Finite Group Databasebeta

← #20279153

Subgroups — #20279153

Order 6784.

546 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 153 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×#13133200C211yesmaximal, characteristic
2#16810561C211yesmaximal, characteristic, complemented
3#16810558C211yesmaximal, characteristic, complemented
4#16810554C211yesmaximal, characteristic, complemented
5#16810549C211yesmaximal, characteristic, complemented
6#16810514C212yesmaximal, complemented
7#16810507C212yesmaximal, complemented
8#16810500C212yesmaximal
9#16810481C212yesmaximal, complemented
10C2×#13133124C211yesmaximal, characteristic, complemented
11C2×#117×C53C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C23×Dic53C2212yes—
14#13133200—24no—
15C2×#11719919C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×#11719921C2212yes—
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2211yescharacteristic
31C23×C4×C53C2211yescharacteristic
32#117×C53—24no—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36C2×C42×C53C2212yes—
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53—24no—
40C22×C4×C53D414yes—
41C22×C4×C53C2312yes—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—22no—
46C2×C22⋊D53—24no—
47C22×Dic53—24no—
48C22×Dic53—24no—
49C22×Dic53—42no—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2312yes—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C22⋊C4×C53—24no—
58C42×C53—24no—
59C22⋊Dic53—22no—
60#11719921—22no—
61#11719921—22no—
62#11719921—24no—
63#11719921—42no—
64#11719919—22no—
65#11719919—22no—
66#11719919—24no—
67#11719919—42no—
68C23×C53—22no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53C2×D412yes—
72C23×C53#4114yes—
73C23×C53C2411yescharacteristic
74C22×D53—21no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—21no—
78C22×D53—21no—
79C2×C4×C53—28no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53C2×D412yes—
89C2×C4×C53C2×D412yes—
90C22⋊D53—44no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—48no—
104C2×Dic53#4114yes—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—24no—
108C22×C53—24no—
109C22×C53#12314yes—
110C22×C53C22×D411yescharacteristic
111C22×C53C2×#4112yes—
112C22×C53C2×#4112yes—
113C22×C53H(5,𝔽2)11yescharacteristic
114C22×C532−1+411yescharacteristic
115C2×D53—41no—
116C2×D53—41no—
117C2×D53—41no—
118C2×D53—42no—
119C2×D53—42no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—42no—
123C4×C53—42no—
124Dic53—24no—
125Dic53—41no—
126Dic53—41no—
127Dic53—41no—
128Dic53—42no—
129#2498—531nomaximal, Sylow
130C2×C53—24no—
131C2×C53C2×#12311yescharacteristic
132C2×C53#53211yescharacteristic
133C2×C53#53711yescharacteristic
134C2×C53#55112yes—
135C2×C53#55212yes—
136D53—81no—
137C2×#123—531no—
138C2×#117D5311yescharacteristic, complemented
139#400—531no—
140#399—531no—
141#398—531no—
142#397—531no—
143#388—532no—
144#387—532no—
145#386—532no—
146#380—532no—
147C2×#95—531no—
148C53#249811yescharacteristic, complemented, Sylow
149C22×D4—531no—
150C23×C4—532no—
151C23×C4C2×D5311yescharacteristic
152#123—1064no—
153#117—24no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4C2×D5311yescharacteristic
159C2×C4⋊C4C2×D5311yescharacteristic
160C2×C22⋊C4—531no—
161C2×C22⋊C4—531no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4C2×D5312yes—
166C2×C42C2×D5312yes—
167#95—532no—
168#95—531no—
169#95—531no—
170#95—532no—
171#95—532no—
172#95—532no—
173#95—532no—
174C24—531no—
175C24C22×D5311yescharacteristic
176C2×D4—1064no—
177C2×D4—1062no—
178C22×C4—24no—
179C22×C4—532no—
180C22×C4—532no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—1064no—
185C22×C4—1064no—
186C22×C4—2122no—
187C22×C4C22⋊D5314yes—
188C22×C4C22×D5312yes—
189C22×C4C22×D5312yes—
190C22×C4C22×D5311yescharacteristic
191C22×C4C22×D5311yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—1064no—
195C4⋊C4—1062no—
196C4⋊C4—1062no—
197C4⋊C4—2122no—
198C22⋊C4—24no—
199C22⋊C4—1064no—
200C22⋊C4—1062no—
201C22⋊C4—1062no—
202C22⋊C4—1062no—
203C22⋊C4—2122no—
204C42—24no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—1061no—
209C23—1062no—
210C23—1062no—
211C23—1061no—
212C23—1061no—
213C23#1171994514yes—
214C23C2×C22⋊D5312yes—
215C23C23×D5311yescharacteristic
216D4—2124no—
217C2×C4—28no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—534no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1061no—
238C2×C4—1061no—
239C2×C4—2128no—
240C2×C4C2×C22⋊D5312yes—
241C2×C4C2×C22⋊D5312yes—
242C22—24no—
243C22—24no—
244C22—22no—
245C22—22no—
246C22—2122no—
247C22—2122no—
248C22—2121no—
249C22—2121no—
250C22—2121no—
251C22#1313320014yes—
252C22C22×C22⋊D5311yescharacteristic
253C22#41×D5312yes—
254C22C2×#1171994512yes—
255C22#1313326311yescharacteristic
256C22#1313326411yescharacteristic
257C4—22no—
258C4—22no—
259C4—42no—
260C4—42no—
261C4—1064no—
262C4—2122no—
263C4—2121no—
264C4—2121no—
265C4—2121no—
266C2—24no—
267C2—4241no—
268C2C2×#1313320011yescharacteristic
269C2#1681116911yescharacteristic
270C2#1681120111yescharacteristic
271C2#1681129712yes—
272C2#1681130612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#117×C53C2
hypercenterC2×#117D53
hypocenterC53#2498
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #16810561C2semidirect
3. #16810558C2semidirect
4. #16810554C2semidirect
5. #16810549C2semidirect
6. #16810514C2semidirect
7. #16810507C2semidirect
9. #16810481C2semidirect
10. C2×#13133124C2semidirect
11. C2×#117×C53C2semidirect
138. C2×#117D53semidirect
148. C53#2498semidirect