Finite Group Databasebeta

← #13122901

Subgroups — #13122901

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×#88152C211yesmaximal, characteristic
2C2×#88142C211yesmaximal, characteristic, complemented
3C13×#400C211yesmaximal, characteristic, complemented
4#11718653C211yesmaximal, characteristic
5#11718647C211yesmaximal, characteristic, complemented
6#11718338C211yesmaximal, characteristic
7#11718336C212yesmaximal
8#11718334C212yesmaximal
9#11718318C212yesmaximal, complemented
10#11718317C212yesmaximal, complemented
11#11718312C211yesmaximal, characteristic, complemented
12C22×Dic26C2212yes—
13C2×C13×C4⋊C4C2211yescharacteristic
14C2×C13×C4⋊C4C2211yescharacteristic
15C2×#6103C2211yescharacteristic
16C2×#6103C2211yescharacteristic
17C2×#6103C2211yescharacteristic
18C2×#6103C2212yes—
19C2×#6103C2212yes—
20C2×C4⋊Dic13C2211yescharacteristic
21C2×C4⋊Dic13C2211yescharacteristic
22C2×C4⋊Dic13C2212yes—
23C2×#6101C2212yes—
24C2×#6101C2211yescharacteristic
25C2×#6101C2212yes—
26C2×C4×Dic13C2211yescharacteristic
27#88152—24no—
28#88142—24no—
29C13×#95C2211yescharacteristic
30C13×#95C2212yes—
31C13×#95C2212yes—
32#88071C2212yes—
33#88071C2212yes—
34#88071C2211yescharacteristic
35#88071C2212yes—
36#88071C2212yes—
37C23×D13C2311yescharacteristic
38C22×C4×C13C2312yes—
39C22×C4×C13C2311yescharacteristic
40C22×C4×C13C2311yescharacteristic
41C22×C4×C13C2312yes—
42C22×C4×C13C2311yescharacteristic
43C22×Dic13C2312yes—
44C22×Dic13C2312yes—
45C22×Dic13C2311yescharacteristic
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2311yescharacteristic
48C2×Dic26—24no—
49C2×Dic26—42no—
50C2×Dic26—42no—
51C13×C4⋊C4—22no—
52C13×C4⋊C4—22no—
53#6103—22no—
54#6103—22no—
55#6103—22no—
56#6103—42no—
57#6103—42no—
58C4⋊Dic13—22no—
59C4⋊Dic13—22no—
60C4⋊Dic13—42no—
61#6101—24no—
62#6101—22no—
63#6101—42no—
64C4×Dic13D414yes—
65#2821—131nomaximal, Sylow
66C23×C13C2411yescharacteristic
67C22×D13—21no—
68C22×D13—22no—
69C22×D13—21no—
70C22×D13—21no—
71C22×D13—22no—
72C2×C4×C13—22no—
73C2×C4×C13—21no—
74C2×C4×C13—22no—
75C2×C4×C13—21no—
76C2×C4×C13—22no—
77C2×C4×C13—22no—
78C2×C4×C13—22no—
79C2×C4×C13—21no—
80C2×C4×C13—21no—
81C2×C4×C13—22no—
82C2×C4×C13—22no—
83C2×C4×C13—22no—
84C2×C4×C13C2×D412yes—
85C2×Dic13—22no—
86C2×Dic13—21no—
87C2×Dic13—21no—
88C2×Dic13—21no—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13C2×D412yes—
97C2×Dic13C2×D412yes—
98Dic26—48no—
99C2×C4⋊Q8—131no—
100C2×#126—131no—
101#400—131no—
102#400D1311yescharacteristic, complemented
103#399—131no—
104#399—131no—
105#398—132no—
106#397—132no—
107#395—132no—
108#393—132no—
109#383—131no—
110C22×C13C22×D411yescharacteristic
111C22×C13H(5,𝔽2)12yes—
112C22×C132−1+411yescharacteristic
113C22×C132−1+411yescharacteristic
114C22×C132−1+412yes—
115C2×D13—41no—
116C2×D13—41no—
117C2×D13—41no—
118C2×D13—42no—
