Finite Group Databasebeta

← #20283990

Subgroups — #20283990

Order 6784.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133263C211yesmaximal, characteristic
2C2×#13133248C211yesmaximal, characteristic, complemented
3C22×#11719924C211yesmaximal, characteristic, complemented
4#16810783C214yesmaximal, complemented
5#16810773C214yesmaximal, complemented
6#16810627C211yesmaximal, characteristic, complemented
7#16810617C211yesmaximal, characteristic, complemented
8#16810590C211yesmaximal, characteristic, complemented
9C53×#536C211yesmaximal, characteristic, complemented
10#13133263—24no—
11#13133263—24no—
12C2×#11719949—21no—
13C2×#11719945C2212yes—
14C2×#11719945C2211yescharacteristic
15C22×D212C2212yes—
16C22×D212C2211yescharacteristic
17C2×D4×D53—22no—
18C2×D4×D53—21no—
19#13133248—24no—
20C2×#11719925C2212yescomplemented
21C2×#11719924—24no—
22C2×#11719924C2212yescomplemented
23C2×#11719924C2214yescomplemented
24C2×#11719917C2211yescharacteristic, complemented
25C22×#88322C2211yescharacteristic, complemented
26#13133140C2212yescomplemented
27#13133137C2212yescomplemented
28#13133137C2214yescomplemented
29C22⋊#88322C2214yescomplemented
30C2×#41×C53C2211yescharacteristic
31C22×Q8×C53C2211yescharacteristic, complemented
32C4⋊Q8×C53—21no—
33#124×C53—21no—
34#123×C53—22no—
35C22⋊Q8×C53C2214yescomplemented
36#117×C53C2211yescharacteristic, complemented
37C23×D53—22no—
38C23×D53—21no—
39#41×C53—22no—
40#41×C53—22no—
41C2×Q8×C53—21no—
42C2×Q8×C53—22no—
43C2×Q8×C53—22no—
44C2×Q8×C53D411yescharacteristic
45C2×Q8×C53C2312yes—
46C2×D4×C53D412yes—
47C2×D4×C53D411yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53C2312yes—
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—22no—
52C2×C22⋊D53—21no—
53#11719949—28no—
54D4×D53—216no—
55D4×D53—28no—
56#11719945—22no—
57#11719945—24no—
58#11719945—24no—
59#11719945D414yes—
60C2×D212—22no—
61C2×D212—21no—
62C2×D212—22no—
63C2×D212—24no—
64C2×D212—24no—
65C2×D212D414yes—
66C2×D212C2311yescharacteristic
67C2×D212C2312yes—
68C2×C4×D53—22no—
69C2×C4×D53—21no—
70C2×Dic106C2311yescharacteristic
71C4⋊C4×C53—24no—
72C4⋊C4×C53C2312yes—
73C22⋊C4×C53—21no—
74C22⋊C4×C53—24no—
75C42×C53—21no—
76#11719925—24no—
77#11719924—28no—
78#11719924—24no—
79#11719917—22no—
80C2×#88322—22no—
81C2×#88322C2312yes—
82C2×#88322C2312yes—
83C23×C53C2×D412yes—
84C23×C53C2×D411yescharacteristic
85C22×D53—22no—
86C22×D53—21no—
87C22×D53—28no—
88C22×D53—24no—
89C22×D53—82no—
90C22×D53—84no—
91Q8×C53—22no—
92Q8×C53—22no—
93Q8×C53—22no—
94Q8×C53—42no—
95D4×C53—24no—
96D4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—42no—
102C2×C4×C53C2×D412yes—
103C2×C4×C53C2×D412yes—
104C2×C4×C53C2×D412yes—
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2411yescharacteristic
107C22⋊D53—28no—
108C22⋊D53—24no—
109C2×Dic53—21no—
110D212—28no—
111D212—24no—
112D212—21no—
113D212—22no—
114D212C2×D412yes—
115D212C2×D414yes—
116C4×D53—28no—
117C4×D53—24no—
118Dic106—21no—
119Dic106C2×D412yes—
120#88322—22no—
121#88322—22no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—24no—
125C22×C53#12012yes—
126C22×C53C22×D411yescharacteristic
127C2×D53—24no—
128C2×D53—22no—
129C2×D53—44no—
130C2×D53—42no—
131C2×D53—84no—
132C2×D53—88no—
133C4×C53—22no—
134C4×C53—42no—
135C4×C53—42no—
136C4×C53#12012yes—
137C4×C53C22×D411yescharacteristic
138C4×C53C22×D411yescharacteristic
139Dic53—22no—
140#3009—531nomaximal, Sylow
141C2×C53—22no—
142C2×C53—42no—
143C2×C53C2×#12011yescharacteristic
144C2×C53#57712yes—
145D53—44no—
