Finite Group Databasebeta

← #13123136

Subgroups — #13123136

Order 1664.

696 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 165 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×#88192C211yesmaximal, characteristic
2C2×#88190C211yesmaximal, characteristic, complemented
3C2×#88151C211yesmaximal, characteristic, complemented
4C2×C42×D13C211yesmaximal, characteristic, complemented
5C13×#390C211yesmaximal, characteristic, complemented
6#11718638C211yesmaximal, characteristic, complemented
7#11718617C214yesmaximal, complemented
8#11718612C214yesmaximal, complemented
9#11718590C211yesmaximal, characteristic, complemented
10C22×D4×C13C2211yescharacteristic
11C22×C22⋊D13C2212yes—
12C22×C4×D13C2212yes—
13C22×C4×D13C2211yescharacteristic
14C22×Dic26C2211yescharacteristic
15C2×C13×C4⋊C4C2211yescharacteristic
16C2×C13×C22⋊C4C2214yes—
17C2×C42×C13C2211yescharacteristic
18C2×C22⋊Dic13C2214yes—
19C2×C22⋊Dic13C2212yes—
20#88192—24no—
21#88190—24no—
22C2×#6103C2212yes—
23C2×C4⋊Dic13C2212yes—
24C2×C4⋊Dic13C2211yescharacteristic
25C2×C4×Dic13C2212yes—
26C2×C4×Dic13C2211yescharacteristic
27#88151—24no—
28C42×D13—24no—
29#88071C2214yes—
30#88071C2214yes—
31C24×C13C2312yes—
32C23×D13C2311yescharacteristic
33C2×D4×C13—22no—
34C2×D4×C13—22no—
35C2×D4×C13—22no—
36C22×C4×C13C2312yes—
37C22×C4×C13C2312yes—
38C22×C4×C13C2311yescharacteristic
39C2×C22⋊D13—24no—
40C2×C22⋊D13—28no—
41C22×Dic13C2312yes—
42C22×Dic13C2312yes—
43C22×Dic13C2312yes—
44C22×Dic13C2311yescharacteristic
45C2×C4×D13—24no—
46C2×C4×D13—22no—
47C2×C4×D13—24no—
48C2×C4×D13—22no—
49C2×C4×D13—24no—
50C2×C4×D13D414yes—
51C2×Dic26—22no—
52C2×Dic26—22no—
53C2×Dic26—22no—
54C13×C4⋊C4—22no—
55C13×C22⋊C4—28no—
56C42×C13—22no—
57C22⋊Dic13—28no—
58C22⋊Dic13—24no—
59#6103—24no—
60C4⋊Dic13—22no—
61C4⋊Dic13—24no—
62C4×Dic13—24no—
63C4×Dic13D414yescomplemented
64#2850—131nomaximal, Sylow
65C23×C13—22no—
66C23×C13—22no—
67C23×C13—24no—
68C23×C13—24no—
69C23×C13—22no—
70C23×C13C2411yescharacteristic
71C22×D13—21no—
72C22×D13—21no—
73C22×D13—22no—
74C22×D13C2×D412yes—
75C22×D13#4114yes—
76D4×C13—44no—
77C2×C4×C13—22no—
78C2×C4×C13—22no—
79C2×C4×C13—22no—
80C2×C4×C13—24no—
81C2×C4×C13C2×D412yes—
82C2×C4×C13C2×D412yes—
83C2×C4×C13#4114yes—
84C2×C4×C13#4112yes—
85C22⋊D13—48no—
86C2×Dic13—21no—
87C2×Dic13—21no—
88C2×Dic13—22no—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—24no—
95C2×Dic13C2×D414yes—
96C2×Dic13C2×D412yes—
97C4×D13—216no—
98C4×D13—28no—
99Dic26—44no—
100C2×#124—131no—
101C2×C22⋊Q8—131no—
102C2×#121—131no—
103C22×C42—131no—
104#391—131no—
105#390D1311yescharacteristic, complemented
106#388—134no—
107#386—134no—
108#382—131no—
109C22×C13—42no—
110C22×C13—44no—
111C22×C13—42no—
112C22×C13—44no—
113C22×C13—42no—
114C22×C13C22×D411yescharacteristic
115C22×C13C22×D411yescharacteristic
116C22×C13C2×#4111yescharacteristic
117C22×C13C2×#4112yes—
118C22×C13C2×#4112yes—
119C2×D13—22no—
120C2×D13—24no—
121C2×D13—24no—
122C2×D13—22no—
123C2×D13#12414yescomplemented
124C4×C13—24no—
125C4×C13—42no—
126C4×C13#12114yes—
