Finite Group Databasebeta

← #13126654

Subgroups — #13126654

Order 1664.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C13×#554C211yesmaximal, characteristic, complemented
2D4×D52C211yesmaximal, characteristic
3#11718749C211yesmaximal, characteristic, complemented
4#11718745C211yesmaximal, characteristic, complemented
5#11718709C212yesmaximal, complemented
6#11718707C212yesmaximal, complemented
7#11718688C211yesmaximal, characteristic, complemented
8C4×#6106C211yesmaximal, characteristic, complemented
9D4×#1055C211yesmaximal, characteristic, complemented
10#11718629C212yesmaximal, complemented
11C2×#88049C212yesmaximal, complemented
12C2×D4×D13—21no—
13C22×D52C2212yes—
14C13×C4⋊Q8C2211yescharacteristic, complemented
15C13×#125—21no—
16C13×C22⋊Q8—22no—
17C13×C22⋊Q8C2212yescomplemented
18C4×Q8×C13C2211yescharacteristic, complemented
19C4×D4×C13—21no—
20C4×D4×C13C2211yescharacteristic
21C2×C13×C4⋊C4C2212yescomplemented
22C2×#6106C2211yescharacteristic
23C2×#6106C2212yescomplemented
24#88184C2212yes—
25C22×#1055C2212yescomplemented
26#88149—21no—
27#88136—22no—
28#88128C2211yescharacteristic
29C4×D52C2211yescharacteristic
30#88075C2211yescharacteristic, complemented
31C22⋊#1055C2212yescomplemented
32#88049C2211yescharacteristic
33#88049C2212yes—
34#88049C2212yescomplemented
35#88049C2212yescomplemented
36#88049C2212yescomplemented
37#88048C2211yescharacteristic, complemented
38#88048C2212yescomplemented
39C4⋊#1055C2211yescharacteristic, complemented
40C4×#1055C2211yescharacteristic, complemented
41C23×D13—22no—
42C2×Q8×C13—21no—
43C2×Q8×C13C2311yescharacteristic
44C2×D4×C13D411yescharacteristic
45C22×C4×C13—22no—
46C22×C4×C13C2312yes—
47C2×C22⋊D13—22no—
48D4×D13—44no—
49C2×D52—24no—
50C2×D52—21no—
51C2×D52—22no—
52C2×D52—21no—
53C2×D52—42no—
54C2×D52C2312yes—
55C2×D52C2311yescharacteristic
56C2×C4×D13—21no—
57C13×C4⋊C4—22no—
58C13×C4⋊C4—21no—
59C13×C4⋊C4—21no—
60C13×C4⋊C4—22no—
61C13×C4⋊C4D411yescharacteristic
62C13×C4⋊C4C2311yescharacteristic
63C13×C4⋊C4C2312yes—
64C13×C22⋊C4—22no—
65C13×C22⋊C4D412yes—
66C42×C13—21no—
67C42×C13C2311yescharacteristic
68#6106—22no—
69#6106—42no—
70#6103—21no—
71#6103—22no—
72C4⋊Dic13C2311yescharacteristic
73C2×#1055—42no—
74C2×#1055C2311yescharacteristic
75C2×#1055C2312yes—
76C2×#1055C2311yescharacteristic
77#3295—131nomaximal, Sylow
78C23×C13C2×D412yes—
79C22×D13—21no—
80C22×D13—22no—
81C22×D13—44no—
82C22×D13—44no—
83C22×D13—82no—
84Q8×C13—21no—
85Q8×C13—21no—
86Q8×C13—81no—
87D4×C13SD1614yescomplemented
88C2×C4×C13—22no—
89C2×C4×C13—22no—
90C2×C4×C13—21no—
91C2×C4×C13—21no—
92C2×C4×C13—22no—
93C2×C4×C13—82no—
94C2×C4×C13C2×D411yescharacteristic
95C2×C4×C13C2×D412yes—
96C2×C4×C13C2×D411yescharacteristic
97C2×C4×C13C2411yescharacteristic
98C22⋊D13—44no—
99C2×Dic13—21no—
100D52—22no—
101D52—21no—
102D52—22no—
103D52—44no—
104D52—42no—
105D52—81no—
106D52#4112yes—
107C4×D13—42no—
108#1055—21no—
109#1055—21no—
110#1055—22no—
111#554D1311yescharacteristic, complemented
112D42—131no—
113#486—131no—
114#481—132no—
115#476—131no—
116#465—132no—
117#460—131no—
118#450—132no—
119C4×SD16—131no—
120C8×D4—131no—
121C2×#102—132no—
122C22×C13—22no—
123C22×C13—22no—
124C22×C13C2×SD1614yes—
125C22×C13C22×D411yescharacteristic
126C2×D13—22no—
127C2×D13—41no—
128C2×D13—41no—
129C2×D13—42no—
130C2×D13—42no—
131C2×D13—42no—
132C2×D13—88no—
133C4×C13—22no—
134C4×C13—21no—
135C4×C13—41no—
136C4×C13—41no—
137C4×C13—42no—
138C4×C13C2×SD1612yes—
139C4×C13C2×#4111yescharacteristic
140C4×C13H(5,𝔽2)11yescharacteristic
141Dic13—41no—
142C22×D4—132no—
143C22×D4—261no—
144C2×SD16—131no—
145C2×SD16—132no—
146C22×C8—132no—
