Finite Group Databasebeta

← #13125678

Subgroups — #13125678

Order 1664.

418 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 117 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#11719496C212yesmaximal, complemented
2C13×#407C211yesmaximal, characteristic, complemented
3#11718798C211yesmaximal, characteristic, complemented
4#11718796C211yesmaximal, characteristic, complemented
5#11718795C212yesmaximal, complemented
6#11718398C212yesmaximal, complemented
7#11718395C212yesmaximal, complemented
8#11718394C212yesmaximal, complemented
9#11718384C212yesmaximal, complemented
10C23×C4×C13C2211yescharacteristic
11C2×#6127C2212yes—
12C2×C13×M16C2211yescharacteristic
13C2×C13×M16C2211yescharacteristic
14#88207—22no—
15#88184C2212yes—
16#88184C2212yes—
17#88183—22no—
18C4×C22⋊D13—22no—
19C4×C22⋊D13—22no—
20#88180C2212yes—
21#88178C2212yes—
22#88178C2212yes—
23C2×Dic52C2211yescharacteristic, complemented
24C2×D104C2211yescharacteristic, complemented
25C2×#6095C2212yescomplemented
26C13×C22⋊C8C2212yes—
27C13×C22⋊C8C2212yes—
28#88061C2212yescomplemented
29#88061C2212yescomplemented
30#88058C2212yescomplemented
31#88057C2211yescharacteristic, complemented
32#88057C2211yescharacteristic, complemented
33#88056C2212yescomplemented
34#88056C2212yescomplemented
35C24×C13D411yescharacteristic
36C22×C4×C13—41no—
37C22×C4×C13—41no—
38C22×C4×C13D411yescharacteristic
39C22×C4×C13D412yes—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C2×C22⋊D13—22no—
43C2×C22⋊D13—22no—
44C2×C22⋊D13—22no—
45#6127—42no—
46#6127—42no—
47C2×D52C2312yes—
48C2×C4×D13—22no—
49C2×Dic26C2312yes—
50C13×M16—41no—
51C13×M16—41no—
52C2×C8×C13C2312yes—
53C2×C8×C13C2311yescharacteristic
54C2×C8×C13C2311yescharacteristic
55C22⋊Dic13—22no—
56C22⋊Dic13—22no—
57C22⋊Dic13—22no—
58#6103—22no—
59#6103—22no—
60C4⋊Dic13C2312yes—
61C4⋊Dic13C2312yes—
62#6101—22no—
63#6101—22no—
64C4×Dic13—22no—
65Dic52—41no—
66D104—41no—
67#6095—42no—
68#3197—131nomaximal, Sylow
69C23×C13—41no—
70C23×C13—42no—
71C23×C13C2×D412yes—
72C23×C13C2×D411yescharacteristic
73C22×D13—22no—
74C2×C4×C13—21no—
75C2×C4×C13—21no—
76C2×C4×C13—22no—
77C2×C4×C13—22no—
78C2×C4×C13—41no—
79C2×C4×C13—42no—
80C2×C4×C13C2×D412yes—
81C2×C4×C13C2×D411yescharacteristic
82C2×C4×C13C2411yescharacteristic
83C22⋊D13—44no—
84C22⋊D13—44no—
85C22⋊D13—44no—
86C2×Dic13—22no—
87C2×Dic13—22no—
88C2×Dic13—22no—
89D52—22no—
90D52—22no—
91C4×D13—44no—
92Dic26—22no—
93Dic26—22no—
94C8×C13—22no—
95C8×C13—21no—
96C8×C13—21no—
97#525—132no—
98#485—132no—
99#482—132no—
100#470—131no—
101#469—131no—
102#468—132no—
103#452—132no—
104#449—132no—
105#407D1311yescharacteristic, complemented
106C22×C13—22no—
107C22×C13—21no—
108C22×C13—22no—
109C22×C13—22no—
110C22×C13—22no—
111C22×C13—44no—
112C22×C13C22×D411yescharacteristic
113C2×D13—42no—
114C2×D13—42no—
115C2×D13—42no—
116C4×C13—22no—
117C4×C13—21no—
118C4×C13H(5,𝔽2)11yescharacteristic
119C4×C13H(5,𝔽2)11yescharacteristic
120Dic13—42no—
121Dic13—42no—
