Finite Group Databasebeta

← #28314194

Subgroups — #28314194

Order 13568.

984 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291566C211yesmaximal, characteristic, complemented
2#20291562C211yesmaximal, characteristic, complemented
3#20289999C212yesmaximal, complemented
4#20289994C212yesmaximal, complemented
5#20289987C212yesmaximal, complemented
6#20289982C212yesmaximal, complemented
7#20289953C212yesmaximal, complemented
8C2×#16811468C212yesmaximal, complemented
9C2×#16811217C211yesmaximal, characteristic
10#16811654—24no—
11D4×#88324—24no—
12#16811643C2212yes—
13#16811639C2212yes—
14#16811593C2212yes—
15C2×C22⋊#88324C2212yes—
16C2×C22⋊#88324C2212yes—
17C2×C22⋊#88324C2212yes—
18#16811469C2214yes—
19#16811468C2214yes—
20#16811468C2214yes—
21#16811467C2214yes—
22#16811217—24no—
23#16811217—24no—
24C2×#13133212C2211yescharacteristic
25C2×#13133210C411yescharacteristic
26C2×#13133208C411yescharacteristic
27C2×#13133207C2211yescharacteristic
28C2×#13133201C2211yescharacteristic
29C2×#13133200C411yescharacteristic
30C2×#13133199C411yescharacteristic
31C2×#13133197C411yescharacteristic
32C2×#13133194C411yescharacteristic
33C2×#13133194C2211yescharacteristic
34C2×#13133189C411yescharacteristic
35C2×D4×Dic53C2211yescharacteristic
36C2×#13133177C2211yescharacteristic
37C22×C4⋊Dic53C2211yescharacteristic
38C2×#123×C53C411yescharacteristic
39C2×#11719940—22no—
40C22×#88324—22no—
41C22×#88324—22no—
42C22×#88324—22no—
43#13133284—22no—
44C4⋊#88324—22no—
45C22⋊#88324—22no—
46C22⋊#88324—22no—
47C22⋊#88324—22no—
48C22⋊#88324—22no—
49C22⋊#88324C2314yes—
50C22⋊#88324C2314yes—
51#13133272—22no—
52C4×#88324—22no—
53C22×Dic106C2×C411yescharacteristic
54C22×C22⋊D53C2×C411yescharacteristic
55C22×C22⋊D53C2×C411yescharacteristic
56C23×Dic53C2×C411yescharacteristic
57C23×Dic53C2311yescharacteristic
58C23×Dic53C2311yescharacteristic
59C22×C4×D53C2×C411yescharacteristic
60#13133212—22no—
61#13133212—22no—
62#13133210—24no—
63#13133208—24no—
64#13133207—22no—
65#13133207—22no—
66#13133201—22no—
67#13133201—22no—
68#13133200—24no—
69#13133199—24no—
70#13133197—24no—
71#13133194—24no—
72#13133194—22no—
73#13133194—22no—
74#13133189—24no—
75D4×Dic53D414yes—
76D4×Dic53D414yes—
77#13133177—22no—
78#13133177—22no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53—22no—
81C2×C4⋊Dic53C2×C411yescharacteristic
82C2×C4⋊Dic53C2×C411yescharacteristic
83C2×C4⋊Dic53C2×C411yescharacteristic
84C2×C4⋊Dic53C2×C411yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×#11719919C2×C411yescharacteristic
88C2×#11719919C2×C411yescharacteristic
89C2×#11719919C2×C411yescharacteristic
90C2×#11719919C2×C411yescharacteristic
91C2×#11719919C2×C411yescharacteristic
92C2×#11719919C2311yescharacteristic
93C2×C22⋊Dic53C2×C411yescharacteristic
94C2×C22⋊Dic53C2×C411yescharacteristic
95C2×C22⋊Dic53C2×C411yescharacteristic
96C2×#11719921C2×C411yescharacteristic
97C2×#11719921C2×C411yescharacteristic
98C2×#11719921C2×C411yescharacteristic
99C2×#11719921C2311yescharacteristic
