Finite Group Databasebeta

← #16788177

Subgroups — #16788177

Order 3328.

390 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 66 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#13130733C211yesmaximal, characteristic
2#13130069C211yesmaximal, characteristic
3#13129831C212yesmaximal
4#13129816C212yesmaximal
5#13120407C211yesmaximal, characteristic
6C2×C4×Q8×C13C411yescharacteristic, complemented
7C2×#88126C2211yescharacteristic
8#11719135—21no—
9#11719119—21no—
10#11718892C2211yescharacteristic
11#11718890C411yescharacteristic, complemented
12#11718654C2211yescharacteristic
13#11718650—21no—
14#11718641C411yescharacteristic, complemented
15#11718639—21no—
16#11718601—21no—
17#11718598C411yescharacteristic, complemented
18#11718594—21no—
19#11718289—21no—
20#11718289C2212yes—
21#11718287C2212yes—
22#11718286—21no—
23C22×Q8×C13C2×C411yescharacteristic
24C22×C4×D13D411yescharacteristic
25C2×C22⋊#261—22no—
26C4×Q8×C13—41no—
27C4×Q8×C13—41no—
28C2×C13×C4⋊C4—21no—
29C2×C13×C4⋊C4C2×C411yescharacteristic
30C2×C42×C13—21no—
31C2×C42×C13C2×C411yescharacteristic
32C2×#6103—21no—
33C2×#6103—21no—
34C2×#6103D411yescharacteristic
35C2×#6103C2311yescharacteristic
36C2×#6101—21no—
37C2×#6101—21no—
38C2×#6101C2×C411yescharacteristic
39C2×#6101D411yescharacteristic
40C2×C4×Dic13D411yescharacteristic
41#88126—41no—
42#88126—41no—
43C22⋊#1057—22no—
44(C2×#261)⋊C4—41no—
45(C2×#261)⋊C4—41no—
46C4⋊#1057—41no—
47C4×#1057—41no—
48#88071—21no—
49#88071—21no—
50#88071—21no—
51#88071—21no—
52#88071—21no—
53#88071—21no—
54#88071C2×C411yescharacteristic
55#88071C2×C411yescharacteristic
56#11842—131nomaximal, Sylow
57C23×D13C2×D411yescharacteristic
58C22×#261—42no—
59C2×Q8×C13—42no—
60C2×Q8×C13—41no—
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13C22⋊C411yescharacteristic
64C22×C4×C13C22⋊C411yescharacteristic
65C22×C4×C13C22×C411yescharacteristic
66C22×Dic13—21no—
67C22×Dic13—21no—
68C22×Dic13C22⋊C411yescharacteristic
69C22×Dic13C22⋊C411yescharacteristic
70C22×Dic13C2×D411yescharacteristic
71C2×C4×D13—22no—
72C2×C4×D13—21no—
73C2×C4×D13—21no—
74C2×C4×D13—41no—
75C22⋊#261—42no—
76C22⋊#261—42no—
77C2×#1057—42no—
78C13×C4⋊C4—41no—
79C13×C4⋊C4—41no—
80C13×C4⋊C4—41no—
81C42×C13—21no—
82C42×C13—21no—
83C42×C13—42no—
84#6103—41no—
85#6103—41no—
86#6103—42no—
87#6103—42no—
88#6101—41no—
89#6101—41no—
90#6101—42no—
91#6101—42no—
92C4×Dic13—21no—
93C4×Dic13—21no—
94#2599—131no—
95#1820—131no—
96#1458—131no—
97#1351—132no—
98#1344—132no—
99C23×C13C2×C22⋊C411yescharacteristic
100C22×D13—21no—
101C22×D13—21no—
102C22×D13—41no—
103C22×D13—41no—
104C22×D13#13512yes—
105C2×#261—82no—
106C2×#261—82no—
107C2×#261—82no—
108Q8×C13—42no—
109Q8×C13—42no—
110C2×C4×C13—21no—
111C2×C4×C13—21no—
112C2×C4×C13—22no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—41no—
116C2×C4×C13—41no—
117C2×C4×C13—41no—
118C2×C4×C13—41no—
119C2×C4×C13—41no—
120C2×C4×C13—41no—
121C2×C4×C13—41no—
122C2×C4×C13#9911yescharacteristic
123C2×C4×C13#9911yescharacteristic
124C2×Dic13—21no—
125C2×Dic13—21no—
126C2×Dic13—21no—
127C2×Dic13—21no—
128C2×Dic13—21no—
129C2×Dic13—41no—
130C2×Dic13—41no—
131C2×Dic13—41no—
132C2×Dic13—41no—
133C2×Dic13—41no—
134C2×Dic13—41no—
135C2×Dic13—41no—
136C2×Dic13—41no—
137C4×D13—42no—
138C4×D13—42no—
139#1057—82no—
140C2×C4×Q8#26111yescharacteristic, complemented
141C2×#117—131no—
142#431—261no—
143#398—131no—
144#397—131no—
145#395—131no—
146#393—131no—
147#388—261no—
148#387—261no—
149#387—261no—
150#385—261no—
151#383—131no—
152#381—261no—
153#342—132no—
154#342—261no—
155#324—261no—
