Finite Group Databasebeta

← #13125284

Subgroups — #13125284

Order 1664.

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×C13×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#11719537C211yesmaximal, characteristic, complemented
3C2×#88203C211yesmaximal, characteristic, complemented
4C2×#88194C211yesmaximal, characteristic, complemented
5C2×#88186C211yesmaximal, characteristic, complemented
6C2×#88162C211yesmaximal, characteristic, complemented
7#11718837C212yesmaximal, complemented
8#11718836C212yesmaximal, complemented
9#11718815C212yesmaximal, complemented
10#11718812C212yesmaximal, complemented
11M16×Dic13C211yesmaximal, characteristic, complemented
12C2×C13×#41C2211yescharacteristic
13C22×D4×C13C2211yescharacteristic
14C22×C22⋊D13—22no—
15C2×#6127C2211yescharacteristic
16C22×D52C2211yescharacteristic
17C13×AGL(1,ℤ/8)—24no—
18C2×C13×SD16C2212yescomplemented
19C2×C13×D8C2212yescomplemented
20C2×C13×M16C2211yescharacteristic
21#88203—24no—
22#88201C2211yescharacteristic, complemented
23#88198C2211yescharacteristic, complemented
24C2×#6106C2212yescomplemented
25#88194—22no—
26#88194C2212yes—
27#88194C2212yes—
28#88194C2211yescharacteristic, complemented
29#88193—22no—
30#88190C2211yescharacteristic, complemented
31C2×#6105C2212yescomplemented
32#88186—24no—
33C2×#6104C2212yescomplemented
34C2×#6104C2212yescomplemented
35C4×C22⋊D13—22no—
36C2×C4×Dic13C2211yescharacteristic
37C2×#6099C2211yescharacteristic
38#88162—24no—
39C2×D104C2212yescomplemented
40C2×#6095C2212yescomplemented
41#88055C2212yescomplemented
42C8×Dic13C2212yescomplemented
43C24×C13—21no—
44C23×D13—21no—
45C13×#41—22no—
46C13×#41—22no—
47C2×Q8×C13C2311yescharacteristic
48C2×D4×C13—21no—
49C2×D4×C13—22no—
50C2×D4×C13—22no—
51C2×D4×C13—21no—
52C2×D4×C13C2311yescharacteristic
53C2×D4×C13C2312yes—
54C22×C4×C13—21no—
55C22×C4×C13C2311yescharacteristic
56C2×C22⋊D13—21no—
57C2×C22⋊D13—22no—
58C2×C22⋊D13—24no—
59C2×C22⋊D13—24no—
60C2×C22⋊D13—48no—
61C22×Dic13D412yes—
62#6127—22no—
63#6127—22no—
64C2×D52—22no—
65C2×D52—22no—
66C2×D52—21no—
67C2×D52C2312yes—
68C2×D52C2311yescharacteristic
69C2×C4×D13—21no—
70C2×Dic26C2311yescharacteristic
71C13×SD16—24no—
72C13×D8—24no—
73C13×M16D414yescomplemented
74C2×C8×C13C2312yes—
75C22⋊Dic13—22no—
76#6106—24no—
77#6105—24no—
78#6104—24no—
79#6104—24no—
80#6103—22no—
81#6101—22no—
82C4×Dic13D412yes—
83C4×Dic13C2311yescharacteristic
84C4×Dic13C2311yescharacteristic
85#6099D414yescomplemented
86C2×#1055C2312yes—
87D104—24no—
88#6095—24no—
89#3151—131nomaximal, Sylow
90C23×C13—21no—
91C23×C13—22no—
92C23×C13—44no—
93C23×C13—42no—
94C23×C13C2×D411yescharacteristic
95C22×D13—21no—
96C22×D13—22no—
97C22×D13—44no—
98C22×D13—42no—
99Q8×C13—21no—
100Q8×C13—21no—
101D4×C13—21no—
102D4×C13—22no—
103D4×C13—21no—
104D4×C13—22no—
105D4×C13—42no—
106D4×C13—42no—
107C2×C4×C13—21no—
108C2×C4×C13—42no—
109C2×C4×C13C2×D411yescharacteristic
110C2×C4×C13C2×D412yes—
111C2×C4×C13C2×D412yes—
112C2×C4×C13C2411yescharacteristic
113C22⋊D13—48no—
114C22⋊D13—42no—
115C22⋊D13—44no—
116C22⋊D13—48no—
117C2×Dic13—24no—
118C2×Dic13—21no—
119C2×Dic13C2×D412yes—
120C2×Dic13C2×D412yes—
121D52—22no—
122D52—21no—
123D52—21no—
124D52—22no—
125D52—42no—
126C4×D13—42no—
127Dic26—21no—
128Dic26—21no—
129C8×C13C2×D414yescomplemented
130#1055C2×D414yescomplemented
131C2×AGL(1,ℤ/8)—131no—
132C2×AGL(1,ℤ/8)—131no—
133C2×AGL(1,ℤ/8)—131no—
134C2×AGL(1,ℤ/8)D1311yescharacteristic, complemented
135#532—131no—
136C2×#127—131no—
137#496—132no—
138#496—132no—
139#493—132no—
140#492—132no—
141C4×M16—131no—
142C22×C13—22no—
143C22×C13—41no—
144C22×C13—42no—
145C22×C13—44no—
146C22×C13—42no—
147C22×C13—84no—
148C22×C13#12712yes—
149C22×C13C22×D411yescharacteristic
150C2×D13—41no—
151C2×D13—42no—
152C2×D13—44no—
153C2×D13—42no—
154C2×D13—84no—
155C4×C13—41no—
156C4×C13#12712yes—
157C4×C13C22×D411yescharacteristic
158C4×C13C22×D411yescharacteristic
159Dic13—22no—
160Dic13—41no—
161Dic13AGL(1,ℤ/8)14yescomplemented
162C2×#41—131no—
163C2×#41C2×D1311yescharacteristic
