Finite Group Databasebeta

← #27369357

Subgroups — #27369357

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#20285876C211yesmaximal, characteristic, complemented
2#20284814C211yesmaximal, characteristic
3#20284723C211yesmaximal, characteristic
4#20274166C212yesmaximal, complemented
5#20274138C212yesmaximal, complemented
6#20274104C212yesmaximal, complemented
7#20274090C212yesmaximal, complemented
8#20273495C211yesmaximal, characteristic, complemented
9#20273423C211yesmaximal, characteristic, complemented
10#20273379C211yesmaximal, characteristic, complemented
11C53×#2972C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16811180C2211yescharacteristic
14#16811028C2211yescharacteristic, complemented
15#16811021C2211yescharacteristic, complemented
16#16810974C2211yescharacteristic, complemented
17#16810974C2211yescharacteristic, complemented
18#16810965—22no—
19#16810958—21no—
20#16810953—21no—
21#16810931—22no—
22#16810913C2212yes—
23#16810903C2212yes—
24#16810828C2212yes—
25#16810828C2212yes—
26#16810774—21no—
27#16810774—21no—
28#16810723—22no—
29C4×#11719923—22no—
30#16810680C2211yescharacteristic
31#16810676C2211yescharacteristic
32#16810376C2212yescomplemented
33#16810374C2212yescomplemented
34#16810371C2212yescomplemented
35#16810370C2212yescomplemented
36#16810368C2212yescomplemented
37#16810368C2212yescomplemented
38C2×C4×D4×C53C2211yescharacteristic
39C8×D4×C53—22no—
40C8×D4×C53C2212yes—
41C8×D4×C53C2212yes—
42C53×#431C2211yescharacteristic
43C53×#431C2211yescharacteristic
44C2×C22⋊C8×C53—22no—
45C2×#11719923—22no—
46C2×#11719923—22no—
47#13133217C2311yescharacteristic
48#13133217C2311yescharacteristic
49#13133217C2311yescharacteristic
50#13133217C2311yescharacteristic
51#13133209—21no—
52#13133209—21no—
53#13133196—21no—
54#13133196—21no—
55#13133196—21no—
56#13133196—21no—
57C4×Dic106—21no—
58C4×Dic106—21no—
59C4×Dic106—21no—
60C4×Dic106—21no—
61#13133176—21no—
62#13133176—21no—
63#13133151—22no—
64#13133150—22no—
65#13133149—22no—
66#13133148—22no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133142—22no—
70#13133142—22no—
71#13133140—22no—
72#13133140—22no—
73#13133138—22no—
74#13133138—22no—
75C22⋊#88322—21no—
76C22⋊#88322—21no—
77#13133128—21no—
78C4×#88322—21no—
79C22×D4×C53D411yescharacteristic
80C23×C4×C53—21no—
81C22×C8×C53—22no—
82C22×C8×C53—22no—
83C4×D4×C53—21no—
84C4×D4×C53—21no—
85C4×D4×C53D412yes—
86C4×D4×C53C2312yes—
87C2×C4⋊C4×C53D411yescharacteristic
88C2×C22⋊C4×C53—21no—
89C2×C42×C53C2311yescharacteristic
90C4⋊C8×C53D412yes—
91C4⋊C8×C53C2312yes—
92C22⋊C8×C53—22no—
93C22⋊C8×C53—22no—
94C22⋊C8×C53—21no—
95C22⋊C8×C53D412yes—
96C4×C8×C53—21no—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99C24×C53—21no—
100C2×D4×C53—24no—
101C2×D4×C53—21no—
102C2×D4×C53C2×D412yes—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—82no—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C2×Dic106—21no—
111C2×Dic106—21no—
112C2×Dic106—21no—
113C2×Dic106—21no—
114C2×C8×C53—22no—
115C2×C8×C53—82no—
116C2×C8×C53—82no—
117C2×C8×C53C2×D412yes—
118C2×C8×C53C2×D412yes—
119C4⋊C4×C53—21no—
120C4⋊C4×C53C2×D412yes—
121C22⋊C4×C53—22no—
122C22⋊C4×C53—22no—
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53#4111yescharacteristic
126C42×C53C2411yescharacteristic
127C22⋊Dic53—41no—
128C22⋊Dic53—41no—
129#11719923—44no—
130#11719923—44no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—21no—
134C4⋊Dic53—21no—
135C4⋊Dic53—21no—
136C4⋊Dic53—21no—
137C4⋊Dic53—21no—
138C4⋊Dic53—21no—
139#11719919—41no—
140#11719919—41no—
141#11719919—41no—
142#11719919—41no—
143C4×Dic53—41no—
144C4×Dic53—41no—
145C2×#88322—22no—
146C2×#88322—22no—
147C23×C53—24no—
148C23×C53—22no—
149C23×C53—81no—
150C23×C53—81no—
151C23×C53#12111yescharacteristic
152D4×C53—24no—
153D4×C53—22no—
154D4×C53#13514yes—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—82no—
162C2×C4×C53—81no—
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53C22×D411yescharacteristic
167C2×C4×C53C22×D411yescharacteristic
168C2×C4×C53C2×#4111yescharacteristic
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173Dic106—42no—
174Dic106—42no—
175Dic106—42no—
176Dic106—81no—
177C8×C53—22no—
178C8×C53—22no—
179C8×C53—42no—
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×C53C2×#13512yes—
197C4×C53#57712yes—
198Dic53—81no—
199Dic53—81no—
200Dic53—81no—
201Dic53—81no—
202#3231—531no—
203#3110—531no—
204#3101—531no—
205#2972D5311yescharacteristic
