Finite Group Databasebeta

← #27368230

Subgroups — #27368230

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285143C211yesmaximal, characteristic, complemented
2#20284866C212yesmaximal, complemented
3#20274209C211yesmaximal, characteristic, complemented
4#20274204C211yesmaximal, characteristic, complemented
5#20274199C211yesmaximal, characteristic, complemented
6#20274194C211yesmaximal, characteristic, complemented
7#20274187C212yesmaximal, complemented
8#20274180C212yesmaximal, complemented
9#20273478C211yesmaximal, characteristic, complemented
10#20273418C212yesmaximal, complemented
11C53×#2965C211yesmaximal, characteristic, complemented
12C2×#13133239—21no—
13C2×#13133239—21no—
14#16811173C2212yes—
15#16811007—21no—
16#16811007—21no—
17#16811003C2211yescharacteristic
18#16811000C2211yescharacteristic
19#16810996C2212yes—
20#16810947—22no—
21#16810908C2212yes—
22#16810903C2212yes—
23#16810823C2212yes—
24#16810818C2212yes—
25#16810768—22no—
26#16810733—22no—
27#16810721—22no—
28#16810680C2212yes—
29#16810376C2212yes—
30#16810374C2211yescharacteristic
31#16810374C2211yescharacteristic
32#16810370C2212yes—
33#16810370C2212yes—
34#16810366C2212yes—
35#16810366C2212yes—
36C4×#41×C53C2211yescharacteristic
37C53×#446C2211yescharacteristic
38C53×#446C2211yescharacteristic
39C53×#436C2211yescharacteristic
40C53×#436C2211yescharacteristic
41C53×#435—21no—
42C53×#435—21no—
43C53×#431—21no—
44C53×#431C2211yescharacteristic
45C4×M16×C53—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×#11719945—22no—
48#13133239—44no—
49#13133239—44no—
50C2×Dic212—21no—
51C2×Dic212—21no—
52C2×#11719913—22no—
53C2×#11719913—22no—
54C2×#11719924—22no—
55C2×D424—21no—
56C2×D424—21no—
57C2×#11719922—22no—
58#13133217C2312yes—
59#13133214C2312yes—
60#13133205—22no—
61#13133191—22no—
62#13133191—22no—
63C4×D212—22no—
64C4×D212—22no—
65#13133150—22no—
66#13133148—22no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133141—22no—
70#13133138—22no—
71#13133137—22no—
72#13133137—22no—
73C22⋊#88322—22no—
74#13133128—22no—
75C2×#41×C53D411yescharacteristic
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C22×C8×C53D411yescharacteristic
79C22×C8×C53D411yescharacteristic
80C4×Q8×C53D411yescharacteristic
81C4×Q8×C53C2311yescharacteristic
82C4×D4×C53—21no—
83C4×D4×C53—21no—
84C4×D4×C53D411yescharacteristic
85C4×D4×C53C2311yescharacteristic
86#117×C53—21no—
87#117×C53D411yescharacteristic
88C2×C42×C53—21no—
89C2×C42×C53C2311yescharacteristic
90C4⋊C8×C53—21no—
91C4⋊C8×C53—21no—
92C4⋊C8×C53C2311yescharacteristic
93C4⋊C8×C53C2311yescharacteristic
94C22⋊C8×C53—21no—
95C22⋊C8×C53—21no—
96C22⋊C8×C53—21no—
97#97×C53—21no—
98#97×C53—21no—
99C4×C8×C53—21no—
100C4×C8×C53D411yescharacteristic
101C4×C8×C53D411yescharacteristic
102C4×C8×C53C2311yescharacteristic
103C4×C8×C53C2311yescharacteristic
104#41×C53—42no—
105C2×Q8×C53C2×D411yescharacteristic
106C2×D4×C53—21no—
107C2×D4×C53C2×D411yescharacteristic
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C2×C22⋊D53—42no—
115#11719945—44no—
116#11719945—44no—
117C2×D212—22no—
118C2×D212—22no—
119C2×D212—22no—
120C2×C4×D53—42no—
121C2×Dic106—22no—
122M16×C53—42no—
123M16×C53—42no—
124C2×C8×C53—22no—
125C2×C8×C53—22no—
126C2×C8×C53—21no—
127C2×C8×C53—21no—
128C2×C8×C53C2×D411yescharacteristic
129C2×C8×C53C2×D411yescharacteristic
130C2×C8×C53C2×D411yescharacteristic
131C2×C8×C53C2×D411yescharacteristic
