Finite Group Databasebeta

← #16800699

Subgroups — #16800699

Order 3328.

1754 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 1754 conjugacy classes of subgroups — too many to draw. Only its 264 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×#11719128C212yesmaximal, complemented
2#13130962C218yesmaximal, complemented
3#13130659C214yesmaximal, complemented
4C22×#88147C211yesmaximal, characteristic
5C23×D52C411yescharacteristic
6C23×C4×D13C2212yes—
7C22×C13×C4⋊C4C411yescharacteristic
8C22×#6103C412yes—
9C22×C4×Dic13C2211yescharacteristic
10C2×#88147—28no—
11C2×#88147C414yes—
12C2×#88147C2214yes—
13C2×C22⋊#1057C2214yes—
14#11719128—28no—
15C2×#88112C2214yescomplemented
16C2×#88111C22116yescomplemented
17C2×#88110C2214yescomplemented
18C24×D13C2×C412yes—
19C23×C4×C13C2×C411yescharacteristic
20C23×C4×C13C2×C412yes—
21C23×Dic13C2×C411yescharacteristic
22C23×Dic13C2311yescharacteristic
23C22×D52—28no—
24C22×D52—24no—
25C22×D52C2×C414yes—
26C22×C4×D13—28no—
27C22×C4×D13—48no—
28C22×C4×D13C2×C414yes—
29C22×C4×D13C2314yes—
30C22×#1057C2318yes—
31C2×C13×C4⋊C4—24no—
32C2×C13×C4⋊C4C2×C414yes—
33C2×#6103—28no—
34C2×#6103C2×C418yes—
35C2×C4×Dic13C2×C412yes—
36C2×C4×Dic13D418yes—
37C2×C4×Dic13C2312yes—
38#88147—216no—
39#88147—28no—
40#88147—28no—
41C22⋊#1057—28no—
42#88112—28no—
43#88111—416no—
44#88110—28no—
45#21790—131nomaximal, Sylow
46C24×C13C22×C411yescharacteristic
47C23×D13—24no—
48C23×D13—24no—
49C23×D13—44no—
50C23×D13—44no—
51C23×D13M1618yes—
52C23×D13C22×C414yes—
53C22×C4×C13—24no—
54C22×C4×C13—24no—
55C22×C4×C13—44no—
56C22×C4×C13C22⋊C414yes—
57C22×C4×C13C22×C414yes—
58C22×C4×C13C22×C412yes—
59C22×Dic13—24no—
60C22×Dic13—42no—
61C22×Dic13C22⋊C414yes—
62C22×Dic13C22×C411yescharacteristic
63C22×Dic13C22×C412yes—
64C22×Dic13C2×D414yes—
65C22×Dic13C2411yescharacteristic
66C2×D52—232no—
67C2×D52—216no—
68C2×D52—28no—
69C2×C4×D13—28no—
70C2×C4×D13—28no—
71C2×C4×D13—216no—
72C2×C4×D13—48no—
73C2×C4×D13—432no—
74C2×#1057—28no—
75C2×#1057—216no—
76C13×C4⋊C4—28no—
77#6103—216no—
78C4×Dic13C22⋊C418yes—
79C4×Dic13C2×D414yes—
80C4×Dic13C2×D414yes—
81C22×C4×D4—131no—
82C2×#432—132no—
83#1955—138no—
84#1779—134no—
85C23×C13—22no—
86C23×C13—22no—
87C23×C13C2×C22⋊C412yes—
88C23×C13C2×M1614yes—
89C23×C13C23×C411yescharacteristic
90C22×D13—24no—
91C22×D13—28no—
92C22×D13—24no—
93C22×D13—28no—
94C22×D13—216no—
95C22×D13—416no—
96C22×D13—48no—
97C22×D13—48no—
98C22×D13—416no—
99C22×D13C2×M1618yes—
100C2×C4×C13—24no—
101C2×C4×C13—28no—
102C2×C4×C13—24no—
103C2×C4×C13—28no—
104C2×C4×C13—48no—
105C2×C4×C13C2×C22⋊C414yes—
106C2×Dic13—22no—
107C2×Dic13—24no—
108C2×Dic13—28no—
109C2×Dic13—44no—
110C2×Dic13C2×C22⋊C412yes—
111C2×Dic13C2×C22⋊C414yes—
112C2×Dic13C22×D412yes—
113C2×Dic13H(5,𝔽2)14yes—
114D52—232no—
115C4×D13—416no—
116C4×D13—416no—
117#1057—48no—
118C23×D4—131no—
119C24×C4—132no—
120C2×C4×D4—134no—
121C2×C4×D4—134no—
122C2×C4×D4—268no—
123C22×C4⋊C4#26111yescharacteristic
124C22×C22⋊C4—132no—
125C22×C42—131no—
126#432—268no—
127C2×C4⋊C8—134no—
128C2×C22⋊C8—134no—
129C2×C22⋊C8—1316no—
130C2×#97—134no—
131C22×C13—22no—
132C22×C13—22no—
133C22×C13—28no—
134C22×C13#40714yes—