156#323—132no—
157C22×C13—21no—
158C22×C13—21no—
159C22×C13C2×#9911yescharacteristic
160C22×C13#41611yescharacteristic
161C22×C13#42111yescharacteristic
162C2×D13—22no—
163C2×D13—41no—
164C2×D13—41no—
165C2×D13—42no—
166C2×D13—81no—
167#261—162no—
168C4×C13—21no—
169C4×C13—21no—
170C4×C13—42no—
171C4×C13—41no—
172C4×C13—41no—
173C4×C13—81no—
174Dic13—41no—
175Dic13—41no—
176Dic13—41no—
177Dic13—81no—
178Dic13—81no—
179C22×Q8C2×#26111yescharacteristic
180C23×C4—131no—
181C4×Q8—41no—
182C4×Q8—41no—
183#117—521no—
184#117—521no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—131no—
187C2×C4⋊C4—131no—
188C2×C4⋊C4—261no—
189C2×C4⋊C4—261no—
190C2×C4⋊C4C2×#26111yescharacteristic
191C2×C22⋊C4—131no—
192C2×C22⋊C4—131no—
193C2×C22⋊C4—261no—
194C2×C22⋊C4—261no—
195C2×C22⋊C4—262no—
196C2×C42—21no—
197C2×C42—131no—
198C2×C42C2×#26111yescharacteristic
199C4⋊C8—521no—
200C22⋊C8—262no—
201C4×C8—521no—
202#95—131no—
203#95—131no—
204#95—261no—
205#95—261no—
206#95—261no—
207#95—261no—
208#95—261no—
209#95—261no—
210#95—521no—
211#95—521no—
212C2×C13—21no—
213C2×C13—21no—
214C2×C13#151611yescharacteristic
215C2×C13#212011yescharacteristic
216C2×C13#212911yescharacteristic
217D13—42no—
218C24—131no—
219C2×Q8—42no—
220C2×Q8—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—131no—
224C22×C4—131no—
225C22×C4—131no—
226C22×C4—262no—
227C22×C4—261no—
228C22×C4—261no—
229C22×C4—261no—
230C22×C4—261no—
231C22×C4—521no—
232C22×C4—522no—
233C22×C4C22⋊#26111yescharacteristic
234C22×C4C22⋊#26111yescharacteristic
235C22×C4C22×#26111yescharacteristic
236C2×C8—522no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—521no—
241C4⋊C4—521no—
242C4⋊C4—522no—
243C4⋊C4—522no—
244C22⋊C4—521no—
245C22⋊C4—521no—
246C22⋊C4—522no—
247C22⋊C4—522no—
248C22⋊C4—522no—
249C22⋊C4—522no—
250C42—21no—
251C42—21no—
252C42—42no—
253C42—261no—
254C42—261no—
255C13#1184211yescharacteristic, complemented, Sylow
256C23—132no—
257C23—261no—
258C23—261no—
259C23—521no—
260C23—521no—
261C23C2×C22⋊#26111yescharacteristic
262Q8—42no—
263Q8—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—261no—
277C2×C4—261no—
278C2×C4—261no—
279C2×C4—261no—
280C2×C4—261no—
281C2×C4—522no—
282C2×C4—522no—
283C2×C4—521no—
284C2×C4—521no—
285C2×C4—521no—
286C2×C4—521no—
287C2×C4—521no—
288C2×C4—521no—
289C2×C4—521no—
290C2×C4—521no—
291C2×C4—1042no—
292C2×C4—1042no—
293C2×C4—1042no—
294C2×C4#8811911yescharacteristic
295C2×C4#8811911yescharacteristic
296C8—1042no—
297C22—21no—
298C22—21no—
299C22—262no—
300C22—522no—
301C22—521no—
302C22—521no—
303C22—1041no—
304C22#1171916511yescharacteristic
305C22#1171916611yescharacteristic
306C22C2×#8811911yescharacteristic
307C4—21no—
308C4—21no—
309C4—42no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—521no—
314C4—521no—
315C4—521no—
316C4—1041no—
317C4—1041no—
318C4—2082no—
319C2—21no—
320C2—21no—
321C2—522no—
322C2#1313026211yescharacteristic
323C2#1313108711yescharacteristic
324C2#1313111811yescharacteristic
subgroupG/H
centerC22C2×#88119
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC2×C4×Q8×C13C4
socleC22×C13C2×#99
prime radicalC23×D13C2×D4
cosocleC2×#6103C23
hypercenterC2×C4×Q8#261
hypocenterC13#11842
LCSG ▷ C22×C4×C13 ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
6. C2×C4×Q8×C13C4semidirect
11. #11718890C4semidirect
14. #11718641C4semidirect
17. #11718598C4semidirect
140. C2×C4×Q8#261semidirect
255. C13#11842semidirect

Maximal subgroups

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

subgroupindex
#131307332
#131300692
#131298312
#131298162
#131204072
#1184213