Finite Group Databasebeta

← #27365377

Subgroups — #27365377

Order 13568.

960 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C2×#16811371C211yesmaximal, characteristic, complemented
2#20285698C212yesmaximal, complemented
3#20274036C214yesmaximal, complemented
4#20274007C214yesmaximal, complemented
5#20273342C211yesmaximal, characteristic, complemented
6#20273312C212yesmaximal, complemented
7C53×#3142C211yesmaximal, characteristic, complemented
8#16811371—24no—
9#16811371—22no—
10#16811371C2214yes—
11C2×#13133205C2212yes—
12C2×#13133202—22no—
13C2×#13133195—22no—
14C2×#13133192—22no—
15#16811010C2214yescomplemented
16#16810985C2214yescomplemented
17#16810838—24no—
18#16810761—22no—
19#16810756—22no—
20#16810680C2212yes—
21#16810365C2214yescomplemented
22#16810362C2218yescomplemented
23C53×#532C2211yescharacteristic
24C2×#125×C53C2211yescharacteristic
25C53×#491C2212yescomplemented
26C53×#490C2212yescomplemented
27C53×#488—22no—
28C53×#417—22no—
29C4×M16×C53C2211yescharacteristic
30C22×Dic106D412yes—
31C22×D212—22no—
32C22×D212D412yes—
33C22×C4×D53—22no—
34C2×#11719923—24no—
35C2×#11719922—24no—
36#13133205—22no—
37#13133205—22no—
38#13133205D414yes—
39#13133205C2314yes—
40#13133202—28no—
41#13133202—28no—
42#13133195—28no—
43#13133195—28no—
44#13133192—28no—
45#13133192—28no—
46C2×#11719919—22no—
47C2×#11719921—22no—
48C2×#11719921—22no—
49C2×#11719921—22no—
50C4⋊#88322—22no—
51#13133140—24no—
52#13133137—24no—
53C22⋊#88322—22no—
54#13133128C2314yes—
55C2×#41×C53—21no—
56C2×M16×C53—21no—
57C4⋊Q8×C53C2311yescharacteristic
58#127×C53C2311yescharacteristic
59#125×C53—22no—
60#125×C53D412yes—
61#125×C53C2312yes—
62#124×C53—21no—
63#121×C53—22no—
64C4×D4×C53—22no—
65C2×C4⋊C4×C53—22no—
66C2×C4⋊C4×C53D412yes—
67C2×C42×C53C2311yescharacteristic
68#103×C53—24no—
69#102×C53—24no—
70#97×C53C2312yes—
71C4×C8×C53—21no—
72C23×D53—22no—
73#41×C53—44no—
74C2×Q8×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—22no—
77C2×D4×C53—41no—
78C22×C4×C53—22no—
79C22×C4×C53—41no—
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D412yes—
82C22×Dic53—22no—
83C2×D212—28no—
84C2×D212—22no—
85C2×D212—82no—
86C2×D212—84no—
87C2×D212C2×D414yes—
88C2×C4×D53—28no—
89C2×C4×D53—84no—
90C2×Dic106—22no—
91C2×Dic106—82no—
92C2×Dic106C2×D414yes—
93M16×C53—42no—
94C2×C8×C53—22no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53C2×D414yes—
100C22⋊C4×C53—42no—
101C42×C53C2×D412yes—
102C42×C53C2×D411yescharacteristic
103C42×C53C2411yescharacteristic
104#11719923—84no—
105#11719922—84no—
106#11719921—28no—
107#11719921—24no—
108#11719921—24no—
109#11719921—24no—
110#11719921—24no—
111#11719919—28no—
112C2×#88322—24no—
113C23×C53—41no—
114C23×C53#12011yescharacteristic
115C22×D53—24no—
116C22×D53—44no—
117C22×D53—44no—
118C22×D53—44no—
119Q8×C53—42no—
120D4×C53—42no—
121D4×C53—44no—
122D4×C53—82no—
123C2×C4×C53—22no—
124C2×C4×C53—24no—
125C2×C4×C53—42no—
126C2×C4×C53—44no—
127C2×C4×C53—41no—
128C2×C4×C53—82no—
129C2×C4×C53#12012yes—
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53C22×D412yes—
132C2×C4×C53C22×D411yescharacteristic
133C2×Dic53—24no—
134C2×Dic53—44no—
135D212—44no—
136D212—82no—
137D212—88no—
138C4×D53—88no—
139Dic106—44no—
140Dic106—82no—
141C8×C53—42no—
142#88322—44no—
143#19001—531nomaximal, Sylow
144C22×C53—22no—
145C22×C53—81no—
146C22×C53—82no—
147C22×C53#45412yes—
148C22×C53C2×#12011yescharacteristic
149C2×D53—44no—
150C2×D53—44no—
151C2×D53—44no—
152C2×D53—88no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—44no—
158C4×C53—81no—
159Dic53—44no—
160C2×#562—531no—
161#3213—532no—
162#3142D5311yescharacteristic, complemented
163#1662—534no—
164#1656—534no—
165#1534—531no—
166#1527—532no—
167C2×C53—22no—
168C2×C53—161no—
