Finite Group Databasebeta

← #20280541

Subgroups — #20280541

Order 6784.

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

normal subgroups only

This group has 488 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×#13133216C211yesmaximal, characteristic, complemented
2C2×#13133173C211yesmaximal, characteristic
3#16810563C211yesmaximal, characteristic, complemented
4#16810561C211yesmaximal, characteristic, complemented
5#16810559C211yesmaximal, characteristic
6#16810543C212yesmaximal, complemented
7#16810535C212yesmaximal
8#16810518C212yesmaximal, complemented
9#16810487C211yesmaximal, characteristic, complemented
10#16810481C212yesmaximal
11C53×#399C211yesmaximal, characteristic
12C22×Dic106C2212yes—
13C23×Dic53C2211yescharacteristic
14#13133216—24no—
15#13133173—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25C2×C4×Dic53C2211yescharacteristic
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×Dic53—22no—
43C22×Dic53—22no—
44C22×Dic53Q814yescomplemented
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2311yescharacteristic
51C2×Dic106—24no—
52C2×Dic106—42no—
53C2×Dic106—42no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—42no—
56C22⋊C4×C53—22no—
57C22⋊C4×C53—42no—
58C22⋊Dic53—22no—
59C22⋊Dic53—42no—
60C4⋊Dic53—22no—
61#11719919—22no—
62#11719919—22no—
63#11719919—24no—
64#11719919—24no—
65#11719919—42no—
66C4×Dic53—22no—
67C4×Dic53D414yes—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53C2×Q812yes—
73C23×C53C2411yescharacteristic
74C2×C4×C53—22no—
75C2×C4×C53—22no—
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×C53C2×D412yes—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—22no—
87C2×Dic53—28no—
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53C2×D412yes—
97C2×Dic53C2×D412yes—
98C2×Dic53C2×Q812yes—
99C2×Dic53C2×Q812yes—
100C2×Dic53#4112yes—
101Dic106—48no—
102C22×C53—22no—
103C22×C53—24no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—42no—
107C22×C53C22×D411yescharacteristic
108C22×C53C22×Q811yescharacteristic
109C22×C53C2×#4111yescharacteristic
110C22×C53H(5,𝔽2)12yes—
111C22×C532−1+412yes—
112C4×C53—22no—
113C4×C53—42no—
114C4×C53—41no—
115C4×C53—42no—
116Dic53—22no—
117Dic53—22no—
118Dic53—42no—
119Dic53—41no—
120Dic53—42no—
121Dic53C22⋊Q814yes—
122#2628—531nomaximal, Sylow
123C2×C53—42no—
124C2×C53C2×C22⋊Q811yescharacteristic
125C2×C53#53511yescharacteristic
126C2×C53#53611yescharacteristic
127C2×C53#54212yes—
128C2×C53#54412yes—
129C2×C4⋊Q8—531no—
130C2×#117—531no—
131#400—531no—
132#400—531no—
133#399—531no—
134#399D5311yescharacteristic
135#395—532no—
136#393—532no—
137#389—532no—
138#382—531no—
139#380—532no—
140C53#262811yescharacteristic, complemented, Sylow
141C22×Q8—532no—
142C23×C4—531no—
143C4⋊Q8—1064no—
144#117—1064no—
145C2×C4⋊C4—531no—
146C2×C4⋊C4—531no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4—532no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4C2×D5311yescharacteristic
152C2×C4⋊C4C2×D5312yes—
153C2×C22⋊C4—531no—
154C2×C22⋊C4—532no—
155C2×C22⋊C4C2×D5311yescharacteristic
156C2×C22⋊C4C2×D5312yes—
157C2×C42—531no—
158C2×C42—531no—
159#95—532no—
160#95—532no—
161#95—532no—
162#95—531no—
163#95—532no—
164#95—532no—
165#95C2×D5311yescharacteristic
166C24C22×D5311yescharacteristic
167C2×Q8—1064no—
168C2×Q8—2122no—
169C2×Q8—2122no—
170C22×C4—531no—
171C22×C4—534no—
172C22×C4—531no—
173C22×C4—531no—
174C22×C4—531no—
175C22×C4—532no—
176C22×C4—532no—
177C22×C4—1062no—
178C22×C4—1062no—
179C22×C4C22×D5312yes—
180C22×C4C22×D5312yes—
181C22×C4C22×D5311yescharacteristic
182C22×C4C22×D5311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—1062no—
186C4⋊C4—1064no—
187C4⋊C4—1064no—
188C4⋊C4—1062no—
189C4⋊C4—1062no—
190C4⋊C4—2122no—
191C22⋊C4—22no—
192C22⋊C4—42no—
193C22⋊C4—1062no—
194C22⋊C4—2122no—
195C42—534no—
196C42—1062no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23Q8×D5312yes—
202C23C23×D5311yescharacteristic
203Q8—2128no—
204C2×C4—22no—
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—532no—
214C2×C4—532no—
215C2×C4—532no—
216C2×C4—532no—
217C2×C4—532no—
218C2×C4—1068no—
219C2×C4—1061no—
220C2×C4—1061no—
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—1062no—
230C2×C4D4×D5312yes—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22C2×D4×D5311yescharacteristic
237C22C2×Q8×D5311yescharacteristic
238C22#41×D5311yescharacteristic
239C22#1313326212yes—
240C22#1313326412yes—
241C4—22no—
242C4—41no—
243C4—42no—
244C4—42no—
245C4—534no—
246C4—1062no—
247C4—1062no—
248C4—2122no—
249C4—2121no—
250C4—2122no—
251C2—42no—
252C2C22⋊Q8×D5311yescharacteristic
253C2#1681118611yescharacteristic
254C2#1681119511yescharacteristic
255C2#1681124112yes—
256C2#1681125212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2628
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
3. #16810563C2semidirect
4. #16810561C2semidirect
6. #16810543C2semidirect
8. #16810518C2semidirect
9. #16810487C2semidirect
44. C22×Dic53Q8semidirect
140. C53#2628semidirect