Finite Group Databasebeta

← #27369349

Subgroups — #27369349

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#20285847C211yesmaximal, characteristic, complemented
2#20284723C211yesmaximal, characteristic, complemented
3#20283739C211yesmaximal, characteristic, complemented
4#20274150C212yesmaximal, complemented
5#20274122C212yesmaximal, complemented
6#20274112C212yesmaximal, complemented
7#20274092C212yesmaximal, complemented
8#20273496C211yesmaximal, characteristic, complemented
9#20273426C211yesmaximal, characteristic, complemented
10#20273377C211yesmaximal, characteristic, complemented
11C53×#3109C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16811026C2211yescharacteristic, complemented
14#16811019C2211yescharacteristic, complemented
15#16810969C2211yescharacteristic, complemented
16#16810969C2211yescharacteristic, complemented
17#16810953—21no—
18#16810944—22no—
19#16810923—22no—
20#16810913C2212yes—
21#16810828C2212yes—
22#16810774—21no—
23C4×#11719923—22no—
24D4×#88322—22no—
25D4×#88322C2212yes—
26D4×#88322C2212yes—
27#16810676C2211yescharacteristic
28#16810676C2211yescharacteristic
29#16810676C2211yescharacteristic
30C2×C22⋊#88322—22no—
31#16810375C2212yescomplemented
32#16810373C2212yescomplemented
33#16810372C2212yescomplemented
34#16810371C2212yescomplemented
35#16810370C2212yescomplemented
36#16810367C2212yescomplemented
37C53×#532C2211yescharacteristic
38C2×C4×D4×C53C2211yescharacteristic
39C53×#481—21no—
40C53×#476C2212yes—
41C53×#459C2212yes—
42C53×#447—21no—
43C53×#442—22no—
44C53×#432C2211yescharacteristic
45C2×#11719923—22no—
46C22×#88322—22no—
47C22×#88322—22no—
48#13133217C2311yescharacteristic
49#13133217C2311yescharacteristic
50#13133209—21no—
51#13133196—21no—
52#13133196—21no—
53C4×Dic106—21no—
54C4×Dic106—21no—
55#13133176—21no—
56#13133150—22no—
57#13133148—22no—
58#13133148—22no—
59C4⋊#88322D412yes—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133140—22no—
63#13133138—22no—
64#13133138—22no—
65#13133138—22no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68C22⋊#88322—21no—
69C22⋊#88322—21no—
70C22⋊#88322D412yes—
71C4×#88322—21no—
72C4×#88322—21no—
73C4×#88322C2311yescharacteristic
74C4×#88322C2311yescharacteristic
75C2×#41×C53—21no—
76C22×D4×C53D411yescharacteristic
77C23×C4×C53—21no—
78C2×D8×C53—22no—
79C4⋊Q8×C53C2311yescharacteristic
80#127×C53C2311yescharacteristic
81#124×C53—21no—
82#121×C53—21no—
83#121×C53—21no—
84C4×D4×C53—21no—
85C4×D4×C53—21no—
86C4×D4×C53—21no—
87C4×D4×C53—21no—
88C4×D4×C53D412yes—
89C4×D4×C53C2312yes—
90C2×C4⋊C4×C53D411yescharacteristic
91C2×C22⋊C4×C53—21no—
92C2×C42×C53C2311yescharacteristic
93#106×C53—22no—
94C4⋊C8×C53C2312yes—
95#102×C53—22no—
96#102×C53—22no—
97C22⋊C8×C53—21no—
98#97×C53—21no—
99C24×C53—21no—
100#41×C53—82no—
101C2×Q8×C53—21no—
102C2×D4×C53—24no—
103C2×D4×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—41no—
108C2×D4×C53C2×D412yes—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—41no—
113C22×C4×C53—82no—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×Dic106—21no—
118C2×Dic106—21no—
119D8×C53—44no—
120C2×C8×C53—22no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53C2×D412yes—
125C22⋊C4×C53—22no—
126C22⋊C4×C53—22no—
127C22⋊C4×C53—41no—
128C22⋊C4×C53—41no—
129C42×C53C2×D411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C42×C53#4111yescharacteristic
132C42×C53C2411yescharacteristic
133C22⋊Dic53—41no—
134#11719923—44no—
135C4⋊Dic53—21no—
136C4⋊Dic53—21no—
137C4⋊Dic53—21no—
138C4⋊Dic53—21no—
139#11719919—41no—
140#11719919—41no—
141C4×Dic53—41no—
142C2×#88322—22no—
143C2×#88322—22no—
144C2×#88322—82no—
145C2×#88322—82no—
146C2×#88322C2×D412yes—
147C2×#88322C2×D412yes—
148C23×C53—24no—
149C23×C53—22no—
150C23×C53—41no—
151C23×C53—81no—
152C23×C53—81no—
153C23×C53#12111yescharacteristic
154Q8×C53—81no—
155D4×C53—24no—
156D4×C53—22no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—81no—
160D4×C53—82no—
161D4×C53#13514yes—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—41no—
169C2×C4×C53—82no—
170C2×C4×C53—82no—
171C2×C4×C53—81no—
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53C22×D411yescharacteristic
176C2×C4×C53C22×D411yescharacteristic
177C2×C4×C53C2×#4111yescharacteristic
178C2×Dic53—41no—
179C2×Dic53—41no—
180Dic106—42no—
181Dic106—81no—
182C8×C53—42no—
183#88322—22no—
184#88322—22no—
185#88322—42no—
186#88322—42no—
187#19394—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—24no—
191C22×C53—21no—
192C22×C53—42no—
193C22×C53—42no—
