Finite Group Databasebeta

← #27371980

Subgroups — #27371980

Order 13568.

836 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285866C211yesmaximal, characteristic, complemented
2#20285630C211yesmaximal, characteristic
3#20285270C211yesmaximal, characteristic, complemented
4C2×#16810997C211yesmaximal, characteristic, complemented
5#20285090C211yesmaximal, characteristic, complemented
6#20274369C212yesmaximal, complemented
7#20274355C212yesmaximal, complemented
8#20274318C212yesmaximal, complemented
9#20274306C212yesmaximal, complemented
10M16×#88322C211yesmaximal, characteristic, complemented
11C53×#3141C211yesmaximal, characteristic, complemented
12C22×#11719924—22no—
13#16811180C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16811026C2212yescomplemented
16#16811019C2211yescharacteristic, complemented
17#16810999C2212yes—
18#16810997—22no—
19#16810997C2212yes—
20#16810997C2212yes—
21#16810995C2211yescharacteristic, complemented
22#16810988—22no—
23#16810975C2211yescharacteristic, complemented
24#16810975C2212yescomplemented
25#16810971C2212yes—
26#16810970C2212yes—
27#16810970C2212yes—
28#16810969C2211yescharacteristic, complemented
29#16810958—22no—
30#16810915—22no—
31#16810850—22no—
32#16810818—22no—
33#16810756—22no—
34C4×#11719923—22no—
35#16810676C2211yescharacteristic
36#16810629—22no—
37C2×C4×#88322C2211yescharacteristic
38#16810344C2212yescomplemented
39C8×#88322C2212yescomplemented
40C53×#532C2211yescharacteristic
41C2×C4⋊Q8×C53C2211yescharacteristic
42C53×#489—22no—
43C53×#487C2212yescomplemented
44C53×#486C2212yescomplemented
45C53×#417—22no—
46C4×M16×C53C2211yescharacteristic
47C22×D212—21no—
48C22×D212—21no—
49C22×D212—21no—
50C2×#11719923—22no—
51C2×#11719924—24no—
52C2×#11719924—24no—
53C2×#11719924—48no—
54C2×#11719922—24no—
55C22×#88322D412yes—
56#13133217C2311yescharacteristic
57#13133217C2311yescharacteristic
58#13133214—21no—
59#13133214—21no—
60#13133214—21no—
61#13133214C2312yes—
62#13133209—21no—
63#13133196—21no—
64#13133196—21no—
65C4×Dic106—21no—
66C4×Dic106—21no—
67#13133176—21no—
68#13133150—24no—
69#13133148—22no—
70C4⋊#88322D412yes—
71#13133140—22no—
72#13133140—24no—
73#13133139—24no—
74#13133139—24no—
75#13133138—22no—
76#13133137—24no—
77C22⋊#88322D412yes—
78C4×#88322D412yes—
79C4×#88322C2311yescharacteristic
80C4×#88322C2312yes—
81C4×#88322C2311yescharacteristic
82C2×#41×C53—21no—
83C22×Q8×C53—21no—
84C2×M16×C53D412yes—
85C4⋊Q8×C53—21no—
86C4⋊Q8×C53—21no—
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53C2312yes—
89C4⋊Q8×C53C2311yescharacteristic
90#127×C53C2311yescharacteristic
91#124×C53—21no—
92#121×C53—21no—
93#121×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C42×C53C2311yescharacteristic
99#103×C53—24no—
100#102×C53—24no—
101#97×C53D412yes—
102C4×C8×C53C2312yes—
103C23×D53—41no—
104#41×C53—44no—
105C2×Q8×C53—21no—
106C2×Q8×C53—22no—
107C2×Q8×C53—44no—
108C2×Q8×C53—41no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—41no—
113C22×C4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C2×D212—22no—
119C2×D212—22no—
120C2×D212—22no—
121C2×D212—44no—
122C2×D212—44no—
123C2×D212—41no—
124C2×D212—41no—
125C2×D212—41no—
126C2×D212—82no—
127C2×Dic106—21no—
128C2×Dic106—21no—
129M16×C53—24no—
130C2×C8×C53C2×D414yes—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C22⋊C4×C53—41no—
138C22⋊C4×C53—41no—
139C42×C53C2×D411yescharacteristic
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C42×C53C2411yescharacteristic
