Finite Group Databasebeta

← #13120415

Subgroups — #13120415

Order 1664.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#88126C211yesmaximal, characteristic
2C2×C4×Dic26C211yesmaximal, characteristic, complemented
3C13×#383C211yesmaximal, characteristic, complemented
4#11718335C211yesmaximal, characteristic
5#11718334C211yesmaximal, characteristic
6#11718331C212yesmaximal
7#11718329C212yesmaximal
8#11718322C211yesmaximal, characteristic, complemented
9#11718317C211yesmaximal, characteristic, complemented
10#11718314C212yesmaximal, complemented
11#11718314C212yesmaximal, complemented
12C22×C4×D13C2211yescharacteristic
13C22×Dic26C2211yescharacteristic
14C2×C42×C13C2211yescharacteristic
15C2×#6103C2211yescharacteristic
16C2×#6103C2211yescharacteristic
17C2×#6103C2212yes—
18C2×#6103C2212yes—
19C2×C4⋊Dic13C2211yescharacteristic
20C2×C4⋊Dic13C2212yes—
21C2×C4⋊Dic13C2212yes—
22C2×#6101C2212yes—
23C2×#6101C2211yescharacteristic
24C2×#6101C2211yescharacteristic
25C2×C4×Dic13C2212yes—
26C2×C4×Dic13C2211yescharacteristic
27#88126—24no—
28C4×Dic26—24no—
29C13×#95C2212yes—
30C13×#95C2212yes—
31C13×#95C2211yescharacteristic
32C13×#95C2211yescharacteristic
33#88071C2212yes—
34#88071C2212yes—
35#88071C2212yes—
36#88071C2212yes—
37C23×D13C2311yescharacteristic
38C22×C4×C13C2312yes—
39C22×C4×C13C2312yes—
40C22×C4×C13C2311yescharacteristic
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C22×Dic13C2311yescharacteristic
44C22×Dic13C2312yes—
45C22×Dic13C2311yescharacteristic
46C22×Dic13C2312yes—
47C22×Dic13C2311yescharacteristic
48C2×C4×D13—22no—
49C2×C4×D13—22no—
50C2×C4×D13—22no—
51C2×Dic26—22no—
52C2×Dic26—24no—
53C42×C13D414yes—
54#6103—24no—
55#6103—22no—
56#6103—22no—
57#6103—24no—
58C4⋊Dic13—22no—
59C4⋊Dic13—24no—
60C4⋊Dic13—24no—
61#6101—24no—
62#6101—22no—
63#6101—22no—
64C4×Dic13—24no—
65C4×Dic13—22no—
66#2599—131nomaximal, Sylow
67C23×C13C2411yescharacteristic
68C22×D13—21no—
69C22×D13—22no—
70C22×D13—21no—
71C22×D13—22no—
72C22×D13#4112yes—
73C2×C4×C13—22no—
74C2×C4×C13—21no—
75C2×C4×C13—21no—
76C2×C4×C13—22no—
77C2×C4×C13—22no—
78C2×C4×C13—22no—
79C2×C4×C13—22no—
80C2×C4×C13—22no—
81C2×C4×C13—22no—
82C2×C4×C13—22no—
83C2×C4×C13C2×D412yes—
84C2×C4×C13C2×D412yes—
85C2×C4×C13C2×D412yes—
86C2×Dic13—22no—
87C2×Dic13—21no—
88C2×Dic13—22no—
89C2×Dic13—21no—
90C2×Dic13—22no—
91C2×Dic13—21no—
92C2×Dic13—21no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—22no—
97C2×Dic13#4112yes—
98C2×Dic13#4114yes—
99C4×D13—28no—
100Dic26—28no—
101C2×C4×Q8—131no—
102C2×#117—131no—
103#398—131no—
104#397—131no—
105#395—131no—
106#393—131no—
107#388—132no—
108#387—132no—
109#387—132no—
110#385—132no—
111#383D1311yescharacteristic, complemented
112C22×C13C22×D411yescharacteristic
113C22×C13C2×#4111yescharacteristic
114C22×C13C2×#4112yes—
115C22×C13C2×#4111yescharacteristic
116C22×C132−1+412yes—
117C2×D13—22no—
118C2×D13—24no—
119C2×D13—41no—
120C2×D13—41no—
