Finite Group Databasebeta

← #20277979

Subgroups — #20277979

Order 6784.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#13133179C211yesmaximal, characteristic
2C2×#13133175C211yesmaximal, characteristic, complemented
3#16810562C211yesmaximal, characteristic, complemented
4#16810560C211yesmaximal, characteristic, complemented
5#16810550C212yesmaximal, complemented
6#16810549C211yesmaximal, characteristic, complemented
7#16810540C212yesmaximal
8C4⋊C4×Dic53C212yesmaximal, complemented
9#16810470C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#398C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15#13133179—24no—
16#13133175—24no—
17C2×C4⋊Dic53C2212yes—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×C4×Dic53C2212yes—
26C2×C4×Dic53C2211yescharacteristic
27C2×C4×Dic53C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31#13133124C2212yes—
32C2×C4⋊C4×C53C2212yes—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×D212—22no—
49C2×D212—22no—
50C2×D212—22no—
51C2×C4×D53—24no—
52C2×C4×D53—22no—
53C2×C4×D53—24no—
54C2×C4×D53D414yes—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—24no—
58#11719921—24no—
59#11719921—22no—
60#11719921—24no—
61#11719921—22no—
62#11719921—24no—
63#11719921—24no—
64C4⋊Dic53—22no—
65C4⋊Dic53—24no—
66C4×Dic53—24no—
67C4×Dic53—22no—
68C4×Dic53—24no—
69C23×C53C2411yescharacteristic
70C22×D53—21no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—21no—
74C22×D53—21no—
75C22×D53—21no—
76C22×D53—22no—
77C22×D53—22no—
78C22×D53C2×D412yes—
79C22×D53#4112yes—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53C2×D412yes—
91C2×C4×C53#4112yes—
92C2×C4×C53#4114yes—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53C2×D412yes—
102C2×Dic53#4114yes—
103D212—28no—
104C4×D53—28no—
105C4×D53—44no—
106C22×C53C22×D411yescharacteristic
107C22×C53C2×#4112yes—
108C22×C53C2×#4111yescharacteristic
109C22×C53C2×#4111yescharacteristic
110C22×C53C2×#4112yes—
111C2×D53—22no—
112C2×D53—22no—
113C2×D53—22no—
114C2×D53—24no—
115C2×D53—22no—
116C2×D53—41no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—42no—
120C2×D53—41no—
121C4×C53—22no—
122C4×C53—22no—
123C4×C53—24no—
124C4×C53—41no—
125C4×C53—42no—
126Dic53—42no—
127Dic53—42no—
128Dic53—41no—
129Dic53#12314yes—
130#2395—531nomaximal, Sylow
131C2×C53C2×#12311yescharacteristic
132C2×C53#52511yescharacteristic
133C2×C53#52912yes—
134C2×C53#52912yes—
135C2×C53#53211yescharacteristic
136D53—42no—
137D53—42no—
138C2×C4×D4—531no—
139C2×#117—531no—
140#400—531no—
141#399—531no—
142#398D5311yescharacteristic, complemented
143#397—531no—
144#397—532no—
145#394—532no—
146C4×C4⋊C4—532no—
147C4×C22⋊C4—532no—
148C4×C22⋊C4—531no—
149C53#239511yescharacteristic, complemented, Sylow
150C22×D4—531no—
151C23×C4—531no—
152C23×C4—531no—
153C4×D4—1064no—
154#117—1064no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4C2×D5311yescharacteristic
158C2×C4⋊C4C2×D5311yescharacteristic
159C2×C4⋊C4C2×D5312yes—
160C2×C22⋊C4—532no—
161C2×C22⋊C4—531no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C42—532no—
167C2×C42—531no—
168C2×C42—532no—
169#95—531no—
170#95—532no—
171#95—532no—
172#95—532no—
173#95C2×D5312yes—
174#95C2×D5311yescharacteristic
175C24—531no—
176C24—531no—
177C2×D4—1062no—
178C2×D4—1062no—
179C2×D4—1062no—
180C22×C4—531no—
181C22×C4—534no—
182C22×C4—532no—
183C22×C4—531no—
184C22×C4—532no—
185C22×C4—1064no—
186C22×C4—1062no—
187C22×C4—1064no—
188C22×C4C22×D5311yescharacteristic
189C22×C4C22×D5312yes—
190C22×C4C22×D5311yescharacteristic
191C22×C4C22×D5311yescharacteristic
192C22×C4C22×D5312yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—1062no—
197C4⋊C4—1064no—
198C22⋊C4—1064no—
199C22⋊C4—1062no—
200C22⋊C4—1064no—
201C22⋊C4—1062no—
202C22⋊C4—1064no—
203C22⋊C4—1064no—
204C42—1064no—
205C42—1062no—
206C42—1064no—
207C23—532no—
208C23—532no—
209C23—1061no—
210C23—1062no—
211C23—1062no—
212C23—1061no—
213C23—1061no—
214C23—1061no—
215C23—1062no—
216C23—1062no—
217C23C23×D5311yescharacteristic
218D4—1068no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—532no—
230C2×C4—534no—
231C2×C4—1068no—
232C2×C4—1061no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—2124no—
241C2×C4D4×D5312yes—
242C2×C4#1171994712yes—
243C2×C4#1171994914yes—
244C22—1062no—
245C22—1062no—
246C22—1062no—
247C22—1064no—
248C22—1062no—
249C22—2121no—
250C22—2122no—
251C22—2122no—
252C22—2122no—
253C22—2121no—
254C22C2×D4×D5311yescharacteristic
255C22#41×D5312yes—
256C22#41×D5311yescharacteristic
257C22C2×#1171994711yescharacteristic
258C22C2×#1171994912yes—
259C4—22no—
260C4—22no—
261C4—24no—
262C4—41no—
263C4—42no—
264C4—534no—
265C4—2122no—
266C4—2121no—
267C4—2122no—
268C2—2122no—
269C2—2122no—
270C2#123×D5311yescharacteristic
271C2#1681113311yescharacteristic
272C2#1681115812yes—
273C2#1681115912yes—
274C2#1681117411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2395
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#13133175C2semidirect
3. #16810562C2semidirect
4. #16810560C2semidirect
5. #16810550C2semidirect
6. #16810549C2semidirect
8. C4⋊C4×Dic53C2semidirect
9. #16810470C2semidirect
10. #16810470C2semidirect
11. C53×#398C2semidirect
142. #398D53semidirect
149. C53#2395semidirect