Finite Group Databasebeta

← #13123120

Subgroups — #13123120

Order 1664.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D13C211yesmaximal, characteristic
2C2×C13×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#88200C212yesmaximal, complemented
4#11718873C211yesmaximal, characteristic, complemented
5#11718650C214yesmaximal, complemented
6#11718648C212yesmaximal, complemented
7#11718616C212yesmaximal
8#11718607C212yesmaximal, complemented
9C24×D13C2211yescharacteristic
10C22×Q8×C13C2211yescharacteristic, complemented
11C23×C4×C13C2211yescharacteristic
12C23×Dic13C2211yescharacteristic
13C22×C4×D13—24no—
14C22×C4×D13—28no—
15C22×C4×D13C2212yes—
16C22×C4×D13C2212yes—
17C13×C22⋊Q8—24no—
18C2×C13×C4⋊C4C2211yescharacteristic, complemented
19C2×C13×C4⋊C4C2212yescomplemented
20C2×C13×C22⋊C4C2212yes—
21C2×C22⋊Dic13C2212yes—
22#88200—28no—
23C2×#6103C2214yes—
24C2×#6103C2214yes—
25C2×C4⋊Dic13C2212yescomplemented
26C2×#6101C2214yescomplemented
27#88071C2214yescomplemented
28#88071C2212yescomplemented
29C24×C13C2311yescharacteristic
30C23×D13—22no—
31C23×D13—22no—
32C23×D13—24no—
33C23×D13—24no—
34C23×D13Q814yescomplemented
35C23×D13C2312yes—
36C2×Q8×C13—24no—
37C2×Q8×C13—22no—
38C22×C4×C13—24no—
39C22×C4×C13D414yes—
40C22×C4×C13C2312yes—
41C22×C4×C13C2312yes—
42C22×C4×C13C2311yescharacteristic
43C22×C4×C13C2311yescharacteristic
44C22×Dic13—22no—
45C22×Dic13—24no—
46C22×Dic13C2314yes—
47C22×Dic13C2312yes—
48C2×C4×D13—232no—
49C2×C4×D13—24no—
50C2×C4×D13—24no—
51C2×C4×D13—24no—
52C2×C4×D13—24no—
53C2×C4×D13—24no—
54C2×C4×D13D418yes—
55C13×C4⋊C4—22no—
56C13×C4⋊C4—24no—
57C13×C22⋊C4—24no—
58C22⋊Dic13—24no—
59#6103—28no—
60#6103—28no—
61C4⋊Dic13—24no—
62#6101—28no—
63#2848—131nomaximal, Sylow
64C23×C13—22no—
65C23×C13—22no—
66C23×C13C2×D412yes—
67C23×C13C2×Q812yes—
68C23×C13#4112yes—
69C23×C13C2411yescharacteristic
70C22×D13—28no—
71C22×D13—216no—
72C22×D13—28no—
73C22×D13—28no—
74C22×D13—22no—
75C22×D13—22no—
76C22×D13—22no—
77C22×D13—22no—
78C22×D13—216no—
79C22×D13C2×D414yes—
80C22×D13C2×Q814yes—
81C22×D13#4114yes—
82Q8×C13—44no—
83C2×C4×C13—22no—
84C2×C4×C13—28no—
85C2×C4×C13—24no—
86C2×C4×C13—22no—
87C2×C4×C13—22no—
88C2×C4×C13—24no—
89C2×C4×C13C2×D412yes—
90C2×C4×C13C2×D414yes—
91C2×C4×C13C2×D412yes—
92C2×Dic13—28no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—24no—
96C2×Dic13—24no—
97C2×Dic13—24no—
98C2×Dic13C2×D414yes—
99C4×D13—216no—
100C4×D13—216no—
101C24×C4—131no—
102C2×C22⋊Q8—132no—
103C2×C22⋊Q8D1311yescharacteristic, complemented
104#385—134no—
105#385—132no—
106#380—131no—
107#380—132no—
108#379—132no—
109C22×C13—22no—
110C22×C13—24no—
111C22×C13—22no—
112C22×C13—24no—
113C22×C13C22⋊Q814yescomplemented
114C22×C13C22×D412yes—
115C22×C13C22×D411yescharacteristic
116C22×C13C22×Q811yescharacteristic
117C22×C13C2×#4111yescharacteristic
118C22×C13C2×#4112yes—
119C2×D13—232no—
120C2×D13—24no—
121C2×D13—24no—
122C2×D13—24no—
123C2×D13—24no—
124C2×D13—24no—
125C2×D13—24no—
126C2×D13C22⋊Q818yescomplemented
127C4×C13—22no—
128C4×C13—42no—
129C4×C13—42no—
130C4×C13#12014yes—
131Dic13—24no—
132Dic13—44no—
133C25—131no—
134C22×Q8C2×D1311yescharacteristic, complemented
135C23×C4—132no—
136C23×C4—131no—
137C23×C4—132no—
138C23×C4—268no—