127Dic13—22no—
128Dic13—24no—
129Dic13—42no—
130Dic13—42no—
131C22×Q8—131no—
132C22×D4—132no—
133C22×D4C2×D1311yescharacteristic
134C23×C4—131no—
135C23×C4—132no—
136#124—264no—
137C22⋊Q8—264no—
138#121—264no—
139C2×C4⋊C4—131no—
140C2×C4⋊C4—132no—
141C2×C4⋊C4C2×D1311yescharacteristic
142C2×C22⋊C4—132no—
143C2×C22⋊C4—134no—
144C2×C22⋊C4—132no—
145C2×C22⋊C4C2×D1314yes—
146C2×C42—131no—
147C2×C42—132no—
148C2×C42—264no—
149C2×C42C2×D1311yescharacteristic
150#95—134no—
151#95—134no—
152C2×C13—82no—
153C2×C13C2×#12111yescharacteristic
154C2×C13#52512yes—
155C2×C13C2×#12411yescharacteristic
156C2×C13#52912yes—
157C2×C13#53211yescharacteristic
158D13—24no—
159C24—131no—
160C24C22×D1312yes—
161C2×Q8—262no—
162C2×Q8—262no—
163C2×Q8—262no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—264no—
168C2×D4—268no—
169C22×C4—132no—
170C22×C4—131no—
171C22×C4—134no—
172C22×C4—132no—
173C22×C4—132no—
174C22×C4—264no—
175C22×C4—264no—
176C22×C4—262no—
177C22×C4—262no—
178C22×C4—264no—
179C22×C4C22×D1312yes—
180C22×C4C22×D1312yes—
181C22×C4C22×D1311yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—264no—
184C4⋊C4—262no—
185C22⋊C4—28no—
186C22⋊C4—264no—
187C22⋊C4—264no—
188C22⋊C4—268no—
189C42—22no—
190C42—134no—
191C42—264no—
192C13#285011yescharacteristic, complemented, Sylow
193C23—22no—
194C23—22no—
195C23—22no—
196C23—24no—
197C23—24no—
198C23—132no—
199C23—134no—
200C23—261no—
201C23—261no—
202C23—262no—
203C23C23×D1311yescharacteristic
204Q8—524no—
205D4—44no—
206D4—528no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—134no—
212C2×C4—132no—
213C2×C4—2616no—
214C2×C4—261no—
215C2×C4—261no—
216C2×C4—262no—
217C2×C4—262no—
218C2×C4—268no—
219C2×C4—262no—
220C2×C4—262no—
221C2×C4—262no—
222C2×C4—262no—
223C2×C4—264no—
224C2×C4#612712yes—
225C2×C4D4×D1312yes—
226C2×C4D4×D1312yes—
227C2×C4#612914yes—
228C22—42no—
229C22—44no—
230C22—44no—
231C22—42no—
232C22—42no—
233C22—134no—
234C22—262no—
235C22—264no—
236C22—264no—
237C22—262no—
238C22C2×#612711yescharacteristic
239C22C2×D4×D1311yescharacteristic
240C22C2×D4×D1311yescharacteristic
241C22C2×#612912yes—
242C22#41×D1312yes—
243C4—24no—
244C4—42no—
245C4—262no—
246C4—264no—
247C4—522no—
248C4—522no—
249C4#8813814yes—
250C2—82no—
251C2—264no—
252C2C2×#8813811yescharacteristic
253C2#1171926212yes—
254C2#1171931112yes—
255C2D13×#12411yescharacteristic
256C2#1171942111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#390C2
prime radicalC2×#88192C2
hypercenter#390D13
hypocenterC13#2850
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#88190C2semidirect
3. C2×#88151C2semidirect
4. C2×C42×D13C2semidirect
5. C13×#390C2semidirect
6. #11718638C2semidirect
7. #11718617C2semidirect
8. #11718612C2semidirect
9. #11718590C2semidirect
63. C4×Dic13D4semidirect
105. #390D13semidirect
123. C2×D13#124semidirect
192. C13#2850semidirect

Maximal subgroups

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

subgroupindex
C2×#881922
C2×#881902
C2×#881512
C2×C42×D132
C13×#3902
#117186382
#117186172
#117186122
#117185902
#285013