Finite Group Databasebeta

← #13123139

Subgroups — #13123139

Order 1664.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×C13×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#88201C211yesmaximal, characteristic, complemented
3C2×#88184C211yesmaximal, characteristic
4C22×C4×Dic13C211yesmaximal, characteristic, complemented
5#11718891C211yesmaximal, characteristic, complemented
6#11718651C214yesmaximal, complemented
7#11718649C211yesmaximal, characteristic, complemented
8#11718638C211yesmaximal, characteristic, complemented
9#11718619C214yesmaximal, complemented
10C22×Q8×C13C2211yescharacteristic
11C23×C4×C13C2211yescharacteristic
12C22×C22⋊D13C2212yes—
13C23×Dic13C2212yes—
14C22×D52C2211yescharacteristic
15C13×C22⋊Q8—24no—
16C2×C13×C4⋊C4C2211yescharacteristic
17C2×C13×C4⋊C4C2212yes—
18C2×C13×C22⋊C4C2212yes—
19#88201—24no—
20#88184—24no—
21C2×#6103C2212yes—
22C2×#6103C2214yes—
23C2×#6103C2214yes—
24C2×C4⋊Dic13C2211yescharacteristic
25C2×C4×Dic13—24no—
26C2×C4×Dic13C2211yescharacteristic
27C2×C4×Dic13C2212yes—
28C2×C4×Dic13C2211yescharacteristic
29#88071C2214yes—
30#88071C2214yes—
31C24×C13C2311yescharacteristic
32C23×D13C2312yes—
33C2×Q8×C13—22no—
34C2×Q8×C13—22no—
35C2×Q8×C13—22no—
36C22×C4×C13—22no—
37C22×C4×C13—22no—
38C22×C4×C13D414yes—
39C22×C4×C13C2312yes—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C2×C22⋊D13—24no—
44C2×C22⋊D13—28no—
45C22×Dic13—24no—
46C22×Dic13—24no—
47C22×Dic13—24no—
48C22×Dic13C2312yes—
49C22×Dic13C2312yes—
50C22×Dic13C2312yes—
51C2×D52—22no—
52C2×D52—22no—
53C2×D52—22no—
54C13×C4⋊C4—22no—
55C13×C4⋊C4—24no—
56C13×C22⋊C4—24no—
57#6103—24no—
58#6103—28no—
59#6103—28no—
60C4⋊Dic13—22no—
61C4×Dic13—22no—
62C4×Dic13—24no—
63C4×Dic13D414yescomplemented
64#2850—131nomaximal, Sylow
65C23×C13—21no—
66C23×C13—22no—
67C23×C13—21no—
68C23×C13C2×D412yes—
69C23×C13#4114yes—
70C23×C13C2411yescharacteristic
71C22×D13—22no—
72C22×D13—22no—
73C22×D13—24no—
74C22×D13—22no—
75C22×D13—24no—
76Q8×C13—44no—
77C2×C4×C13—21no—
78C2×C4×C13—21no—
79C2×C4×C13—28no—
80C2×C4×C13—22no—
81C2×C4×C13—22no—
82C2×C4×C13—22no—
83C2×C4×C13—24no—
84C2×C4×C13—22no—
85C2×C4×C13C2×D412yes—
86C2×C4×C13C2×D412yes—
87C2×C4×C13C2×D412yes—
88C2×C4×C13#4112yes—
89C22⋊D13—48no—
90C2×Dic13—216no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—24no—
97C2×Dic13C2×D414yes—
98C2×Dic13#4114yes—
99D52—44no—
100C2×#124—131no—
101C2×C22⋊Q8D1311yescharacteristic, complemented
102C2×#121—131no—
103C22×C42—131no—
104#391—131no—
105#390—131no—
106#388—134no—
107#386—134no—
108#382—131no—
109C22×C13—22no—
110C22×C13—24no—
111C22×C13—24no—
112C22×C13—22no—
113C22×C13#12414yescomplemented
114C22×C13C22×D411yescharacteristic
115C22×C13C22×D411yescharacteristic
116C22×C13C2×#4111yescharacteristic
117C22×C13C2×#4112yes—
118C22×C13C2×#4112yes—
119C2×D13—42no—
120C2×D13—44no—
121C2×D13—42no—
122C2×D13—44no—
123C2×D13—42no—
124C4×C13—22no—
125C4×C13—42no—
126C4×C13—42no—
127C4×C13#12114yes—
128Dic13—24no—
129Dic13—24no—
130Dic13—42no—
131C22×Q8C2×D1311yescharacteristic
