Finite Group Databasebeta

← #13121853

Subgroups — #13121853

Order 1664.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#88132C212yesmaximal, complemented
2C13×#390C211yesmaximal, characteristic
3#11718881C212yesmaximal
4#11718645C212yesmaximal, complemented
5#11718611C212yesmaximal, complemented
6#11718610C212yesmaximal, complemented
7#11718609C212yesmaximal, complemented
8#11718590C211yesmaximal, characteristic, complemented
9#11718589C211yesmaximal, characteristic, complemented
10C22×D4×C13C2211yescharacteristic
11C23×Dic13C2212yes—
12C22×Dic26C2211yescharacteristic
13C2×C13×C4⋊C4C2211yescharacteristic
14C2×C13×C22⋊C4C2212yes—
15C2×C13×C22⋊C4C2212yes—
16C2×C42×C13C2211yescharacteristic
17C2×C22⋊Dic13C2212yes—
18C2×C22⋊Dic13C2212yes—
19C2×C22⋊Dic13C2212yes—
20C2×C4⋊Dic13C2212yes—
21C2×C4⋊Dic13C2211yescharacteristic
22C2×C4⋊Dic13C2211yescharacteristic
23C2×C4⋊Dic13C2211yescharacteristic
24C2×#6101C2212yes—
25C2×#6101C2212yes—
26C2×#6101C2212yes—
27#88132—28no—
28#88071C2212yes—
29#88071C2212yes—
30#88071C2212yes—
31#88071C2212yes—
32C24×C13C2312yes—
33C2×D4×C13—41no—
34C2×D4×C13—41no—
35C2×D4×C13Q814yescomplemented
36C22×C4×C13C2312yes—
37C22×C4×C13C2311yescharacteristic
38C22×C4×C13C2311yescharacteristic
39C22×C4×C13C2311yescharacteristic
40C22×Dic13—24no—
41C22×Dic13—42no—
42C22×Dic13—42no—
43C22×Dic13C2312yes—
44C22×Dic13C2312yes—
45C22×Dic13C2312yes—
46C22×Dic13C2312yes—
47C2×Dic26—41no—
48C2×Dic26—41no—
49C2×Dic26D414yescomplemented
50C13×C4⋊C4—41no—
51C13×C22⋊C4—24no—
52C13×C22⋊C4—24no—
53C42×C13—41no—
54C22⋊Dic13—24no—
55C22⋊Dic13—42no—
56C22⋊Dic13—42no—
57C4⋊Dic13—22no—
58C4⋊Dic13—24no—
59C4⋊Dic13—41no—
60C4⋊Dic13—41no—
61#6101—24no—
62#6101—24no—
63#6101—42no—
64#2732—131nomaximal, Sylow
65C23×C13—22no—
66C23×C13—22no—
67C23×C13—22no—
68C23×C13—22no—
69C23×C13—22no—
70C23×C13—22no—
71C23×C13C2×Q814yes—
72C23×C13C2411yescharacteristic
73D4×C13—44no—
74C2×C4×C13—21no—
75C2×C4×C13—21no—
76C2×C4×C13—21no—
77C2×C4×C13—21no—
78C2×C4×C13—21no—
79C2×C4×C13—21no—
80C2×C4×C13—22no—
81C2×C4×C13—22no—
82C2×C4×C13—22no—
83C2×C4×C13C2×D412yes—
84C2×C4×C13C2×Q812yes—
85C2×C4×C13#4112yes—
86C2×Dic13—22no—
87C2×Dic13—22no—
88C2×Dic13—22no—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—22no—
97C2×Dic13—48no—
98C2×Dic13C2×D414yes—
99Dic26—44no—
100C2×C22⋊Q8—132no—
101#399—132no—
102#398—132no—
103#396—132no—
104#391—131no—
105#390D1311yescharacteristic
106#389—131no—
107#386—132no—
108#385—132no—
109C22×C13—24no—
110C22×C13—24no—
111C22×C13—24no—
112C22×C13—42no—
113C22×C13—42no—
114C22×C13—42no—
115C22×C13—42no—
116C22×C13C22×D411yescharacteristic
117C22×C13C22×Q811yescharacteristic
118C22×C13C2×#4111yescharacteristic
119C22×C13H(5,𝔽2)11yescharacteristic
120C22×C13H(5,𝔽2)11yescharacteristic
121C22×C13H(5,𝔽2)11yescharacteristic
122C22×C132−1+411yescharacteristic
123C4×C13—22no—
124C4×C13—22no—
125C4×C13—22no—
126C4×C13—42no—
127Dic13—24no—
128Dic13—42no—
129Dic13—42no—
130Dic13—42no—
131C22×Q8—131no—
132C22×D4C2×D1311yescharacteristic
133C23×C4—132no—
134C22⋊Q8—268no—
