Finite Group Databasebeta

← #15778091

Subgroups — #15778091

Order 3328.

564 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 125 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#13117549C211yesmaximal, characteristic
2C2×#11718133C211yesmaximal, characteristic
3#99×Dic13C212yesmaximal
4#13112960C212yesmaximal
5C2×C13×#354C211yesmaximal, characteristic
6C2×C13×#124C2211yescharacteristic
7C2×Q8×Dic13C411yescharacteristic, complemented
8C2×D4×Dic13C2211yescharacteristic
9C2×C13×#99C2212yes—
10#11718891C411yescharacteristic, complemented
11#11718882C2211yescharacteristic
12C2×#88074C2212yes—
13#11718612—22no—
14#11718611—22no—
15C22⋊C4×Dic13—22no—
16#11718596C411yescharacteristic, complemented
17#11718593C411yescharacteristic, complemented
18C13×#354C414yescomplemented
19#11718133C414yescomplemented
20#11718119—22no—
21C22×Q8×C13C2×C411yescharacteristic
22C22×D4×C13C2311yescharacteristic
23C23×Dic13D412yes—
24C13×#124—22no—
25C13×#124C2×C412yes—
26C13×#124C2×C412yes—
27C2×C13×C22⋊C4—22no—
28C2×C13×C22⋊C4—22no—
29C2×C42×C13C2×C411yescharacteristic
30C2×C22⋊Dic13—22no—
31C2×C22⋊Dic13—22no—
32C2×C22⋊Dic13D412yes—
33Q8×Dic13—42no—
34D4×Dic13—42no—
35C2×C4⋊Dic13—21no—
36C2×C4⋊Dic13—22no—
37C2×C4⋊Dic13C2×C411yescharacteristic
38C2×C4⋊Dic13C2×C411yescharacteristic
39C2×C4×Dic13—21no—
40C2×C4×Dic13—42no—
41C2×C4×Dic13C2×C411yescharacteristic
42C2×C4×Dic13C2×C411yescharacteristic
43C13×#99—22no—
44C13×#99C2×C414yes—
45#88074—22no—
46#88074C2×C414yes—
47#88071—21no—
48#88071—21no—
49#88071—22no—
50#88071—42no—
51#12627—131nomaximal, Sylow
52C24×C13C2×D412yes—
53C2×Q8×C13—21no—
54C2×Q8×C13—41no—
55C2×Q8×C13—41no—
56C2×Q8×C13C4212yescomplemented
57C2×D4×C13—41no—
58C2×D4×C13—41no—
59C2×D4×C13C22×C411yescharacteristic
60C2×D4×C13C22×C411yescharacteristic
61C2×D4×C13#4112yes—
62C22×C4×C13—21no—
63C22×C4×C13—21no—
64C22×C4×C13—42no—
65C22×C4×C13C22×C411yescharacteristic
66C22×Dic13—22no—
67C22×Dic13—22no—
68C22×Dic13—21no—
69C22×Dic13—21no—
70C22×Dic13—42no—
71C22×Dic13—82no—
72C22×Dic13C22⋊C411yescharacteristic
73C22×Dic13C22⋊C412yes—
74C22×Dic13C22⋊C411yescharacteristic
75C13×C22⋊C4—24no—
76C13×C22⋊C4—24no—
77C13×C22⋊C4—24no—
78C13×C22⋊C4—42no—
79C42×C13—21no—
80C42×C13C4212yescomplemented
81C22⋊Dic13—24no—
82C22⋊Dic13—42no—
83C22⋊Dic13—42no—
84C22⋊Dic13—82no—
85C4⋊Dic13—41no—
86C4⋊Dic13—42no—
87C4⋊Dic13—41no—
88C4⋊Dic13—82no—
89C4×Dic13—22no—
90C4×Dic13—42no—
91C4×Dic13—41no—
92C4×Dic13—84no—
93#2326—131no—
94C2×#354—131no—
95C2×#354D1311yescharacteristic
96C4×#99—132no—
97#1393—132no—
98C23×C13—22no—
99C23×C13—22no—
100C23×C13—42no—
101C23×C13—42no—
102C23×C13C2×C22⋊C411yescharacteristic
103C23×C13C4×D412yes—
104C23×C13C4×D412yes—
105Q8×C13—42no—
106Q8×C13—42no—
107D4×C13—41no—
108D4×C13—41no—
109D4×C13—42no—
110C2×C4×C13—21no—
111C2×C4×C13—21no—
112C2×C4×C13—22no—
113C2×C4×C13—22no—
114C2×C4×C13—41no—
115C2×C4×C13—41no—
116C2×C4×C13—41no—
117C2×C4×C13—41no—
118C2×C4×C13—44no—
119C2×C4×C13—82no—
120C2×C4×C13—82no—
121C2×C4×C13C2×C4211yescharacteristic
122C2×C4×C13#11711yescharacteristic
123C2×Dic13—21no—
124C2×Dic13—22no—
125C2×Dic13—41no—
126C2×Dic13—44no—
127C2×Dic13—42no—
128C2×Dic13—41no—
129C2×Dic13—42no—
130C2×Dic13—44no—
131C2×Dic13—41no—
132C2×Dic13—41no—
133C2×Dic13—41no—
134C2×Dic13—41no—
135C2×Dic13—82no—
136C2×Dic13—82no—
137C2×Dic13—82no—
138C2×Dic13#9912yes—
139C2×#124C2×D1311yescharacteristic
140C2×C4×Q8—131no—
141C2×C4×D4—131no—
142C2×#99—132no—
143C2×#99C2×D1312yes—
144#391—131no—
145#390—131no—
146#388—262no—
147#385—262no—
148#384—131no—
149C4×C22⋊C4—262no—
150#376—131no—
151#354—134no—
152#354Dic1314yescomplemented
153#342—262no—
154C22×C13—21no—
155C22×C13—21no—
156C22×C13—22no—
