Finite Group Databasebeta

← #20285542

Subgroups — #20285542

Order 6784.

414 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 113 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#16811133C211yesmaximal, characteristic
2C2×#13133193C211yesmaximal, characteristic, complemented
3#16810963C212yesmaximal, complemented
4#16810959C211yesmaximal, characteristic, complemented
5#16810939C211yesmaximal, characteristic, complemented
6#16810880C212yesmaximal, complemented
7#16810866C212yesmaximal, complemented
8#16810782C212yesmaximal, complemented
9#16810779C211yesmaximal, characteristic, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#448C211yesmaximal, characteristic, complemented
12C2×#11719947C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C2×#11719925C2212yescomplemented
15C2×#11719912C2212yes—
16#13133199—21no—
17#13133197C2211yescharacteristic
18#13133196C2211yescharacteristic
19#13133193—21no—
20#13133193—21no—
21#13133193C2212yes—
22#13133193C2212yes—
23#13133188C2211yescharacteristic
24#13133185—21no—
25D4×Dic53—21no—
26#13133177—21no—
27#13133175C2211yescharacteristic
28C2×C4⋊Dic53C2211yescharacteristic
29C2×#11719919—21no—
30C2×#11719921—21no—
31#13133151C2212yescomplemented
32#13133148C2212yescomplemented
33#13133142C2212yescomplemented
34#13133141C2212yescomplemented
35#13133138C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yes—
38C22×Q8×C53C2211yescharacteristic
39C2×SD16×C53C2212yescomplemented
40#121×C53C2211yescharacteristic
41#103×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53D411yescharacteristic
44C2×Q8×C53—21no—
45C2×Q8×C53—42no—
46C2×Q8×C53C2312yes—
47C2×D4×C53—21no—
48C2×D4×C53C2311yescharacteristic
49C22×C4×C53—21no—
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—21no—
52C2×C22⋊D53—21no—
53C22×Dic53—21no—
54C22×Dic53—21no—
55C22×Dic53D411yescharacteristic
56#11719947—42no—
57C2×C4×D53—42no—
58C2×C4×D53D411yescharacteristic
59C2×C4×D53D411yescharacteristic
60C2×C4×D53C2311yescharacteristic
61C2×C4×D53C2311yescharacteristic
62C2×Dic106C2311yescharacteristic
63SD16×C53—42no—
64C2×C8×C53C2312yes—
65C4⋊C4×C53C2311yescharacteristic
66C22⋊C4×C53—21no—
67C22⋊Dic53—21no—
68C22⋊Dic53—21no—
69#11719925—42no—
70#11719921—22no—
71#11719921—22no—
72#11719921—21no—
73#11719921—21no—
74C4⋊Dic53—21no—
75C4⋊Dic53—21no—
76C4⋊Dic53C2312yes—
77C4⋊Dic53C2311yescharacteristic
78#11719919—21no—
79#11719919—22no—
80#11719919—22no—
81C4×Dic53—21no—
82C2×#88322C2312yes—
83#11719912—42no—
84C23×C53—21no—
85C23×C53C2×D411yescharacteristic
86C22×D53—22no—
87C22×D53—41no—
88C22×D53—41no—
89C22×D53C2×D411yescharacteristic
90C22×D53C2×D411yescharacteristic
91Q8×C53—22no—
92Q8×C53—22no—
93Q8×C53—42no—
94D4×C53—22no—
95D4×C53—42no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—42no—
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2411yescharacteristic
102C22⋊D53—42no—
103C22⋊D53—42no—
104C2×Dic53—21no—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—41no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53C2×D411yescharacteristic
111C2×Dic53C2×D411yescharacteristic
112C4×D53—22no—
113C4×D53—22no—
114C4×D53—41no—
115C4×D53—41no—
116Dic106—22no—
117C8×C53—22no—
118#88322—22no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53#13712yes—
125C22×C53C22×D411yescharacteristic
126C2×D53—21no—
127C2×D53—21no—
128C2×D53—21no—
129C2×D53—21no—
130C2×D53—21no—
131C2×D53—21no—
132C2×D53—44no—
133C4×C53—21no—
134C4×C53—42no—
135C4×C53—41no—
136C4×C53H(5,𝔽2)12yes—
137Dic53—21no—
138Dic53—21no—
139Dic53—41no—
140Dic53—42no—
141Dic53—41no—
