Finite Group Databasebeta

← #20283986

Subgroups — #20283986

Order 6784.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133263C211yesmaximal, characteristic, complemented
2C2×#13133242C211yesmaximal, characteristic, complemented
3C22×D424C211yesmaximal, characteristic, complemented
4#16811192C211yesmaximal, characteristic, complemented
5#16810771C214yesmaximal, complemented
6#16810758C214yesmaximal, complemented
7#16810627C211yesmaximal, characteristic, complemented
8#16810589C211yesmaximal, characteristic, complemented
9C53×#416C211yesmaximal, characteristic, complemented
10#13133263—24no—
11#13133263—24no—
12C2×#11719949C2212yescomplemented
13C2×#11719945—21no—
14C2×#11719945C2211yescharacteristic
15C22×D212—21no—
16C22×D212C2211yescharacteristic
17C22×D212C2211yescharacteristic, complemented
18C2×D4×D53—22no—
19C2×D4×D53C2212yescomplemented
20#13133242—24no—
21C2×#11719924C2212yescomplemented
22C2×D424—24no—
23C2×D424C2214yescomplemented
24C2×#11719922C2212yescomplemented
25C2×#11719917C2211yescharacteristic, complemented
26#13133214—21no—
27#13133205—21no—
28#13133192C2214yescomplemented
29#13133187—22no—
30#13133137C2214yescomplemented
31#13133129C2214yescomplemented
32C2×#41×C53C2211yescharacteristic
33C22×C8×C53C2211yescharacteristic, complemented
34#117×C53C2211yescharacteristic, complemented
35#103×C53C2212yescomplemented
36#102×C53C2212yescomplemented
37C23×D53—21no—
38C23×D53—22no—
39C23×D53—21no—
40#41×C53—22no—
41#41×C53D414yescomplemented
42C2×Q8×C53C2311yescharacteristic
43C2×D4×C53—21no—
44C2×D4×C53C2311yescharacteristic
45C22×C4×C53—21no—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—22no—
48C2×C22⋊D53D412yes—
49#11719949—24no—
50#11719949—24no—
51D4×D53—216no—
52D4×D53—24no—
53D4×D53—24no—
54#11719945—28no—
55#11719945—22no—
56#11719945—22no—
57C2×D212—28no—
58C2×D212—22no—
59C2×D212—21no—
60C2×D212—22no—
61C2×D212—22no—
62C2×D212—22no—
63C2×D212—21no—
64C2×D212—24no—
65C2×D212D411yescharacteristic
66C2×D212D414yescomplemented
67C2×D212C2312yes—
68C2×D212C2312yes—
69C2×C4×D53—22no—
70C2×C4×D53C2312yes—
71C2×Dic106D411yescharacteristic
72C2×C8×C53—22no—
73C2×C8×C53C2312yes—
74C4⋊C4×C53C2312yes—
75C22⋊C4×C53—21no—
76C42×C53—21no—
77#11719924—24no—
78#11719922—24no—
79#11719921—24no—
80#11719917—22no—
81C2×#88322C2312yes—
82D424—28no—
83C23×C53—21no—
84C23×C53C2×D411yescharacteristic
85C22×D53—22no—
86C22×D53—28no—
87C22×D53—24no—
88C22×D53—22no—
89C22×D53—42no—
90C22×D53—84no—
91C22×D53—82no—
92C22×D53—82no—
93C22×D53C2×D412yes—
94Q8×C53—21no—
95Q8×C53C2×D412yes—
96D4×C53—24no—
97D4×C53—21no—
98D4×C53C2×D412yes—
99C2×C4×C53—21no—
100C2×C4×C53—24no—
101C2×C4×C53—22no—
102C2×C4×C53C2×D412yes—
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2×D412yes—
105C2×C4×C53C2411yescharacteristic
106C22⋊D53—28no—
107C22⋊D53—24no—
108C2×Dic53C2×D412yes—
109D212—28no—
110D212—22no—
111D212—24no—
112D212—22no—
113D212—22no—
114D212—22no—
115D212—42no—
116D212—84no—
117D212C2×D414yes—
118C4×D53—28no—
119C4×D53—22no—
120C4×D53—22no—
121Dic106—22no—
122C8×C53—22no—
123#88322—22no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—42no—
127C22×C53#12012yes—
128C22×C53C22×D411yescharacteristic
129C2×D53—22no—
130C2×D53—24no—
131C2×D53—24no—
132C2×D53—44no—
133C2×D53—42no—
134C2×D53—44no—
135C2×D53—84no—
136C2×D53—88no—
137C2×D53—84no—
138C4×C53—22no—
139C4×C53—42no—
140C4×C53#12012yes—
141C4×C53C22×D411yescharacteristic
142C4×C53C22×D411yescharacteristic
143Dic53—22no—
144#3008—531nomaximal, Sylow
145C2×C53—22no—
146C2×C53—42no—
147C2×C53C2×#12011yescharacteristic
148C2×C53#57612yes—
149D53—42no—
150D53—44no—
151D53—82no—
152C2×H(5,𝔽2)—531no—
153C2×AGL(1,ℤ/8)—531no—
154C22×D8—531no—
155#535—531no—
