Finite Group Databasebeta

← #16778488

Subgroups — #16778488

Order 3328.

730 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 104 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#13130499C212yesmaximal, complemented
2#13129656C212yesmaximal, complemented
3#13129652C212yesmaximal, complemented
4#13117327C211yesmaximal, characteristic
5C22×C13×#41C411yescharacteristic
6C22×#6122—22no—
7C2×#88180C411yescharacteristic
8C22×C4×Dic13C2211yescharacteristic
9#11718891C411yescharacteristic
10#11718891C2211yescharacteristic
11#11718882C411yescharacteristic
12#11718882C2211yescharacteristic
13#11718881—22no—
14C4×C22⋊Dic13—22no—
15#11718290—22no—
16#11718290C2214yes—
17C2×C13×#41—22no—
18C2×C13×#41—22no—
19C2×C13×#41—42no—
20C2×C13×#41—42no—
21C22×Q8×C13C2×C411yescharacteristic
22C22×D4×C13—21no—
23C22×D4×C13C2×C411yescharacteristic
24C23×C4×C13—21no—
25C23×C4×C13C2×C411yescharacteristic
26C23×Dic13D412yes—
27C2×#6122—28no—
28C2×#6122—44no—
29C2×#6122—44no—
30C22×#1057—24no—
31C2×C22⋊Dic13—21no—
32C2×C22⋊Dic13—22no—
33C2×C22⋊Dic13—22no—
34#88180—42no—
35C2×C4⋊Dic13C2×C411yescharacteristic
36C2×C4⋊Dic13C2×C411yescharacteristic
37C2×C4×Dic13—21no—
38C2×C4×Dic13—42no—
39C2×C4×Dic13C2×C411yescharacteristic
40C2×C4×Dic13D412yes—
41C2×C4×Dic13C2311yescharacteristic
42#88071—22no—
43#88071—22no—
44#7631—131nomaximal, Sylow
45C24×C13—21no—
46C24×C13C22⋊C411yescharacteristic
47C13×#41—48no—
48C13×#41—48no—
49C2×Q8×C13—21no—
50C2×Q8×C13—41no—
51C2×Q8×C13—41no—
52C2×Q8×C13M1612yescomplemented
53C2×D4×C13—24no—
54C2×D4×C13—21no—
55C2×D4×C13—41no—
56C2×D4×C13—41no—
57C2×D4×C13—42no—
58C2×D4×C13—42no—
59C2×D4×C13M1612yescomplemented
60C22×C4×C13—22no—
61C22×C4×C13—22no—
62C22×C4×C13—21no—
63C22×C4×C13—21no—
64C22×C4×C13—24no—
65C22×C4×C13—21no—
66C22×C4×C13—42no—
67C22×C4×C13—42no—
68C22×C4×C13C22⋊C411yescharacteristic
69C22×C4×C13C22×C411yescharacteristic
70C22×Dic13—24no—
71C22×Dic13—24no—
72C22×Dic13—22no—
73C22×Dic13—22no—
74C22×Dic13—21no—
75C22×Dic13—21no—
76C22×Dic13C22⋊C412yes—
77C22×Dic13C2×D412yes—
78#6122—416no—
79C2×#1057—28no—
80C2×#1057—44no—
81C2×#1057—44no—
82C22⋊Dic13—42no—
83C22⋊Dic13—44no—
84C22⋊Dic13—42no—
85C22⋊Dic13—42no—
86C4⋊Dic13—41no—
87C4⋊Dic13—41no—
88C4×Dic13—41no—
89C4×Dic13—42no—
90C4×Dic13—41no—
91C4×Dic13—42no—
92#2297—131no—
93#1728—132no—
94#1272—132no—
95#1270—132no—
96C23×C13—21no—
97C23×C13—21no—
98C23×C13—22no—
99C23×C13—22no—
100C23×C13—22no—
101C23×C13—41no—
102C23×C13—41no—
103C23×C13—42no—
104C23×C13—42no—
105C23×C13#9912yes—
106C23×C13C2×C22⋊C411yescharacteristic
107Q8×C13—42no—
108Q8×C13—42no—
109D4×C13—42no—
110D4×C13—42no—
111D4×C13—48no—
112C2×C4×C13—24no—
113C2×C4×C13—24no—
114C2×C4×C13—21no—
115C2×C4×C13—21no—
116C2×C4×C13—21no—
117C2×C4×C13—21no—
118C2×C4×C13—22no—
119C2×C4×C13—48no—
120C2×C4×C13—41no—
121C2×C4×C13—41no—
122C2×C4×C13#10012yes—
123C2×C4×C13C2×M1611yescharacteristic
124C2×C4×C13C2×M1611yescharacteristic
125C2×Dic13—28no—
126C2×Dic13—28no—
127C2×Dic13—22no—
128C2×Dic13—22no—
129C2×Dic13—22no—
130C2×Dic13—22no—
131C2×Dic13—41no—
132C2×Dic13—41no—
133C2×Dic13—42no—
134C2×Dic13—42no—
135C2×Dic13#10414yes—
136C2×Dic13AGL(1,ℤ/8)12yes—
137C2×Dic13#13712yes—
138#1057—48no—
139C22×#41#26111yescharacteristic
140C22×M16—262no—
141C2×#117—131no—
142C22×C42—131no—
143#391—131no—
144#391—131no—
145#390—131no—
146#390—131no—
147#386—262no—
148C4×C22⋊C4—262no—
149#336—134no—
150#336—262no—
151C22×C13—22no—
152C22×C13—24no—
153C22×C13—41no—
154C22×C13—42no—
155C22×C13—41no—
156C22×C13—42no—
157C22×C13—44no—
158C22×C13—81no—
159C22×C13—82no—
160C22×C13—81no—