142#3196—531nomaximal, Sylow
143C2×C53—22no—
144C2×C53—81no—
145C2×C53#53411yescharacteristic
146C2×C53C2×#13711yescharacteristic
147C2×C53#57511yescharacteristic
148D53—41no—
149D53—41no—
150#525—531no—
151C2×C22⋊Q8—531no—
152#485—532no—
153#484—531no—
154#481—531no—
155#470—532no—
156#468—532no—
157#452—532no—
158#451—531no—
159#448D5311yescharacteristic, complemented
160#407—531no—
161C53#319611yescharacteristic, complemented, Sylow
162C2×#41—531no—
163C22×Q8C2×D5311yescharacteristic
164C23×C4—531no—
165C2×Q16—532no—
166C2×SD16C2×D5312yescomplemented
167C2×M16—532no—
168#123—1061no—
169C22⋊Q8—532no—
170C22⋊Q8—532no—
171C22⋊Q8—531no—
172C22⋊Q8—531no—
173C22⋊Q8—1061no—
174C22⋊Q8—1061no—
175#121—531no—
176#121C2×D5311yescharacteristic
177#120—1061no—
178C4×D4—1061no—
179C4×D4—1061no—
180#117—531no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—1061no—
183C2×C22⋊C4—1061no—
184#107—532no—
185#106—532no—
186#103—532no—
187#103—532no—
188#103C2×D5312yescomplemented
189#102—532no—
190C22⋊C8—531no—
191C22⋊C8—532no—
192C22⋊C8C2×D5311yescharacteristic
193C24—531no—
194#41—2122no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—531no—
198C2×Q8C22×D5312yes—
199C2×D4—21no—
200C2×D4—1061no—
201C2×D4—1061no—
202C2×D4C22×D5311yescharacteristic
203C22×C4—21no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—531no—
207C22×C4—531no—
208C22×C4—531no—
209C22×C4—1061no—
210C22×C4—1061no—
211C22×C4—2122no—
212C22×C4C22×D5311yescharacteristic
213Q16—2122no—
214SD16—42no—
215M16—2122no—
216C2×C8—532no—
217C2×C8C22×D5312yes—
218C4⋊C4—532no—
219C4⋊C4—531no—
220C4⋊C4—1062no—
221C4⋊C4—1062no—
222C4⋊C4—1061no—
223C4⋊C4—1061no—
224C4⋊C4—1061no—
225C4⋊C4C22×D5311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1062no—
228C22⋊C4—1062no—
229C22⋊C4—1061no—
230C22⋊C4—1061no—
231C22⋊C4—1061no—
232C22⋊C4—1061no—
233C42—1061no—
234C23—21no—
235C23—531no—
236C23—531no—
237C23—1062no—
238C23—2121no—
239C23—2121no—
240C23D4×D5311yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—1062no—
245D4—22no—
246D4—42no—
247D4—2122no—
248D4—2122no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—531no—
254C2×C4—531no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1061no—
258C2×C4—1062no—
259C2×C4—1061no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4D4×D5311yescharacteristic
266C2×C4C23×D5311yescharacteristic
267C8—22no—
268C8—1062no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—1061no—
275C22—1061no—
276C22—1061no—
277C22—1061no—
278C22—1061no—
279C22—1061no—
280C22—2124no—
281C22#1313324612yes—
282C22C2×D4×D5311yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—1061no—
287C4—1061no—
288C4—2122no—
289C4—2121no—
290C4—2121no—
291C4#1313326212yes—
292C2—22no—
293C2—81no—
294C2—2121no—
295C2—2121no—
296C2#1681118311yescharacteristic
297C2C2×#1313324611yescharacteristic
298C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#448C2
socleC22×C53C22×D4
hypercenter#448D53
hypocenterC53#3196
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C2×#13133193C2semidirect
3. #16810963C2semidirect
4. #16810959C2semidirect
5. #16810939C2semidirect
6. #16810880C2semidirect
7. #16810866C2semidirect
8. #16810782C2semidirect
9. #16810779C2semidirect
10. #16810585C2semidirect
11. C53×#448C2semidirect
14. C2×#11719925C22semidirect
31. #13133151C22semidirect
32. #13133148C22semidirect
33. #13133142C22semidirect
34. #13133141C22semidirect
35. #13133138C22semidirect
39. C2×SD16×C53C22semidirect
41. #103×C53C22semidirect
159. #448D53semidirect
161. C53#3196semidirect
166. C2×SD16C2×D53semidirect
188. #103C2×D53semidirect