Finite Group Databasebeta

← #15753982

Subgroups — #15753982

Order 3328.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123812C211yesmaximal, characteristic
2#13114589C211yesmaximal, characteristic, complemented
3#13113322C211yesmaximal, characteristic
4M16×#1055C211yesmaximal, characteristic, complemented
5#13112735C211yesmaximal, characteristic, complemented
6#13112730C211yesmaximal, characteristic, complemented
7C13×#1524C211yesmaximal, characteristic
8C13×#529C2211yescharacteristic
9C4×C13×M16C2211yescharacteristic
10C2×C13×#97C2211yescharacteristic
11#11718685C411yescharacteristic, complemented
12Q8×#1055C411yescharacteristic, complemented
13#11718673C411yescharacteristic, complemented
14#11718672—21no—
15D4×#1055—21no—
16D4×#1055C411yescharacteristic, complemented
17#11718661—21no—
18#11718658—21no—
19#11718593C2211yescharacteristic
20#11718587C2211yescharacteristic
21#11718586C2211yescharacteristic
22C2×C4×#1055C2211yescharacteristic
23C13×#330C412yescomplemented
24C13×#329C412yescomplemented
25#11718142—21no—
26#11718142—21no—
27#11718130C412yes—
28#11718129C412yes—
29#11718047—21no—
30C8×#1055—21no—
31C2×C13×M16—21no—
32C22×C8×C13—21no—
33C13×#126—21no—
34C13×#125C2×C411yescharacteristic
35C13×#124C2×C411yescharacteristic
36C13×#123—21no—
37C13×C22⋊Q8C2×C411yescharacteristic
38C13×#121C2×C411yescharacteristic
39C4×Q8×C13C2×C411yescharacteristic
40C4×D4×C13—21no—
41C4×D4×C13C2×C411yescharacteristic
42C13×#117—21no—
43C2×C42×C13C2311yescharacteristic
44C2×C4×Dic13—21no—
45C2×#6099—21no—
46C22×#1055—21no—
47C22×#1055D411yescharacteristic
48C22×#1055D411yescharacteristic
49C13×#97—21no—
50C13×#97C2×C412yes—
51C13×#97C2×C412yes—
52C4×C8×C13—21no—
53#88071—21no—
54#88071C2×C412yes—
55C22⋊#1055—21no—
56C22⋊#1055—21no—
57C22⋊#1055—21no—
58C22⋊#1055C2×C412yes—
59C4⋊#1055—21no—
60C4⋊#1055—21no—
61C4⋊#1055—21no—
62C4⋊#1055C2×C411yescharacteristic
63C4⋊#1055C2×C411yescharacteristic
64#88043—21no—
65#88043—21no—
66C4×#1055—21no—
67C4×#1055—21no—
68C4×#1055C2×C411yescharacteristic
69C4×#1055C2×C411yescharacteristic
70C4×#1055D411yescharacteristic
71C4×#1055D411yescharacteristic
72#8250—131nomaximal, Sylow
73C2×Q8×C13C4211yescharacteristic
74C2×D4×C13—21no—
75C2×D4×C13C4211yescharacteristic
76C22×C4×C13—21no—
77C22×C4×C13C22×C411yescharacteristic
78C22×C4×C13C2×D411yescharacteristic
79C22×C4×C13#4111yescharacteristic
80C22×Dic13—22no—
81C13×M16—24no—
82C2×C8×C13—22no—
83C2×C8×C13—22no—
84C2×C8×C13—41no—
85C2×C8×C13—41no—
86C13×C4⋊C4—21no—
87C13×C4⋊C4—21no—
88C13×C4⋊C4—21no—
89C13×C4⋊C4—21no—
90C13×C4⋊C4C4211yescharacteristic
91C13×C4⋊C4C4211yescharacteristic
92C13×C22⋊C4—21no—
93C13×C22⋊C4—21no—
94C13×C22⋊C4—21no—
95C13×C22⋊C4—21no—
96C13×C22⋊C4—21no—
97C42×C13—21no—
98C42×C13C22×C411yescharacteristic
