Finite Group Databasebeta

← #20279026

Subgroups — #20279026

Order 6784.

584 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 173 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×C4×C22⋊D53C211yesmaximal, characteristic
2#16810561C212yesmaximal, complemented
3#16810558C212yesmaximal, complemented
4#16810519C212yesmaximal, complemented
5#16810502C212yesmaximal
6#16810496C212yesmaximal, complemented
7#16810483C211yesmaximal, characteristic, complemented
8#16810480C212yesmaximal, complemented
9C22×C4⋊C4×C53C211yesmaximal, characteristic, complemented
10C22×C22⋊D53C2211yescharacteristic
11C22×C4×D53C2211yescharacteristic
12C4×C22⋊D53—24no—
13C2×C4⋊Dic53C2212yes—
14C2×#11719919C2211yescharacteristic
15C2×#11719919C2214yes—
16C2×C22⋊Dic53C2211yescharacteristic
17C2×C22⋊Dic53C2212yes—
18C2×#11719921C2211yescharacteristic
19C2×#11719921C2214yes—
20C2×C4×Dic53C2211yescharacteristic
21#13133124C2214yes—
22#13133124C2214yes—
23#13133124C2212yes—
24C23×C4×C53C2211yescharacteristic
25C23×C4×C53C2212yes—
26C2×C4⋊C4×C53—24no—
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2212yes—
29C24×C53C2311yescharacteristic
30C23×D53C2311yescharacteristic
31C22×C4×C53—24no—
32C22×C4×C53—24no—
33C22×C4×C53—44no—
34C22×C4×C53D414yes—
35C22×C4×C53C2314yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—22no—
39C2×C22⋊D53—22no—
40C2×C22⋊D53Q814yes—
41C22×Dic53C2311yescharacteristic
42C22×Dic53C2314yes—
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2311yescharacteristic
45C2×C4×D53—22no—
46C2×C4×D53—24no—
47C4⋊C4×C53—24no—
48C4⋊C4×C53—24no—
49C22⋊Dic53—22no—
50C22⋊Dic53—24no—
51#11719921—22no—
52#11719921—44no—
53C4⋊Dic53—24no—
54#11719919—22no—
55#11719919—44no—
56C4×Dic53—22no—
57C23×C53—22no—
58C23×C53—22no—
59C23×C53C2×D412yes—
60C23×C53C2×Q812yes—
61C23×C53#4112yes—
62C23×C53C2411yescharacteristic
63C22×D53—21no—
64C22×D53—22no—
65C22×D53—21no—
66C22×D53—22no—
67C22×D53C2×Q812yes—
68C2×C4×C53—22no—
69C2×C4×C53—28no—
70C2×C4×C53—24no—
71C2×C4×C53—24no—
72C2×C4×C53—24no—
73C2×C4×C53—48no—
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2×D412yes—
76C2×C4×C53#4114yes—
77C22⋊D53—28no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—22no—
83C2×Dic53—24no—
84C2×Dic53—24no—
85C2×Dic53—22no—
86C2×Dic53—24no—
87C2×Dic53C2×Q812yes—
88C4×D53—44no—
89C22×C53—22no—
90C22×C53—22no—
91C22×C53—24no—
92C22×C53—24no—
93C22×C53C22⋊Q814yes—
94C22×C53C22×D411yescharacteristic
95C22×C53C22×Q811yescharacteristic
96C22×C53C2×#4112yes—
97C22×C53C2×#4111yescharacteristic
98C22×C53H(5,𝔽2)12yes—
99C2×D53—22no—
100C2×D53—24no—
101C2×D53—41no—
102C2×D53—41no—
103C2×D53—42no—
104C4×C53—22no—
105C4×C53—44no—
106C4×C53#12314yes—
107Dic53—22no—
108Dic53—41no—
109Dic53—41no—
110Dic53—44no—
111#2487—531nomaximal, Sylow
112C2×C53—24no—
113C2×C53C2×C22⋊Q811yescharacteristic
114C2×C53C2×#12311yescharacteristic
115C2×C53#53511yescharacteristic
116C2×C53#55312yes—
117C2×C53#55412yes—
118D53—42no—
119C2×C4×D4—531no—
120C22×C4⋊C4D5311yescharacteristic, complemented
121#400—532no—
122#399—532no—
123#389—532no—
124#386—532no—
125#385—532no—
126#380—531no—
127#380—532no—
128C53#248711yescharacteristic, complemented, Sylow
129C22×D4—531no—
130C23×C4—531no—
131C23×C4C2×D5311yescharacteristic
132C23×C4C2×D5312yes—
133C4×D4—1064no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4—534no—
138C2×C4⋊C4C2×D5312yes—
139C2×C4⋊C4C2×D5312yes—
140C2×C22⋊C4—531no—
141C2×C22⋊C4—532no—
142C2×C22⋊C4—531no—
143C2×C22⋊C4—534no—
144C2×C42—531no—
145#95—534no—
146#95—532no—
147#95—534no—
148C24—531no—
149C24C22×D5311yescharacteristic
150C2×D4—534no—
151C2×D4—1062no—
152C2×D4—1062no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—44no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—534no—
159C22×C4—531no—
160C22×C4—1064no—
161C22×C4—1062no—
162C22×C4C22⋊D5314yes—
163C22×C4C22×D5314yes—
164C22×C4C22×D5311yescharacteristic
165C22×C4C22×D5311yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—1064no—
169C4⋊C4—1062no—
170C4⋊C4—2124no—
171C22⋊C4—1064no—
172C22⋊C4—1062no—
173C22⋊C4—1062no—
174C22⋊C4—2124no—
175C42—1062no—
176C23—22no—
177C23—22no—
178C23—532no—
179C23—1061no—
180C23—1062no—
181C23—1061no—
182C23—1062no—
183C23Q8×D5312yes—
184C23#1171994912yes—
185C23C2×C22⋊D5312yes—
186C23C23×D5311yescharacteristic
187D4—1068no—
188C2×C4—22no—
189C2×C4—28no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—48no—
194C2×C4—532no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—1064no—
200C2×C4—1062no—
201C2×C4—1064no—
202C2×C4—1064no—
203C2×C4—1062no—
204C2×C4—2124no—
205C2×C4#1171994514yes—
206C2×C4C2×C22⋊D5312yes—
207C2×C4C2×C22⋊D5312yes—
208C22—24no—
209C22—24no—
210C22—22no—
211C22—22no—
212C22—1064no—
213C22—1062no—
214C22—2122no—
215C22—2121no—
216C22—2121no—
217C22#1313319314yes—
218C22C22×C22⋊D5311yescharacteristic
219C22C2×Q8×D5311yescharacteristic
220C22C2×#1171994512yes—
221C22C2×#1171994911yescharacteristic
222C22#1313326212yes—
223C4—22no—
224C4—44no—
225C4—1062no—
226C4—2124no—
227C4—2121no—
228C4—2121no—
229C4#1313320014yes—
230C2—24no—
231C2—2122no—
232C2C2×#1313319311yescharacteristic
233C2C2×#1313320011yescharacteristic
234C2#1681119011yescharacteristic
235C2#1681132112yes—
236C2#1681132812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C4⋊C4×C53C2
hypercenterC22×C4⋊C4D53
hypocenterC53#2487
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16810561C2semidirect
3. #16810558C2semidirect
4. #16810519C2semidirect
6. #16810496C2semidirect
7. #16810483C2semidirect
8. #16810480C2semidirect
9. C22×C4⋊C4×C53C2semidirect
120. C22×C4⋊C4D53semidirect
128. C53#2487semidirect