Finite Group Databasebeta

← #13121215

Subgroups — #13121215

Order 1664.

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

normal subgroups only

This group has 688 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×#88190C211yesmaximal, characteristic, complemented
3C13×#386C211yesmaximal, characteristic, complemented
4#11718874C212yesmaximal, complemented
5#11718622C212yesmaximal, complemented
6#11718616C212yesmaximal, complemented
7#11718614C212yesmaximal, complemented
8#11718334C212yesmaximal, complemented
9#11718331C212yesmaximal, complemented
10C24×D13C2211yescharacteristic
11C22×D4×C13C2211yescharacteristic
12C23×C4×C13C2211yescharacteristic, complemented
13C22×C22⋊D13C2212yes—
14C22×Dic26C2211yescharacteristic, complemented
15C2×C13×C22⋊C4C2212yes—
16C2×C13×C22⋊C4C2211yescharacteristic, complemented
17C2×C22⋊Dic13C2212yes—
18C2×C22⋊Dic13C2211yescharacteristic
19C2×C22⋊Dic13C2212yescomplemented
20#88191—24no—
21#88190—24no—
22C2×#6103C2212yes—
23C2×#6103C2212yes—
24C2×#6103C2212yes—
25C2×#6103C2212yes—
26C2×#6103C2212yes—
27C2×C4⋊Dic13C2212yescomplemented
28C2×C4×Dic13C2211yescharacteristic, complemented
29C13×#95C2212yescomplemented
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×C13—41no—
45C22×C4×C13—42no—
46C22×C4×C13C2312yes—
47C22×C4×C13C2312yes—
48C22×C4×C13C2311yescharacteristic
49C2×C22⋊D13—24no—
50C2×C22⋊D13—24no—
51C2×C22⋊D13—42no—
52C22×Dic13C2312yes—
53C22×Dic13C2312yes—
54C22×Dic13C2312yes—
55C2×Dic26—22no—
56C2×Dic26—41no—
57C2×Dic26—41no—
58C13×C22⋊C4—42no—
59C13×C22⋊C4—41no—
60C22⋊Dic13—24no—
61C22⋊Dic13—24no—
62C22⋊Dic13—22no—
63#6103—24no—
64#6103—24no—
65#6103—42no—
66#6103—42no—
67#6103—42no—
68C4⋊Dic13—42no—
69C4×Dic13—22no—
70#2684—131nomaximal, Sylow
71C23×C13—22no—
72C23×C13—22no—
73C23×C13—21no—
74C23×C13—22no—
75C23×C13—22no—
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×C13—44no—
104C2×C4×C13C2×D412yes—
105C22⋊D13—48no—
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—22no—
115Dic26—44no—
116C2×#124—131no—
117C2×#120—131no—
118#397—132no—
119#394—132no—
120#393—132no—
121#388—132no—
122#386—132no—
123#386D1311yescharacteristic, complemented
124#379—132no—
125C22×C13—22no—
126C22×C13—22no—
127C22×C13—24no—
128C22×C13—24no—
129C22×C13—42no—
130C22×C13—42no—
131C22×C13—41no—
132C22×C13—41no—
133C22×C13—41no—
134C22×C13—41no—
135C22×C13C22×D411yescharacteristic
136C22×C13C2×#4112yes—
137C22×C13H(5,𝔽2)11yescharacteristic
138C22×C13H(5,𝔽2)11yescharacteristic
139C22×C13H(5,𝔽2)12yes—
140C2×D13—24no—
141C2×D13—24no—
142C2×D13—24no—
143C2×D13—416no—
144C2×D13—42no—
145C2×D13—42no—
146C2×D13—42no—
147C2×D13—42no—
148C4×C13—22no—
149C4×C13—42no—
150C4×C13—42no—
151Dic13—42no—
152Dic13—42no—
153Dic13—42no—
154C25—131no—
155C22×Q8—131no—
156C22×D4—132no—
157C22×D4C2×D1311yescharacteristic
158C23×C4C2×D1311yescharacteristic, complemented
159#124—264no—
160#120—264no—
161C2×C4⋊C4—132no—
162C2×C22⋊C4—131no—
163C2×C22⋊C4—132no—
164C2×C22⋊C4—132no—
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×D1311yescharacteristic, complemented