122Dic13—42no—
123C2×#41—132no—
124C23×C4C2×D1311yescharacteristic
125C2×Q16—131no—
126C2×SD16—132no—
127C2×D8—131no—
128C2×M16C2×D1311yescharacteristic
129C2×M16C2×D1311yescharacteristic
130#123—262no—
131C22⋊Q8—132no—
132C22⋊Q8—132no—
133#121—132no—
134#121—132no—
135#120—262no—
136C4×D4—262no—
137C4×D4—262no—
138#117—132no—
139#107—132no—
140#106—131no—
141#106—131no—
142#103—132no—
143#103—132no—
144#102—132no—
145#102—132no—
146C22⋊C8C2×D1312yes—
147C22⋊C8C2×D1312yes—
148C2×C13—42no—
149C2×C13—41no—
150C2×C13#53411yescharacteristic
151C2×C13#57512yes—
152D13—82no—
153C24D5211yescharacteristic
154#41—522no—
155#41—522no—
156C2×Q8—132no—
157C2×D4—132no—
158C2×D4—262no—
159C2×D4—262no—
160C2×D4—262no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—262no—
164C22×C4D5211yescharacteristic
165C22×C4D5212yes—
166C22×C4C22×D1312yes—
167C22×C4C22×D1311yescharacteristic
168Q16—521no—
169SD16—522no—
170D8—521no—
171M16—41no—
172M16—41no—
173C2×C8C22×D1311yescharacteristic
174C2×C8C22×D1312yes—
175C2×C8C22×D1311yescharacteristic
176C4⋊C4—132no—
177C4⋊C4—132no—
178C4⋊C4—262no—
179C4⋊C4—262no—
180C22⋊C4—262no—
181C22⋊C4—262no—
182C22⋊C4—262no—
183C22⋊C4—262no—
184C22⋊C4—262no—
185C42—262no—
186C13#319711yescharacteristic, complemented, Sylow
187C23—41no—
188C23—42no—
189C23—262no—
190C23C2×D5211yescharacteristic
191C23C2×D5212yes—
192Q8—262no—
193Q8—262no—
194D4—262no—
195D4—262no—
196D4—524no—
197D4—524no—
198D4—524no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—262no—
206C2×C4—262no—
207C2×C4—262no—
208C2×C4—524no—
209C2×C4C2×D5212yes—
210C2×C4C2×D5211yescharacteristic
211C2×C4C23×D1311yescharacteristic
212C8—21no—
213C8—22no—
214C8—21no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—21no—
220C22—44no—
221C22—522no—
222C22—522no—
223C22—522no—
224C22C22×D5211yescharacteristic
225C4—22no—
226C4—21no—
227C4—522no—
228C4—522no—
229C4—522no—
230C4#8825111yescharacteristic
231C4#8825111yescharacteristic
232C2—41no—
233C2—42no—
234C2—1042no—
235C2#1171919811yescharacteristic
236C2#1171945312yes—
subgroupG/H
centerC22C22×D52
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#407C2
socleC22×C13C22×D4
hypercenter#407D13
hypocenterC13#3197
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #11719496C2semidirect
2. C13×#407C2semidirect
3. #11718798C2semidirect
4. #11718796C2semidirect
5. #11718795C2semidirect
6. #11718398C2semidirect
7. #11718395C2semidirect
8. #11718394C2semidirect
9. #11718384C2semidirect
23. C2×Dic52C22semidirect
24. C2×D104C22semidirect
25. C2×#6095C22semidirect
28. #88061C22semidirect
29. #88061C22semidirect
30. #88058C22semidirect
31. #88057C22semidirect
32. #88057C22semidirect
33. #88056C22semidirect
34. #88056C22semidirect
105. #407D13semidirect
186. C13#3197semidirect

Maximal subgroups

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

subgroupindex
#117194962
C13×#4072
#117187982
#117187962
#117187952
#117183982
#117183952
#117183942
#117183842
#319713