Finite Group Databasebeta

← #27541929

Subgroups — #27541929

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287704C211yesmaximal, characteristic, complemented
2#20276962C211yesmaximal, characteristic, complemented
3#20276954C211yesmaximal, characteristic, complemented
4#20276949C211yesmaximal, characteristic, complemented
5#20276876C211yesmaximal, characteristic, complemented
6#20276869C211yesmaximal, characteristic, complemented
7#20276854C211yesmaximal, characteristic, complemented
8#20276770C211yesmaximal, characteristic
9#20276770C211yesmaximal, characteristic
10#20276760C211yesmaximal, characteristic, complemented
11#20276624C211yesmaximal, characteristic, complemented
12#20276589C211yesmaximal, characteristic
13#20276579C211yesmaximal, characteristic, complemented
14C4×#13133161C211yesmaximal, characteristic, complemented
15C53×#3380C211yesmaximal, characteristic, complemented
16#16811346—21no—
17C2×#13133161C2211yescharacteristic
18C2×#13133161C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20#16811007C2211yescharacteristic
21#16811007C2211yescharacteristic
22#16811003C2211yescharacteristic
23#16810846—21no—
24#16810846—21no—
25#16810829—21no—
26C4×Dic212C2211yescharacteristic
27C4×Dic212C2211yescharacteristic
28C4×D424C2211yescharacteristic
29#16810450C2211yescharacteristic
30#16810450C2211yescharacteristic
31#16810448C2211yescharacteristic
32C4⋊#11719908C2211yescharacteristic
33C4⋊#11719908C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810426C2211yescharacteristic
36#16810426C2211yescharacteristic
37#16810426C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45#16810399C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#554—21no—
49C53×#500C2211yescharacteristic
50C53×#499C2211yescharacteristic
51C53×#499C2211yescharacteristic
52C53×#478—21no—
53C53×#478—21no—
54C53×#476—21no—
55C8×Q8×C53C2211yescharacteristic
56C4×D8×C53C2211yescharacteristic
57C4×D8×C53C2211yescharacteristic
58C4×D8×C53C2211yescharacteristic
59C2×Dic212—21no—
60C2×Dic212C2311yescharacteristic
61C2×Dic212C2311yescharacteristic
62C2×#11719913—21no—
63C2×#11719913—21no—
64C2×D424C2311yescharacteristic
65#13133217—21no—
66#13133212—41no—
67#13133212—41no—
68#13133205—21no—
69#13133205—21no—
70#13133193—41no—
71C4×Dic106—21no—
72C4×Dic106—21no—
73C4×D212—21no—
74#13133175—41no—
75#13133161—22no—
76#13133161—22no—
77#13133158—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133142—21no—
84#13133142—21no—
85#13133139—21no—
86C2×D8×C53C2311yescharacteristic
87C4⋊Q8×C53—21no—
88#125×C53—21no—
89#125×C53—21no—
90C22⋊Q8×C53—41no—
91C22⋊Q8×C53—41no—
92C22⋊Q8×C53—41no—
93C4×Q8×C53D411yescharacteristic
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C2×C4⋊C4×C53—41no—
98#107×C53—21no—
99#107×C53C2311yescharacteristic
100#107×C53C2311yescharacteristic
101#107×C53C2311yescharacteristic
102#106×C53—21no—
103#106×C53—21no—
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106C4⋊C8×C53D411yescharacteristic
107#102×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110C4×C8×C53C2311yescharacteristic
111C4×C8×C53C2311yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C2×Q8×C53—41no—
114C2×Q8×C53D811yescharacteristic
115C2×D4×C53—21no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C2×D212—21no—
120C2×C4×D53—41no—
121C2×Dic106—21no—
122C2×Dic106—21no—
123D8×C53—21no—
124D8×C53#4112yes—
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53C2×D411yescharacteristic
128C2×C8×C53C2411yescharacteristic
