Finite Group Databasebeta

← #20281667

Subgroups — #20281667

Order 6784.

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

normal subgroups only

This group has 416 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×#13133207C212yesmaximal
2#16810555C212yesmaximal
3#16810553C212yesmaximal
4#16810553C212yesmaximal
5#16810522C212yesmaximal
6#16810521C211yesmaximal, characteristic
7#16810521C211yesmaximal, characteristic
8#16810508C212yesmaximal
9C53×#389C211yesmaximal, characteristic
10#13133207—28no—
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2212yes—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×C4×Dic53C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C42×C53C2211yescharacteristic
31#95×C53C2212yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C4⋊C4×C53—24no—
42C4⋊C4×C53—41no—
43C4⋊C4×C53D414yes—
44C42×C53—41no—
45C4⋊Dic53—24no—
46C4⋊Dic53—22no—
47C4⋊Dic53—41no—
48C4⋊Dic53—41no—
49C4⋊Dic53—42no—
50C4⋊Dic53Q814yes—
51#11719919—24no—
52#11719919—24no—
53#11719919—24no—
54#11719919—24no—
55C4×Dic53—24no—
56C23×C53C2411yescharacteristic
57C2×C4×C53—22no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53C2×D414yes—
69C2×C4×C53C2×D412yes—
70C2×C4×C53C2×Q812yes—
71C2×C4×C53#4112yes—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53C2×Q814yes—
84C22×C53C22×D411yescharacteristic
85C22×C53C22×Q811yescharacteristic
86C22×C53C2×#4111yescharacteristic
87C22×C53H(5,𝔽2)11yescharacteristic
88C22×C532−1+411yescharacteristic
89C22×C532−1+411yescharacteristic
90C22×C532−1+411yescharacteristic
91C4×C53—22no—
92C4×C53—22no—
93C4×C53—22no—
94C4×C53—24no—
95C4×C53—42no—
96Dic53—24no—
97Dic53—42no—
98Dic53—42no—
99Dic53—42no—
100#2726—531nomaximal, Sylow
101C2×C53#53711yescharacteristic
102C2×C53#53911yescharacteristic
103C2×C53#54411yescharacteristic
104C2×C53#54711yescharacteristic
105C2×C53D4×Q811yescharacteristic
106C2×C53#55711yescharacteristic
107C2×C53#56411yescharacteristic
108C2×#125—532no—
109#398—532no—
110#398—532no—
111#398—532no—
112#389—532no—
113#389—531no—
114#389—531no—
115#389D5311yescharacteristic
116#387—532no—
117C53#272611yescharacteristic, complemented, Sylow
118#125—1068no—
119C2×C4⋊C4—531no—
120C2×C4⋊C4—532no—
121C2×C4⋊C4—532no—
122C2×C4⋊C4—532no—
123C2×C4⋊C4—532no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—531no—
126C2×C4⋊C4—531no—
127C2×C4⋊C4—532no—
128C2×C4⋊C4—531no—
129C2×C4⋊C4C2×D5312yes—
130C2×C4⋊C4C2×D5311yescharacteristic
131C2×C4⋊C4C2×D5311yescharacteristic
132C2×C42—532no—
133C2×C42C2×D5311yescharacteristic
134#95—532no—
135#95—532no—
136#95—532no—
137#95—532no—
138#95—532no—
139#95C2×D5312yes—
140C22×C4—532no—
141C22×C4—532no—
142C22×C4—532no—
143C22×C4—532no—
144C22×C4C22×D5312yes—
145C22×C4C22×D5312yes—
146C22×C4C22×D5311yescharacteristic
147C22×C4C22×D5311yescharacteristic
148C22×C4C22×D5311yescharacteristic
149C4⋊C4—24no—
150C4⋊C4—41no—
151C4⋊C4—534no—
152C4⋊C4—1064no—
153C4⋊C4—1064no—
154C4⋊C4—1064no—
155C4⋊C4—1064no—
156C4⋊C4—1064no—
157C4⋊C4—1062no—
158C4⋊C4—2121no—
159C4⋊C4—2121no—
160C4⋊C4—2122no—
161C4⋊C4D21214yes—
162C42—41no—
163C42—1064no—
164C23C23×D5311yescharacteristic
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—534no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4—1062no—
180C2×C4—1062no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4C2×D21214yes—
189C2×C4C2×D21212yes—
190C2×C4#1171994712yes—
191C2×C4Q8×D5312yes—
192C22C22×D21211yescharacteristic
193C22C2×Q8×D5311yescharacteristic
194C22C2×#1171994711yescharacteristic
195C22#1313326211yescharacteristic
196C22#1313326411yescharacteristic
197C22#1313326411yescharacteristic
198C22#1313326511yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—22no—
203C4—42no—
204C4—1064no—
205C4—2122no—
206C4—2122no—
207C4—2122no—
208C2#1681120411yescharacteristic
209C2#1681121811yescharacteristic
210C2#1681125211yescharacteristic
211C2#1681127711yescharacteristic
212C2Q8×D21211yescharacteristic
213C2#1681134811yescharacteristic
214C2C53⋊#56411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#389C2
hypercenter#389D53
hypocenterC53#2726
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
117. C53#2726semidirect