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53C2×D411yescharacteristic
135C4⋊C4×C53C2×D411yescharacteristic
136C22⋊C4×C53—21no—
137C22⋊C4×C53—21no—
138C22⋊C4×C53—21no—
139C42×C53—21no—
140C42×C53—21no—
141C42×C53—21no—
142C42×C53C2×D411yescharacteristic
143C42×C53C2×D411yescharacteristic
144C42×C53#4111yescharacteristic
145C42×C53C2411yescharacteristic
146#11719924—44no—
147#11719922—44no—
148#11719921—42no—
149#11719921—42no—
150C4⋊Dic53—22no—
151C4⋊Dic53—22no—
152C2×#88322—22no—
153C2×#88322—22no—
154Dic212—42no—
155Dic212—42no—
156D424—42no—
157D424—42no—
158#11719913—44no—
159#11719913—44no—
160C23×C53—21no—
161C23×C53#12111yescharacteristic
162C22×D53—42no—
163Q8×C53—22no—
164D4×C53—22no—
165D4×C53—42no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—22no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C2×C4×C53C2×#4111yescharacteristic
183C22⋊D53—84no—
184C2×Dic53—42no—
185D212—42no—
186D212—42no—
187D212—42no—
188D212—42no—
189D212—82no—
190C4×D53—84no—
191Dic106—42no—
192Dic106—42no—
193C8×C53—21no—
194C8×C53—21no—
195C8×C53—41no—
196C8×C53—41no—
197C8×C53AGL(1,ℤ/8)12yes—
198C8×C53AGL(1,ℤ/8)12yes—
199#88322—42no—
200#88322—42no—
201#19294—531nomaximal, Sylow
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53C2×#12111yescharacteristic
209C2×D53—82no—
210C2×D53—82no—
211C2×D53—82no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—41no—
217C4×C53—41no—
218C4×C53—41no—
219C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
220C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
221C4×C53#57512yes—
222Dic53—82no—
223#3147—531no—
224#3115—532no—
225#2965D5311yescharacteristic, complemented
226#1690—531no—
227#1689—531no—
228#1688—531no—
229#1687—531no—
230#1686—532no—
231#1685—532no—
232#1568—531no—
233#1553—532no—
234C2×C53—41no—
235C2×C53—81no—
236C2×C53#303811yescharacteristic
237C2×C53#305411yescharacteristic
238C2×C53#306711yescharacteristic
239D53—162no—
240C2×#135—1061no—
241C2×#135—1061no—
242#532—532no—
243C4×#41C2×D5311yescharacteristic
244#495—1061no—
245#495—1061no—
246#494—531no—
247#493—531no—
248#492—532no—
249#482—1062no—
250#475—532no—
251#474—532no—
252#460—532no—
253#459—532no—
254#449—1062no—
255#446C2×D5311yescharacteristic
256#446C2×D5311yescharacteristic
257#442—1062no—
258#440—1062no—
259#436C2×D5311yescharacteristic
260#436C2×D5311yescharacteristic
261#435—21no—
262#435—21no—
263#432—532no—
264#431—21no—
265#431C2×D5311yescharacteristic
266C4×M16—21no—
267C2×C4×C8C2×D5311yescharacteristic
268#335—532no—
269#334—531no—
270#334—531no—
271#332—532no—
272#332—532no—
273#331—532no—
274#331—532no—
275C53#1929411yescharacteristic, complemented, Sylow
276C2×#41—1062no—
277C2×#41D21211yescharacteristic
278#135—2124no—
279#135—2124no—
280C2×Q16—1061no—
281C2×Q16—1061no—
282C2×SD16—1062no—
283C2×SD16—1062no—
284C2×SD16—1062no—
285C2×D8—1062no—
286C2×D8—1061no—
287C2×D8—1061no—
288C2×M16—21no—
289C2×M16—21no—
290C22×C8C22⋊D5311yescharacteristic
291C22×C8C22⋊D5311yescharacteristic
292C4⋊Q8—532no—
293#127—532no—
294#124—1062no—
295#121—1062no—
296#121—1062no—
297C4×Q8D21211yescharacteristic
298C4×Q8C22×D5311yescharacteristic
299C4×D4—21no—
300C4×D4—21no—
301C4×D4—1062no—
302C4×D4—1062no—
303C4×D4D21211yescharacteristic
304C4×D4C22×D5311yescharacteristic
305#117—21no—
306#117D21211yescharacteristic
307C2×C42—21no—
308C2×C42C22×D5311yescharacteristic
