Finite Group Databasebeta

← #20278985

Subgroups — #20278985

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#16810558C212yesmaximal, complemented
3#16810549C212yesmaximal, complemented
4#16810514C214yesmaximal, complemented
5#16810495C212yesmaximal, complemented
6#16810486C211yesmaximal, characteristic, complemented
7#16810478C212yesmaximal
8C2×#117×C53C211yesmaximal, characteristic, complemented
9C24×D53C2211yescharacteristic
10C23×Dic53C2211yescharacteristic
11C2×#11719919C2214yescomplemented
12C2×C22⋊Dic53C2212yes—
13C2×#11719921—24no—
14C2×#11719921C2212yes—
15C2×#11719921C2214yes—
16C2×#11719921C2212yes—
17C2×#11719921C2214yes—
18#13133124C2212yescomplemented
19#13133124C2214yescomplemented
20#13133124C2212yescomplemented
21C23×C4×C53C2211yescharacteristic
22#117×C53—24no—
23C2×C4⋊C4×C53C2211yescharacteristic, complemented
24C2×C4⋊C4×C53C2211yescharacteristic, complemented
25C2×C22⋊C4×C53C2212yes—
26C2×C42×C53C2212yescomplemented
27C24×C53C2311yescharacteristic
28C23×D53—22no—
29C23×D53—24no—
30C23×D53—24no—
31C23×D53—42no—
32C23×D53C2312yes—
33C22×C4×C53—24no—
34C22×C4×C53D414yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×Dic53—22no—
40C22×Dic53—42no—
41C22×Dic53C2314yes—
42C22×Dic53C2312yes—
43C4⋊C4×C53—22no—
44C4⋊C4×C53—22no—
45C22⋊C4×C53—24no—
46C42×C53—24no—
47C22⋊Dic53—24no—
48#11719921—28no—
49#11719921—24no—
50#11719921—24no—
51#11719921—44no—
52#11719919—44no—
53C23×C53—22no—
54C23×C53—22no—
55C23×C53C2×D412yes—
56C23×C53#4114yes—
57C23×C53C2411yescharacteristic
58C22×D53—28no—
59C22×D53—22no—
60C22×D53—22no—
61C22×D53—24no—
62C22×D53—24no—
63C22×D53—416no—
64C22×D53—44no—
65C22×D53—44no—
66C22×D53#4114yes—
67C2×C4×C53—28no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—24no—
73C2×C4×C53—24no—
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2×D412yes—
76C2×Dic53—22no—
77C2×Dic53—24no—
78C2×Dic53—24no—
79C2×Dic53—24no—
80C2×Dic53—24no—
81C2×Dic53—44no—
82C22×C53—22no—
83C22×C53—22no—
84C22×C53—28no—
85C22×C53#12314yes—
86C22×C53C22×D411yescharacteristic
87C22×C53C2×#4112yes—
88C22×C53C2×#4112yes—
89C22×C53H(5,𝔽2)12yes—
90C2×D53—24no—
91C2×D53—28no—
92C2×D53—416no—
93C2×D53—42no—
94C2×D53—42no—
95C2×D53—44no—
96C4×C53—22no—
97C4×C53—22no—
98C4×C53—42no—
99C4×C53—42no—
100Dic53—42no—
101Dic53—44no—
102#2482—531nomaximal, Sylow
103C2×C53—24no—
104C2×C53C2×#12311yescharacteristic
105C2×C53#52511yescharacteristic
106C2×C53#53511yescharacteristic
107C2×C53#55114yes—
108D53—44no—
109C2×#117D5311yescharacteristic, complemented
110C22×C22⋊C4—531no—
111#399—532no—
112#397—532no—
113#388—534no—
114#385—532no—
115#381—531no—
116#379—532no—
117C53#248211yescharacteristic, complemented, Sylow
118C25—531no—
119C23×C4—531no—
120C23×C4C2×D5311yescharacteristic
121#117—24no—
122C2×C4⋊C4—534no—
123C2×C4⋊C4C2×D5311yescharacteristic, complemented
124C2×C4⋊C4C2×D5311yescharacteristic, complemented
125C2×C22⋊C4—532no—
126C2×C22⋊C4—532no—
127C2×C22⋊C4—532no—
128C2×C22⋊C4—534no—
129C2×C22⋊C4—534no—
130C2×C22⋊C4—1064no—
131C2×C22⋊C4C2×D5312yes—
132C2×C42C2×D5312yescomplemented
133#95—532no—
134#95—532no—
135#95—534no—
136C24—532no—
137C24—1064no—
138C24—1062no—
139C24—1064no—
140C24—2122no—
141C24C22×D5311yescharacteristic
142C22×C4—24no—
143C22×C4—534no—
144C22×C4—532no—
145C22×C4—1062no—
146C22×C4—2122no—
147C22×C4C22⋊D5314yes—
148C22×C4C22×D5312yes—
149C22×C4C22×D5312yes—
150C22×C4C22×D5311yescharacteristic
151C22×C4C22×D5311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—2124no—
155C22⋊C4—24no—
156C22⋊C4—1068no—
157C22⋊C4—1064no—
158C22⋊C4—1064no—
159C22⋊C4—1064no—
160C22⋊C4—2124no—
161C42—24no—
162C23—22no—
163C23—22no—
164C23—534no—
165C23—1068no—
166C23—1062no—
167C23—1062no—
168C23—1064no—
169C23—1064no—
170C23—21216no—
171C23—2124no—
172C23—2124no—
173C23#1171994514yes—
174C23C2×C22⋊D5312yes—
175C23C23×D5311yescharacteristic
176C2×C4—28no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—1062no—
184C2×C4—1064no—
185C2×C4—1064no—
186C2×C4—1064no—
187C2×C4—1064no—
188C2×C4—2124no—
189C2×C4C2×C22⋊D5312yes—
190C2×C4C2×C22⋊D5312yes—
191C22—28no—
192C22—22no—
193C22—22no—
194C22—1068no—
195C22—1064no—
196C22—21216no—
197C22—2124no—
198C22—2122no—
199C22—2122no—
200C22#1313320014yes—
201C22C22×C22⋊D5311yescharacteristic
202C22#41×D5312yes—
203C22C2×#1171994512yes—
204C22#1313326312yes—
205C4—22no—
206C4—22no—
207C4—42no—
208C4—42no—
209C4—2124no—
210C4—2122no—
211C2—24no—
212C2—2124no—
213C2C2×#1313320011yescharacteristic
214C2#1681113011yescharacteristic
215C2#1681118711yescharacteristic
216C2#1681129714yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#117×C53C2
hypercenterC2×#117D53
hypocenterC53#2482
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #16810558C2semidirect
3. #16810549C2semidirect
4. #16810514C2semidirect
5. #16810495C2semidirect
6. #16810486C2semidirect
8. C2×#117×C53C2semidirect
11. C2×#11719919C22semidirect
18. #13133124C22semidirect
19. #13133124C22semidirect
20. #13133124C22semidirect
23. C2×C4⋊C4×C53C22semidirect
24. C2×C4⋊C4×C53C22semidirect
26. C2×C42×C53C22semidirect
109. C2×#117D53semidirect
117. C53#2482semidirect
123. C2×C4⋊C4C2×D53semidirect
124. C2×C4⋊C4C2×D53semidirect
132. C2×C42C2×D53semidirect