Finite Group Databasebeta

← #20281086

Subgroups — #20281086

Order 6784.

592 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133197C212yesmaximal, complemented
2C2×#13133195C212yesmaximal, complemented
3#16810545C212yesmaximal, complemented
4#16810545C212yesmaximal, complemented
5#16810544C212yesmaximal, complemented
6#16810525C211yesmaximal, characteristic
7#16810521C211yesmaximal, characteristic, complemented
8#16810501C212yesmaximal
9C53×#389C211yesmaximal, characteristic, complemented
10C22×Dic106C2212yes—
11C22×D212C2211yescharacteristic
12C22×C4×D53C2212yes—
13#13133197—28no—
14#13133195—28no—
15C2×C4⋊Dic53C2212yes—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28C2×C4⋊C4×C53C2212yes—
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C4⋊C4×C53C2211yescharacteristic
31C2×C42×C53C2211yescharacteristic
32#95×C53C2212yes—
33C23×D53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C2×D212—41no—
43C2×D212—41no—
44C2×D212Q814yescomplemented
45C2×C4×D53—24no—
46C2×C4×D53—24no—
47C2×C4×D53—42no—
48C2×Dic106—24no—
49C2×Dic106—24no—
50C2×Dic106—42no—
51C4⋊C4×C53—24no—
52C4⋊C4×C53—41no—
53C4⋊C4×C53D414yescomplemented
54C42×C53—41no—
55#11719921—24no—
56#11719921—24no—
57#11719921—24no—
58#11719921—24no—
59#11719921—42no—
60C4⋊Dic53—24no—
61C4⋊Dic53—41no—
62C4⋊Dic53—41no—
63C4⋊Dic53D414yescomplemented
64#11719919—24no—
65#11719919—24no—
66C23×C53C2411yescharacteristic
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53C2×Q814yes—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53C2×D414yes—
86C2×C4×C53C2×D412yes—
87C2×C4×C53C2×D412yes—
88C2×C4×C53C2×Q812yes—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53C2×D414yes—
98D212—44no—
99C4×D53—48no—
100Dic106—48no—
101C22×C53C22×D411yescharacteristic
102C22×C53C22×D411yescharacteristic
103C22×C53C22×Q811yescharacteristic
104C22×C53H(5,𝔽2)11yescharacteristic
105C22×C53H(5,𝔽2)11yescharacteristic
106C22×C532−1+411yescharacteristic
107C22×C532−1+411yescharacteristic
108C2×D53—24no—
109C2×D53—24no—
110C2×D53—24no—
111C2×D53—42no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53—42no—
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—22no—
118C4×C53—24no—
119C4×C53—42no—
120Dic53—24no—
121Dic53—42no—
122Dic53—42no—
123#2683—531nomaximal, Sylow
124C2×C53#53711yescharacteristic
125C2×C53#53711yescharacteristic
126C2×C53#54311yescharacteristic
127C2×C53D4211yescharacteristic
128C2×C53D4×Q811yescharacteristic
129C2×C53D4×Q811yescharacteristic
130C2×C53#56311yescharacteristic
131D53—44no—
132C2×C22⋊Q8—532no—
133C2×C22⋊Q8—532no—
134#396—532no—
135#396—532no—
136#395—532no—
137#390—531no—
138#389—531no—
139#389D5311yescharacteristic, complemented
140#386—532no—
141C53#268311yescharacteristic, complemented, Sylow
142C22×Q8—532no—
143C22×D4—531no—
144C23×C4—532no—
145C22⋊Q8—1068no—
146C22⋊Q8—1068no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—532no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4C2×D5312yes—
154C2×C4⋊C4C2×D5311yescharacteristic
155C2×C4⋊C4C2×D5311yescharacteristic
156C2×C22⋊C4—532no—
157C2×C22⋊C4—532no—
158C2×C22⋊C4—532no—
159C2×C22⋊C4—532no—
160C2×C22⋊C4—532no—
161C2×C42C2×D5311yescharacteristic
162#95—532no—
163#95—532no—
164#95C2×D5312yes—
165C24—532no—
166C2×Q8—1064no—
167C2×Q8—1064no—
168C2×Q8—2122no—
169C2×D4—534no—
170C2×D4—2121no—
171C2×D4—2121no—
172C22×C4—532no—
173C22×C4—532no—
174C22×C4—532no—
175C22×C4—1064no—
176C22×C4—1064no—
177C22×C4—2122no—
178C22×C4C22×D5312yes—
179C22×C4C22×D5312yes—
180C22×C4C22×D5311yescharacteristic
181C22×C4C22×D5311yescharacteristic
182C22×C4C22×D5311yescharacteristic
183C4⋊C4—24no—
184C4⋊C4—41no—
185C4⋊C4—534no—
186C4⋊C4—1064no—
187C4⋊C4—1064no—
188C4⋊C4—1064no—
189C4⋊C4—2121no—
190C4⋊C4—2121no—
191C4⋊C4D21214yescomplemented
192C22⋊C4—1064no—
193C22⋊C4—1064no—
194C22⋊C4—1064no—
195C22⋊C4—1064no—
196C22⋊C4—2122no—
197C42—41no—
198C23—534no—
199C23—1062no—
200C23—1062no—
201C23—1062no—
202C23—1062no—
203C23—1062no—
204C23—1062no—
205C23C23×D5311yescharacteristic
206Q8—2128no—
207D4—2124no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—534no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—2128no—
229C2×C4C2×D21212yes—
230C2×C4C2×D21214yes—
231C2×C4D4×D5312yes—
232C2×C4Q8×D5312yes—
233C22—1064no—
234C22—1064no—
235C22—1064no—
236C22—2122no—
237C22—2122no—
238C22—2122no—
239C22—2122no—
240C22C2×D4×D5311yescharacteristic
241C22C22×D21211yescharacteristic
242C22C2×Q8×D5311yescharacteristic
243C22#1313326211yescharacteristic
244C22#1313326311yescharacteristic
245C22#1313326411yescharacteristic
246C22#1313326511yescharacteristic
247C4—22no—
248C4—22no—
249C4—22no—
250C4—24no—
251C4—42no—
252C4—1064no—
253C4—2122no—
254C4—2122no—
255C2—2124no—
256C2#1681120011yescharacteristic
257C2#1681120411yescharacteristic
258C2#1681124811yescharacteristic
259C2D4×D21211yescharacteristic
260C2#1681128811yescharacteristic
261C2Q8×D21211yescharacteristic
262C2#1681137711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#389C2
hypercenter#389D53
hypocenterC53#2683
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#13133197C2semidirect
2. C2×#13133195C2semidirect
3. #16810545C2semidirect
4. #16810545C2semidirect
5. #16810544C2semidirect
7. #16810521C2semidirect
9. C53×#389C2semidirect
44. C2×D212Q8semidirect
53. C4⋊C4×C53D4semidirect
63. C4⋊Dic53D4semidirect
139. #389D53semidirect
141. C53#2683semidirect
191. C4⋊C4D212semidirect