99C42×C13C22×C411yescharacteristic
100C42×C13C2×D411yescharacteristic
101C42×C13#4111yescharacteristic
102C4×Dic13—41no—
103C4×Dic13—41no—
104#6099—42no—
105C2×#1055—21no—
106C2×#1055—21no—
107C2×#1055—22no—
108C2×#1055—41no—
109C2×#1055—41no—
110C2×#1055—42no—
111C2×#1055C22⋊C411yescharacteristic
112C2×#1055C22⋊C411yescharacteristic
113C2×#1055C22⋊C411yescharacteristic
114C2×#1055C22⋊C411yescharacteristic
115#2966—131no—
116#1746—131no—
117#1524D1311yescharacteristic
118#1500—131no—
119C8×M16—131no—
120#1302—131no—
121#1300—131no—
122C23×C13—21no—
123C23×C13C4×D411yescharacteristic
124Q8×C13—41no—
125D4×C13—41no—
126D4×C13—42no—
127C2×C4×C13—22no—
128C2×C4×C13—21no—
129C2×C4×C13—21no—
130C2×C4×C13—21no—
131C2×C4×C13—21no—
132C2×C4×C13—21no—
133C2×C4×C13—41no—
134C2×C4×C13—41no—
135C2×C4×C13—41no—
136C2×C4×C13C2×C4211yescharacteristic
137C2×C4×C13C2×C22⋊C411yescharacteristic
138C2×C4×C13#11711yescharacteristic
139C2×C4×C13C4×D411yescharacteristic
140C2×C4×C13C4×D411yescharacteristic
141C2×C4×C13C4×D411yescharacteristic
142C2×Dic13—42no—
143C2×Dic13—42no—
144C2×Dic13—42no—
145C8×C13—24no—
146C8×C13—42no—
147#1055—21no—
148#1055—21no—
149#1055—21no—
150#1055—21no—
151#1055—42no—
152#1055—41no—
153#1055—41no—
154#529C2×D1311yescharacteristic
155#446—131no—
156C8×Q8—131no—
157#436—131no—
158#435—261no—
159C8×D4—131no—
160C8×D4—261no—
161#433—261no—
162#432—131no—
163#431—131no—
164#405—261no—
165C4×M16C2×D1311yescharacteristic
166C2×#97C2×D1311yescharacteristic
167C2×C4×C8—131no—
168#376—131no—
169#336—261no—
170#336—261no—
171#330Dic1312yescomplemented
172#329Dic1312yescomplemented
173#328—132no—
174#327—132no—
175#322—261no—
176C82—261no—
177C22×C13—21no—
178C22×C13—21no—
179C22×C13—41no—
180C22×C13—41no—
181C22×C13—41no—
182C22×C13C4×C22⋊C411yescharacteristic
183C22×C13#44312yes—
184C4×C13—21no—
185C4×C13—21no—
186C4×C13—21no—
187C4×C13—21no—
188C4×C13—21no—
189C4×C13—41no—
190C4×C13—41no—
191C4×C13—41no—
192C4×C13#42111yescharacteristic
193C4×C13#42111yescharacteristic
194Dic13—82no—
195C2×M16—21no—
196C2×M16—261no—
197C22×C8—21no—
198C22×C8—131no—
199C22×C8—131no—
200C22×C8—261no—
201#126—21no—
202#125C2×Dic1311yescharacteristic
203#124C2×Dic1311yescharacteristic
204#123—21no—
205C22⋊Q8C4×D1311yescharacteristic
206#121C4×D1311yescharacteristic
207C4×Q8C4×D1311yescharacteristic
208C4×D4—21no—
209C4×D4C4×D1311yescharacteristic
210#117—21no—
211C2×C42—261no—
212C2×C42C22×D1311yescharacteristic
213C4⋊C8—131no—
214C4⋊C8—131no—
215C4⋊C8—261no—
216C4⋊C8—261no—
217C4⋊C8—261no—
218C22⋊C8—132no—
219C22⋊C8—261no—
220C22⋊C8—261no—
221C22⋊C8—261no—
222#97—21no—
223#97—261no—
224#97—261no—
225#97C2×Dic1312yes—
226#97C2×Dic1312yes—
227C4×C8—21no—
228C4×C8—131no—
229C4×C8—131no—
230C4×C8—131no—
