Finite Group Databasebeta

← #13124314

Subgroups — #13124314

Order 1664.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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#41×Dic13C211yesmaximal, characteristic, complemented
3C2×#88203C211yesmaximal, characteristic, complemented
4C2×#88194C211yesmaximal, characteristic, complemented
5#11718876C211yesmaximal, characteristic, complemented
6#11718824C214yesmaximal, complemented
7#11718811C214yesmaximal, complemented
8#11718792C211yesmaximal, characteristic, complemented
9#11718779C211yesmaximal, characteristic, complemented
10C2×C13×#41C2211yescharacteristic
11C22×D4×C13C2211yescharacteristic
12C22×C22⋊D13—22no—
13C22×D52C2211yescharacteristic
14C13×AGL(1,ℤ/8)—24no—
15C2×C13×SD16C2212yescomplemented
16C2×C13×D8C2212yescomplemented
17C2×C13×M16C2211yescharacteristic
18#88203—24no—
19Q8×Dic13C2212yescomplemented
20C2×#6106C2212yescomplemented
21#88194—21no—
22#88194—21no—
23#88194C2214yes—
24D4×Dic13—22no—
25D4×Dic13C2212yescomplemented
26C2×#6104C2212yescomplemented
27#88180—21no—
28#88180C2211yescharacteristic
29C2×C4×Dic13—21no—
30C2×C4×Dic13C2211yescharacteristic
31C2×#6099C2211yescharacteristic
32#88072C2214yescomplemented
33#88061C2214yescomplemented
34#88054C2214yescomplemented
35C24×C13—21no—
36C23×D13—21no—
37C13×#41—22no—
38C13×#41D414yescomplemented
39C2×Q8×C13C2311yescharacteristic
40C2×D4×C13—21no—
41C2×D4×C13—22no—
42C2×D4×C13—21no—
43C2×D4×C13—22no—
44C2×D4×C13C2312yes—
45C2×D4×C13C2311yescharacteristic
46C22×C4×C13—21no—
47C22×C4×C13C2311yescharacteristic
48C2×C22⋊D13—24no—
49C2×C22⋊D13—24no—
50C2×C22⋊D13—48no—
51C22×Dic13—22no—
52C22×Dic13D412yes—
53C2×D52—22no—
54C2×D52—22no—
55C2×D52—21no—
56C2×D52C2312yes—
57C13×SD16—24no—
58C13×D8—24no—
59C13×M16—22no—
60C2×C8×C13C2312yes—
61C22⋊Dic13—21no—
62C22⋊Dic13—22no—
63#6106—24no—
64#6104—24no—
65C4⋊Dic13—22no—
66C4⋊Dic13C2312yes—
67C4×Dic13—21no—
68C4×Dic13—22no—
69C4×Dic13D411yescharacteristic
70C4×Dic13D411yescharacteristic
71C4×Dic13C2312yes—
72#6099—22no—
73C2×#1055C2312yes—
74#3038—131nomaximal, Sylow
75C23×C13—21no—
76C23×C13—22no—
77C23×C13—44no—
78C23×C13—42no—
79C23×C13C2×D411yescharacteristic
80C22×D13—22no—
81C22×D13—44no—
82C22×D13—42no—
83Q8×C13—21no—
84Q8×C13C2×D412yescomplemented
85D4×C13—24no—
86D4×C13—22no—
87D4×C13—21no—
88D4×C13—22no—
89D4×C13—42no—
90D4×C13C2×D412yescomplemented
91C2×C4×C13—21no—
92C2×C4×C13—24no—
93C2×C4×C13C2×D411yescharacteristic
94C2×C4×C13C2×D412yes—
95C2×C4×C13#4112yes—
96C2×C4×C13C2411yescharacteristic
97C22⋊D13—48no—
98C22⋊D13—48no—
99C2×Dic13—24no—
100C2×Dic13—22no—
101C2×Dic13—44no—
102C2×Dic13C2×D412yes—
103C2×Dic13C2×D412yes—
104D52—22no—
105D52—22no—
106D52—42no—
107C8×C13—22no—
108#1055—22no—
109C2×AGL(1,ℤ/8)—131no—
110C2×AGL(1,ℤ/8)D1311yescharacteristic, complemented
111C2×#127—131no—
112C4×#41—131no—
113#460—134no—
114#459—134no—
115#423—131no—
116#418—131no—
117#418—131no—
118C22×C13—22no—
119C22×C13—22no—
120C22×C13—42no—
121C22×C13—42no—
122C22×C13—44no—
123C22×C13—84no—
124C22×C13#12112yes—
125C22×C13C22×D411yescharacteristic
126C2×D13—42no—
127C2×D13—44no—
128C2×D13—84no—
129C4×C13—22no—
130C4×C13#12112yes—
131C4×C13C22×D411yescharacteristic
132C4×C13C2×#4111yescharacteristic
133Dic13—22no—
134Dic13—42no—
135Dic13AGL(1,ℤ/8)14yescomplemented
136C2×#41C2×D1311yescharacteristic
