Finite Group Databasebeta

← #27370014

Subgroups — #27370014

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#20285658C211yesmaximal, characteristic, complemented
2#20284814C211yesmaximal, characteristic, complemented
3C2×#16810818C211yesmaximal, characteristic
4#20274276C212yesmaximal, complemented
5#20274180C212yesmaximal, complemented
6#20274152C212yesmaximal, complemented
7#20274056C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273428C211yesmaximal, characteristic, complemented
10#20273416C211yesmaximal, characteristic, complemented
11C53×#2974C211yesmaximal, characteristic, complemented
12C22×#11719922—21no—
13#16811180C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16810996C2212yescomplemented
16#16810970C2212yescomplemented
17#16810958—21no—
18#16810903C2211yescharacteristic, complemented
19#16810903C2211yescharacteristic, complemented
20#16810828C2211yescharacteristic, complemented
21#16810828C2211yescharacteristic, complemented
22#16810824—22no—
23#16810818—21no—
24#16810818—21no—
25#16810818C2212yes—
26#16810818C2212yes—
27#16810776—21no—
28#16810776—21no—
29#16810774—21no—
30#16810723—21no—
31#16810723—21no—
32#16810680C2211yescharacteristic
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133137—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810370C2212yescomplemented
38#16810368C2212yescomplemented
39#16810366C2212yescomplemented
40#16810366C2212yescomplemented
41C2×C4×D4×C53C2211yescharacteristic
42C53×#436C2212yescomplemented
43C53×#435—22no—
44C8×D4×C53C2212yescomplemented
45C53×#432C2211yescharacteristic
46C53×#431C2211yescharacteristic
47C53×#407—21no—
48C2×C22⋊C8×C53—21no—
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52C2×#11719913—22no—
53C2×#11719913—22no—
54C2×#11719923—21no—
55C2×#11719923—21no—
56C2×#11719922—22no—
57C2×#11719922—22no—
58C2×#11719922—44no—
59C22×#88322—21no—
60#13133217C2311yescharacteristic
61#13133217C2311yescharacteristic
62#13133214—21no—
63#13133214—21no—
64#13133214—21no—
65#13133214C2312yes—
66#13133209—21no—
67#13133196—21no—
68#13133196—21no—
69C4×Dic106—21no—
70C4×Dic106—21no—
71#13133176—21no—
72#13133150—21no—
73#13133150—21no—
74C4⋊#88322—21no—
75C4⋊#88322C2312yes—
76C4⋊#88322C2311yescharacteristic
77C4⋊#88322C2311yescharacteristic
78#13133140—21no—
79#13133140—21no—
80#13133139—22no—
81#13133139—22no—
82#13133138—21no—
83#13133138—21no—
84#13133137—22no—
85#13133137—22no—
86C22⋊#88322—21no—
87#13133128—21no—
88C22×D4×C53D411yescharacteristic
89C23×C4×C53—21no—
90C2×M16×C53—22no—
91C22×C8×C53—22no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53D411yescharacteristic
95C4×D4×C53D411yescharacteristic
96C4×D4×C53C2311yescharacteristic
97C4×D4×C53C2311yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C42×C53C2311yescharacteristic
101C4⋊C8×C53D412yes—
102C4⋊C8×C53C2312yes—
103C22⋊C8×C53—22no—
104C22⋊C8×C53—22no—
105C22⋊C8×C53—21no—
106C22⋊C8×C53D411yescharacteristic
107C22⋊C8×C53D411yescharacteristic
108#97×C53—21no—
109C4×C8×C53C2312yes—
110C24×C53—21no—
111C23×D53—41no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—41no—
115C2×D4×C53C2×D411yescharacteristic
116C2×D4×C53C2×D411yescharacteristic
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—82no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C2×D212—22no—
127C2×D212—22no—
