Finite Group Databasebeta

← #20280095

Subgroups — #20280095

Order 6784.

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

normal subgroups only

This group has 544 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×#13133196C211yesmaximal, characteristic
2#16810553C212yesmaximal, complemented
3#16810543C212yesmaximal, complemented
4#16810518C214yesmaximal, complemented
5#16810496C212yesmaximal
6#16810496C212yesmaximal
7C4×C22⋊Dic53C211yesmaximal, characteristic
8C22×C4⋊C4×C53C211yesmaximal, characteristic
9C22×Dic106C2211yescharacteristic
10#13133196—24no—
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919C2214yes—
13C2×#11719919C2212yes—
14C2×#11719919C2214yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C4×Dic53C2212yes—
18#13133124C2214yes—
19#13133124C2214yes—
20#13133124C2212yes—
21C23×C4×C53C2211yescharacteristic
22C23×C4×C53C2211yescharacteristic
23C23×C4×C53C2211yescharacteristic
24C2×C4⋊C4×C53—24no—
25C2×C4⋊C4×C53C2212yes—
26C2×C4⋊C4×C53C2212yes—
27C24×C53C2311yescharacteristic
28C22×C4×C53—22no—
29C22×C4×C53—24no—
30C22×C4×C53—22no—
31C22×C4×C53—42no—
32C22×C4×C53—42no—
33C22×C4×C53D414yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2314yes—
41C2×Dic106—22no—
42C2×Dic106—42no—
43C4⋊C4×C53—24no—
44C4⋊C4×C53—24no—
45C22⋊Dic53—24no—
46C22⋊Dic53Q818yescomplemented
47C4⋊Dic53—22no—
48#11719919—24no—
49#11719919—28no—
50#11719919—44no—
51C4×Dic53—42no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53C2×D412yes—
55C23×C53C2×Q814yes—
56C23×C53C2411yescharacteristic
57C2×C4×C53—28no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—24no—
64C2×C4×C53—44no—
65C2×C4×C53—44no—
66C2×C4×C53C2×D412yes—
67C2×C4×C53C2×D412yes—
68C2×C4×C53#4114yes—
69C2×Dic53—24no—
70C2×Dic53—24no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—24no—
74C2×Dic53—24no—
75C2×Dic53C2×Q818yes—
76Dic106—44no—
77C22×C53—22no—
78C22×C53—22no—
79C22×C53—28no—
80C22×C53C4⋊Q814yes—
81C22×C53C22×D411yescharacteristic
82C22×C53C22×Q812yes—
83C22×C53C2×#4112yes—
84C22×C53H(5,𝔽2)12yes—
85C4×C53—24no—
86C4×C53—22no—
87C4×C53—22no—
88C4×C53—42no—
89Dic53—28no—
90Dic53—42no—
91Dic53—42no—
92#2583—531nomaximal, Sylow
93C2×C53—24no—
94C2×C53#52511yescharacteristic
95C2×C53C2×C4⋊Q811yescharacteristic
96C2×C53#53411yescharacteristic
97C2×C53#55414yes—
98C2×C22⋊Q8—531no—
99C22×C4⋊C4D5311yescharacteristic
100#398—532no—
101#395—532no—
102#389—534no—
103#385—532no—
104#385—532no—
105C4×C22⋊C4—531no—
106C53#258311yescharacteristic, complemented, Sylow
107C22×Q8—531no—
108C23×C4C2×D5311yescharacteristic
109C23×C4C2×D5311yescharacteristic
110C23×C4C2×D5311yescharacteristic
111C22⋊Q8—1064no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—532no—
114C2×C4⋊C4—531no—
115C2×C4⋊C4—534no—
116C2×C4⋊C4—534no—
117C2×C4⋊C4C2×D5312yes—
118C2×C4⋊C4C2×D5312yes—
119C2×C22⋊C4—532no—
120C2×C22⋊C4—532no—
121C2×C42—532no—
122#95—532no—
123#95—534no—
124#95—534no—
125C24C22×D5311yescharacteristic
126C2×Q8—1062no—
127C2×Q8—2122no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—534no—
134C22×C4—532no—
135C22×C4—532no—
136C22×C4C22⋊D5314yes—
137C22×C4C22×D5312yes—
138C22×C4C22×D5312yes—
139C22×C4C22×D5311yescharacteristic
140C22×C4C22×D5311yescharacteristic
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—1068no—
144C4⋊C4—1064no—
145C4⋊C4—1062no—
146C4⋊C4—2124no—
147C22⋊C4—538no—
148C22⋊C4—1064no—
149C42—2122no—
150C23—22no—
151C23—22no—
152C23Q8×D5314yes—
153C23C2×C22⋊D5312yes—
154C23C23×D5311yescharacteristic
155Q8—2124no—
156C2×C4—28no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—538no—
166C2×C4—1064no—
167C2×C4—1064no—
168C2×C4—1062no—
169C2×C4—1062no—
170C2×C4—1064no—
171C2×C4—1064no—
172C2×C4#1171994514yes—
173C2×C4C2×C22⋊D5312yes—
174C2×C4C2×C22⋊D5312yes—
175C22—28no—
176C22—22no—
177C22—22no—
178C22#1313321514yes—
179C22C22×C22⋊D5311yescharacteristic
180C22C2×Q8×D5312yes—
181C22C2×#1171994512yes—
182C22#1313326212yes—
183C4—24no—
184C4—22no—
185C4—22no—
186C4—42no—
187C4—1068no—
188C4—2122no—
189C4—2122no—
190C2—24no—
191C2#1681113411yescharacteristic
192C2C2×#1313321511yescharacteristic
193C2#1681118211yescharacteristic
194C2#1681132814yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C4⋊C4×C53C2
hypercenterC22×C4⋊C4D53
hypocenterC53#2583
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16810553C2semidirect
3. #16810543C2semidirect
4. #16810518C2semidirect
46. C22⋊Dic53Q8semidirect
106. C53#2583semidirect