Finite Group Databasebeta

← #15778089

Subgroups — #15778089

Order 3328.

564 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 135 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×#11718133C212yesmaximal
2C4×#88074C212yesmaximal
3#13112961C212yesmaximal
4C13×#2326C211yesmaximal, characteristic
5C2×C13×#124C2211yescharacteristic
6C2×C4×Q8×C13C411yescharacteristic, complemented
7C2×C4×D4×C13C2211yescharacteristic
8C13×#391C411yescharacteristic, complemented
9C13×#390C2211yescharacteristic
10C13×#388—22no—
11C13×#385—22no—
12C13×#384C411yescharacteristic, complemented
13C13×#376C411yescharacteristic, complemented
14C2×#88074C2214yes—
15C4×C22⋊Dic13—22no—
16#11718133C418yescomplemented
17#11718118—22no—
18C22×Q8×C13C2×C411yescharacteristic
19C22×D4×C13C2311yescharacteristic
20C23×C4×C13D412yes—
21C13×#124—22no—
22C13×#124C2×C414yes—
23C4×Q8×C13—42no—
24C4×D4×C13—42no—
25C2×C13×C4⋊C4—21no—
26C2×C13×C4⋊C4—22no—
27C2×C13×C4⋊C4C2×C411yescharacteristic
28C2×C13×C4⋊C4C2×C411yescharacteristic
29C2×C13×C22⋊C4—22no—
30C2×C13×C22⋊C4—22no—
31C2×C13×C22⋊C4D412yes—
32C2×C42×C13—21no—
33C2×C42×C13C2×C411yescharacteristic
34C2×C42×C13C2×C411yescharacteristic
35C2×C42×C13C2×C411yescharacteristic
36C2×C22⋊Dic13—24no—
37C2×C4×Dic13—42no—
38C13×#95—21no—
39C13×#95—22no—
40C13×#95—21no—
41#88074—24no—
42#88074C2×C418yes—
43#88071—42no—
44#12627—131nomaximal, Sylow
45C24×C13C2×D412yes—
46C2×Q8×C13—21no—
47C2×Q8×C13—41no—
48C2×Q8×C13—41no—
49C2×Q8×C13C4212yescomplemented
50C2×D4×C13—41no—
51C2×D4×C13—41no—
52C2×D4×C13C22×C412yes—
53C2×D4×C13#4112yes—
54C22×C4×C13—21no—
55C22×C4×C13—21no—
56C22×C4×C13—22no—
57C22×C4×C13—22no—
58C22×C4×C13—21no—
59C22×C4×C13—21no—
60C22×C4×C13—82no—
61C22×C4×C13C22⋊C411yescharacteristic
62C22×C4×C13C22⋊C412yes—
63C22×C4×C13C22⋊C411yescharacteristic
64C22×C4×C13C22×C411yescharacteristic
65C22×Dic13—44no—
66C13×C4⋊C4—41no—
67C13×C4⋊C4—41no—
68C13×C4⋊C4—42no—
69C13×C4⋊C4—82no—
70C13×C22⋊C4—28no—
71C13×C22⋊C4—42no—
72C13×C22⋊C4—82no—
73C42×C13—22no—
74C42×C13—21no—
75C42×C13—41no—
76C42×C13—42no—
77C42×C13C4212yescomplemented
78C22⋊Dic13—28no—
79C22⋊Dic13—44no—
80C4×Dic13—84no—
81#2326D1311yescharacteristic
82C2×#354—132no—
83C4×#99—132no—
84#1393—132no—
85C23×C13—22no—
86C23×C13—22no—
87C23×C13—42no—
88C23×C13—42no—
89C23×C13C2×C22⋊C411yescharacteristic
90C23×C13C4×D414yes—
91Q8×C13—42no—
92Q8×C13—42no—
93D4×C13—42no—
94D4×C13—42no—
95C2×C4×C13—21no—
96C2×C4×C13—21no—
97C2×C4×C13—22no—
98C2×C4×C13—22no—
99C2×C4×C13—21no—
100C2×C4×C13—22no—
101C2×C4×C13—41no—
102C2×C4×C13—44no—
103C2×C4×C13—41no—
104C2×C4×C13—41no—
105C2×C4×C13—41no—
106C2×C4×C13—41no—
107C2×C4×C13—42no—
108C2×C4×C13—41no—
109C2×C4×C13—41no—
110C2×C4×C13—42no—
111C2×C4×C13—41no—
112C2×C4×C13—41no—
113C2×C4×C13—41no—
114C2×C4×C13—82no—
115C2×C4×C13#9912yes—
116C2×C4×C13C2×C4211yescharacteristic
117C2×C4×C13#11711yescharacteristic
118C2×Dic13—48no—
119C2×Dic13—84no—
120C2×Dic13—84no—
121C2×#124C2×D1311yescharacteristic
122C2×C4×Q8Dic1311yescharacteristic, complemented
123C2×C4×D4C2×D1311yescharacteristic
124C2×#99—134no—
125#391Dic1311yescharacteristic, complemented
126#390C2×D1311yescharacteristic
127#388—22no—
128#385—22no—
129#384Dic1311yescharacteristic, complemented
130C4×C22⋊C4—262no—
131#376Dic1311yescharacteristic, complemented
132#354—138no—
133#342—262no—
134C22×C13—21no—
