Finite Group Databasebeta

← #13120414

Subgroups — #13120414

Order 1664.

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

normal subgroups only

This group has 492 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
1C2×#88141C211yesmaximal, characteristic, complemented
2C2×#88131C211yesmaximal, characteristic
3C13×#397C211yesmaximal, characteristic
4#11718641C211yesmaximal, characteristic, complemented
5#11718612C212yesmaximal
6#11718608C211yesmaximal, characteristic
7#11718607C212yesmaximal
8#11718317C211yesmaximal, characteristic, complemented
9#11718315C212yesmaximal, complemented
10#11718313C212yesmaximal, complemented
11#11718312C211yesmaximal, characteristic, complemented
12C23×Dic13C2211yescharacteristic
13C22×Dic26C2211yescharacteristic
14C2×C13×C4⋊C4C2211yescharacteristic
15C2×C13×C22⋊C4C2211yescharacteristic
16C2×C13×C22⋊C4C2212yes—
17C2×C22⋊Dic13C2211yescharacteristic
18C2×C22⋊Dic13C2212yes—
19C2×C4⋊Dic13C2212yes—
20C2×#6101C2211yescharacteristic
21C2×#6101C2211yescharacteristic
22C2×#6101C2212yes—
23C2×#6101C2212yes—
24C2×C4×Dic13C2211yescharacteristic
25C2×C4×Dic13C2212yes—
26C2×C4×Dic13C2211yescharacteristic
27#88141—24no—
28#88131—24no—
29C13×#95C2211yescharacteristic
30C13×#95C2212yes—
31#88071C2212yes—
32#88071C2212yes—
33#88071C2211yescharacteristic
34#88071C2212yes—
35#88071C2212yes—
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C22×C4×C13C2312yes—
39C22×C4×C13C2311yescharacteristic
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×Dic13—22no—
43C22×Dic13—22no—
44C22×Dic13—22no—
45C22×Dic13C2312yes—
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2311yescharacteristic
48C22×Dic13C2311yescharacteristic
49C22×Dic13C2312yes—
50C22×Dic13C2311yescharacteristic
51C2×Dic26—22no—
52C2×Dic26—24no—
53C13×C4⋊C4—22no—
54C13×C22⋊C4—24no—
55C13×C22⋊C4—22no—
56C22⋊Dic13—22no—
57C22⋊Dic13—24no—
58C4⋊Dic13—24no—
59#6101—22no—
60#6101—22no—
61#6101—24no—
62#6101—24no—
63C4×Dic13—22no—
64C4×Dic13—24no—
65C4×Dic13D414yes—
66#2599—131nomaximal, Sylow
67C23×C13—21no—
68C23×C13—22no—
69C23×C13—21no—
70C23×C13—22no—
71C23×C13#4112yes—
72C23×C13C2411yescharacteristic
73C2×C4×C13—22no—
74C2×C4×C13—21no—
75C2×C4×C13—22no—
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×C13C2×D412yes—
83C2×Dic13—28no—
84C2×Dic13—22no—
85C2×Dic13—21no—
86C2×Dic13—22no—
87C2×Dic13—21no—
88C2×Dic13—21no—
89C2×Dic13—21no—
90C2×Dic13—22no—
91C2×Dic13—21no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13C2×D412yes—
97C2×Dic13C2×D412yes—
98C2×Dic13#4112yes—
99C2×Dic13#4114yes—
100Dic26—28no—
101C2×C4×Q8—131no—
102C2×#117—131no—
103#398—131no—
104#397D1311yescharacteristic
105#395—131no—
106#393—131no—
107#388—132no—
108#387—132no—
109#387—132no—
110#385—132no—
111#383—131no—
112C22×C13—22no—
113C22×C13—24no—
114C22×C13—41no—
115C22×C13—41no—
116C22×C13—42no—
117C22×C13C22×D411yescharacteristic
118C22×C13C2×#4111yescharacteristic
119C22×C13C2×#4112yes—
