Finite Group Databasebeta

← #13124632

Subgroups — #13124632

Order 1664.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#11719222C211yesmaximal, characteristic, complemented
2C2×#88147C211yesmaximal, characteristic
3C13×#428C211yesmaximal, characteristic, complemented
4#11718792C211yesmaximal, characteristic
5#11718779C211yesmaximal, characteristic, complemented
6#11718662C211yesmaximal, characteristic
7#11718625C211yesmaximal, characteristic, complemented
8#11718552C212yesmaximal, complemented
9#11718551C212yesmaximal, complemented
10#11718532C212yesmaximal, complemented
11#11718531C212yesmaximal, complemented
12C22×D52C2211yescharacteristic
13C22×C4×D13—21no—
14C2×C13×M16C2211yescharacteristic
15C13×#117C2211yescharacteristic
16C2×C13×C4⋊C4C2211yescharacteristic
17C2×#6103—21no—
18#88180C2211yescharacteristic
19C2×C4×Dic13C2211yescharacteristic
20C2×#6099C2211yescharacteristic
21#88147—21no—
22#88147—21no—
23#88147C2212yes—
24#88147C2212yes—
25#88144C2211yescharacteristic, complemented
26#88144C2211yescharacteristic, complemented
27#88142C2211yescharacteristic, complemented
28#88142C2211yescharacteristic, complemented
29#88132—21no—
30#88132—21no—
31C4×Dic26—21no—
32C4×Dic26—21no—
33C13×#107C2212yescomplemented
34C13×#106C2212yescomplemented
35#88061C2212yes—
36#88061C2212yes—
37#88054C2212yescomplemented
38#88054C2212yescomplemented
39#88049C2212yes—
40#88049C2212yes—
41#88048C2212yescomplemented
42#88047C2212yescomplemented
43C23×D13—21no—
44C22×C4×C13—21no—
45C22×C4×C13C2311yescharacteristic
46C22×Dic13D411yescharacteristic
47C22×Dic13D411yescharacteristic
48C2×D52—21no—
49C2×D52—22no—
50C2×D52Q814yescomplemented
51C2×D52C2312yes—
52C2×C4×D13—22no—
53C2×C4×D13—22no—
54C2×C4×D13—44no—
55C2×Dic26—21no—
56C2×Dic26—21no—
57C13×M16—22no—
58C2×C8×C13C2312yes—
59C13×C4⋊C4—21no—
60C13×C4⋊C4C2311yescharacteristic
61C13×C4⋊C4C2312yes—
62C13×C4⋊C4C2311yescharacteristic
63C13×C22⋊C4—21no—
64C42×C13—21no—
65C22⋊Dic13—21no—
66#6103—22no—
67#6103—22no—
68C4⋊Dic13—21no—
69C4⋊Dic13—21no—
70C4⋊Dic13C2311yescharacteristic
71C4⋊Dic13C2311yescharacteristic
72#6101—21no—
73#6101—21no—
74#6101—21no—
75#6101—21no—
76C4×Dic13—21no—
77C4×Dic13D411yescharacteristic
78C4×Dic13D411yescharacteristic
79C4×Dic13C2311yescharacteristic
80C4×Dic13C2311yescharacteristic
81#6099—22no—
82C2×#1055C2312yes—
83#3077—131nomaximal, Sylow
84C23×C13C2×D411yescharacteristic
85C22×D13—24no—
86C22×D13—22no—
87C22×D13—42no—
88C22×D13—42no—
89C2×C4×C13—21no—
90C2×C4×C13—21no—
91C2×C4×C13—22no—
92C2×C4×C13—42no—
93C2×C4×C13C2×D411yescharacteristic
94C2×C4×C13C2×Q812yes—
95C2×C4×C13#4112yes—
96C2×C4×C13C2411yescharacteristic
97C2×Dic13—22no—
98C2×Dic13—21no—
99C2×Dic13—21no—
100C2×Dic13—41no—
101C2×Dic13C2×D411yescharacteristic
102C2×Dic13C2×D411yescharacteristic
103C2×Dic13C2×D411yescharacteristic
104C2×Dic13C2×D411yescharacteristic
105D52—24no—
106D52—22no—
107D52C2×Q814yes—
108C4×D13—44no—
109C4×D13—44no—
110Dic26—42no—
111Dic26—42no—
112C8×C13—22no—
113#1055—22no—
114#533—131no—
115C2×C4×D4—131no—
116#478—132no—
117#478—132no—
118#476—132no—
119#474—132no—
120#428—131no—
121#428D1311yescharacteristic, complemented
122#423—131no—
123#418—131no—
124#418—131no—
125C22×C13—21no—
126C22×C13—21no—
127C22×C13C22⋊Q812yes—
128C22×C13C22×D411yescharacteristic
129C2×D13—24no—
130C2×D13—48no—
131C2×D13—42no—
132C2×D13—42no—
133C4×C13—41no—
134C4×C13—42no—
135C4×C13—41no—
136C4×C13C22⋊Q812yes—
137C4×C13C22×Q811yescharacteristic
138C4×C13C2×#4111yescharacteristic
139Dic13—21no—
140Dic13—21no—
