Finite Group Databasebeta

← #15772029

Subgroups — #15772029

Order 3328.

594 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 131 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×#11718120C212yesmaximal, complemented
2#13112770C212yesmaximal, complemented
3#13112768C212yesmaximal, complemented
4C13×#2311C211yesmaximal, characteristic
5C22×C13×#41C411yescharacteristic
6C2×C13×#117C411yescharacteristic
7C22×C13×C4⋊C4C2211yescharacteristic
8C13×#391C411yescharacteristic
9C13×#391C2211yescharacteristic
10C13×#390C411yescharacteristic
11C13×#390C2211yescharacteristic
12C13×#386—22no—
13C13×#380—22no—
14C2×C22⋊#1055—22no—
15#11718120—22no—
16#11718120C2214yes—
17C2×C13×#41—22no—
18C2×C13×#41—42no—
19C2×C13×#41—42no—
20C2×C13×#41C814yescomplemented
21C22×Q8×C13C2×C411yescharacteristic
22C22×D4×C13—21no—
23C22×D4×C13C2×C411yescharacteristic
24C23×C4×C13—21no—
25C23×C4×C13C2×C411yescharacteristic
26C23×C4×C13D412yes—
27C13×#117—42no—
28C2×C13×C4⋊C4—21no—
29C2×C13×C4⋊C4—42no—
30C2×C13×C4⋊C4C2×C411yescharacteristic
31C2×C13×C4⋊C4D412yes—
32C2×C13×C4⋊C4C2311yescharacteristic
33C2×C13×C22⋊C4—21no—
34C2×C13×C22⋊C4—22no—
35C2×C13×C22⋊C4—22no—
36C2×C42×C13C2×C411yescharacteristic
37C2×C42×C13C2×C411yescharacteristic
38C22×#1055—42no—
39C13×#95—22no—
40C13×#95—22no—
41C22⋊#1055—24no—
42C22⋊#1055—42no—
43#11432—131nomaximal, Sylow
44C24×C13—21no—
45C24×C13C22⋊C411yescharacteristic
46C13×#41—48no—
47C13×#41—48no—
48C2×Q8×C13—21no—
49C2×Q8×C13—41no—
50C2×Q8×C13—41no—
51C2×Q8×C13C2×C812yescomplemented
52C2×D4×C13—24no—
53C2×D4×C13—21no—
54C2×D4×C13—41no—
55C2×D4×C13—41no—
56C2×D4×C13—42no—
57C2×D4×C13—42no—
58C2×D4×C13C2×C812yescomplemented
59C22×C4×C13—22no—
60C22×C4×C13—22no—
61C22×C4×C13—22no—
62C22×C4×C13—21no—
63C22×C4×C13—22no—
64C22×C4×C13—24no—
65C22×C4×C13—21no—
66C22×C4×C13—21no—
67C22×C4×C13—42no—
68C22×C4×C13—42no—
69C22×C4×C13—82no—
70C22×C4×C13C22⋊C411yescharacteristic
71C22×C4×C13C22⋊C412yes—
72C22×C4×C13C2×C812yes—
73C22×C4×C13M1612yes—
74C22×C4×C13C22×C411yescharacteristic
75C22×C4×C13C2×D412yes—
76C13×C4⋊C4—41no—
77C13×C4⋊C4—42no—
78C13×C4⋊C4—41no—
79C13×C4⋊C4—42no—
80C13×C22⋊C4—42no—
81C13×C22⋊C4—82no—
82C13×C22⋊C4—82no—
83C42×C13—41no—
84C42×C13—41no—
85C2×#1055—44no—
86C2×#1055—84no—
87#2311D1311yescharacteristic
88C2×#324—132no—
89#1319—132no—
90#1318—132no—
91C23×C13—22no—
92C23×C13—41no—
93C23×C13—41no—
94C23×C13—42no—
95C23×C13—42no—
96C23×C13—41no—
97C23×C13—41no—
98C23×C13C22⋊C812yes—
99C23×C13#9912yes—
100C23×C13#10012yes—
101C23×C13C2×C22⋊C411yescharacteristic
102Q8×C13—42no—
103Q8×C13—42no—
104D4×C13—42no—
105D4×C13—42no—
106D4×C13—48no—
107C2×C4×C13—24no—
108C2×C4×C13—24no—
109C2×C4×C13—21no—
110C2×C4×C13—21no—
111C2×C4×C13—21no—
112C2×C4×C13—21no—
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—48no—
117C2×C4×C13—44no—
118C2×C4×C13—41no—
119C2×C4×C13—41no—
120C2×C4×C13—42no—
121C2×C4×C13—42no—
122C2×C4×C13—41no—
123C2×C4×C13—41no—
124C2×C4×C13—41no—
125C2×C4×C13—41no—
126C2×C4×C13—41no—
127C2×C4×C13—41no—
128C2×C4×C13—82no—
129C2×C4×C13C22⋊C812yes—
130C2×C4×C13C22×C811yescharacteristic
131C2×C4×C13C2×M1611yescharacteristic
132#1055—84no—
133C22×#41Dic1311yescharacteristic
134C2×#117Dic1311yescharacteristic
135C22×C4⋊C4C2×D1311yescharacteristic
136C2×C22⋊C8—262no—
137#391Dic1311yescharacteristic
138#391C2×D1311yescharacteristic
139#390Dic1311yescharacteristic
140#390C2×D1311yescharacteristic
141#386—22no—
142#380—22no—
143#324—134no—
144#324—262no—
145C22×C13—21no—
