Finite Group Databasebeta

← #13121063

Subgroups — #13121063

Order 1664.

796 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 141 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×#88191C212yesmaximal, complemented
2C2×#88139C212yesmaximal, complemented
3C13×#392C211yesmaximal, characteristic, complemented
4#11718884C212yesmaximal, complemented
5#11718882C212yesmaximal, complemented
6#11718622C212yesmaximal, complemented
7#11718616C211yesmaximal, characteristic, complemented
8#11718608C211yesmaximal, characteristic, complemented
9#11718331C212yesmaximal, complemented
10C24×D13C2211yescharacteristic
11C22×D4×C13C2212yes—
12C22×D4×C13C2211yescharacteristic, complemented
13C22×C22⋊D13C2212yes—
14C22×C22⋊D13C2212yes—
15C22×Dic26C2211yescharacteristic, complemented
16C2×C13×C22⋊C4C2211yescharacteristic
17C2×C13×C22⋊C4C2212yescomplemented
18C2×C22⋊Dic13C2212yes—
19C2×C22⋊Dic13C2211yescharacteristic
20C2×C22⋊Dic13C2212yescomplemented
21C2×C22⋊Dic13C2212yescomplemented
22#88191—28no—
23C2×#6103C2212yes—
24C2×#6103C2212yes—
25C2×#6103C2212yes—
26C2×#6103C2212yes—
27C2×C4⋊Dic13C2212yescomplemented
28C2×C4×Dic13C2212yescomplemented
29#88139—28no—
30C13×#95C2211yescharacteristic, complemented
31#88071C2212yescomplemented
32#88071C2211yescharacteristic, complemented
33C24×C13C2312yes—
34C24×C13C2311yescharacteristic
35C23×D13—22no—
36C23×D13—24no—
37C23×D13—42no—
38C23×D13—41no—
39C23×D13—41no—
40C23×D13C2312yes—
41C2×D4×C13—24no—
42C2×D4×C13—24no—
43C2×D4×C13—42no—
44C2×D4×C13—42no—
45C2×D4×C13—41no—
46C22×C4×C13C2311yescharacteristic
47C22×C4×C13C2311yescharacteristic
48C22×C4×C13C2312yes—
49C2×C22⋊D13—24no—
50C2×C22⋊D13—24no—
51C2×C22⋊D13—24no—
52C2×C22⋊D13—42no—
53C2×C22⋊D13—42no—
54C2×C22⋊D13—42no—
55C22×Dic13C2312yes—
56C22×Dic13C2312yes—
57C22×Dic13C2312yes—
58C2×Dic26—24no—
59C2×Dic26—41no—
60C13×C22⋊C4—24no—
61C13×C22⋊C4—41no—
62C22⋊Dic13—24no—
63C22⋊Dic13—24no—
64C22⋊Dic13—41no—
65C22⋊Dic13—42no—
66#6103—24no—
67#6103—24no—
68#6103—42no—
69#6103D418yescomplemented
70C4⋊Dic13—42no—
71C4×Dic13—24no—
72#2670—131nomaximal, Sylow
73C23×C13—22no—
74C23×C13—21no—
75C23×C13—21no—
76C23×C13—22no—
77C23×C13—22no—
78C23×C13—22no—
79C23×C13—22no—
80C23×C13—22no—
81C23×C13—22no—
82C23×C13—22no—
83C23×C13—22no—
84C23×C13#4112yes—
85C23×C13C2411yescharacteristic
86C22×D13—22no—
87C22×D13—22no—
88C22×D13—22no—
89C22×D13—22no—
90C22×D13—22no—
91C22×D13—22no—
92C22×D13—44no—
93C22×D13—48no—
94C22×D13—48no—
95C22×D13—44no—
96C22×D13—44no—
97C22×D13C2×D414yes—
98D4×C13—48no—
99D4×C13—44no—
100C2×C4×C13—22no—
101C2×C4×C13—22no—
102C2×C4×C13—22no—
103C2×C4×C13—22no—
104C2×C4×C13—21no—
105C2×C4×C13—21no—
106C2×C4×C13C2×D414yes—
107C22⋊D13—48no—
108C22⋊D13—48no—
109C2×Dic13—22no—
110C2×Dic13—22no—
111C2×Dic13—22no—
112C2×Dic13—22no—
113C2×Dic13—22no—
114C2×Dic13—22no—
115C2×Dic13—22no—
116C2×Dic13—22no—
117C2×Dic13C2×D414yes—
118Dic26—44no—
119C2×#124—132no—
120C2×#120—132no—
121#394—132no—
122#393—131no—
123#392—132no—
124#392D1311yescharacteristic, complemented
125#390—132no—
126#388—132no—
127#379—131no—
128C22×C13—22no—
129C22×C13—24no—
130C22×C13—42no—
131C22×C13—41no—
132C22×C13—41no—
133C22×C13—42no—
134C22×C13—42no—
135C22×C13—42no—
136C22×C13—42no—
137C22×C13—42no—
138C22×C13—42no—
139C22×C13—42no—
140C22×C13C22×D412yes—
141C22×C13C2×#4111yescharacteristic
142C22×C13H(5,𝔽2)12yes—
143C22×C13H(5,𝔽2)11yescharacteristic
144C22×C13H(5,𝔽2)11yescharacteristic
145C2×D13—24no—
146C2×D13—24no—
147C2×D13—24no—
148C2×D13—416no—
149C2×D13—42no—
150C2×D13—42no—
151C2×D13—42no—
152C2×D13—42no—
153C4×C13—24no—
