Finite Group Databasebeta

← #13124518

Subgroups — #13124518

Order 1664.

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

normal subgroups only

This group has 504 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
1C13×#537C211yesmaximal, characteristic, complemented
2#11719535C211yesmaximal, characteristic, complemented
3C2×#88203C211yesmaximal, characteristic, complemented
4C2×#88202C211yesmaximal, characteristic, complemented
5#11718894C211yesmaximal, characteristic, complemented
6#11718710C212yesmaximal, complemented
7#11718709C212yesmaximal, complemented
8#11718700C212yesmaximal, complemented
9#11718699C212yesmaximal, complemented
10C2×#88049C211yesmaximal, characteristic, complemented
11#11718625C211yesmaximal, characteristic, complemented
12C2×C13×#41C2211yescharacteristic
13C2×C13×#41C2211yescharacteristic
14C22×D52C2211yescharacteristic
15C13×C22⋊Q8—21no—
16C13×C22⋊Q8C2212yescomplemented
17C13×#121—21no—
18C13×#121—22no—
19C13×#121C2212yescomplemented
20C2×C13×C4⋊C4C2211yescharacteristic
21#88203—24no—
22#88202—24no—
23#88201—21no—
24C2×#6106C2212yes—
25#88194—21no—
26#88191—22no—
27C2×#6104C2212yes—
28#88184C2212yes—
29#88184C2212yes—
30#88180C2211yescharacteristic
31C2×#6099C2211yescharacteristic
32C2×#6099C2212yescomplemented
33C22×#1055C2211yescharacteristic
34C22×#1055C2212yescomplemented
35#88075C2211yescharacteristic, complemented
36#88075C2211yescharacteristic, complemented
37#88072C2211yescharacteristic, complemented
38#88072C2211yescharacteristic, complemented
39#88049C2212yes—
40#88049C2212yes—
41#88048C2212yescomplemented
42#88047C2212yescomplemented
43C23×D13—21no—
44C13×#41—22no—
45C13×#41—22no—
46C13×#41—22no—
47C13×#41—22no—
48C2×Q8×C13D411yescharacteristic
49C2×Q8×C13C2311yescharacteristic
50C2×D4×C13—21no—
51C2×D4×C13—22no—
52C2×D4×C13D412yes—
53C2×D4×C13D411yescharacteristic
54C2×D4×C13C2311yescharacteristic
55C22×C4×C13—21no—
56C22×C4×C13—21no—
57C22×C4×C13C2312yes—
58C22×C4×C13C2311yescharacteristic
59C2×C22⋊D13—22no—
60C2×C22⋊D13—22no—
61C2×D52—22no—
62C2×D52—22no—
63C2×D52—21no—
64C2×D52C2312yes—
65C13×C4⋊C4—21no—
66C13×C4⋊C4—22no—
67C13×C4⋊C4C2312yes—
68C13×C22⋊C4—22no—
69C13×C22⋊C4—22no—
70C22⋊Dic13—21no—
71#6106—24no—
72#6104—24no—
73#6103—22no—
74#6103—22no—
75C4⋊Dic13C2311yescharacteristic
76C4⋊Dic13C2311yescharacteristic
77C4×Dic13—21no—
78#6099—24no—
79#6099D414yescomplemented
80C2×#1055—22no—
81C2×#1055—24no—
82C2×#1055C2311yescharacteristic
83C2×#1055C2311yescharacteristic
84C2×#1055C2312yes—
85#3062—131nomaximal, Sylow
86C23×C13—21no—
87C23×C13C2×D412yes—
88C23×C13C2×D411yescharacteristic
89C22×D13—22no—
90C22×D13—42no—
91C22×D13—81no—
92C22×D13—81no—
93Q8×C13—22no—
94Q8×C13—21no—
95Q8×C13—21no—
96D4×C13—24no—
97D4×C13—22no—
98D4×C13—21no—
99D4×C13—21no—
100D4×C13—42no—
101D4×C13—82no—
102C2×C4×C13—22no—
103C2×C4×C13—22no—
104C2×C4×C13—22no—
105C2×C4×C13—21no—
106C2×C4×C13—42no—
107C2×C4×C13—81no—
108C2×C4×C13C2×D412yes—
109C2×C4×C13C2×D411yescharacteristic
110C2×C4×C13C2×D412yes—
111C2×C4×C13#4112yes—
112C2×C4×C13C2411yescharacteristic
113C22⋊D13—82no—
114C22⋊D13—82no—
115C2×Dic13—21no—
116C2×Dic13—21no—
117D52—22no—
118D52—22no—
119D52—42no—
120#1055—21no—
121#1055—21no—
122#1055C2×D414yes—
123C2×AGL(1,ℤ/8)—131no—
124C2×#131—131no—
125#537D1311yescharacteristic, complemented
126#535—131no—
127#469—132no—
128#468—132no—
129#466—132no—
130#465—132no—
131#428—131no—
132#416—131no—
133C2×#102—131no—
134C22×C13—21no—
135C22×C13—21no—
136C22×C13—22no—
137C22×C13—22no—
138C22×C13—22no—
139C22×C13—41no—
140C22×C13—41no—
141C22×C13—41no—
142C22×C13#12112yes—
143C22×C13C22×D411yescharacteristic
144C2×D13—42no—
145C2×D13—42no—
146C2×D13—42no—
147C2×D13—84no—
148C4×C13—22no—
149C4×C13—41no—
150C4×C13—42no—
151C4×C13#12112yes—