156#449—534no—
157#447—534no—
158#418—531no—
159#416D5311yescharacteristic, complemented
160#408—531no—
161C53#300811yescharacteristic, complemented, Sylow
162H(5,𝔽2)—1064no—
163H(5,𝔽2)—1064no—
164C2×#41—531no—
165C2×#41—532no—
166C2×#41—1061no—
167C2×#41C2×D5311yescharacteristic
168C22×D4—532no—
169C22×D4—531no—
170C22×D4—531no—
171C22×D4—1061no—
172C22×D4—1062no—
173AGL(1,ℤ/8)—1064no—
174C2×SD16—532no—
175C2×D8—532no—
176C2×D8—534no—
177C2×D8—1064no—
178C2×M16—531no—
179C22×C8C2×D5311yescharacteristic, complemented
180#127—1061no—
181#124—1061no—
182#121—534no—
183#120—1062no—
184#117C2×D5311yescharacteristic, complemented
185#103C2×D5312yescomplemented
186#102—534no—
187#102C2×D5312yescomplemented
188C22⋊C8—534no—
189C24—1062no—
190C24—1061no—
191C24—1061no—
192#41—22no—
193#41—1068no—
194#41—1064no—
195#41—1064no—
196#41—1062no—
197#41—1062no—
198#41D21214yescomplemented
199C2×Q8—531no—
200C2×Q8C22×D5311yescharacteristic
201C2×D4—21no—
202C2×D4—532no—
203C2×D4—534no—
204C2×D4—531no—
205C2×D4—532no—
206C2×D4—532no—
207C2×D4—1062no—
208C2×D4—1068no—
209C2×D4—1064no—
210C2×D4—1062no—
211C2×D4—1064no—
212C2×D4—10616no—
213C2×D4—1064no—
214C2×D4—1062no—
215C2×D4—1062no—
216C2×D4—1061no—
217C2×D4—1062no—
218C2×D4—1061no—
219C2×D4C22×D5311yescharacteristic
220C22×C4—21no—
221C22×C4—532no—
222C22×C4—1062no—
223C22×C4C22×D5311yescharacteristic
224SD16—1064no—
225D8—1068no—
226D8—1064no—
227M16—1062no—
228C2×C8—22no—
229C2×C8—532no—
230C2×C8C22×D5312yes—
231C4⋊C4C22×D5312yes—
232C22⋊C4—21no—
233C22⋊C4—1064no—
234C42—21no—
235C23—21no—
236C23—532no—
237C23—1068no—
238C23—1064no—
239C23—1062no—
240C23—1062no—
241C23—4242no—
242C23—4242no—
243C23—4244no—
244C23—2122no—
245C23D4×D5311yescharacteristic
246Q8—21no—
247Q8—1062no—
248Q8C2×D21212yes—
249D4—24no—
250D4—21no—
251D4—534no—
252D4—1068no—
253D4—1068no—
254D4—1064no—
255D4—1064no—
256D4—1062no—
257D4—1062no—
258D4—1062no—
259D4—1062no—
260D4—4244no—
261D4—2122no—
262D4C2×D21212yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—532no—
267C2×C4—1068no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4C2×D21212yes—
271C2×C4D4×D5311yescharacteristic
272C2×C4D4×D5312yes—
273C2×C4C23×D5311yescharacteristic
274C8—22no—
275C8—1062no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—1064no—
280C22—1062no—
281C22—1064no—
282C22—4244no—
283C22—4244no—
284C22—4248no—
285C22—2122no—
286C22—2124no—
287C22—2124no—
288C22#1313318712yes—
289C22C2×D4×D5311yescharacteristic
290C4—22no—
291C4—42no—
292C4—1062no—
293C4#1313318712yes—
294C4C2×D4×D5311yescharacteristic
295C4C22×D21211yescharacteristic
296C2—22no—
297C2—42no—
298C2—4242no—
299C2—2122no—
300C2—2124no—
301C2C2×#1313318711yescharacteristic
302C2#1681142512yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#416C2
socleC22×C53C22×D4
hypercenter#416D53
hypocenterC53#3008
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. C2×#13133242C2semidirect
3. C22×D424C2semidirect
4. #16811192C2semidirect
5. #16810771C2semidirect
6. #16810758C2semidirect
7. #16810627C2semidirect
8. #16810589C2semidirect
9. C53×#416C2semidirect
12. C2×#11719949C22semidirect
17. C22×D212C22semidirect
19. C2×D4×D53C22semidirect
21. C2×#11719924C22semidirect
23. C2×D424C22semidirect
24. C2×#11719922C22semidirect
25. C2×#11719917C22semidirect
28. #13133192C22semidirect
30. #13133137C22semidirect
31. #13133129C22semidirect
33. C22×C8×C53C22semidirect
34. #117×C53C22semidirect
35. #103×C53C22semidirect
36. #102×C53C22semidirect
41. #41×C53D4semidirect
66. C2×D212D4semidirect
159. #416D53semidirect
161. C53#3008semidirect
179. C22×C8C2×D53semidirect
184. #117C2×D53semidirect
185. #103C2×D53semidirect
187. #102C2×D53semidirect
198. #41D212semidirect