146C22×C13—22no—
147C22×C13—21no—
148C22×C13—22no—
149C22×C13—42no—
150C22×C13—42no—
151C22×C13—44no—
152C22×C13—42no—
153C22×C13—82no—
154C22×C13—81no—
155C22×C13—81no—
156C22×C13#32314yes—
157C22×C13C2×C22⋊C811yescharacteristic
158C22×C13C2×#9911yescharacteristic
159C22×C13C2×#10011yescharacteristic
160C4×C13—22no—
161C4×C13—21no—
162C4×C13—21no—
163C4×C13—42no—
164C4×C13—42no—
165C4×C13—42no—
166C4×C13—81no—
167C4×C13—81no—
168C2×#41—22no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41#105514yescomplemented
172C22×Q8C2×Dic1311yescharacteristic
173C22×D4—21no—
174C22×D4C2×Dic1311yescharacteristic
175C23×C4—21no—
176C23×C4C2×Dic1311yescharacteristic
177C23×C4C22⋊D1312yes—
178C22×C8—522no—
179#117—42no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C2×Dic1311yescharacteristic
183C2×C4⋊C4C22⋊D1312yes—
184C2×C4⋊C4C22×D1311yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C42C2×Dic1311yescharacteristic
189C2×C42C2×Dic1311yescharacteristic
190C22⋊C8—264no—
191C22⋊C8—522no—
192#95—22no—
193#95—22no—
194C2×C13—21no—
195C2×C13—21no—
196C2×C13—24no—
197C2×C13—82no—
198C2×C13C2×#32311yescharacteristic
199C2×C13#212711yescharacteristic
200C2×C13#213311yescharacteristic
201C24—21no—
202C24C22⋊Dic1311yescharacteristic
203#41—48no—
204#41—48no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8C2×#105512yescomplemented
209C2×D4—24no—
210C2×D4—21no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4C2×#105512yescomplemented
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—82no—
227C22×C4C2×#105512yes—
228C22×C4#609912yes—
229C22×C4C22⋊Dic1311yescharacteristic
230C22×C4C22⋊Dic1312yes—
231C22×C4C22×Dic1311yescharacteristic
232C22×C4C2×C22⋊D1312yes—
233C2×C8—524no—
234C2×C8—1044no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C42—41no—
243C42—41no—
244C13#1143211yescharacteristic, complemented, Sylow
245C23—22no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23—41no—
252C23C22⋊#105512yes—
253C23#8807312yes—
254C23#8807412yes—
255C23C2×C22⋊Dic1311yescharacteristic
256Q8—42no—
257Q8—42no—
258D4—42no—
259D4—42no—
260D4—48no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—82no—
283C2×C4C22⋊#105512yes—
284C2×C4C22×#105511yescharacteristic
285C2×C4C2×#609911yescharacteristic
286C8—1044no—
287C22—21no—
288C22—22no—
289C22—21no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—44no—
294C22—42no—
295C22—82no—
296C22—81no—
297C22—81no—
298C22#1171811714yes—
299C22C2×C22⋊#105511yescharacteristic
300C22C2×#8807311yescharacteristic
301C22C2×#8807411yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2—24no—
313C2—82no—
314C2C2×#1171811711yescharacteristic
315C2#1311650511yescharacteristic
316C2#1311652511yescharacteristic
subgroupG/H
centerC22C2×#88074
commutatorC2×C4×C13C22×C8
FrattiniC2×C4⋊C4C22×D13
FittingC13×#2311C2
socleC22×C13C2×#99
prime radicalC22×C13×#41C4
cosocleC2×C13×C4⋊C4C23
hypercenter#2311D13
hypocenterC13#11432
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2311

Complements

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

NH ≅ G/Ntype
1. C2×#11718120C2semidirect
2. #13112770C2semidirect
3. #13112768C2semidirect
20. C2×C13×#41C8semidirect
51. C2×Q8×C13C2×C8semidirect
58. C2×D4×C13C2×C8semidirect
171. C2×#41#1055semidirect
208. C2×Q8C2×#1055semidirect
215. C2×D4C2×#1055semidirect
244. C13#11432semidirect

Maximal subgroups

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

subgroupindex
C2×#117181202
#131127702
#131127682
C13×#23112
#1143213