Finite Group Databasebeta

← #20283955

Subgroups — #20283955

Order 6784.

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

normal subgroups only

This group has 758 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
1C22×#11719945C211yesmaximal, characteristic, complemented
2C2×#13133247C211yesmaximal, characteristic, complemented
3C2×#13133242C211yesmaximal, characteristic, complemented
4#16810783C214yesmaximal, complemented
5#16810766C214yesmaximal, complemented
6#16810631C211yesmaximal, characteristic, complemented
7#16810627C211yesmaximal, characteristic, complemented
8#16810587C211yesmaximal, characteristic, complemented
9C53×#525C211yesmaximal, characteristic, complemented
10C2×#11719945—22no—
11C2×#11719945—24no—
12C2×#11719945—24no—
13C2×#11719945C2214yes—
14C22×Dic106C2211yescharacteristic
15C22×D212C2211yescharacteristic
16C22×C22⋊D53—21no—
17C22×C4×D53—21no—
18#13133247—24no—
19#13133242—24no—
20C2×#11719925C2212yescomplemented
21C2×#11719923C2212yescomplemented
22C2×#11719924C2212yescomplemented
23C2×#11719922C2212yescomplemented
24C2×#11719917C2211yescharacteristic
25C2×#11719917C2211yescharacteristic
26#13133140C2214yescomplemented
27#13133137C2214yescomplemented
28C22⋊#88322C2214yescomplemented
29C2×#41×C53C2211yescharacteristic
30C23×C4×C53C2211yescharacteristic
31#123×C53—21no—
32C22⋊Q8×C53C2212yescomplemented
33#121×C53C2212yescomplemented
34#120×C53—21no—
35C4×D4×C53—22no—
36#117×C53C2211yescharacteristic
37C24×C53D411yescharacteristic
38C23×D53—21no—
39#41×C53—22no—
40#41×C53—22no—
41C2×Q8×C53C2311yescharacteristic
42C2×D4×C53—22no—
43C2×D4×C53—21no—
44C2×D4×C53C2311yescharacteristic
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—21no—
48C22×C4×C53D412yes—
49C22×C4×C53D411yescharacteristic
50C22×C4×C53C2312yes—
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—28no—
53C2×C22⋊D53—24no—
54C22×Dic53—21no—
55#11719945—216no—
56#11719945—28no—
57#11719945—28no—
58C2×D212—21no—
59C2×D212—22no—
60C2×D212D414yescomplemented
61C2×D212C2312yes—
62C2×C4×D53—28no—
63C2×C4×D53—24no—
64C2×Dic106—21no—
65C2×Dic106—22no—
66C2×Dic106D414yescomplemented
67C2×Dic106C2312yes—
68C4⋊C4×C53—22no—
69C4⋊C4×C53C2312yes—
70C22⋊C4×C53—21no—
71C22⋊C4×C53—22no—
72C22⋊C4×C53—22no—
73C42×C53—21no—
74#11719925—24no—
75#11719924—24no—
76#11719923—24no—
77#11719922—24no—
78#11719917—22no—
79#11719917—22no—
80C2×#88322C2312yes—
81C2×#88322C2312yes—
82C23×C53—22no—
83C23×C53—21no—
84C23×C53—42no—
85C23×C53C2×D412yes—
86C23×C53C2×D411yescharacteristic
87C22×D53—24no—
88C22×D53—22no—
89C22×D53—44no—
90Q8×C53—21no—
91Q8×C53—21no—
92D4×C53—21no—
93D4×C53—21no—
94D4×C53—44no—
95D4×C53—42no—
96C2×C4×C53—22no—
97C2×C4×C53—24no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—42no—
103C2×C4×C53C2×D412yes—
104C2×C4×C53C2×D412yes—
105C2×C4×C53C2×D412yes—
106C2×C4×C53C2×D411yescharacteristic
107C2×C4×C53C2411yescharacteristic
108C22⋊D53—216no—
109C2×Dic53—24no—
110C2×Dic53—22no—
111D212—24no—
112D212—22no—
113D212C2×D414yes—
114C4×D53—216no—
115Dic106—24no—
116Dic106—22no—
117Dic106C2×D414yes—
118#88322—22no—
119#88322—22no—
120C22×C53—22no—
121C22×C53—22no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—44no—
125C22×C53—41no—
126C22×C53—42no—
127C22×C53#12012yes—
128C22×C53C22×D411yescharacteristic
129C2×D53—24no—
130C2×D53—48no—
131C2×D53—44no—
132C4×C53—22no—
133C4×C53—41no—
134C4×C53—42no—
135C4×C53#12012yes—
136C4×C53C22×D411yescharacteristic
137C4×C53C22×D411yescharacteristic