132C22×D4—132no—
133C22×D4—131no—
134C23×C4—132no—
135C23×C4C2×D1311yescharacteristic
136#124—264no—
137C22⋊Q8—24no—
138#121—264no—
139C2×C4⋊C4—131no—
140C2×C4⋊C4C2×D1311yescharacteristic
141C2×C4⋊C4C2×D1312yes—
142C2×C22⋊C4—132no—
143C2×C22⋊C4—134no—
144C2×C22⋊C4—134no—
145C2×C22⋊C4C2×D1312yes—
146C2×C42—131no—
147C2×C42—132no—
148C2×C42—131no—
149C2×C42—264no—
150#95—134no—
151#95—134no—
152C2×C13—24no—
153C2×C13C2×#12111yescharacteristic
154C2×C13#52512yes—
155C2×C13C2×#12411yescharacteristic
156C2×C13#52912yes—
157C2×C13#53211yescharacteristic
158D13—82no—
159C24—132no—
160C24C22×D1311yescharacteristic
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×D4—268no—
165C2×D4—264no—
166C2×D4—262no—
167C2×D4—262no—
168C2×D4—262no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—132no—
172C22×C4—132no—
173C22×C4—132no—
174C22×C4—264no—
175C22×C4—264no—
176C22×C4—264no—
177C22×C4C22⋊D1314yes—
178C22×C4C22×D1312yes—
179C22×C4C22×D1312yes—
180C22×C4C22×D1311yescharacteristic
181C22×C4C22×D1311yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—262no—
185C22⋊C4—24no—
186C22⋊C4—268no—
187C22⋊C4—268no—
188C22⋊C4—264no—
189C42—134no—
190C42—264no—
191C42—262no—
192C13#285011yescharacteristic, complemented, Sylow
193C23—21no—
194C23—21no—
195C23—22no—
196C23—262no—
197C23—262no—
198C23—264no—
199C23—264no—
200C23—262no—
201C23#613114yes—
202C23C2×C22⋊D1312yes—
203C23C23×D1311yescharacteristic
204Q8—44no—
205D4—528no—
206D4—524no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—134no—
216C2×C4—134no—
217C2×C4—2616no—
218C2×C4—262no—
219C2×C4—262no—
220C2×C4—262no—
221C2×C4—262no—
222C2×C4—264no—
223C2×C4—262no—
224C2×C4D4×D1312yes—
225C2×C4#612912yes—
226C2×C4C2×C22⋊D1312yes—
227C2×C4C2×C22⋊D1312yes—
228C22—22no—
229C22—22no—
230C22—24no—
231C22—24no—
232C22—522no—
233C22—522no—
234C22—522no—
235C22—524no—
236C22—524no—
237C22#8820114yescomplemented
238C22C2×D4×D1311yescharacteristic
239C22C2×#612911yescharacteristic
240C22C2×#613112yes—
241C22#41×D1312yes—
242C22C22×C22⋊D1311yescharacteristic
243C4—22no—
244C4—42no—
245C4—42no—
246C4—264no—
247C4—264no—
248C4—522no—
249C4#8819314yes—
250C2—24no—
251C2—1042no—
252C2#1171932912yes—
253C2#1171933212yes—
254C2C2×#8819311yescharacteristic
255C2C2×#8820111yescharacteristic
256C2#1171953711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC2×C13×C22⋊Q8C2
prime radicalC2×#88184C2
hypercenterC2×C22⋊Q8D13
hypocenterC13#2850
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C13×C22⋊Q8C2semidirect
2. C2×#88201C2semidirect
4. C22×C4×Dic13C2semidirect
5. #11718891C2semidirect
6. #11718651C2semidirect
7. #11718649C2semidirect
8. #11718638C2semidirect
9. #11718619C2semidirect
63. C4×Dic13D4semidirect
101. C2×C22⋊Q8D13semidirect
113. C22×C13#124semidirect
192. C13#2850semidirect
237. C22#88201semidirect

Maximal subgroups

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

subgroupindex
C2×C13×C22⋊Q82
C2×#882012
C2×#881842
C22×C4×Dic132
#117188912
#117186512
#117186492
#117186382
#117186192
#285013