135C22×C13C22×C22⋊C411yescharacteristic
136C22×C13#51812yes—
137C22×C13C22×M1612yes—
138C22×C13#56812yes—
139C2×D13—28no—
140C2×D13—216no—
141C2×D13—432no—
142C2×D13—44no—
143C2×D13—44no—
144C2×D13—48no—
145C4×C13—24no—
146C4×C13—44no—
147Dic13—22no—
148Dic13—24no—
149Dic13#53514yes—
150C25—132no—
151C22×D4—134no—
152C22×D4—268no—
153C22×D4—264no—
154C23×C4—131no—
155C23×C4—131no—
156C23×C4—134no—
157C23×C4—134no—
158C23×C4—268no—
159C23×C4—528no—
160C23×C4C2×#26111yescharacteristic
161C23×C4C2×#26112yes—
162C22×C8—138no—
163C4×D4—268no—
164C4×D4—268no—
165C4×D4—2616no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4C2×#26114yes—
168C2×C22⋊C4—138no—
169C2×C22⋊C4—268no—
170C2×C42—138no—
171C2×C42—132no—
172C2×C42—132no—
173C4⋊C8—268no—
174C22⋊C8—268no—
175C22⋊C8—5216no—
176#97—268no—
177C2×C13—24no—
178C2×C13#230911yescharacteristic
179C2×C13C2×#40711yescharacteristic
180C2×C13#287911yescharacteristic
181C2×C13#297314yes—
182D13—48no—
183C24—134no—
184C24—138no—
185C24—264no—
186C24—264no—
187C24—524no—
188C24—524no—
189C24C22×#26111yescharacteristic
190C2×D4—2616no—
191C2×D4—2632no—
192C2×D4—268no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—44no—
196C22×C4—131no—
197C22×C4—134no—
198C22×C4—132no—
199C22×C4—131no—
200C22×C4—134no—
201C22×C4—2616no—
202C22×C4—268no—
203C22×C4—268no—
204C22×C4—264no—
205C22×C4—528no—
206C22×C4—5232no—
207C22×C4—522no—
208C22×C4C22⋊#26114yes—
209C22×C4C22×#26114yes—
210C22×C4C22×#26112yes—
211C2×C8—268no—
212C2×C8—2616no—
213C4⋊C4—28no—
214C22⋊C4—2616no—
215C42—134no—
216C42—134no—
217C42—138no—
218C13#2179011yescharacteristic, complemented, Sylow
219C23—22no—
220C23—22no—
221C23—138no—
222C23—264no—
223C23—268no—
224C23—268no—
225C23—264no—
226C23—2616no—
227C23—5216no—
228C23—528no—
229C23—528no—
230C23—5216no—
231C23D13⋊M1614yes—
232C23C2×C22⋊#26112yes—
233C23C23×#26111yescharacteristic
234D4—2632no—
235C2×C4—24no—
236C2×C4—28no—
237C2×C4—28no—
238C2×C4—24no—
239C2×C4—48no—
240C2×C4—134no—
241C2×C4—132no—
242C2×C4—134no—
243C2×C4—132no—
244C2×C4—268no—
245C2×C4—262no—
246C2×C4—264no—
247C2×C4—5216no—
248C2×C4—5216no—
249C2×C4—524no—
250C2×C4C2×C22⋊#26114yes—
251C8—528no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—268no—
256C22—2616no—
257C22—5232no—
258C22—524no—
259C22—524no—
260C22—528no—
261C22#1171912614yes—
262C22C2×D13⋊M1612yes—
263C22#1171962312yes—
264C22#1171962512yes—
265C22C22×C22⋊#26111yescharacteristic
266C4—24no—
267C4—44no—
268C4—134no—
269C4—262no—
270C4—264no—
271C2—24no—
272C2—528no—
273C2#1313130911yescharacteristic
274C2C2×#1171912611yescharacteristic
275C2#1313142311yescharacteristic
276C2#1313169114yes—
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC22×C13×C4⋊C4C4
prime radicalC23×D52C4
cosocleC22×Dic13C24
hypercenterC22×C4⋊C4#261
hypocenterC13#21790
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11719128C2semidirect
2. #13130962C2semidirect
3. #13130659C2semidirect
15. C2×#88112C22semidirect
16. C2×#88111C22semidirect
17. C2×#88110C22semidirect
218. C13#21790semidirect

Maximal subgroups

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

subgroupindex
C2×#117191282
#131309622
#131306592
C22×#881472
#2179013