Finite Group Databasebeta

← #13118463

Subgroups — #13118463

Order 1664.

836 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 169 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×#6131C211yesmaximal, characteristic
2C2×#88198C211yesmaximal, characteristic, complemented
3C2×#88151C211yesmaximal, characteristic, complemented
4C2×#88129C211yesmaximal, characteristic, complemented
5C13×#391C211yesmaximal, characteristic, complemented
6#11718649C211yesmaximal, characteristic, complemented
7C4×#6103C211yesmaximal, characteristic, complemented
8#11718333C214yesmaximal, complemented
9#11718329C214yesmaximal, complemented
10C22×Q8×C13C2211yescharacteristic
11C2×#6131—24no—
12C2×#6131—24no—
13C2×#6131—24no—
14C22×D52C2212yes—
15C22×D52C2211yescharacteristic
16C22×C4×D13C2212yes—
17C22×C4×D13C2211yescharacteristic
18C22×Dic26C2211yescharacteristic
19C2×C13×C4⋊C4C2211yescharacteristic
20C2×C42×C13C2211yescharacteristic
21#88198—24no—
22C2×#6103C2214yes—
23C2×#6103C2212yes—
24C2×#6103C2214yes—
25C2×#6103C2212yes—
26C2×C4⋊Dic13C2212yes—
27C2×#6101C2214yes—
28C2×C4×Dic13C2211yescharacteristic
29#88151—24no—
30#88129—24no—
31C13×#95C2214yes—
32#88071C2212yes—
33C23×D13C2312yes—
34C23×D13C2311yescharacteristic
35C2×Q8×C13—22no—
36C2×Q8×C13—22no—
37C2×Q8×C13D414yes—
38C22×C4×C13C2312yes—
39C22×C4×C13C2312yes—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×Dic13C2312yes—
43C22×Dic13C2311yescharacteristic
44C22×Dic13C2312yes—
45#6131—232no—
46C2×D52—24no—
47C2×D52—24no—
48C2×D52—24no—
49C2×D52—22no—
50C2×D52—22no—
51C2×D52D414yes—
52C2×C4×D13—24no—
53C2×C4×D13—24no—
54C2×C4×D13—22no—
55C2×C4×D13—22no—
56C2×C4×D13—24no—
57C2×C4×D13D414yes—
58C2×Dic26—22no—
59C2×Dic26—22no—
60C2×Dic26—22no—
61C13×C4⋊C4—22no—
62C42×C13—22no—
63#6103—28no—
64#6103—24no—
65#6103—28no—
66#6103—24no—
67C4⋊Dic13—24no—
68#6101—28no—
69C4×Dic13—22no—
70#2432—131nomaximal, Sylow
71C23×C13C2411yescharacteristic
72C22×D13—22no—
73C22×D13—21no—
74C22×D13—22no—
75C22×D13—21no—
76C22×D13—22no—
77C22×D13—24no—
78C22×D13—22no—
79C22×D13—24no—
80C22×D13C2×D414yes—
81C22×D13C2×D412yes—
82Q8×C13—28no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13—22no—
87C2×C4×C13—24no—
88C2×C4×C13—22no—
89C2×C4×C13—22no—
90C2×C4×C13C2×D414yes—
91C2×C4×C13C2×D412yes—
92C2×C4×C13C2×D412yes—
93C2×C4×C13C2×D412yes—
94C2×Dic13—22no—
95C2×Dic13—21no—
96C2×Dic13—21no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13C2×D412yes—
102C2×Dic13#4114yescomplemented
103D52—216no—
104D52—28no—
105C4×D13—216no—
106C4×D13—28no—
107Dic26—44no—
108C22×#41—131no—
109C2×C4⋊Q8—131no—
110C2×#124—131no—
111C2×C22⋊Q8—131no—
112#394—134no—
113#391D1311yescharacteristic, complemented
114#390—131no—
115#385—134no—
116C4×C22⋊C4—131no—
117C22×C13C22×D411yescharacteristic
118C22×C13C22×D411yescharacteristic
119C22×C13C22×D411yescharacteristic
120C22×C13C2×#4112yes—
121C22×C132−1+412yes—
122C2×D13—24no—
123C2×D13—22no—
124C2×D13—22no—
125C2×D13—22no—
126C2×D13—28no—
127C2×D13—42no—
128C2×D13—42no—
129C2×D13—42no—
130C2×D13—42no—
131C2×D13—44no—
