Finite Group Databasebeta

← #27360361

Subgroups — #27360361

Order 13568.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285172C211yesmaximal, characteristic, complemented
2#20284806C212yesmaximal, complemented
3#20273860C214yesmaximal, complemented
4#20273798C214yesmaximal, complemented
5#20273441C211yesmaximal, characteristic, complemented
6#20273158C212yesmaximal, complemented
7C2×C53×#555C211yesmaximal, characteristic, complemented
8C2×#13133248—22no—
9#16811173C2211yescharacteristic
10#16811015—22no—
11#16811002C2212yescomplemented
12#16810997C2212yescomplemented
13#16810941—22no—
14#16810913C2214yescomplemented
15#16810818C2214yescomplemented
16#16810757—22no—
17#16810733—24no—
18#16810680C2212yes—
19C4×#11719917C2211yescharacteristic
20#16810372C2214yescomplemented
21D4⋊#88322C2218yescomplemented
22C53×#555—22no—
23C53×#555—24no—
24C53×#555C2214yes—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×#124×C53—22no—
27C2×#121×C53—22no—
28C2×C4×D4×C53C2212yes—
29C2×#117×C53—22no—
30C2×#11719945—21no—
31#13133248—48no—
32C2×#11719925—24no—
33C2×#11719924—24no—
34C2×#11719922—24no—
35C2×#11719917—21no—
36#13133217C2311yescharacteristic
37#13133214C2311yescharacteristic
38#13133205—21no—
39#13133191—22no—
40C4×D212—22no—
41#13133148—24no—
42C4⋊#88322C2314yes—
43#13133138—24no—
44#13133137—24no—
45C22⋊#88322—22no—
46#13133128—21no—
47#13133128—22no—
48C4×#88322C2312yes—
49C22×Q8×C53D412yes—
50C22×D4×C53—22no—
51C22×D4×C53D412yes—
52C23×C4×C53—22no—
53C4⋊Q8×C53—22no—
54C4⋊Q8×C53D412yes—
55C4⋊Q8×C53C2312yes—
56#124×C53—28no—
57#124×C53—28no—
58#121×C53—28no—
59#121×C53—28no—
60C4×D4×C53—22no—
61C4×D4×C53—22no—
62C4×D4×C53D414yes—
63C4×D4×C53C2314yes—
64#117×C53—28no—
65#117×C53—28no—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53D412yes—
68C2×C22⋊C4×C53—24no—
69C2×C22⋊C4×C53—22no—
70C2×C42×C53—22no—
71C2×C42×C53C2311yescharacteristic
72C24×C53—22no—
73C2×Q8×C53—22no—
74C2×Q8×C53—44no—
75C2×Q8×C53C2×D414yes—
76C2×D4×C53—28no—
77C2×D4×C53—22no—
78C2×D4×C53—44no—
79C2×D4×C53—84no—
80C2×D4×C53C2×D414yes—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—24no—
84C22×C4×C53—24no—
85C22×C4×C53—48no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D412yes—
88C2×C22⋊D53—41no—
89#11719945—44no—
90C2×D212—22no—
91C2×D212—21no—
92C2×C4×D53—41no—
93C2×Dic106—21no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53C2×D414yes—
98C22⋊C4×C53—216no—
99C22⋊C4×C53—24no—
100C22⋊C4×C53—24no—
101C42×C53—28no—
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D412yes—
104C42×C53C2411yescharacteristic
105#11719925—48no—
106#11719924—48no—
107#11719922—48no—
108#11719921—42no—
109C4⋊Dic53—22no—
110#11719917—24no—
111C2×#88322—22no—
112C2×#88322—24no—
113C23×C53—24no—
114C23×C53—44no—
115C23×C53—44no—
116C23×C53—44no—
117C23×C53#12011yescharacteristic
118C22×D53—41no—
119Q8×C53—44no—
120Q8×C53—82no—
121D4×C53—24no—
122D4×C53—24no—
123D4×C53—44no—
124D4×C53—88no—
125C2×C4×C53—22no—
126C2×C4×C53—24no—
127C2×C4×C53—24no—
128C2×C4×C53—24no—
129C2×C4×C53—48no—
130C2×C4×C53—44no—
131C2×C4×C53—44no—
132C2×C4×C53—48no—
133C2×C4×C53#12011yescharacteristic
134C2×C4×C53#12012yes—
135C2×C4×C53C22×D411yescharacteristic
136C2×C4×C53C22×D412yes—
137C22⋊D53—82no—
138C2×Dic53—41no—
139D212—42no—
140D212—44no—
141C4×D53—82no—
142Dic106—42no—
143#88322—24no—
144#88322—44no—
145#18547—531nomaximal, Sylow
146C22×C53—22no—
147C22×C53—44no—
148C22×C53—44no—
149C22×C53—44no—
150C22×C53—88no—
151C22×C53#45312yes—
152C22×C53C2×#12011yescharacteristic
153C2×D53—82no—
154C2×D53—81no—
155C4×C53—22no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—44no—
159C4×C53—44no—
160C4×C53#57514yes—
161Dic53—81no—
162C2×#555D5311yescharacteristic, complemented
