Finite Group Databasebeta

← #27369342

Subgroups — #27369342

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284802C211yesmaximal, characteristic
2#20284719C211yesmaximal, characteristic, complemented
3#20283738C211yesmaximal, characteristic, complemented
4#20274149C212yesmaximal, complemented
5#20274121C212yesmaximal, complemented
6#20274111C212yesmaximal, complemented
7#20274087C212yesmaximal, complemented
8#20273495C211yesmaximal, characteristic, complemented
9#20273425C211yesmaximal, characteristic, complemented
10#20273377C211yesmaximal, characteristic, complemented
11C53×#3228C211yesmaximal, characteristic, complemented
12C2×C4×D212C2211yescharacteristic
13#16810951—21no—
14#16810939—21no—
15#16810910C2212yes—
16#16810910C2212yes—
17#16810825C2212yes—
18#16810815C2212yes—
19#16810773—21no—
20#16810756—21no—
21#16810733—22no—
22C4×#11719924—22no—
23C8×D212C2212yes—
24C8×D212C2212yes—
25#16810688—22no—
26#16810680C2211yescharacteristic
27#16810676C2211yescharacteristic
28#16810676C2211yescharacteristic
29C22⋊C8×D53—22no—
30#16810376C2212yescomplemented
31#16810374C2212yescomplemented
32#16810372C2212yescomplemented
33#16810372C2212yescomplemented
34#16810369C2212yescomplemented
35#16810367C2212yescomplemented
36C53×#533C2211yescharacteristic
37C53×#532C2211yescharacteristic
38C53×#500C2211yescharacteristic, complemented
39C53×#498C2211yescharacteristic, complemented
40C53×#486C2211yescharacteristic, complemented
41C53×#486C2211yescharacteristic, complemented
42C53×#482—22no—
43C53×#478—22no—
44C53×#431C2211yescharacteristic
45C22×D212D411yescharacteristic
46C22×C4×D53—21no—
47C2×#11719924—22no—
48C2×#11719922—22no—
49C2×C8×D53—22no—
50C2×C8×D53—22no—
51C4×D212—21no—
52C4×D212—21no—
53C4×D212D412yes—
54C4×D212C2312yes—
55C2×C4⋊Dic53D411yescharacteristic
56C2×#11719921—21no—
57#13133148—22no—
58#13133148—22no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133141—22no—
63#13133138—22no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322—21no—
67C22⋊#88322—21no—
68C22⋊#88322—21no—
69#13133129—22no—
70#13133129—22no—
71#13133128—21no—
72C4×#88322—21no—
73C4×#88322—21no—
74C2×#41×C53—21no—
75C4⋊Q8×C53C2311yescharacteristic
76C4⋊Q8×C53C2311yescharacteristic
77C4⋊Q8×C53C2311yescharacteristic
78#127×C53C2311yescharacteristic
79#125×C53—21no—
80#124×C53—21no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53—21no—
83#121×C53—21no—
84#121×C53—21no—
85C4×Q8×C53—21no—
86C4×Q8×C53—21no—
87C4×D4×C53—21no—
88C4×D4×C53—21no—
89#117×C53—21no—
90C2×C42×C53C2311yescharacteristic
91#107×C53—22no—
92#106×C53—22no—
93C4⋊C8×C53D412yes—
94#102×C53—22no—
95#102×C53—22no—
96C22⋊C8×C53D412yes—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99C23×D53—21no—
100#41×C53—82no—
101C2×Q8×C53—21no—
102C2×Q8×C53—21no—
103C2×Q8×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—41no—
108C22×C4×C53—41no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×Dic53—21no—
113C2×D212—21no—
114C2×D212—24no—
115C2×D212C2×D412yes—
116C2×C4×D53—22no—
117C2×C4×D53—22no—
118C2×C4×D53—82no—
119C2×C8×C53C2×D412yes—
120C2×C8×C53C2×D412yes—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53—41no—
129C22⋊C4×C53—41no—
130C22⋊C4×C53—41no—
131C22⋊C4×C53—41no—