132C4×C13—24no—
133C4×C13—42no—
134C4×C13—42no—
135C4×C13#12014yes—
136Dic13—22no—
137Dic13—24no—
138Dic13—42no—
139C2×#41—264no—
140C2×#41—264no—
141C2×#41—264no—
142C22×Q8—131no—
143C22×Q8C2×D1311yescharacteristic
144C22×D4—131no—
145C22×D4—132no—
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—132no—
155C2×C22⋊C4—134no—
156C2×C22⋊C4—132no—
157C2×C22⋊C4—134no—
158C2×C42—131no—
159C2×C42C2×D1311yescharacteristic
160#95—132no—
161#95C2×D1314yes—
162C2×C13C2×#12011yescharacteristic
163C2×C13#53211yescharacteristic
164C2×C13#53611yescharacteristic
165C2×C13#54712yes—
166C2×C13#54812yes—
167D13—42no—
168D13—44no—
169C24—131no—
170C24—132no—
171#41—2632no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—262no—
175C2×Q8—262no—
176C2×Q8—262no—
177C2×Q8D5214yes—
178C2×D4—134no—
179C2×D4—264no—
180C2×D4—264no—
181C2×D4—264no—
182C2×D4—262no—
183C2×D4—262no—
184C22×C4—131no—
185C22×C4—132no—
186C22×C4—134no—
187C22×C4—132no—
188C22×C4—264no—
189C22×C4—264no—
190C22×C4—264no—
191C22×C4—262no—
192C22×C4—262no—
193C22×C4C22×D1312yes—
194C22×C4C22×D1312yes—
195C22×C4C22×D1312yes—
196C22×C4C22×D1311yescharacteristic
197C4⋊C4—22no—
198C4⋊C4—264no—
199C4⋊C4—268no—
200C22⋊C4—264no—
201C22⋊C4—268no—
202C22⋊C4—264no—
203C22⋊C4—268no—
204C42—22no—
205C42—262no—
206C13#243211yescharacteristic, complemented, Sylow
207C23—132no—
208C23—134no—
209C23—262no—
210C23—261no—
211C23—262no—
212C23—261no—
213C23—262no—
214C23—264no—
215C23—264no—
216C23—262no—
217C23C23×D1311yescharacteristic
218Q8—28no—
219Q8—524no—
220D4—2616no—
221D4—268no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—132no—
230C2×C4—134no—
231C2×C4—2616no—
232C2×C4—262no—
233C2×C4—261no—
234C2×C4—261no—
235C2×C4—262no—
236C2×C4—268no—
237C2×C4—262no—
238C2×C4—262no—
239C2×C4—262no—
240C2×C4C2×D5214yes—
241C2×C4C2×D5212yes—
242C2×C4D4×D1312yes—
243C2×C4D4×D1312yes—
244C22—262no—
245C22—264no—
246C22—262no—
247C22—268no—
248C22—262no—
249C22—522no—
250C22—522no—
251C22—524no—
252C22—522no—
253C22—522no—
254C22C22×D5211yescharacteristic
255C22C2×D4×D1311yescharacteristic
256C22C2×D4×D1311yescharacteristic
257C22#8825412yes—
258C22#41×D1312yes—
259C4—24no—
260C4—42no—
261C4—42no—
262C4—262no—
263C4—264no—
264C4—522no—
265C4#8813614yes—
266C2—522no—
267C2—524no—
268C2C2×#8813611yescharacteristic
269C2#1171928112yes—
270C2#1171933612yes—
271C2#1171937311yescharacteristic
272C2#1171942911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#391C2
prime radicalC22×#6131C2
hypercenter#391D13
hypocenterC13#2432
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#88198C2semidirect
3. C2×#88151C2semidirect
4. C2×#88129C2semidirect
5. C13×#391C2semidirect
6. #11718649C2semidirect
7. C4×#6103C2semidirect
8. #11718333C2semidirect
9. #11718329C2semidirect
102. C2×Dic13#41semidirect
113. #391D13semidirect
206. C13#2432semidirect

Maximal subgroups

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

subgroupindex
C22×#61312
C2×#881982
C2×#881512
C2×#881292
C13×#3912
#117186492
C4×#61032
#117183332
#117183292
#243213