Finite Group Databasebeta

← #20280529

Subgroups — #20280529

Order 6784.

508 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 143 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×#13133197C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic
3#16810561C211yesmaximal, characteristic, complemented
4#16810557C212yesmaximal, complemented
5#16810555C212yesmaximal, complemented
6#16810544C211yesmaximal, characteristic, complemented
7#16810543C211yesmaximal, characteristic, complemented
8#16810497C212yesmaximal
9#16810485C212yesmaximal
10#16810482C211yesmaximal, characteristic
11C53×#400C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133197—24no—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2212yes—
19C2×C4⋊Dic53C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C4⋊C4×D53—24no—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34#95×C53C2212yes—
35#95×C53C2211yescharacteristic
36#95×C53C2212yes—
37C23×D53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53—22no—
51C2×C4×D53—42no—
52C2×C4×D53—42no—
53C2×C4×D53D414yes—
54C2×C4×D53Q814yescomplemented
55C2×Dic106—22no—
56C2×Dic106—42no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—22no—
59#11719921—24no—
60#11719921—24no—
61C4⋊Dic53—22no—
62C4⋊Dic53—24no—
63C4⋊Dic53—42no—
64#11719919—22no—
65#11719919—22no—
66#11719919—42no—
67#11719919—42no—
68C23×C53C2411yescharacteristic
69C22×D53—22no—
70C22×D53—22no—
71C22×D53C2×D412yes—
72C22×D53C2×Q812yes—
73C22×D53#4112yes—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—24no—
83C2×C4×C53—22no—
84C2×C4×C53C2×D412yes—
85C2×C4×C53C2×Q812yes—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53C2×D412yes—
98C2×Dic53C2×Q812yes—
99C4×D53—28no—
100C4×D53—44no—
101C4×D53—44no—
102Dic106—44no—
103C22×C53C22×D411yescharacteristic
104C22×C53C22×Q811yescharacteristic
105C22×C53C2×#4111yescharacteristic
106C22×C53H(5,𝔽2)12yes—
107C22×C532−1+412yes—
108C2×D53—22no—
109C2×D53—22no—
110C2×D53—24no—
111C2×D53—24no—
112C2×D53C22⋊Q814yes—
113C4×C53—22no—
114C4×C53—22no—
115C4×C53—41no—
116C4×C53—42no—
117C4×C53—42no—
118Dic53—22no—
119Dic53—22no—
120Dic53—42no—
121Dic53—41no—
122Dic53—42no—
123#2627—531nomaximal, Sylow
124C2×C53C2×C22⋊Q811yescharacteristic
125C2×C53#53411yescharacteristic
126C2×C53#53611yescharacteristic
127C2×C53#53912yes—
128C2×C53#54412yes—
129D53—24no—
130C2×C22⋊Q8—531no—
131C22×C4⋊C4—531no—
132#400—531no—
133#400D5311yescharacteristic, complemented
134#398—532no—
135#398—532no—
136#395—531no—
137#395—531no—
138#385—532no—
139#381—532no—
140#380—531no—
141C53#262711yescharacteristic, complemented, Sylow
142C22×Q8—531no—
143C23×C4—531no—
144C23×C4—531no—
145C23×C4—531no—
146C22⋊Q8—1064no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4—531no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4—532no—
154C2×C4⋊C4—1064no—
155C2×C4⋊C4C2×D5311yescharacteristic
156C2×C4⋊C4C2×D5311yescharacteristic
157C2×C22⋊C4—532no—
158C2×C22⋊C4—532no—
159#95—531no—
160#95—532no—
161#95—532no—
162#95—532no—
163#95—532no—
164#95C2×D5311yescharacteristic
165#95C2×D5312yes—
166#95C2×D5312yes—
167C24—531no—
168C2×Q8—1062no—
169C2×Q8—2122no—
170C22×C4—531no—
171C22×C4—531no—
172C22×C4—534no—
173C22×C4—534no—
174C22×C4—532no—
175C22×C4—532no—
176C22×C4—531no—
177C22×C4—1062no—
178C22×C4—1062no—
179C22×C4—1062no—
180C22×C4—2122no—
181C22×C4—2122no—
182C22×C4C22×D5312yes—
183C22×C4C22×D5312yes—
184C22×C4C22×D5311yescharacteristic
185C22×C4C22×D5311yescharacteristic
186C22×C4C22×D5311yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—1062no—
190C4⋊C4—1062no—
191C4⋊C4—1064no—
192C4⋊C4—1062no—
193C4⋊C4—2122no—
194C4⋊C4—2122no—
195C4⋊C4—2122no—
196C22⋊C4—1064no—
197C22⋊C4—1064no—
198C23—532no—
199C23—532no—
200C23—532no—
201C23—1062no—
202C23—1062no—
203C23C23×D5311yescharacteristic
204Q8—2124no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—532no—
216C2×C4—532no—
217C2×C4—1068no—
218C2×C4—1061no—
219C2×C4—1061no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1061no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—2124no—
230C2×C4—2124no—
231C2×C4D4×D5312yes—
232C2×C4Q8×D5312yes—
233C22—534no—
234C22—1062no—
235C22—1062no—
236C22—1064no—
237C22—1064no—
238C22C2×D4×D5311yescharacteristic
239C22C2×Q8×D5311yescharacteristic
240C22#41×D5311yescharacteristic
241C22#1313326312yes—
242C22#1313326412yes—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—42no—
247C4—42no—
248C4—1062no—
249C4—1062no—
250C4—2121no—
251C4—2122no—
252C4—2122no—
253C2—1064no—
254C2C22⋊Q8×D5311yescharacteristic
255C2#1681118411yescharacteristic
256C2#1681119311yescharacteristic
257C2#1681121412yes—
258C2#1681125312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2627
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #16810561C2semidirect
4. #16810557C2semidirect
5. #16810555C2semidirect
6. #16810544C2semidirect
7. #16810543C2semidirect
11. C53×#400C2semidirect
54. C2×C4×D53Q8semidirect
133. #400D53semidirect
141. C53#2627semidirect