161C22×C13#40711yescharacteristic
162C22×C13C2×#9911yescharacteristic
163C22×C13C2×#10011yescharacteristic
164C22×C13#41712yes—
165C22×C13C2×#10412yes—
166C4×C13—22no—
167C4×C13—21no—
168C4×C13—21no—
169C4×C13—42no—
170Dic13—22no—
171Dic13—24no—
172Dic13—22no—
173Dic13—81no—
174Dic13—81no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—42no—
178C2×#41—42no—
179C22×Q8C2×#26111yescharacteristic
180C22×D4—21no—
181C22×D4C2×#26111yescharacteristic
182C23×C4—21no—
183C23×C4—132no—
184C23×C4C2×#26111yescharacteristic
185C2×M16—268no—
186C2×M16—524no—
187C2×M16—524no—
188C22×C8—264no—
189#117—522no—
190C2×C4⋊C4—131no—
191C2×C4⋊C4—131no—
192C2×C22⋊C4—262no—
193C2×C22⋊C4—262no—
194C2×C22⋊C4—261no—
195C2×C42—132no—
196C2×C42—131no—
197C2×C42—131no—
198C2×C42—261no—
199C2×C42—522no—
200#95—262no—
201#95—262no—
202C2×C13—42no—
203C2×C13—82no—
204C2×C13#146211yescharacteristic
205C2×C13#148912yes—
206C2×C13#150412yes—
207C2×C13#152511yescharacteristic
208C2×C13#156011yescharacteristic
209C24—21no—
210C24C22⋊#26111yescharacteristic
211#41—48no—
212#41—48no—
213C2×Q8—21no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8#612212yescomplemented
217C2×D4—24no—
218C2×D4—21no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4#612212yescomplemented
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—132no—
233C22×C4—132no—
234C22×C4—264no—
235C22×C4—262no—
236C22×C4—262no—
237C22×C4—264no—
238C22×C4—261no—
239C22×C4—261no—
240C22×C4C22⋊#26111yescharacteristic
241C22×C4C22×#26111yescharacteristic
242M16—5216no—
243C2×C8—268no—
244C2×C8—524no—
245C2×C8—524no—
246C4⋊C4—521no—
247C4⋊C4—521no—
248C22⋊C4—524no—
249C22⋊C4—522no—
250C22⋊C4—522no—
251C22⋊C4—522no—
252C42—522no—
253C42—521no—
254C42—521no—
255C42—522no—
256C13#763111yescharacteristic, complemented, Sylow
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—41no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23#8811912yes—
267C23C2×C22⋊#26111yescharacteristic
268Q8—42no—
269Q8—42no—
270D4—42no—
271D4—42no—
272D4—48no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—48no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—134no—
284C2×C4—132no—
285C2×C4—132no—
286C2×C4—268no—
287C2×C4—268no—
288C2×C4—262no—
289C2×C4—262no—
290C2×C4—262no—
291C2×C4—262no—
292C2×C4—521no—
293C2×C4—521no—
294C2×C4—522no—
295C2×C4—522no—
296C2×C4#8812112yes—
297C2×C4C2×#612211yescharacteristic
298C2×C4C2×#612211yescharacteristic
299C8—528no—
300C22—22no—
301C22—24no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—42no—
306C22—44no—
307C22—81no—
308C22—81no—
309C22—82no—
310C22D13⋊#10412yes—
311C22#1171916412yes—
312C22C2×#8811911yescharacteristic
313C22#1171916911yescharacteristic
314C22C2×#8812111yescharacteristic
315C4—22no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—262no—
320C4—264no—
321C4—262no—
322C4—1041no—
323C4—1041no—
324C2—42no—
325C2—82no—
326C2#1313008111yescharacteristic
327C2#1313017112yes—
328C2#1313022212yes—
329C2#1313029011yescharacteristic
330C2#1313037311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
Fitting, prime radicalC22×C13×#41C4
socleC23×C13C2×C22⋊C4
cosocleC2×C4×Dic13C23
hypercenterC22×#41#261
hypocenterC13#7631
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #13130499C2semidirect
2. #13129656C2semidirect
3. #13129652C2semidirect
52. C2×Q8×C13M16semidirect
59. C2×D4×C13M16semidirect
216. C2×Q8#6122semidirect
223. C2×D4#6122semidirect
256. C13#7631semidirect

Maximal subgroups

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

subgroupindex
#131304992
#131296562
#131296522
#131173272
#763113