163#3152—531no—
164#3109—532no—
165#1634—534no—
166#1624—534no—
167#1558—531no—
168#1490—532no—
169C2×C53—22no—
170C2×C53—84no—
171C2×C53#300512yes—
172C2×C53C2×#45311yescharacteristic
173D53—161no—
174C2×#137—1062no—
175#555—22no—
176#555—24no—
177#555C2×D5314yes—
178#532—531no—
179C2×C4⋊Q8C2×D5311yescharacteristic
180C2×#124—22no—
181C2×#121—22no—
182C2×C4×D4C2×D5312yes—
183C2×#117—22no—
184#497—1062no—
185#494—532no—
186#492—532no—
187#481—1062no—
188#476—534no—
189#459—534no—
190#447—1062no—
191#442—1064no—
192#432—532no—
193C4×M16—531no—
194#333—534no—
195D4⋊C8—538no—
196C53#1854711yescharacteristic, complemented, Sylow
197C2×#41—1061no—
198C22×Q8C22⋊D5312yes—
199C22×D4—22no—
200C22×D4C22⋊D5312yes—
201C23×C4—22no—
202#137—2128no—
203C2×Q16—1064no—
204C2×SD16—1064no—
205C2×D8—1064no—
206C2×M16—1061no—
207C4⋊Q8—22no—
208C4⋊Q8—531no—
209C4⋊Q8C22⋊D5312yes—
210C4⋊Q8C22×D5312yes—
211#127—531no—
212#124—28no—
213#124—28no—
214#124—1061no—
215#121—28no—
216#121—28no—
217#121—1062no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—1062no—
221C4×D4C22⋊D5314yes—
222C4×D4C22×D5314yes—
223#117—28no—
224#117—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C22⋊D5312yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C42—22no—
230C2×C42C22×D5311yescharacteristic
231#106—1064no—
232C4⋊C8—534no—
233#102—1064no—
234#102—1064no—
235C22⋊C8—1062no—
236#97—1061no—
237#97—1062no—
238C4×C8—532no—
239C24—22no—
240#41—2124no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—1061no—
244C2×Q8C2×C22⋊D5314yes—
245C2×D4—28no—
246C2×D4—22no—
247C2×D4—44no—
248C2×D4—84no—
249C2×D4—1062no—
250C2×D4—1061no—
251C2×D4—2121no—
252C2×D4C2×C22⋊D5314yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—48no—
258C22×C4—2121no—
259C22×C4C2×C22⋊D5312yes—
260C22×C4C2×C22⋊D5311yescharacteristic
261Q16—2128no—
262SD16—2128no—
263D8—2128no—
264M16—1064no—
265C2×C8—1062no—
266C2×C8—1064no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—1062no—
271C4⋊C4C2×C22⋊D5314yes—
272C22⋊C4—216no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—2122no—
276C42—28no—
277C42C2×C22⋊D5311yescharacteristic
278C42C2×C22⋊D5312yes—
279C42C23×D5311yescharacteristic
280C23—24no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—2121no—
285C23#1313318611yescharacteristic
286Q8—44no—
287Q8—82no—
288Q8—2122no—
289D4—24no—
290D4—24no—
291D4—44no—
292D4—88no—
293D4—4242no—
294D4—2122no—
295D4—2124no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—4242no—
305C2×C4—2121no—
306C2×C4#1313318611yescharacteristic
307C2×C4#1313318612yes—
308C2×C4C22×C22⋊D5311yescharacteristic
309C2×C4C22×C22⋊D5312yes—
310C8—1064no—
311C8—2124no—
312C22—22no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—88no—
317C22—4242no—
318C22—4241no—
319C22#1681079112yes—
320C22C2×#1313318611yescharacteristic
321C4—22no—
322C4—21no—
323C4—21no—
324C4—44no—
325C4—44no—
326C4—4241no—
327C4#1681142114yes—
328C2—22no—
329C2—84no—
330C2—8481no—
331C2#2028395712yes—
332C2C2×#1681079111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#555C2
socleC22×C53C2×#120
hypercenterC2×#555D53
hypocenterC53#18547
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #20285172C2semidirect
2. #20284806C2semidirect
3. #20273860C2semidirect
4. #20273798C2semidirect
5. #20273441C2semidirect
6. #20273158C2semidirect
7. C2×C53×#555C2semidirect
11. #16811002C22semidirect
12. #16810997C22semidirect
14. #16810913C22semidirect
15. #16810818C22semidirect
20. #16810372C22semidirect
21. D4⋊#88322C22semidirect
162. C2×#555D53semidirect
196. C53#18547semidirect

Maximal subgroups

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

subgroupindex
#202851722
#202848062
#202738602
#202737982
#202734412
#202731582
C2×C53×#5552
#1854753