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53D811yescharacteristic
138C4⋊C4×C53D811yescharacteristic
139C4⋊C4×C53D811yescharacteristic
140C22⋊C4×C53—41no—
141C22⋊C4×C53—41no—
142C22⋊C4×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146#11719921—41no—
147#11719921—41no—
148#11719921—41no—
149#11719921—41no—
150#11719921—41no—
151C4⋊Dic53—21no—
152C4⋊Dic53—41no—
153C4⋊Dic53—41no—
154#11719919—41no—
155#11719919—41no—
156C4×Dic53—41no—
157C4×Dic53—41no—
158Dic212—21no—
159Dic212—21no—
160Dic212—21no—
161Dic212—21no—
162Dic212—81no—
163D424—21no—
164D424—21no—
165#11719913—81no—
166#11719913—81no—
167#11719908—21no—
168#11719908—21no—
169#11719908—21no—
170#11719908—21no—
171C23×C53—41no—
172C22×D53—41no—
173Q8×C53—41no—
174Q8×C53—161no—
175D4×C53—22no—
176D4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—82no—
181C2×C4×C53—161no—
182C2×C4×C53—161no—
183C2×C4×C53C2×D811yescharacteristic
184C2×C4×C53C2×D811yescharacteristic
185C2×C4×C53C2×D811yescharacteristic
186C2×C4×C53C2×D811yescharacteristic
187C2×C4×C53C2×D811yescharacteristic
188C2×C4×C53C2×D811yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193D212—41no—
194D212—41no—
195C4×D53—82no—
196Dic106—41no—
197Dic106—41no—
198Dic106—41no—
199Dic106—41no—
200C8×C53—21no—
201C8×C53—21no—
202C8×C53—21no—
203C8×C53C2×#4111yescharacteristic
204C8×C532−1+411yescharacteristic
205#33943—531nomaximal, Sylow
206C22×C53—42no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53C22×D811yescharacteristic
210C2×D53—81no—
211C2×D53—81no—
212C2×D53—81no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—81no—
220C4×C53—81no—
221C4×C53—81no—
222C4×C53#54811yescharacteristic
223C4×C53#57711yescharacteristic
224Dic53—81no—
225Dic53—81no—
226Dic53—81no—
227#3393—531no—
228#3380D5311yescharacteristic, complemented
229#2245—531no—
230#2243—531no—
231#2242—531no—
232#2229—531no—
233#2228—531no—
234#2225—531no—
235#2210—531no—
236#2210—531no—
237#2208—531no—
238#2183—531no—
239#2177—531no—
240#2175—531no—
241C4×SD32—531no—
242C2×C53—82no—
243C2×C53#335811yescharacteristic
244C2×C53#341411yescharacteristic
245C2×C53#341611yescharacteristic
246D53—161no—
247#556—1061no—
248#554—21no—
249C2×SD32—531no—
250C2×SD32—531no—
251C2×D16—531no—
252#500C2×D5311yescharacteristic
253#499C2×D5311yescharacteristic
254#499C2×D5311yescharacteristic
255#495—531no—
256#495—531no—
257#494—531no—
258#478—21no—
259#478—21no—
260#476—21no—
261#464—1061no—
262#464—1061no—
263#461—1061no—
264C8×Q8C2×D5311yescharacteristic
265C4×Q16—531no—
266C4×Q16—531no—
267C4×D8—531no—
268C4×D8C2×D5311yescharacteristic
269C4×D8C2×D5311yescharacteristic
270C4×D8C2×D5311yescharacteristic
271#367—531no—
272#367—531no—
273#366—531no—
274C4⋊C16—531no—
275C4⋊C16—531no—
276C4⋊C16—531no—
277#358—531no—
278#358—531no—
279#358—531no—
280#358—531no—
281#358—531no—
282#358—531no—
283#357—531no—
284#357—531no—
285#357—531no—
286#357—531no—
287#346—531no—
288#346—531no—
289C4×C16—531no—
290C53#3394311yescharacteristic, complemented, Sylow
291C2×Q16—531no—
292C2×Q16—531no—
293C2×Q16—1061no—
294C2×SD16—1061no—
295C2×SD16—1061no—
296C2×D8—531no—
297C2×D8C22×D5311yescharacteristic
298C4⋊Q8—21no—
299C4⋊Q8—1061no—
300#126—2121no—
301#126—2121no—
302#125—21no—
303#125—21no—
304#124—1061no—
305#124—1061no—
306C22⋊Q8—41no—
307C22⋊Q8—41no—
308C22⋊Q8—41no—
309C22⋊Q8—2121no—
310C4×Q8—1061no—
311C4×Q8—1061no—
312C4×Q8C22⋊D5311yescharacteristic
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—1061no—
317#117—2121no—
318C2×C4⋊C4—41no—
319SD32—1062no—