143C22⋊Dic53—41no—
144#11719924—48no—
145#11719924—48no—
146#11719923—44no—
147#11719922—84no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152#11719919—41no—
153#11719919—41no—
154C4×Dic53—41no—
155C2×#88322—24no—
156C2×#88322C2×D412yes—
157C2×#88322C2×D414yes—
158C2×#88322C2×D412yes—
159C23×C53—41no—
160C23×C53#12711yescharacteristic
161C22×D53—42no—
162C22×D53—82no—
163C22×D53—82no—
164C22×D53—82no—
165Q8×C53—42no—
166Q8×C53—44no—
167Q8×C53—82no—
168D4×C53—42no—
169D4×C53—42no—
170D4×C53—42no—
171D4×C53—82no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—41no—
175C2×C4×C53—42no—
176C2×C4×C53—82no—
177C2×C4×C53—82no—
178C2×C4×C53#12711yescharacteristic
179C2×C4×C53#12711yescharacteristic
180C2×C4×C53#12711yescharacteristic
181C2×C4×C53#13514yes—
182C2×C4×C53C22×D411yescharacteristic
183C2×C4×C53C22×D411yescharacteristic
184C2×C4×C53C22×D411yescharacteristic
185C2×Dic53—41no—
186C2×Dic53—41no—
187D212—44no—
188D212—44no—
189D212—44no—
190D212—82no—
191D212—82no—
192D212—82no—
193Dic106—42no—
194Dic106—42no—
195C8×C53—24no—
196#88322—22no—
197#88322—24no—
198#88322AGL(1,ℤ/8)14yes—
199#19625—531nomaximal, Sylow
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53#49512yes—
206C22×C53C2×#12711yescharacteristic
207C2×D53—82no—
208C2×D53—82no—
209C2×D53—82no—
210C2×D53—164no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—81no—
214C4×C53—82no—
215C4×C53#49512yes—
216C4×C53C2×#13512yes—
217C4×C53C2×AGL(1,ℤ/8)12yes—
218C4×C53#57712yes—
219Dic53—81no—
220Dic53—81no—
221#3230—531no—
222#3205—531no—
223#3167—531no—
224C2×#492—531no—
225#3141D5311yescharacteristic, complemented
226#3138—531no—
227#1719—532no—
228#1717—532no—
229#1708—532no—
230#1706—532no—
231C8×M16—531no—
232C2×C53—22no—
233C2×C53—161no—
234C2×C53C2×#49511yescharacteristic
235C2×C53#315111yescharacteristic
236C2×C53#315411yescharacteristic
237D53—162no—
238C22×SD16—1062no—
239#533—531no—
240#532C2×D5311yescharacteristic
241C2×C4⋊Q8C2×D5311yescharacteristic
242C2×#127—531no—
243#500—532no—
244#498—531no—
245#493—532no—
246#492—532no—
247#492—532no—
248#492—531no—
249#492—1062no—
250#490—1062no—
251#489—22no—
252#487—531no—
253#487—532no—
254#487C2×D5312yescomplemented
255#486—532no—
256#486—531no—
257#486—532no—
258#486—532no—
259#486C2×D5312yescomplemented
260#484—1062no—
261#477—1062no—
262#465—1062no—
263#459—1062no—
264#447—1062no—
265C4×SD16—1062no—
266#431—531no—
267#418—1062no—
268#417—22no—
269C4×M16C2×D5311yescharacteristic
270C2×C4×C8—531no—
271#322—532no—
272C82—532no—
273C53#1962511yescharacteristic, complemented, Sylow
274C2×#41—21no—
275C22×Q8—21no—
276C22×D4—1061no—
277C22×D4—1061no—
278C22×D4—1061no—
279C2×SD16—1062no—
280C2×SD16—1064no—
281C2×SD16—1064no—
282C2×SD16—2128no—
283C2×D8—1064no—
284C2×M16C22⋊D5312yes—
285C22×C8—532no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—531no—
290C4⋊Q8—531no—
291C4⋊Q8C22×D5312yes—
292C4⋊Q8C22×D5311yescharacteristic
293#127—532no—
294#127—1061no—
295#127—1061no—
296#127—1061no—
297#127C22×D5311yescharacteristic
298#125—1061no—
299#124—21no—
300C22⋊Q8—1061no—
301C22⋊Q8—1061no—
302#121—21no—
303#121—21no—
304C4×Q8—1061no—
305C4×Q8—1061no—
306C4×D4—21no—
307C4×D4—21no—
308#117—1061no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C42C22×D5311yescharacteristic
312#107—1064no—
313#106—1062no—
314C4⋊C8—532no—
315#103—24no—
316#103—1062no—
317#103—1064no—
318#102—24no—
319#102—1064no—