100C2×C4×Dic53C2×C411yescharacteristic
101C2×C4×Dic53C2311yescharacteristic
102C22×D4×C53C2×C411yescharacteristic
103C23×C4×C53C2×C411yescharacteristic
104#123×C53—24no—
105C2×C4⋊C4×C53C2×C411yescharacteristic
106C2×C4⋊C4×C53C2×C411yescharacteristic
107C2×C22⋊C4×C53C2×C411yescharacteristic
108C2×C22⋊C4×C53C2×C411yescharacteristic
109C2×C22⋊C4×C53C2×C411yescharacteristic
110C24×C53C22×C411yescharacteristic
111C24×C53C22×C411yescharacteristic
112C23×D53C22×C411yescharacteristic
113C2×D4×C53—24no—
114C2×D4×C53C22⋊C414yes—
115C22×C4×C53—22no—
116C22×C4×C53—24no—
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C22×C4×C53C22×C411yescharacteristic
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C411yescharacteristic
122C2×C22⋊D53—22no—
123C2×C22⋊D53—24no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—81no—
126C22×Dic53—22no—
127C22×Dic53—81no—
128C22×Dic53—81no—
129C22×Dic53—81no—
130C22×Dic53C22×C411yescharacteristic
131C22×Dic53C22×C411yescharacteristic
132C22×Dic53C22×C411yescharacteristic
133C22×Dic53C22×C411yescharacteristic
134C22×Dic53C22×C411yescharacteristic
135C22×Dic53C22×C411yescharacteristic
136C22×Dic53C2×D412yes—
137C22×Dic53C2×D412yes—
138C22×Dic53C2×D412yes—
139C22×Dic53C2×D412yes—
140C22×Dic53C2411yescharacteristic
141C2×C4×D53—22no—
142C2×C4×D53—81no—
143C2×Dic106—22no—
144C2×Dic106—81no—
145#11719940—82no—
146C2×#88324—24no—
147C2×#88324—24no—
148C2×#88324—82no—
149C2×#88324—82no—
150C2×#88324—82no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C22⋊Dic53—22no—
157C22⋊Dic53C22⋊C414yes—
158C22⋊Dic53C22⋊C414yes—
159#11719921—22no—
160#11719921—22no—
161#11719921—22no—
162#11719921—21no—
163#11719921—21no—
164C4⋊Dic53—22no—
165C4⋊Dic53—22no—
166C4⋊Dic53—22no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53C22⋊C414yes—
172#11719919—22no—
173#11719919—22no—
174#11719919—22no—
175#11719919—22no—
176#11719919—22no—
177#11719919—21no—
178#11719919—21no—
179C4×Dic53—22no—
180C4×Dic53C2×D412yes—
181C4×Dic53C2×D412yes—
182C23×C53—22no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53C2×C22⋊C412yes—
188C23×C53C2×C22⋊C412yes—
189C23×C53#13114yes—
190C23×C53C23×C411yescharacteristic
191C22×D53—21no—
192C22×D53—22no—
193C22×D53—41no—
194C22×D53—41no—
195D4×C53—28no—
196C2×C4×C53—22no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—44no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53C2×C22⋊C412yes—
207C22⋊D53—44no—
208C22⋊D53—82no—
209C2×Dic53—21no—
210C2×Dic53—21no—
211C2×Dic53—21no—
212C2×Dic53—22no—
213C2×Dic53—42no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—41no—
223C2×Dic53—82no—
224C2×Dic53C2×C22⋊C412yes—
225C2×Dic53C2×C22⋊C412yes—
226C2×Dic53C2×C22⋊C412yes—
227C2×Dic53C22×D411yescharacteristic
228C2×Dic53C22×D411yescharacteristic
229C4×D53—82no—
230Dic106—82no—
231#88324—44no—
232#88324—44no—
233#17387—531nomaximal, Sylow
234C22×C53—21no—
235C22×C53—22no—
236C22×C53—21no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—44no—
242C22×C53—82no—
243C22×C53#40814yes—
