Finite Group Databasebeta

← #13120650

Subgroups — #13120650

Order 1664.

508 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 143 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×#88151C211yesmaximal, characteristic
2C2×C4⋊C4×D13C211yesmaximal, characteristic
3C13×#400C211yesmaximal, characteristic, complemented
4#11718643C211yesmaximal, characteristic, complemented
5#11718640C211yesmaximal, characteristic, complemented
6#11718329C212yesmaximal
7#11718328C211yesmaximal, characteristic
8#11718325C212yesmaximal
9#11718322C212yesmaximal, complemented
10#11718318C212yesmaximal, complemented
11#11718317C211yesmaximal, characteristic, complemented
12C22×C4×D13C2211yescharacteristic
13C22×C4×D13C2211yescharacteristic
14C22×C4×D13C2211yescharacteristic
15C22×Dic26C2211yescharacteristic
16C2×C13×C4⋊C4C2211yescharacteristic
17C2×C13×C4⋊C4C2211yescharacteristic
18C2×#6103C2212yes—
19C2×#6103C2212yes—
20C2×C4⋊Dic13C2211yescharacteristic
21C2×C4⋊Dic13C2212yes—
22C2×C4⋊Dic13C2212yes—
23C2×#6101C2211yescharacteristic
24C2×#6101C2211yescharacteristic
25C2×#6101C2212yes—
26C2×#6101C2212yes—
27#88151—24no—
28C4⋊C4×D13—24no—
29C13×#95C2211yescharacteristic
30C13×#95C2212yes—
31C13×#95C2212yes—
32#88071C2212yes—
33#88071C2212yes—
34#88071C2212yes—
35#88071C2212yes—
36#88071C2211yescharacteristic
37C23×D13C2311yescharacteristic
38C22×C4×C13C2312yes—
39C22×C4×C13C2312yes—
40C22×C4×C13C2311yescharacteristic
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C22×Dic13C2312yes—
44C22×Dic13C2311yescharacteristic
45C22×Dic13C2312yes—
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2311yescharacteristic
48C2×C4×D13—22no—
49C2×C4×D13—22no—
50C2×C4×D13—22no—
51C2×C4×D13—42no—
52C2×C4×D13—42no—
53C2×C4×D13D414yes—
54C2×C4×D13Q814yescomplemented
55C2×Dic26—22no—
56C2×Dic26—42no—
57C13×C4⋊C4—22no—
58C13×C4⋊C4—22no—
59#6103—24no—
60#6103—24no—
61C4⋊Dic13—22no—
62C4⋊Dic13—24no—
63C4⋊Dic13—42no—
64#6101—22no—
65#6101—22no—
66#6101—42no—
67#6101—42no—
68#2627—131nomaximal, Sylow
69C23×C13C2411yescharacteristic
70C22×D13—22no—
71C22×D13—22no—
72C22×D13C2×D412yes—
73C22×D13C2×Q812yes—
74C22×D13#4112yes—
75C2×C4×C13—21no—
76C2×C4×C13—21no—
77C2×C4×C13—22no—
78C2×C4×C13—21no—
79C2×C4×C13—22no—
80C2×C4×C13—22no—
81C2×C4×C13—22no—
82C2×C4×C13—22no—
83C2×C4×C13—24no—
84C2×C4×C13—22no—
85C2×C4×C13C2×D412yes—
86C2×C4×C13C2×Q812yes—
87C2×Dic13—21no—
88C2×Dic13—21no—
89C2×Dic13—22no—
90C2×Dic13—21no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13C2×D412yes—
99C2×Dic13C2×Q812yes—
100C4×D13—28no—
101C4×D13—44no—
102C4×D13—44no—
103Dic26—44no—
104C2×C22⋊Q8—131no—
105C22×C4⋊C4—131no—
106#400—131no—
107#400D1311yescharacteristic, complemented
108#398—132no—
109#398—132no—
110#395—131no—
111#395—131no—
112#385—132no—
113#381—132no—
114#380—131no—
115C22×C13C22×D411yescharacteristic
116C22×C13C22×Q811yescharacteristic
117C22×C13C2×#4111yescharacteristic
118C22×C13H(5,𝔽2)12yes—
119C22×C132−1+412yes—
120C2×D13—22no—
121C2×D13—22no—
122C2×D13—24no—
123C2×D13—24no—
124C2×D13C22⋊Q814yes—
