Finite Group Databasebeta

← #15760729

Subgroups — #15760729

Order 3328.

734 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#13117189C211yesmaximal, characteristic
2#13114935C212yesmaximal, complemented
3#13113163C211yesmaximal, characteristic
4#13112688C212yesmaximal, complemented
5C13×#2482C211yesmaximal, characteristic
6C2×C13×#117C2211yescharacteristic
7C22×C13×C22⋊C4C2211yescharacteristic
8C22×C22⋊Dic13—21no—
9C13×#399C412yescomplemented
10C13×#397C412yescomplemented
11C13×#388—22no—
12C13×#385—21no—
13C13×#381C2211yescharacteristic
14C13×#379—21no—
15C4×C22⋊Dic13—21no—
16C4×C22⋊Dic13C2212yes—
17#11718868C2212yes—
18C22⋊C4×Dic13—24no—
19#11718124—24no—
20#11718117—22no—
21C25×C13D411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×C4×C13C2311yescharacteristic
24C23×Dic13—22no—
25C13×#117—22no—
26C13×#117—22no—
27C2×C13×C4⋊C4—22no—
28C2×C13×C4⋊C4C2×C411yescharacteristic, complemented
29C2×C13×C4⋊C4C2×C411yescharacteristic, complemented
30C2×C13×C22⋊C4—22no—
31C2×C13×C22⋊C4—22no—
32C2×C13×C22⋊C4—22no—
33C2×C13×C22⋊C4—21no—
34C2×C13×C22⋊C4—21no—
35C2×C13×C22⋊C4—22no—
36C2×C13×C22⋊C4C2×C412yes—
37C2×C13×C22⋊C4D412yes—
38C2×C42×C13—21no—
39C2×C22⋊Dic13—28no—
40C2×C22⋊Dic13—22no—
41C2×C22⋊Dic13—22no—
42C2×C4×Dic13—22no—
43C2×C4×Dic13D414yes—
44C2×#6099—22no—
45C13×#95—22no—
46C13×#95—21no—
47C13×#95C2×C412yescomplemented
48#88071—22no—
49#88071—22no—
50C22⋊#1055—22no—
51C22⋊#1055—22no—
52#9477—131nomaximal, Sylow
53C24×C13—21no—
54C24×C13—24no—
55C24×C13—21no—
56C24×C13—42no—
57C24×C13—81no—
58C24×C13C22⋊C412yes—
59C24×C13C2×D411yescharacteristic
60C22×C4×C13—21no—
61C22×C4×C13—21no—
62C22×C4×C13—22no—
63C22×C4×C13—21no—
64C22×C4×C13—21no—
65C22×C4×C13—42no—
66C22×C4×C13—81no—
67C22×C4×C13C22⋊C412yes—
68C22×C4×C13C22×C411yescharacteristic
69C22×C4×C13C2×D412yes—
70C22×C4×C13C2×D411yescharacteristic
71C22×C4×C13#4112yes—
72C22×Dic13—24no—
73C22×Dic13—28no—
74C22×Dic13—82no—
75C22×Dic13—82no—
76C13×C4⋊C4—22no—
77C13×C4⋊C4—22no—
78C13×C4⋊C4—82no—
79C13×C22⋊C4—24no—
80C13×C22⋊C4—24no—
81C13×C22⋊C4—42no—
82C13×C22⋊C4—44no—
83C13×C22⋊C4—41no—
84C13×C22⋊C4—41no—
85C13×C22⋊C4—82no—
86C42×C13—42no—
87C22⋊Dic13—28no—
88C22⋊Dic13—28no—
89C4×Dic13—44no—
90C4×Dic13—82no—
91#6099—44no—
92C2×#1055—42no—
93#2482D1311yescharacteristic
94#2273—131no—
95#1826—132no—
96#1462—131no—
97#1284—132no—
98C23×C13—22no—
99C23×C13—22no—
100C23×C13—24no—
101C23×C13—22no—
102C23×C13—22no—
103C23×C13—22no—
104C23×C13—22no—
105C23×C13—41no—
106C23×C13—42no—
107C23×C13—42no—
108C23×C13—42no—
109C23×C13—42no—
110C23×C13—81no—
111C23×C13—81no—
112C23×C13—88no—
113C23×C13#10414yes—
114C23×C13C2×C22⋊C411yescharacteristic
115C23×C13#12011yescharacteristic
116C23×C13#12311yescharacteristic
117C2×C4×C13—24no—
118C2×C4×C13—22no—
119C2×C4×C13—41no—
120C2×C4×C13—41no—
121C2×C4×C13—42no—
122C2×C4×C13—42no—
123C2×C4×C13—42no—
124C2×C4×C13—42no—
125C2×C4×C13—42no—
126C2×C4×C13—42no—
127C2×C4×C13—41no—
128C2×C4×C13—41no—
129C2×C4×C13—42no—
130C2×C4×C13—42no—
131C2×C4×C13—81no—
132C2×C4×C13C4×D412yes—
133C2×C4×C13#12112yes—
134C2×Dic13—28no—
135C2×Dic13—44no—
136C2×Dic13—44no—
137C2×Dic13—88no—
138#1055—82no—
139C2×#117C2×D1311yescharacteristic
140C22×C22⋊C4—261no—
141C22×C22⋊C4C2×D1311yescharacteristic
142#407—132no—
143#399Dic1312yescomplemented
144#397Dic1312yescomplemented
145#388—22no—
146#385—21no—
147#381C2×D1311yescharacteristic
148#379—21no—
149C4×C22⋊C4—132no—
150C4×C22⋊C4—264no—
151C4×C22⋊C4—261no—