146D53—42no—
147C2×H(5,𝔽2)—531no—
148C2×#137—531no—
149C22×SD16—531no—
150#536D5311yescharacteristic, complemented
151#452—534no—
152#450—534no—
153#418—531no—
154#416—531no—
155#408—531no—
156C53#300911yescharacteristic, complemented, Sylow
157H(5,𝔽2)—1064no—
158H(5,𝔽2)—1064no—
159C2×#41—531no—
160C2×#41—532no—
161C2×#41—1061no—
162C2×#41C2×D5311yescharacteristic
163C22×Q8C2×D5311yescharacteristic, complemented
164C22×D4—531no—
165C22×D4—532no—
166C22×D4—1062no—
167C22×D4—1061no—
168#137—1064no—
169C2×Q16—532no—
170C2×SD16—534no—
171C2×SD16—532no—
172C2×SD16—1064no—
173C2×M16—531no—
174C22×C8—531no—
175C4⋊Q8—21no—
176#124—21no—
177#123—22no—
178C22⋊Q8C2×D5314yescomplemented
179#117C2×D5311yescharacteristic, complemented
180#103—532no—
181#102—532no—
182#102—534no—
183C22⋊C8—534no—
184C24—1062no—
185C24—1061no—
186#41—22no—
187#41—22no—
188#41—534no—
189#41—1064no—
190#41—1064no—
191#41—1068no—
192#41—1062no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—531no—
197C2×Q8C22⋊D5311yescharacteristic
198C2×Q8C22×D5312yes—
199C2×D4—534no—
200C2×D4—532no—
201C2×D4—531no—
202C2×D4—1064no—
203C2×D4—1064no—
204C2×D4—10616no—
205C2×D4—1068no—
206C2×D4—1062no—
207C2×D4—1061no—
208C2×D4—1062no—
209C2×D4—1062no—
210C2×D4—1061no—
211C2×D4C22⋊D5312yes—
212C2×D4C22⋊D5311yescharacteristic
213C22×C4—21no—
214C22×C4—1062no—
215C22×C4—1061no—
216C22×C4C22×D5312yes—
217C22×C4C22×D5311yescharacteristic
218Q16—1064no—
219SD16—1068no—
220SD16—1064no—
221M16—1062no—
222C2×C8—532no—
223C2×C8—532no—
224C2×C8—1062no—
225C4⋊C4—24no—
226C4⋊C4C22×D5312yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—1068no—
231C23—1064no—
232C23—1062no—
233C23—1061no—
234C23—4242no—
235C23—4244no—
236C23C2×C22⋊D5312yes—
237C23C2×C22⋊D5311yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—532no—
243Q8—1061no—
244D4—24no—
245D4—22no—
246D4—534no—
247D4—532no—
248D4—1068no—
249D4—1068no—
250D4—1064no—
251D4—1064no—
252D4—1061no—
253D4—1062no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—1068no—
260C2×C4—1061no—
261C2×C4—1064no—
262C2×C4D4×D5312yes—
263C2×C4D4×D5312yes—
264C2×C4C2×C22⋊D5312yes—
265C2×C4C2×C22⋊D5311yescharacteristic
266C2×C4C23×D5311yescharacteristic
267C8—1062no—
268C8—1062no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—1064no—
273C22—1062no—
274C22—4244no—
275C22—4248no—
276C22—2124no—
277C22—2122no—
278C22#1313318512yes—
279C22C22×C22⋊D5311yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—1062no—
284C4#1313318512yes—
285C4C2×D4×D5311yescharacteristic
286C4C2×D4×D5311yescharacteristic
287C2—22no—
288C2—42no—
289C2—2124no—
290C2—2122no—
291C2C2×#1313318511yescharacteristic
292C2#1681143112yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#536C2
socleC22×C53C22×D4
hypercenter#536D53
hypocenterC53#3009
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#13133248C2semidirect
3. C22×#11719924C2semidirect
4. #16810783C2semidirect
5. #16810773C2semidirect
6. #16810627C2semidirect
7. #16810617C2semidirect
8. #16810590C2semidirect
9. C53×#536C2semidirect
20. C2×#11719925C22semidirect
22. C2×#11719924C22semidirect
23. C2×#11719924C22semidirect
24. C2×#11719917C22semidirect
25. C22×#88322C22semidirect
26. #13133140C22semidirect
27. #13133137C22semidirect
28. #13133137C22semidirect
29. C22⋊#88322C22semidirect
31. C22×Q8×C53C22semidirect
35. C22⋊Q8×C53C22semidirect
36. #117×C53C22semidirect
150. #536D53semidirect
156. C53#3009semidirect
163. C22×Q8C2×D53semidirect
178. C22⋊Q8C2×D53semidirect
179. #117C2×D53semidirect