Finite Group Databasebeta

← #20280522

Subgroups — #20280522

Order 6784.

668 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 157 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×#11719921C211yesmaximal, characteristic
2#16810558C212yesmaximal, complemented
3#16810545C214yesmaximal, complemented
4#16810534C212yesmaximal, complemented
5#16810495C212yesmaximal, complemented
6#16810483C211yesmaximal, characteristic, complemented
7#16810478C212yesmaximal
8C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
9C24×D53C2211yescharacteristic
10C23×Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2212yescomplemented
12C2×#11719919C2214yescomplemented
13C2×C22⋊Dic53C2212yes—
14C2×#11719921—24no—
15C2×#11719921C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2214yes—
18C2×#11719921C2214yes—
19#13133124C2214yescomplemented
20#13133124C2212yescomplemented
21C22×Q8×C53C2211yescharacteristic, complemented
22C23×C4×C53C2211yescharacteristic
23C22⋊Q8×C53—24no—
24C2×C4⋊C4×C53C2212yescomplemented
25C2×C4⋊C4×C53C2211yescharacteristic, complemented
26C2×C22⋊C4×C53C2212yes—
27C24×C53C2311yescharacteristic
28C23×D53—22no—
29C23×D53—22no—
30C23×D53—42no—
31C23×D53—42no—
32C23×D53Q814yescomplemented
33C23×D53C2312yes—
34C2×Q8×C53—22no—
35C2×Q8×C53—42no—
36C22×C4×C53—42no—
37C22×C4×C53D414yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×Dic53—22no—
43C22×Dic53—42no—
44C22×Dic53C2314yes—
45C22×Dic53C2312yes—
46C4⋊C4×C53—24no—
47C4⋊C4×C53—22no—
48C22⋊C4×C53—24no—
49C22⋊Dic53—24no—
50#11719921—24no—
51#11719921—24no—
52#11719921—44no—
53#11719921—44no—
54C4⋊Dic53—42no—
55#11719919—44no—
56C23×C53—22no—
57C23×C53—22no—
58C23×C53C2×D412yes—
59C23×C53C2×Q812yes—
60C23×C53#4112yes—
61C23×C53C2411yescharacteristic
62C22×D53—28no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—24no—
66C22×D53—24no—
67C22×D53—48no—
68C22×D53—44no—
69C22×D53—44no—
70C22×D53—48no—
71C22×D53C2×Q814yes—
72Q8×C53—44no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—24no—
78C2×C4×C53—24no—
79C2×C4×C53—44no—
80C2×C4×C53C2×D412yes—
81C2×C4×C53C2×D412yes—
82C2×Dic53—22no—
83C2×Dic53—24no—
84C2×Dic53—24no—
85C2×Dic53—24no—
86C2×Dic53—24no—
87C2×Dic53—44no—
88C22×C53—22no—
89C22×C53—22no—
90C22×C53—24no—
91C22×C53—24no—
92C22×C53C22⋊Q814yes—
93C22×C53C22×D411yescharacteristic
94C22×C53C22×Q811yescharacteristic
95C22×C53C2×#4111yescharacteristic
96C22×C53H(5,𝔽2)12yes—
97C22×C53H(5,𝔽2)12yes—
98C2×D53—24no—
99C2×D53—28no—
100C2×D53—416no—
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—44no—
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—42no—
107C4×C53—42no—
108Dic53—42no—
109Dic53—44no—
110#2626—531nomaximal, Sylow
111C2×C53—24no—
112C2×C53C2×C22⋊Q811yescharacteristic
113C2×C53#53411yescharacteristic
114C2×C53#53511yescharacteristic
115C2×C53#53812yes—
116C2×C53#54312yes—
117D53—44no—
118C2×C22⋊Q8D5311yescharacteristic, complemented