141Dic13—21no—
142Dic13—41no—
143Dic13—41no—
144Dic13AGL(1,ℤ/8)12yes—
145C22×D4—131no—
146C23×C4—261no—
147C2×M16—131no—
148C2×M16C2×D1311yescharacteristic
149C4⋊Q8—131no—
150C4⋊Q8—131no—
151#125—131no—
152#125—131no—
153C22⋊Q8—261no—
154C22⋊Q8—261no—
155C4×Q8—261no—
156C4×Q8—261no—
157C4×D4—132no—
158C4×D4—132no—
159C4×D4—261no—
160C4×D4—261no—
161#117—131no—
162#117C2×D1311yescharacteristic
163C2×C4⋊C4C2×D1311yescharacteristic
164C2×C22⋊C4—261no—
165C2×C42—131no—
166#107—132no—
167#107C2×D1312yescomplemented
168#106—132no—
169#106C2×D1312yescomplemented
170C4⋊C8—132no—
171C4⋊C8—132no—
172#102—132no—
173#102—132no—
174#102—132no—
175#102—132no—
176C2×C13—22no—
177C2×C13C2×C22⋊Q811yescharacteristic
178C2×C13C2×AGL(1,ℤ/8)11yescharacteristic
179C2×C13#57511yescharacteristic
180D13—44no—
181C24—261no—
182C2×Q8—261no—
183C2×Q8—261no—
184C2×D4—134no—
185C2×D4—132no—
186C2×D4—262no—
187C2×D4—261no—
188C22×C4—21no—
189C22×C4—131no—
190C22×C4—131no—
191C22×C4—262no—
192C22×C4—262no—
193C22×C4—524no—
194C22×C4C22×D1311yescharacteristic
195M16—22no—
196M16—262no—
197C2×C8—132no—
198C2×C8C22×D1312yes—
199C4⋊C4—21no—
200C4⋊C4—131no—
201C4⋊C4—131no—
202C4⋊C4—261no—
203C4⋊C4—261no—
204C4⋊C4—261no—
205C4⋊C4—261no—
206C4⋊C4—261no—
207C4⋊C4—261no—
208C4⋊C4C22×D1311yescharacteristic
209C4⋊C4C22×D1311yescharacteristic
210C4⋊C4C22×D1312yes—
211C22⋊C4—21no—
212C22⋊C4—262no—
213C22⋊C4—262no—
214C22⋊C4—261no—
215C42—21no—
216C42—131no—
217C42—131no—
218C42—131no—
219C42—131no—
220C42—261no—
221C13#307711yescharacteristic, complemented, Sylow
222C23—264no—
223C23—262no—
224C23—522no—
225C23—522no—
226C23D4×D1311yescharacteristic
227Q8—522no—
228Q8—522no—
229D4—134no—
230D4—264no—
231D4—262no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—131no—
237C2×C4—131no—
238C2×C4—131no—
239C2×C4—131no—
240C2×C4—262no—
241C2×C4—261no—
242C2×C4—261no—
243C2×C4—524no—
244C2×C4—524no—
245C2×C4—521no—
246C2×C4#612712yes—
247C2×C4D4×D1311yescharacteristic
248C2×C4Q8×D1312yes—
249C2×C4C23×D1311yescharacteristic
250C8—22no—
251C8—262no—
252C22—21no—
253C22—21no—
254C22—264no—
255C22—528no—
256C22—522no—
257C22—522no—
258C22#8815012yes—
259C22C2×D4×D1311yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—132no—
264C4—261no—
265C4—261no—
266C4—261no—
267C4—521no—
268C4—521no—
269C4#8815012yes—
270C4C2×#612711yescharacteristic
271C4C2×Q8×D1311yescharacteristic
272C2—22no—
273C2—524no—
274C2C2×#8815011yescharacteristic
275C2D13×AGL(1,ℤ/8)11yescharacteristic
276C2#1171948211yescharacteristic
subgroupG/H
centerC22C2×D4×D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#428C2
socleC22×C13C22×D4
prime radicalC22×D52C22
hypercenter#428D13
hypocenterC13#3077
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #11719222C2semidirect
3. C13×#428C2semidirect
5. #11718779C2semidirect
7. #11718625C2semidirect
8. #11718552C2semidirect
9. #11718551C2semidirect
10. #11718532C2semidirect
11. #11718531C2semidirect
25. #88144C22semidirect
26. #88144C22semidirect
27. #88142C22semidirect
28. #88142C22semidirect
33. C13×#107C22semidirect
34. C13×#106C22semidirect
37. #88054C22semidirect
38. #88054C22semidirect
41. #88048C22semidirect
42. #88047C22semidirect
50. C2×D52Q8semidirect
121. #428D13semidirect
167. #107C2×D13semidirect
169. #106C2×D13semidirect
221. C13#3077semidirect

Maximal subgroups

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

subgroupindex
#117192222
C2×#881472
C13×#4282
#117187922
#117187792
#117186622
#117186252
#117185522
#117185512
#117185322
#117185312
#307713