Finite Group Databasebeta

← #20279873

Subgroups — #20279873

Order 6784.

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

normal subgroups only

This group has 672 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
1C2×#13133196C211yesmaximal, characteristic, complemented
2#16810548C212yesmaximal, complemented
3#16810535C212yesmaximal, complemented
4#16810517C214yesmaximal, complemented
5#16810486C212yesmaximal, complemented
6#16810479C212yesmaximal
7C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C53C211yesmaximal, characteristic
9C22×Dic106C2211yescharacteristic, complemented
10#13133196—24no—
11C2×C4⋊Dic53C2211yescharacteristic, complemented
12C2×#11719919C2212yescomplemented
13C2×C22⋊Dic53C2214yes—
14C2×C22⋊Dic53C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2214yes—
17C2×C4×Dic53C2212yescomplemented
18#13133124C2214yescomplemented
19#13133124C2214yescomplemented
20#13133124C2212yescomplemented
21C25×C53C2211yescharacteristic
22C23×C4×C53C2211yescharacteristic
23C23×C4×C53C2211yescharacteristic
24C2×C22⋊C4×C53—24no—
25C2×C22⋊C4×C53C2212yes—
26C2×C22⋊C4×C53C2212yes—
27C24×C53—22no—
28C24×C53—24no—
29C24×C53—24no—
30C24×C53—42no—
31C24×C53C2312yes—
32C24×C53C2311yescharacteristic
33C22×C4×C53—22no—
34C22×C4×C53—24no—
35C22×C4×C53—42no—
36C22×C4×C53D414yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×Dic53C2314yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C2×Dic106—22no—
44C2×Dic106—42no—
45C22⋊C4×C53—24no—
46C22⋊C4×C53—24no—
47C22⋊Dic53—24no—
48C22⋊Dic53—28no—
49C22⋊Dic53—24no—
50C22⋊Dic53—44no—
51C4⋊Dic53—22no—
52#11719919—24no—
53C4×Dic53—42no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53—28no—
57C23×C53—22no—
58C23×C53—24no—
59C23×C53—22no—
60C23×C53—24no—
61C23×C53—416no—
62C23×C53—44no—
63C23×C53—44no—
64C23×C53C2×D412yes—
65C23×C53#4114yes—
66C23×C53#4114yes—
67C23×C53C2411yescharacteristic
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—24no—
72C2×C4×C53—44no—
73C2×C4×C53—44no—
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2×D412yes—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—24no—
79C2×Dic53—24no—
80C2×Dic53—24no—
81C2×Dic53—24no—
82C2×Dic53—24no—
83Dic106—44no—
84C22×C53—22no—
85C22×C53—22no—
86C22×C53—28no—
87C22×C53—24no—
88C22×C53—28no—
89C22×C53—416no—
90C22×C53—42no—
91C22×C53—42no—
92C22×C53—44no—
93C22×C53#12414yes—
94C22×C53C22×D411yescharacteristic
95C22×C53C2×#4112yes—
96C22×C53C2×#4112yes—
97C22×C53H(5,𝔽2)12yes—
98C4×C53—22no—
99C4×C53—22no—
100C4×C53—42no—
101Dic53—42no—
102Dic53—42no—
103Dic53—44no—
104#2561—531nomaximal, Sylow
105C2×C53—24no—
106C2×C53—44no—
107C2×C53#52511yescharacteristic
108C2×C53C2×#12411yescharacteristic
109C2×C53#53411yescharacteristic
110C2×C53#55114yes—
111C2×C22⋊Q8—531no—
112C22×C22⋊C4D5311yescharacteristic
113#397—532no—
114#393—532no—
115#388—534no—
116#381—532no—
117#379—532no—
118C4×C22⋊C4—531no—
119C53#256111yescharacteristic, complemented, Sylow
120C25C2×D5311yescharacteristic
121C22×Q8—531no—
122C23×C4C2×D5311yescharacteristic
123C23×C4C2×D5311yescharacteristic
124C22⋊Q8—1064no—
125C2×C4⋊C4—532no—
126C2×C4⋊C4—531no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—532no—
129C2×C22⋊C4—532no—
130C2×C22⋊C4—534no—
131C2×C22⋊C4—534no—
132C2×C22⋊C4C2×D5312yes—
133C2×C22⋊C4C2×D5312yes—
134C2×C42—532no—
135#95—532no—
136#95—534no—
137#95—534no—
138C24—24no—
139C24—22no—
140C24—24no—
141C24—42no—
142C24C22×D5312yes—
143C24C22×D5311yescharacteristic
144C2×Q8—1062no—
145C2×Q8—2122no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—534no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4C22⋊D5314yes—
153C22×C4C22×D5312yes—
154C22×C4C22×D5311yescharacteristic
155C22×C4C22×D5311yescharacteristic
156C4⋊C4—1064no—
157C4⋊C4—1062no—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—1068no—
161C22⋊C4—1064no—
162C22⋊C4—1064no—
163C22⋊C4—2124no—
164C42—2122no—
165C23—22no—
166C23—22no—
167C23—28no—
168C23—22no—
169C23—24no—
170C23—22no—
171C23—24no—
172C23—416no—
173C23—44no—
174C23—44no—
175C23#1171994514yes—
176C23#1171994714yes—
177C23C2×C22⋊D5312yes—
178C23C23×D5311yescharacteristic
179Q8—2124no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1064no—
189C2×C4—1064no—
190C2×C4—1064no—
191C2×C4—1064no—
192C2×C4—1064no—
193C2×C4C2×C22⋊D5312yes—
194C2×C4C2×C22⋊D5312yes—
195C22—28no—
196C22—22no—
197C22—22no—
198C22—28no—
199C22—24no—
200C22—416no—
201C22—44no—
202C22—42no—
203C22—42no—
204C22#1313320414yes—
205C22C22×C22⋊D5311yescharacteristic
206C22C2×#1171994712yes—
207C22C2×#1171994512yes—
208C22#1313326212yes—
209C4—22no—
210C4—22no—
211C4—42no—
212C4—2124no—
213C4—2122no—
214C4—2122no—
215C2—24no—
216C2—44no—
217C2#1681113411yescharacteristic
218C2C2×#1313320411yescharacteristic
219C2#1681118211yescharacteristic
220C2#1681129914yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C22⋊C4×C53C2
hypercenterC22×C22⋊C4D53
hypocenterC53#2561
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13133196C2semidirect
2. #16810548C2semidirect
3. #16810535C2semidirect
4. #16810517C2semidirect
5. #16810486C2semidirect
7. C4×C22⋊Dic53C2semidirect
9. C22×Dic106C22semidirect
11. C2×C4⋊Dic53C22semidirect
12. C2×#11719919C22semidirect
17. C2×C4×Dic53C22semidirect
18. #13133124C22semidirect
19. #13133124C22semidirect
20. #13133124C22semidirect
119. C53#2561semidirect