147C4⋊Q8C2×D1311yescharacteristic, complemented
148#127—131no—
149#125—21no—
150C22⋊Q8—22no—
151C22⋊Q8C2×D1312yescomplemented
152#121—132no—
153#121—261no—
154#120—262no—
155C4×Q8C2×D1311yescharacteristic, complemented
156C4×D4—21no—
157C4×D4—131no—
158C4×D4C2×D1311yescharacteristic
159C2×C4⋊C4C2×D1312yescomplemented
160#106—131no—
161#106—132no—
162C4⋊C8—131no—
163#103—131no—
164#102—131no—
165#102—132no—
166#102—132no—
167#102—132no—
168#102—132no—
169C22⋊C8—132no—
170C4×C8—131no—
171C2×C13—24no—
172C2×C13#54611yescharacteristic
173C2×C13C22×SD1611yescharacteristic
174C2×C13#57611yescharacteristic
175D13—42no—
176D13—82no—
177C24—262no—
178C2×Q8—21no—
179C2×Q8C22×D1311yescharacteristic
180C2×D4—131no—
181C2×D4—132no—
182C2×D4—264no—
183C2×D4—262no—
184C2×D4—262no—
185C2×D4—261no—
186C2×D4—261no—
187C2×D4—522no—
188C2×D4—524no—
189C2×D4C22⋊D1311yescharacteristic
190C22×C4—22no—
191C22×C4—261no—
192C22×C4C22×D1312yes—
193SD16—262no—
194SD16—522no—
195C2×C8—131no—
196C2×C8—131no—
197C2×C8—132no—
198C2×C8—522no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—131no—
204C4⋊C4C22⋊D1311yescharacteristic
205C4⋊C4C22×D1311yescharacteristic
206C4⋊C4C22×D1312yes—
207C22⋊C4—22no—
208C22⋊C4—261no—
209C22⋊C4—262no—
210C22⋊C4C22⋊D1312yes—
211C42—21no—
212C42C22×D1311yescharacteristic
213C13#329511yescharacteristic, complemented, Sylow
214C23—261no—
215C23—262no—
216C23—524no—
217C23—524no—
218C23—1042no—
219C23C2×C22⋊D1312yes—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223D4—132no—
224D4—262no—
225D4—261no—
226D4—262no—
227D4—524no—
228D4—524no—
229D4—522no—
230D4—1041no—
231D4#610614yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—82no—
238C2×C4—261no—
239C2×C4—522no—
240C2×C4C2×C22⋊D1311yescharacteristic
241C2×C4C2×C22⋊D1312yes—
242C2×C4C2×C22⋊D1311yescharacteristic
243C2×C4C23×D1311yescharacteristic
244C8—261no—
245C8—261no—
246C8—262no—
247C22—22no—
248C22—22no—
249C22—262no—
250C22—521no—
251C22—521no—
252C22—522no—
253C22—522no—
254C22—522no—
255C22—1048no—
256C22C2×#610614yes—
257C22C22×C22⋊D1311yescharacteristic
258C4—22no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—42no—
263C4—521no—
264C4C2×#610612yes—
265C4#41×D1311yescharacteristic
266C4#8825611yescharacteristic
267C2—24no—
268C2—522no—
269C2—1042no—
270C2C22×#610611yescharacteristic
271C2#1171953411yescharacteristic
272C2#1171954011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#554C2
socleC22×C13C22×D4
prime radicalD4×D52C2
hypercenter#554D13
hypocenterC13#3295
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
1. C13×#554C2semidirect
3. #11718749C2semidirect
4. #11718745C2semidirect
5. #11718709C2semidirect
6. #11718707C2semidirect
7. #11718688C2semidirect
8. C4×#6106C2semidirect
9. D4×#1055C2semidirect
10. #11718629C2semidirect
11. C2×#88049C2semidirect
14. C13×C4⋊Q8C22semidirect
17. C13×C22⋊Q8C22semidirect
18. C4×Q8×C13C22semidirect
21. C2×C13×C4⋊C4C22semidirect
23. C2×#6106C22semidirect
25. C22×#1055C22semidirect
30. #88075C22semidirect
31. C22⋊#1055C22semidirect
34. #88049C22semidirect
35. #88049C22semidirect
36. #88049C22semidirect
37. #88048C22semidirect
38. #88048C22semidirect
39. C4⋊#1055C22semidirect
40. C4×#1055C22semidirect
87. D4×C13SD16semidirect
111. #554D13semidirect
147. C4⋊Q8C2×D13semidirect
151. C22⋊Q8C2×D13semidirect
155. C4×Q8C2×D13semidirect
159. C2×C4⋊C4C2×D13semidirect
213. C13#3295semidirect
231. D4#6106semidirect

Maximal subgroups

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

subgroupindex
C13×#5542
D4×D522
#117187492
#117187452
#117187092
#117187072
#117186882
C4×#61062
D4×#10552
#117186292
C2×#880492
#329513