Finite Group Databasebeta

← #13121398

Subgroups — #13121398

Order 1664.

680 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 137 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×#88191C211yesmaximal, characteristic
2C2×#88189C211yesmaximal, characteristic, complemented
3C13×#390C211yesmaximal, characteristic, complemented
4#11718655C212yesmaximal, complemented
5#11718622C212yesmaximal, complemented
6#11718619C212yesmaximal, complemented
7#11718616C212yesmaximal, complemented
8#11718614C212yesmaximal, complemented
9#11718601C212yesmaximal, complemented
10C24×D13C2211yescharacteristic
11C22×D4×C13C2211yescharacteristic
12C22×C22⋊D13C2212yes—
13C23×Dic13C2211yescharacteristic, complemented
14C2×C13×C4⋊C4C2211yescharacteristic, complemented
15C2×C13×C22⋊C4C2212yes—
16C2×C13×C22⋊C4C2212yescomplemented
17C2×C42×C13C2211yescharacteristic, complemented
18C2×C22⋊Dic13C2212yes—
19C2×C22⋊Dic13C2211yescharacteristic
20C2×C22⋊Dic13C2211yescharacteristic, complemented
21#88191—24no—
22#88189—24no—
23C2×#6103C2212yes—
24C2×#6103C2212yes—
25C2×#6103C2212yes—
26C2×#6103C2212yes—
27C2×#6103C2212yes—
28C2×C4⋊Dic13C2212yescomplemented
29C2×#6101C2212yescomplemented
30#88071C2212yescomplemented
31#88071C2212yescomplemented
32#88071C2212yescomplemented
33C24×C13C2311yescharacteristic
34C24×C13C2311yescharacteristic
35C23×D13—22no—
36C23×D13—24no—
37C23×D13—41no—
38C23×D13—41no—
39C23×D13—42no—
40C23×D13C2312yes—
41C2×D4×C13—41no—
42C2×D4×C13—41no—
43C2×D4×C13D414yes—
44C22×C4×C13C2312yes—
45C22×C4×C13C2312yes—
46C22×C4×C13C2311yescharacteristic
47C2×C22⋊D13—24no—
48C2×C22⋊D13—24no—
49C2×C22⋊D13—42no—
50C22×Dic13—22no—
51C22×Dic13—41no—
52C22×Dic13—41no—
53C22×Dic13C2312yes—
54C22×Dic13C2312yes—
55C22×Dic13C2312yes—
56C13×C4⋊C4—41no—
57C13×C22⋊C4—42no—
58C13×C22⋊C4—42no—
59C42×C13—41no—
60C22⋊Dic13—24no—
61C22⋊Dic13—22no—
62C22⋊Dic13—22no—
63#6103—24no—
64#6103—24no—
65#6103—42no—
66#6103—42no—
67#6103—42no—
68C4⋊Dic13—42no—
69#6101—24no—
70#2697—131nomaximal, Sylow
71C23×C13—22no—
72C23×C13—22no—
73C23×C13—22no—
74C23×C13—22no—
75C23×C13—21no—
76C23×C13—21no—
77C23×C13—21no—
78C23×C13—21no—
79C23×C13C2×D412yes—
80C23×C13C2×D412yes—
81C23×C13C2411yescharacteristic
82C22×D13—22no—
83C22×D13—22no—
84C22×D13—22no—
85C22×D13—22no—
86C22×D13—22no—
87C22×D13—22no—
88C22×D13—44no—
89C22×D13—48no—
90C22×D13—48no—
91C22×D13—44no—
92C22×D13—44no—
93C22×D13#4114yes—
94D4×C13—44no—
95C2×C4×C13—21no—
96C2×C4×C13—21no—
97C2×C4×C13—22no—
98C2×C4×C13—22no—
99C2×C4×C13—22no—
100C2×C4×C13—22no—
101C2×C4×C13—22no—
102C2×C4×C13—22no—
103C2×C4×C13C2×D412yes—
104C22⋊D13—48no—
105C2×Dic13—22no—
106C2×Dic13—22no—
107C2×Dic13—22no—
108C2×Dic13—22no—
109C2×Dic13—22no—
110C2×Dic13—22no—
111C2×Dic13—22no—
112C2×Dic13—22no—
113C2×Dic13—22no—
114C2×Dic13—44no—
115C2×#123—131no—
116C2×#120—131no—
117#397—132no—
118#396—132no—
119#394—132no—
120#390D1311yescharacteristic, complemented
121#388—132no—
122#386—132no—
123#379—132no—
124C22×C13—22no—
125C22×C13—22no—
126C22×C13—24no—
127C22×C13—24no—
128C22×C13—42no—
129C22×C13—42no—
130C22×C13—41no—
131C22×C13—41no—
132C22×C13—41no—
133C22×C13—41no—
134C22×C13C22×D411yescharacteristic
135C22×C13C2×#4112yes—
136C22×C13H(5,𝔽2)11yescharacteristic
137C22×C13H(5,𝔽2)11yescharacteristic
138C22×C13H(5,𝔽2)12yes—
139C2×D13—24no—
140C2×D13—24no—
141C2×D13—24no—
142C2×D13—416no—
143C2×D13—42no—
144C2×D13—42no—
145C2×D13—42no—
146C2×D13—42no—
147C4×C13—22no—
148C4×C13—42no—
149C4×C13—42no—
150Dic13—42no—
151Dic13—42no—
