Finite Group Databasebeta

← #20280483

Subgroups — #20280483

Order 6784.

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

normal subgroups only

This group has 528 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×#13133195C212yesmaximal
3C2×#13133175C211yesmaximal, characteristic
4#16810556C212yesmaximal, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810552C212yesmaximal
7#16810550C211yesmaximal, characteristic
8#16810549C211yesmaximal, characteristic
9#16810527C212yesmaximal, complemented
10#16810487C211yesmaximal, characteristic, complemented
11C53×#398C211yesmaximal, characteristic, complemented
12C22×Dic106C2212yes—
13C22×C4×D53C2211yescharacteristic
14#13133216—24no—
15#13133195—28no—
16#13133175—24no—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×C4×Dic53C2211yescharacteristic
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32#13133124C2212yes—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2212yes—
36#95×C53C2212yes—
37#95×C53C2211yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C2×C4×D53—22no—
50C2×C4×D53—22no—
51C2×C4×D53D414yes—
52C2×Dic106—24no—
53C2×Dic106—24no—
54C2×Dic106—24no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—24no—
57C4⋊C4×C53—22no—
58#11719921—24no—
59#11719921—24no—
60#11719921—22no—
61#11719921—22no—
62C4⋊Dic53—22no—
63C4⋊Dic53—22no—
64#11719919—24no—
65#11719919—24no—
66#11719919—24no—
67C4×Dic53—22no—
68C4×Dic53D414yes—
69C23×C53C2411yescharacteristic
70C22×D53—21no—
71C22×D53—21no—
72C22×D53—22no—
73C22×D53—22no—
74C22×D53C2×D412yes—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53C2×D412yes—
87C2×C4×C53C2×D412yes—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53C2×D412yes—
98C2×Dic53C2×D412yes—
99C2×Dic53C2×D412yes—
100C2×Dic53#4112yes—
101C4×D53—28no—
102Dic106—48no—
103C22×C53C22×D411yescharacteristic
104C22×C53C22×D411yescharacteristic
105C22×C53C2×#4111yescharacteristic
106C22×C532−1+412yes—
107C22×C532−1+412yes—
108C2×D53—22no—
109C2×D53—24no—
110C2×D53—41no—
111C2×D53—41no—
112C2×D53—42no—
113C4×C53—22no—
114C4×C53—22no—
115C4×C53—42no—
116C4×C53—41no—
117C4×C53—42no—
118Dic53—22no—
119Dic53—42no—
120Dic53—42no—
121Dic53—41no—
122Dic53#12114yes—
123#2621—531nomaximal, Sylow
124C2×C53C2×#12111yescharacteristic
125C2×C53#53611yescharacteristic
126C2×C53#53612yes—
127C2×C53#53611yescharacteristic
128C2×C53#54112yes—
129D53—42no—
130C2×C4⋊Q8—531no—
131C2×C22⋊Q8—532no—
132C2×#117—531no—
133#398—531no—
134#398—532no—
135#398D5311yescharacteristic, complemented
136#397—531no—
137#397—531no—
138#397—532no—
139#391—532no—
140#382—531no—
141C53#262111yescharacteristic, complemented, Sylow
142C22×Q8—532no—
143C23×C4—531no—
144C4⋊Q8—1064no—
145C22⋊Q8—1068no—
146#117—1064no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—532no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4C2×D5311yescharacteristic
153C2×C4⋊C4C2×D5312yes—
154C2×C4⋊C4C2×D5311yescharacteristic
155C2×C22⋊C4—531no—
156C2×C22⋊C4—531no—
157C2×C22⋊C4—532no—
158C2×C22⋊C4—532no—
159C2×C42—531no—
160C2×C42—531no—
161#95—532no—
162#95—531no—
163#95—532no—
164#95—532no—
165#95—532no—
166#95C2×D5311yescharacteristic
167#95C2×D5312yes—
168C24—531no—
169C2×Q8—1064no—
170C2×Q8—1064no—
171C2×Q8—1064no—
172C22×C4—531no—
173C22×C4—531no—
174C22×C4—534no—
175C22×C4—531no—
176C22×C4—532no—
177C22×C4—532no—
178C22×C4—1062no—
179C22×C4—1062no—
180C22×C4C22×D5311yescharacteristic
181C22×C4C22×D5312yes—
182C22×C4C22×D5312yes—
183C22×C4C22×D5311yescharacteristic
184C22×C4C22×D5311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—24no—
187C4⋊C4—22no—
188C4⋊C4—1062no—
189C4⋊C4—1064no—
190C4⋊C4—1064no—
191C4⋊C4—1064no—
192C4⋊C4—1062no—
193C22⋊C4—1062no—
194C22⋊C4—1064no—
195C22⋊C4—1064no—
196C22⋊C4—1062no—
197C42—534no—
198C42—1062no—
199C23—532no—
200C23—1061no—
201C23—1061no—
202C23—1062no—
203C23—1062no—
204C23C23×D5311yescharacteristic
205Q8—2128no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—532no—
218C2×C4—532no—
219C2×C4—532no—
220C2×C4—532no—
221C2×C4—1068no—
222C2×C4—1062no—
223C2×C4—1061no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4D4×D5312yes—
232C2×C4D4×D5312yes—
233C22—1062no—
234C22—1064no—
235C22—2121no—
236C22—2121no—
237C22—2122no—
238C22C2×D4×D5311yescharacteristic
239C22C2×D4×D5311yescharacteristic
240C22#41×D5311yescharacteristic
241C22#1313326412yes—
242C22#1313326512yes—
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—42no—
248C4—534no—
249C4—1062no—
250C4—2122no—
251C4—2122no—
252C4—2121no—
253C2—2122no—
254C2#121×D5311yescharacteristic
255C2#1681119511yescharacteristic
256C2#1681119612yes—
257C2#1681119811yescharacteristic
258C2#1681123212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2621
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
4. #16810556C2semidirect
5. #16810553C2semidirect
9. #16810527C2semidirect
10. #16810487C2semidirect
11. C53×#398C2semidirect
135. #398D53semidirect
141. C53#2621semidirect