Finite Group Databasebeta

← #27369360

Subgroups — #27369360

Order 13568.

628 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285870C211yesmaximal, characteristic, complemented
2#20284723C211yesmaximal, characteristic
3#20283739C211yesmaximal, characteristic
4#20274164C212yesmaximal, complemented
5#20274136C212yesmaximal, complemented
6#20274102C212yesmaximal, complemented
7#20274088C212yesmaximal, complemented
8#20273496C211yesmaximal, characteristic, complemented
9#20273426C211yesmaximal, characteristic, complemented
10#20273380C211yesmaximal, characteristic, complemented
11C53×#3110C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16811026C2211yescharacteristic, complemented
14#16811019C2211yescharacteristic, complemented
15#16810973C2211yescharacteristic, complemented
16#16810973C2211yescharacteristic, complemented
17#16810961—22no—
18#16810953—21no—
19#16810930—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#16810371C2212yescomplemented
34#16810370C2212yescomplemented
35#16810369C2212yescomplemented
36#16810369C2212yescomplemented
37C53×#533C2211yescharacteristic
38C2×C4×D4×C53C2211yescharacteristic
39C53×#484—21no—
40C53×#474C2212yes—
41C53×#461C2212yes—
42C53×#450—21no—
43C53×#440—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#13133141—22no—
63#13133141—22no—
64#13133140—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
75C22×D4×C53D411yescharacteristic
76C23×C4×C53—21no—
77C2×SD16×C53—22no—
78C4⋊Q8×C53C2311yescharacteristic
79C4⋊Q8×C53C2311yescharacteristic
80#125×C53—21no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53—21no—
83C4×Q8×C53—21no—
84C4×Q8×C53—21no—
85C4×D4×C53—21no—
86C4×D4×C53—21no—
87C4×D4×C53D412yes—
88C4×D4×C53C2312yes—
89#117×C53—21no—
90C2×C4⋊C4×C53D411yescharacteristic
91C2×C22⋊C4×C53—21no—
92C2×C42×C53C2311yescharacteristic
93#107×C53—22no—
94C4⋊C8×C53C2312yes—
95#103×C53—22no—
96#102×C53—22no—
97C22⋊C8×C53—21no—
98#97×C53—21no—
99C24×C53—21no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×D4×C53—24no—
103C2×D4×C53—21no—
104C2×D4×C53C2×D412yes—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—82no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×Dic106—21no—
113C2×Dic106—21no—
114SD16×C53—44no—
115C2×C8×C53—22no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53C2×D412yes—
124C22⋊C4×C53—22no—
125C22⋊C4×C53—22no—
126C22⋊C4×C53—41no—
127C42×C53—41no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53C2411yescharacteristic
132C22⋊Dic53—41no—
133#11719923—44no—
134C4⋊Dic53—21no—
135C4⋊Dic53—21no—
136C4⋊Dic53—21no—
137C4⋊Dic53—21no—
138#11719919—41no—
139#11719919—41no—
140C4×Dic53—41no—
141C2×#88322—22no—
142C2×#88322—22no—
143C2×#88322—82no—
144C2×#88322—82no—
145C2×#88322C2×D412yes—
146C2×#88322C2×D412yes—
147C23×C53—24no—
148C23×C53—22no—
149C23×C53—81no—
150C23×C53—81no—
151C23×C53#12111yescharacteristic
152Q8×C53—42no—
153Q8×C53—42no—
154D4×C53—24no—
155D4×C53—22no—
156D4×C53#13514yes—
157C2×C4×C53—22no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—82no—
166C2×C4×C53—81no—
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53C22×D411yescharacteristic
171C2×C4×C53C22×D411yescharacteristic
172C2×C4×C53C2×#4111yescharacteristic
173C2×Dic53—41no—
174C2×Dic53—41no—
175Dic106—42no—
176Dic106—81no—
177C8×C53—42no—
178#88322—22no—
179#88322—22no—
180#88322—42no—
181#88322—42no—
182#19395—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—24no—
186C22×C53—21no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—84no—
190C22×C53C2×#12111yescharacteristic
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—42no—
196C4×C53—81no—
197C4×C53—81no—
198C4×C53C2×#13512yes—
199C4×C53#57712yes—
200Dic53—81no—
201Dic53—81no—
202#3231—531no—
203#3110D5311yescharacteristic
204#3101—531no—
205#2972—531no—
206#1683—532no—
207#1679—532no—
208#1673—532no—
209#1671—532no—
210#1572—531no—
211#1554—531no—
212#1544—531no—
213C2×C53—41no—
214C2×C53—44no—
215C2×C53#304211yescharacteristic
216C2×C53#305011yescharacteristic
217C2×C53#306911yescharacteristic
218#533—531no—
219#533C2×D5311yescharacteristic
220C2×C4×D4C2×D5311yescharacteristic
221#500—531no—
222#498—531no—
223#487—531no—
