Finite Group Databasebeta

← #20285163

Subgroups — #20285163

Order 6784.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#13133243C211yesmaximal, characteristic, complemented
3C2×#13133242C211yesmaximal, characteristic, complemented
4#16811169C211yesmaximal, characteristic, complemented
5C2×#13133213C211yesmaximal, characteristic, complemented
6#16811011C212yesmaximal, complemented
7#16811009C212yesmaximal, complemented
8#16810998C212yesmaximal, complemented
9#16810994C212yesmaximal, complemented
10M16×Dic53C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C53C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C22×D212C2211yescharacteristic
14C22×C22⋊D53—22no—
15#13133245—24no—
16#13133243—24no—
17#13133242—24no—
18C2×#11719913C2212yescomplemented
19C2×#11719923C2212yescomplemented
20C2×#11719924C2212yescomplemented
21C2×D424C2212yescomplemented
22C2×#11719922C2212yescomplemented
23C2×#11719922C2212yescomplemented
24C2×#11719917C2211yescharacteristic
25#13133215C2211yescharacteristic, complemented
26#13133213—22no—
27#13133213C2212yes—
28#13133213C2212yes—
29#13133213C2211yescharacteristic, complemented
30#13133206C2211yescharacteristic, complemented
31#13133204C2211yescharacteristic, complemented
32#13133189—22no—
33C4×C22⋊D53—22no—
34C2×C4×Dic53C2211yescharacteristic
35#13133127C2212yescomplemented
36C8×Dic53C2212yescomplemented
37C2×#41×C53C2211yescharacteristic
38C22×D4×C53C2211yescharacteristic
39AGL(1,ℤ/8)×C53—24no—
40C2×SD16×C53C2212yescomplemented
41C2×D8×C53C2212yescomplemented
42C2×M16×C53C2211yescharacteristic
43C24×C53—21no—
44C23×D53—21no—
45#41×C53—22no—
46#41×C53—22no—
47C2×Q8×C53C2311yescharacteristic
48C2×D4×C53—21no—
49C2×D4×C53—22no—
50C2×D4×C53—22no—
51C2×D4×C53—21no—
52C2×D4×C53C2311yescharacteristic
53C2×D4×C53C2312yes—
54C22×C4×C53—21no—
55C22×C4×C53C2311yescharacteristic
56C2×C22⋊D53—21no—
57C2×C22⋊D53—22no—
58C2×C22⋊D53—24no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—48no—
61C22×Dic53D412yes—
62#11719945—22no—
63#11719945—22no—
64C2×D212—22no—
65C2×D212—22no—
66C2×D212—21no—
67C2×D212C2311yescharacteristic
68C2×D212C2312yes—
69C2×C4×D53—21no—
70C2×Dic106C2311yescharacteristic
71SD16×C53—24no—
72D8×C53—24no—
73M16×C53D414yescomplemented
74C2×C8×C53C2312yes—
75C22⋊Dic53—22no—
76#11719924—24no—
77#11719923—24no—
78#11719922—24no—
79#11719922—24no—
80#11719921—22no—
81#11719919—22no—
82C4×Dic53D412yes—
83C4×Dic53C2311yescharacteristic
84C4×Dic53C2311yescharacteristic
85#11719917D414yescomplemented
86C2×#88322C2312yes—
87D424—24no—
88#11719913—24no—
89C23×C53—22no—
90C23×C53—21no—
91C23×C53—44no—
92C23×C53—42no—
93C23×C53C2×D411yescharacteristic
94C22×D53—21no—
95C22×D53—22no—
96C22×D53—44no—
97C22×D53—42no—
98Q8×C53—21no—
99Q8×C53—21no—
100D4×C53—22no—
101D4×C53—22no—
102D4×C53—21no—
103D4×C53—21no—
104D4×C53—42no—
105D4×C53—42no—
106C2×C4×C53—21no—
107C2×C4×C53—42no—
108C2×C4×C53C2×D411yescharacteristic
109C2×C4×C53C2×D412yes—
110C2×C4×C53C2×D412yes—
111C2×C4×C53C2411yescharacteristic
112C22⋊D53—48no—
113C22⋊D53—42no—
114C22⋊D53—44no—
115C22⋊D53—48no—
116C2×Dic53—24no—
117C2×Dic53—21no—
118C2×Dic53C2×D412yes—
119C2×Dic53C2×D412yes—
120D212—22no—
121D212—22no—
122D212—21no—
123D212—21no—
124D212—42no—
125C4×D53—42no—
126Dic106—21no—
127Dic106—21no—
128C8×C53C2×D414yescomplemented
129#88322C2×D414yescomplemented
130C22×C53—22no—
131C22×C53—41no—
132C22×C53—42no—
133C22×C53—44no—
134C22×C53—42no—
135C22×C53—84no—
136C22×C53#12712yes—
137C22×C53C22×D411yescharacteristic
138C2×D53—41no—
139C2×D53—42no—
140C2×D53—44no—
141C2×D53—42no—
142C2×D53—84no—
143C4×C53—41no—
144C4×C53#12712yes—
145C4×C53C22×D411yescharacteristic
146C4×C53C22×D411yescharacteristic
147Dic53—22no—
148Dic53—41no—
149Dic53AGL(1,ℤ/8)14yescomplemented
150#3151—531nomaximal, Sylow
151C2×C53—22no—
152C2×C53—81no—
153C2×C53—82no—
154C2×C53C2×#12711yescharacteristic
155C2×C53C2×AGL(1,ℤ/8)12yes—
156D53—81no—
157D53—82no—
158C2×AGL(1,ℤ/8)—531no—
