Finite Group Databasebeta

← #20283975

Subgroups — #20283975

Order 6784.

754 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 169 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×#13133247C212yesmaximal, complemented
2C2×#13133196C211yesmaximal, characteristic, complemented
3#16810785C214yesmaximal, complemented
4#16810779C212yesmaximal, complemented
5#16810774C212yesmaximal, complemented
6#16810624C212yesmaximal, complemented
7#16810587C211yesmaximal, characteristic, complemented
8C22×#41×C53C211yesmaximal, characteristic
9C22×Dic106C2211yescharacteristic, complemented
10#13133247—28no—
11C2×#11719925C2214yescomplemented
12C2×#11719923C2214yescomplemented
13C2×#11719917C2212yescomplemented
14#13133196—21no—
15#13133196—21no—
16#13133196C2212yes—
17#13133196C2212yescomplemented
18C2×C4⋊Dic53C2211yescharacteristic, complemented
19C2×#11719919—21no—
20C2×C22⋊Dic53—21no—
21#13133141C2214yescomplemented
22#13133138C2214yescomplemented
23C22⋊#88322C2212yes—
24C22⋊#88322C2212yescomplemented
25C2×#41×C53—28no—
26C2×#41×C53—21no—
27C2×#41×C53—21no—
28C2×#41×C53C2212yes—
29C2×#41×C53C2212yes—
30C22×Q8×C53C2211yescharacteristic
31C22×D4×C53—21no—
32C22×D4×C53C2211yescharacteristic
33C23×C4×C53—21no—
34C23×C4×C53C2211yescharacteristic
35C24×C53—21no—
36C24×C53D411yescharacteristic
37#41×C53—28no—
38#41×C53—28no—
39#41×C53—24no—
40#41×C53—24no—
41#41×C53—24no—
42#41×C53D418yes—
43C2×Q8×C53—24no—
44C2×Q8×C53—21no—
45C2×Q8×C53C2312yes—
46C2×D4×C53—28no—
47C2×D4×C53—22no—
48C2×D4×C53—22no—
49C2×D4×C53—24no—
50C2×D4×C53—21no—
51C2×D4×C53C2312yes—
52C22×C4×C53—28no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—24no—
57C22×C4×C53D411yescharacteristic
58C22×C4×C53D411yescharacteristic
59C22×C4×C53D411yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C2×Dic106—24no—
66C2×Dic106—21no—
67C2×Dic106C2312yes—
68C22⋊Dic53—22no—
69C22⋊Dic53—22no—
70#11719925—28no—
71#11719923—28no—
72C4⋊Dic53—21no—
73C4⋊Dic53C2312yes—
74#11719919—22no—
75#11719919—22no—
76#11719917—24no—
77C2×#88322C2314yes—
78C23×C53—24no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—41no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53C2×D411yescharacteristic
86C23×C53C2×D411yescharacteristic
87C23×C53C2×D411yescharacteristic
88Q8×C53—24no—
89Q8×C53—22no—
90Q8×C53C2×D414yes—
91D4×C53—28no—
92D4×C53—24no—
93D4×C53—28no—
94D4×C53—22no—
95D4×C53C2×D414yes—
96C2×C4×C53—24no—
97C2×C4×C53—28no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—28no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2×D411yescharacteristic
107C2×C4×C53C2×D411yescharacteristic
108C2×C4×C53C2×D414yes—
109C2×C4×C53C2411yescharacteristic
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114Dic106—24no—
115Dic106—42no—
116#88322—24no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—24no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—44no—
126C22×C53—48no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53#12012yes—
130C22×C53#13712yescomplemented
131C22×C53C22×D411yescharacteristic
132C4×C53—21no—
133C4×C53—24no—
134C4×C53—21no—
135C4×C53#12012yes—
136C4×C53C22×D412yes—
137Dic53—42no—
138Dic53—42no—
139#3007—531nomaximal, Sylow
140C2×C53—22no—
141C2×C53—22no—
142C2×C53—41no—
143C2×C53—44no—
144C2×C53C2×#12011yescharacteristic
145C2×C53C2×#13711yescharacteristic
