Finite Group Databasebeta

← #20281642

Subgroups — #20281642

Order 6784.

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

normal subgroups only

This group has 504 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×#13133202C212yesmaximal
2#16810553C211yesmaximal, characteristic, complemented
3#16810552C212yesmaximal, complemented
4#16810552C212yesmaximal, complemented
5#16810524C211yesmaximal, characteristic
6#16810518C212yesmaximal, complemented
7#16810497C212yesmaximal, complemented
8#16810494C212yesmaximal, complemented
9C53×#398C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C4×D53C2212yes—
12#13133202—28no—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#13133124C2212yes—
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2211yescharacteristic
32C23×D53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C2×D212—41no—
42C2×D212—41no—
43C2×D212D414yes—
44C2×C4×D53—24no—
45C2×C4×D53—42no—
46C2×C4×D53—42no—
47C4⋊C4×C53—24no—
48C4⋊C4×C53—41no—
49C4⋊C4×C53—41no—
50#11719921—24no—
51#11719921—24no—
52#11719921—24no—
53#11719921—24no—
54#11719921—24no—
55C4⋊Dic53—41no—
56#11719919—24no—
57#11719919—24no—
58#11719919—42no—
59C4×Dic53—41no—
60C23×C53C2411yescharacteristic
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53—22no—
67C22×D53C2×D414yes—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D412yes—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53#4114yes—
90D212—44no—
91C4×D53—48no—
92C22×C53C22×D411yescharacteristic
93C22×C53C2×#4112yes—
94C22×C53H(5,𝔽2)11yescharacteristic
95C22×C53H(5,𝔽2)11yescharacteristic
96C22×C532−1+412yes—
97C2×D53—24no—
98C2×D53—24no—
99C2×D53—24no—
100C2×D53—42no—
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—42no—
104C4×C53—22no—
105C4×C53—42no—
106C4×C53—42no—
107C4×C53—41no—
108C4×C53—41no—
109Dic53—24no—
110Dic53—42no—
111Dic53—42no—
112#2724—531nomaximal, Sylow
113C2×C53#53411yescharacteristic
114C2×C53#53912yes—
115C2×C53#54712yes—
116C2×C53#55911yescharacteristic
117C2×C53#56311yescharacteristic
118D53—44no—
119C2×#123—532no—
120#398—531no—
121#398D5311yescharacteristic, complemented
122#397—532no—
123#397—532no—
124#390—531no—
125#389—532no—
126#385—532no—
127#385—532no—
128C53#272411yescharacteristic, complemented, Sylow
129C22×D4—531no—
130C23×C4—532no—
131#123—1068no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4C2×D5312yes—
137C2×C4⋊C4C2×D5311yescharacteristic
138C2×C4⋊C4C2×D5311yescharacteristic
139C2×C22⋊C4—532no—
140C2×C22⋊C4—532no—
141C2×C22⋊C4—532no—
142C2×C22⋊C4—532no—
143C2×C22⋊C4—532no—
144C2×C42—531no—
145#95—532no—
146#95—532no—
147#95—532no—
148#95—531no—
149#95C2×D5312yes—
150#95C2×D5311yescharacteristic
151C24—532no—
152C2×D4—534no—
153C2×D4—2121no—
154C2×D4—2121no—
155C22×C4—532no—
156C22×C4—532no—
157C22×C4—532no—
158C22×C4—1064no—
159C22×C4—2122no—
160C22×C4—2122no—
161C22×C4C22×D5311yescharacteristic
162C22×C4C22×D5312yes—
163C22×C4C22×D5311yescharacteristic
164C22×C4C22×D5312yes—
165C22×C4C22×D5311yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—1064no—
170C4⋊C4—1064no—
171C4⋊C4—2122no—
172C4⋊C4—2121no—
173C22⋊C4—1064no—
174C22⋊C4—1064no—
175C22⋊C4—1064no—
176C22⋊C4—1064no—
177C22⋊C4—1064no—
178C42—2121no—
179C23—534no—
180C23—1062no—
181C23—1062no—
182C23—1062no—
183C23—1062no—
184C23—1062no—
185C23—1062no—
186C23C23×D5311yescharacteristic
187D4—2124no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—534no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—2128no—
210C2×C4D4×D5312yes—
211C22—1064no—
212C22—1064no—
213C22—1064no—
214C22—2122no—
215C22—2122no—
216C22—2122no—
217C22—2122no—
218C22C2×D4×D5311yescharacteristic
219C22#41×D5312yes—
220C22#1313326211yescharacteristic
221C22#1313326311yescharacteristic
222C22#1313326512yes—
223C4—22no—
224C4—42no—
225C4—41no—
226C4—41no—
227C4—42no—
228C4—1064no—
229C4—2122no—
230C4—2122no—
231C2—2124no—
232C2#1681118111yescharacteristic
233C2#1681122112yes—
234C2#1681127512yes—
235C2#1681135711yescharacteristic
236C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2724
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #16810553C2semidirect
3. #16810552C2semidirect
4. #16810552C2semidirect
6. #16810518C2semidirect
7. #16810497C2semidirect
8. #16810494C2semidirect
9. C53×#398C2semidirect
121. #398D53semidirect
128. C53#2724semidirect