Finite Group Databasebeta

← #13118466

Subgroups — #13118466

Order 1664.

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

normal subgroups only

This group has 836 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×#6129C211yesmaximal, characteristic
2C2×#88190C211yesmaximal, characteristic, complemented
3C2×#88151C211yesmaximal, characteristic, complemented
4C2×#88123C211yesmaximal, characteristic, complemented
5C13×#390C211yesmaximal, characteristic, complemented
6#11718635C211yesmaximal, characteristic, complemented
7#11718622C214yesmaximal, complemented
8#11718611C214yesmaximal, complemented
9C4×#6103C211yesmaximal, characteristic, complemented
10C22×D4×C13C2211yescharacteristic
11C22×C22⋊D13C2212yes—
12C23×Dic13C2212yes—
13C2×#6129—24no—
14C2×#6129—24no—
15C2×#6129—24no—
16C22×C4×D13C2211yescharacteristic
17C22×Dic26C2211yescharacteristic
18C22×Dic26C2211yescharacteristic
19C2×C13×C4⋊C4C2211yescharacteristic
20C2×C13×C22⋊C4C2214yes—
21C2×C42×C13C2211yescharacteristic
22C2×C22⋊Dic13C2214yes—
23#88190—24no—
24C2×#6103C2212yes—
25C2×#6103C2212yes—
26C2×C4⋊Dic13C2214yes—
27C2×C4⋊Dic13C2212yes—
28C2×C4×Dic13C2211yescharacteristic
29#88151—24no—
30#88123—24no—
31#88071C2212yes—
32#88071C2214yes—
33C24×C13C2312yes—
34C23×D13C2311yescharacteristic
35C2×D4×C13—22no—
36C2×D4×C13—22no—
37C2×D4×C13D414yes—
38C22×C4×C13C2312yes—
39C22×C4×C13C2312yes—
40C22×C4×C13C2311yescharacteristic
41C2×C22⋊D13—24no—
42C2×C22⋊D13—24no—
43C2×C22⋊D13—24no—
44C22×Dic13—24no—
45C22×Dic13—24no—
46C22×Dic13—24no—
47C22×Dic13C2312yes—
48C22×Dic13C2312yes—
49C22×Dic13C2312yes—
50C22×Dic13C2311yescharacteristic
51#6129—232no—
52C2×C4×D13—22no—
53C2×C4×D13—22no—
54C2×C4×D13D414yes—
55C2×Dic26—22no—
56C2×Dic26—22no—
57C2×Dic26—22no—
58C2×Dic26—22no—
59C2×Dic26—22no—
60C2×Dic26D414yes—
61C13×C4⋊C4—22no—
62C13×C22⋊C4—28no—
63C42×C13—22no—
64C22⋊Dic13—28no—
65#6103—24no—
66#6103—24no—
67C4⋊Dic13—28no—
68C4⋊Dic13—24no—
69C4×Dic13—22no—
70#2432—131nomaximal, Sylow
71C23×C13—22no—
72C23×C13—22no—
73C23×C13—24no—
74C23×C13—24no—
75C23×C13C2×D414yes—
76C23×C13C2411yescharacteristic
77C22×D13—21no—
78C22×D13—21no—
79C22×D13—22no—
80C22×D13—22no—
81C22×D13C2×D412yes—
82D4×C13—28no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13—24no—
87C2×C4×C13C2×D412yes—
88C2×C4×C13C2×D412yes—
89C2×C4×C13C2×D412yes—
90C2×C4×C13#4114yescomplemented
91C22⋊D13—216no—
92C2×Dic13—216no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—21no—
96C2×Dic13—21no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13—22no—
102C2×Dic13—22no—
103C2×Dic13C2×D414yes—
104C2×Dic13C2×D412yes—
105C4×D13—28no—
106Dic26—28no—
107Dic26—44no—
108C22×#41—131no—
109C2×C4⋊Q8—131no—
110C2×#124—131no—
111C2×C22⋊Q8—131no—
112#394—134no—
113#391—131no—
114#390D1311yescharacteristic, complemented
115#385—134no—
116C4×C22⋊C4—131no—
117C22×C13—24no—
118C22×C13—28no—
119C22×C13—42no—
120C22×C13—42no—
121C22×C13—44no—
122C22×C13C22×D411yescharacteristic
123C22×C13C22×D411yescharacteristic
124C22×C13C22×D411yescharacteristic
125C22×C13C2×#4112yes—
126C22×C132−1+412yes—
127C2×D13—22no—
128C2×D13—22no—
129C2×D13—22no—
130C2×D13—42no—
131C2×D13—42no—
132C4×C13—24no—