146C2×C53#57511yescharacteristic
147C22×#41D5311yescharacteristic
148C2×#137—532no—
149C2×C22⋊Q8—531no—
150#452—534no—
151#451—532no—
152#450—532no—
153#417—532no—
154#407—531no—
155C53#300711yescharacteristic, complemented, Sylow
156C2×#41—28no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41C2×D5312yes—
160C2×#41C2×D5312yes—
161C22×Q8—531no—
162C22×Q8C2×D5311yescharacteristic
163C22×D4—21no—
164C22×D4C2×D5311yescharacteristic
165C23×C4—21no—
166C23×C4C2×D5311yescharacteristic
167#137—1068no—
168C2×Q16—534no—
169C2×SD16—534no—
170C2×M16—532no—
171C22⋊Q8—532no—
172C22⋊Q8—532no—
173C22⋊Q8—1061no—
174C22⋊Q8—1061no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—1061no—
177C2×C22⋊C4—1061no—
178#103—534no—
179#102—534no—
180C22⋊C8—532no—
181C22⋊C8—532no—
182C24—21no—
183C24C22⋊D5311yescharacteristic
184#41—28no—
185#41—28no—
186#41—24no—
187#41—24no—
188#41—24no—
189#41C22⋊D5318yes—
190C2×Q8—24no—
191C2×Q8—21no—
192C2×Q8—532no—
193C2×Q8—1061no—
194C2×Q8—1064no—
195C2×Q8C22×D5312yes—
196C2×D4—28no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—21no—
201C2×D4C22×D5312yes—
202C22×C4—28no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4C22⋊D5311yescharacteristic
210C22×C4C22⋊D5311yescharacteristic
211C22×C4C22⋊D5311yescharacteristic
212C22×C4C22×D5311yescharacteristic
213C22×C4C22×D5311yescharacteristic
214C22×C4C22×D5311yescharacteristic
215Q16—1068no—
216SD16—1068no—
217M16—1064no—
218C2×C8—534no—
219C4⋊C4—532no—
220C4⋊C4—1061no—
221C4⋊C4—1062no—
222C4⋊C4—1062no—
223C22⋊C4—1062no—
224C22⋊C4—1062no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23C2×C22⋊D5311yescharacteristic
233C23C2×C22⋊D5311yescharacteristic
234C23C2×C22⋊D5311yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—1064no—
238Q8—2122no—
239Q8C2×C22⋊D5314yes—
240D4—28no—
241D4—24no—
242D4—28no—
243D4—22no—
244D4C2×C22⋊D5314yes—
245C2×C4—24no—
246C2×C4—28no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—28no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4C2×C22⋊D5311yescharacteristic
259C2×C4C2×C22⋊D5311yescharacteristic
260C2×C4C2×C22⋊D5311yescharacteristic
261C2×C4C2×C22⋊D5314yes—
262C2×C4C23×D5311yescharacteristic
263C8—1064no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—24no—
270C22—21no—
271C22—21no—
272C22—44no—
273C22—48no—
274C22—42no—
275C22—42no—
276C22#1313318612yes—
277C22#1313324712yescomplemented
278C22C22×C22⋊D5311yescharacteristic
279C4—21no—
280C4—24no—
281C4—21no—
282C4—2122no—
283C4—2122no—
284C4#1313318612yes—
285C4C22×C22⋊D5312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—44no—
290C2C2×#1313318611yescharacteristic
291C2C2×#1313324711yescharacteristic
292C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC22×#41×C53C2
socleC22×C53C22×D4
hypercenterC22×#41D53
hypocenterC53#3007
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#13133247C2semidirect
2. C2×#13133196C2semidirect
3. #16810785C2semidirect
4. #16810779C2semidirect
5. #16810774C2semidirect
6. #16810624C2semidirect
7. #16810587C2semidirect
9. C22×Dic106C22semidirect
11. C2×#11719925C22semidirect
12. C2×#11719923C22semidirect
13. C2×#11719917C22semidirect
17. #13133196C22semidirect
18. C2×C4⋊Dic53C22semidirect
21. #13133141C22semidirect
22. #13133138C22semidirect
24. C22⋊#88322C22semidirect
130. C22×C53#137semidirect
155. C53#3007semidirect
277. C22#13133247semidirect