244C22×C53C22×C22⋊C411yescharacteristic
245C22×C53C2×#13112yes—
246C2×D53—41no—
247C2×D53—41no—
248C2×D53—41no—
249C2×D53—82no—
250C4×C53—22no—
251C4×C53—41no—
252C4×C53—41no—
253C4×C53—41no—
254C4×C53—41no—
255Dic53—21no—
256Dic53—21no—
257Dic53—22no—
258Dic53—22no—
259Dic53—22no—
260Dic53—41no—
261Dic53—41no—
262Dic53—41no—
263C2×#539—531no—
264#2973—531no—
265#2972—531no—
266#1473—532no—
267#1472—532no—
268#1470—532no—
269#1468—532no—
270#1459—532no—
271C2×#324—532no—
272C2×C53—22no—
273C2×C53—24no—
274C2×C53—42no—
275C2×C53C2×#40811yescharacteristic
276C2×C53#288311yescharacteristic
277C2×C53#288911yescharacteristic
278D53—81no—
279#539—1064no—
280#539—1064no—
281C2×#126—531no—
282C2×#126—531no—
283C2×#125—531no—
284C2×#125—531no—
285C2×#123—531no—
286C2×#123—531no—
287C2×#123—531no—
288C2×#123#615711yescharacteristic
289C2×C22⋊Q8—531no—
290C2×C22⋊Q8—531no—
291C2×C22⋊Q8—531no—
292C2×#121—531no—
293C2×C4×D4—531no—
294C2×C4×D4—531no—
295C22×C4⋊C4—531no—
296#435—1064no—
297C8×D4—1064no—
298#432—532no—
299#431—532no—
300#407—532no—
301C2×C22⋊C8—532no—
302C2×C22⋊C8—532no—
303C2×C22⋊C8—532no—
304#324—534no—
305#324—534no—
306#324—534no—
307#323—534no—
308C53#1738711yescharacteristic, complemented, Sylow
309C22×Q8—531no—
310C22×D4—531no—
311C22×D4—531no—
312C22×D4C2×#615711yescharacteristic
313C23×C4—531no—
314C23×C4—531no—
315C23×C4—531no—
316C23×C4—531no—
317C23×C4C2×#615711yescharacteristic
318C2×M16—1062no—
319C22×C8—1062no—
320C22×C8—1062no—
321C22×C8—1062no—
322#126—1062no—
323#126—1062no—
324#126—1064no—
325#125—1062no—
326#125—1062no—
327#125—1064no—
328#123—24no—
329#123—1064no—
330#123—1062no—
331#123—1062no—
332#123—1064no—
333C22⋊Q8—1064no—
334C22⋊Q8—1062no—
335C22⋊Q8—1062no—
336C22⋊Q8—1064no—
337#121—1064no—
338C4×D4—534no—
339C4×D4—534no—
340C4×D4—1062no—
341C4×D4—1062no—
342C2×C4⋊C4—531no—
343C2×C4⋊C4—531no—
344C2×C4⋊C4—531no—
345C2×C4⋊C4—531no—
346C2×C4⋊C4—531no—
347C2×C4⋊C4—531no—
348C2×C4⋊C4—531no—
349C2×C4⋊C4—531no—
350C2×C4⋊C4—531no—
351C2×C4⋊C4—531no—
352C2×C4⋊C4—531no—
353C2×C4⋊C4—531no—
354C2×C4⋊C4—1062no—
355C2×C4⋊C4—1062no—
356C2×C4⋊C4C2×#615711yescharacteristic
357C2×C4⋊C4C2×#615711yescharacteristic
358C2×C22⋊C4—531no—
359C2×C22⋊C4—531no—
360C2×C22⋊C4—531no—
361C2×C22⋊C4—531no—
362C2×C22⋊C4—531no—
363C2×C22⋊C4—531no—
364C2×C22⋊C4—531no—
365C2×C22⋊C4C2×#615711yescharacteristic
366C2×C22⋊C4C2×#615711yescharacteristic
367C2×C22⋊C4C2×#615711yescharacteristic
368C2×C42—531no—
369C2×C42—531no—
370C4⋊C8—1062no—
371C4⋊C8—1062no—
372C22⋊C8—534no—
373C22⋊C8—534no—
374C22⋊C8—1062no—
375C22⋊C8—1062no—
376C22⋊C8—1062no—
377C22⋊C8—1062no—
378#97—1062no—
379C4×C8—1062no—
380C24—531no—
381C24C22×#615711yescharacteristic
382C24C22×#615711yescharacteristic
383C2×Q8—1062no—
384C2×Q8—4241no—
385C2×D4—24no—
386C2×D4—1062no—
387C2×D4—1062no—
388C2×D4—1064no—
389C2×D4—4241no—
390C2×D4C22⋊#615714yes—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—532no—
394C22×C4—532no—
395C22×C4—532no—
396C22×C4—532no—
397C22×C4—531no—