152Dic13—42no—
153C25—131no—
154C22×D4—132no—
155C22×D4C2×D1311yescharacteristic
156C23×C4—131no—
157#123—264no—
158#120—264no—
159C2×C4⋊C4—132no—
160C2×C4⋊C4—132no—
161C2×C4⋊C4C2×D1311yescharacteristic, complemented
162C2×C22⋊C4—131no—
163C2×C22⋊C4—132no—
164C2×C22⋊C4—131no—
165C2×C22⋊C4—132no—
166C2×C22⋊C4—132no—
167C2×C22⋊C4—132no—
168C2×C22⋊C4—132no—
169C2×C22⋊C4—132no—
170C2×C22⋊C4C2×D1312yes—
171C2×C22⋊C4C2×D1312yescomplemented
172C2×C42C2×D1311yescharacteristic, complemented
173#95—132no—
174#95—132no—
175#95—132no—
176C2×C13—42no—
177C2×C13—42no—
178C2×C13#53411yescharacteristic
179C2×C13#53812yes—
180C2×C13#54612yes—
181C2×C13#55111yescharacteristic
182C2×C13#56011yescharacteristic
183D13—44no—
184C24—132no—
185C24—264no—
186C24—262no—
187C24—522no—
188C24—521no—
189C24—521no—
190C24C22×D1311yescharacteristic
191C24C22×D1311yescharacteristic
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—264no—
195C2×D4—264no—
196C2×D4—522no—
197C2×D4D5214yes—
198C22×C4—132no—
199C22×C4—132no—
200C22×C4—132no—
201C22×C4—262no—
202C22×C4—521no—
203C22×C4—521no—
204C22×C4C22×D1312yes—
205C22×C4C22×D1312yes—
206C22×C4C22×D1311yescharacteristic
207C4⋊C4—41no—
208C4⋊C4—264no—
209C4⋊C4—522no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—264no—
213C22⋊C4—262no—
214C22⋊C4—262no—
215C22⋊C4—264no—
216C22⋊C4—264no—
217C22⋊C4—522no—
218C22⋊C4—522no—
219C22⋊C4—522no—
220C42—41no—
221C13#269711yescharacteristic, complemented, Sylow
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—134no—
231C23—262no—
232C23—262no—
233C23—262no—
234C23—262no—
235C23—262no—
236C23—262no—
237C23—528no—
238C23—524no—
239C23—524no—
240C23—528no—
241C23—524no—
242C23C2×D5212yes—
243C23C2×D5212yes—
244C23C23×D1311yescharacteristic
245D4—44no—
246D4—528no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—262no—
256C2×C4—262no—
257C2×C4—262no—
258C2×C4—262no—
259C2×C4—262no—
260C2×C4—262no—
261C2×C4—262no—
262C2×C4—262no—
263C2×C4—262no—
264C2×C4—524no—
265C2×C4C2×D5212yes—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22—264no—
277C22—264no—
278C22—264no—
279C22—5216no—
280C22—522no—
281C22—522no—
282C22—522no—
283C22—522no—
284C22C22×D5211yescharacteristic
285C22#8825111yescharacteristic
286C22#8825112yes—
287C22#8825111yescharacteristic
288C22#41×D1312yes—
289C4—22no—
290C4—42no—
291C4—42no—
292C4—522no—
293C4—522no—
294C4—522no—
295C2—42no—
296C2—42no—
297C2—524no—
298C2#1171919811yescharacteristic
299C2#1171925912yes—
300C2#1171931812yes—
301C2#1171938411yescharacteristic
302C2#1171942511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#390C2
prime radicalC2×#88191C2
hypercenter#390D13
hypocenterC13#2697
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#88189C2semidirect
3. C13×#390C2semidirect
4. #11718655C2semidirect
5. #11718622C2semidirect
6. #11718619C2semidirect
7. #11718616C2semidirect
8. #11718614C2semidirect
9. #11718601C2semidirect
13. C23×Dic13C22semidirect
14. C2×C13×C4⋊C4C22semidirect
16. C2×C13×C22⋊C4C22semidirect
17. C2×C42×C13C22semidirect
20. C2×C22⋊Dic13C22semidirect
28. C2×C4⋊Dic13C22semidirect
29. C2×#6101C22semidirect
30. #88071C22semidirect
31. #88071C22semidirect
32. #88071C22semidirect
120. #390D13semidirect
161. C2×C4⋊C4C2×D13semidirect
171. C2×C22⋊C4C2×D13semidirect
172. C2×C42C2×D13semidirect
221. C13#2697semidirect

Maximal subgroups

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

subgroupindex
C2×#881912
C2×#881892
C13×#3902
#117186552
#117186222
#117186192
#117186162
#117186142
#117186012
#269713