Finite Group Databasebeta

← #13121000

Subgroups — #13121000

Order 1664.

434 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 133 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×#88144C211yesmaximal, characteristic, complemented
2C2×#88133C211yesmaximal, characteristic
3C13×#399C211yesmaximal, characteristic
4#11718646C212yesmaximal, complemented
5#11718643C211yesmaximal, characteristic, complemented
6#11718641C212yesmaximal, complemented
7#11718614C211yesmaximal, characteristic
8#11718612C212yesmaximal
9#11718609C212yesmaximal
10#11718322C211yesmaximal, characteristic, complemented
11#11718310C211yesmaximal, characteristic, complemented
12C2×C13×C4⋊C4C2211yescharacteristic
13C2×C13×C4⋊C4C2212yes—
14C2×C13×C22⋊C4C2211yescharacteristic
15C2×C13×C22⋊C4C2212yes—
16C2×C22⋊Dic13C2211yescharacteristic
17C2×C22⋊Dic13C2211yescharacteristic
18C2×C22⋊Dic13C2212yes—
19C2×C4⋊Dic13C2211yescharacteristic
20C2×C4⋊Dic13C2211yescharacteristic
21C2×C4⋊Dic13C2212yes—
22C2×C4⋊Dic13C2212yes—
23C2×#6101C2211yescharacteristic
24C2×#6101C2211yescharacteristic
25C2×#6101C2212yes—
26C2×#6101C2212yes—
27C2×C4×Dic13C2212yes—
28C2×C4×Dic13C2211yescharacteristic
29#88144—24no—
30#88133—24no—
31C13×#95C2211yescharacteristic
32#88071C2211yescharacteristic
33#88071C2212yes—
34#88071C2212yes—
35#88071C2212yes—
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C22×C4×C13C2311yescharacteristic
39C22×C4×C13C2312yes—
40C22×C4×C13C2311yescharacteristic
41C22×C4×C13C2312yes—
42C22×Dic13C2312yes—
43C22×Dic13C2311yescharacteristic
44C22×Dic13C2312yes—
45C22×Dic13C2311yescharacteristic
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2311yescharacteristic
48C13×C4⋊C4—22no—
49C13×C4⋊C4—24no—
50C13×C22⋊C4—22no—
51C13×C22⋊C4—24no—
52C22⋊Dic13—22no—
53C22⋊Dic13—22no—
54C22⋊Dic13—42no—
55C4⋊Dic13—22no—
56C4⋊Dic13—22no—
57C4⋊Dic13—24no—
58C4⋊Dic13—42no—
59#6101—22no—
60#6101—24no—
61#6101—22no—
62#6101—24no—
63C4×Dic13—42no—
64C4×Dic13D414yes—
65#2664—131nomaximal, Sylow
66C23×C13—21no—
67C23×C13—22no—
68C23×C13—21no—
69C23×C13—22no—
70C23×C13—21no—
71C23×C13C2411yescharacteristic
72C2×C4×C13—22no—
73C2×C4×C13—22no—
74C2×C4×C13—22no—
75C2×C4×C13—21no—
76C2×C4×C13—22no—
77C2×C4×C13—22no—
78C2×C4×C13—22no—
79C2×C4×C13—22no—
80C2×C4×C13C2×D412yes—
81C2×C4×C13#4114yes—
82C2×Dic13—22no—
83C2×Dic13—21no—
84C2×Dic13—21no—
85C2×Dic13—21no—
86C2×Dic13—22no—
87C2×Dic13—21no—
88C2×Dic13—22no—
89C2×Dic13—22no—
90C2×Dic13—21no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—21no—
95C2×Dic13—22no—
96C2×Dic13C2×D412yes—
97C2×Dic13C2×D412yes—
98C2×#126—131no—
99C2×#125—131no—
100#400—131no—
101#399D1311yescharacteristic
102#398—132no—
103#398—131no—
104#397—131no—
105#396—132no—
106#389—132no—
107#388—132no—
108C4×C4⋊C4—131no—
109C22×C13—41no—
110C22×C13—41no—
111C22×C13—41no—
112C22×C13—42no—
113C22×C13—42no—
114C22×C13C22×D411yescharacteristic
115C22×C13C2×#4112yes—
