Finite Group Databasebeta

← #20278993

Subgroups — #20278993

Order 6784.

700 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 165 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×C22⋊Dic53C211yesmaximal, characteristic
2#16810551C214yesmaximal, complemented
3#16810496C212yesmaximal, complemented
4#16810495C212yesmaximal, complemented
5#16810486C212yesmaximal, complemented
6#16810484C212yesmaximal, complemented
7#16810479C211yesmaximal, characteristic
8C22×C22⋊C4×C53C211yesmaximal, characteristic
9C23×Dic53C2211yescharacteristic
10C23×Dic53C2211yescharacteristic
11C2×#11719919C2214yescomplemented
12C2×C22⋊Dic53—24no—
13C2×C22⋊Dic53C2212yes—
14C2×C22⋊Dic53C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18#13133124C2214yescomplemented
19#13133124C2214yescomplemented
20#13133124C2214yescomplemented
21C25×C53C2211yescharacteristic
22C23×C4×C53C2211yescharacteristic
23C23×C4×C53C2211yescharacteristic
24C2×C22⋊C4×C53—24no—
25C2×C22⋊C4×C53C2212yes—
26C2×C22⋊C4×C53C2212yes—
27C24×C53—24no—
28C24×C53—24no—
29C24×C53—42no—
30C24×C53D414yes—
31C24×C53C2311yescharacteristic
32C24×C53C2311yescharacteristic
33C24×C53C2311yescharacteristic
34C22×C4×C53—22no—
35C22×C4×C53—22no—
36C22×C4×C53—24no—
37C22×C4×C53—42no—
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×Dic53—22no—
41C22×Dic53—22no—
42C22×Dic53—24no—
43C22×Dic53—42no—
44C22×Dic53C2312yes—
45C22×Dic53C2314yes—
46C22×Dic53C2312yes—
47C22⋊C4×C53—24no—
48C22⋊C4×C53—24no—
49C22⋊Dic53—24no—
50C22⋊Dic53—24no—
51C22⋊Dic53—24no—
52C22⋊Dic53—24no—
53C22⋊Dic53—42no—
54#11719919—44no—
55C23×C53—22no—
56C23×C53—22no—
57C23×C53—22no—
58C23×C53—22no—
59C23×C53—28no—
60C23×C53—28no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—44no—
65C23×C53—416no—
66C23×C53C2×D412yes—
67C23×C53C2×D412yes—
68C23×C53C2×D412yes—
69C23×C53#4114yes—
70C23×C53#4114yes—
71C23×C53C2411yescharacteristic
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—24no—
75C2×C4×C53—24no—
76C2×C4×C53—44no—
77C2×C4×C53—44no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—24no—
81C2×Dic53—24no—
82C2×Dic53—24no—
83C2×Dic53—24no—
84C2×Dic53—24no—
85C2×Dic53—44no—
86C2×Dic53—44no—
87C22×C53—22no—
88C22×C53—22no—
89C22×C53—22no—
90C22×C53—22no—
91C22×C53—28no—
92C22×C53—28no—
93C22×C53—22no—
94C22×C53—22no—
95C22×C53—22no—
96C22×C53—416no—
97C22×C53—44no—
98C22×C53#12314yes—
99C22×C53#12314yes—
100C22×C53C22×D411yescharacteristic
101C22×C53C2×#4112yes—
102C22×C53C2×#4112yes—
103C22×C53H(5,𝔽2)12yes—
104C4×C53—42no—
105C4×C53—42no—
106Dic53—42no—
107Dic53—42no—
108Dic53—44no—
109#2483—531nomaximal, Sylow
110C2×C53—24no—
111C2×C53—24no—
112C2×C53—42no—
113C2×C53C2×#12311yescharacteristic
114C2×C53C2×#12311yescharacteristic
115C2×C53#53411yescharacteristic
116C2×C53#55114yes—
117C22×C22⋊C4—531no—
118C22×C22⋊C4D5311yescharacteristic
119#397—534no—
120#385—532no—
121#385—532no—
122#381—532no—
123#381—532no—
124#379—531no—
125C53#248311yescharacteristic, complemented, Sylow
126C25C2×D5311yescharacteristic
127C23×C4—531no—
128C23×C4—531no—
129C23×C4C2×D5311yescharacteristic
130C23×C4C2×D5311yescharacteristic
131C2×C4⋊C4—534no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—532no—
134C2×C22⋊C4—532no—
135C2×C22⋊C4—532no—
136C2×C22⋊C4—532no—
137C2×C22⋊C4—532no—
138C2×C22⋊C4—1064no—
139C2×C22⋊C4C2×D5312yes—
140C2×C22⋊C4C2×D5312yes—
141#95—534no—
142#95—534no—
143#95—534no—
144C24—24no—
145C24—24no—
146C24—42no—
147C24C22⋊D5314yes—
148C24C22×D5311yescharacteristic
149C24C22×D5311yescharacteristic
150C24C22×D5311yescharacteristic
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4—532no—
156C22×C4—534no—
157C22×C4—532no—
158C22×C4—1064no—
159C22×C4—1062no—
160C22×C4—1062no—
161C22×C4—2122no—
162C22×C4C22×D5312yes—
163C22×C4C22×D5312yes—
164C4⋊C4—2124no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—1064no—
168C22⋊C4—1064no—
169C22⋊C4—1064no—
170C22⋊C4—1064no—
171C22⋊C4—2122no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—28no—
177C23—28no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—44no—
182C23—416no—
183C23#1171994514yes—
184C23#1171994714yes—
185C23C2×C22⋊D5312yes—
186C23C2×C22⋊D5312yes—
187C23C2×C22⋊D5312yes—
188C23C23×D5311yescharacteristic
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1064no—
198C2×C4—1064no—
199C2×C4—1064no—
200C2×C4—1064no—
201C2×C4—1064no—
202C2×C4—2124no—
203C2×C4—2124no—
204C22—28no—
205C22—28no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—416no—
214C22—44no—
215C22#1313319814yes—
216C22#1313320014yes—
217C22C22×C22⋊D5311yescharacteristic
218C22C2×#1171994712yes—
219C22C2×#1171994512yes—
220C22#1313326212yes—
221C4—42no—
222C4—42no—
223C4—2124no—
224C4—2122no—
225C4—2122no—
226C2—24no—
227C2—24no—
228C2—42no—
229C2C2×#1313319811yescharacteristic
230C2C2×#1313320011yescharacteristic
231C2#1681118211yescharacteristic
232C2#1681129914yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C22⋊C4×C53C2
hypercenterC22×C22⋊C4D53
hypocenterC53#2483
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #16810551C2semidirect
3. #16810496C2semidirect
4. #16810495C2semidirect
5. #16810486C2semidirect
6. #16810484C2semidirect
11. C2×#11719919C22semidirect
18. #13133124C22semidirect
19. #13133124C22semidirect
20. #13133124C22semidirect
125. C53#2483semidirect