Finite Group Databasebeta

← #20273216

Subgroups — #20273216

Order 6784.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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
2#16810585C211yesmaximal, characteristic, complemented
3#95×D53C211yesmaximal, characteristic, complemented
4#16810361C211yesmaximal, characteristic, complemented
5#16810359C211yesmaximal, characteristic, complemented
6#16810357C211yesmaximal, characteristic, complemented
7C53×#327C211yesmaximal, characteristic, complemented
8C2×#11719947C411yescharacteristic, complemented
9C22×C4×D53—21no—
10C22×C4×D53C2211yescharacteristic
11C2×#11719912—21no—
12#13133199—21no—
13#13133197C411yescharacteristic, complemented
14#13133196C2211yescharacteristic
15#13133188C411yescharacteristic, complemented
16#13133185—21no—
17D4×Dic53—21no—
18#13133177—21no—
19#13133175C411yescharacteristic, complemented
20C22⋊#88322C2211yescharacteristic
21#13133129—21no—
22#13133124—21no—
23#13133124C2211yescharacteristic
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C23×D53D411yescharacteristic
28C2×D4×C53—21no—
29C2×D4×C53C2×C411yescharacteristic, complemented
30C22×C4×C53—21no—
31C22×C4×C53C2311yescharacteristic
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C22×Dic53—21no—
35C22×Dic53—21no—
36C22×Dic53D411yescharacteristic
37#11719947—42no—
38C2×C4×D53—24no—
39C2×C4×D53—42no—
40C2×C4×D53—42no—
41C2×C4×D53—42no—
42C2×C4×D53C2×C411yescharacteristic
43C2×C4×D53C2×C411yescharacteristic
44C2×C4×D53D411yescharacteristic
45C2×C4×D53D411yescharacteristic
46C2×Dic106C2×C411yescharacteristic, complemented
47C2×C8×C53—21no—
48C4⋊C4×C53C2×C411yescharacteristic, complemented
49C22⋊C4×C53—21no—
50C22⋊Dic53—21no—
51C22⋊Dic53—21no—
52#11719921—21no—
53#11719921—21no—
54C4⋊Dic53—21no—
55C4⋊Dic53C2×C411yescharacteristic, complemented
56#11719919—21no—
57C4×Dic53—21no—
58C2×#88322—21no—
59#11719912—42no—
60C23×C53—21no—
61C23×C53C2×D411yescharacteristic
62C22×D53—21no—
63C22×D53—21no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53C22⋊C411yescharacteristic
67C22×D53C22⋊C411yescharacteristic
68D4×C53—41no—
69D4×C53—41no—
70D4×C53—41no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53C22×C411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C22⋊D53—42no—
79C22⋊D53—42no—
80C2×Dic53—22no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—41no—
84C2×Dic53—42no—
85C2×Dic53—41no—
86C2×Dic53—41no—
87C2×Dic53C22⋊C411yescharacteristic
88C2×Dic53C22⋊C411yescharacteristic
89C4×D53—22no—
90C4×D53—22no—
91C4×D53—41no—
92C4×D53—41no—
93C4×D53—48no—
94Dic106—41no—
95C8×C53—41no—
96#88322—41no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53C2×C22⋊C411yescharacteristic
103C22×C53AGL(1,ℤ/8)11yescharacteristic
104C22×C53#13711yescharacteristic
105C2×D53—24no—
106C2×D53—24no—
107C2×D53—21no—
108C2×D53—21no—
109C2×D53—21no—
110C2×D53—21no—
111C2×D53#9912yes—
112C2×D53#10412yes—
113C4×C53—21no—
114C4×C53—21no—
115C4×C53—42no—
116C4×C53—41no—
117Dic53—21no—
118Dic53—21no—
119Dic53—42no—
120Dic53—41no—
121Dic53—41no—
122#1504—531nomaximal, Sylow
123C2×C53—21no—
