Finite Group Databasebeta

← #20281622

Subgroups — #20281622

Order 6784.

448 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 129 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×#13133210C212yesmaximal
2#16810557C212yesmaximal, complemented
3#16810556C212yesmaximal, complemented
4#16810548C211yesmaximal, characteristic
5#16810520C212yesmaximal, complemented
6#16810505C212yesmaximal, complemented
7#16810495C211yesmaximal, characteristic
8#16810481C212yesmaximal
9C53×#397C211yesmaximal, characteristic
10C23×Dic53C2211yescharacteristic
11#13133210—28no—
12C2×C4⋊Dic53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C4×Dic53C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#13133124C2212yes—
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2211yescharacteristic
30#95×C53C2211yescharacteristic
31#95×C53C2212yes—
32C24×C53C2311yescharacteristic
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×Dic53—24no—
38C22×Dic53—41no—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C4⋊C4×C53—41no—
44C22⋊C4×C53—24no—
45C22⋊C4×C53—41no—
46C22⋊Dic53—24no—
47C22⋊Dic53D414yes—
48C4⋊Dic53—24no—
49C4⋊Dic53—41no—
50#11719919—24no—
51#11719919—24no—
52#11719919—24no—
53#11719919—24no—
54#11719919—42no—
55C4×Dic53—24no—
56C23×C53—22no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—22no—
60C23×C53C2×D412yes—
61C23×C53C2411yescharacteristic
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
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—44no—
83C2×Dic53C2×D414yes—
84C2×Dic53#4114yes—
85C22×C53—22no—
86C22×C53—24no—
87C22×C53—42no—
88C22×C53—41no—
89C22×C53—41no—
90C22×C53C22×D411yescharacteristic
91C22×C53C2×#4112yes—
92C22×C53H(5,𝔽2)11yescharacteristic
93C22×C532−1+411yescharacteristic
94C22×C532−1+412yes—
95C4×C53—42no—
96C4×C53—42no—
97C4×C53—41no—
98C4×C53—41no—
99Dic53—24no—
100Dic53—24no—
101Dic53—42no—
102Dic53—42no—
103#2722—531nomaximal, Sylow
104C2×C53—42no—
105C2×C53#53711yescharacteristic
106C2×C53#54112yes—
107C2×C53#54712yes—
108C2×C53#55111yescharacteristic
109C2×C53#56311yescharacteristic
110C2×#126—532no—
111#398—532no—
112#398—532no—
113#397—531no—
114#397D5311yescharacteristic
115#389—532no—
116#387—532no—
117#385—531no—
118#380—532no—
119C53#272211yescharacteristic, complemented, Sylow
120C23×C4—531no—
121#126—1068no—
122C2×C4⋊C4—532no—
123C2×C4⋊C4—532no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—532no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—532no—
128C2×C4⋊C4—531no—
129C2×C4⋊C4C2×D5311yescharacteristic
130C2×C22⋊C4—531no—
131C2×C22⋊C4—532no—
132C2×C22⋊C4C2×D5312yes—
133C2×C22⋊C4C2×D5311yescharacteristic
134C2×C42—532no—
135#95—532no—
136#95—532no—
137#95—532no—
138#95—532no—
139#95—531no—
140#95C2×D5311yescharacteristic
141#95C2×D5312yes—
142C24C22×D5311yescharacteristic
143C22×C4—532no—
144C22×C4—532no—
145C22×C4—532no—
146C22×C4—532no—
147C22×C4—1064no—
148C22×C4—2121no—
149C22×C4C22×D5311yescharacteristic
150C22×C4C22×D5312yes—
151C22×C4C22×D5311yescharacteristic
152C22×C4C22×D5312yes—
153C4⋊C4—41no—
154C4⋊C4—1064no—
155C4⋊C4—1064no—
156C4⋊C4—1064no—
157C4⋊C4—1064no—
158C4⋊C4—1064no—
159C4⋊C4—2122no—
160C4⋊C4—2121no—
161C22⋊C4—24no—
162C22⋊C4—41no—
163C22⋊C4—534no—
164C22⋊C4—1064no—
165C42—1064no—
166C23—22no—
167C23—21no—
168C23—22no—
169C23—21no—
170C23D4×D5312yes—
171C23C23×D5311yescharacteristic
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—534no—
183C2×C4—534no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—2124no—
195C22—24no—
196C22—22no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22C2×D4×D5311yescharacteristic
201C22#41×D5312yes—
202C22#1313326211yescharacteristic
203C22#1313326412yes—
204C22#1313326411yescharacteristic
205C4—42no—
206C4—41no—
207C4—41no—
208C4—42no—
209C4—1064no—
210C4—1064no—
211C4—2122no—
212C4—2122no—
213C2—42no—
214C2#1681120011yescharacteristic
215C2#1681123012yes—
216C2#1681127112yes—
217C2#1681130311yescharacteristic
218C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2722
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #16810557C2semidirect
3. #16810556C2semidirect
5. #16810520C2semidirect
6. #16810505C2semidirect
119. C53#2722semidirect