Finite Group Databasebeta

← #20285535

Subgroups — #20285535

Order 6784.

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

normal subgroups only

This group has 410 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
1C22⋊Q8×D53C212yesmaximal
2#16810955C212yesmaximal, complemented
3#16810952C212yesmaximal, complemented
4#16810881C211yesmaximal, characteristic, complemented
5#16810879C212yesmaximal, complemented
6#16810780C212yesmaximal, complemented
7#16810777C212yesmaximal, complemented
8#16810586C211yesmaximal, characteristic
9C53×#470C211yesmaximal, characteristic, complemented
10C2×Q8×D53C2212yes—
11C22×C4×D53C2211yescharacteristic
12C2×Dic212C2211yescharacteristic, complemented
13C2×#11719925C2212yescomplemented
14C2×#11719917C2211yescharacteristic
15#13133197C2212yes—
16#13133196C2212yes—
17#13133195—22no—
18#13133194—22no—
19#13133193C2212yes—
20C4⋊C4×D53—22no—
21C4⋊C4×D53C2212yes—
22C22⋊C4×D53—22no—
23#13133149C2211yescharacteristic, complemented
24#13133148C2212yescomplemented
25#13133142C2212yescomplemented
26#13133141C2212yescomplemented
27#13133140C2212yescomplemented
28#13133129C2212yes—
29#13133129C2212yes—
30C2×Q16×C53C2211yescharacteristic, complemented
31C2×M16×C53C2211yescharacteristic
32C22⋊Q8×C53C2212yes—
33#106×C53C2211yescharacteristic, complemented
34#103×C53C2212yescomplemented
35C23×D53D411yescharacteristic
36C2×Q8×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×Dic53D411yescharacteristic
39Q8×D53—42no—
40Q8×D53—42no—
41C2×C4×D53—22no—
42C2×C4×D53—22no—
43C2×C4×D53—41no—
44C2×C4×D53—41no—
45C2×C4×D53D412yes—
46C2×C4×D53C2312yes—
47C2×Dic106—22no—
48C2×Dic106C2312yes—
49Q16×C53—41no—
50M16×C53—41no—
51C2×C8×C53C2311yescharacteristic
52C2×C8×C53C2311yescharacteristic
53C4⋊C4×C53—22no—
54C4⋊C4×C53C2312yes—
55C22⋊C4×C53—22no—
56C22⋊Dic53—22no—
57#11719925—42no—
58#11719921—22no—
59#11719921—22no—
60C4⋊Dic53—22no—
61C4⋊Dic53C2312yes—
62#11719919—22no—
63#11719919—22no—
64#11719919—22no—
65#11719917—41no—
66C2×#88322C2312yes—
67Dic212—41no—
68C23×C53C2×D411yescharacteristic
69C22×D53—24no—
70C22×D53—41no—
71C22×D53C2×D412yes—
72Q8×C53—22no—
73Q8×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2411yescharacteristic
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—41no—
83C2×Dic53C2×D412yes—
84C4×D53—22no—
85C4×D53—22no—
86C4×D53—42no—
87C4×D53—44no—
88C4×D53—44no—
89Dic106—22no—
90Dic106—22no—
91Dic106—44no—
92C8×C53—21no—
93C8×C53—21no—
94#88322—22no—
95C22×C53—21no—
96C22×C53—22no—
97C22×C53C22×D411yescharacteristic
98C2×D53—22no—
99C2×D53—22no—
100C2×D53—44no—
101C2×D53#13714yes—
102C4×C53—21no—
103C4×C53—42no—
104C4×C53—42no—
105C4×C53H(5,𝔽2)11yescharacteristic
106C4×C53H(5,𝔽2)11yescharacteristic
107Dic53—22no—
108Dic53—42no—
109Dic53—42no—
110#3195—531nomaximal, Sylow
111C2×C53—41no—
112C2×C53#53411yescharacteristic
113C2×C53C2×#13712yes—
114D53—24no—
115C2×C22⋊Q8—532no—
116#483—532no—
117#483—532no—
118#470—531no—
119#470—532no—
120#470D5311yescharacteristic, complemented
121#451—532no—
122#451—532no—
123#407—531no—
124C53#319511yescharacteristic, complemented, Sylow
125C22×Q8—532no—
126C23×C4—531no—
127C2×Q16—532no—
128C2×Q16—531no—
129C2×Q16C2×D5311yescharacteristic, complemented
130C2×M16—531no—
131C2×M16C2×D5311yescharacteristic
132C22⋊Q8—532no—
133C22⋊Q8—532no—
134C22⋊Q8—532no—
135C22⋊Q8—1062no—
136C22⋊Q8—1062no—
137C22⋊Q8C2×D5312yes—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—1062no—
140C2×C22⋊C4—1062no—
141#106—531no—
142#106—532no—
143#106C2×D5311yescharacteristic, complemented
144#103—532no—
145#103—532no—
146#103—532no—
147#103C2×D5312yescomplemented
148C22⋊C8—532no—
149C22⋊C8—532no—
150C24—531no—
151C2×Q8—532no—
152C2×Q8—1062no—
153C2×Q8—2122no—
154C2×Q8—2122no—
155C2×Q8C22×D5312yes—
156C22×C4—531no—
157C22×C4—532no—
158C22×C4—532no—
159C22×C4—1062no—
160C22×C4—1062no—
161C22×C4—2121no—
162C22×C4—2121no—
163C22×C4C22×D5311yescharacteristic
164Q16—41no—
165Q16—2121no—
166Q16—2122no—
167M16—41no—
168M16—2121no—
169C2×C8—532no—
170C2×C8C22×D5311yescharacteristic
171C2×C8C22×D5311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—532no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4C22×D5312yes—
179C22⋊C4—22no—
180C22⋊C4—1062no—
181C22⋊C4—1062no—
182C22⋊C4—1062no—
183C23—532no—
184C23—1064no—
185C23—2121no—
186C23D4×D5311yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—1062no—
190Q8—1062no—
191Q8—2124no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—532no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—2122no—
202C2×C4—2124no—
203C2×C4—2124no—
204C2×C4—2121no—
205C2×C4D4×D5311yescharacteristic
206C2×C4C23×D5311yescharacteristic
207C8—21no—
208C8—21no—
209C8—1062no—
210C22—22no—
211C22—21no—
212C22—534no—
213C22—1062no—
214C22—1062no—
215C22—2124no—
216C22C2×D4×D5311yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—1062no—
221C4—2122no—
222C4—2122no—
223C4#1313326211yescharacteristic
224C4#1313326311yescharacteristic
225C2—41no—
226C2—1064no—
227C2#1681118111yescharacteristic
228C2#137×D5312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#470C2
socleC22×C53C22×D4
hypercenter#470D53
hypocenterC53#3195
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #16810955C2semidirect
3. #16810952C2semidirect
4. #16810881C2semidirect
5. #16810879C2semidirect
6. #16810780C2semidirect
7. #16810777C2semidirect
9. C53×#470C2semidirect
12. C2×Dic212C22semidirect
13. C2×#11719925C22semidirect
23. #13133149C22semidirect
24. #13133148C22semidirect
25. #13133142C22semidirect
26. #13133141C22semidirect
27. #13133140C22semidirect
30. C2×Q16×C53C22semidirect
33. #106×C53C22semidirect
34. #103×C53C22semidirect
120. #470D53semidirect
124. C53#3195semidirect
129. C2×Q16C2×D53semidirect
143. #106C2×D53semidirect
147. #103C2×D53semidirect