138Dic53—24no—
139#3005—531nomaximal, Sylow
140C2×C53—22no—
141C2×C53—42no—
142C2×C53—81no—
143C2×C53C2×#12011yescharacteristic
144C2×C53#57512yes—
145D53—44no—
146C22×#41—531no—
147C2×#137—531no—
148C2×AGL(1,ℤ/8)—531no—
149#525D5311yescharacteristic, complemented
150#452—534no—
151#449—534no—
152#419—531no—
153#418—531no—
154#407—531no—
155C53#300511yescharacteristic, complemented, Sylow
156C2×#41—534no—
157C2×#41—1064no—
158C2×#41—1064no—
159C2×#41—1062no—
160C2×#41C2×D5311yescharacteristic
161C22×Q8—531no—
162C22×D4—531no—
163C22×D4—1061no—
164C23×C4—1061no—
165C23×C4C2×D5311yescharacteristic
166#137—1064no—
167AGL(1,ℤ/8)—1064no—
168C2×Q16—532no—
169C2×SD16—532no—
170C2×SD16—532no—
171C2×D8—532no—
172C2×M16—531no—
173C2×M16—531no—
174#123—21no—
175C22⋊Q8C2×D5312yescomplemented
176#121C2×D5312yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D5311yescharacteristic
180#103—534no—
181#102—534no—
182C22⋊C8—534no—
183C24—1061no—
184C24C22⋊D5311yescharacteristic
185#41—22no—
186#41—22no—
187#41—1068no—
188#41—1068no—
189#41—10616no—
190C2×Q8—534no—
191C2×Q8—532no—
192C2×Q8—1062no—
193C2×Q8—1061no—
194C2×Q8C22×D5311yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—534no—
198C2×D4—532no—
199C2×D4—1068no—
200C2×D4—1064no—
201C2×D4—1062no—
202C2×D4—1061no—
203C2×D4C22×D5311yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—1068no—
208C22×C4—1061no—
209C22×C4—1064no—
210C22×C4C22⋊D5312yes—
211C22×C4C22⋊D5311yescharacteristic
212C22×C4C22×D5312yes—
213C22×C4C22×D5311yescharacteristic
214Q16—1064no—
215SD16—1064no—
216SD16—1064no—
217D8—1064no—
218M16—1062no—
219M16—1062no—
220C2×C8—532no—
221C2×C8—532no—
222C4⋊C4—22no—
223C4⋊C4C22×D5312yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—1064no—
232C23—1062no—
233C23—2124no—
234C23C2×C22⋊D5312yes—
235C23C2×C22⋊D5311yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—534no—
239Q8—1064no—
240Q8—1062no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—534no—
246D4—10616no—
247D4—1064no—
248D4—1062no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—10616no—
257C2×C4—1064no—
258C2×C4—1062no—
259C2×C4D4×D5312yes—
260C2×C4D4×D5312yes—
261C2×C4C2×C22⋊D5312yes—
262C2×C4C2×C22⋊D5311yescharacteristic
263C2×C4C23×D5311yescharacteristic
264C8—1062no—
265C8—1062no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—1064no—
274C22—2128no—
275C22—2124no—
276C22#1313318512yes—
277C22C22×C22⋊D5311yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—1064no—
282C4#1313318512yes—
283C4C2×D4×D5311yescharacteristic
284C4C2×D4×D5311yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—2124no—
289C2C2×#1313318511yescharacteristic
290C2#1681142112yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#525C2
socleC22×C53C22×D4
hypercenter#525D53
hypocenterC53#3005
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#11719945C2semidirect
2. C2×#13133247C2semidirect
3. C2×#13133242C2semidirect
4. #16810783C2semidirect
5. #16810766C2semidirect
6. #16810631C2semidirect
7. #16810627C2semidirect
8. #16810587C2semidirect
9. C53×#525C2semidirect
20. C2×#11719925C22semidirect
21. C2×#11719923C22semidirect
22. C2×#11719924C22semidirect
23. C2×#11719922C22semidirect
26. #13133140C22semidirect
27. #13133137C22semidirect
28. C22⋊#88322C22semidirect
32. C22⋊Q8×C53C22semidirect
33. #121×C53C22semidirect
60. C2×D212D4semidirect
66. C2×Dic106D4semidirect
149. #525D53semidirect
155. C53#3005semidirect
175. C22⋊Q8C2×D53semidirect
176. #121C2×D53semidirect