Finite Group Databasebeta

← #13122900

Subgroups — #13122900

Order 1664.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88198C211yesmaximal, characteristic, complemented
2C2×#88152C211yesmaximal, characteristic
3C13×#395C211yesmaximal, characteristic, complemented
4#11718892C211yesmaximal, characteristic
5#11718653C212yesmaximal
6#11718645C212yesmaximal, complemented
7#11718640C211yesmaximal, characteristic, complemented
8#11718336C212yesmaximal
9#11718334C211yesmaximal, characteristic
10#11718319C212yesmaximal, complemented
11#11718312C211yesmaximal, characteristic, complemented
12C22×Q8×C13C2211yescharacteristic
13C22×Dic26C2211yescharacteristic
14C2×C13×C4⋊C4C2211yescharacteristic
15C2×C13×C4⋊C4C2212yes—
16#88198—24no—
17C2×#6103C2211yescharacteristic
18C2×#6103C2211yescharacteristic
19C2×#6103C2211yescharacteristic
20C2×#6103C2212yes—
21C2×#6103C2212yes—
22C2×C4⋊Dic13C2211yescharacteristic
23C2×#6101C2211yescharacteristic
24C2×#6101C2212yes—
25C2×#6101C2212yes—
26C2×#6101C2212yes—
27C2×C4×Dic13C2211yescharacteristic
28#88152—24no—
29C13×#95C2211yescharacteristic
30C13×#95C2212yes—
31#88071C2212yes—
32#88071C2212yes—
33#88071C2211yescharacteristic
34#88071C2212yes—
35#88071C2212yes—
36#88071C2212yes—
37C23×D13C2311yescharacteristic
38C2×Q8×C13—22no—
39C2×Q8×C13—42no—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2312yes—
43C22×C4×C13C2311yescharacteristic
44C22×C4×C13C2311yescharacteristic
45C22×Dic13C2312yes—
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2312yes—
48C22×Dic13C2311yescharacteristic
49C22×Dic13C2311yescharacteristic
50C2×Dic26—22no—
51C2×Dic26—42no—
52C13×C4⋊C4—22no—
53C13×C4⋊C4—42no—
54#6103—22no—
55#6103—22no—
56#6103—22no—
57#6103—42no—
58#6103—42no—
59C4⋊Dic13—22no—
60#6101—22no—
61#6101—24no—
62#6101—24no—
63#6101—42no—
64C4×Dic13D414yes—
65#2821—131nomaximal, Sylow
66C23×C13C2411yescharacteristic
67C22×D13—21no—
68C22×D13—22no—
69C22×D13—22no—
70C22×D13—21no—
71C22×D13—21no—
72Q8×C13—44no—
73C2×C4×C13—22no—
74C2×C4×C13—22no—
75C2×C4×C13—21no—
76C2×C4×C13—21no—
77C2×C4×C13—22no—
78C2×C4×C13—22no—
79C2×C4×C13—22no—
80C2×C4×C13—22no—
81C2×C4×C13—21no—
82C2×C4×C13—21no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13C2×D412yes—
86C2×Dic13—21no—
87C2×Dic13—22no—
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×Dic13C2×D412yes—
98C2×Dic13C2×D412yes—
99Dic26—44no—
100C2×C4⋊Q8—131no—
101C2×#126—131no—
102#400—132no—
103#399—132no—
104#398—132no—
105#397—132no—
106#395—131no—
107#395D1311yescharacteristic, complemented
108#393—131no—
109#393—131no—
110#383—131no—
111C22×C13C22×D411yescharacteristic
112C22×C13H(5,𝔽2)12yes—
113C22×C132−1+412yes—
114C22×C132−1+411yescharacteristic
115C22×C132−1+411yescharacteristic
116C2×D13—41no—
117C2×D13—41no—
118C2×D13—41no—
119C2×D13—42no—
120C2×D13—42no—
