Finite Group Databasebeta

← #20279129

Subgroups — #20279129

Order 6784.

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

normal subgroups only

This group has 412 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#16810565C212yesmaximal, complemented
2#16810550C212yesmaximal
3#16810522C211yesmaximal, characteristic, complemented
4#16810521C211yesmaximal, characteristic, complemented
5#16810512C212yesmaximal
6#16810490C212yesmaximal, complemented
7#16810480C211yesmaximal, characteristic
8#16810480C211yesmaximal, characteristic
9C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
10C4×#11719921C211yesmaximal, characteristic
11C53×#382C211yesmaximal, characteristic, complemented
12C22×C4×D53C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2212yes—
16C2×#11719919C2212yes—
17C2×#11719921C2211yescharacteristic
18C2×#11719921C2212yes—
19C2×#11719921C2211yescharacteristic
20C2×#11719921C2212yes—
21C2×C4×Dic53C2212yes—
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30C2×C4⋊C4×C53C2211yescharacteristic
31C2×C4⋊C4×C53C2211yescharacteristic
32C2×C42×C53C2211yescharacteristic
33#95×C53C2212yes—
34#95×C53C2212yes—
35C23×D53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×Dic53C2311yescharacteristic
42C22×Dic53C2312yes—
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C2×C4×D53—22no—
47C2×C4×D53—22no—
48C2×C4×D53—22no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—22no—
51C42×C53—22no—
52#11719921—24no—
53#11719921—22no—
54#11719921—22no—
55#11719921—42no—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53—42no—
59#11719919—42no—
60C4×Dic53—22no—
61C4×Dic53—24no—
62C23×C53C2411yescharacteristic
63C22×D53—21no—
64C22×D53—21no—
65C22×D53—21no—
66C22×D53—21no—
67C22×D53—21no—
68C22×D53—21no—
69C22×D53#4112yes—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53#4112yes—
83C2×C4×C53#4112yes—
84C2×C4×C53#4112yes—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—21no—
99C2×Dic53#4112yes—
100C4×D53—44no—
101C22×C53C2×#4111yescharacteristic
102C22×C53C2×#4111yescharacteristic
103C22×C53C2×#4111yescharacteristic
104C22×C53C2×#4111yescharacteristic
105C22×C53C2×#4111yescharacteristic
106C22×C53H(5,𝔽2)11yescharacteristic
107C22×C532−1+411yescharacteristic
108C2×D53—22no—
109C2×D53—22no—
110C2×D53—22no—
111C2×D53—41no—
112C2×D53—41no—
113C2×D53—41no—
114C2×D53—41no—
115C4×C53—42no—
116C4×C53—42no—
117C4×C53—41no—
118C4×C53—41no—
119C4×C53#12614yes—
120Dic53—22no—
121Dic53—41no—
122Dic53—41no—
123Dic53—42no—
124Dic53—42no—
125#2495—531nomaximal, Sylow
126C2×C53C2×#12611yescharacteristic
127C2×C53#52911yescharacteristic
128C2×C53#54211yescharacteristic
129C2×C53#55211yescharacteristic
130C2×C53#55211yescharacteristic
131C2×C53#55311yescharacteristic
132C2×C53#55511yescharacteristic
133D53—42no—
134#400—532no—
135#397—532no—
136#389—531no—
137#389—531no—
138#388—532no—
139#383—532no—
140#382D5311yescharacteristic, complemented
141#380—531no—
142#380—531no—
143C4×C4⋊C4—531no—
144C4×C22⋊C4—531no—
145C53#249511yescharacteristic, complemented, Sylow
146C23×C4—531no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4—532no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4C2×D5311yescharacteristic
152C2×C4⋊C4C2×D5311yescharacteristic
153C2×C22⋊C4—531no—
154C2×C22⋊C4—531no—
155C2×C22⋊C4—532no—
156C2×C22⋊C4—532no—
157C2×C42—531no—
158C2×C42—532no—
159C2×C42C2×D5311yescharacteristic
160#95—531no—
161#95—531no—
162#95—532no—
163#95—532no—
164#95—532no—
165#95—532no—
166#95—532no—
167#95C2×D5312yes—
168#95C2×D5312yes—
169C24—531no—
170C22×C4—531no—
171C22×C4—531no—
172C22×C4—532no—
173C22×C4—531no—
174C22×C4—532no—
175C22×C4—1062no—
176C22×C4—1062no—
177C22×C4—1062no—
178C22×C4C22×D5311yescharacteristic
179C22×C4C22×D5312yes—
180C22×C4C22×D5312yes—
181C22×C4C22×D5311yescharacteristic
182C22×C4C22×D5311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—1062no—
186C4⋊C4—1062no—
187C4⋊C4—2122no—
188C4⋊C4—2122no—
189C22⋊C4—1062no—
190C22⋊C4—1062no—
191C22⋊C4—1064no—
192C22⋊C4—2122no—
193C42—22no—
194C42—1064no—
195C42—1062no—
196C23—532no—
197C23—1061no—
198C23—1061no—
199C23—1061no—
200C23—1061no—
201C23—1061no—
202C23—1061no—
203C23C23×D5311yescharacteristic
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—532no—
217C2×C4—1061no—
218C2×C4—1061no—
219C2×C4—1062no—
220C2×C4—1061no—
221C2×C4—1062no—
222C2×C4—1061no—
223C2×C4—1062no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1062no—
227C2×C4—1061no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1061no—
231C2×C4—2124no—
232C2×C4#1171994512yes—
233C2×C4#1171994712yes—
234C2×C4#1171994912yes—
235C22—1062no—
236C22—1062no—
237C22—1062no—
238C22—2121no—
239C22—2121no—
240C22—2121no—
241C22—2121no—
242C22#41×D5311yescharacteristic
243C22#41×D5311yescharacteristic
244C22C2×#1171994711yescharacteristic
245C22C2×#1171994511yescharacteristic
246C22C2×#1171994911yescharacteristic
247C22#1313326311yescharacteristic
248C22#1313326411yescharacteristic
249C4—41no—
250C4—41no—
251C4—42no—
252C4—42no—
253C4—1062no—
254C4—2121no—
255C4—2122no—
256C4—2122no—
257C4—2121no—
258C4#1313321214yes—
259C2—2122no—
260C2C2×#1313321211yescharacteristic
261C2#1681115211yescharacteristic
262C2#1681124211yescharacteristic
263C2#1681131111yescharacteristic
264C2#1681131211yescharacteristic
265C2#1681132611yescharacteristic
266C2#1681133611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#382C2
hypercenter#382D53
hypocenterC53#2495
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #16810565C2semidirect
3. #16810522C2semidirect
4. #16810521C2semidirect
6. #16810490C2semidirect
9. C4⋊C4×Dic53C2semidirect
11. C53×#382C2semidirect
140. #382D53semidirect
145. C53#2495semidirect