398C22×C4—531no—
399C22×C4—531no—
400C22×C4—531no—
401C22×C4—531no—
402C22×C4—531no—
403C22×C4—531no—
404C22×C4—1062no—
405C22×C4—1062no—
406C22×C4—4241no—
407C22×C4—4241no—
408C22×C4—4241no—
409C22×C4—4241no—
410C22×C4C22×#615711yescharacteristic
411C22×C4C22×#615711yescharacteristic
412C22×C4C22×#615711yescharacteristic
413C22×C4C22×#615711yescharacteristic
414C22×C4C22×#615711yescharacteristic
415M16—4242no—
416C2×C8—1064no—
417C2×C8—1064no—
418C2×C8—4242no—
419C2×C8—4242no—
420C2×C8—4242no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—534no—
424C4⋊C4—1062no—
425C4⋊C4—1061no—
426C4⋊C4—1061no—
427C4⋊C4—1062no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C4⋊C4—1061no—
432C4⋊C4—1062no—
433C4⋊C4—1062no—
434C4⋊C4—1062no—
435C4⋊C4—1062no—
436C4⋊C4—1062no—
437C4⋊C4—1062no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—534no—
442C22⋊C4—534no—
443C22⋊C4—1061no—
444C22⋊C4—1061no—
445C22⋊C4—1062no—
446C22⋊C4—1062no—
447C22⋊C4—1062no—
448C22⋊C4—1062no—
449C42—532no—
450C42—532no—
451C42—1062no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—1062no—
458C23—1061no—
459C23—2121no—
460C23—2121no—
461C23C2×C22⋊#615712yes—
462C23C2×C22⋊#615712yes—
463C23#1313330414yes—
464C23C23×#615711yescharacteristic
465Q8—4242no—
466D4—28no—
467D4—4242no—
468D4—2124no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—532no—
480C2×C4—532no—
481C2×C4—532no—
482C2×C4—531no—
483C2×C4—531no—
484C2×C4—1061no—
485C2×C4—1061no—
486C2×C4—1061no—
487C2×C4—1062no—
488C2×C4—4242no—
489C2×C4—4242no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2122no—
493C2×C4—2122no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4—2121no—
497C2×C4—2121no—
498C2×C4—2121no—
499C2×C4—2121no—
500C2×C4C2×C22⋊#615712yes—
501C8—2124no—
502C8—2124no—
503C22—21no—
504C22—22no—
505C22—21no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—44no—
511C22—82no—
512C22—4242no—
513C22—2121no—
514C22—2121no—
515C22—2121no—
516C22#1681159614yes—
517C22C22×C22⋊#615711yescharacteristic
518C22C2×#1313330412yes—
519C4—22no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—1061no—
525C4—1061no—
526C4—1062no—
527C4—1062no—
528C4—1062no—
529C4—2121no—
530C4—2121no—
531C4—2121no—
532C2—22no—
533C2—24no—
534C2—42no—
535C2—4241no—
536C2C2×#1681159611yescharacteristic
537C2#2029131511yescharacteristic
538C2#2029132711yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×#123×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133189C4
cosocleC22×Dic53C24
hypercenterC2×#123#6157
hypocenterC53#17387
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #20291566C2semidirect
2. #20291562C2semidirect
3. #20289999C2semidirect
4. #20289994C2semidirect
5. #20289987C2semidirect
6. #20289982C2semidirect
7. #20289953C2semidirect
8. C2×#16811468C2semidirect
308. C53#17387semidirect

Maximal subgroups

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

subgroupindex
#202915662
#202915622
#202899992
#202899942
#202899872
#202899822
#202899532
C2×#168114682
C2×#168112172
#1738753