169C2×C53C2×#45411yescharacteristic
170C2×C53#301912yes—
171D53—84no—
172#562—534no—
173#562—1064no—
174#562—1062no—
175#532C2×D5311yescharacteristic
176C2×#125C2×D5311yescharacteristic
177C2×#124—532no—
178C2×#123—1062no—
179C2×C22⋊Q8—1062no—
180C2×#121—1062no—
181#496—534no—
182#491C2×D5312yescomplemented
183#490C2×D5312yescomplemented
184#489—534no—
185#488—22no—
186#463—1064no—
187#448—1062no—
188#447—1062no—
189#432—532no—
190#417—22no—
191C4×M16C2×D5311yescharacteristic
192#330—534no—
193#329—538no—
194C53#1900111yescharacteristic, complemented, Sylow
195C2×#41—21no—
196C22×Q8—532no—
197C22×D4—532no—
198C22×D4—1062no—
199C23×C4—1062no—
200C2×SD16—1064no—
201C2×D8—1064no—
202C2×M16—21no—
203C4⋊Q8C22×D5311yescharacteristic
204#127C22×D5311yescharacteristic
205#125—22no—
206#125C22⋊D5312yes—
207#125C22×D5312yes—
208#124—21no—
209#124—534no—
210#124—534no—
211#124—1062no—
212#124—1062no—
213#123—1068no—
214#123—1068no—
215C22⋊Q8—1068no—
216C22⋊Q8—1068no—
217#121—22no—
218#121—1068no—
219#121—1068no—
220C4×D4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4C22⋊D5312yes—
224C2×C22⋊C4—1062no—
225C2×C22⋊C4—1062no—
226C2×C22⋊C4—1062no—
227C2×C42C22×D5311yescharacteristic
228C4⋊C8—1062no—
229#103—24no—
230#103—1064no—
231#102—24no—
232#102—1064no—
233C22⋊C8—1062no—
234#97—534no—
235#97C22×D5312yes—
236C4×C8—21no—
237C24—1062no—
238#41—44no—
239C2×Q8—21no—
240C2×Q8—534no—
241C2×Q8—1062no—
242C2×Q8—4242no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—534no—
247C2×D4—1068no—
248C2×D4—1062no—
249C2×D4—4244no—
250C2×D4—4242no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—1068no—
254C22×C4—1062no—
255C22×C4—4244no—
256C22×C4D4×D5312yes—
257C22×C4C2×C22⋊D5311yescharacteristic
258SD16—4244no—
259D8—4244no—
260M16—42no—
261C2×C8—22no—
262C2×C8—1064no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—1068no—
268C4⋊C4C2×C22⋊D5314yes—
269C22⋊C4—42no—
270C22⋊C4—1068no—
271C22⋊C4—1064no—
272C22⋊C4—1064no—
273C22⋊C4—1064no—
274C22⋊C4—1064no—
275C42D4×D5312yes—
276C42C2×C22⋊D5311yescharacteristic
277C42C23×D5311yescharacteristic
278C23—41no—
279C23—1064no—
280C23—2124no—
281C23—2124no—
282C23—2124no—
283C23#1313318511yescharacteristic
284Q8—42no—
285Q8—4242no—
286Q8—2124no—
287D4—44no—
288D4—42no—
289D4—82no—
290D4—4242no—
291D4—4248no—
292D4—2124no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—41no—
298C2×C4—82no—
299C2×C4—1064no—
300C2×C4—4248no—
301C2×C4—2124no—
302C2×C4#1313318511yescharacteristic
303C2×C4#1313318512yes—
304C2×C4C2×D4×D5312yes—
305C2×C4C22×C22⋊D5311yescharacteristic
306C8—42no—
307C8—2124no—
308C22—22no—
309C22—81no—
310C22—82no—
311C22—4248no—
312C22—2124no—
313C22—2124no—
314C22—2124no—
315C22#1681079512yes—
316C22C2×#1313318511yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—44no—
322C4—81no—
323C4—2124no—
324C2—22no—
325C2—161no—
326C2—4244no—
327C2C2×#1681079511yescharacteristic
328C2#2028407012yes—
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3142C2
socleC22×C53C2×#120
hypercenter#3142D53
hypocenterC53#19001
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
1. C2×#16811371C2semidirect
2. #20285698C2semidirect
3. #20274036C2semidirect
4. #20274007C2semidirect
5. #20273342C2semidirect
6. #20273312C2semidirect
7. C53×#3142C2semidirect
15. #16811010C22semidirect
16. #16810985C22semidirect
21. #16810365C22semidirect
22. #16810362C22semidirect
25. C53×#491C22semidirect
26. C53×#490C22semidirect
162. #3142D53semidirect
182. #491C2×D53semidirect
183. #490C2×D53semidirect
194. C53#19001semidirect

Maximal subgroups

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

subgroupindex
C2×#168113712
#202856982
#202740362
#202740072
#202733422
#202733122
C53×#31422
#1900153