Finite Group Databasebeta

← #13125321

Subgroups — #13125321

Order 1664.

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

normal subgroups only

This group has 596 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×C13×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#11719535C211yesmaximal, characteristic, complemented
3C2×#88204C211yesmaximal, characteristic, complemented
4#11719510C211yesmaximal, characteristic, complemented
5C22×#6104C211yesmaximal, characteristic, complemented
6C2×#88162C211yesmaximal, characteristic, complemented
7#11718837C212yesmaximal, complemented
8#11718829C212yesmaximal, complemented
9#11718815C212yesmaximal, complemented
10#11718812C212yesmaximal, complemented
11#11718783C211yesmaximal, characteristic, complemented
12C2×C13×#41C2211yescharacteristic
13C22×D4×C13C2211yescharacteristic
14C2×#6127C2211yescharacteristic
15C22×D52C2211yescharacteristic
16C13×AGL(1,ℤ/8)—24no—
17C2×C13×SD16C2212yescomplemented
18C2×C13×D8C2212yescomplemented
19C2×C13×M16C2211yescharacteristic
20#88207—22no—
21#88204—24no—
22#88201C2212yescomplemented
23C2×#6107C2211yescharacteristic, complemented
24C2×#6106C2211yescharacteristic, complemented
25#88194C2212yescomplemented
26#88194C2212yescomplemented
27#88192—22no—
28#88191—22no—
29#88190C2212yescomplemented
30C2×#6105C2211yescharacteristic, complemented
31C2×#6104—24no—
32C2×#6104C2212yes—
33C2×#6104C2212yes—
34C2×#6104C2211yescharacteristic, complemented
35#88184—22no—
36#88180C2211yescharacteristic
37C22×#1055C2211yescharacteristic
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×C13C2312yes—
53C2×D4×C13C2311yescharacteristic
54C22×C4×C13—21no—
55C22×C4×C13C2311yescharacteristic
56C2×C22⋊D13—21no—
57C2×C22⋊D13—24no—
58C2×C22⋊D13—24no—
59#6127—22no—
60#6127—22no—
61C2×D52—22no—
62C2×D52—22no—
63C2×D52—21no—
64C2×D52C2312yes—
65C2×D52C2311yescharacteristic
66C2×C4×D13—21no—
67C2×Dic26C2311yescharacteristic
68C13×SD16—24no—
69C13×D8—24no—
70C13×M16D414yescomplemented
71C2×C8×C13C2312yes—
72C22⋊Dic13—24no—
73C22⋊Dic13D412yes—
74#6107—22no—
75#6106—22no—
76#6105—22no—
77#6104—24no—
78#6104—24no—
79#6104—22no—
80#6103—24no—
81C4⋊Dic13D412yes—
82C4×Dic13C2312yes—
83C2×#1055D414yescomplemented
84C2×#1055C2311yescharacteristic
85C2×#1055C2311yescharacteristic
86D104—24no—
87#6095—24no—
88#3155—131nomaximal, Sylow
89C23×C13—22no—
90C23×C13—21no—
91C23×C13—42no—
92C23×C13—82no—
93C23×C13C2×D411yescharacteristic
94C22×D13—22no—
95C22×D13—21no—
96C22×D13—42no—
97C22×D13—82no—
98Q8×C13—21no—
99Q8×C13—21no—
100D4×C13—22no—
101D4×C13—21no—
102D4×C13—21no—
103D4×C13—22no—
104D4×C13—42no—
105D4×C13—42no—
106C2×C4×C13—21no—
107C2×C4×C13—42no—
108C2×C4×C13C2×D411yescharacteristic
109C2×C4×C13C2×D412yes—
110C2×C4×C13C2×D412yes—
111C2×C4×C13C2411yescharacteristic
112C22⋊D13—42no—
113C22⋊D13—84no—
114C22⋊D13—84no—
115C2×Dic13—21no—
116C2×Dic13C2×D414yes—
117D52—22no—
118D52—21no—
119D52—22no—
120D52—21no—
121D52—42no—
122C4×D13—42no—
123Dic26—21no—
124Dic26—21no—
125C8×C13C2×D414yes—
126#1055C2×D412yes—
127#1055C2×D412yes—
128C2×AGL(1,ℤ/8)—131no—
129C2×AGL(1,ℤ/8)D1311yescharacteristic, complemented
130C2×#135—131no—
131C22×D8—131no—
132#535—131no—
133#535—131no—
134#496—132no—
135#496—132no—
136#495—132no—
137#493—132no—
138#405—131no—
139C22×C13—22no—
140C22×C13—41no—
141C22×C13—42no—
142C22×C13—42no—
143C22×C13—44no—
144C22×C13—84no—
145C22×C13#12712yes—
146C22×C13C22×D411yescharacteristic
147C2×D13—41no—
148C2×D13—42no—
149C2×D13—42no—
150C2×D13—44no—
151C2×D13—84no—
152C4×C13—41no—
153C4×C13#12712yes—
154C4×C13C22×D411yescharacteristic
155C4×C13C22×D411yescharacteristic
156Dic13—24no—
157Dic13—41no—