231C4×C8—131no—
232C4×C8—261no—
233C4×C8—261no—
234#95—132no—
235#95—261no—
236C2×C13—22no—
237C2×C13—81no—
238C2×C13#175711yescharacteristic
239C2×C13#175911yescharacteristic
240C2×C13#176411yescharacteristic
241C2×Q8C4×Dic1311yescharacteristic
242C2×D4—21no—
243C2×D4C4×Dic1311yescharacteristic
244C22×C4—21no—
245C22×C4—262no—
246C22×C4C2×C4×D1311yescharacteristic
247C22×C4D4×D1311yescharacteristic
248C22×C4#612911yescharacteristic
249M16—24no—
250M16—522no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—131no—
256C2×C8—131no—
257C2×C8—131no—
258C2×C8—131no—
259C2×C8—261no—
260C2×C8—261no—
261C2×C8—262no—
262C2×C8—521no—
263C2×C8—521no—
264C2×C8—522no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4C4×Dic1311yescharacteristic
270C4⋊C4C4×Dic1311yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C42—21no—
277C42—521no—
278C42—521no—
279C42C2×C4×D1311yescharacteristic
280C42D4×D1311yescharacteristic
281C42#612911yescharacteristic
282C42C22×Dic1311yescharacteristic
283C13#825011yescharacteristic, complemented, Sylow
284C23—21no—
285C23#8813511yescharacteristic
286Q8—41no—
287D4—41no—
288D4—42no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—522no—
299C2×C4—522no—
300C2×C4—522no—
301C2×C4#8813111yescharacteristic
302C2×C4C22⋊C4×D1311yescharacteristic
303C2×C4#8813511yescharacteristic
304C2×C4C2×C4×Dic1311yescharacteristic
305C2×C4D4×Dic1311yescharacteristic
306C2×C4D4×Dic1311yescharacteristic
307C8—24no—
308C8—42no—
309C8—261no—
310C8—261no—
311C8—261no—
312C8—261no—
313C8—522no—
314C8—521no—
315C8—521no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22#1171848512yes—
322C22C22⋊C4×Dic1311yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—1042no—
332C4#1171848211yescharacteristic
333C4#1171848211yescharacteristic
334C2—22no—
335C2—81no—
336C2#1311463911yescharacteristic
337C2#1311464811yescharacteristic
338C2#1311467411yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D13
commutatorC2×C4×C13C2×C42
FrattiniC2×C42C22×D13
FittingC13×#1524C2
socleC22×C13C4×C22⋊C4
prime radicalC13×#121C2×C4
cosocleC2×C42×C13C23
hypercenter#1524D13
hypocenterC13#8250
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1524

Complements

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

NH ≅ G/Ntype
2. #13114589C2semidirect
4. M16×#1055C2semidirect
5. #13112735C2semidirect
6. #13112730C2semidirect
11. #11718685C4semidirect
12. Q8×#1055C4semidirect
13. #11718673C4semidirect
16. D4×#1055C4semidirect
23. C13×#330C4semidirect
24. C13×#329C4semidirect
171. #330Dic13semidirect
172. #329Dic13semidirect
283. C13#8250semidirect

Maximal subgroups

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

subgroupindex
#131238122
#131145892
#131133222
M16×#10552
#131127352
#131127302
C13×#15242
#825013