320#102—1062no—
321#102—1064no—
322#102—1064no—
323C22⋊C8—532no—
324#97C22⋊D5312yes—
325C4×C8—532no—
326C4×C8—531no—
327C4×C8—532no—
328C4×C8—531no—
329C4×C8C22×D5312yes—
330C24—2121no—
331#41—44no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—44no—
335C2×Q8—41no—
336C2×Q8—1061no—
337C2×Q8—1061no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—41no—
342C2×D4—1062no—
343C2×D4—1062no—
344C2×D4—1062no—
345C2×D4—4242no—
346C2×D4—2124no—
347C2×D4—2121no—
348C2×D4—2121no—
349C2×D4—2121no—
350C2×D4—2124no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4D4×D5311yescharacteristic
354C22×C4D4×D5311yescharacteristic
355C22×C4C2×C22⋊D5311yescharacteristic
356SD16—2124no—
357SD16—2128no—
358SD16—2128no—
359D8—4244no—
360M16—24no—
361C2×C8—532no—
362C2×C8—532no—
363C2×C8—534no—
364C2×C8—1064no—
365C2×C8C2×C22⋊D5314yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—1061no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C4⋊C4—2121no—
377C4⋊C4—2121no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—2121no—
381C42—2121no—
382C42D4×D5311yescharacteristic
383C42D4×D5311yescharacteristic
384C42C2×C22⋊D5311yescharacteristic
385C42C23×D5311yescharacteristic
386C23—41no—
387C23—4242no—
388C23—4242no—
389C23—4242no—
390C23—2122no—
391C23#1313321311yescharacteristic
392Q8—42no—
393Q8—44no—
394Q8—82no—
395Q8—2122no—
396Q8—2122no—
397D4—42no—
398D4—42no—
399D4—42no—
400D4—82no—
401D4—4242no—
402D4—4242no—
403D4—4242no—
404D4—2124no—
405D4—2124no—
406D4—2124no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4#1313321311yescharacteristic
416C2×C4#1313321311yescharacteristic
417C2×C4#1313321311yescharacteristic
418C2×C4#1313323714yes—
419C2×C4C2×D4×D5311yescharacteristic
420C2×C4C2×D4×D5311yescharacteristic
421C2×C4C22×C22⋊D5311yescharacteristic
422C8—24no—
423C8—534no—
424C8—1062no—
425C8—1064no—
426C22—21no—
427C22—21no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—8484no—
432C22—4242no—
433C22—4242no—
434C22—4242no—
435C22#1681100512yes—
436C22C2×#1313321311yescharacteristic
437C4—21no—
438C4—21no—
439C4—81no—
440C4—82no—
441C4—4241no—
442C4—4241no—
443C4#1681100512yes—
444C4C2×#1313323712yes—
445C4AGL(1,ℤ/8)×D5312yes—
446C4#1681143212yes—
447C2—22no—
448C2—161no—
449C2—8482no—
450C2C2×#1681100511yescharacteristic
451C2#2028516311yescharacteristic
452C2#2028519411yescharacteristic
subgroupG/H
centerC22C2×#13133213
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3141C2
socleC22×C53C2×#127
prime radical#20285630C2
hypercenter#3141D53
hypocenterC53#19625
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
1. #20285866C2semidirect
3. #20285270C2semidirect
4. C2×#16810997C2semidirect
5. #20285090C2semidirect
6. #20274369C2semidirect
7. #20274355C2semidirect
8. #20274318C2semidirect
9. #20274306C2semidirect
10. M16×#88322C2semidirect
11. C53×#3141C2semidirect
15. #16811026C22semidirect
16. #16811019C22semidirect
21. #16810995C22semidirect
23. #16810975C22semidirect
24. #16810975C22semidirect
28. #16810969C22semidirect
38. #16810344C22semidirect
39. C8×#88322C22semidirect
43. C53×#487C22semidirect
44. C53×#486C22semidirect
225. #3141D53semidirect
254. #487C2×D53semidirect
259. #486C2×D53semidirect
273. C53#19625semidirect

Maximal subgroups

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

subgroupindex
#202858662
#202856302
#202852702
C2×#168109972
#202850902
#202743692
#202743552
#202743182
#202743062
M16×#883222
C53×#31412
#1962553