132C42×C53—41no—
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53#4111yescharacteristic
136C42×C53C2411yescharacteristic
137#11719924—44no—
138#11719922—44no—
139#11719921—22no—
140#11719921—22no—
141C4⋊Dic53—21no—
142C4⋊Dic53C2×D412yes—
143C2×#88322—22no—
144C2×#88322—22no—
145C2×#88322—22no—
146C8×D53—82no—
147C8×D53—82no—
148C23×C53—41no—
149C23×C53#12111yescharacteristic
150C22×D53—22no—
151C22×D53—24no—
152C22×D53—81no—
153C22×D53—81no—
154Q8×C53—42no—
155Q8×C53—81no—
156Q8×C53—81no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—81no—
160D4×C53—82no—
161C2×C4×C53—22no—
162C2×C4×C53—22no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—82no—
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53C22×D411yescharacteristic
172C2×C4×C53C22×D411yescharacteristic
173C2×C4×C53C2×#4111yescharacteristic
174C2×Dic53—22no—
175C2×Dic53—81no—
176D212—24no—
177D212—22no—
178D212#13514yes—
179C4×D53—42no—
180C4×D53—42no—
181C4×D53—82no—
182C8×C53—22no—
183C8×C53—22no—
184#88322—42no—
185#88322—42no—
186#88322—42no—
187#19394—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53C2×#12111yescharacteristic
195C2×D53—24no—
196C2×D53—42no—
197C2×D53—42no—
198C2×D53—84no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—81no—
204C4×C53—81no—
205C4×C53—81no—
206C4×C53C2×#13512yes—
207C4×C53#57712yes—
208Dic53—42no—
209#3228D5311yescharacteristic, complemented
210#3109—531no—
211#3101—531no—
212#2972—531no—
213#1681—532no—
214#1677—532no—
215#1675—532no—
216#1671—532no—
217#1572—531no—
218#1554—531no—
219#1543—531no—
220C2×C53—41no—
221C2×C53—161no—
222C2×C53#304211yescharacteristic
223C2×C53#305011yescharacteristic
224C2×C53#306811yescharacteristic
225D53—44no—
226#533C2×D5311yescharacteristic
227#532C2×D5311yescharacteristic
228C2×C4×D4—531no—
229#500C2×D5311yescharacteristic, complemented
230#498C2×D5311yescharacteristic, complemented
231#486C2×D5311yescharacteristic, complemented
232#486C2×D5311yescharacteristic, complemented
233#483—1061no—
234#482—22no—
235#481—1061no—
236#478—22no—
237#476—532no—
238#476—532no—
239#461—532no—
240#459—532no—
241#450—1061no—
242#447—1061no—
243#442—1062no—
244C4×SD16—1062no—
245C8×D4—532no—
246C8×D4—532no—
247C8×D4—1062no—
248#432—531no—
249#431—531no—
250#431—531no—
251#431C2×D5311yescharacteristic
252C2×C22⋊C8—1062no—
253#335—532no—
254#334—532no—
255#333—532no—
256#333—532no—
257#332—532no—
258#331—532no—
259C53#1939411yescharacteristic, complemented, Sylow
260C2×#41—21no—
261C22×D4—531no—
262C23×C4—1061no—
263C2×SD16—1062no—
264C2×D8—1062no—
265C22×C8—1062no—
266C22×C8—1062no—
267C4⋊Q8C22×D5311yescharacteristic
268C4⋊Q8C22×D5311yescharacteristic
269C4⋊Q8C22×D5311yescharacteristic
270#127C22×D5311yescharacteristic
271#125—21no—
272#124—21no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275#121—21no—
276#121—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—532no—
282C4×D4—532no—
283C4×D4—1061no—
284C4×D4—1061no—
285#117—21no—
286C2×C4⋊C4—531no—
287C2×C22⋊C4—1061no—
288C2×C42C22×D5311yescharacteristic
289#107—22no—
290#106—22no—
291#106—1062no—
292#106—1062no—
293C4⋊C8—532no—
294C4⋊C8—532no—
295C4⋊C8—532no—
296C4⋊C8C22⋊D5312yes—
297#103—1062no—
298#102—22no—
299#102—22no—
300#102—1062no—
301#102—1062no—