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#533—531no—
220C2×C4×D4C2×D5311yescharacteristic
221#500—531no—
222#498—531no—
223#487—531no—
224#487—531no—
225#485—1062no—
226#484—1061no—
227#483—1061no—
228#479—1062no—
229#476—532no—
230#474—532no—
231#461—532no—
232#461—532no—
233#450—1061no—
234#450—1061no—
235#440—1062no—
236C4×SD16—1062no—
237C8×D4—22no—
238C8×D4C2×D5312yes—
239C8×D4C2×D5312yes—
240#432—531no—
241#431—531no—
242#431C2×D5311yescharacteristic
243#431C2×D5311yescharacteristic
244C2×C22⋊C8—22no—
245#335—532no—
246#334—532no—
247#333—532no—
248#332—532no—
249#332—532no—
250#332—532no—
251C53#1939511yescharacteristic, complemented, Sylow
252C22×D4D21211yescharacteristic
253C23×C4—21no—
254C2×SD16—1062no—
255C2×SD16—1062no—
256C22×C8—22no—
257C22×C8—22no—
258C4⋊Q8—531no—
259C4⋊Q8—531no—
260C4⋊Q8—531no—
261C4⋊Q8—531no—
262#125—1061no—
263#125—1061no—
264C22⋊Q8—1061no—
265C22⋊Q8—1061no—
266C22⋊Q8—1061no—
267C22⋊Q8—1061no—
268C4×Q8—1061no—
269C4×Q8—1061no—
270C4×Q8—1061no—
271C4×Q8—1061no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4D21212yes—
275C4×D4C22×D5312yes—
276#117—1061no—
277#117—1061no—
278C2×C4⋊C4D21211yescharacteristic
279C2×C22⋊C4—21no—
280C2×C42C22×D5311yescharacteristic
281#107—1062no—
282#107—1062no—
283#106—1062no—
284#106—1062no—
285C4⋊C8—532no—
286C4⋊C8—532no—
287C4⋊C8C22⋊D5312yes—
288C4⋊C8C22×D5312yes—
289#103—1062no—
290#103—1062no—
291#103—1062no—
292#103—1062no—
293#102—1062no—
294#102—1062no—
295C22⋊C8—21no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—1061no—
299C22⋊C8—1061no—
300C22⋊C8C22⋊D5312yes—
301#97—1061no—
302C4×C8—21no—
303C4×C8—1061no—
304C4×C8C22×D5311yescharacteristic
305C4×C8C22×D5311yescharacteristic
306C24—21no—
307C2×Q8—1061no—
308C2×Q8—1061no—
309C2×Q8—1061no—
310C2×Q8—1061no—
311C2×D4—24no—
312C2×D4—21no—
313C2×D4C2×D21212yes—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—82no—
318C22×C4C2×D21211yescharacteristic
319C22×C4#1171994511yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321SD16—2124no—
322SD16—2124no—
323C2×C8—22no—
324C2×C8—82no—
325C2×C8—82no—
326C2×C8—1062no—
327C2×C8—1062no—
328C2×C8C2×C22⋊D5312yes—
329C2×C8C2×C22⋊D5312yes—
330C4⋊C4—21no—
331C4⋊C4—1061no—
332C4⋊C4—1061no—
333C4⋊C4—1061no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C4⋊C4—1061no—
338C4⋊C4—1061no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C4⋊C4—2121no—
343C4⋊C4C2×D21212yes—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C42—2121no—
349C42—2121no—
350C42C2×D21211yescharacteristic
351C42#1171994511yescharacteristic
352C42C2×C22⋊D5311yescharacteristic
353C42C23×D5311yescharacteristic
354C23—24no—
355C23—22no—
356C23—81no—
357C23—81no—
358C23#1313319111yescharacteristic
359Q8—4241no—
360Q8—2122no—
361Q8—2122no—
362Q8—2122no—
363D4—24no—
364D4—22no—
365D4#1313323914yes—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4#1313319111yescharacteristic
379C2×C4#1313319111yescharacteristic
380C2×C4#1313319111yescharacteristic
381C2×C4C22×C22⋊D5311yescharacteristic
382C2×C4C22×D21211yescharacteristic
383C2×C4C2×#1171994511yescharacteristic
384C8—22no—
385C8—22no—
386C8—42no—
387C8—2122no—
388C8—2122no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—84no—
396C22C2×#1313319111yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—4241no—
403C4—4241no—
404C4—4241no—
405C4—4241no—
406C4C2×#1313323912yes—
407C4#1681143412yes—
408C2—41no—
409C2—44no—
410C2#2028422711yescharacteristic
411C2#2028429611yescharacteristic
412C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2972C2
socleC22×C53C2×#121
prime radicalC22×D4×C53D4
hypercenter#2972D53
hypocenterC53#19395
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #20285876C2semidirect
4. #20274166C2semidirect
5. #20274138C2semidirect
6. #20274104C2semidirect
7. #20274090C2semidirect
8. #20273495C2semidirect
9. #20273423C2semidirect
10. #20273379C2semidirect
14. #16811028C22semidirect
15. #16811021C22semidirect
16. #16810974C22semidirect
17. #16810974C22semidirect
32. #16810376C22semidirect
33. #16810374C22semidirect
34. #16810371C22semidirect
35. #16810370C22semidirect
36. #16810368C22semidirect
37. #16810368C22semidirect
251. C53#19395semidirect

Maximal subgroups

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

subgroupindex
#202858762
#202848142
#202847232
#202741662
#202741382
#202741042
#202740902
#202734952
#202734232
#202733792
C53×#29722
#1939553