Finite Group Databasebeta

← #13124098

Subgroups — #13124098

Order 1664.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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×#6127C211yesmaximal, characteristic
2C2×#88178C211yesmaximal, characteristic, complemented
3C2×#88163C212yesmaximal, complemented
4C13×#407C211yesmaximal, characteristic, complemented
5#11718794C212yesmaximal, complemented
6#11718400C212yesmaximal, complemented
7#11718394C214yesmaximal, complemented
8#11718392C212yesmaximal, complemented
9C23×C4×C13C2211yescharacteristic
10C22×C22⋊D13—21no—
11C2×#6127—21no—
12C2×#6127—21no—
13C2×#6127—28no—
14C2×#6127C2212yes—
15C2×#6127C2212yes—
16C22×D52C2211yescharacteristic
17C22×C4×D13—21no—
18C22×Dic26C2211yescharacteristic
19C22×Dic26C2211yescharacteristic, complemented
20C2×C13×M16C2212yescomplemented
21C2×C22⋊Dic13—21no—
22C2×C4⋊Dic13C2211yescharacteristic, complemented
23#88178—21no—
24#88178—21no—
25#88178C2212yes—
26#88178C2212yescomplemented
27C2×#6101—21no—
28#88163—28no—
29C2×Dic52C2214yescomplemented
30C2×#6095C2214yescomplemented
31C13×C22⋊C8C2212yes—
32C13×C22⋊C8C2212yescomplemented
33#88061C2214yescomplemented
34#88056C2214yescomplemented
35C24×C13D411yescharacteristic
36C23×D13—21no—
37C22×C4×C13—24no—
38C22×C4×C13D411yescharacteristic
39C22×C4×C13D411yescharacteristic
40C22×C4×C13D411yescharacteristic
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C22×C4×C13C2311yescharacteristic
44C2×C22⋊D13—28no—
45C2×C22⋊D13—22no—
46C2×C22⋊D13—22no—
47C22×Dic13—21no—
48C22×Dic13—21no—
49C22×Dic13—21no—
50#6127—28no—
51#6127—28no—
52#6127—24no—
53#6127—24no—
54#6127—24no—
55#6127D418yes—
56C2×D52—21no—
57C2×D52—24no—
58C2×D52C2312yes—
59C2×C4×D13—28no—
60C2×C4×D13—22no—
61C2×C4×D13—22no—
62C2×Dic26—21no—
63C2×Dic26—24no—
64C2×Dic26—24no—
65C2×Dic26—21no—
66C2×Dic26C2312yes—
67C2×Dic26C2312yes—
68C13×M16—24no—
69C2×C8×C13C2314yes—
70C22⋊Dic13—22no—
71C22⋊Dic13—22no—
72C4⋊Dic13—21no—
73C4⋊Dic13C2312yes—
74#6101—22no—
75#6101—22no—
76Dic52—28no—
77#6095—28no—
78#3007—131nomaximal, Sylow
79C23×C13—22no—
80C23×C13—22no—
81C23×C13—41no—
82C23×C13C2×D411yescharacteristic
83C23×C13C2×D411yescharacteristic
84C23×C13C2×D411yescharacteristic
85C22×D13—24no—
86C22×D13—22no—
87C22×D13—42no—
88C22×D13—42no—
89C2×C4×C13—22no—
90C2×C4×C13—22no—
91C2×C4×C13—22no—
92C2×C4×C13—22no—
93C2×C4×C13—22no—
94C2×C4×C13C2×D411yescharacteristic
95C2×C4×C13C2×D411yescharacteristic
96C2×C4×C13C2×D411yescharacteristic
97C2×C4×C13C2×D414yes—
98C2×C4×C13C2411yescharacteristic
99C22⋊D13—28no—
100C22⋊D13—28no—
101C2×Dic13—24no—
102C2×Dic13—22no—
103C2×Dic13—22no—
104C2×Dic13—22no—
105C2×Dic13—42no—
106C2×Dic13—42no—
107D52—24no—
108D52—22no—
109D52C2×D414yes—
110C4×D13—28no—
111C4×D13—28no—
112Dic26—24no—
113Dic26—22no—
114Dic26—24no—
115Dic26—42no—
116Dic26C2×D414yes—
117C8×C13—24no—
118C22×#41—131no—
119C2×#137—132no—
120C2×C22⋊Q8—131no—
121#452—134no—
122#451—132no—
123#450—132no—
124#417—132no—
125#407D1311yescharacteristic, complemented
126C22×C13—21no—
127C22×C13—21no—
128C22×C13—21no—
129C22×C13—21no—
130C22×C13—21no—
131C22×C13—21no—
132C22×C13—21no—
133C22×C13—44no—
134C22×C13#12012yes—
135C22×C13#13712yescomplemented
136C22×C13C22×D411yescharacteristic
137C2×D13—24no—
138C2×D13—48no—
139C2×D13—42no—
140C2×D13—42no—
141C4×C13—21no—
142C4×C13—21no—
143C4×C13#12012yes—
144C4×C13C22×D412yes—
145Dic13—24no—
146Dic13—42no—
147Dic13—42no—
148C2×#41—132no—
149C2×#41—132no—
150C2×#41—268no—
151C2×#41—261no—
152C2×#41—261no—
153C22×Q8—131no—