164C22×D4—131no—
165C22×D4—262no—
166C22×D4C2×D1311yescharacteristic
167AGL(1,ℤ/8)—24no—
168AGL(1,ℤ/8)—264no—
169AGL(1,ℤ/8)—264no—
170AGL(1,ℤ/8)—264no—
171C2×SD16—132no—
172C2×SD16—132no—
173C2×SD16—132no—
174C2×SD16C2×D1312yescomplemented
175C2×D8—132no—
176C2×D8—132no—
177C2×D8—132no—
178C2×D8C2×D1312yescomplemented
179C2×M16—131no—
180C2×M16C2×D1311yescharacteristic
181C4⋊Q8—131no—
182#127—132no—
183#127—132no—
184#127—131no—
185#127—262no—
186#124—131no—
187#124—131no—
188#121—262no—
189C4×D4—262no—
190C2×C42—131no—
191#97—132no—
192C4×C8—132no—
193C2×C13—22no—
194C2×C13—81no—
195C2×C13—82no—
196C2×C13C2×#12711yescharacteristic
197C2×C13C2×AGL(1,ℤ/8)12yes—
198D13—81no—
199D13—82no—
200C24—21no—
201C24—261no—
202#41—22no—
203#41—22no—
204#41—262no—
205#41—262no—
206C2×Q8—131no—
207C2×Q8C22×D1311yescharacteristic
208C2×D4—21no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—22no—
212C2×D4—131no—
213C2×D4—132no—
214C2×D4—262no—
215C2×D4—264no—
216C2×D4—264no—
217C2×D4—261no—
218C2×D4—262no—
219C2×D4—261no—
220C2×D4—262no—
221C2×D4—528no—
222C2×D4C22×D1311yescharacteristic
223C2×D4C22×D1312yes—
224C22×C4—21no—
225C22×C4—132no—
226C22×C4—261no—
227C22×C4C22×D1311yescharacteristic
228SD16—24no—
229SD16—264no—
230SD16—264no—
231SD16—264no—
232D8—24no—
233D8—264no—
234D8—264no—
235D8—264no—
236M16—134no—
237M16C22⋊D1314yescomplemented
238C2×C8—132no—
239C2×C8C22×D1312yes—
240C4⋊C4—262no—
241C22⋊C4—262no—
242C22⋊C4—262no—
243C42—132no—
244C42—131no—
245C42—131no—
246C13#315111yescharacteristic, complemented, Sylow
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—261no—
252C23—262no—
253C23—524no—
254C23—522no—
255C23D4×D1311yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—261no—
259Q8—261no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—261no—
267D4—262no—
268D4—261no—
269D4—262no—
270D4—524no—
271D4—528no—
272D4—528no—
273D4—522no—
274D4—522no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—132no—
278C2×C4—132no—
279C2×C4—264no—
280C2×C4—261no—
281C2×C4—522no—
282C2×C4D4×D1311yescharacteristic
283C2×C4D4×D1312yes—
284C2×C4C2×C22⋊D1312yes—
285C2×C4C23×D1311yescharacteristic
286C8—134no—
287C8C2×C22⋊D1314yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—524no—
295C22—522no—
296C22—521no—
297C22—522no—
298C22—1044no—
299C22#8819412yes—
300C22C2×D4×D1311yescharacteristic
301C4—41no—
302C4—134no—
303C4—262no—
304C4—521no—
305C4#8819412yes—
306C4C2×D4×D1311yescharacteristic
307C4C22×C22⋊D1311yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—1042no—
312C2—1041no—
313C2D13×AGL(1,ℤ/8)12yes—
314C2C2×#8819411yescharacteristic
subgroupG/H
centerC22C2×D4×D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC2×C13×AGL(1,ℤ/8)C2
socleC22×C13C22×D4
hypercenterC2×AGL(1,ℤ/8)D13
hypocenterC13#3151
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
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×C13×AGL(1,ℤ/8)C2semidirect
2. #11719537C2semidirect
3. C2×#88203C2semidirect
4. C2×#88194C2semidirect
5. C2×#88186C2semidirect
6. C2×#88162C2semidirect
7. #11718837C2semidirect
8. #11718836C2semidirect
9. #11718815C2semidirect
10. #11718812C2semidirect
11. M16×Dic13C2semidirect
18. C2×C13×SD16C22semidirect
19. C2×C13×D8C22semidirect
22. #88201C22semidirect
23. #88198C22semidirect
24. C2×#6106C22semidirect
28. #88194C22semidirect
30. #88190C22semidirect
31. C2×#6105C22semidirect
33. C2×#6104C22semidirect
34. C2×#6104C22semidirect
39. C2×D104C22semidirect
40. C2×#6095C22semidirect
41. #88055C22semidirect
42. C8×Dic13C22semidirect
73. C13×M16D4semidirect
85. #6099D4semidirect
129. C8×C13C2×D4semidirect
130. #1055C2×D4semidirect
134. C2×AGL(1,ℤ/8)D13semidirect
161. Dic13AGL(1,ℤ/8)semidirect
174. C2×SD16C2×D13semidirect
178. C2×D8C2×D13semidirect
237. M16C22⋊D13semidirect
246. C13#3151semidirect
287. C8C2×C22⋊D13semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C13×AGL(1,ℤ/8)2
#117195372
C2×#882032
C2×#881942
C2×#881862
C2×#881622
#117188372
#117188362
#117188152
#117188122
M16×Dic132
#315113