159C2×AGL(1,ℤ/8)—531no—
160C2×AGL(1,ℤ/8)—531no—
161C2×AGL(1,ℤ/8)D5311yescharacteristic, complemented
162#532—531no—
163C2×#127—531no—
164#496—532no—
165#496—532no—
166#493—532no—
167#492—532no—
168C4×M16—531no—
169C53#315111yescharacteristic, complemented, Sylow
170C2×#41—531no—
171C2×#41C2×D5311yescharacteristic
172C22×D4—531no—
173C22×D4—1062no—
174C22×D4C2×D5311yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—1064no—
177AGL(1,ℤ/8)—1064no—
178AGL(1,ℤ/8)—1064no—
179C2×SD16—532no—
180C2×SD16—532no—
181C2×SD16—532no—
182C2×SD16C2×D5312yescomplemented
183C2×D8—532no—
184C2×D8—532no—
185C2×D8—532no—
186C2×D8C2×D5312yescomplemented
187C2×M16—531no—
188C2×M16C2×D5311yescharacteristic
189C4⋊Q8—531no—
190#127—532no—
191#127—532no—
192#127—531no—
193#127—1062no—
194#124—531no—
195#124—531no—
196#121—1062no—
197C4×D4—1062no—
198C2×C42—531no—
199#97—532no—
200C4×C8—532no—
201C24—21no—
202C24—1061no—
203#41—22no—
204#41—22no—
205#41—1062no—
206#41—1062no—
207C2×Q8—531no—
208C2×Q8C22×D5311yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—531no—
214C2×D4—532no—
215C2×D4—1062no—
216C2×D4—1064no—
217C2×D4—1064no—
218C2×D4—1061no—
219C2×D4—1062no—
220C2×D4—1061no—
221C2×D4—1062no—
222C2×D4—2128no—
223C2×D4C22×D5311yescharacteristic
224C2×D4C22×D5312yes—
225C22×C4—21no—
226C22×C4—532no—
227C22×C4—1061no—
228C22×C4C22×D5311yescharacteristic
229SD16—24no—
230SD16—1064no—
231SD16—1064no—
232SD16—1064no—
233D8—24no—
234D8—1064no—
235D8—1064no—
236D8—1064no—
237M16—534no—
238M16C22⋊D5314yescomplemented
239C2×C8—532no—
240C2×C8C22×D5312yes—
241C4⋊C4—1062no—
242C22⋊C4—1062no—
243C22⋊C4—1062no—
244C42—532no—
245C42—531no—
246C42—531no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—1061no—
252C23—1062no—
253C23—2124no—
254C23—2122no—
255C23D4×D5311yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—1061no—
259Q8—1061no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—1061no—
267D4—1062no—
268D4—1061no—
269D4—1062no—
270D4—2124no—
271D4—2128no—
272D4—2128no—
273D4—2122no—
274D4—2122no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—532no—
278C2×C4—532no—
279C2×C4—1064no—
280C2×C4—1061no—
281C2×C4—2122no—
282C2×C4D4×D5311yescharacteristic
283C2×C4D4×D5312yes—
284C2×C4C2×C22⋊D5312yes—
285C2×C4C23×D5311yescharacteristic
286C8—534no—
287C8C2×C22⋊D5314yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—4244no—
295C22—2124no—
296C22—2122no—
297C22—2121no—
298C22—2122no—
299C22#1313321312yes—
300C22C2×D4×D5311yescharacteristic
301C4—41no—
302C4—534no—
303C4—1062no—
304C4—2121no—
305C4#1313321312yes—
306C4C2×D4×D5311yescharacteristic
307C4C22×C22⋊D5311yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—4242no—
312C2—4241no—
313C2C2×#1313321311yescharacteristic
314C2AGL(1,ℤ/8)×D5312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×AGL(1,ℤ/8)×C53C2
socleC22×C53C22×D4
hypercenterC2×AGL(1,ℤ/8)D53
hypocenterC53#3151
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#13133245C2semidirect
2. C2×#13133243C2semidirect
3. C2×#13133242C2semidirect
4. #16811169C2semidirect
5. C2×#13133213C2semidirect
6. #16811011C2semidirect
7. #16811009C2semidirect
8. #16810998C2semidirect
9. #16810994C2semidirect
10. M16×Dic53C2semidirect
11. C2×AGL(1,ℤ/8)×C53C2semidirect
18. C2×#11719913C22semidirect
19. C2×#11719923C22semidirect
20. C2×#11719924C22semidirect
21. C2×D424C22semidirect
22. C2×#11719922C22semidirect
23. C2×#11719922C22semidirect
25. #13133215C22semidirect
29. #13133213C22semidirect
30. #13133206C22semidirect
31. #13133204C22semidirect
35. #13133127C22semidirect
36. C8×Dic53C22semidirect
40. C2×SD16×C53C22semidirect
41. C2×D8×C53C22semidirect
73. M16×C53D4semidirect
85. #11719917D4semidirect
128. C8×C53C2×D4semidirect
129. #88322C2×D4semidirect
149. Dic53AGL(1,ℤ/8)semidirect
161. C2×AGL(1,ℤ/8)D53semidirect
169. C53#3151semidirect
182. C2×SD16C2×D53semidirect
186. C2×D8C2×D53semidirect
238. M16C22⋊D53semidirect
287. C8C2×C22⋊D53semidirect