Finite Group Databasebeta

← #20285194

Subgroups — #20285194

Order 6784.

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

normal subgroups only

This group has 564 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×#13133245C211yesmaximal, characteristic, complemented
2C2×#13133237C211yesmaximal, characteristic, complemented
3C22×#11719924C211yesmaximal, characteristic, complemented
4#16811197C211yesmaximal, characteristic, complemented
5#16811187C211yesmaximal, characteristic, complemented
6#16811016C212yesmaximal, complemented
7#16811009C212yesmaximal, complemented
8#16811008C212yesmaximal, complemented
9#16810994C212yesmaximal, complemented
10#16810581C211yesmaximal, characteristic, complemented
11C2×#137×C53C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C22×D212C2211yescharacteristic
14#13133245—24no—
15#13133237—24no—
16C2×#11719925C2211yescharacteristic, complemented
17C2×#11719913C2212yes—
18C2×#11719923C2211yescharacteristic, complemented
19C2×#11719924—24no—
20C2×#11719924C2212yes—
21C2×#11719924C2212yes—
22C2×#11719924C2211yescharacteristic, complemented
23C2×D424C2212yes—
24C2×#11719922C2211yescharacteristic, complemented
25C22×#88322C2211yescharacteristic
26#13133215C2212yescomplemented
27#13133213C2212yes—
28#13133206C2212yes—
29#13133206C2212yescomplemented
30#13133200—22no—
31#13133193—22no—
32#13133191—22no—
33#13133185—22no—
34#13133176C2211yescharacteristic
35#13133127C2212yescomplemented
36C8×Dic53C2212yescomplemented
37C2×#41×C53C2211yescharacteristic
38C22×Q8×C53C2211yescharacteristic
39#137×C53—24no—
40C2×Q16×C53C2212yescomplemented
41C2×SD16×C53C2212yescomplemented
42C2×M16×C53C2211yescharacteristic
43C23×D53—21no—
44#41×C53—22no—
45#41×C53—22no—
46C2×Q8×C53—21no—
47C2×Q8×C53—22no—
48C2×Q8×C53—22no—
49C2×Q8×C53C2312yes—
50C2×Q8×C53C2311yescharacteristic
51C2×D4×C53—21no—
52C2×D4×C53C2311yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53C2311yescharacteristic
56C2×C22⋊D53—21no—
57C2×C22⋊D53—24no—
58#11719945—22no—
59#11719945—22no—
60C2×D212—22no—
61C2×D212—22no—
62C2×D212—21no—
63C2×D212C2312yes—
64C2×D212C2311yescharacteristic
65C2×C4×D53—21no—
66C2×Dic106C2311yescharacteristic
67Q16×C53—24no—
68SD16×C53—24no—
69M16×C53D414yescomplemented
70C2×C8×C53C2312yes—
71C22⋊Dic53D412yes—
72#11719925—22no—
73#11719924—24no—
74#11719924—24no—
75#11719924—22no—
76#11719923—22no—
77#11719922—22no—
78#11719921—24no—
79#11719921—24no—
80C4⋊Dic53D412yes—
81#11719919—24no—
82C4×Dic53C2312yes—
83C2×#88322D414yescomplemented
84C2×#88322C2311yescharacteristic
85C2×#88322C2311yescharacteristic
86D424—24no—
87#11719913—24no—
88C23×C53—21no—
89C23×C53C2×D411yescharacteristic
90C22×D53—21no—
91C22×D53—22no—
92C22×D53—42no—
93C22×D53—82no—
94Q8×C53—22no—
95Q8×C53—22no—
96Q8×C53—21no—
97Q8×C53—21no—
98Q8×C53—42no—
99D4×C53—21no—
100D4×C53—21no—
101D4×C53—42no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53C2×D411yescharacteristic
107C2×C4×C53C2×D412yes—
108C2×C4×C53C2×D412yes—
109C2×C4×C53C2411yescharacteristic
110C22⋊D53—42no—
111C22⋊D53—84no—
112C2×Dic53—21no—
113C2×Dic53C2×D414yes—
114D212—22no—
115D212—22no—
116D212—21no—
117D212—21no—
118D212—42no—
119C4×D53—42no—
120Dic106—21no—
121Dic106—21no—
122C8×C53C2×D414yes—
123#88322C2×D412yes—
124#88322C2×D412yes—
125C22×C53—22no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53#12712yes—
129C22×C53C22×D411yescharacteristic
130C2×D53—41no—
131C2×D53—42no—
132C2×D53—44no—
133C2×D53—42no—
134C2×D53—84no—
135C4×C53—41no—
136C4×C53—42no—
137C4×C53#12712yes—
138C4×C53C22×D411yescharacteristic
139C4×C53C22×D411yescharacteristic
140Dic53—24no—
141Dic53—41no—
142#3154—531nomaximal, Sylow
143C2×C53—22no—
144C2×C53—81no—
