Finite Group Databasebeta

← #20283982

Subgroups — #20283982

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
2C22×#11719922C211yesmaximal, characteristic, complemented
3#16811190C211yesmaximal, characteristic, complemented
4#16810767C214yesmaximal, complemented
5#16810755C214yesmaximal, complemented
6#16810629C211yesmaximal, characteristic, complemented
7#16810619C211yesmaximal, characteristic, complemented
8#16810591C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C53C211yesmaximal, characteristic, complemented
10#13133263—24no—
11#13133263—24no—
12C2×#11719949C2212yescomplemented
13C2×#11719945—21no—
14C2×#11719945C2211yescharacteristic
15C22×D212—21no—
16C22×D212C2211yescharacteristic
17C2×D4×D53—22no—
18C2×D4×D53C2212yescomplemented
19C2×#11719922—24no—
20C2×#11719922C2214yescomplemented
21C22×#88322C2211yescharacteristic, complemented
22#13133213—21no—
23#13133204—21no—
24#13133188C2214yescomplemented
25#13133186—22no—
26#13133176C2211yescharacteristic, complemented
27#13133141C2212yescomplemented
28#13133139C2214yescomplemented
29#13133138C2212yescomplemented
30#13133129C2214yescomplemented
31C2×#41×C53C2211yescharacteristic
32C22×D4×C53C2211yescharacteristic, complemented
33AGL(1,ℤ/8)×C53—24no—
34C2×SD16×C53C2212yescomplemented
35C2×D8×C53C2212yescomplemented
36C2×M16×C53C2211yescharacteristic, complemented
37C24×C53—21no—
38C23×D53—21no—
39C23×D53—22no—
40#41×C53—22no—
41#41×C53D414yescomplemented
42C2×Q8×C53C2311yescharacteristic
43C2×D4×C53—21no—
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53—21no—
47C2×D4×C53C2312yes—
48C2×D4×C53C2311yescharacteristic
49C22×C4×C53—21no—
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—24no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53D412yes—
54#11719949—24no—
55#11719949—24no—
56D4×D53—216no—
57D4×D53—24no—
58D4×D53—24no—
59#11719945—28no—
60#11719945—22no—
61#11719945—22no—
62C2×D212—28no—
63C2×D212—22no—
64C2×D212—21no—
65C2×D212—22no—
66C2×D212D411yescharacteristic
67C2×D212D414yescomplemented
68C2×D212C2312yes—
69C2×C4×D53—22no—
70C2×C4×D53C2312yes—
71C2×Dic106D411yescharacteristic
72SD16×C53—24no—
73D8×C53—24no—
74M16×C53—22no—
75C2×C8×C53C2312yes—
76C22⋊Dic53—21no—
77C22⋊Dic53—24no—
78#11719922—28no—
79C4⋊Dic53C2312yes—
80C4×Dic53—21no—
81C2×#88322—22no—
82C2×#88322C2312yes—
83C23×C53—21no—
84C23×C53—22no—
85C23×C53—42no—
86C23×C53—82no—
87C23×C53C2×D411yescharacteristic
88C22×D53—22no—
89C22×D53—28no—
90C22×D53—24no—
91C22×D53—84no—
92C22×D53—82no—
93C22×D53C2×D412yes—
94Q8×C53—21no—
95Q8×C53C2×D412yes—
96D4×C53—24no—
97D4×C53—21no—
98D4×C53—22no—
99D4×C53—22no—
100D4×C53—42no—
101D4×C53C2×D412yes—
102C2×C4×C53—21no—
103C2×C4×C53—24no—
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53C2×D412yes—
106C2×C4×C53C2×D412yes—
107C2×C4×C53C2411yescharacteristic
108C22⋊D53—28no—
109C22⋊D53—24no—
110C22⋊D53—84no—
111C2×Dic53—22no—
112C2×Dic53C2×D412yes—
113D212—28no—
114D212—22no—
115D212—24no—
116D212—22no—
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—42no—
128C22×C53—44no—
129C22×C53—84no—
130C22×C53#12012yes—
131C22×C53C22×D411yescharacteristic
132C2×D53—22no—
133C2×D53—24no—
134C2×D53—24no—
135C2×D53—44no—
136C2×D53—84no—
137C2×D53—88no—
138C4×C53—22no—
139C4×C53#12012yes—
140C4×C53C22×D411yescharacteristic
141C4×C53C22×D411yescharacteristic
142Dic53—22no—
143Dic53—42no—
144#3008—531nomaximal, Sylow
145C2×C53—22no—
146C2×C53—42no—
147C2×C53—82no—
148C2×C53C2×#12011yescharacteristic
149C2×C53#57612yes—
150D53—42no—
151D53—44no—
152C2×H(5,𝔽2)—531no—