194C22×C53—84no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53C2×#12111yescharacteristic
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—42no—
204C4×C53—81no—
205C4×C53C2×#13512yes—
206C4×C53#57712yes—
207Dic53—81no—
208Dic53—81no—
209#3228—531no—
210#3109D5311yescharacteristic
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—44no—
222C2×C53—161no—
223C2×C53#304211yescharacteristic
224C2×C53#305011yescharacteristic
225C2×C53#306811yescharacteristic
226#533—531no—
227#532C2×D5311yescharacteristic
228C2×C4×D4C2×D5311yescharacteristic
229#500—531no—
230#498—531no—
231#486—531no—
232#486—531no—
233#483—1061no—
234#482—1062no—
235#481—21no—
236#478—1062no—
237#476—532no—
238#476C2×D5312yes—
239#461—532no—
240#459C2×D5312yes—
241#450—1061no—
242#447—21no—
243#442—22no—
244C4×SD16—1062no—
245C8×D4—532no—
246C8×D4—532no—
247C8×D4—1062no—
248#432C2×D5311yescharacteristic
249#431—531no—
250#431—531no—
251#431—531no—
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×D4C22⋊D5311yescharacteristic
262C23×C4—21no—
263C2×SD16—1062no—
264C2×D8—22no—
265C22×C8—1062no—
266C22×C8—1062no—
267C4⋊Q8—531no—
268C4⋊Q8—531no—
269C4⋊Q8C22×D5311yescharacteristic
270#127C22×D5311yescharacteristic
271#125—1061no—
272#124—21no—
273C22⋊Q8—1061no—
274C22⋊Q8—1061no—
275#121—21no—
276#121—21no—
277C4×Q8—1061no—
278C4×Q8—1061no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4C22⋊D5312yes—
284C4×D4C22×D5312yes—
285#117—1061no—
286C2×C4⋊C4C22⋊D5311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C42C22×D5311yescharacteristic
289#107—1062no—
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—21no—
304C22⋊C8—532no—
305C22⋊C8—1062no—
306C22⋊C8—1062no—
307C22⋊C8—1061no—
308C22⋊C8—1061no—
309#97—21no—
310C4×C8—531no—
311C4×C8—531no—
312C4×C8—1061no—
313C4×C8—1061no—
314C24—21no—
315#41—82no—
316C2×Q8—21no—
317C2×Q8—1061no—
318C2×Q8—1061no—
319C2×D4—24no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—41no—
325C2×D4C2×C22⋊D5312yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—82no—
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334SD16—2124no—
335D8—44no—
336C2×C8—22no—
337C2×C8—532no—
338C2×C8—532no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8—4242no—
342C2×C8—4242no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—1061no—
347C4⋊C4—1061no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C4⋊C4—2121no—
351C4⋊C4—2121no—
352C4⋊C4C2×C22⋊D5312yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—2121no—
358C42—2121no—
359C42D4×D5311yescharacteristic
360C42#1171994711yescharacteristic
361C42C2×C22⋊D5311yescharacteristic
362C42C23×D5311yescharacteristic
363C23—24no—
364C23—22no—
365C23—41no—
366C23—81no—
367C23—81no—
368C23#1313318911yescharacteristic
369Q8—81no—
370Q8—4241no—
371Q8—2122no—
372D4—24no—
373D4—22no—
374D4—42no—
375D4—42no—
376D4—81no—
377D4—82no—
378D4#1313323714yes—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4#1313318911yescharacteristic
392C2×C4#1313318911yescharacteristic
393C2×C4#1313318911yescharacteristic
394C2×C4C2×D4×D5311yescharacteristic
395C2×C4C22×C22⋊D5311yescharacteristic
396C2×C4C2×#1171994711yescharacteristic
397C8—42no—
398C8—1062no—
399C8—1062no—
400C8—2122no—
401C8—2122no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—24no—
406C22—42no—
407C22—42no—
408C22—84no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22C2×#1313318911yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—81no—
419C4—4241no—
420C4—4241no—
421C4C2×#1313323712yes—
422C4#1681143312yes—
423C2—41no—
424C2—44no—
425C2—161no—
426C2#2028422911yescharacteristic
427C2#2028429311yescharacteristic
428C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3109C2
socleC22×C53C2×#121
hypercenter#3109D53
hypocenterC53#19394
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #20285847C2semidirect
2. #20284723C2semidirect
3. #20283739C2semidirect
4. #20274150C2semidirect
5. #20274122C2semidirect
6. #20274112C2semidirect
7. #20274092C2semidirect
8. #20273496C2semidirect
9. #20273426C2semidirect
10. #20273377C2semidirect
13. #16811026C22semidirect
14. #16811019C22semidirect
15. #16810969C22semidirect
16. #16810969C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810372C22semidirect
34. #16810371C22semidirect
35. #16810370C22semidirect
36. #16810367C22semidirect
259. C53#19394semidirect

Maximal subgroups

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

subgroupindex
#202858472
#202847232
#202837392
#202741502
#202741222
#202741122
#202740922
#202734962
#202734262
#202733772
C53×#31092
#1939453