120C22×C13C2×#4111yescharacteristic
121C22×C132−1+412yes—
122C4×C13—22no—
123C4×C13—42no—
124C4×C13—41no—
125C4×C13—42no—
126Dic13—24no—
127Dic13—22no—
128Dic13—22no—
129Dic13—22no—
130Dic13—41no—
131Dic13—42no—
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—132no—
140C2×C4⋊C4—132no—
141C2×C4⋊C4C2×D1311yescharacteristic
142C2×C22⋊C4—131no—
143C2×C22⋊C4—132no—
144C2×C22⋊C4C2×D1311yescharacteristic
145C2×C22⋊C4C2×D1312yes—
146C2×C42—131no—
147C2×C42—131no—
148C2×C42—132no—
149#95—132no—
150#95—132no—
151#95—131no—
152#95—132no—
153#95—132no—
154#95—132no—
155#95C2×D1311yescharacteristic
156#95C2×D1312yes—
157C2×C13—42no—
158C2×C13#52511yescharacteristic
159C2×C13#52511yescharacteristic
160C2×C13#53611yescharacteristic
161C2×C13#55212yes—
162C2×C13#55612yes—
163C24C22×D1311yescharacteristic
164C2×Q8—264no—
165C2×Q8—262no—
166C22×C4—131no—
167C22×C4—131no—
168C22×C4—132no—
169C22×C4—131no—
170C22×C4—132no—
171C22×C4—131no—
172C22×C4—262no—
173C22×C4—262no—
174C22×C4—262no—
175C22×C4C22×D1312yes—
176C22×C4C22×D1311yescharacteristic
177C22×C4C22×D1312yes—
178C22×C4C22×D1311yescharacteristic
179C4⋊C4—22no—
180C4⋊C4—262no—
181C4⋊C4—264no—
182C4⋊C4—264no—
183C4⋊C4—264no—
184C4⋊C4—262no—
185C22⋊C4—24no—
186C22⋊C4—22no—
187C22⋊C4—262no—
188C22⋊C4—264no—
189C42—134no—
190C42—262no—
191C42—264no—
192C13#259911yescharacteristic, complemented, Sylow
193C23—21no—
194C23—22no—
195C23—21no—
196C23—22no—
197C23#612912yes—
198C23C23×D1311yescharacteristic
199Q8—268no—
200C2×C4—22no—
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—132no—
210C2×C4—132no—
211C2×C4—134no—
212C2×C4—132no—
213C2×C4—268no—
214C2×C4—262no—
215C2×C4—261no—
216C2×C4—261no—
217C2×C4—261no—
218C2×C4—261no—
219C2×C4—262no—
220C2×C4—262no—
221C2×C4—261no—
222C2×C4—262no—
223C2×C4—262no—
224C2×C4—262no—
225C2×C4—262no—
226C2×C4D4×D1312yes—
227C22—22no—
228C22—24no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22C2×D4×D1311yescharacteristic
233C22C2×#612911yescharacteristic
234C22#41×D1311yescharacteristic
235C22#41×D1312yes—
236C22#8825712yes—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—41no—
241C4—262no—
242C4—262no—
243C4—264no—
244C4—262no—
245C4—521no—
246C4—522no—
247C2—42no—
248C2#1171931111yescharacteristic
249C2#1171935512yes—
250C2#1171936111yescharacteristic
251C2#1171938011yescharacteristic
252C2#1171940712yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#397C2
prime radicalC24×C13C23
hypercenter#397D13
hypocenterC13#2599
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#88141C2semidirect
4. #11718641C2semidirect
8. #11718317C2semidirect
9. #11718315C2semidirect
10. #11718313C2semidirect
11. #11718312C2semidirect
192. C13#2599semidirect

Maximal subgroups

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

subgroupindex
C2×#881412
C2×#881312
C13×#3972
#117186412
#117186122
#117186082
#117186072
#117183172
#117183152
#117183132
#117183122
#259913