—82no—
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—4242no—
396SD16—4242no—
397SD16—2122no—
398SD16—2122no—
399D8—44no—
400D8—44no—
401M16—4242no—
402C2×C8—22no—
403C2×C8—42no—
404C2×C8—532no—
405C2×C8—532no—
406C2×C8—1062no—
407C2×C8—1061no—
408C2×C8—1061no—
409C2×C8—4242no—
410C4⋊C4—21no—
411C4⋊C4—1061no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1061no—
415C4⋊C4—2121no—
416C4⋊C4—2121no—
417C4⋊C4C2×C22⋊D5311yescharacteristic
418C4⋊C4C2×C22⋊D5311yescharacteristic
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—2121no—
424C42—2121no—
425C42D4×D5311yescharacteristic
426C42#1171994711yescharacteristic
427C42C2×C22⋊D5311yescharacteristic
428C42C23×D5311yescharacteristic
429C23—21no—
430C23—21no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—81no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23#1313318911yescharacteristic
439Q8—2121no—
440Q8—2121no—
441Q8—2122no—
442D4—21no—
443D4—21no—
444D4—21no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—82no—
451D4—82no—
452D4—82no—
453D4—82no—
454D4#1313324312yes—
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—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—2121no—
467C2×C4—2121no—
468C2×C4#1313318911yescharacteristic
469C2×C4#1313318911yescharacteristic
470C2×C4#1313318911yescharacteristic
471C2×C4C2×D4×D5311yescharacteristic
472C2×C4C22×C22⋊D5311yescharacteristic
473C2×C4C2×#1171994711yescharacteristic
474C8—42no—
475C8—1062no—
476C8—1062no—
477C8—2121no—
478C8—2121no—
479C8—2122no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—164no—
493C22#1681088412yes—
494C22C2×#1313318911yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—41no—
500C4—41no—
501C4—4241no—
502C4—4241no—
503C4C2×#1313324311yescharacteristic
504C4#1681142111yescharacteristic
505C4#1681142412yes—
506C2—22no—
507C2—42no—
508C2—81no—
509C2—162no—
510C2#2028421511yescharacteristic
511C2#2028433011yescharacteristic
512C2C2×#1681088411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×C53×#459C2
socleC22×C53C2×#121
hypercenterC2×#459D53
hypocenterC53#19456
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #20285656C2semidirect
2. #20284814C2semidirect
3. #20283750C2semidirect
4. #20274274C2semidirect
5. #20274178C2semidirect
6. #20274150C2semidirect
7. #20274054C2semidirect
8. #20273498C2semidirect
9. #20273429C2semidirect
10. #20273413C2semidirect
13. #16810995C22semidirect
14. #16810969C22semidirect
16. #16810903C22semidirect
17. #16810903C22semidirect
18. #16810828C22semidirect
19. #16810828C22semidirect
26. #16810701C22semidirect
28. D4×#88322C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810370C22semidirect
37. #16810368C22semidirect
38. #16810367C22semidirect
39. #16810367C22semidirect
305. C53#19456semidirect

Maximal subgroups

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

subgroupindex
#202856562
#202848142
#202837502
#202742742
#202741782
#202741502
#202740542
#202734982
#202734292
#202734132
C2×C53×#4592
#1945653