152#339—264no—
153#323—262no—
154C22×C13—21no—
155C22×C13—21no—
156C22×C13—21no—
157C22×C13—21no—
158C22×C13—21no—
159C22×C13—24no—
160C22×C13—21no—
161C22×C13—21no—
162C22×C13—44no—
163C22×C13—44no—
164C22×C13—44no—
165C22×C13—44no—
166C22×C13—42no—
167C22×C13—42no—
168C22×C13—82no—
169C22×C13—84no—
170C22×C13#38611yescharacteristic
171C22×C13C2×#10412yes—
172C22×C13#45812yes—
173C4×C13—22no—
174C4×C13—22no—
175C4×C13—42no—
176C4×C13—82no—
177C4×C13—81no—
178C4×C13—81no—
179Dic13—48no—
180C25C22⋊D1311yescharacteristic
181C23×C4—262no—
182C23×C4C22⋊D1311yescharacteristic
183C23×C4C22×D1311yescharacteristic
184C2×M16—262no—
185#117—22no—
186#117—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C2×Dic1311yescharacteristic, complemented
189C2×C4⋊C4C2×Dic1311yescharacteristic, complemented
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—262no—
197C2×C22⋊C4—268no—
198C2×C22⋊C4—262no—
199C2×C22⋊C4C2×Dic1312yes—
200C2×C22⋊C4C22⋊D1312yes—
201C2×C42—21no—
202C2×C42—134no—
203C2×C42—262no—
204C22⋊C8—262no—
205C22⋊C8—262no—
206#95—22no—
207#95—21no—
208#95—262no—
209#95—262no—
210#95C2×Dic1312yescomplemented
211C2×C13—22no—
212C2×C13—22no—
213C2×C13—41no—
214C2×C13—44no—
215C2×C13#189011yescharacteristic
216C2×C13#189611yescharacteristic
217C2×C13#191111yescharacteristic
218C24—21no—
219C24—24no—
220C24—21no—
221C24—42no—
222C24—81no—
223C24C22⋊Dic1312yes—
224C24C2×C22⋊D1311yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—81no—
232C22×C4—264no—
233C22×C4—268no—
234C22×C4—1042no—
235C22×C4—1042no—
236C22×C4C22⋊Dic1312yes—
237C22×C4D4×D1312yes—
238C22×C4#612912yes—
239C22×C4C22×Dic1311yescharacteristic
240C22×C4C2×C22⋊D1311yescharacteristic
241M16—524no—
242C2×C8—522no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—82no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—82no—
253C22⋊C4—268no—
254C22⋊C4—268no—
255C42—42no—
256C42—524no—
257C42—1042no—
258C13#947711yescharacteristic, complemented, Sylow
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—81no—
272C23—81no—
273C23—88no—
274C23#8807714yes—
275C23#8818911yescharacteristic
276C23#8819111yescharacteristic
277C23C2×C22⋊Dic1311yescharacteristic
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—268no—
294C2×C4—524no—
295C2×C4—524no—
296C2×C4—1048no—
297C2×C4D4×Dic1312yes—
298C2×C4#8819312yes—
299C8—1042no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22#1171871412yes—
317C22#1171888111yescharacteristic
318C22C2×#8807712yes—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—82no—
323C4—81no—
324C4—81no—
325C4—528no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2#1311525011yescharacteristic
331C2#1311528011yescharacteristic
332C2#1311536911yescharacteristic
subgroupG/H
centerC22#11718881
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2482C2
socleC22×C13#386
prime radicalC25×C13D4
cosocleC23×C4×C13C23
hypercenter#2482D13
hypocenterC13#9477
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
2. #13114935C2semidirect
4. #13112688C2semidirect
9. C13×#399C4semidirect
10. C13×#397C4semidirect
28. C2×C13×C4⋊C4C2×C4semidirect
29. C2×C13×C4⋊C4C2×C4semidirect
47. C13×#95C2×C4semidirect
143. #399Dic13semidirect
144. #397Dic13semidirect
188. C2×C4⋊C4C2×Dic13semidirect
189. C2×C4⋊C4C2×Dic13semidirect
210. #95C2×Dic13semidirect
258. C13#9477semidirect

Maximal subgroups

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

subgroupindex
#131171892
#131149352
#131131632
#131126882
C13×#24822
#947713