135C2×C4⋊C4—132no—
136C2×C4⋊C4—132no—
137C2×C4⋊C4—132no—
138C2×C4⋊C4—131no—
139C2×C4⋊C4—132no—
140C2×C4⋊C4—131no—
141C2×C4⋊C4—131no—
142C2×C4⋊C4C2×D1311yescharacteristic
143C2×C22⋊C4—132no—
144C2×C22⋊C4—132no—
145C2×C22⋊C4—132no—
146C2×C22⋊C4C2×D1312yes—
147C2×C22⋊C4C2×D1312yes—
148C2×C42C2×D1311yescharacteristic
149#95—132no—
150#95—132no—
151#95—132no—
152#95—132no—
153C2×C13—44no—
154C2×C13#53411yescharacteristic
155C2×C13#53911yescharacteristic
156C2×C13#54311yescharacteristic
157C2×C13D4×Q811yescharacteristic
158C2×C13#55011yescharacteristic
159C2×C13#55411yescharacteristic
160C2×C13#56211yescharacteristic
161C24C22×D1312yes—
162C2×Q8—134no—
163C2×Q8—521no—
164C2×Q8—521no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4Dic2614yescomplemented
168C22×C4—132no—
169C22×C4—132no—
170C22×C4—132no—
171C22×C4—132no—
172C22×C4—264no—
173C22×C4—522no—
174C22×C4—522no—
175C22×C4C22×D1312yes—
176C22×C4C22×D1311yescharacteristic
177C22×C4C22×D1311yescharacteristic
178C22×C4C22×D1311yescharacteristic
179C4⋊C4—41no—
180C4⋊C4—264no—
181C4⋊C4—264no—
182C4⋊C4—264no—
183C4⋊C4—262no—
184C4⋊C4—522no—
185C4⋊C4—521no—
186C4⋊C4—521no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—264no—
190C22⋊C4—522no—
191C22⋊C4—522no—
192C42—41no—
193C13#273211yescharacteristic, complemented, Sylow
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23C2×Dic2614yes—
201C23C23×D1311yescharacteristic
202Q8—524no—
203D4—44no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—134no—
214C2×C4—262no—
215C2×C4—262no—
216C2×C4—262no—
217C2×C4—262no—
218C2×C4—262no—
219C2×C4—262no—
220C2×C4—262no—
221C2×C4—262no—
222C2×C4—262no—
223C2×C4—262no—
224C2×C4—262no—
225C2×C4—528no—
226C2×C4C2×Dic2612yes—
227C2×C4D4×D1312yes—
228C2×C4#612912yes—
229C22—24no—
230C22—24no—
231C22—24no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22C22×Dic2611yescharacteristic
237C22C2×D4×D1311yescharacteristic
238C22C2×#612911yescharacteristic
239C22#8825111yescharacteristic
240C22#8825111yescharacteristic
241C22#8825611yescharacteristic
242C22#8825711yescharacteristic
243C4—22no—
244C4—22no—
245C4—22no—
246C4—42no—
247C4—264no—
248C4—522no—
249C4—522no—
250C4—522no—
251C2—44no—
252C2#1171918811yescharacteristic
253C2D4×Dic2611yescharacteristic
254C2#1171924811yescharacteristic
255C2#1171930711yescharacteristic
256C2#1171931011yescharacteristic
257C2#1171938911yescharacteristic
258C2#1171942311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#390C2
prime radicalC22×D4×C13C22
hypercenter#390D13
hypocenterC13#2732
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#88132C2semidirect
4. #11718645C2semidirect
5. #11718611C2semidirect
6. #11718610C2semidirect
7. #11718609C2semidirect
8. #11718590C2semidirect
9. #11718589C2semidirect
35. C2×D4×C13Q8semidirect
49. C2×Dic26D4semidirect
167. C2×D4Dic26semidirect
193. C13#2732semidirect

Maximal subgroups

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

subgroupindex
C2×#881322
C13×#3902
#117188812
#117186452
#117186112
#117186102
#117186092
#117185902
#117185892
#273213