154C4×C13—41no—
155C4×C13—41no—
156Dic13—24no—
157Dic13—42no—
158Dic13—42no—
159C25—131no—
160C22×Q8—131no—
161C22×D4—132no—
162C22×D4—132no—
163C22×D4C2×D1312yes—
164C22×D4C2×D1311yescharacteristic, complemented
165#124—268no—
166#120—268no—
167C2×C4⋊C4—132no—
168C2×C22⋊C4—132no—
169C2×C22⋊C4—132no—
170C2×C22⋊C4—132no—
171C2×C22⋊C4—132no—
172C2×C22⋊C4—132no—
173C2×C22⋊C4—131no—
174C2×C22⋊C4—132no—
175C2×C22⋊C4—132no—
176C2×C22⋊C4C2×D1311yescharacteristic
177C2×C22⋊C4C2×D1312yescomplemented
178C2×C42—132no—
179#95—132no—
180#95—131no—
181#95C2×D1311yescharacteristic, complemented
182C2×C13—42no—
183C2×C13—82no—
184C2×C13#53512yes—
185C2×C13#53811yescharacteristic
186C2×C13D4211yescharacteristic
187C2×C13#54612yes—
188C2×C13H(3,𝔽4)11yescharacteristic
189D13—44no—
190C24—132no—
191C24—264no—
192C24—262no—
193C24—521no—
194C24—522no—
195C24—521no—
196C24C22×D1312yes—
197C24C22×D1311yescharacteristic
198C2×Q8—264no—
199C2×Q8—521no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—42no—
205C2×D4—264no—
206C2×D4—264no—
207C2×D4—264no—
208C2×D4—522no—
209C2×D4—522no—
210C2×D4—522no—
211C22×C4—132no—
212C22×C4—132no—
213C22×C4—132no—
214C22×C4C22×D1311yescharacteristic
215C22×C4C22×D1311yescharacteristic
216C22×C4C22×D1312yes—
217C4⋊C4—522no—
218C22⋊C4—24no—
219C22⋊C4—41no—
220C22⋊C4—138no—
221C22⋊C4—264no—
222C22⋊C4—264no—
223C22⋊C4—264no—
224C22⋊C4—264no—
225C22⋊C4—522no—
226C22⋊C4—521no—
227C22⋊C4—522no—
228C42—264no—
229C13#267011yescharacteristic, complemented, Sylow
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—134no—
242C23—262no—
243C23—262no—
244C23—262no—
245C23—262no—
246C23—262no—
247C23—262no—
248C23—528no—
249C23—524no—
250C23—524no—
251C23—528no—
252C23—524no—
253C23#612912yes—
254C23C23×D1311yescharacteristic
255Q8—524no—
256D4—48no—
257D4—44no—
258D4—528no—
259D4—528no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—134no—
267C2×C4—262no—
268C2×C4—262no—
269C2×C4—262no—
270C2×C4—262no—
271C2×C4—262no—
272C2×C4—262no—
273C2×C4—262no—
274C2×C4—262no—
275C2×C4D4×D1314yes—
276C22—24no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—264no—
289C22—264no—
290C22—264no—
291C22—5216no—
292C22—522no—
293C22—522no—
294C22—522no—
295C22—522no—
296C22C2×D4×D1312yes—
297C22C2×#612911yescharacteristic
298C22#8825112yes—
299C22#8825111yescharacteristic
300C22#8825611yescharacteristic
301C4—24no—
302C4—41no—
303C4—41no—
304C4—264no—
305C4—522no—
306C4—522no—
307C2—42no—
308C2—82no—
309C2—524no—
310C2#1171929612yes—
311C2#1171929911yescharacteristic
312C2#1171931212yes—
313C2#1171932211yescharacteristic
314C2#1171942211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#392C2
hypercenter#392D13
hypocenterC13#2670
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#88191C2semidirect
2. C2×#88139C2semidirect
3. C13×#392C2semidirect
4. #11718884C2semidirect
5. #11718882C2semidirect
6. #11718622C2semidirect
7. #11718616C2semidirect
8. #11718608C2semidirect
9. #11718331C2semidirect
12. C22×D4×C13C22semidirect
15. C22×Dic26C22semidirect
17. C2×C13×C22⋊C4C22semidirect
20. C2×C22⋊Dic13C22semidirect
21. C2×C22⋊Dic13C22semidirect
27. C2×C4⋊Dic13C22semidirect
28. C2×C4×Dic13C22semidirect
30. C13×#95C22semidirect
31. #88071C22semidirect
32. #88071C22semidirect
69. #6103D4semidirect
124. #392D13semidirect
164. C22×D4C2×D13semidirect
177. C2×C22⋊C4C2×D13semidirect
181. #95C2×D13semidirect
229. C13#2670semidirect

Maximal subgroups

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

subgroupindex
C2×#881912
C2×#881392
C13×#3922
#117188842
#117188822
#117186222
#117186162
#117186082
#117183312
#267013