Finite Group Databasebeta

← #16792510

Subgroups — #16792510

Order 3328.

1334 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 272 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×#11719060C212yesmaximal
2#13130025C214yesmaximal
3#13130021C218yesmaximal
4C2×D13×#125C211yesmaximal, characteristic
5C22×C4⋊#261C2214yes—
6C2×C13×#125C411yescharacteristic, complemented
7D13×#125—28no—
8C2×C4⋊C4×D13C2214yes—
9C2×C4⋊C4×D13C2212yes—
10C2×#88144C412yescomplemented
11C2×#88143C414yescomplemented
12C2×C42×D13C2211yescharacteristic
13C2×#88124C411yescharacteristic, complemented
14C2×(C2×#261)⋊C4C2214yes—
15C2×(C2×#261)⋊C4C2214yes—
16C2×(C2×#261)⋊C4C22116yes—
17#11719060—28no—
18C23×#261C2318yes—
19C22×C4×D13C2314yes—
20C22×C4×D13C2312yes—
21C22×C4×D13C2311yescharacteristic
22C2×C4⋊#261—28no—
23C2×C4⋊#261—48no—
24C13×#125—24no—
25C2×C13×C4⋊C4C2×C414yes—
26C2×C13×C4⋊C4C2×C412yes—
27C2×C42×C13C2×C411yescharacteristic
28C2×C4⋊Dic13C2×C412yes—
29C2×C4⋊Dic13C2×C414yes—
30C2×#6101C2×C414yes—
31C2×#6101C2×C414yes—
32C2×#6101C2×C414yes—
33C2×C4×Dic13C2×C412yes—
34C2×C4×Dic13C2×C411yescharacteristic
35C4⋊C4×D13—28no—
36C4⋊C4×D13—28no—
37C4⋊C4×D13—24no—
38C4⋊C4×D13—24no—
39#88144—28no—
40#88143—216no—
41C42×D13D418yes—
42#88124—24no—
43(C2×#261)⋊C4—28no—
44(C2×#261)⋊C4—28no—
45(C2×#261)⋊C4—416no—
46#14664—131nomaximal, Sylow
47C23×D13C2411yescharacteristic
48C22×#261—28no—
49C22×#261—28no—
50C22×#261—216no—
51C22×#261—216no—
52C22×#261—28no—
53C22×C4×C13C22×C414yes—
54C22×C4×C13C22×C412yes—
55C22×C4×C13C22×C411yescharacteristic
56C22×Dic13C22×C414yes—
57C22×Dic13C22×C411yescharacteristic
58C22×Dic13C22×C412yes—
59C2×C4×D13—24no—
60C2×C4×D13—24no—
61C2×C4×D13—24no—
62C2×C4×D13—24no—
63C2×C4×D13—24no—
64C2×C4×D13—28no—
65C2×C4×D13—28no—
66C2×C4×D13C2×D414yes—
67C2×C4×D13C2×D414yes—
68C2×C4×D13C2×D414yes—
69C4⋊#261—416no—
70C13×C4⋊C4—28no—
71C13×C4⋊C4—24no—
72C42×C13C22⋊C414yes—
73C4⋊Dic13—24no—
74C4⋊Dic13—28no—
75#6101—28no—
76#6101—28no—
77#6101—28no—
78C4×Dic13C22⋊C418yes—
79C4×Dic13C22⋊C414yes—
80C22×#125—131no—
81C2×#382—132no—
82#1445—134no—
83#1441—138no—
84C23×C13C23×C411yescharacteristic
85C22×D13C22×D412yes—
86C22×D13H(5,𝔽2)14yes—
87C22×D132−1+414yes—
88C22×D132−1+414yes—
89C2×#261—48no—
90C2×#261—48no—
91C2×#261—48no—
92C2×#261—416no—
93C2×#261—416no—
94C2×C4×C13—22no—
95C2×C4×C13—24no—
96C2×C4×C13—24no—
97C2×C4×C13—28no—
98C2×C4×C13C2×C22⋊C414yes—
99C2×C4×C13C2×C22⋊C412yes—
100C2×Dic13—22no—
101C2×Dic13—24no—
102C2×Dic13—24no—
103C2×Dic13—28no—
104C2×Dic13C2×C22⋊C412yes—
105C2×Dic13C2×C22⋊C414yes—
106C4×D13—24no—
107C4×D13—28no—
108C4×D13—24no—
109C4×D13—28no—
110C4×D13—44no—
111C4×D13—44no—
112C4×D13—48no—
113C2×#125—134no—
114C2×#125—132no—
115C2×#125—131no—
116C2×#125—268no—
117C2×#125#26111yescharacteristic, complemented
118C22×C4⋊C4—134no—
119C22×C4⋊C4—132no—
120C22×C4⋊C4—134no—
121C22×C42—131no—
122#382—268no—
123C2×#95—134no—
124C2×#95—134no—
125C2×#95—1316no—
126C22×C13C22×C22⋊C411yescharacteristic
