Finite Group Databasebeta

← #13118155

Subgroups — #13118155

Order 1664.

1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#6127C211yesmaximal, characteristic, complemented
2C22×C13×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×#88207C212yesmaximal, complemented
4#11718874C212yesmaximal, complemented
5#11718872C211yesmaximal, characteristic, complemented
6#11718621C212yesmaximal, complemented
7#11718617C214yesmaximal, complemented
8#11718608C212yesmaximal, complemented
9C25×C13C2211yescharacteristic
10C23×C4×C13C2211yescharacteristic
11C23×C4×C13C2211yescharacteristic
12C22×C22⋊D13C2214yes—
13C22×C22⋊D13C2212yes—
14C2×#6127—24no—
15C2×#6127—28no—
16C22×D52C2211yescharacteristic, complemented
17C22×C4×D13C2212yescomplemented
18C22×Dic26C2211yescharacteristic, complemented
19C2×C13×C22⋊C4—24no—
20C2×C13×C22⋊C4C2212yes—
21C2×C13×C22⋊C4C2212yes—
22#88207—28no—
23C2×C22⋊Dic13C2214yes—
24C2×C22⋊Dic13C2212yes—
25C2×#6103C2214yescomplemented
26#88071C2212yescomplemented
27#88071C2214yescomplemented
28#88071C2212yescomplemented
29C24×C13—22no—
30C24×C13—24no—
31C24×C13—24no—
32C24×C13—42no—
33C24×C13C2312yes—
34C24×C13C2311yescharacteristic
35C23×D13C2312yes—
36C22×C4×C13—24no—
37C22×C4×C13—22no—
38C22×C4×C13—42no—
39C22×C4×C13D414yescomplemented
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C2×C22⋊D13—24no—
44C2×C22⋊D13—24no—
45C2×C22⋊D13—28no—
46C2×C22⋊D13—28no—
47C2×C22⋊D13—44no—
48C2×C22⋊D13D418yescomplemented
49C22×Dic13C2314yes—
50C22×Dic13C2312yes—
51#6127—232no—
52C2×D52—24no—
53C2×D52—22no—
54C2×C4×D13—24no—
55C2×C4×D13—24no—
56C2×C4×D13—24no—
57C2×Dic26—22no—
58C2×Dic26—24no—
59C13×C22⋊C4—24no—
60C13×C22⋊C4—24no—
61C22⋊Dic13—24no—
62C22⋊Dic13—28no—
63#6103—44no—
64#2404—131nomaximal, Sylow
65C23×C13—22no—
66C23×C13—22no—
67C23×C13—28no—
68C23×C13—24no—
69C23×C13—22no—
70C23×C13—22no—
71C23×C13—24no—
72C23×C13—416no—
73C23×C13—44no—
74C23×C13—44no—
75C23×C13C2×D414yes—
76C23×C13C2×D412yes—
77C23×C13#4114yescomplemented
78C23×C13C2411yescharacteristic
79C22×D13—22no—
80C22×D13—22no—
81C22×D13—24no—
82C22×D13—24no—
83C22×D13C2×D414yes—
84C2×C4×C13—28no—
85C2×C4×C13—22no—
86C2×C4×C13—22no—
87C2×C4×C13—22no—
88C2×C4×C13—24no—
89C2×C4×C13—44no—
90C2×C4×C13C2×D412yes—
91C2×C4×C13C2×D412yes—
92C22⋊D13—216no—
93C22⋊D13—416no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—24no—
97C2×Dic13—24no—
98C2×Dic13—24no—
99C2×Dic13C2×D414yes—
100D52—28no—
101C4×D13—216no—
102Dic26—28no—
103C22×#41—131no—
104C2×#120—132no—
105C22×C22⋊C4D1311yescharacteristic, complemented
106#393—132no—
107#392—132no—
108#386—132no—
109#386—134no—
110#381—131no—
111C22×C13—22no—
112C22×C13—22no—
113C22×C13—28no—
114C22×C13—24no—
115C22×C13—28no—
116C22×C13—416no—
117C22×C13—42no—
118C22×C13—42no—
119C22×C13—44no—
120C22×C13#12014yes—
121C22×C13C22×D412yes—
122C22×C13C22×D411yescharacteristic
123C22×C13C2×#4112yes—
124C22×C13H(5,𝔽2)12yes—
125C2×D13—24no—
126C2×D13—28no—
127C2×D13—42no—
128C2×D13—42no—
129C2×D13—44no—
130C4×C13—22no—
131C4×C13—22no—
132C4×C13—42no—
133Dic13—24no—
134Dic13—44no—
135C25C2×D1311yescharacteristic
136C2×#41—268no—
137C2×#41—264no—
138C22×Q8—131no—
139C22×D4—132no—
140C22×D4—134no—
141C22×D4—131no—