133C4×C13—42no—
134C4×C13#12014yes—
135Dic13—24no—
136Dic13—22no—
137Dic13—42no—
138Dic13—42no—
139C2×#41—264no—
140C2×#41—264no—
141C2×#41—264no—
142C22×Q8—131no—
143C22×Q8—131no—
144C22×D4—132no—
145C22×D4C2×D1311yescharacteristic
146C23×C4—131no—
147C23×C4—132no—
148C4⋊Q8—264no—
149#124—264no—
150C22⋊Q8—264no—
151C2×C4⋊C4—132no—
152C2×C4⋊C4—134no—
153C2×C4⋊C4C2×D1311yescharacteristic
154C2×C22⋊C4—134no—
155C2×C22⋊C4—132no—
156C2×C22⋊C4—132no—
157C2×C22⋊C4C2×D1314yes—
158C2×C42—131no—
159C2×C42C2×D1311yescharacteristic
160#95—132no—
161#95—134no—
162C2×C13—44no—
163C2×C13C2×#12011yescharacteristic
164C2×C13#53211yescharacteristic
165C2×C13#53611yescharacteristic
166C2×C13#54712yes—
167C2×C13#54812yes—
168D13—42no—
169C24—131no—
170C24C22×D1312yes—
171#41—2632no—
172C2×Q8—134no—
173C2×Q8—262no—
174C2×Q8—262no—
175C2×Q8—262no—
176C2×Q8—262no—
177C2×Q8—262no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—264no—
181C2×D4—264no—
182C2×D4—264no—
183C2×D4D5214yes—
184C22×C4—131no—
185C22×C4—132no—
186C22×C4—134no—
187C22×C4—132no—
188C22×C4—132no—
189C22×C4—264no—
190C22×C4—264no—
191C22×C4—264no—
192C22×C4—262no—
193C22×C4—262no—
194C22×C4C22×D1312yes—
195C22×C4C22×D1312yes—
196C22×C4C22×D1311yescharacteristic
197C4⋊C4—22no—
198C4⋊C4—264no—
199C4⋊C4—268no—
200C22⋊C4—28no—
201C22⋊C4—264no—
202C22⋊C4—268no—
203C22⋊C4—264no—
204C42—22no—
205C42—262no—
206C13#243211yescharacteristic, complemented, Sylow
207C23—22no—
208C23—22no—
209C23—24no—
210C23—24no—
211C23—132no—
212C23—261no—
213C23—261no—
214C23—262no—
215C23—262no—
216C23C2×D5214yes—
217C23C23×D1311yescharacteristic
218Q8—268no—
219Q8—524no—
220D4—28no—
221D4—2616no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—132no—
227C2×C4—134no—
228C2×C4—2616no—
229C2×C4—261no—
230C2×C4—262no—
231C2×C4—261no—
232C2×C4—262no—
233C2×C4—268no—
234C2×C4—262no—
235C2×C4—262no—
236C2×C4—262no—
237C2×C4—262no—
238C2×C4—262no—
239C2×C4—262no—
240C2×C4C2×D5212yes—
241C2×C4D4×D1312yes—
242C2×C4D4×D1312yes—
243C2×C4#612914yescomplemented
244C22—24no—
245C22—28no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—262no—
250C22—262no—
251C22—262no—
252C22—522no—
253C22—522no—
254C22C22×D5211yescharacteristic
255C22C2×D4×D1311yescharacteristic
256C22C2×D4×D1311yescharacteristic
257C22C2×#612912yes—
258C22#8825712yes—
259C4—24no—
260C4—42no—
261C4—262no—
262C4—264no—
263C4—522no—
264C4—522no—
265C4#8813614yes—
266C2—44no—
267C2—522no—
268C2C2×#8813611yescharacteristic
269C2#1171926012yes—
270C2#1171930312yes—
271C2#1171938011yescharacteristic
272C2#1171942111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#390C2
prime radicalC22×#6129C2
hypercenter#390D13
hypocenterC13#2432
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×#88123C2semidirect
5. C13×#390C2semidirect
6. #11718635C2semidirect
7. #11718622C2semidirect
8. #11718611C2semidirect
9. C4×#6103C2semidirect
90. C2×C4×C13#41semidirect
114. #390D13semidirect
206. C13#2432semidirect
243. C2×C4#6129semidirect

Maximal subgroups

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

subgroupindex
C22×#61292
C2×#881902
C2×#881512
C2×#881232
C13×#3902
#117186352
#117186222
#117186112
C4×#61032
#243213