153C2×AGL(1,ℤ/8)D5311yescharacteristic, complemented
154C22×D8—531no—
155#535—531no—
156#449—534no—
157#447—534no—
158#418—531no—
159#416—531no—
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—1061no—
171C22×D4—1062no—
172C22×D4C2×D5311yescharacteristic, complemented
173AGL(1,ℤ/8)—24no—
174C2×SD16C2×D5312yescomplemented
175C2×D8—534no—
176C2×D8—1064no—
177C2×D8C2×D5312yescomplemented
178C2×M16C2×D5311yescharacteristic, complemented
179C22×C8—531no—
180#127—1061no—
181#124—1061no—
182#121—534no—
183#120—1062no—
184#117—531no—
185#103—532no—
186#102—532no—
187#102—534no—
188C22⋊C8—534no—
189C24—21no—
190C24—1062no—
191C24—1061no—
192#41—22no—
193#41—1068no—
194#41—1064no—
195#41—1064no—
196#41—1062no—
197#41—1062no—
198#41C22⋊D5314yescomplemented
199C2×Q8—531no—
200C2×Q8C22×D5311yescharacteristic
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—532no—
206C2×D4—534no—
207C2×D4—531no—
208C2×D4—532no—
209C2×D4—1062no—
210C2×D4—1068no—
211C2×D4—1062no—
212C2×D4—1064no—
213C2×D4—10616no—
214C2×D4—1064no—
215C2×D4—1064no—
216C2×D4—1061no—
217C2×D4—1062no—
218C2×D4C22×D5312yes—
219C2×D4C22×D5311yescharacteristic
220C22×C4—21no—
221C22×C4—532no—
222C22×C4—1062no—
223C22×C4C22×D5311yescharacteristic
224SD16—24no—
225D8—24no—
226D8—1068no—
227M16—22no—
228C2×C8—532no—
229C2×C8—1062no—
230C2×C8C22×D5312yes—
231C4⋊C4—532no—
232C22⋊C4—1064no—
233C22⋊C4—1061no—
234C42—1061no—
235C23—21no—
236C23—22no—
237C23—42no—
238C23—82no—
239C23—532no—
240C23—1068no—
241C23—1064no—
242C23—1062no—
243C23—4242no—
244C23—4244no—
245C23D4×D5311yescharacteristic
246Q8—21no—
247Q8—1062no—
248Q8C2×C22⋊D5312yes—
249D4—24no—
250D4—21no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—534no—
255D4—1068no—
256D4—1068no—
257D4—1064no—
258D4—1064no—
259D4—1062no—
260D4—1062no—
261D4—4244no—
262D4C2×C22⋊D5312yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—532no—
266C2×C4—1068no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4D4×D5311yescharacteristic
271C2×C4D4×D5312yes—
272C2×C4C2×C22⋊D5312yes—
273C2×C4C23×D5311yescharacteristic
274C8—22no—
275C8—1062no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—1064no—
283C22—1062no—
284C22—1064no—
285C22—4244no—
286C22—4248no—
287C22—2124no—
288C22#1313318512yes—
289C22C2×D4×D5311yescharacteristic
290C4—22no—
291C4—1062no—
292C4—2122no—
293C4#1313318512yes—
294C4C2×D4×D5311yescharacteristic
295C4C22×C22⋊D5311yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—2122no—
300C2—2124no—
301C2C2×#1313318511yescharacteristic
302C2#1681142712yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×AGL(1,ℤ/8)×C53C2
socleC22×C53C22×D4
hypercenterC2×AGL(1,ℤ/8)D53
hypocenterC53#3008
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. C22×#11719922C2semidirect
3. #16811190C2semidirect
4. #16810767C2semidirect
5. #16810755C2semidirect
6. #16810629C2semidirect
7. #16810619C2semidirect
8. #16810591C2semidirect
9. C2×AGL(1,ℤ/8)×C53C2semidirect
12. C2×#11719949C22semidirect
18. C2×D4×D53C22semidirect
20. C2×#11719922C22semidirect
21. C22×#88322C22semidirect
24. #13133188C22semidirect
26. #13133176C22semidirect
27. #13133141C22semidirect
28. #13133139C22semidirect
29. #13133138C22semidirect
30. #13133129C22semidirect
32. C22×D4×C53C22semidirect
34. C2×SD16×C53C22semidirect
35. C2×D8×C53C22semidirect
36. C2×M16×C53C22semidirect
41. #41×C53D4semidirect
67. C2×D212D4semidirect
153. C2×AGL(1,ℤ/8)D53semidirect
161. C53#3008semidirect
172. C22×D4C2×D53semidirect
174. C2×SD16C2×D53semidirect
177. C2×D8C2×D53semidirect
178. C2×M16C2×D53semidirect
198. #41C22⋊D53semidirect