121C2×D13—42no—
122C4×C13—22no—
123C4×C13—22no—
124C4×C13—22no—
125C4×C13—42no—
126C4×C13—42no—
127Dic13—24no—
128Dic13—22no—
129Dic13—41no—
130Dic13—42no—
131Dic13—41no—
132C22×Q8—131no—
133C23×C4—131no—
134C4×Q8—264no—
135#117—264no—
136C2×C4⋊C4—131no—
137C2×C4⋊C4—131no—
138C2×C4⋊C4—132no—
139C2×C4⋊C4—131no—
140C2×C4⋊C4—132no—
141C2×C4⋊C4—132no—
142C2×C22⋊C4—131no—
143C2×C22⋊C4—131no—
144C2×C22⋊C4—132no—
145C2×C22⋊C4—132no—
146C2×C42—131no—
147C2×C42—132no—
148C2×C42C2×D1311yescharacteristic
149#95—132no—
150#95—132no—
151#95—132no—
152#95—132no—
153#95C2×D1312yes—
154#95C2×D1312yes—
155#95C2×D1311yescharacteristic
156#95C2×D1311yescharacteristic
157C2×C13#52511yescharacteristic
158C2×C13#52511yescharacteristic
159C2×C13#53611yescharacteristic
160C2×C13#55212yes—
161C2×C13#55612yes—
162D13—42no—
163C24—131no—
164C2×Q8—264no—
165C2×Q8—262no—
166C22×C4—132no—
167C22×C4—131no—
168C22×C4—131no—
169C22×C4—131no—
170C22×C4—132no—
171C22×C4—262no—
172C22×C4—262no—
173C22×C4—262no—
174C22×C4C22×D1312yes—
175C22×C4C22×D1312yes—
176C22×C4C22×D1311yescharacteristic
177C22×C4C22×D1311yescharacteristic
178C22×C4C22×D1311yescharacteristic
179C4⋊C4—264no—
180C4⋊C4—264no—
181C4⋊C4—262no—
182C4⋊C4—262no—
183C4⋊C4—264no—
184C4⋊C4—262no—
185C22⋊C4—264no—
186C22⋊C4—264no—
187C22⋊C4—262no—
188C22⋊C4—262no—
189C42—262no—
190C42—264no—
191C42D5214yes—
192C13#259911yescharacteristic, complemented, Sylow
193C23—132no—
194C23—261no—
195C23—262no—
196C23—261no—
197C23—262no—
198C23C23×D1311yescharacteristic
199Q8—268no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—134no—
211C2×C4—132no—
212C2×C4—268no—
213C2×C4—262no—
214C2×C4—261no—
215C2×C4—261no—
216C2×C4—262no—
217C2×C4—261no—
218C2×C4—261no—
219C2×C4—262no—
220C2×C4—262no—
221C2×C4—262no—
222C2×C4—262no—
223C2×C4—262no—
224C2×C4C2×D5212yes—
225C2×C4C2×D5212yes—
226C2×C4C2×D5212yes—
227C22—262no—
228C22—264no—
229C22—521no—
230C22—521no—
231C22—522no—
232C22C22×D5211yescharacteristic
233C22#41×D1311yescharacteristic
234C22#41×D1312yes—
235C22#41×D1311yescharacteristic
236C22#8825712yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—262no—
243C4—264no—
244C4—521no—
245C4—522no—
246C4—521no—
247C2—522no—
248C2#1171922911yescharacteristic
249C2#1171922911yescharacteristic
250C2#1171923111yescharacteristic
251C2#1171940712yes—
252C2#1171941312yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#383C2
prime radicalC23×D13C23
hypercenter#383D13
hypocenterC13#2599
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic26C2semidirect
3. C13×#383C2semidirect
8. #11718322C2semidirect
9. #11718317C2semidirect
10. #11718314C2semidirect
11. #11718314C2semidirect
111. #383D13semidirect
192. C13#2599semidirect

Maximal subgroups

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

subgroupindex
C2×#881262
C2×C4×Dic262
C13×#3832
#117183352
#117183342
#117183312
#117183292
#117183222
#117183172
#117183142
#259913