Finite Group Databasebeta

← #20280495

Subgroups — #20280495

Order 6784.

526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133200C211yesmaximal, characteristic
2#16810561C212yesmaximal, complemented
3#16810558C212yesmaximal, complemented
4#16810554C212yesmaximal, complemented
5#16810549C212yesmaximal, complemented
6#16810502C212yesmaximal
7#16810496C212yesmaximal, complemented
8#16810486C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C53C211yesmaximal, characteristic, complemented
10C22×C22⋊D53C2211yescharacteristic
11#13133200—24no—
12C2×#11719919C2211yescharacteristic
13C2×#11719919C2211yescharacteristic
14C2×#11719919C2214yes—
15C2×C22⋊Dic53C2211yescharacteristic
16C2×C22⋊Dic53C2212yes—
17C2×#11719921C2211yescharacteristic
18C2×#11719921C2211yescharacteristic
19C2×#11719921C2214yes—
20#13133124C2214yes—
21#13133124C2214yes—
22#13133124C2212yes—
23#13133124C2212yes—
24C23×C4×C53C2211yescharacteristic
25C23×C4×C53C2212yes—
26C2×C4⋊C4×C53—24no—
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2212yes—
29C24×C53C2311yescharacteristic
30C23×D53C2311yescharacteristic
31C22×C4×C53—24no—
32C22×C4×C53—24no—
33C22×C4×C53—44no—
34C22×C4×C53D414yes—
35C22×C4×C53C2314yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—22no—
39C2×C22⋊D53—24no—
40C22×Dic53C2311yescharacteristic
41C22×Dic53C2314yes—
42C22×Dic53C2311yescharacteristic
43C22×Dic53C2311yescharacteristic
44C4⋊C4×C53—24no—
45C4⋊C4×C53—24no—
46C22⋊Dic53—22no—
47C22⋊Dic53—24no—
48#11719921—22no—
49#11719921—22no—
50#11719921—44no—
51#11719919—22no—
52#11719919—22no—
53#11719919—44no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53C2×D412yes—
57C23×C53#4114yes—
58C23×C53C2411yescharacteristic
59C22×D53—21no—
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—22no—
63C2×C4×C53—28no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—24no—
67C2×C4×C53—24no—
68C2×C4×C53—24no—
69C2×C4×C53—48no—
70C2×C4×C53C2×D412yes—
71C2×C4×C53C2×D412yes—
72C22⋊D53—44no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—22no—
77C2×Dic53—24no—
78C2×Dic53—22no—
79C2×Dic53—24no—
80C2×Dic53—22no—
81C2×Dic53—24no—
82C22×C53—22no—
83C22×C53—22no—
84C22×C53—28no—
85C22×C53#12314yes—
86C22×C53C22×D411yescharacteristic
87C22×C53C2×#4112yes—
88C22×C53H(5,𝔽2)12yes—
89C22×C532−1+412yes—
90C2×D53—41no—
91C2×D53—41no—
92C2×D53—41no—
93C2×D53—44no—
94C4×C53—22no—
95C4×C53—22no—
96C4×C53—44no—
97Dic53—41no—
98Dic53—41no—
99Dic53—41no—
100Dic53—44no—
101#2623—531nomaximal, Sylow
102C2×C53—24no—
103C2×C53C2×#12311yescharacteristic
104C2×C53#53511yescharacteristic
105C2×C53#53611yescharacteristic
106C2×C53#53914yes—
107D53—81no—
108C2×#123—531no—
109C22×C4⋊C4D5311yescharacteristic, complemented
110#400—532no—
111#399—532no—
112#398—532no—
113#397—532no—
114#386—532no—
115#385—532no—
116#381—531no—
117C53#262311yescharacteristic, complemented, Sylow
118C22×D4—531no—
119C23×C4C2×D5311yescharacteristic
120C23×C4C2×D5312yes—
121#123—1064no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—531no—
124C2×C4⋊C4—531no—
125C2×C4⋊C4—534no—
126C2×C4⋊C4C2×D5312yes—
127C2×C4⋊C4C2×D5312yes—
128C2×C22⋊C4—531no—
129C2×C22⋊C4—532no—
130C2×C22⋊C4—531no—
131C2×C22⋊C4—531no—
132C2×C22⋊C4—534no—
133#95—532no—
134#95—532no—
135#95—534no—
136#95—534no—
137C24—531no—
138C24C22×D5311yescharacteristic
139C2×D4—1064no—
140C2×D4—1062no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—44no—
144C22×C4—531no—
145C22×C4—534no—
146C22×C4—531no—
147C22×C4—531no—
148C22×C4C22⋊D5314yes—
149C22×C4C22×D5314yes—
150C22×C4C22×D5311yescharacteristic
151C22×C4C22×D5311yescharacteristic
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—1062no—
155C4⋊C4—1062no—
156C4⋊C4—2124no—
157C22⋊C4—1064no—
158C22⋊C4—1062no—
159C22⋊C4—1062no—
160C22⋊C4—1062no—
161C22⋊C4—2124no—
162C23—22no—
163C23—22no—
164C23—1061no—
165C23—1062no—
166C23—1062no—
167C23—1062no—
168C23#1171994514yes—
169C23C2×C22⋊D5312yes—
170C23C23×D5311yescharacteristic
171D4—2124no—
172C2×C4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—48no—
179C2×C4—1061no—
180C2×C4—1061no—
181C2×C4—1061no—
182C2×C4—1064no—
183C2×C4—1062no—
184C2×C4—1064no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1064no—
188C2×C4C2×C22⋊D5312yes—
189C2×C4C2×C22⋊D5312yes—
190C22—22no—
191C22—22no—
192C22—28no—
193C22—2121no—
194C22—2121no—
195C22—2121no—
196C22—2124no—
197C22#1313320014yes—
198C22C22×C22⋊D5311yescharacteristic
199C22C2×#1171994512yes—
200C22#1313326212yes—
201C22#1313326512yes—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—2121no—
206C4—2121no—
207C4—2121no—
208C4—2124no—
209C2—24no—
210C2—4241no—
211C2C2×#1313320011yescharacteristic
212C2#1681119011yescharacteristic
213C2#1681119711yescharacteristic
214C2#1681122014yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C4⋊C4×C53C2
hypercenterC22×C4⋊C4D53
hypocenterC53#2623
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16810561C2semidirect
3. #16810558C2semidirect
4. #16810554C2semidirect
5. #16810549C2semidirect
7. #16810496C2semidirect
8. #16810486C2semidirect
9. C22×C4⋊C4×C53C2semidirect
109. C22×C4⋊C4D53semidirect
117. C53#2623semidirect