Finite Group Databasebeta

← #20285182

Subgroups — #20285182

Order 6784.

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

normal subgroups only

This group has 572 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×#13133248C211yesmaximal, characteristic, complemented
2C2×#13133245C211yesmaximal, characteristic, complemented
3C2×#13133242C211yesmaximal, characteristic, complemented
4#16811169C211yesmaximal, characteristic, complemented
5C2×#13133206C211yesmaximal, characteristic, complemented
6#16811016C212yesmaximal, complemented
7#16811009C212yesmaximal, complemented
8#16811008C212yesmaximal, complemented
9#16811004C212yesmaximal, complemented
10M16×Dic53C211yesmaximal, characteristic, complemented
11C2×#137×C53C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C22×D212C2211yescharacteristic
14#13133248—24no—
15#13133245—24no—
16#13133242—24no—
17C2×#11719925C2212yescomplemented
18C2×#11719913C2212yes—
19C2×#11719924C2212yes—
20C2×#11719924C2212yescomplemented
21C2×D424C2212yes—
22C2×#11719922C2212yes—
23C2×#11719917C2211yescharacteristic
24#13133215C2211yescharacteristic, complemented
25#13133213C2211yescharacteristic, complemented
26#13133206—22no—
27#13133206C2212yes—
28#13133206C2212yes—
29#13133206C2211yescharacteristic, complemented
30#13133204C2211yescharacteristic, complemented
31#13133189—22no—
32C4×C22⋊D53—22no—
33C2×#11719921—22no—
34C2×C4×Dic53C2211yescharacteristic
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—22no—
58C22×Dic53D412yes—
59#11719945—22no—
60#11719945—22no—
61C2×D212—22no—
62C2×D212—22no—
63C2×D212—21no—
64C2×D212C2312yes—
65C2×D212C2311yescharacteristic
66C2×C4×D53—21no—
67C2×Dic106C2311yescharacteristic
68Q16×C53—24no—
69SD16×C53—24no—
70M16×C53D414yescomplemented
71C2×C8×C53C2312yes—
72C22⋊Dic53—22no—
73#11719925—24no—
74#11719924—24no—
75#11719924—24no—
76#11719922—24no—
77#11719921—24no—
78#11719921—24no—
79#11719921—22no—
80#11719919—22no—
81C4×Dic53D412yes—
82C4×Dic53C2311yescharacteristic
83C4×Dic53C2311yescharacteristic
84#11719917D414yescomplemented
85C2×#88322C2312yes—
86D424—24no—
87#11719913—24no—
88C23×C53—21no—
89C23×C53C2×D411yescharacteristic
90C22×D53—21no—
91C22×D53—22no—
92C22×D53—44no—
93C22×D53—42no—
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—44no—
111C22⋊D53—42no—
112C2×Dic53—21no—
113C2×Dic53—42no—
114C2×Dic53C2×D412yes—
115C2×Dic53C2×D412yes—
116D212—22no—
117D212—22no—
118D212—21no—
119D212—21no—
120D212—42no—
121C4×D53—42no—
122Dic106—21no—
123Dic106—21no—
124C8×C53C2×D414yes—
125#88322C2×D414yes—
126C22×C53—22no—
127C22×C53—41no—
128C22×C53—42no—
129C22×C53#12712yes—
130C22×C53C22×D411yescharacteristic
131C2×D53—41no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—44no—
135C2×D53—84no—
136C4×C53—41no—
137C4×C53—42no—
138C4×C53#12712yes—
139C4×C53C22×D411yescharacteristic
140C4×C53C22×D411yescharacteristic
141Dic53—22no—
142Dic53—22no—
143Dic53—41no—
144#3153—531nomaximal, Sylow
145C2×C53—22no—
146C2×C53—81no—
147C2×C53C2×#12711yescharacteristic
148C2×C53#57512yes—
149D53—81no—
150D53—82no—
151C2×#137—531no—
152C2×#137D5311yescharacteristic, complemented
153C2×AGL(1,ℤ/8)—531no—
154C2×AGL(1,ℤ/8)—531no—
155#532—531no—
156C2×#124—531no—
157#497—532no—
158#496—532no—
159#495—532no—
160#495—532no—
161C4×M16—531no—
162C53#315311yescharacteristic, complemented, Sylow
163C2×#41—531no—