119C2×D13—42no—
120C4×C13—22no—
121C4×C13—42no—
122C4×C13—41no—
123C4×C13—41no—
124C4×C13—42no—
125Dic13—22no—
126Dic13—22no—
127Dic13—42no—
128Dic13—41no—
129Dic13—42no—
130C22×Q8—132no—
131C4⋊Q8—264no—
132#126—264no—
133C2×C4⋊C4—131no—
134C2×C4⋊C4—131no—
135C2×C4⋊C4—132no—
136C2×C4⋊C4—131no—
137C2×C4⋊C4—132no—
138C2×C4⋊C4—132no—
139C2×C4⋊C4C2×D1311yescharacteristic
140C2×C4⋊C4C2×D1311yescharacteristic
141C2×C22⋊C4—131no—
142C2×C22⋊C4—131no—
143C2×C22⋊C4—131no—
144C2×C22⋊C4—132no—
145C2×C22⋊C4—132no—
146C2×C42—131no—
147#95—132no—
148#95—132no—
149#95—131no—
150#95—132no—
151#95—132no—
152#95C2×D1311yescharacteristic
153#95C2×D1312yes—
154#95C2×D1312yes—
155C2×C13#53511yescharacteristic
156C2×C13#53611yescharacteristic
157C2×C13#53611yescharacteristic
158C2×C13#56312yes—
159C2×C13#56312yes—
160D13—81no—
161C24—131no—
162C2×Q8—264no—
163C2×Q8—522no—
164C2×Q8—522no—
165C22×C4—131no—
166C22×C4—131no—
167C22×C4—132no—
168C22×C4—132no—
169C22×C4—131no—
170C22×C4C22×D1312yes—
171C22×C4C22×D1311yescharacteristic
172C22×C4C22×D1311yescharacteristic
173C22×C4C22×D1312yes—
174C22×C4C22×D1311yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—262no—
178C4⋊C4—262no—
179C4⋊C4—264no—
180C4⋊C4—262no—
181C4⋊C4—522no—
182C4⋊C4—522no—
183C22⋊C4—262no—
184C22⋊C4—262no—
185C22⋊C4—262no—
186C22⋊C4—522no—
187C22⋊C4—522no—
188C42—134no—
189C13#282111yescharacteristic, complemented, Sylow
190C23—261no—
191C23—261no—
192C23—262no—
193C23—261no—
194C23—262no—
195C23C23×D1311yescharacteristic
196Q8—528no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—132no—
210C2×C4—132no—
211C2×C4—261no—
212C2×C4—262no—
213C2×C4—261no—
214C2×C4—261no—
215C2×C4—262no—
216C2×C4—262no—
217C2×C4—262no—
218C2×C4—262no—
219C2×C4—262no—
220C2×C4—262no—
221C2×C4—262no—
222C2×C4D4×D1312yes—
223C22—521no—
224C22—521no—
225C22—521no—
226C22—522no—
227C22—522no—
228C22C2×D4×D1311yescharacteristic
229C22#8825411yescharacteristic
230C22#8825612yes—
231C22#8825711yescharacteristic
232C22#8825712yes—
233C4—22no—
234C4—41no—
235C4—41no—
236C4—42no—
237C4—42no—
238C4—262no—
239C4—262no—
240C4—521no—
241C4—522no—
242C4—522no—
243C2—1041no—
244C2#1171933311yescharacteristic
245C2#1171935311yescharacteristic
246C2#1171937911yescharacteristic
247C2#1171940412yes—
248C2#1171941712yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#400C2
prime radicalC23×D13C23
hypercenter#400D13
hypocenterC13#2821
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#88142C2semidirect
3. C13×#400C2semidirect
5. #11718647C2semidirect
9. #11718318C2semidirect
10. #11718317C2semidirect
11. #11718312C2semidirect
102. #400D13semidirect
189. C13#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#881522
C2×#881422
C13×#4002
#117186532
#117186472
#117183382
#117183362
#117183342
#117183182
#117183172
#117183122
#282113