154C22×Q8—131no—
155C22×D4—131no—
156C22×D4—261no—
157C23×C4—261no—
158C23×C4C2×D1311yescharacteristic
159#137—268no—
160C2×Q16—134no—
161C2×SD16—134no—
162C2×M16C2×D1312yescomplemented
163C22⋊Q8—132no—
164C22⋊Q8—132no—
165C22⋊Q8—261no—
166C22⋊Q8—261no—
167C2×C4⋊C4—131no—
168C2×C4⋊C4—261no—
169C2×C22⋊C4—261no—
170#103—134no—
171#102—134no—
172C22⋊C8C2×D1312yes—
173C22⋊C8C2×D1312yescomplemented
174C2×C13—22no—
175C2×C13—22no—
176C2×C13—41no—
177C2×C13C2×#12011yescharacteristic
178C2×C13C2×#13711yescharacteristic
179C2×C13#57511yescharacteristic
180D13—44no—
181C24—261no—
182C24D5211yescharacteristic
183#41—138no—
184#41—264no—
185#41—264no—
186#41—268no—
187#41—268no—
188#41—264no—
189C2×Q8—132no—
190C2×Q8—132no—
191C2×Q8—264no—
192C2×Q8—261no—
193C2×Q8—261no—
194C2×Q8—264no—
195C2×D4—132no—
196C2×D4—264no—
197C2×D4—268no—
198C2×D4—262no—
199C2×D4—262no—
200C2×D4—261no—
201C22×C4—24no—
202C22×C4—268no—
203C22×C4—261no—
204C22×C4—262no—
205C22×C4—262no—
206C22×C4—261no—
207C22×C4—261no—
208C22×C4D5211yescharacteristic
209C22×C4D5211yescharacteristic
210C22×C4D5211yescharacteristic
211C22×C4C22×D1311yescharacteristic
212C22×C4C22×D1311yescharacteristic
213C22×C4C22×D1311yescharacteristic
214Q16—268no—
215SD16—268no—
216M16—24no—
217C2×C8C22×D1314yes—
218C4⋊C4—132no—
219C4⋊C4—261no—
220C4⋊C4—262no—
221C4⋊C4—262no—
222C22⋊C4—262no—
223C22⋊C4—262no—
224C13#300711yescharacteristic, complemented, Sylow
225C23—22no—
226C23—22no—
227C23—41no—
228C23—264no—
229C23—262no—
230C23—522no—
231C23—522no—
232C23C2×D5211yescharacteristic
233C23C2×D5211yescharacteristic
234C23C2×D5211yescharacteristic
235Q8—134no—
236Q8—264no—
237Q8—262no—
238Q8—264no—
239Q8—522no—
240D4—134no—
241D4—264no—
242D4—268no—
243D4—268no—
244D4—262no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—268no—
251C2×C4—264no—
252C2×C4—262no—
253C2×C4—268no—
254C2×C4—262no—
255C2×C4—262no—
256C2×C4—522no—
257C2×C4—522no—
258C2×C4C2×D5211yescharacteristic
259C2×C4C2×D5211yescharacteristic
260C2×C4C2×D5211yescharacteristic
261C2×C4D4×D1314yes—
262C2×C4C23×D1311yescharacteristic
263C8—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—264no—
273C22—528no—
274C22—522no—
275C22—522no—
276C22#8813612yes—
277C22#8816312yescomplemented
278C22C22×D5211yescharacteristic
279C4—21no—
280C4—21no—
281C4—264no—
282C4—522no—
283C4—522no—
284C4#8813612yes—
285C4C2×D4×D1312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—524no—
290C2C2×#8813611yescharacteristic
291C2C2×#8816311yescharacteristic
292C2#1171945311yescharacteristic
subgroupG/H
centerC22C22×D52
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#407C2
socleC22×C13C22×D4
prime radicalC22×#6127C2
hypercenter#407D13
hypocenterC13#3007
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#88178C2semidirect
3. C2×#88163C2semidirect
4. C13×#407C2semidirect
5. #11718794C2semidirect
6. #11718400C2semidirect
7. #11718394C2semidirect
8. #11718392C2semidirect
19. C22×Dic26C22semidirect
20. C2×C13×M16C22semidirect
22. C2×C4⋊Dic13C22semidirect
26. #88178C22semidirect
29. C2×Dic52C22semidirect
30. C2×#6095C22semidirect
32. C13×C22⋊C8C22semidirect
33. #88061C22semidirect
34. #88056C22semidirect
125. #407D13semidirect
135. C22×C13#137semidirect
162. C2×M16C2×D13semidirect
173. C22⋊C8C2×D13semidirect
224. C13#3007semidirect
277. C22#88163semidirect

Maximal subgroups

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

subgroupindex
C22×#61272
C2×#881782
C2×#881632
C13×#4072
#117187942
#117184002
#117183942
#117183922
#300713