142C23×C4—132no—
143C23×C4C2×D1311yescharacteristic
144C23×C4C2×D1311yescharacteristic
145#120—268no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—132no—
148C2×C22⋊C4—134no—
149C2×C22⋊C4—134no—
150C2×C22⋊C4C2×D1312yes—
151C2×C22⋊C4C2×D1312yes—
152#95—132no—
153#95—132no—
154#95—134no—
155C2×C13—24no—
156C2×C13—44no—
157C2×C13C2×#12011yescharacteristic
158C2×C13#52511yescharacteristic
159C2×C13#53511yescharacteristic
160C2×C13#54614yes—
161D13—44no—
162C24—24no—
163C24—22no—
164C24—24no—
165C24—42no—
166C24—132no—
167C24C22×D1312yes—
168C24C22×D1311yescharacteristic
169#41—2632no—
170C2×Q8—264no—
171C2×Q8—262no—
172C2×D4—138no—
173C2×D4—264no—
174C2×D4—264no—
175C2×D4—264no—
176C2×D4—268no—
177C2×D4—268no—
178C2×D4—262no—
179C2×D4—524no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—132no—
184C22×C4—134no—
185C22×C4—264no—
186C22×C4—264no—
187C22×C4—264no—
188C22×C4C22⋊D1314yescomplemented
189C22×C4C22×D1312yes—
190C22×C4C22×D1311yescharacteristic
191C22×C4C22×D1311yescharacteristic
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—268no—
195C22⋊C4—264no—
196C22⋊C4—524no—
197C13#240411yescharacteristic, complemented, Sylow
198C23—22no—
199C23—22no—
200C23—28no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—22no—
205C23—416no—
206C23—44no—
207C23—44no—
208C23—134no—
209C23—264no—
210C23—262no—
211C23—264no—
212C23—262no—
213C23#612714yescomplemented
214C23D4×D1314yes—
215C23C2×C22⋊D1312yes—
216C23C23×D1311yescharacteristic
217Q8—268no—
218D4—268no—
219D4—2616no—
220D4—5216no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—134no—
228C2×C4—2616no—
229C2×C4—262no—
230C2×C4—262no—
231C2×C4—264no—
232C2×C4—264no—
233C2×C4—264no—
234C2×C4C2×C22⋊D1312yes—
235C2×C4C2×C22⋊D1312yes—
236C22—28no—
237C22—22no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—416no—
242C22—44no—
243C22—42no—
244C22—42no—
245C22—268no—
246C22—264no—
247C22—524no—
248C22—522no—
249C22—522no—
250C22#8819114yes—
251C22C2×#612712yes—
252C22C2×D4×D1312yes—
253C22#8825112yes—
254C22C22×C22⋊D1311yescharacteristic
255C4—22no—
256C4—22no—
257C4—42no—
258C4—264no—
259C4—524no—
260C2—24no—
261C2—44no—
262C2—524no—
263C2#1171919514yes—
264C2#1171949611yescharacteristic
265C2C2×#8819111yescharacteristic
266C2#1171951011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC22×C13×C22⋊C4C2
hypercenterC22×C22⋊C4D13
hypocenterC13#2404
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6127C2semidirect
2. C22×C13×C22⋊C4C2semidirect
3. C2×#88207C2semidirect
4. #11718874C2semidirect
5. #11718872C2semidirect
6. #11718621C2semidirect
7. #11718617C2semidirect
8. #11718608C2semidirect
16. C22×D52C22semidirect
17. C22×C4×D13C22semidirect
18. C22×Dic26C22semidirect
25. C2×#6103C22semidirect
26. #88071C22semidirect
27. #88071C22semidirect
28. #88071C22semidirect
39. C22×C4×C13D4semidirect
48. C2×C22⋊D13D4semidirect
77. C23×C13#41semidirect
105. C22×C22⋊C4D13semidirect
188. C22×C4C22⋊D13semidirect
197. C13#2404semidirect
213. C23#6127semidirect

Maximal subgroups

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

subgroupindex
C22×#61272
C22×C13×C22⋊C42
C2×#882072
#117188742
#117188722
#117186212
#117186172
#117186082
#240413