119C22×C22⋊C4—531no—
120#399—532no—
121#396—534no—
122#393—532no—
123#385—532no—
124#380—531no—
125#379—532no—
126C53#262611yescharacteristic, complemented, Sylow
127C25—531no—
128C22×Q8C2×D5311yescharacteristic, complemented
129C23×C4—531no—
130C23×C4C2×D5311yescharacteristic
131C22⋊Q8—24no—
132C2×C4⋊C4—534no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4C2×D5312yescomplemented
135C2×C4⋊C4C2×D5311yescharacteristic, complemented
136C2×C22⋊C4—532no—
137C2×C22⋊C4—532no—
138C2×C22⋊C4—532no—
139C2×C22⋊C4—534no—
140C2×C22⋊C4—534no—
141C2×C22⋊C4—1064no—
142C2×C22⋊C4C2×D5312yes—
143#95—532no—
144#95—534no—
145C24—532no—
146C24—534no—
147C24—1062no—
148C24—1062no—
149C24—2122no—
150C24—2122no—
151C24C22×D5311yescharacteristic
152C2×Q8—22no—
153C2×Q8—42no—
154C22×C4—42no—
155C22×C4—534no—
156C22×C4—532no—
157C22×C4—1062no—
158C22×C4—2122no—
159C22×C4C22⋊D5314yes—
160C22×C4C22×D5312yes—
161C22×C4C22×D5312yes—
162C22×C4C22×D5311yescharacteristic
163C22×C4C22×D5311yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—24no—
166C4⋊C4—2124no—
167C4⋊C4—2122no—
168C22⋊C4—24no—
169C22⋊C4—1064no—
170C22⋊C4—1064no—
171C22⋊C4—1064no—
172C22⋊C4—2124no—
173C22⋊C4—2124no—
174C23—22no—
175C23—22no—
176C23—534no—
177C23—1068no—
178C23—1062no—
179C23—1062no—
180C23—1064no—
181C23—1064no—
182C23—2128no—
183C23—2124no—
184C23—2124no—
185C23—2128no—
186C23Q8×D5312yes—
187C23#1171994912yes—
188C23C2×C22⋊D5312yes—
189C23C23×D5311yescharacteristic
190Q8—44no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—1062no—
199C2×C4—1064no—
200C2×C4—1064no—
201C2×C4—1064no—
202C2×C4—1064no—
203C2×C4—2124no—
204C2×C4C2×C22⋊D5312yes—
205C2×C4C2×C22⋊D5312yes—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—24no—
210C22—1064no—
211C22—1068no—
212C22—21216no—
213C22—2122no—
214C22—2122no—
215C22—2124no—
216C22#1313319314yes—
217C22C22×C22⋊D5311yescharacteristic
218C22C2×Q8×D5311yescharacteristic
219C22C2×#1171994911yescharacteristic
220C22#1313326212yes—
221C22#1313326312yes—
222C4—22no—
223C4—22no—
224C4—42no—
225C4—42no—
226C4—2122no—
227C4—2124no—
228C2—24no—
229C2—2124no—
230C2C2×#1313319311yescharacteristic
231C2#1681118211yescharacteristic
232C2#1681118711yescharacteristic
233C2#1681121312yes—
234C2#1681124812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2626
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810558C2semidirect
3. #16810545C2semidirect
4. #16810534C2semidirect
5. #16810495C2semidirect
6. #16810483C2semidirect
8. C2×C22⋊Q8×C53C2semidirect
11. C2×C4⋊Dic53C22semidirect
12. C2×#11719919C22semidirect
19. #13133124C22semidirect
20. #13133124C22semidirect
21. C22×Q8×C53C22semidirect
24. C2×C4⋊C4×C53C22semidirect
25. C2×C4⋊C4×C53C22semidirect
32. C23×D53Q8semidirect
118. C2×C22⋊Q8D53semidirect
126. C53#2626semidirect
128. C22×Q8C2×D53semidirect
134. C2×C4⋊C4C2×D53semidirect
135. C2×C4⋊C4C2×D53semidirect