Finite Group Databasebeta

← #20279003

Subgroups — #20279003

Order 6784.

512 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 139 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×#13133175C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic
3#16810561C211yesmaximal, characteristic, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810510C212yesmaximal, complemented
7#16810506C212yesmaximal, complemented
8#16810497C212yesmaximal
9#16810485C212yesmaximal
10#16810485C211yesmaximal, characteristic
11C53×#400C211yesmaximal, characteristic, complemented
12C22×C4×D53C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15#13133175—24no—
16C2×C4⋊Dic53C2212yes—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C4⋊C4×D53—24no—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×C4×Dic53C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34#95×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×Dic53C2312yes—
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×C4×D53—22no—
49C2×C4×D53—24no—
50C2×C4×D53—22no—
51C2×C4×D53—22no—
52C2×C4×D53—22no—
53C2×C4×D53—24no—
54C2×C4×D53D414yes—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57#11719921—24no—
58#11719921—24no—
59C4⋊Dic53—22no—
60C4⋊Dic53—22no—
61C4⋊Dic53—24no—
62#11719919—22no—
63#11719919—24no—
64#11719919—22no—
65C4×Dic53—24no—
66C23×C53C2411yescharacteristic
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—22no—
70C22×D53C2×D412yes—
71C22×D53#4114yes—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—24no—
81C2×C4×C53—22no—
82C2×C4×C53C2×D412yes—
83C2×C4×C53#4112yes—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53C2×D412yes—
96C2×Dic53#4112yes—
97C4×D53—28no—
98C4×D53—28no—
99C4×D53—44no—
100C22×C53C22×D411yescharacteristic
101C22×C53C2×#4112yes—
102C22×C53C2×#4111yescharacteristic
103C22×C53C2×#4111yescharacteristic
104C22×C532−1+412yes—
105C2×D53—22no—
106C2×D53—22no—
107C2×D53—24no—
108C2×D53—24no—
109C2×D53#12314yes—
110C4×C53—22no—
111C4×C53—22no—
112C4×C53—41no—
113C4×C53—42no—
114C4×C53—42no—
115Dic53—22no—
116Dic53—22no—
117Dic53—41no—
118Dic53—42no—
119Dic53—42no—
120#2484—531nomaximal, Sylow
121C2×C53C2×#12311yescharacteristic
122C2×C53#52511yescharacteristic
123C2×C53#53611yescharacteristic
124C2×C53#55212yes—
125C2×C53#55212yes—
126D53—24no—
127C2×#117—531no—
128C22×C4⋊C4—531no—
129#400—531no—
130#400D5311yescharacteristic, complemented
131#398—531no—
132#398—531no—
133#387—532no—
134#387—532no—
135#385—532no—
136#381—531no—
137#381—532no—
138C53#248411yescharacteristic, complemented, Sylow
139C23×C4—531no—
140C23×C4—531no—
141C23×C4—531no—
142#117—1064no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—531no—
145C2×C4⋊C4—531no—
146C2×C4⋊C4—531no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—1064no—
150C2×C4⋊C4C2×D5311yescharacteristic
151C2×C4⋊C4C2×D5311yescharacteristic
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C42—532no—
155#95—532no—
156#95—532no—
157#95—532no—
158#95—532no—
159#95—531no—
160#95—532no—
161#95C2×D5312yes—
162#95C2×D5311yescharacteristic
163#95C2×D5312yes—
164C24—531no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4—534no—
168C22×C4—532no—
169C22×C4—531no—
170C22×C4—532no—
171C22×C4—1062no—
172C22×C4—1062no—
173C22×C4—1064no—
174C22×C4—1062no—
175C22×C4—1064no—
176C22×C4—1062no—
177C22×C4C22×D5312yes—
178C22×C4C22×D5312yes—
179C22×C4C22×D5311yescharacteristic
180C22×C4C22×D5311yescharacteristic
181C22×C4C22×D5311yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—1062no—
185C4⋊C4—1062no—
186C4⋊C4—1064no—
187C4⋊C4—1062no—
188C4⋊C4—1062no—
189C4⋊C4—1064no—
190C22⋊C4—1064no—
191C22⋊C4—1064no—
192C42—1064no—
193C23—532no—
194C23—534no—
195C23—1061no—
196C23—1061no—
197C23—1062no—
198C23C23×D5311yescharacteristic
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—532no—
210C2×C4—532no—
211C2×C4—1068no—
212C2×C4—1068no—
213C2×C4—1061no—
214C2×C4—1062no—
215C2×C4—1061no—
216C2×C4—1062no—
217C2×C4—1061no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—2124no—
225C2×C4D4×D5312yes—
226C2×C4#1171994912yes—
227C22—534no—
228C22—1064no—
229C22—1064no—
230C22—1062no—
231C22—1062no—
232C22C2×D4×D5311yescharacteristic
233C22#41×D5311yescharacteristic
234C22#41×D5312yes—
235C22C2×#1171994911yescharacteristic
236C22#1313326412yes—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—41no—
241C4—42no—
242C4—1062no—
243C4—1062no—
244C4—2122no—
245C4—2121no—
246C4—2122no—
247C2—1064no—
248C2#123×D5311yescharacteristic
249C2#1681113611yescharacteristic
250C2#1681119311yescharacteristic
251C2#1681130912yes—
252C2#1681131112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2484
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #16810561C2semidirect
4. #16810555C2semidirect
5. #16810553C2semidirect
6. #16810510C2semidirect
7. #16810506C2semidirect
11. C53×#400C2semidirect
130. #400D53semidirect
138. C53#2484semidirect