145C2×C53C2×#12711yescharacteristic
146C2×C53#57712yes—
147D53—81no—
148D53—82no—
149C2×#137D5311yescharacteristic, complemented
150C2×AGL(1,ℤ/8)—531no—
151C2×#135—531no—
152C22×SD16—531no—
153#536—531no—
154#535—531no—
155#497—532no—
156#496—532no—
157#495—532no—
158#492—532no—
159#405—531no—
160C53#315411yescharacteristic, complemented, Sylow
161C2×#41—531no—
162C2×#41C2×D5311yescharacteristic
163C22×Q8C2×D5311yescharacteristic
164C22×D4—531no—
165#137—24no—
166AGL(1,ℤ/8)—1064no—
167#135—1064no—
168C2×Q16—531no—
169C2×Q16C2×D5312yescomplemented
170C2×SD16—532no—
171C2×SD16—532no—
172C2×SD16—531no—
173C2×SD16—531no—
174C2×SD16—532no—
175C2×SD16—1064no—
176C2×SD16C2×D5312yescomplemented
177C2×D8—531no—
178C2×D8—532no—
179C2×M16C2×D5311yescharacteristic
180C22×C8—531no—
181C4⋊Q8—532no—
182#127—532no—
183#124—532no—
184#124—532no—
185#123—1062no—
186C22⋊Q8—1062no—
187#121—1062no—
188#120—1062no—
189#117—531no—
190#97—532no—
191C4×C8—532no—
192C24—1061no—
193#41—22no—
194#41—22no—
195#41—1062no—
196#41—1062no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—531no—
201C2×Q8C22×D5311yescharacteristic
202C2×Q8C22×D5312yes—
203C2×D4—21no—
204C2×D4—531no—
205C2×D4—532no—
206C2×D4—1064no—
207C2×D4—1061no—
208C2×D4—1062no—
209C2×D4—1061no—
210C2×D4—1062no—
211C2×D4C22×D5311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—1061no—
215C22×C4C22×D5311yescharacteristic
216Q16—24no—
217Q16—1062no—
218SD16—24no—
219SD16—1062no—
220SD16—1064no—
221SD16—1064no—
222SD16—1062no—
223SD16—1064no—
224D8—1062no—
225D8—1064no—
226M16C22⋊D5314yescomplemented
227C2×C8—534no—
228C2×C8—531no—
229C2×C8—531no—
230C2×C8C22×D5312yes—
231C4⋊C4—532no—
232C4⋊C4—1064no—
233C22⋊C4—532no—
234C22⋊C4—1064no—
235C22⋊C4—1064no—
236C42—532no—
237C23—21no—
238C23—1061no—
239C23—1062no—
240C23—4242no—
241C23—2122no—
242C23D4×D5311yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—1061no—
249Q8—1061no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—1061no—
254D4—1062no—
255D4—1061no—
256D4—1062no—
257D4—4244no—
258D4—2122no—
259D4—2122no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—534no—
265C2×C4—1061no—
266C2×C4—2122no—
267C2×C4D4×D5311yescharacteristic
268C2×C4D4×D5312yes—
269C2×C4C2×C22⋊D5312yes—
270C2×C4C23×D5311yescharacteristic
271C8—532no—
272C8—532no—
273C8C2×C22⋊D5314yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—4244no—
278C22—2121no—
279C22—2122no—
280C22—2124no—
281C22—2122no—
282C22#1313321312yes—
283C22C2×D4×D5311yescharacteristic
284C4—41no—
285C4—42no—
286C4—1064no—
287C4—2121no—
288C4#1313321312yes—
289C4C2×D4×D5311yescharacteristic
290C4C22×C22⋊D5311yescharacteristic
291C2—22no—
292C2—81no—
293C2—4242no—
294C2—4241no—
295C2C2×#1313321311yescharacteristic
296C2#1681143212yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#137×C53C2
socleC22×C53C22×D4
hypercenterC2×#137D53
hypocenterC53#3154
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#13133245C2semidirect
2. C2×#13133237C2semidirect
3. C22×#11719924C2semidirect
4. #16811197C2semidirect
5. #16811187C2semidirect
6. #16811016C2semidirect
7. #16811009C2semidirect
8. #16811008C2semidirect
9. #16810994C2semidirect
10. #16810581C2semidirect
11. C2×#137×C53C2semidirect
16. C2×#11719925C22semidirect
18. C2×#11719923C22semidirect
22. C2×#11719924C22semidirect
24. C2×#11719922C22semidirect
26. #13133215C22semidirect
29. #13133206C22semidirect
35. #13133127C22semidirect
36. C8×Dic53C22semidirect
40. C2×Q16×C53C22semidirect
41. C2×SD16×C53C22semidirect
69. M16×C53D4semidirect
83. C2×#88322D4semidirect
149. C2×#137D53semidirect
160. C53#3154semidirect
169. C2×Q16C2×D53semidirect
176. C2×SD16C2×D53semidirect
226. M16C22⋊D53semidirect