Finite Group Databasebeta

← #20283944

Subgroups — #20283944

Order 6784.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D53C211yesmaximal, characteristic, complemented
2C2×#13133245C211yesmaximal, characteristic, complemented
3C2×#13133243C211yesmaximal, characteristic, complemented
4#16811135C211yesmaximal, characteristic, complemented
5#16810775C214yesmaximal, complemented
6#16810758C214yesmaximal, complemented
7#16810627C211yesmaximal, characteristic, complemented
8#16810586C211yesmaximal, characteristic, complemented
9C53×#418C211yesmaximal, characteristic, complemented
10C24×D53—21no—
11C2×#11719945C2211yescharacteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53—21no—
14C2×D4×D53—24no—
15C2×D4×D53—24no—
16C2×D4×D53—22no—
17C2×D4×D53C2214yes—
18C22×C4×D53C2211yescharacteristic
19#13133245—24no—
20#13133243—24no—
21C2×#11719913C2212yescomplemented
22C2×#11719923C2212yescomplemented
23C2×D424C2212yescomplemented
24C2×#11719922C2212yescomplemented
25C2×#11719917C2211yescharacteristic, complemented
26#13133201—21no—
27#13133197C2212yescomplemented
28#13133192C2212yescomplemented
29#13133187—21no—
30C4×D212—22no—
31#13133137C2214yescomplemented
32#13133129C2214yescomplemented
33C22×D4×C53C2211yescharacteristic
34C2×M16×C53C2211yescharacteristic, complemented
35#117×C53C2211yescharacteristic, complemented
36#102×C53C2214yescomplemented
37C24×C53—21no—
38C23×D53—28no—
39C23×D53—21no—
40C23×D53—24no—
41C23×D53—48no—
42C23×D53D411yescharacteristic
43C2×D4×C53—21no—
44C2×D4×C53—22no—
45C2×D4×C53D414yescomplemented
46C2×D4×C53C2312yes—
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—28no—
49C2×C22⋊D53—21no—
50C2×C22⋊D53—24no—
51C22×Dic53D411yescharacteristic
52D4×D53—216no—
53D4×D53—28no—
54D4×D53—28no—
55#11719945—22no—
56#11719945—22no—
57C2×D212—21no—
58C2×D212—22no—
59C2×D212—22no—
60C2×D212D414yescomplemented
61C2×D212C2312yes—
62C2×D212C2311yescharacteristic, complemented
63C2×C4×D53—22no—
64C2×C4×D53—22no—
65C2×C4×D53—21no—
66C2×C4×D53D412yes—
67C2×C4×D53C2312yes—
68C2×Dic106C2311yescharacteristic, complemented
69M16×C53—22no—
70C2×C8×C53C2312yescomplemented
71C4⋊C4×C53C2312yescomplemented
72C22⋊C4×C53—21no—
73C42×C53—21no—
74#11719923—24no—
75#11719922—24no—
76#11719921—22no—
77#11719921—22no—
78C4⋊Dic53—22no—
79#11719917—22no—
80C2×#88322C2312yescomplemented
81D424—24no—
82#11719913—24no—
83C23×C53—24no—
84C23×C53—22no—
85C23×C53—44no—
86C23×C53C2×D411yescharacteristic
87C22×D53—22no—
88C22×D53—24no—
89C22×D53—216no—
90C22×D53—24no—
91C22×D53—22no—
92C22×D53—21no—
93C22×D53—432no—
94C22×D53—48no—
95C22×D53—44no—
96C22×D53—48no—
97C22×D53C2×D412yes—
98D4×C53—24no—
99D4×C53—22no—
100D4×C53C2×D414yes—
101C2×C4×C53—22no—
102C2×C4×C53C2×D412yes—
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2×D412yes—
105C2×C4×C53C2411yescharacteristic
106C22⋊D53—216no—
107C22⋊D53—42no—
108C2×Dic53—22no—
109C2×Dic53—21no—
110C2×Dic53C2×D412yes—
111D212—24no—
112D212—22no—
113D212—21no—
114D212—21no—
115D212—44no—
116D212C2×D414yes—
117C4×D53—24no—
118C4×D53—22no—
119C4×D53—22no—
120C4×D53—42no—
121Dic106—21no—
122Dic106—21no—
123C8×C53—22no—
124#88322—22no—
125C22×C53—22no—
126C22×C53—24no—
127C22×C53—48no—
128C22×C53—44no—
129C22×C53#12012yes—
130C22×C53C22×D411yescharacteristic
131C2×D53—28no—
132C2×D53—22no—
133C2×D53—22no—
134C2×D53—24no—
135C2×D53—432no—
136C2×D53—48no—
137C2×D53—44no—
138C2×D53—41no—
139C2×D53—42no—
140C2×D53AGL(1,ℤ/8)14yescomplemented
141C4×C53—42no—
142C4×C53#12012yes—
143C4×C53C22×D411yescharacteristic
144C4×C53C22×D411yescharacteristic
145Dic53—22no—
146Dic53—41no—
147#3003—531nomaximal, Sylow
148C2×C53—22no—
149C2×C53—44no—
150C2×C53C2×#12011yescharacteristic
151C2×C53C2×AGL(1,ℤ/8)12yes—
152D53—24no—
153D53—44no—
154D53—81no—
155C23×D4—531no—
156C2×AGL(1,ℤ/8)—531no—
157C2×AGL(1,ℤ/8)—531no—
158#525—531no—
159#450—534no—
160#447—534no—
161#418—531no—