137C22×D4—131no—
138C22×D4—262no—
139C22×D4C2×D1311yescharacteristic
140AGL(1,ℤ/8)—24no—
141AGL(1,ℤ/8)—264no—
142C2×SD16—132no—
143C2×SD16C2×D1312yescomplemented
144C2×D8—132no—
145C2×D8C2×D1312yescomplemented
146C2×M16—131no—
147C2×M16C2×D1311yescharacteristic
148#127—134no—
149#127—261no—
150#127—261no—
151C4×Q8—132no—
152C4×D4—132no—
153C4×D4—262no—
154#117—131no—
155#117—261no—
156C2×C42—131no—
157C2×C42—261no—
158C4⋊C8—134no—
159#102—134no—
160#102—134no—
161C2×C13—22no—
162C2×C13—42no—
163C2×C13—82no—
164C2×C13C2×#12111yescharacteristic
165C2×C13C2×AGL(1,ℤ/8)12yes—
166D13—82no—
167C24—21no—
168C24—261no—
169#41—22no—
170#41C22⋊D1314yescomplemented
171C2×Q8C22×D1311yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—132no—
177C2×D4—264no—
178C2×D4—264no—
179C2×D4—261no—
180C2×D4—262no—
181C2×D4—262no—
182C2×D4—528no—
183C2×D4C22×D1312yes—
184C2×D4C22×D1311yescharacteristic
185C22×C4—21no—
186C22×C4—132no—
187C22×C4—262no—
188C22×C4C22×D1311yescharacteristic
189SD16—24no—
190SD16—264no—
191D8—24no—
192D8—264no—
193M16—22no—
194M16—262no—
195C2×C8—132no—
196C2×C8C22×D1312yes—
197C4⋊C4—132no—
198C4⋊C4—262no—
199C22⋊C4—262no—
200C22⋊C4—261no—
201C42—131no—
202C42—131no—
203C42—132no—
204C42—262no—
205C42—261no—
206C13#303811yescharacteristic, complemented, Sylow
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—262no—
212C23—524no—
213C23—522no—
214C23D4×D1311yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D1312yescomplemented
217D4—24no—
218D4—21no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—262no—
223D4—262no—
224D4—528no—
225D4—528no—
226D4—522no—
227D4C2×C22⋊D1312yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—132no—
231C2×C4—132no—
232C2×C4—264no—
233C2×C4—262no—
234C2×C4—524no—
235C2×C4D4×D1311yescharacteristic
236C2×C4#612912yes—
237C2×C4C2×C22⋊D1312yes—
238C2×C4C23×D1311yescharacteristic
239C8—22no—
240C8—262no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—44no—
245C22—42no—
246C22—84no—
247C22—524no—
248C22—522no—
249C22—1044no—
250C22#8819312yes—
251C22C2×D4×D1311yescharacteristic
252C4—22no—
253C4—134no—
254C4—262no—
255C4—522no—
256C4#8819312yes—
257C4C2×#612911yescharacteristic
258C4C22×C22⋊D1311yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—1042no—
263C2D13×AGL(1,ℤ/8)12yes—
264C2C2×#8819311yescharacteristic
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#3038
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. #41×Dic13C2semidirect
3. C2×#88203C2semidirect
4. C2×#88194C2semidirect
5. #11718876C2semidirect
6. #11718824C2semidirect
7. #11718811C2semidirect
8. #11718792C2semidirect
9. #11718779C2semidirect
15. C2×C13×SD16C22semidirect
16. C2×C13×D8C22semidirect
19. Q8×Dic13C22semidirect
20. C2×#6106C22semidirect
25. D4×Dic13C22semidirect
26. C2×#6104C22semidirect
32. #88072C22semidirect
33. #88061C22semidirect
34. #88054C22semidirect
38. C13×#41D4semidirect
84. Q8×C13C2×D4semidirect
90. D4×C13C2×D4semidirect
110. C2×AGL(1,ℤ/8)D13semidirect
135. Dic13AGL(1,ℤ/8)semidirect
143. C2×SD16C2×D13semidirect
145. C2×D8C2×D13semidirect
170. #41C22⋊D13semidirect
206. C13#3038semidirect
216. Q8C2×C22⋊D13semidirect
227. D4C2×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
#41×Dic132
C2×#882032
C2×#881942
#117188762
#117188242
#117188112
#117187922
#117187792
#303813