158C2×#41—131no—
159C2×#41C2×D1311yescharacteristic
160C22×D4—131no—
161C22×D4C2×D1311yescharacteristic
162AGL(1,ℤ/8)—24no—
163AGL(1,ℤ/8)—264no—
164#135—264no—
165C2×Q16—131no—
166C2×SD16—131no—
167C2×SD16—131no—
168C2×SD16—132no—
169C2×SD16C2×D1312yescomplemented
170C2×D8—132no—
171C2×D8—132no—
172C2×D8—131no—
173C2×D8—132no—
174C2×D8—264no—
175C2×D8C2×D1312yescomplemented
176C2×M16C2×D1311yescharacteristic
177C22×C8—131no—
178#127—132no—
179#127—132no—
180#124—132no—
181#124—132no—
182#121—262no—
183#121—262no—
184#120—262no—
185#120—262no—
186#117—131no—
187#97—132no—
188C4×C8—132no—
189C2×C13—22no—
190C2×C13—81no—
191C2×C13—82no—
192C2×C13C2×#12711yescharacteristic
193C2×C13#57612yes—
194D13—81no—
195D13—82no—
196C24—21no—
197C24—261no—
198#41—22no—
199#41—22no—
200#41—262no—
201#41—262no—
202C2×Q8—131no—
203C2×Q8C22×D1311yescharacteristic
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—132no—
209C2×D4—131no—
210C2×D4—264no—
211C2×D4—264no—
212C2×D4—261no—
213C2×D4—262no—
214C2×D4—261no—
215C2×D4—262no—
216C2×D4C22×D1312yes—
217C2×D4C22×D1311yescharacteristic
218C22×C4—21no—
219C22×C4—261no—
220C22×C4C22×D1311yescharacteristic
221Q16—262no—
222SD16—24no—
223SD16—262no—
224SD16—262no—
225SD16—264no—
226D8—24no—
227D8—264no—
228D8—264no—
229D8—262no—
230D8—264no—
231M16C22⋊D1314yescomplemented
232C2×C8—134no—
233C2×C8—131no—
234C2×C8—131no—
235C2×C8C22×D1312yes—
236C4⋊C4—132no—
237C22⋊C4—132no—
238C22⋊C4—264no—
239C22⋊C4—264no—
240C42—132no—
241C13#315511yescharacteristic, complemented, Sylow
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—262no—
247C23—261no—
248C23—522no—
249C23—1042no—
250C23D4×D1311yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—261no—
254Q8—261no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—262no—
262D4—261no—
263D4—261no—
264D4—262no—
265D4—522no—
266D4—522no—
267D4—1044no—
268D4—1044no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—134no—
272C2×C4—261no—
273C2×C4—522no—
274C2×C4D4×D1311yescharacteristic
275C2×C4D4×D1312yes—
276C2×C4C2×C22⋊D1312yes—
277C2×C4C23×D1311yescharacteristic
278C8—132no—
279C8—132no—
280C8C2×C22⋊D1314yes—
281C22—22no—
282C22—42no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—84no—
287C22—522no—
288C22—524no—
289C22—521no—
290C22—522no—
291C22—1044no—
292C22#8819412yes—
293C22C2×D4×D1311yescharacteristic
294C4—41no—
295C4—264no—
296C4—521no—
297C4#8819412yes—
298C4C2×D4×D1311yescharacteristic
299C4C22×C22⋊D1311yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—1042no—
304C2—1041no—
305C2#1171947912yes—
306C2C2×#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#3155
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. #11719535C2semidirect
3. C2×#88204C2semidirect
4. #11719510C2semidirect
5. C22×#6104C2semidirect
6. C2×#88162C2semidirect
7. #11718837C2semidirect
8. #11718829C2semidirect
9. #11718815C2semidirect
10. #11718812C2semidirect
11. #11718783C2semidirect
17. C2×C13×SD16C22semidirect
18. C2×C13×D8C22semidirect
22. #88201C22semidirect
23. C2×#6107C22semidirect
24. C2×#6106C22semidirect
25. #88194C22semidirect
26. #88194C22semidirect
29. #88190C22semidirect
30. C2×#6105C22semidirect
34. C2×#6104C22semidirect
39. C2×D104C22semidirect
40. C2×#6095C22semidirect
41. #88055C22semidirect
42. C8×Dic13C22semidirect
70. C13×M16D4semidirect
83. C2×#1055D4semidirect
129. C2×AGL(1,ℤ/8)D13semidirect
169. C2×SD16C2×D13semidirect
175. C2×D8C2×D13semidirect
231. M16C22⋊D13semidirect
241. C13#3155semidirect

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
#117195352
C2×#882042
#117195102
C22×#61042
C2×#881622
#117188372
#117188292
#117188152
#117188122
#117187832
#315513