309#107—1062no—
310#106—1062no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—532no—
314C4⋊C8—532no—
315C4⋊C8C22×D5311yescharacteristic
316C4⋊C8C22×D5311yescharacteristic
317#103—1062no—
318#102—1062no—
319#102—1062no—
320#102—1062no—
321C22⋊C8—21no—
322C22⋊C8—21no—
323C22⋊C8—21no—
324C22⋊C8—1062no—
325#97—21no—
326#97—21no—
327#97—1062no—
328C4×C8—21no—
329C4×C8C22⋊D5311yescharacteristic
330C4×C8C22⋊D5311yescharacteristic
331C4×C8C22×D5311yescharacteristic
332C4×C8C22×D5311yescharacteristic
333#41—42no—
334#41—2124no—
335#41—2124no—
336C2×Q8—1062no—
337C2×Q8C2×D21211yescharacteristic
338C2×D4—21no—
339C2×D4—1062no—
340C2×D4—1062no—
341C2×D4—1062no—
342C2×D4—2122no—
343C2×D4C2×D21211yescharacteristic
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—2122no—
348C22×C4C2×D21211yescharacteristic
349C22×C4#1171994511yescharacteristic
350C22×C4C2×C22⋊D5311yescharacteristic
351Q16—2122no—
352Q16—2122no—
353SD16—2124no—
354SD16—2124no—
355SD16—2124no—
356D8—2122no—
357D8—2124no—
358D8—2122no—
359M16—42no—
360M16—42no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—1062no—
366C2×C8—1062no—
367C2×C8C2×C22⋊D5311yescharacteristic
368C2×C8C2×C22⋊D5311yescharacteristic
369C2×C8C2×C22⋊D5311yescharacteristic
370C2×C8C2×C22⋊D5311yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—1062no—
374C4⋊C4—1062no—
375C4⋊C4C2×D21211yescharacteristic
376C4⋊C4C2×D21211yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—2122no—
381C22⋊C4—2122no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42C2×D21211yescharacteristic
386C42#1171994511yescharacteristic
387C42C2×C22⋊D5311yescharacteristic
388C42C23×D5311yescharacteristic
389C23—21no—
390C23—2122no—
391C23#1313319111yescharacteristic
392Q8—22no—
393Q8—2122no—
394Q8—2122no—
395D4—22no—
396D4—42no—
397D4—4242no—
398D4—4244no—
399D4—2122no—
400D4—2122no—
401D4—2122no—
402D4—2122no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—4244no—
415C2×C4—2122no—
416C2×C4#1313319111yescharacteristic
417C2×C4#1313319111yescharacteristic
418C2×C4#1313319111yescharacteristic
419C2×C4C22×C22⋊D5311yescharacteristic
420C2×C4C22×D21211yescharacteristic
421C2×C4C2×#1171994511yescharacteristic
422C8—21no—
423C8—21no—
424C8—41no—
425C8—41no—
426C8—2122no—
427C8—2122no—
428C8#1313324212yes—
429C8#1313324212yes—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—4242no—
437C22—4242no—
438C22—4242no—
439C22C2×#1313319111yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—4242no—
448C4C2×#1313324211yescharacteristic
449C4C2×#1313324211yescharacteristic
450C4#1681142312yes—
451C2—41no—
452C2—81no—
453C2—8482no—
454C2#2028419511yescharacteristic
455C2#2028432411yescharacteristic
456C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2965C2
socleC22×C53C2×#121
hypercenter#2965D53
hypocenterC53#19294
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #20285143C2semidirect
2. #20284866C2semidirect
3. #20274209C2semidirect
4. #20274204C2semidirect
5. #20274199C2semidirect
6. #20274194C2semidirect
7. #20274187C2semidirect
8. #20274180C2semidirect
9. #20273478C2semidirect
10. #20273418C2semidirect
11. C53×#2965C2semidirect
225. #2965D53semidirect
275. C53#19294semidirect

Maximal subgroups

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

subgroupindex
#202851432
#202848662
#202742092
#202742042
#202741992
#202741942
#202741872
#202741802
#202734782
#202734182
C53×#29652
#1929453