164C2×#41C2×D5311yescharacteristic
165C22×Q8C2×D5311yescharacteristic
166C22×D4—531no—
167#137—24no—
168#137—1064no—
169AGL(1,ℤ/8)—1064no—
170AGL(1,ℤ/8)—1064no—
171C2×Q16—532no—
172C2×Q16C2×D5312yescomplemented
173C2×SD16—532no—
174C2×SD16—532no—
175C2×SD16—532no—
176C2×SD16C2×D5312yescomplemented
177C2×D8—532no—
178C2×D8—532no—
179C2×M16—531no—
180C2×M16C2×D5311yescharacteristic
181C4⋊Q8—531no—
182#127—531no—
183#124—532no—
184#124—532no—
185#124—531no—
186#124—531no—
187#124—1062no—
188#121—1062no—
189C4×D4—1062no—
190C2×C22⋊C4—1062no—
191C2×C42—531no—
192#97—532no—
193C4×C8—532no—
194C24—1061no—
195#41—22no—
196#41—22no—
197#41—1062no—
198#41—1062no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—531no—
203C2×Q8C22×D5312yes—
204C2×Q8C22×D5311yescharacteristic
205C2×D4—21no—
206C2×D4—532no—
207C2×D4—531no—
208C2×D4—1062no—
209C2×D4—1061no—
210C2×D4—1062no—
211C2×D4—1061no—
212C2×D4—1062no—
213C2×D4C22×D5311yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—532no—
217C22×C4—1061no—
218C22×C4C22×D5311yescharacteristic
219Q16—24no—
220Q16—1064no—
221SD16—24no—
222SD16—1064no—
223SD16—1064no—
224SD16—1064no—
225D8—1064no—
226D8—1064no—
227M16—534no—
228M16C22⋊D5314yescomplemented
229C2×C8—532no—
230C2×C8C22×D5312yes—
231C4⋊C4—1062no—
232C22⋊C4—1064no—
233C22⋊C4—1064no—
234C22⋊C4—1062no—
235C22⋊C4—1062no—
236C42—532no—
237C42—531no—
238C42—531no—
239C23—21no—
240C23—1062no—
241C23—1061no—
242C23—2124no—
243C23—2122no—
244C23D4×D5311yescharacteristic
245Q8—21no—
246Q8—22no—
247Q8—22no—
248Q8—21no—
249Q8—42no—
250Q8—1061no—
251Q8—1061no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—1062no—
256D4—1061no—
257D4—1061no—
258D4—1062no—
259D4—2124no—
260D4—2122no—
261D4—2122no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—532no—
267C2×C4—532no—
268C2×C4—1061no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4D4×D5311yescharacteristic
272C2×C4D4×D5312yes—
273C2×C4C2×C22⋊D5312yes—
274C2×C4C23×D5311yescharacteristic
275C8—534no—
276C8C2×C22⋊D5314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—4244no—
281C22—2121no—
282C22—2122no—
283C22—2122no—
284C22—2124no—
285C22#1313321312yes—
286C22C2×D4×D5311yescharacteristic
287C4—41no—
288C4—42no—
289C4—1062no—
290C4—1062no—
291C4—2121no—
292C4#1313321312yes—
293C4C2×D4×D5311yescharacteristic
294C4C22×C22⋊D5311yescharacteristic
295C2—22no—
296C2—81no—
297C2—4242no—
298C2—4241no—
299C2C2×#1313321311yescharacteristic
300C2#1681142212yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#137×C53C2
socleC22×C53C22×D4
hypercenterC2×#137D53
hypocenterC53#3153
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×#13133248C2semidirect
2. C2×#13133245C2semidirect
3. C2×#13133242C2semidirect
4. #16811169C2semidirect
5. C2×#13133206C2semidirect
6. #16811016C2semidirect
7. #16811009C2semidirect
8. #16811008C2semidirect
9. #16811004C2semidirect
10. M16×Dic53C2semidirect
11. C2×#137×C53C2semidirect
17. C2×#11719925C22semidirect
20. C2×#11719924C22semidirect
24. #13133215C22semidirect
25. #13133213C22semidirect
29. #13133206C22semidirect
30. #13133204C22semidirect
35. #13133127C22semidirect
36. C8×Dic53C22semidirect
40. C2×Q16×C53C22semidirect
41. C2×SD16×C53C22semidirect
70. M16×C53D4semidirect
84. #11719917D4semidirect
152. C2×#137D53semidirect
162. C53#3153semidirect
172. C2×Q16C2×D53semidirect
176. C2×SD16C2×D53semidirect
228. M16C22⋊D53semidirect