128C2×D212—22no—
129C2×D212—42no—
130C2×D212—42no—
131C2×D212—41no—
132C2×D212—41no—
133C2×D212—41no—
134C2×D212—82no—
135C2×D212—82no—
136C2×Dic106—21no—
137C2×Dic106—21no—
138M16×C53—82no—
139C2×C8×C53—22no—
140C2×C8×C53—82no—
141C2×C8×C53C2×D412yes—
142C2×C8×C53C2×D412yes—
143C4⋊C4×C53—21no—
144C4⋊C4×C53C2×D411yescharacteristic
145C4⋊C4×C53C2×D411yescharacteristic
146C22⋊C4×C53—21no—
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154C22⋊Dic53—41no—
155#11719923—42no—
156#11719923—42no—
157#11719922—44no—
158#11719922—44no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163#11719919—41no—
164#11719919—41no—
165C4×Dic53—41no—
166C2×#88322—21no—
167C2×#88322—21no—
168C2×#88322—22no—
169C2×#88322—42no—
170#11719913—82no—
171#11719913—82no—
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—
182D4×C53—21no—
183D4×C53—21no—
184D4×C53—21no—
185D4×C53—42no—
186D4×C53AGL(1,ℤ/8)12yes—
187C2×C4×C53—22no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—22no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—81no—
196C2×C4×C53—81no—
197C2×C4×C53—81no—
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C2×#4111yescharacteristic
204C2×Dic53—41no—
205C2×Dic53—41no—
206D212—42no—
207D212—42no—
208D212—42no—
209D212—42no—
210D212—82no—
211D212—82no—
212D212—82no—
213D212—82no—
214Dic106—41no—
215Dic106—41no—
216Dic106—42no—
217C8×C53—22no—
218C8×C53—22no—
219C8×C53—42no—
220#88322—42no—
221#88322—41no—
222#88322—41no—
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×C53C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C53#57511yescharacteristic
247C4×C53#57612yes—
248Dic53—81no—
249Dic53—81no—
250#3208—531no—
251#3110—531no—
252C2×#459—531no—
253#2974D5311yescharacteristic, complemented
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#533—531no—
270C2×#127—531no—
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—532no—
281#459—532no—
282#459—1061no—
283#459—1061no—
284#450—1061no—
285#450—1061no—
286#450—1061no—
287#440—1061no—
288#440—1061no—
289#436C2×D5312yescomplemented
290#435—22no—
291C8×D4C2×D5312yescomplemented
292#432—531no—
293#432C2×D5311yescharacteristic
294#431C2×D5311yescharacteristic
295C2×C4⋊C8—531no—
296C2×#102—1061no—
297#407—21no—
298C2×C22⋊C8—21no—
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×D4D21211yescharacteristic
310C23×C4—21no—
311C2×SD16—1061no—
312C2×SD16—1061no—
313C2×SD16—1062no—
314C2×SD16—1062no—
315C2×D8—1062no—
316C2×D8—1062no—
317C2×D8—2124no—
318C2×M16—22no—
319C22×C8—22no—
320C22×C8—1061no—
321C4⋊Q8—531no—
322C4⋊Q8—531no—
323#127—532no—
324#127—1061no—
325#127—1061no—
326#127—1061no—
327#125—1061no—
328C22⋊Q8—1061no—
329C22⋊Q8—1061no—
330C4×Q8—1061no—
331C4×Q8—1061no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4D21211yescharacteristic
335C4×D4D21211yescharacteristic
336C4×D4C22×D5311yescharacteristic
337C4×D4C22×D5311yescharacteristic
338#117—1061no—
339C2×C4⋊C4D21211yescharacteristic
340C2×C22⋊C4—21no—
341C2×C42C22×D5311yescharacteristic
342#107—1061no—
343#107—1061no—
344C4⋊C8—531no—
345C4⋊C8—531no—
346C4⋊C8—532no—
347C4⋊C8—1061no—
348C4⋊C8C22⋊D5312yes—
349C4⋊C8C22×D5312yes—
350#103—1061no—
351#103—1061no—
352#102—1061no—
353#102—1061no—
354#102—1062no—
355#102—1062no—
356#102—1062no—
357#102—1062no—
358C22⋊C8—21no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—1061no—