152C4×C13C22×D411yescharacteristic
153C4×C13C2×#4111yescharacteristic
154Dic13—41no—
155Dic13—41no—
156C2×#41C2×D1311yescharacteristic
157C2×#41C2×D1311yescharacteristic
158C22×D4—131no—
159AGL(1,ℤ/8)—264no—
160C2×SD16—132no—
161C2×D8—132no—
162#131—264no—
163C2×M16—131no—
164C2×M16—132no—
165C22×C8—131no—
166C22×C8—132no—
167#127—261no—
168#124—261no—
169C22⋊Q8—21no—
170C22⋊Q8C2×D1312yescomplemented
171#121—21no—
172#121—22no—
173#121—132no—
174#121—132no—
175#121C2×D1312yescomplemented
176#120—262no—
177#117—131no—
178C2×C4⋊C4C2×D1311yescharacteristic
179#107—132no—
180#106—132no—
181#103—131no—
182#103—131no—
183#102—131no—
184#102—131no—
185#102—132no—
186#102—132no—
187C2×C13—22no—
188C2×C13—42no—
189C2×C13—81no—
190C2×C13C2×#12111yescharacteristic
191C2×C13#57611yescharacteristic
192C2×C13#57711yescharacteristic
193D13—82no—
194C24—261no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41—22no—
199C2×Q8C22⋊D1311yescharacteristic
200C2×Q8C22×D1311yescharacteristic
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—132no—
204C2×D4—262no—
205C2×D4—262no—
206C2×D4—262no—
207C2×D4—262no—
208C2×D4—261no—
209C2×D4C22⋊D1312yes—
210C2×D4C22⋊D1311yescharacteristic
211C2×D4C22×D1311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4C22×D1312yes—
215C22×C4C22×D1311yescharacteristic
216SD16—264no—
217D8—264no—
218M16—134no—
219M16—264no—
220C2×C8—132no—
221C2×C8—131no—
222C2×C8—131no—
223C2×C8—264no—
224C2×C8—262no—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—131no—
228C4⋊C4—131no—
229C4⋊C4C22×D1312yes—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—261no—
233C22⋊C4—262no—
234C22⋊C4—262no—
235C42—261no—
236C13#306211yescharacteristic, complemented, Sylow
237C23—21no—
238C23—262no—
239C23—522no—
240C23—1041no—
241C23—1041no—
242C23C2×C22⋊D1312yes—
243C23C2×C22⋊D1311yescharacteristic
244Q8—22no—
245Q8—21no—
246Q8—21no—
247D4—24no—
248D4—22no—
249D4—21no—
250D4—21no—
251D4—42no—
252D4—82no—
253D4—262no—
254D4—262no—
255D4—522no—
256D4—1042no—
257D4—1042no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—261no—
265C2×C4—261no—
266C2×C4D4×D1312yes—
267C2×C4#612912yes—
268C2×C4C2×C22⋊D1312yes—
269C2×C4C2×C22⋊D1311yescharacteristic
270C2×C4C23×D1311yescharacteristic
271C8—134no—
272C8—261no—
273C8—261no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—522no—
283C22—522no—
284C22—522no—
285C22—1044no—
286C22#8819312yes—
287C22C22×C22⋊D1311yescharacteristic
288C4—22no—
289C4—41no—
290C4—42no—
291C4—521no—
292C4—521no—
293C4#8819312yes—
294C4C2×D4×D1311yescharacteristic
295C4C2×#612911yescharacteristic
296C2—22no—
297C2—42no—
298C2—81no—
299C2—1042no—
300C2C2×#8819311yescharacteristic
301C2#1171954011yescharacteristic
302C2#1171954211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#537C2
socleC22×C13C22×D4
hypercenter#537D13
hypocenterC13#3062
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C13×#537C2semidirect
2. #11719535C2semidirect
3. C2×#88203C2semidirect
4. C2×#88202C2semidirect
5. #11718894C2semidirect
6. #11718710C2semidirect
7. #11718709C2semidirect
8. #11718700C2semidirect
9. #11718699C2semidirect
10. C2×#88049C2semidirect
11. #11718625C2semidirect
16. C13×C22⋊Q8C22semidirect
19. C13×#121C22semidirect
32. C2×#6099C22semidirect
34. C22×#1055C22semidirect
35. #88075C22semidirect
36. #88075C22semidirect
37. #88072C22semidirect
38. #88072C22semidirect
41. #88048C22semidirect
42. #88047C22semidirect
79. #6099D4semidirect
125. #537D13semidirect
170. C22⋊Q8C2×D13semidirect
175. #121C2×D13semidirect
236. C13#3062semidirect

Maximal subgroups

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

subgroupindex
C13×#5372
#117195352
C2×#882032
C2×#882022
#117188942
#117187102
#117187092
#117187002
#117186992
C2×#880492
#117186252
#306213