Finite Group Databasebeta

← #20278999

Subgroups — #20278999

Order 6784.

512 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13133173C211yesmaximal, characteristic
2C22×#11719919C211yesmaximal, characteristic
3#16810564C212yesmaximal, complemented
4#16810557C212yesmaximal, complemented
5#16810506C212yesmaximal, complemented
6#16810505C212yesmaximal, complemented
7#16810498C211yesmaximal, characteristic
8#16810495C211yesmaximal, characteristic
9#16810486C211yesmaximal, characteristic
10#16810484C211yesmaximal, characteristic
11C53×#381C211yesmaximal, characteristic
12C23×Dic53C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14#13133173—24no—
15C2×C4⋊Dic53C2212yes—
16C2×#11719919—24no—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C4×Dic53C2211yescharacteristic
25C2×C4×Dic53C2211yescharacteristic
26#13133124C2212yes—
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×C53—22no—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×Dic53—22no—
46C22×Dic53—22no—
47C22×Dic53—22no—
48C22×Dic53—24no—
49C22×Dic53D414yes—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2312yes—
56C22⋊C4×C53—22no—
57C22⋊C4×C53—22no—
58C22⋊Dic53—22no—
59C22⋊Dic53—22no—
60C4⋊Dic53—24no—
61#11719919—24no—
62#11719919—24no—
63#11719919—22no—
64#11719919—22no—
65#11719919—24no—
66C4×Dic53—22no—
67C4×Dic53—22no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53C2×D412yes—
71C23×C53#4112yes—
72C23×C53#4112yes—
73C23×C53C2411yescharacteristic
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—44no—
85C2×Dic53—28no—
86C2×Dic53—28no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53C2×D412yes—
101C2×Dic53C2×D412yes—
102C2×Dic53#4114yes—
103C22×C53—22no—
104C22×C53—22no—
105C22×C53—24no—
106C22×C53—24no—
107C22×C53#12314yes—
108C22×C53C22×D411yescharacteristic
109C22×C53C2×#4112yes—
110C22×C53C2×#4111yescharacteristic
111C22×C53C2×#4111yescharacteristic
112C22×C532−1+412yes—
113C4×C53—42no—
114C4×C53—41no—
115C4×C53—41no—
116C4×C53—42no—
117Dic53—22no—
118Dic53—22no—
119Dic53—24no—
120Dic53—41no—
121Dic53—41no—
122Dic53—42no—
123#2484—531nomaximal, Sylow
124C2×C53—24no—
125C2×C53C2×#12311yescharacteristic
126C2×C53#52511yescharacteristic
127C2×C53#53611yescharacteristic
128C2×C53#55212yes—
129C2×C53#55212yes—
130C2×#117—531no—
131C22×C4⋊C4—531no—
132#400—532no—
133#398—532no—
134#387—532no—
135#387—532no—
136#385—531no—
137#385—531no—
138#381—531no—
139#381—531no—
140#381D5311yescharacteristic
141C53#248411yescharacteristic, complemented, Sylow
142C23×C4—531no—
143C23×C4—531no—
144C23×C4C2×D5311yescharacteristic
145#117—1064no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4—531no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4—1064no—
153C2×C22⋊C4—531no—
154C2×C22⋊C4—531no—
155C2×C22⋊C4C2×D5311yescharacteristic
156C2×C22⋊C4C2×D5311yescharacteristic
157C2×C42—531no—
158C2×C42—531no—
159#95—532no—
160#95—532no—
161#95—532no—
162#95—532no—
163#95—532no—
164#95—532no—
165#95C2×D5312yes—
166#95C2×D5312yes—
167C24C22×D5311yescharacteristic
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—534no—
172C22×C4—531no—
173C22×C4—531no—
174C22×C4—532no—
175C22×C4—532no—
176C22×C4—531no—
177C22×C4—531no—
178C22×C4—1064no—
179C22×C4—1062no—
180C22×C4—1062no—
181C22×C4—1062no—
182C22×C4C22×D5312yes—
183C22×C4C22×D5311yescharacteristic
184C22×C4C22×D5312yes—
185C22×C4C22×D5311yescharacteristic
186C4⋊C4—1064no—
187C4⋊C4—1064no—
188C4⋊C4—1062no—
189C4⋊C4—1062no—
190C4⋊C4—1064no—
191C4⋊C4—1064no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—1062no—
195C22⋊C4—1062no—
196C42—1062no—
197C42—1062no—
198C23—22no—
199C23—22no—
200C23#1171994512yes—
201C23D4×D5312yes—
202C23#1171994712yes—
203C23C23×D5311yescharacteristic
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—532no—
216C2×C4—532no—
217C2×C4—534no—
218C2×C4—1068no—
219C2×C4—1068no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—22no—
237C22#1313319914yes—
238C22C2×D4×D5311yescharacteristic
239C22#41×D5312yes—
240C22C2×#1171994711yescharacteristic
241C22C2×#1171994511yescharacteristic
242C22#1313326412yes—
243C4—42no—
244C4—42no—
245C4—41no—
246C4—41no—
247C4—1062no—
248C4—1062no—
249C4—1064no—
250C4—2122no—
251C4—2121no—
252C4—2121no—
253C2—24no—
254C2C2×#1313319911yescharacteristic
255C2#1681113111yescharacteristic
256C2#1681119511yescharacteristic
257C2#1681130612yes—
258C2#1681131012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#381C2
hypercenter#381D53
hypocenterC53#2484
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #16810564C2semidirect
4. #16810557C2semidirect
5. #16810506C2semidirect
6. #16810505C2semidirect
141. C53#2484semidirect