Finite Group Databasebeta

← #20285468

Subgroups — #20285468

Order 6784.

468 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13133191C211yesmaximal, characteristic
2#16810951C212yesmaximal, complemented
3#16810941C212yesmaximal, complemented
4#16810854C214yesmaximal, complemented
5#16810773C212yesmaximal, complemented
6#16810762C212yesmaximal, complemented
7C2×C22⋊#88322C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
9C22×D212C2211yescharacteristic
10C22×C22⋊D53—21no—
11C2×#11719924C2214yescomplemented
12C22×#88322C2212yescomplemented
13#13133191—21no—
14#13133191—21no—
15#13133191C2212yes—
16#13133191C2212yes—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719921—21no—
19#13133148C2214yescomplemented
20#13133141C2214yescomplemented
21#13133137C2214yescomplemented
22C22⋊#88322C2212yes—
23C22⋊#88322C2212yescomplemented
24C22×Q8×C53C2211yescharacteristic, complemented
25C23×C4×C53C2211yescharacteristic
26C22⋊Q8×C53—21no—
27C22⋊Q8×C53—21no—
28C22⋊Q8×C53C2212yes—
29C22⋊Q8×C53C2212yescomplemented
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53C2211yescharacteristic, complemented
32C2×C22⋊C4×C53—21no—
33C24×C53D411yescharacteristic
34C23×D53—21no—
35C2×Q8×C53—21no—
36C2×Q8×C53—42no—
37C2×Q8×C53C2312yes—
38C22×C4×C53—21no—
39C22×C4×C53—21no—
40C22×C4×C53—42no—
41C22×C4×C53D411yescharacteristic
42C22×C4×C53D411yescharacteristic
43C22×C4×C53D411yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—44no—
50C22×Dic53—21no—
51C2×D212—21no—
52C2×D212—42no—
53C2×D212C2312yes—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—21no—
57C4⋊C4×C53C2312yes—
58C22⋊C4×C53—22no—
59C22⋊C4×C53—22no—
60#11719924—44no—
61#11719921—22no—
62#11719921—22no—
63C4⋊Dic53—21no—
64C4⋊Dic53C2312yes—
65C2×#88322—42no—
66C2×#88322C2314yes—
67C23×C53—41no—
68C23×C53—41no—
69C23×C53SD1614yescomplemented
70C23×C53C2×D411yescharacteristic
71C23×C53C2×D411yescharacteristic
72C23×C53C2×D411yescharacteristic
73C22×D53—22no—
74C22×D53—42no—
75C22×D53—42no—
76C22×D53—42no—
77Q8×C53—24no—
78Q8×C53—42no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53C2×D411yescharacteristic
90C2×C4×C53C2×D411yescharacteristic
91C2×C4×C53C2×D411yescharacteristic
92C2×C4×C53C2411yescharacteristic
93C22⋊D53—44no—
94C22⋊D53—44no—
95C2×Dic53—22no—
96C2×Dic53—42no—
97D212—24no—
98D212—42no—
99#88322—24no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—21no—
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—21no—
106C22×C53—44no—
107C22×C53C2×SD1612yes—
108C22×C53C2×SD1612yes—
109C22×C53C2×SD1612yes—
110C22×C53C22×D411yescharacteristic
111C2×D53—42no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53—84no—
115C4×C53—21no—
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—42no—
119C4×C53—42no—
120C4×C53H(5,𝔽2)12yes—
121Dic53—42no—
122#3187—531nomaximal, Sylow
123C2×C53—22no—
124C2×C53—22no—
125C2×C53—22no—
126C2×C53#53411yescharacteristic
127C2×C53C22×SD1611yescharacteristic
128C2×C53#57511yescharacteristic
129D53—82no—
130C2×C22⋊Q8D5311yescharacteristic, complemented
131C2×#121—531no—
132#483—532no—
133#481—532no—
134#465—534no—
135#450—532no—
