Finite Group Databasebeta

← #20283052

Subgroups — #20283052

Order 6784.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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#16810527C212yesmaximal, complemented
4#16810520C212yesmaximal, complemented
5#16810510C212yesmaximal, complemented
6#16810506C212yesmaximal, 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⋊Dic53C2212yes—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C4×Dic53—24no—
25C2×C4×Dic53C2212yes—
26C2×C4×Dic53C2212yes—
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×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×Dic53—22no—
46C22×Dic53—24no—
47C22×Dic53—24no—
48C22×Dic53D414yes—
49C22×Dic53C2312yes—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2312yes—
54C22×Dic53C2311yescharacteristic
55C2×Dic106—22no—
56C2×Dic106—24no—
57C22⋊C4×C53—22no—
58C22⋊C4×C53—22no—
59C22⋊Dic53—22no—
60C22⋊Dic53—22no—
61C4⋊Dic53—22no—
62C4⋊Dic53—24no—
63#11719919—22no—
64#11719919—22no—
65#11719919—24no—
66#11719919—24no—
67C4×Dic53—24no—
68C4×Dic53—24no—
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—28no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—28no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53C2×D412yes—
99C2×Dic53C2×D412yes—
100C2×Dic53#4114yes—
101C2×Dic53#4114yes—
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×#4112yes—
111C22×C53C2×#4112yes—
112C22×C53C2×#4111yescharacteristic
113C4×C53—24no—
114C4×C53—41no—
115C4×C53—41no—
116C4×C53—42no—
117Dic53—22no—
118Dic53—24no—
119Dic53—41no—
120Dic53—41no—
121Dic53—42no—
122Dic53#12314yes—
123#2855—531nomaximal, Sylow
124C2×C53—24no—
125C2×C53C2×C22⋊Q811yescharacteristic
126C2×C53C2×#12311yescharacteristic
127C2×C53#52511yescharacteristic
128C2×C53#52912yes—
129C2×C53#53312yes—
130C2×C22⋊Q8—531no—
131C22×C42—531no—
132#391—532no—
133#389—532no—
134#387—532no—
135#387—532no—
136#385—531no—
137#385—531no—
138#381—531no—
139#381D5311yescharacteristic
140#380—531no—
141C53#285511yescharacteristic, complemented, Sylow
142C22×Q8—531no—
143C23×C4—531no—
144C23×C4—531no—
145C23×C4C2×D5311yescharacteristic
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×C22⋊C4—531no—
154C2×C22⋊C4—531no—
155C2×C22⋊C4C2×D5311yescharacteristic
156C2×C22⋊C4C2×D5311yescharacteristic
157C2×C42—532no—
158C2×C42—532no—
159C2×C42—1064no—
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—1064no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—531no—
173C22×C4—534no—
174C22×C4—531no—
175C22×C4—532no—
176C22×C4—532no—
177C22×C4—531no—
178C22×C4—531no—
179C22×C4—1064no—
180C22×C4—1064no—
181C22×C4—1062no—
182C22×C4Dic10614yescomplemented
183C22×C4C22×D5312yes—
184C22×C4C22×D5311yescharacteristic
185C22×C4C22×D5311yescharacteristic
186C22×C4C22×D5312yes—
187C4⋊C4—1062no—
188C4⋊C4—1062no—
189C4⋊C4—1062no—
190C4⋊C4—1064no—
191C4⋊C4—1064no—
192C4⋊C4—1064no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—1062no—
196C22⋊C4—1062no—
197C42—1064no—
198C42—1064no—
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—534no—
218C2×C4—532no—
219C2×C4—534no—
220C2×C4—1068no—
221C2×C4—1068no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4C2×Dic10614yes—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22#1313319414yes—
238C22C2×D4×D5311yescharacteristic
239C22C22×Dic10611yescharacteristic
240C22#41×D5312yes—
241C22#41×D5312yes—
242C22C2×#1171994711yescharacteristic
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—534no—
248C4—1062no—
249C4—1064no—
250C4—2121no—
251C4—2121no—
252C4—2122no—
253C2—24no—
254C2C2×#1313319411yescharacteristic
255C2#123×D5311yescharacteristic
256C2#1681113111yescharacteristic
257C2#1681115012yes—
258C2#1681117612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#381C2
hypercenter#381D53
hypocenterC53#2855
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #16810527C2semidirect
4. #16810520C2semidirect
5. #16810510C2semidirect
6. #16810506C2semidirect
40. C22×C4×C53Q8semidirect
141. C53#2855semidirect
182. C22×C4Dic106semidirect