Finite Group Databasebeta

← #15760654

Subgroups — #15760654

Order 3328.

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

normal subgroups only

This group has 700 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#13117422C211yesmaximal, characteristic
2#13114760C211yesmaximal, characteristic
3#13113369C212yesmaximal
4#13112689C212yesmaximal
5C13×#2434C211yesmaximal, characteristic
6C23×Q8×C13C2211yescharacteristic
7C2×C13×C4⋊Q8C412yescomplemented
8C2×C13×C22⋊Q8C2211yescharacteristic
9C2×#88180—21no—
10C13×#395—22no—
11C13×#391C412yescomplemented
12C13×#385—21no—
13C13×#385—21no—
14C4×C13×C22⋊C4C2211yescharacteristic
15#11718891—24no—
16C2×#88076—22no—
17#11718872—21no—
18#11718872C2212yes—
19#11718868C2212yes—
20#11718130—24no—
21C22×Q8×C13—22no—
22C22×Q8×C13—22no—
23C22×Q8×C13—44no—
24C22×Q8×C13C2×C411yescharacteristic
25C22×Q8×C13C2×C412yes—
26C22×Q8×C13D412yes—
27C23×C4×C13D411yescharacteristic
28C23×C4×C13D411yescharacteristic
29C23×C4×C13C2311yescharacteristic
30C13×C4⋊Q8—44no—
31C13×C22⋊Q8—22no—
32C13×C22⋊Q8—22no—
33C2×C13×C4⋊C4—22no—
34C2×C13×C4⋊C4—22no—
35C2×C13×C4⋊C4—21no—
36C2×C13×C4⋊C4C2×C411yescharacteristic
37C2×C13×C22⋊C4—21no—
38C2×C13×C22⋊C4—21no—
39C2×C42×C13C2×C412yes—
40C2×C22⋊Dic13—22no—
41#88180—44no—
42C2×C4⋊Dic13—22no—
43C2×C4×Dic13—22no—
44C2×#6099—22no—
45C13×#95—22no—
46C13×#95—22no—
47C13×#95—21no—
48#88076—44no—
49#88071—24no—
50#88071—22no—
51#88071D414yes—
52C22⋊#1055—22no—
53C22⋊#1055—22no—
54#9462—131nomaximal, Sylow
55C24×C13C2×D411yescharacteristic
56C2×Q8×C13—24no—
57C2×Q8×C13—24no—
58C2×Q8×C13—24no—
59C2×Q8×C13—24no—
60C2×Q8×C13—22no—
61C2×Q8×C13—42no—
62C2×Q8×C13—42no—
63C2×Q8×C13—44no—
64C2×Q8×C13—44no—
65C2×Q8×C13—48no—
66C2×Q8×C13—42no—
67C22×C4×C13—21no—
68C22×C4×C13—21no—
69C22×C4×C13—24no—
70C22×C4×C13—22no—
71C22×C4×C13—21no—
72C22×C4×C13—21no—
73C22×C4×C13—21no—
74C22×C4×C13—21no—
75C22×C4×C13—42no—
76C22×C4×C13—42no—
77C22×C4×C13C22⋊C412yes—
78C22×C4×C13C22⋊C412yes—
79C22×C4×C13C22×C411yescharacteristic
80C22×C4×C13C2×D412yes—
81C22×C4×C13C2×D411yescharacteristic
82C22×C4×C13#4112yes—
83C22×Dic13—24no—
84C13×C4⋊C4—22no—
85C13×C4⋊C4—44no—
86C13×C4⋊C4—44no—
87C13×C4⋊C4—42no—
88C13×C22⋊C4—24no—
89C13×C22⋊C4—41no—
90C13×C22⋊C4—41no—
91C42×C13—42no—
92C22⋊Dic13—44no—
93C4⋊Dic13—44no—
94C4×Dic13—44no—
95#6099—44no—
96C2×#1055—42no—
97#2434D1311yescharacteristic
98#2310—131no—
99#1784—131no—
100#1511—132no—
101#1284—132no—
102C23×C13—22no—
103C23×C13—22no—
104C23×C13—41no—
105C23×C13C2×C22⋊C411yescharacteristic
106C23×C13#12011yescharacteristic
107C23×C13#12311yescharacteristic
108Q8×C13—48no—
109Q8×C13—48no—
110Q8×C13—82no—
111C2×C4×C13—22no—
112C2×C4×C13—22no—
113C2×C4×C13—24no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—22no—
117C2×C4×C13—22no—
118C2×C4×C13—24no—
119C2×C4×C13—22no—
120C2×C4×C13—22no—
121C2×C4×C13—22no—
122C2×C4×C13—42no—
123C2×C4×C13—42no—
124C2×C4×C13—42no—
125C2×C4×C13—42no—
126C2×C4×C13—42no—
127C2×C4×C13—42no—
128C2×C4×C13—42no—
129C2×C4×C13—41no—
130C2×C4×C13—41no—
131C2×C4×C13—42no—
132C2×C4×C13#9914yes—
133C2×C4×C13C4×D412yes—
134C2×C4×C13#12112yes—
135C2×Dic13—44no—
136C2×Dic13—44no—
137C2×Dic13—44no—
138#1055—82no—
139C23×Q8C2×D1311yescharacteristic