302#102—1062no—
303C22⋊C8—1061no—
304C22⋊C8—1061no—
305C22⋊C8—1061no—
306C22⋊C8—1062no—
307C22⋊C8—1062no—
308C22⋊C8C22⋊D5312yes—
309#97—1061no—
310C4×C8—1061no—
311C4×C8—1061no—
312C4×C8C22×D5311yescharacteristic
313C4×C8C22×D5311yescharacteristic
314C24—1061no—
315#41—82no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—41no—
323C2×D4—532no—
324C2×D4—1064no—
325C2×D4—1061no—
326C22×C4—41no—
327C22×C4—1061no—
328C22×C4—1062no—
329C22×C4—1062no—
330C22×C4—4242no—
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334SD16—2124no—
335D8—2124no—
336C2×C8—1062no—
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8—4242no—
340C2×C8—4242no—
341C2×C8C2×C22⋊D5312yes—
342C2×C8C2×C22⋊D5312yes—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—532no—
352C4⋊C4—1061no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—1062no—
357C22⋊C4—1062no—
358C42—41no—
359C42D4×D5311yescharacteristic
360C42#1171994711yescharacteristic
361C42C2×C22⋊D5311yescharacteristic
362C42C23×D5311yescharacteristic
363C23—41no—
364C23—1064no—
365C23—1062no—
366C23—4241no—
367C23—4241no—
368C23#1313318811yescharacteristic
369Q8—42no—
370Q8—81no—
371Q8—81no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—82no—
376D4—534no—
377D4—1064no—
378D4—1062no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—1062no—
387C2×C4—4242no—
388C2×C4—4241no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4#1313318811yescharacteristic
392C2×C4#1313318811yescharacteristic
393C2×C4#1313318811yescharacteristic
394C2×C4C2×D4×D5311yescharacteristic
395C2×C4C22×C22⋊D5311yescharacteristic
396C2×C4C2×#1171994711yescharacteristic
397C8—22no—
398C8—22no—
399C8—2122no—
400C8—2122no—
401C8—2122no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—1064no—
409C22—4244no—
410C22—2122no—
411C22—2122no—
412C22C2×#1313318811yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—2122no—
421C4#135×D5312yes—
422C4#1681143112yes—
423C2—41no—
424C2—161no—
425C2—2124no—
426C2#2028422311yescharacteristic
427C2#2028429211yescharacteristic
428C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3228C2
socleC22×C53C2×#121
prime radical#20284802C2
hypercenter#3228D53
hypocenterC53#19394
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3228

Complements

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

NH ≅ G/Ntype
2. #20284719C2semidirect
3. #20283738C2semidirect
4. #20274149C2semidirect
5. #20274121C2semidirect
6. #20274111C2semidirect
7. #20274087C2semidirect
8. #20273495C2semidirect
9. #20273425C2semidirect
10. #20273377C2semidirect
11. C53×#3228C2semidirect
30. #16810376C22semidirect
31. #16810374C22semidirect
32. #16810372C22semidirect
33. #16810372C22semidirect
34. #16810369C22semidirect
35. #16810367C22semidirect
38. C53×#500C22semidirect
39. C53×#498C22semidirect
40. C53×#486C22semidirect
41. C53×#486C22semidirect
209. #3228D53semidirect
229. #500C2×D53semidirect
230. #498C2×D53semidirect
231. #486C2×D53semidirect
232. #486C2×D53semidirect
259. C53#19394semidirect

Maximal subgroups

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

subgroupindex
#202848022
#202847192
#202837382
#202741492
#202741212
#202741112
#202740872
#202734952
#202734252
#202733772
C53×#32282
#1939453