Finite Group Databasebeta

← #20280532

Subgroups — #20280532

Order 6784.

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

normal subgroups only

This group has 508 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×#13133194C211yesmaximal, characteristic
2C22×C4⋊Dic53C211yesmaximal, characteristic
3#16810565C212yesmaximal, complemented
4#16810557C212yesmaximal, complemented
5#16810555C212yesmaximal, complemented
6#16810544C212yesmaximal, complemented
7#16810498C211yesmaximal, characteristic
8#16810495C211yesmaximal, characteristic
9#16810486C211yesmaximal, characteristic
10#16810481C211yesmaximal, characteristic
11C53×#381C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15#13133194—24no—
16C2×C4⋊Dic53—24no—
17C2×C4⋊Dic53C2212yes—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2212yes—
20C2×C4⋊Dic53C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C23×C4×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35#95×C53C2212yes—
36#95×C53C2212yes—
37C24×C53C2311yescharacteristic
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53Q814yescomplemented
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×Dic53—22no—
46C22×Dic53—42no—
47C22×Dic53—42no—
48C22×Dic53D414yes—
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C2×Dic106—22no—
56C2×Dic106—42no—
57C22⋊C4×C53—22no—
58C22⋊C4×C53—22no—
59C22⋊Dic53—22no—
60C22⋊Dic53—22no—
61C4⋊Dic53—24no—
62C4⋊Dic53—24no—
63C4⋊Dic53—22no—
64C4⋊Dic53—42no—
65#11719919—22no—
66#11719919—22no—
67#11719919—42no—
68#11719919—42no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53C2×D412yes—
72C23×C53C2×Q812yes—
73C23×C53#4112yes—
74C23×C53C2411yescharacteristic
75C2×C4×C53—28no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53C2×Q814yes—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—21no—
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×Dic53—22no—
98C2×Dic53—44no—
99C2×Dic53—44no—
100C2×Dic53C2×D412yes—
101C2×Dic53C2×D412yes—
102Dic106—44no—
103C22×C53—22no—
104C22×C53—22no—
105C22×C53—24no—
106C22×C53—24no—
107C22×C53C22⋊Q814yes—
108C22×C53C22×D411yescharacteristic
109C22×C53C22×Q811yescharacteristic
110C22×C53C2×#4111yescharacteristic
111C22×C53H(5,𝔽2)12yes—
112C22×C532−1+412yes—
113C4×C53—24no—
114C4×C53—41no—
115C4×C53—41no—
116C4×C53—42no—
117Dic53—22no—
118Dic53—22no—
119Dic53—41no—
120Dic53—42no—
121Dic53—41no—
122Dic53—42no—
123#2627—531nomaximal, Sylow
124C2×C53—24no—
125C2×C53C2×C22⋊Q811yescharacteristic
126C2×C53#53411yescharacteristic
127C2×C53#53611yescharacteristic
128C2×C53#53912yes—
129C2×C53#54412yes—
130C2×C22⋊Q8—531no—
131C22×C4⋊C4—531no—
132#400—532no—
133#398—532no—
134#398—532no—
135#395—532no—
136#385—531no—
137#385—531no—
138#381—531no—
139#381D5311yescharacteristic
140#380—531no—
141C53#262711yescharacteristic, complemented, Sylow
142C22×Q8—531no—
143C23×C4—531no—
144C23×C4—531no—
145C23×C4C2×D5311yescharacteristic
146C22⋊Q8—1064no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—531no—
150C2×C4⋊C4—531no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4—532no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—1064no—
156C2×C22⋊C4—531no—
157C2×C22⋊C4—531no—
158C2×C22⋊C4C2×D5311yescharacteristic
159C2×C22⋊C4C2×D5311yescharacteristic
160#95—532no—
161#95—532no—
162#95—532no—
163#95—532no—
164#95—532no—
165#95C2×D5312yes—
166#95C2×D5312yes—
167C24C22×D5311yescharacteristic
168C2×Q8—1062no—
169C2×Q8—2122no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—531no—
173C22×C4—531no—
174C22×C4—534no—
175C22×C4—531no—
176C22×C4—532no—
177C22×C4—531no—
178C22×C4—532no—
179C22×C4—1062no—
180C22×C4—2122no—
181C22×C4—2122no—
182C22×C4Dic10614yescomplemented
183C22×C4C22×D5311yescharacteristic
184C22×C4C22×D5312yes—
185C22×C4C22×D5311yescharacteristic
186C22×C4C22×D5312yes—
187C4⋊C4—1064no—
188C4⋊C4—1064no—
189C4⋊C4—1062no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—2122no—
193C4⋊C4—2122no—
194C4⋊C4—2122no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—1062no—
198C22⋊C4—1062no—
199C23—22no—
200C23—22no—
201C23C2×Dic10612yes—
202C23D4×D5312yes—
203C23#1171994712yes—
204C23C23×D5311yescharacteristic
205Q8—2124no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—532no—
217C2×C4—532no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—2124no—
231C2×C4—2124no—
232C2×C4C2×Dic10614yes—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22#1313319414yes—
238C22C2×D4×D5311yescharacteristic
239C22C22×Dic10611yescharacteristic
240C22C2×#1171994711yescharacteristic
241C22#1313326312yes—
242C22#1313326412yes—
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—1062no—
248C4—1062no—
249C4—2122no—
250C4—2121no—
251C4—2121no—
252C4—2122no—
253C2—24no—
254C2C2×#1313319411yescharacteristic
255C2#1681118411yescharacteristic
256C2#1681119511yescharacteristic
257C2#1681121712yes—
258C2#1681125112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#381C2
hypercenter#381D53
hypocenterC53#2627
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #16810565C2semidirect
4. #16810557C2semidirect
5. #16810555C2semidirect
6. #16810544C2semidirect
40. C22×C4×C53Q8semidirect
141. C53#2627semidirect
182. C22×C4Dic106semidirect