121C4×C13—22no—
122C4×C13—41no—
123C4×C13—42no—
124C4×C13—41no—
125C4×C13—42no—
126Dic13—22no—
127Dic13—22no—
128Dic13—42no—
129Dic13—41no—
130Dic13—42no—
131C22×Q8—131no—
132C22×Q8C2×D1311yescharacteristic
133C4⋊Q8—264no—
134#126—264no—
135C2×C4⋊C4—131no—
136C2×C4⋊C4—131no—
137C2×C4⋊C4—132no—
138C2×C4⋊C4—132no—
139C2×C4⋊C4—132no—
140C2×C4⋊C4C2×D1311yescharacteristic
141C2×C4⋊C4C2×D1312yes—
142C2×C22⋊C4—131no—
143C2×C22⋊C4—131no—
144C2×C22⋊C4—131no—
145C2×C22⋊C4—132no—
146C2×C22⋊C4—132no—
147C2×C42—131no—
148#95—131no—
149#95—132no—
150#95—132no—
151#95—132no—
152#95—132no—
153#95—132no—
154#95C2×D1311yescharacteristic
155#95C2×D1312yes—
156C2×C13#53511yescharacteristic
157C2×C13#53611yescharacteristic
158C2×C13#53611yescharacteristic
159C2×C13#56312yes—
160C2×C13#56312yes—
161D13—81no—
162C24—131no—
163C2×Q8—22no—
164C2×Q8—42no—
165C2×Q8—262no—
166C2×Q8—522no—
167C22×C4—131no—
168C22×C4—131no—
169C22×C4—132no—
170C22×C4—131no—
171C22×C4—132no—
172C22×C4C22×D1312yes—
173C22×C4C22×D1311yescharacteristic
174C22×C4C22×D1312yes—
175C22×C4C22×D1311yescharacteristic
176C22×C4C22×D1311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—262no—
180C4⋊C4—262no—
181C4⋊C4—264no—
182C4⋊C4—264no—
183C4⋊C4—522no—
184C22⋊C4—262no—
185C22⋊C4—262no—
186C22⋊C4—262no—
187C22⋊C4—522no—
188C22⋊C4—522no—
189C42—134no—
190C13#282111yescharacteristic, complemented, Sylow
191C23—261no—
192C23—262no—
193C23—262no—
194C23—261no—
195C23—261no—
196C23C23×D1311yescharacteristic
197Q8—44no—
198Q8—524no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—132no—
212C2×C4—132no—
213C2×C4—262no—
214C2×C4—261no—
215C2×C4—261no—
216C2×C4—261no—
217C2×C4—262no—
218C2×C4—262no—
219C2×C4—262no—
220C2×C4—262no—
221C2×C4—262no—
222C2×C4—262no—
223C2×C4—262no—
224C2×C4D4×D1312yes—
225C22—521no—
226C22—522no—
227C22—521no—
228C22—521no—
229C22—522no—
230C22C2×D4×D1311yescharacteristic
231C22#8825112yes—
232C22#8825412yes—
233C22#8825411yescharacteristic
234C22#8825711yescharacteristic
235C4—22no—
236C4—41no—
237C4—42no—
238C4—41no—
239C4—42no—
240C4—262no—
241C4—262no—
242C4—521no—
243C4—522no—
244C4—522no—
245C2—1041no—
246C2#1171929711yescharacteristic
247C2#1171933311yescharacteristic
248C2#1171935112yes—
249C2#1171942911yescharacteristic
250C2#1171944012yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#395C2
prime radicalC23×D13C23
hypercenter#395D13
hypocenterC13#2821
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#88198C2semidirect
3. C13×#395C2semidirect
6. #11718645C2semidirect
7. #11718640C2semidirect
10. #11718319C2semidirect
11. #11718312C2semidirect
107. #395D13semidirect
190. C13#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#881982
C2×#881522
C13×#3952
#117188922
#117186532
#117186452
#117186402
#117183362
#117183342
#117183192
#117183122
#282113