116C22×C13H(5,𝔽2)11yescharacteristic
117C22×C132−1+411yescharacteristic
118C22×C132−1+412yes—
119C4×C13—22no—
120C4×C13—24no—
121C4×C13—41no—
122C4×C13—42no—
123Dic13—22no—
124Dic13—22no—
125Dic13—41no—
126Dic13—42no—
127Dic13—41no—
128Dic13—42no—
129#126—264no—
130#125—264no—
131C2×C4⋊C4—132no—
132C2×C4⋊C4—132no—
133C2×C4⋊C4—131no—
134C2×C4⋊C4—131no—
135C2×C4⋊C4—131no—
136C2×C4⋊C4—131no—
137C2×C4⋊C4—132no—
138C2×C4⋊C4—132no—
139C2×C4⋊C4C2×D1311yescharacteristic
140C2×C4⋊C4C2×D1312yes—
141C2×C22⋊C4—131no—
142C2×C22⋊C4—131no—
143C2×C22⋊C4—132no—
144C2×C22⋊C4C2×D1311yescharacteristic
145C2×C22⋊C4C2×D1312yes—
146C2×C42—131no—
147C2×C42—132no—
148#95—131no—
149#95—132no—
150#95—132no—
151#95—132no—
152#95—132no—
153#95C2×D1311yescharacteristic
154C2×C13—81no—
155C2×C13#53211yescharacteristic
156C2×C13#53611yescharacteristic
157C2×C13#53711yescharacteristic
158C2×C13#53912yes—
159C2×C13#54112yes—
160C24C22×D1311yescharacteristic
161C22×C4—131no—
162C22×C4—131no—
163C22×C4—132no—
164C22×C4—131no—
165C22×C4—131no—
166C22×C4—132no—
167C22×C4C22×D1311yescharacteristic
168C22×C4C22×D1312yes—
169C22×C4C22×D1311yescharacteristic
170C22×C4C22×D1312yes—
171C4⋊C4—22no—
172C4⋊C4—24no—
173C4⋊C4—264no—
174C4⋊C4—264no—
175C4⋊C4—262no—
176C4⋊C4—262no—
177C4⋊C4—262no—
178C4⋊C4—262no—
179C4⋊C4—264no—
180C4⋊C4—522no—
181C22⋊C4—22no—
182C22⋊C4—24no—
183C22⋊C4—262no—
184C22⋊C4—262no—
185C22⋊C4—522no—
186C42—134no—
187C42—522no—
188C13#266411yescharacteristic, complemented, Sylow
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23C23×D1311yescharacteristic
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—132no—
204C2×C4—132no—
205C2×C4—262no—
206C2×C4—261no—
207C2×C4—261no—
208C2×C4—261no—
209C2×C4—261no—
210C2×C4—262no—
211C2×C4—262no—
212C2×C4—261no—
213C2×C4—262no—
214C2×C4—262no—
215C2×C4—262no—
216C2×C4—261no—
217C2×C4—262no—
218C2×C4—262no—
219C2×C4D4×D1312yes—
220C2×C4#612914yes—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—41no—
226C22C2×D4×D1311yescharacteristic
227C22C2×#612912yes—
228C22#8825411yescharacteristic
229C22#8825611yescharacteristic
230C22#8825712yes—
231C4—22no—
232C4—24no—
233C4—41no—
234C4—42no—
235C4—262no—
236C4—262no—
237C4—521no—
238C4—522no—
239C4—521no—
240C4—522no—
241C2—81no—
242C2#1171933012yes—
243C2#1171933411yescharacteristic
244C2#1171935311yescharacteristic
245C2#1171937012yes—
246C2#1171942111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#399C2
prime radicalC24×C13C23
hypercenter#399D13
hypocenterC13#2664
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#88144C2semidirect
4. #11718646C2semidirect
5. #11718643C2semidirect
6. #11718641C2semidirect
10. #11718322C2semidirect
11. #11718310C2semidirect
188. C13#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#881442
C2×#881332
C13×#3992
#117186462
#117186432
#117186412
#117186142
#117186122
#117186092
#117183222
#117183102
#266413