124C2×C53—21no—
125C2×C53—81no—
126C2×C53C2×#9911yescharacteristic
127C2×C53#41711yescharacteristic
128C2×C53C2×#10411yescharacteristic
129D53—22no—
130D53—22no—
131#525—531no—
132#407—531no—
133C2×#95—531no—
134#328—531no—
135#328—531no—
136#327—531no—
137#327D5311yescharacteristic, complemented
138C53#150411yescharacteristic, complemented, Sylow
139C2×#41—531no—
140C23×C4—531no—
141C23×C4—1061no—
142C2×M16—1061no—
143#123—1061no—
144C22⋊Q8—531no—
145C22⋊Q8—531no—
146#121—531no—
147#121C2×D5311yescharacteristic
148#120—1061no—
149C4×D4—1061no—
150C4×D4—1061no—
151#117—531no—
152C22⋊C8—531no—
153C22⋊C8—1061no—
154C22⋊C8C2×D5311yescharacteristic
155#95—531no—
156#95—1061no—
157#95C2×D5311yescharacteristic
158C24—531no—
159#41—2122no—
160C2×Q8—531no—
161C2×D4—21no—
162C2×D4—1061no—
163C2×D4—1061no—
164C2×D4C4×D5311yescharacteristic, complemented
165C22×C4—21no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—531no—
171C22×C4—1064no—
172C22×C4—1061no—
173C22×C4—1061no—
174C22×C4—2122no—
175C22×C4—2122no—
176C22×C4—2122no—
177C22×C4C22×D5311yescharacteristic
178M16—2122no—
179C2×C8—21no—
180C2×C8—1061no—
181C4⋊C4—531no—
182C4⋊C4—1061no—
183C4⋊C4—1061no—
184C4⋊C4C4×D5311yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—1061no—
187C22⋊C4—1061no—
188C22⋊C4—1061no—
189C22⋊C4—1061no—
190C42—1061no—
191C23—21no—
192C23—531no—
193C23—531no—
194C23—1062no—
195C23—1062no—
196C23—1061no—
197C23—1061no—
198C23D4×D5311yescharacteristic
199Q8—2121no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—2122no—
204D4—2122no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—531no—
211C2×C4—531no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1061no—
216C2×C4—1061no—
217C2×C4—2121no—
218C2×C4—2121no—
219C2×C4—2128no—
220C2×C4—2121no—
221C2×C4—2122no—
222C2×C4—2121no—
223C2×C4—2121no—
224C2×C4C2×C4×D5311yescharacteristic
225C2×C4D4×D5311yescharacteristic
226C8—41no—
227C8—2121no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—532no—
234C22—532no—
235C22—1064no—
236C22—1064no—
237C22—1061no—
238C22—1061no—
239C22—1061no—
240C22—1061no—
241C22C22⋊C4×D5311yescharacteristic
242C22#1313324111yescharacteristic
243C22#1313324611yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—1061no—
249C4—1061no—
250C4—2122no—
251C4—2121no—
252C4—2121no—
253C2—21no—
254C2—21no—
255C2—81no—
256C2—1062no—
257C2—1062no—
258C2#99×D5311yescharacteristic
259C2#1681062111yescharacteristic
260C2#104×D5311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53C2×C22⋊C4
hypercenter#327D53
hypocenterC53#1504
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #16810585C2semidirect
3. #95×D53C2semidirect
4. #16810361C2semidirect
5. #16810359C2semidirect
6. #16810357C2semidirect
7. C53×#327C2semidirect
8. C2×#11719947C4semidirect
13. #13133197C4semidirect
15. #13133188C4semidirect
19. #13133175C4semidirect
29. C2×D4×C53C2×C4semidirect
46. C2×Dic106C2×C4semidirect
48. C4⋊C4×C53C2×C4semidirect
55. C4⋊Dic53C2×C4semidirect
137. #327D53semidirect
138. C53#1504semidirect
164. C2×D4C4×D53semidirect
184. C4⋊C4C4×D53semidirect