320SD32—1062no—
321D16—1062no—
322C2×C16—531no—
323C2×C16—531no—
324C2×C16—531no—
325C2×C16—531no—
326#107—21no—
327#107—531no—
328#107C22×D5311yescharacteristic
329#107C22×D5311yescharacteristic
330#107C22×D5311yescharacteristic
331#106—21no—
332#106—21no—
333C4⋊C8C22⋊D5311yescharacteristic
334C4⋊C8C22⋊D5311yescharacteristic
335C4⋊C8C22⋊D5311yescharacteristic
336#103—1061no—
337#103—1061no—
338#102—21no—
339#102—21no—
340#102—21no—
341#102—1061no—
342C4×C8C22×D5311yescharacteristic
343C4×C8C22×D5311yescharacteristic
344C4×C8C22×D5311yescharacteristic
345C2×Q8—41no—
346C2×Q8—1061no—
347C2×Q8—1061no—
348C2×Q8#1171992211yescharacteristic
349C2×D4—21no—
350C2×D4—1061no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—2121no—
355Q16—1061no—
356Q16—1061no—
357Q16—1061no—
358Q16—1061no—
359Q16—4241no—
360SD16—4241no—
361SD16—4241no—
362D8—21no—
363D8—1061no—
364D8—1061no—
365D8#1171994912yes—
366C2×C8C2×C22⋊D5311yescharacteristic
367C2×C8C2×C22⋊D5311yescharacteristic
368C2×C8C2×C22⋊D5311yescharacteristic
369C2×C8C23×D5311yescharacteristic
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—1061no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C4⋊C4—2121no—
382C4⋊C4—2121no—
383C4⋊C4#1171992211yescharacteristic
384C4⋊C4#1171992211yescharacteristic
385C4⋊C4#1171992211yescharacteristic
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C22⋊C4—2121no—
392C22⋊C4—2121no—
393C22⋊C4—2121no—
394C42—2121no—
395C42—2121no—
396C42C2×C22⋊D5311yescharacteristic
397C42C2×C22⋊D5311yescharacteristic
398C42C2×C22⋊D5311yescharacteristic
399C16—1061no—
400C16—1061no—
401C16—1061no—
402C16—1061no—
403C23—41no—
404C23—2121no—
405Q8—41no—
406Q8—161no—
407Q8—2121no—
408Q8—2121no—
409Q8—2121no—
410Q8—2121no—
411D4—22no—
412D4—41no—
413D4—2121no—
414D4—2121no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—82no—
419C2×C4—161no—
420C2×C4—161no—
421C2×C4—4242no—
422C2×C4—2121no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4C2×#1171992211yescharacteristic
426C2×C4C2×#1171992211yescharacteristic
427C2×C4C2×#1171992211yescharacteristic
428C2×C4C2×#1171992211yescharacteristic
429C2×C4C2×#1171992211yescharacteristic
430C2×C4C2×#1171992211yescharacteristic
431C2×C4C22×C22⋊D5311yescharacteristic
432C8—21no—
433C8—21no—
434C8—21no—
435C8C2×#1171994911yescharacteristic
436C8#1313326511yescharacteristic
437C22—42no—
438C22—81no—
439C22—81no—
440C22—4241no—
441C22—4241no—
442C22—4241no—
443C22C22×#1171992211yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—4241no—
454C4—4241no—
455C4—4241no—
456C4#1681128411yescharacteristic
457C4#1681143111yescharacteristic
458C2—82no—
459C2—8481no—
460C2#2028735011yescharacteristic
461C2#2028784911yescharacteristic
462C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3380C2
socleC22×C53C22×D8
prime radicalC2×#13133158C22
hypercenter#3380D53
hypocenterC53#33943
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #20287704C2semidirect
2. #20276962C2semidirect
3. #20276954C2semidirect
4. #20276949C2semidirect
5. #20276876C2semidirect
6. #20276869C2semidirect
7. #20276854C2semidirect
10. #20276760C2semidirect
11. #20276624C2semidirect
13. #20276579C2semidirect
14. C4×#13133161C2semidirect
15. C53×#3380C2semidirect
228. #3380D53semidirect
290. C53#33943semidirect

Maximal subgroups

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

subgroupindex
#202877042
#202769622
#202769542
#202769492
#202768762
#202768692
#202768542
#202767702
#202767602
#202766242
#202765892
#202765792
C4×#131331612
C53×#33802
#3394353