157C22×C13—22no—
158C22×C13—42no—
159C22×C13—42no—
160C22×C13—42no—
161C22×C13—42no—
162C22×C13—82no—
163C22×C13C4×C22⋊C411yescharacteristic
164C22×C13C2×#9911yescharacteristic
165C22×C13#41011yescharacteristic
166C4×C13—21no—
167C4×C13—21no—
168C4×C13—42no—
169C4×C13—42no—
170C4×C13—84no—
171Dic13—22no—
172Dic13—42no—
173Dic13—41no—
174Dic13—81no—
175Dic13—84no—
176Dic13—81no—
177C22×Q8C4×D1311yescharacteristic
178C22×D4C22×D1311yescharacteristic
179C23×C4—132no—
180#124—22no—
181#124C4×D1312yes—
182#124C2×Dic1312yes—
183C4×Q8—522no—
184C4×D4—522no—
185C2×C4⋊C4—131no—
186C2×C4⋊C4—131no—
187C2×C4⋊C4—261no—
188C2×C4⋊C4—262no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—132no—
192C2×C22⋊C4—262no—
193C2×C22⋊C4—262no—
194C2×C42—131no—
195C2×C42—131no—
196C2×C42—261no—
197C2×C42—522no—
198C2×C42C4×D1311yescharacteristic
199#99—22no—
200#99—134no—
201#99—262no—
202#99C2×Dic1314yes—
203#95—261no—
204#95—261no—
205#95—262no—
206#95—522no—
207C2×C13—21no—
208C2×C13—21no—
209C2×C13—42no—
210C2×C13—42no—
211C2×C13C4×#9911yescharacteristic
212C2×C13#212711yescharacteristic
213C2×C13#212911yescharacteristic
214C24D4×D1312yes—
215C2×Q8—21no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8C4×Dic1312yescomplemented
219C2×D4—41no—
220C2×D4—41no—
221C2×D4C2×C4×D1311yescharacteristic
222C2×D4#612912yes—
223C2×D4C22×Dic1311yescharacteristic
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—131no—
228C22×C4—131no—
229C22×C4—132no—
230C22×C4—262no—
231C22×C4—262no—
232C22×C4—261no—
233C22×C4—261no—
234C22×C4—522no—
235C22×C4—1042no—
236C22×C4C2×C4×D1311yescharacteristic
237C4⋊C4—521no—
238C4⋊C4—521no—
239C4⋊C4—522no—
240C4⋊C4—1042no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—42no—
245C22⋊C4—264no—
246C22⋊C4—522no—
247C22⋊C4—522no—
248C22⋊C4—1042no—
249C42—21no—
250C42—262no—
251C42—521no—
252C42—522no—
253C42—1044no—
254C42C4×Dic1312yescomplemented
255C13#1262711yescharacteristic, complemented, Sylow
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23C22⋊C4×D1311yescharacteristic
261C23#8813512yes—
262C23D4×Dic1312yes—
263Q8—42no—
264Q8—42no—
265D4—41no—
266D4—42no—
267D4—41no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—132no—
280C2×C4—261no—
281C2×C4—262no—
282C2×C4—521no—
283C2×C4—524no—
284C2×C4—522no—
285C2×C4—521no—
286C2×C4—524no—
287C2×C4—521no—
288C2×C4—522no—
289C2×C4—521no—
290C2×C4—521no—
291C2×C4—521no—
292C2×C4—1042no—
293C2×C4—1042no—
294C2×C4—1042no—
295C2×C4#8813111yescharacteristic
296C2×C4C2×C4×Dic1311yescharacteristic
297C22—21no—
298C22—21no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22#1171840111yescharacteristic
307C22D13×#9911yescharacteristic
308C22C22⋊C4×Dic1311yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—84no—
314C4—262no—
315C4—522no—
316C4—521no—
317C4—1041no—
318C4—1044no—
319C4—1041no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#99×Dic1311yescharacteristic
325C2#1311650411yescharacteristic
326C2#1311650911yescharacteristic
subgroupG/H
centerC22D13×#99
commutatorC2×C4×C13C2×C42
FrattiniC22×D4C22×D13
FittingC2×C13×#354C2
socleC22×C13C2×#99
prime radical, cosocleC22×D4×C13C23
hypercenterC2×#354D13
hypocenterC13#12627
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
7. C2×Q8×Dic13C4semidirect
10. #11718891C4semidirect
16. #11718596C4semidirect
17. #11718593C4semidirect
18. C13×#354C4semidirect
19. #11718133C4semidirect
56. C2×Q8×C13C42semidirect
80. C42×C13C42semidirect
152. #354Dic13semidirect
218. C2×Q8C4×Dic13semidirect
254. C42C4×Dic13semidirect
255. C13#12627semidirect

Maximal subgroups

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

subgroupindex
#131175492
C2×#117181332
#99×Dic132
#131129602
C2×C13×#3542
#1262713