136#448—532no—
137C2×C22⋊C8—531no—
138C53#318711yescharacteristic, complemented, Sylow
139C22×Q8C2×D5311yescharacteristic, complemented
140C22×D4—531no—
141C22×D4—1061no—
142C23×C4C2×D5311yescharacteristic
143C2×SD16—534no—
144C22×C8—532no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8C2×D5312yes—
148C22⋊Q8C2×D5312yescomplemented
149#121—532no—
150#121—532no—
151#121—1061no—
152#121—1061no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4C2×D5311yescharacteristic, complemented
156C2×C22⋊C4—21no—
157C2×C22⋊C4—1061no—
158#106—534no—
159#103—534no—
160#102—534no—
161C22⋊C8—532no—
162C22⋊C8—532no—
163C24—1061no—
164C24C22⋊D5311yescharacteristic
165C2×Q8—21no—
166C2×Q8—42no—
167C2×Q8C22×D5312yes—
168C2×D4—532no—
169C2×D4—1062no—
170C2×D4—1062no—
171C2×D4—1061no—
172C2×D4—2122no—
173C2×D4—2124no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4—1061no—
178C22×C4C22⋊D5311yescharacteristic
179C22×C4C22⋊D5311yescharacteristic
180C22×C4C22⋊D5311yescharacteristic
181C22×C4C22×D5311yescharacteristic
182C22×C4C22×D5311yescharacteristic
183C22×C4C22×D5311yescharacteristic
184SD16—2124no—
185C2×C8—534no—
186C2×C8—2122no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—532no—
191C4⋊C4—1061no—
192C4⋊C4C22×D5312yes—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—1062no—
196C22⋊C4—1062no—
197C23—41no—
198C23—41no—
199C23—1062no—
200C23—2122no—
201C23—2122no—
202C23—2122no—
203C23#1171992414yescomplemented
204C23C2×C22⋊D5311yescharacteristic
205C23C2×C22⋊D5311yescharacteristic
206C23C2×C22⋊D5311yescharacteristic
207Q8—24no—
208Q8—42no—
209D4—1064no—
210D4—2122no—
211D4—2124no—
212D4—2124no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—1062no—
224C2×C4—2122no—
225C2×C4C2×C22⋊D5311yescharacteristic
226C2×C4C2×C22⋊D5311yescharacteristic
227C2×C4C2×C22⋊D5311yescharacteristic
228C2×C4C23×D5311yescharacteristic
229C8—1064no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—44no—
237C22—4244no—
238C22—2122no—
239C22—2122no—
240C22—2122no—
241C22C2×#1171992412yes—
242C22C2×#1171992412yes—
243C22C2×#1171992412yes—
244C22C22×C22⋊D5311yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—2122no—
251C4#1313326212yes—
252C2—22no—
253C2—22no—
254C2—22no—
255C2—4242no—
256C2#1681118211yescharacteristic
257C2C22×#1171992411yescharacteristic
258C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×C22⋊Q8×C53C2
socleC22×C53C22×D4
hypercenterC2×C22⋊Q8D53
hypocenterC53#3187
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810951C2semidirect
3. #16810941C2semidirect
4. #16810854C2semidirect
5. #16810773C2semidirect
6. #16810762C2semidirect
7. C2×C22⋊#88322C2semidirect
8. C2×C22⋊Q8×C53C2semidirect
11. C2×#11719924C22semidirect
12. C22×#88322C22semidirect
19. #13133148C22semidirect
20. #13133141C22semidirect
21. #13133137C22semidirect
23. C22⋊#88322C22semidirect
24. C22×Q8×C53C22semidirect
29. C22⋊Q8×C53C22semidirect
31. C2×C4⋊C4×C53C22semidirect
69. C23×C53SD16semidirect
130. C2×C22⋊Q8D53semidirect
138. C53#3187semidirect
139. C22×Q8C2×D53semidirect
148. C22⋊Q8C2×D53semidirect
155. C2×C4⋊C4C2×D53semidirect
203. C23#11719924semidirect