127C22×C13#51912yes—
128C22×C13#52014yes—
129C2×D13#53512yes—
130C2×D13#53614yes—
131C2×D13#53612yes—
132C2×D13#53714yes—
133C2×D13#56214yes—
134C2×D13#56318yes—
135C2×D13#56414yes—
136#261—88no—
137C4×C13—22no—
138C4×C13—24no—
139C4×C13—44no—
140Dic13—22no—
141Dic13—24no—
142Dic13—44no—
143C23×C4—132no—
144C23×C4—131no—
145C23×C4—134no—
146C23×C4—138no—
147#125—24no—
148#125—2616no—
149#125—268no—
150#125—264no—
151C2×C4⋊C4—134no—
152C2×C4⋊C4—134no—
153C2×C4⋊C4—134no—
154C2×C4⋊C4—134no—
155C2×C4⋊C4—132no—
156C2×C4⋊C4—264no—
157C2×C4⋊C4—264no—
158C2×C4⋊C4—268no—
159C2×C4⋊C4—268no—
160C2×C4⋊C4—268no—
161C2×C4⋊C4—528no—
162C2×C4⋊C4C2×#26114yes—
163C2×C4⋊C4C2×#26112yes—
164C2×C42—132no—
165C2×C42—131no—
166C2×C42—138no—
167C2×C42C2×#26111yescharacteristic
168#95—268no—
169#95—268no—
170#95—5216no—
171C2×C13#231011yescharacteristic
172C2×C13#231112yes—
173C2×C13#238214yes—
174D13#282014yes—
175D13#282214yes—
176C24—131no—
177C22×C4—131no—
178C22×C4—132no—
179C22×C4—134no—
180C22×C4—134no—
181C22×C4—134no—
182C22×C4—134no—
183C22×C4—264no—
184C22×C4—264no—
185C22×C4—264no—
186C22×C4—268no—
187C22×C4—264no—
188C22×C4—264no—
189C22×C4—268no—
190C22×C4—268no—
191C22×C4—268no—
192C22×C4—2616no—
193C22×C4—2616no—
194C22×C4—268no—
195C22×C4C22×#26114yes—
196C22×C4C22×#26111yescharacteristic
197C22×C4C22×#26112yes—
198C4⋊C4—28no—
199C4⋊C4—24no—
200C4⋊C4—268no—
201C4⋊C4—268no—
202C4⋊C4—268no—
203C4⋊C4—268no—
204C4⋊C4—264no—
205C4⋊C4—5216no—
206C42—138no—
207C42—134no—
208C42C22⋊#26114yes—
209C13#1466411yescharacteristic, complemented, Sylow
210C23—134no—
211C23—134no—
212C23—134no—
213C23—132no—
214C23C23×#26111yescharacteristic
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—28no—
218C2×C4—24no—
219C2×C4—132no—
220C2×C4—134no—
221C2×C4—264no—
222C2×C4—268no—
223C2×C4—264no—
224C2×C4—268no—
225C2×C4—262no—
226C2×C4—264no—
227C2×C4—268no—
228C2×C4—264no—
229C2×C4—524no—
230C2×C4—524no—
231C2×C4—528no—
232C2×C4—528no—
233C2×C4—528no—
234C2×C4—528no—
235C2×C4—5216no—
236C2×C4—5216no—
237C2×C4C2×C22⋊#26112yes—
238C2×C4C2×C22⋊#26114yes—
239C22—134no—
240C22—132no—
241C22—132no—
242C22—134no—
243C22—134no—
244C22—138no—
245C22—134no—
246C22#1171962712yes—
247C22#1171963114yes—
248C22C22×C22⋊#26111yescharacteristic
249C4—22no—
250C4—24no—
251C4—44no—
252C4—262no—
253C4—264no—
254C4—524no—
255C4—1048no—
256C2—134no—
257C2—134no—
258C2#1313131011yescharacteristic
259C2#1313131112yes—
260C2#1313138214yes—
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC2×C13×#125C4
prime radical, cosocleC23×D13C24
hypercenterC2×#125#261
hypocenterC13#14664
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
6. C2×C13×#125C4semidirect
10. C2×#88144C4semidirect
11. C2×#88143C4semidirect
13. C2×#88124C4semidirect
117. C2×#125#261semidirect
209. C13#14664semidirect

Maximal subgroups

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

subgroupindex
C2×#117190602
#131300252
#131300212
C2×D13×#1252
#1466413