139C23×C4—264no—
140C23×C4C2×D1311yescharacteristic
141C22⋊Q8—24no—
142C22⋊Q8—268no—
143C2×C4⋊C4—134no—
144C2×C4⋊C4—132no—
145C2×C4⋊C4C2×D1311yescharacteristic, complemented
146C2×C4⋊C4C2×D1312yescomplemented
147C2×C22⋊C4—132no—
148C2×C22⋊C4—134no—
149C2×C22⋊C4—134no—
150C2×C22⋊C4C2×D1312yes—
151#95—134no—
152#95—132no—
153C2×C13—24no—
154C2×C13C2×#12011yescharacteristic
155C2×C13C2×C22⋊Q811yescharacteristic
156C2×C13C2×C22⋊Q812yes—
157C2×C13#52511yescharacteristic
158C2×C13#52512yes—
159D13—28no—
160C24—132no—
161C24—134no—
162C24—264no—
163C24—264no—
164C24—262no—
165C24—262no—
166C24C22×D1311yescharacteristic
167C2×Q8—24no—
168C2×Q8—22no—
169C22×C4—24no—
170C22×C4—132no—
171C22×C4—138no—
172C22×C4—134no—
173C22×C4—264no—
174C22×C4—264no—
175C22×C4—2632no—
176C22×C4—264no—
177C22×C4—262no—
178C22×C4—264no—
179C22×C4—264no—
180C22×C4—264no—
181C22×C4C22⋊D1314yes—
182C22×C4C22×D1312yes—
183C22×C4C22×D1312yes—
184C22×C4C22×D1311yescharacteristic
185C22×C4C22×D1311yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—24no—
188C4⋊C4—268no—
189C4⋊C4—264no—
190C22⋊C4—24no—
191C22⋊C4—268no—
192C22⋊C4—268no—
193C22⋊C4—264no—
194C13#284811yescharacteristic, complemented, Sylow
195C23—22no—
196C23—22no—
197C23—134no—
198C23—134no—
199C23—134no—
200C23—2616no—
201C23—268no—
202C23—268no—
203C23—2616no—
204C23—268no—
205C23—262no—
206C23—262no—
207C23—262no—
208C23—262no—
209C23Q8×D1312yes—
210C23#613112yes—
211C23C2×C22⋊D1312yes—
212C23C23×D1311yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—134no—
221C2×C4—2616no—
222C2×C4—268no—
223C2×C4—262no—
224C2×C4—262no—
225C2×C4—2616no—
226C2×C4—264no—
227C2×C4—264no—
228C2×C4—264no—
229C2×C4D4×D1314yes—
230C2×C4C2×C22⋊D1312yes—
231C2×C4C2×C22⋊D1312yes—
232C22—22no—
233C22—22no—
234C22—24no—
235C22—24no—
236C22—138no—
237C22—2632no—
238C22—264no—
239C22—264no—
240C22—264no—
241C22—264no—
242C22—264no—
243C22—264no—
244C22#8820014yescomplemented
245C22C2×D4×D1312yes—
246C22C2×Q8×D1311yescharacteristic
247C22C2×#613111yescharacteristic
248C22#41×D1312yes—
249C22C22×C22⋊D1311yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—264no—
254C4—524no—
255C4#8819114yes—
256C2—24no—
257C2—268no—
258C2D13×C22⋊Q812yes—
259C2#1171933212yes—
260C2C2×#8819111yescharacteristic
261C2C2×#8820011yescharacteristic
262C2#1171953311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC2×C13×C22⋊Q8C2
prime radicalC24×D13C22
hypercenterC2×C22⋊Q8D13
hypocenterC13#2848
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C13×C22⋊Q8C2semidirect
3. C2×#88200C2semidirect
4. #11718873C2semidirect
5. #11718650C2semidirect
6. #11718648C2semidirect
8. #11718607C2semidirect
10. C22×Q8×C13C22semidirect
18. C2×C13×C4⋊C4C22semidirect
19. C2×C13×C4⋊C4C22semidirect
25. C2×C4⋊Dic13C22semidirect
26. C2×#6101C22semidirect
27. #88071C22semidirect
28. #88071C22semidirect
34. C23×D13Q8semidirect
103. C2×C22⋊Q8D13semidirect
113. C22×C13C22⋊Q8semidirect
126. C2×D13C22⋊Q8semidirect
134. C22×Q8C2×D13semidirect
145. C2×C4⋊C4C2×D13semidirect
146. C2×C4⋊C4C2×D13semidirect
194. C13#2848semidirect
244. C22#88200semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D132
C2×C13×C22⋊Q82
C2×#882002
#117188732
#117186502
#117186482
#117186162
#117186072
#284813