362C22⋊C8C22⋊D5311yescharacteristic
363C22⋊C8C22⋊D5311yescharacteristic
364#97—21no—
365#97—1061no—
366C4×C8C22×D5312yes—
367C24—21no—
368C24—2121no—
369C2×Q8—1061no—
370C2×Q8—1061no—
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×D21211yescharacteristic
385C2×D4C2×D21211yescharacteristic
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—82no—
392C22×C4C2×D21211yescharacteristic
393C22×C4#1171994511yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395SD16—4242no—
396SD16—4242no—
397SD16—2122no—
398SD16—2122no—
399D8—2124no—
400D8—2124no—
401M16—82no—
402C2×C8—22no—
403C2×C8—82no—
404C2×C8—1061no—
405C2×C8—1061no—
406C2×C8—1062no—
407C2×C8—2122no—
408C2×C8C2×C22⋊D5312yes—
409C2×C8C2×C22⋊D5312yes—
410C4⋊C4—21no—
411C4⋊C4—1061no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1061no—
415C4⋊C4—2121no—
416C4⋊C4—2121no—
417C4⋊C4C2×D21211yescharacteristic
418C4⋊C4C2×D21211yescharacteristic
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—2121no—
424C42—2121no—
425C42C2×D21211yescharacteristic
426C42#1171994511yescharacteristic
427C42C2×C22⋊D5311yescharacteristic
428C42C23×D5311yescharacteristic
429C23—21no—
430C23—21no—
431C23—42no—
432C23—42no—
433C23—81no—
434C23—4242no—
435C23—4242no—
436C23—4242no—
437C23—2122no—
438C23#1313319111yescharacteristic
439Q8—2121no—
440Q8—2121no—
441Q8—2122no—
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#1313324512yes—
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#1313319111yescharacteristic
469C2×C4#1313319111yescharacteristic
470C2×C4#1313319111yescharacteristic
471C2×C4C22×C22⋊D5311yescharacteristic
472C2×C4C22×D21211yescharacteristic
473C2×C4C2×#1171994511yescharacteristic
474C8—22no—
475C8—22no—
476C8—42no—
477C8—2122no—
478C8—2121no—
479C8—2121no—
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#1681088612yes—
494C22C2×#1313319111yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—41no—
500C4—41no—
501C4—4241no—
502C4—4241no—
503C4C2×#1313324511yescharacteristic
504C4#1681142311yescharacteristic
505C4#1681142612yes—
506C2—22no—
507C2—42no—
508C2—81no—
509C2—8482no—
510C2#2028421811yescharacteristic
511C2#2028433411yescharacteristic
512C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2974C2
socleC22×C53C2×#121
prime radicalC2×#16810818C2
hypercenter#2974D53
hypocenterC53#19456
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. #20285658C2semidirect
2. #20284814C2semidirect
4. #20274276C2semidirect
5. #20274180C2semidirect
6. #20274152C2semidirect
7. #20274056C2semidirect
8. #20273500C2semidirect
9. #20273428C2semidirect
10. #20273416C2semidirect
11. C53×#2974C2semidirect
15. #16810996C22semidirect
16. #16810970C22semidirect
18. #16810903C22semidirect
19. #16810903C22semidirect
20. #16810828C22semidirect
21. #16810828C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810370C22semidirect
38. #16810368C22semidirect
39. #16810366C22semidirect
40. #16810366C22semidirect
42. C53×#436C22semidirect
44. C8×D4×C53C22semidirect
253. #2974D53semidirect
289. #436C2×D53semidirect
291. C8×D4C2×D53semidirect
305. C53#19456semidirect

Maximal subgroups

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

subgroupindex
#202856582
#202848142
C2×#168108182
#202742762
#202741802
#202741522
#202740562
#202735002
#202734282
#202734162
C53×#29742
#1945653