162#418D5311yescharacteristic, complemented
163#407—531no—
164C53#300311yescharacteristic, complemented, Sylow
165C25—1061no—
166C2×#41—531no—
167C22×D4—534no—
168C22×D4—531no—
169C22×D4—1062no—
170C22×D4—1061no—
171C22×D4—1064no—
172C22×D4—1064no—
173C22×D4C2×D5311yescharacteristic
174C23×C4—531no—
175AGL(1,ℤ/8)—1064no—
176AGL(1,ℤ/8)—1064no—
177C2×SD16—532no—
178C2×SD16—532no—
179C2×D8—532no—
180C2×D8—532no—
181C2×M16—531no—
182C2×M16C2×D5311yescharacteristic, complemented
183#123—1061no—
184C22⋊Q8—532no—
185#121—532no—
186#120—1061no—
187C4×D4—1062no—
188#117C2×D5311yescharacteristic, complemented
189#102—534no—
190#102C2×D5314yescomplemented
191C22⋊C8—534no—
192C24—21no—
193C24—531no—
194C24—1064no—
195C24—1068no—
196C24—1061no—
197C24—2128no—
198#41—1062no—
199#41—1062no—
200C2×Q8—531no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—534no—
204C2×D4—532no—
205C2×D4—531no—
206C2×D4—1062no—
207C2×D4—1064no—
208C2×D4—1068no—
209C2×D4—10616no—
210C2×D4—1068no—
211C2×D4—1068no—
212C2×D4—1061no—
213C2×D4—1062no—
214C2×D4—1061no—
215C2×D4D21214yescomplemented
216C2×D4C22×D5312yes—
217C22×C4—531no—
218C22×C4—532no—
219C22×C4—532no—
220C22×C4—1062no—
221C22×C4—1062no—
222C22×C4—1061no—
223C22×C4C22×D5311yescharacteristic
224SD16—1064no—
225SD16—1064no—
226D8—1064no—
227D8—1064no—
228M16—22no—
229M16—1062no—
230C2×C8—532no—
231C2×C8C22×D5312yescomplemented
232C4⋊C4—1062no—
233C4⋊C4C22×D5312yescomplemented
234C22⋊C4—21no—
235C22⋊C4—1062no—
236C22⋊C4—1062no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—532no—
242C23—1064no—
243C23—10616no—
244C23—1062no—
245C23—1064no—
246C23—1062no—
247C23—1061no—
248C23—2124no—
249C23—2128no—
250C23—21232no—
251C23—2128no—
252C23D4×D5311yescharacteristic
253Q8—1061no—
254Q8—1061no—
255D4—24no—
256D4—22no—
257D4—534no—
258D4—10616no—
259D4—1064no—
260D4—1061no—
261D4—1062no—
262D4—1061no—
263D4—2124no—
264D4—2122no—
265D4C2×D21214yes—
266C2×C4—22no—
267C2×C4—532no—
268C2×C4—1062no—
269C2×C4—1064no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—1061no—
273C2×C4—2122no—
274C2×C4C2×D21212yes—
275C2×C4D4×D5311yescharacteristic
276C2×C4D4×D5312yes—
277C2×C4C23×D5311yescharacteristic
278C8—22no—
279C8—1062no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—534no—
285C22—1068no—
286C22—1062no—
287C22—1062no—
288C22—1064no—
289C22—2128no—
290C22—21232no—
291C22—2121no—
292C22—2122no—
293C22—2124no—
294C22#1313318712yes—
295C22C2×D4×D5311yescharacteristic
296C4—42no—
297C4—1062no—
298C4—2121no—
299C4#1313318712yes—
300C4C2×D4×D5311yescharacteristic
301C4C22×D21211yescharacteristic
302C2—22no—
303C2—44no—
304C2—1064no—
305C2—4241no—
306C2—2124no—
307C2C2×#1313318711yescharacteristic
308C2AGL(1,ℤ/8)×D5312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#418C2
socleC22×C53C22×D4
hypercenter#418D53
hypocenterC53#3003
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D53C2semidirect
2. C2×#13133245C2semidirect
3. C2×#13133243C2semidirect
4. #16811135C2semidirect
5. #16810775C2semidirect
6. #16810758C2semidirect
7. #16810627C2semidirect
8. #16810586C2semidirect
9. C53×#418C2semidirect
11. C2×#11719945C22semidirect
21. C2×#11719913C22semidirect
22. C2×#11719923C22semidirect
23. C2×D424C22semidirect
24. C2×#11719922C22semidirect
25. C2×#11719917C22semidirect
27. #13133197C22semidirect
28. #13133192C22semidirect
31. #13133137C22semidirect
32. #13133129C22semidirect
34. C2×M16×C53C22semidirect
35. #117×C53C22semidirect
36. #102×C53C22semidirect
45. C2×D4×C53D4semidirect
60. C2×D212D4semidirect
62. C2×D212C23semidirect
68. C2×Dic106C23semidirect
70. C2×C8×C53C23semidirect
71. C4⋊C4×C53C23semidirect
80. C2×#88322C23semidirect
140. C2×D53AGL(1,ℤ/8)semidirect
162. #418D53semidirect
164. C53#3003semidirect
182. C2×M16C2×D53semidirect
188. #117C2×D53semidirect
190. #102C2×D53semidirect
215. C2×D4D212semidirect
231. C2×C8C22×D53semidirect
233. C4⋊C4C22×D53semidirect