140C2×C4⋊Q8Dic1312yescomplemented
141C2×C22⋊Q8C2×D1311yescharacteristic
142C2×#117—261no—
143C2×#101—262no—
144#407—132no—
145#395—22no—
146#391—264no—
147#391Dic1312yescomplemented
148#385—21no—
149#385—21no—
150#381—132no—
151#381—261no—
152C4×C22⋊C4C2×D1311yescharacteristic
153#328—264no—
154C22×C13—21no—
155C22×C13—21no—
156C22×C13—21no—
157C22×C13—21no—
158C22×C13—21no—
159C22×C13—21no—
160C22×C13—21no—
161C22×C13—44no—
162C22×C13#38611yescharacteristic
163C22×C13C2×#9912yes—
164C22×C13#45612yes—
165C4×C13—22no—
166C4×C13—24no—
167C4×C13—22no—
168C4×C13—44no—
169C4×C13—42no—
170C4×C13—81no—
171C4×C13—81no—
172Dic13—84no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—44no—
176C22×Q8C2×Dic1311yescharacteristic
177C22×Q8C2×Dic1312yes—
178C22×Q8C22⋊D1312yes—
179C23×C4C22⋊D1311yescharacteristic
180C23×C4C22⋊D1311yescharacteristic
181C23×C4C22×D1311yescharacteristic
182C2×M16—262no—
183C4⋊Q8—44no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186#117—524no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—262no—
191C2×C4⋊C4C2×Dic1311yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—262no—
195C2×C42—262no—
196C2×C42C2×Dic1312yes—
197#101—524no—
198C22⋊C8—262no—
199C22⋊C8—262no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—134no—
204#95—264no—
205#95—262no—
206C2×C13—22no—
207C2×C13—22no—
208C2×C13—41no—
209C2×C13#188611yescharacteristic
210C2×C13#189811yescharacteristic
211C2×C13#191211yescharacteristic
212C24C2×C22⋊D1311yescharacteristic
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—22no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—42no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—264no—
235C22×C4C22⋊Dic1312yes—
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—44no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—524no—
248C22⋊C4—24no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—524no—
252C42—42no—
253C42—524no—
254C13#946211yescharacteristic, complemented, Sylow
255C23—22no—
256C23—22no—
257C23—41no—
258C23#8818911yescharacteristic
259C23#8819111yescharacteristic
260C23C2×C22⋊Dic1311yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—82no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—524no—
286C2×C4—524no—
287C2×C4—524no—
288C2×C4#8807414yes—
289C2×C4D4×Dic1312yes—
290C2×C4#8819312yes—
291C8—1042no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#1171875612yes—
301C22C2×#8807412yes—
302C22#1171888111yescharacteristic
303C4—22no—
304C4—24no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—1044no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#1311523011yescharacteristic
315C2#1311529011yescharacteristic
316C2#1311537411yescharacteristic
subgroupG/H
centerC22#11718881
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2434C2
socleC22×C13#386
prime radicalC24×C13C2×D4
cosocleC23×C4×C13C23
hypercenter#2434D13
hypocenterC13#9462
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
7. C2×C13×C4⋊Q8C4semidirect
11. C13×#391C4semidirect
140. C2×C4⋊Q8Dic13semidirect
147. #391Dic13semidirect
254. C13#9462semidirect

Maximal subgroups

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

subgroupindex
#131174222
#131147602
#131133692
#131126892
C13×#24342
#946213