125C4×C13—22no—
126C4×C13—22no—
127C4×C13—41no—
128C4×C13—42no—
129C4×C13—42no—
130Dic13—22no—
131Dic13—22no—
132Dic13—42no—
133Dic13—41no—
134Dic13—42no—
135C22×Q8—131no—
136C23×C4—131no—
137C23×C4—131no—
138C23×C4—131no—
139C22⋊Q8—264no—
140C2×C4⋊C4—131no—
141C2×C4⋊C4—131no—
142C2×C4⋊C4—131no—
143C2×C4⋊C4—132no—
144C2×C4⋊C4—132no—
145C2×C4⋊C4—132no—
146C2×C4⋊C4—132no—
147C2×C4⋊C4—264no—
148C2×C4⋊C4C2×D1311yescharacteristic
149C2×C4⋊C4C2×D1311yescharacteristic
150C2×C22⋊C4—132no—
151C2×C22⋊C4—132no—
152#95—131no—
153#95—132no—
154#95—132no—
155#95—132no—
156#95—132no—
157#95C2×D1311yescharacteristic
158#95C2×D1312yes—
159#95C2×D1312yes—
160C2×C13C2×C22⋊Q811yescharacteristic
161C2×C13#53411yescharacteristic
162C2×C13#53611yescharacteristic
163C2×C13#53912yes—
164C2×C13#54412yes—
165D13—24no—
166C24—131no—
167C2×Q8—262no—
168C2×Q8—522no—
169C22×C4—131no—
170C22×C4—131no—
171C22×C4—134no—
172C22×C4—134no—
173C22×C4—132no—
174C22×C4—132no—
175C22×C4—131no—
176C22×C4—262no—
177C22×C4—262no—
178C22×C4—262no—
179C22×C4—522no—
180C22×C4—522no—
181C22×C4C22×D1312yes—
182C22×C4C22×D1312yes—
183C22×C4C22×D1311yescharacteristic
184C22×C4C22×D1311yescharacteristic
185C22×C4C22×D1311yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—262no—
189C4⋊C4—262no—
190C4⋊C4—264no—
191C4⋊C4—262no—
192C4⋊C4—522no—
193C4⋊C4—522no—
194C4⋊C4—522no—
195C22⋊C4—264no—
196C22⋊C4—264no—
197C13#262711yescharacteristic, complemented, Sylow
198C23—132no—
199C23—132no—
200C23—132no—
201C23—262no—
202C23—262no—
203C23C23×D1311yescharacteristic
204Q8—524no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—132no—
216C2×C4—132no—
217C2×C4—268no—
218C2×C4—261no—
219C2×C4—261no—
220C2×C4—262no—
221C2×C4—262no—
222C2×C4—261no—
223C2×C4—262no—
224C2×C4—262no—
225C2×C4—262no—
226C2×C4—262no—
227C2×C4—262no—
228C2×C4—262no—
229C2×C4—524no—
230C2×C4—524no—
231C2×C4D4×D1312yes—
232C2×C4Q8×D1312yes—
233C22—134no—
234C22—262no—
235C22—262no—
236C22—264no—
237C22—264no—
238C22C2×D4×D1311yescharacteristic
239C22C2×Q8×D1311yescharacteristic
240C22#41×D1311yescharacteristic
241C22#8825612yes—
242C22#8825712yes—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—42no—
247C4—42no—
248C4—262no—
249C4—262no—
250C4—521no—
251C4—522no—
252C4—522no—
253C2—264no—
254C2D13×C22⋊Q811yescharacteristic
255C2#1171935811yescharacteristic
256C2#1171938011yescharacteristic
257C2#1171939412yes—
258C2#1171941212yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#400C2
prime radicalC23×D13C23
hypercenter#400D13
hypocenterC13#2627
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C13×#400C2semidirect
4. #11718643C2semidirect
5. #11718640C2semidirect
9. #11718322C2semidirect
10. #11718318C2semidirect
11. #11718317C2semidirect
54. C2×C4×D13Q8semidirect
107. #400D13semidirect
197. C13#2627semidirect

Maximal subgroups

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

subgroupindex
C2×#881512
C2×C4⋊C4×D132
C13×#4002
#117186432
#117186402
#117183292
#117183282
#117183252
#117183222
#117183182
#117183172
#262713