Finite Group Databasebeta

← #20280912

Subgroups — #20280912

Order 6784.

408 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#16810559C212yesmaximal
2#16810556C212yesmaximal, complemented
3#16810556C212yesmaximal, complemented
4#16810546C212yesmaximal
5#16810520C212yesmaximal, complemented
6#16810506C212yesmaximal, complemented
7#16810486C211yesmaximal, characteristic
8C4×C22⋊Dic53C211yesmaximal, characteristic
9C53×#380C211yesmaximal, characteristic
10C2×C4⋊Dic53C2212yes—
11C2×C4⋊Dic53C2212yes—
12C2×#11719919C2212yes—
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C4×Dic53C2212yes—
19#13133124C2211yescharacteristic
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2211yescharacteristic
23#13133124C2211yescharacteristic
24#13133124C2211yescharacteristic
25#13133124C2212yes—
26C23×C4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2212yes—
28C2×C22⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2211yescharacteristic
30#95×C53C2212yes—
31C24×C53C2311yescharacteristic
32C22×C4×C53—42no—
33C22×C4×C53Q814yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C4⋊C4×C53—42no—
43C22⋊C4×C53—41no—
44C22⋊C4×C53—41no—
45C22⋊Dic53—24no—
46C22⋊Dic53—42no—
47C4⋊Dic53—24no—
48C4⋊Dic53—42no—
49#11719919—24no—
50#11719919—42no—
51#11719919—42no—
52#11719919—42no—
53C4×Dic53—42no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53C2×Q812yes—
59C23×C53C2411yescharacteristic
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—44no—
69C2×C4×C53C2×Q812yes—
70C2×C4×C53C2×Q812yes—
71C2×Dic53—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#4114yes—
83C22×C53—22no—
84C22×C53—22no—
85C22×C53—22no—
86C22×C53—42no—
87C22×C53—42no—
88C22×C53C22×Q811yescharacteristic
89C22×C53C2×#4112yes—
90C22×C53H(5,𝔽2)12yes—
91C22×C53H(5,𝔽2)11yescharacteristic
92C22×C532−1+411yescharacteristic
93C4×C53—22no—
94C4×C53—22no—
95C4×C53—42no—
96C4×C53—42no—
97Dic53—24no—
98Dic53—42no—
99Dic53—42no—
100Dic53—42no—
101#2667—531nomaximal, Sylow
102C2×C53—42no—
103C2×C53#53311yescharacteristic
104C2×C53#53912yes—
105C2×C53#54012yes—
106C2×C53#54311yescharacteristic
107C2×C53#54411yescharacteristic
108#399—532no—
109#398—532no—
110#398—532no—
111#396—532no—
112#389—532no—
113#387—532no—
114#381—531no—
115#380D5311yescharacteristic
116C4×C22⋊C4—531no—
117C53#266711yescharacteristic, complemented, Sylow
118C23×C4C2×D5311yescharacteristic
119C2×C4⋊C4—532no—
120C2×C4⋊C4—532no—
121C2×C4⋊C4—532no—
122C2×C4⋊C4—532no—
123C2×C4⋊C4—532no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4C2×D5312yes—
126C2×C22⋊C4—532no—
127C2×C22⋊C4—532no—
128C2×C22⋊C4C2×D5311yescharacteristic
129C2×C22⋊C4C2×D5311yescharacteristic
130C2×C42—532no—
131#95—531no—
132#95—531no—
133#95—531no—
134#95—531no—
135#95—532no—
136#95—532no—
137#95—532no—
138#95C2×D5312yes—
139C24C22×D5311yescharacteristic
140C22×C4—42no—
141C22×C4—532no—
142C22×C4—532no—
143C22×C4—532no—
144C22×C4—532no—
145C22×C4Dic10614yes—
146C22×C4C22×D5312yes—
147C22×C4C22×D5312yes—
148C22×C4C22×D5311yescharacteristic
149C22×C4C22×D5311yescharacteristic
150C4⋊C4—42no—
151C4⋊C4—1064no—
152C4⋊C4—1064no—
153C4⋊C4—2122no—
154C4⋊C4—2122no—
155C4⋊C4—2122no—
156C4⋊C4—2122no—
157C22⋊C4—41no—
158C22⋊C4—41no—
159C22⋊C4—1064no—
160C22⋊C4—2122no—
161C42—2122no—
162C23—22no—
163C23—21no—
164C23—22no—
165C23—21no—
166C23C2×Dic10612yes—
167C23C23×D5311yescharacteristic
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—44no—
177C2×C4—534no—
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×C4—1062no—
189C2×C4C2×Dic10612yes—
190C2×C4C2×Dic10612yes—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22C22×Dic10611yescharacteristic
197C22#41×D5312yes—
198C22#1313326212yes—
199C22#1313326211yescharacteristic
200C22#1313326511yescharacteristic
201C4—22no—
202C4—22no—
203C4—42no—
204C4—42no—
205C4—1064no—
206C4—2122no—
207C4—2122no—
208C4—2122no—
209C2—42no—
210C2#1681117611yescharacteristic
211C2#1681121912yes—
212C2#1681122312yes—
213C2#1681124911yescharacteristic
214C2#1681125411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#380C2
hypercenter#380D53
hypocenterC53#2667
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #16810556C2semidirect
3. #16810556C2semidirect
5. #16810520C2semidirect
6. #16810506C2semidirect
117. C53#2667semidirect