172C2×C42—131no—
173#95—132no—
174#95—132no—
175#95—132no—
176#95C2×D1312yescomplemented
177C2×C13—42no—
178C2×C13—42no—
179C2×C13#53411yescharacteristic
180C2×C13#53812yes—
181C2×C13#54612yes—
182C2×C13#55111yescharacteristic
183C2×C13H(3,𝔽4)11yescharacteristic
184D13—44no—
185C24—132no—
186C24—264no—
187C24—262no—
188C24—522no—
189C24—521no—
190C24—521no—
191C24C22×D1311yescharacteristic
192C24C22×D1311yescharacteristic
193C2×Q8—262no—
194C2×Q8—521no—
195C2×Q8—521no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—264no—
199C2×D4—264no—
200C2×D4—522no—
201C2×D4D5214yes—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—132no—
205C22×C4—132no—
206C22×C4—132no—
207C22×C4C22×D1312yes—
208C22×C4C22×D1312yes—
209C22×C4C22×D1311yescharacteristic
210C4⋊C4—522no—
211C22⋊C4—42no—
212C22⋊C4—41no—
213C22⋊C4—264no—
214C22⋊C4—262no—
215C22⋊C4—264no—
216C22⋊C4—264no—
217C22⋊C4—264no—
218C22⋊C4—522no—
219C22⋊C4—522no—
220C22⋊C4—522no—
221C42—262no—
222C13#268411yescharacteristic, complemented, Sylow
223C23—21no—
224C23—22no—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—134no—
232C23—262no—
233C23—262no—
234C23—262no—
235C23—262no—
236C23—262no—
237C23—262no—
238C23—528no—
239C23—524no—
240C23—524no—
241C23—528no—
242C23—524no—
243C23C2×D5212yes—
244C23C2×D5212yes—
245C23C23×D1311yescharacteristic
246Q8—524no—
247D4—44no—
248D4—528no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—262no—
259C2×C4—262no—
260C2×C4—262no—
261C2×C4—262no—
262C2×C4—262no—
263C2×C4—262no—
264C2×C4—262no—
265C2×C4—262no—
266C2×C4—262no—
267C2×C4C2×D5212yes—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—264no—
279C22—264no—
280C22—264no—
281C22—5216no—
282C22—522no—
283C22—522no—
284C22—522no—
285C22—522no—
286C22C22×D5211yescharacteristic
287C22#8825111yescharacteristic
288C22#8825111yescharacteristic
289C22#8825112yes—
290C22#41×D1312yes—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—522no—
295C4—522no—
296C4—522no—
297C2—42no—
298C2—42no—
299C2—524no—
300C2#1171919811yescharacteristic
301C2#1171925912yes—
302C2#1171929911yescharacteristic
303C2#1171931812yes—
304C2#1171936411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#386C2
prime radicalC2×#88191C2
hypercenter#386D13
hypocenterC13#2684
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#88190C2semidirect
3. C13×#386C2semidirect
4. #11718874C2semidirect
5. #11718622C2semidirect
6. #11718616C2semidirect
7. #11718614C2semidirect
8. #11718334C2semidirect
9. #11718331C2semidirect
12. C23×C4×C13C22semidirect
14. C22×Dic26C22semidirect
16. C2×C13×C22⋊C4C22semidirect
19. C2×C22⋊Dic13C22semidirect
27. C2×C4⋊Dic13C22semidirect
28. C2×C4×Dic13C22semidirect
29. C13×#95C22semidirect
30. #88071C22semidirect
31. #88071C22semidirect
32. #88071C22semidirect
123. #386D13semidirect
158. C23×C4C2×D13semidirect
171. C2×C22⋊C4C2×D13semidirect
176. #95C2×D13semidirect
222. C13#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#881912
C2×#881902
C13×#3862
#117188742
#117186222
#117186162
#117186142
#117183342
#117183312
#268413