Finite Group Databasebeta

← #20279874

Subgroups — #20279874

Order 6784.

672 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 153 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#16810549C212yesmaximal, complemented
3#16810534C212yesmaximal, complemented
4#16810513C214yesmaximal, complemented
5#16810484C212yesmaximal, complemented
6#16810478C212yesmaximal
7C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
9C24×D53C2211yescharacteristic
10C23×Dic53C2211yescharacteristic
11C2×C22⋊Dic53C2212yes—
12C2×#11719921—24no—
13C2×#11719921C2212yes—
14C2×#11719921C2214yes—
15C2×#11719921C2212yes—
16C2×#11719921C2214yes—
17C2×C4×Dic53C2212yescomplemented
18#13133124C2214yescomplemented
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—24no—
30C23×D53—24no—
31C23×D53—42no—
32C23×D53C2312yes—
33C2×Q8×C53—22no—
34C2×Q8×C53—42no—
35C22×C4×C53—42no—
36C22×C4×C53D414yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×Dic53—22no—
42C22×Dic53—24no—
43C22×Dic53C2312yes—
44C22×Dic53C2314yes—
45C4⋊C4×C53—24no—
46C4⋊C4×C53—22no—
47C22⋊C4×C53—24no—
48C22⋊Dic53—24no—
49#11719921—28no—
50#11719921—24no—
51#11719921—24no—
52#11719921—44no—
53C4×Dic53—42no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53C2×D412yes—
57C23×C53#4114yes—
58C23×C53C2411yescharacteristic
59C22×D53—28no—
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—24no—
63C22×D53—24no—
64C22×D53—416no—
65C22×D53—44no—
66C22×D53—44no—
67C22×D53#4114yes—
68Q8×C53—44no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—24no—
74C2×C4×C53—24no—
75C2×C4×C53—44no—
76C2×C4×C53C2×D412yes—
77C2×C4×C53C2×D412yes—
78C2×Dic53—22no—
79C2×Dic53—24no—
80C2×Dic53—24no—
81C2×Dic53—24no—
82C2×Dic53—24no—
83C2×Dic53—44no—
84C22×C53—22no—
85C22×C53—22no—
86C22×C53—28no—
87C22×C53#12414yes—
88C22×C53C22×D411yescharacteristic
89C22×C53C2×#4112yes—
90C22×C53C2×#4112yes—
91C22×C53H(5,𝔽2)12yes—
92C2×D53—24no—
93C2×D53—28no—
94C2×D53—416no—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—44no—
98C4×C53—22no—
99C4×C53—22no—
100C4×C53—42no—
101C4×C53—42no—
102Dic53—42no—
103Dic53—44no—
104#2561—531nomaximal, Sylow
105C2×C53—24no—
106C2×C53#52511yescharacteristic
107C2×C53C2×#12411yescharacteristic
108C2×C53#53411yescharacteristic
109C2×C53#55114yes—
110D53—44no—
111C2×C22⋊Q8D5311yescharacteristic, complemented
112C22×C22⋊C4—531no—
113#397—532no—
114#393—532no—
115#388—534no—
116#381—532no—
117#379—532no—
118C4×C22⋊C4—531no—
119C53#256111yescharacteristic, complemented, Sylow
120C25—531no—
121C22×Q8C2×D5311yescharacteristic, complemented
122C23×C4—531no—
123C23×C4C2×D5311yescharacteristic
124C22⋊Q8—24no—
125C2×C4⋊C4C2×D5312yescomplemented
126C2×C4⋊C4C2×D5311yescharacteristic, complemented
127C2×C22⋊C4—532no—
128C2×C22⋊C4—532no—
129C2×C22⋊C4—532no—
130C2×C22⋊C4—534no—
131C2×C22⋊C4—534no—
132C2×C22⋊C4—1064no—
133C2×C22⋊C4C2×D5312yes—
134C2×C42—532no—
135#95—532no—
136#95—534no—
137#95—534no—
138C24—532no—
139C24—1064no—
140C24—1062no—
141C24—1064no—
142C24—2122no—
143C24C22×D5311yescharacteristic
144C2×Q8—22no—
145C2×Q8—42no—
146C22×C4—42no—
147C22×C4—532no—
148C22×C4—534no—
149C22×C4—1064no—
150C22×C4—1062no—
151C22×C4C22⋊D5314yes—
152C22×C4C22×D5312yes—
153C22×C4C22×D5312yes—
154C22×C4C22×D5311yescharacteristic
155C22×C4C22×D5311yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—24no—
158C22⋊C4—24no—
159C22⋊C4—1068no—
160C22⋊C4—1064no—
161C22⋊C4—1064no—
162C22⋊C4—1064no—
163C22⋊C4—2124no—
164C42—2122no—
165C23—22no—
166C23—22no—
167C23—534no—
168C23—1068no—
169C23—1062no—
170C23—1062no—
171C23—1064no—
172C23—1064no—
173C23—21216no—
174C23—2124no—
175C23—2124no—
176C23#1171994914yes—
177C23C2×C22⋊D5312yes—
178C23C23×D5311yescharacteristic
179Q8—44no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—44no—
187C2×C4—1062no—
188C2×C4—1064no—
189C2×C4—1064no—
190C2×C4—1064no—
191C2×C4—1064no—
192C2×C4—2124no—
193C2×C4C2×C22⋊D5312yes—
194C2×C4C2×C22⋊D5312yes—
195C22—28no—
196C22—22no—
197C22—22no—
198C22—1068no—
199C22—1064no—
200C22—21216no—
201C22—2124no—
202C22—2122no—
203C22—2122no—
204C22#1313320614yes—
205C22C22×C22⋊D5311yescharacteristic
206C22#41×D5312yes—
207C22C2×#1171994912yes—
208C22#1313326212yes—
209C4—22no—
210C4—22no—
211C4—42no—
212C4—42no—
213C4—2124no—
214C4—2122no—
215C2—24no—
216C2—2124no—
217C2#1681113011yescharacteristic
218C2C2×#1313320611yescharacteristic
219C2#1681118211yescharacteristic
220C2#1681130014yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2561
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810549C2semidirect
3. #16810534C2semidirect
4. #16810513C2semidirect
5. #16810484C2semidirect
7. C4×C22⋊Dic53C2semidirect
8. C2×C22⋊Q8×C53C2semidirect
17. C2×C4×Dic53C22semidirect
18. #13133124C22semidirect
19. #13133124C22semidirect
20. #13133124C22semidirect
21. C22×Q8×C53C22semidirect
24. C2×C4⋊C4×C53C22semidirect
25. C2×C4⋊C4×C53C22semidirect
111. C2×C22⋊Q8D53semidirect
119. C53#2561semidirect
121. C22×Q8C2×D53semidirect
125. C2×C4⋊C4C2×D53semidirect
126. C2×C4⋊C4C2×D53semidirect