Finite Group Databasebeta

← #20280582

Subgroups — #20280582

Order 6784.

512 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133207C211yesmaximal, characteristic, complemented
2C2×D4×Dic53C211yesmaximal, characteristic
3#16810559C212yesmaximal, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810551C212yesmaximal, complemented
7#16810537C211yesmaximal, characteristic
8#16810498C212yesmaximal, complemented
9#16810495C212yesmaximal, complemented
10#16810490C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic
12C23×Dic53C2212yes—
13#13133207—24no—
14D4×Dic53—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2212yes—
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2212yes—
26C2×C4×Dic53C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32C22×D4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2312yes—
38C2×D4×C53—24no—
39C2×D4×C53—22no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×Dic53—24no—
45C22×Dic53—42no—
46C22×Dic53—42no—
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2312yes—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C4⋊C4×C53—22no—
54C22⋊C4×C53—24no—
55C22⋊C4×C53—42no—
56C22⋊Dic53—24no—
57C22⋊Dic53—24no—
58C22⋊Dic53—42no—
59C4⋊Dic53—22no—
60C4⋊Dic53—22no—
61C4⋊Dic53—42no—
62#11719919—22no—
63#11719919—22no—
64#11719919—22no—
65#11719919—22no—
66#11719919—42no—
67C4×Dic53D414yes—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53#4114yes—
75C23×C53C2411yescharacteristic
76D4×C53—28no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53C2×D412yes—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53—48no—
102C2×Dic53C2×D412yes—
103C2×Dic53C2×D412yes—
104C22×C53—24no—
105C22×C53—24no—
106C22×C53—24no—
107C22×C53—42no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53—42no—
111C22×C53C22×D411yescharacteristic
112C22×C53C2×#4112yes—
113C22×C53H(5,𝔽2)11yescharacteristic
114C22×C53H(5,𝔽2)11yescharacteristic
115C22×C532−1+412yes—
116C4×C53—22no—
117C4×C53—41no—
118C4×C53—41no—
119C4×C53—42no—
120Dic53—22no—
121Dic53—22no—
122Dic53—41no—
123Dic53—41no—
124Dic53—42no—
125Dic53—42no—
126#2633—531nomaximal, Sylow
127C2×C53—44no—
128C2×C53#52511yescharacteristic
129C2×C53#53511yescharacteristic
130C2×C53#53611yescharacteristic
131C2×C53#53912yes—
132C2×C53#53912yes—
133C2×#125—531no—
134C2×C4×D4—531no—
135#399—532no—
136#398—531no—
137#398—531no—
138#397—532no—
139#394—531no—
140#394D5311yescharacteristic
141#385—532no—
142#385—532no—
143#383—531no—
144C53#263311yescharacteristic, complemented, Sylow
145C22×D4C2×D5311yescharacteristic
146C23×C4—532no—
147#125—1064no—
148C4×D4—1064no—
149C2×C4⋊C4—531no—
150C2×C4⋊C4—531no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4C2×D5311yescharacteristic
158C2×C22⋊C4—532no—
159C2×C22⋊C4—532no—
160C2×C22⋊C4—532no—
161C2×C22⋊C4C2×D5312yes—
162C2×C22⋊C4C2×D5312yes—
163C2×C42—531no—
164#95—532no—
165#95—531no—
166#95—532no—
167#95—532no—
168#95—532no—
169#95C2×D5311yescharacteristic
170C24C22×D5312yes—
171C2×D4—24no—
172C2×D4—22no—
173C22×C4—531no—
174C22×C4—531no—
175C22×C4—532no—
176C22×C4—531no—
177C22×C4—532no—
178C22×C4—531no—
179C22×C4—1064no—
180C22×C4—2122no—
181C22×C4—2122no—
182C22×C4C22×D5312yes—
183C22×C4C22×D5311yescharacteristic
184C22×C4C22×D5311yescharacteristic
185C22×C4C22×D5311yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—1062no—
188C4⋊C4—1062no—
189C4⋊C4—1062no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—1062no—
193C4⋊C4—2122no—
194C4⋊C4—2122no—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C22⋊C4—1064no—
198C22⋊C4—1064no—
199C22⋊C4—2122no—
200C42—534no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23#1171994714yes—
208C23C23×D5311yescharacteristic
209D4—28no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—532no—
220C2×C4—532no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1061no—
230C2×C4—1062no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—2128no—
236C2×C4D4×D5312yes—
237C22—24no—
238C22—24no—
239C22—24no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D5311yescharacteristic
245C22C2×#1171994712yes—
246C22#1313326211yescharacteristic
247C22#1313326311yescharacteristic
248C22#1313326412yes—
249C4—22no—
250C4—41no—
251C4—41no—
252C4—42no—
253C4—1062no—
254C4—1062no—
255C4—2121no—
256C4—2122no—
257C4—2121no—
258C4—2122no—
259C2—44no—
260C2#1681112911yescharacteristic
261C2#1681118911yescharacteristic
262C2#1681119311yescharacteristic
263C2#1681121712yes—
264C2#1681121812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2633
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133207C2semidirect
3. #16810559C2semidirect
4. #16810555C2semidirect
5. #16810553C2semidirect
6. #16810551C2semidirect
8. #16810498C2semidirect
9. #16810495C2semidirect
10. #16810490C2semidirect
144. C53#2633semidirect