135C22×C13—21no—
136C22×C13—24no—
137C22×C13—44no—
138C22×C13—42no—
139C22×C13—42no—
140C22×C13—82no—
141C22×C13C4×C22⋊C411yescharacteristic
142C22×C13C2×#9911yescharacteristic
143C22×C13#41011yescharacteristic
144C4×C13—22no—
145C4×C13—21no—
146C4×C13—21no—
147C4×C13—42no—
148C4×C13—42no—
149C4×C13—41no—
150C4×C13—42no—
151C4×C13—81no—
152C4×C13—81no—
153Dic13—88no—
154C22×Q8C2×Dic1311yescharacteristic
155C22×D4C22×D1311yescharacteristic
156C23×C4C22⋊D1312yes—
157#124—22no—
158#124C4×D1314yes—
159C4×Q8—42no—
160C4×D4—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4C2×Dic1311yescharacteristic
164C2×C4⋊C4C2×Dic1311yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—264no—
168C2×C22⋊C4C22⋊D1312yes—
169C2×C42—21no—
170C2×C42—522no—
171C2×C42C2×Dic1311yescharacteristic
172C2×C42C2×Dic1311yescharacteristic
173C2×C42C2×Dic1311yescharacteristic
174#99—138no—
175#99—264no—
176#95—21no—
177#95—22no—
178#95—21no—
179#95—522no—
180C2×C13—21no—
181C2×C13—21no—
182C2×C13—44no—
183C2×C13C4×#9911yescharacteristic
184C2×C13#212711yescharacteristic
185C2×C13#212911yescharacteristic
186C24C2×C22⋊D1312yes—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8C4×Dic1312yescomplemented
191C2×D4—41no—
192C2×D4—41no—
193C2×D4C2×C4×D1312yes—
194C2×D4#612712yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—82no—
202C22×C4—524no—
203C22×C4C22⋊Dic1311yescharacteristic
204C22×C4C22⋊Dic1312yes—
205C22×C4C22⋊Dic1311yescharacteristic
206C22×C4C22×Dic1311yescharacteristic
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—82no—
211C22⋊C4—28no—
212C22⋊C4—42no—
213C22⋊C4—82no—
214C22⋊C4—268no—
215C22⋊C4—524no—
216C42—22no—
217C42—21no—
218C42—41no—
219C42—42no—
220C42—1044no—
221C42C4×Dic1312yescomplemented
222C13#1262711yescharacteristic, complemented, Sylow
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23C4×C22⋊D1314yes—
228C23C2×C22⋊Dic1311yescharacteristic
229Q8—42no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—528no—
254C2×C4—1044no—
255C2×C4—1044no—
256C2×C4#8807412yes—
257C2×C4C2×C4×Dic1311yescharacteristic
258C2×C4#8818011yescharacteristic
259C22—21no—
260C22—21no—
261C22—24no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22—82no—
266C22C4×C22⋊Dic1311yescharacteristic
267C22C2×#8807411yescharacteristic
268C22#1171890111yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—42no—
273C4—42no—
274C4—42no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—1048no—
279C2—21no—
280C2—21no—
281C2—44no—
282C2C4×#8807411yescharacteristic
283C2#1311650511yescharacteristic
284C2#1311651111yescharacteristic
subgroupG/H
centerC22C2×#88074
commutatorC2×C4×C13C2×C42
FrattiniC22×D4C22×D13
FittingC13×#2326C2
socleC22×C13C2×#99
prime radical, cosocleC22×D4×C13C23
hypercenter#2326D13
hypocenterC13#12627
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
6. C2×C4×Q8×C13C4semidirect
8. C13×#391C4semidirect
12. C13×#384C4semidirect
13. C13×#376C4semidirect
16. #11718133C4semidirect
49. C2×Q8×C13C42semidirect
77. C42×C13C42semidirect
122. C2×C4×Q8Dic13semidirect
125. #391Dic13semidirect
129. #384Dic13semidirect
131. #376Dic13semidirect
190. C2×Q8C4×Dic13semidirect
221. C42C4×Dic13semidirect
222. C13#12627semidirect

Maximal subgroups

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

subgroupindex
C2×#117181332
C4×#880742
#131129612
C13×#23262
#1262713