224#487—531no—
225#485—1062no—
226#484—21no—
227#483—1061no—
228#479—1062no—
229#476—532no—
230#474C2×D5312yes—
231#461—532no—
232#461C2×D5312yes—
233#450—21no—
234#450—1061no—
235#440—22no—
236C4×SD16—1062no—
237C8×D4—532no—
238C8×D4—532no—
239C8×D4—1062no—
240#432C2×D5311yescharacteristic
241#431—531no—
242#431—531no—
243#431—531no—
244C2×C22⋊C8—1062no—
245#335—532no—
246#334—532no—
247#333—532no—
248#332—532no—
249#332—532no—
250#332—532no—
251C53#1939511yescharacteristic, complemented, Sylow
252C22×D4C22⋊D5311yescharacteristic
253C23×C4—21no—
254C2×SD16—22no—
255C2×SD16—1062no—
256C22×C8—1062no—
257C22×C8—1062no—
258C4⋊Q8—531no—
259C4⋊Q8—531no—
260C4⋊Q8C22×D5311yescharacteristic
261C4⋊Q8C22×D5311yescharacteristic
262#125—21no—
263#125—1061no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—1061no—
267C22⋊Q8—1061no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—1061no—
271C4×Q8—1061no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4C22⋊D5312yes—
275C4×D4C22×D5312yes—
276#117—21no—
277#117—1061no—
278C2×C4⋊C4C22⋊D5311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C42C22×D5311yescharacteristic
281#107—22no—
282#107—1062no—
283#106—1062no—
284#106—1062no—
285C4⋊C8—532no—
286C4⋊C8—532no—
287C4⋊C8—532no—
288C4⋊C8C22×D5312yes—
289#103—22no—
290#103—1062no—
291#103—1062no—
292#103—1062no—
293#102—22no—
294#102—1062no—
295C22⋊C8—21no—
296C22⋊C8—532no—
297C22⋊C8—1062no—
298C22⋊C8—1062no—
299C22⋊C8—1061no—
300C22⋊C8—1061no—
301#97—21no—
302C4×C8—531no—
303C4×C8—531no—
304C4×C8—1061no—
305C4×C8—1061no—
306C24—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—1061no—
310C2×Q8—1061no—
311C2×D4—24no—
312C2×D4—21no—
313C2×D4C2×C22⋊D5312yes—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—82no—
318C22×C4D4×D5311yescharacteristic
319C22×C4#1171994711yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321SD16—44no—
322SD16—2124no—
323C2×C8—22no—
324C2×C8—532no—
325C2×C8—532no—
326C2×C8—1062no—
327C2×C8—1062no—
328C2×C8—4242no—
329C2×C8—4242no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—1061no—
338C4⋊C4—1061no—
339C4⋊C4—1061no—
340C4⋊C4—1061no—
341C4⋊C4—2121no—
342C4⋊C4—2121no—
343C4⋊C4C2×C22⋊D5312yes—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—41no—
347C22⋊C4—2121no—
348C42—41no—
349C42—2121no—
350C42D4×D5311yescharacteristic
351C42#1171994711yescharacteristic
352C42C2×C22⋊D5311yescharacteristic
353C42C23×D5311yescharacteristic
354C23—24no—
355C23—22no—
356C23—81no—
357C23—81no—
358C23#1313318911yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—4241no—
362Q8—2122no—
363D4—24no—
364D4—22no—
365D4#1313323714yes—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4#1313318911yescharacteristic
379C2×C4#1313318911yescharacteristic
380C2×C4#1313318911yescharacteristic
381C2×C4C2×D4×D5311yescharacteristic
382C2×C4C22×C22⋊D5311yescharacteristic
383C2×C4C2×#1171994711yescharacteristic
384C8—42no—
385C8—1062no—
386C8—1062no—
387C8—2122no—
388C8—2122no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—84no—
396C22C2×#1313318911yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—4241no—
405C4—4241no—
406C4C2×#1313323712yes—
407C4#1681143312yes—
408C2—41no—
409C2—44no—
410C2#2028422911yescharacteristic
411C2#2028429311yescharacteristic
412C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3110C2
socleC22×C53C2×#121
prime radicalC22×D4×C53D4
hypercenter#3110D53
hypocenterC53#19395
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #20285870C2semidirect
4. #20274164C2semidirect
5. #20274136C2semidirect
6. #20274102C2semidirect
7. #20274088C2semidirect
8. #20273496C2semidirect
9. #20273426C2semidirect
10. #20273380C2semidirect
13. #16811026C22semidirect
14. #16811019C22semidirect
15. #16810973C22semidirect
16. #16810973C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810371C22semidirect
34. #16810370C22semidirect
35. #16810369C22semidirect
36. #16810369C22semidirect
251. C53#19395semidirect

Maximal subgroups

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

subgroupindex
#202858702
#202847232
#202837392
#202741642
#202741362
#202741022
#202740882
#202734962
#202734262
#202733802
C53×#31102
#1939553