Finite Group Databasebeta

← C22×#11718855 #16769404

Subgroups — C22×#11718855 #16769404

Order 3328.

2118 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88158C211yesmaximal, characteristic, complemented
2C2×#11718855C2124yesmaximal, direct product factor
3C22×#88069C211yesmaximal, characteristic, complemented
4C22×#88067C211yesmaximal, characteristic, complemented
5C22×#88068C211yesmaximal, characteristic, complemented
6C22×#88066C211yesmaximal, characteristic, complemented
7C23×#6090C211yesmaximal, characteristic, complemented
8C22×C13×#135C211yesmaximal, characteristic, complemented
9C22×C13×#41—21no—
10C22×#6127—21no—
11C2×C13×#135—26no—
12C2×C13×#135C22112yescomplemented
13C22×C13×Q16C2211yescharacteristic
14C22×C13×SD16—21no—
15C22×C13×D8C2211yescharacteristic
16C23×C8×C13C2211yescharacteristic
17C22×Dic52C2211yescharacteristic
18C2×#88158—26no—
19C2×#88158C22112yescomplemented
20C22×D104C2211yescharacteristic
21C22×#6095—21no—
22#11718855C22164yesdirect product factor
23C2×#88069C22112yescomplemented
24C2×#88068C22112yescomplemented
25C2×#88067C22112yescomplemented
26C2×#88066C22112yescomplemented
27C22×#6090C2211yescharacteristic
28C22×#6090C2211yescharacteristic
29C22×#6090C22112yescomplemented
30C2×C13×#41—212no—
31C2×C13×#41—46no—
32C22×Q8×C13—21no—
33C22×D4×C13—21no—
34C22×D4×C13—41no—
35C23×C4×C13—41no—
36C23×C4×C13D411yescharacteristic
37C22×C22⋊D13—41no—
38C2×#6127—212no—
39C2×#6127—46no—
40C22×D52—21no—
41C22×C4×D13—41no—
42C22×Dic26—21no—
43C13×#135—224no—
44C13×#135C23116yescomplemented
45C2×C13×Q16—23no—
46C2×C13×Q16C2316yes—
47C2×C13×SD16—212no—
48C2×C13×D8—23no—
49C2×C13×D8C2316yes—
50C22×C8×C13D411yescharacteristic
51C22×C8×C13D416yes—
52C22×C8×C13C2311yescharacteristic
53C22×C8×C13C2316yes—
54C2×Dic52—23no—
55C2×Dic52C2316yes—
56#88158—224no—
57#88158C23116yescomplemented
58C2×D104—23no—
59C2×D104C2316yes—
60C2×#6095—212no—
61#88069C23116yescomplemented
62#88068C23116yescomplemented
63#88067C23116yescomplemented
64#88066C23116yescomplemented
65C2×#6090C2316yes—
66C2×#6090C2316yes—
67C2×#6090C23116yescomplemented
68C22×#508—131nomaximal, Sylow
69C24×C13—41no—
70C24×C13D811yescharacteristic
71C23×D13—41no—
72C13×#41—216no—
73C13×#41—424no—
74C2×Q8×C13—26no—
75C2×Q8×C13—43no—
76C2×D4×C13—26no—
77C2×D4×C13—412no—
78C2×D4×C13—43no—
79C22×C4×C13—41no—
80C22×C4×C13—412no—
81C22×C4×C13D816yes—
82C22×C4×C13D811yescharacteristic
83C22×C4×C13C2×D411yescharacteristic
84C22×C4×C13C2×D416yes—
85C2×C22⋊D13—412no—
86C22×Dic13—41no—
87#6127—216no—
88#6127—424no—
89C2×D52—26no—
90C2×D52—43no—
91C2×C4×D13—412no—
92C2×Dic26—26no—
93C2×Dic26—43no—
94C13×Q16—26no—
95C13×Q16C2414yes—
96C13×SD16—216no—
97C13×D8—26no—
98C13×D8C2414yes—
99C2×C8×C13C2×D416yes—
100C2×C8×C13C2×D413yes—
101C2×C8×C13C2×D4112yes—
102C2×C8×C13C2413yes—
103C2×C8×C13C2414yes—
104Dic52—26no—
105Dic52C2414yes—
106D104—26no—
107D104C2414yes—
108#6095—216no—
109#6090C2414yes—
110#6090C2414yes—
111C22×#135—131no—
112C22×#135D1311yescharacteristic, complemented
113C2×#508—1324no—
114C22×Q32—131no—
115C22×SD32—131no—
116C22×SD32—131no—
117C22×D16—131no—
118C23×C16—131no—
119C23×C13—24no—
120C23×C13—46no—
121C23×C13—84no—
122C23×C13C2×D816yes—
123C23×C13C2×D811yescharacteristic
124C22×D13—46no—
125C22×D13—84no—
126Q8×C13—24no—
127Q8×C13—46no—
128D4×C13—24no—
129D4×C13—416no—
130D4×C13—46no—
131C2×C4×C13—46no—
132C2×C4×C13—416no—
133C2×C4×C13C2×D813yes—
134C2×C4×C13C2×D816yes—
135C2×C4×C13C2×D8112yes—
136C2×C4×C13C22×D413yes—
137C2×C4×C13C22×D414yes—
138C22⋊D13—416no—
139C2×Dic13—46no—
140D52—24no—
141D52—46no—
142C4×D13—416no—
143Dic26—24no—
144Dic26—46no—
145C8×C13C22×D414yes—
146C8×C13C22×D413yes—
147C8×C13C2511yescharacteristic
148C22×#41—21no—
149C22×#41—261no—
150C2×#135—26no—
151C2×#135—1312no—
152C2×#135—266no—
153C2×#135C2×D13112yescomplemented
154C22×Q16—131no—
155C22×Q16C2×D1311yescharacteristic
156C22×SD16—21no—
157C22×SD16—261no—
158C22×D8—131no—
159C22×D8C2×D1311yescharacteristic
160C23×C8C2×D1311yescharacteristic
161#508—1364no—
162C2×Q32—1312no—
163C2×SD32—1312no—
164C2×SD32—1312no—
165C2×D16—1312no—
166C22×C16—131no—
167C22×C16—131no—
168C22×C16—1312no—
169C22×C13—212no—
170C22×C13—44no—
171C22×C13—812no—
172C22×C13#50814yescomplemented
173C22×C13C22×D814yes—
174C22×C13C22×D813yes—
175C2×D13—44no—
176C2×D13—812no—
177C4×C13—44no—
178C4×C13C22×D813yes—
179C4×C13C22×D814yes—
180C4×C13C23×D411yescharacteristic
181Dic13—44no—
182C2×#41—212no—
183C2×#41—46no—
184C2×#41—2612no—
185C2×#41—526no—
186C22×Q8—21no—
187C22×Q8—261no—
188C22×D4—21no—
189C22×D4—41no—
190C22×D4—261no—
191C22×D4—521no—
192C23×C4—41no—
193C23×C4—521no—
194C23×C4C22⋊D1311yescharacteristic
195#135—224no—
196#135—1316no—
197#135—2624no—
198#135C22×D13116yescomplemented
199C2×Q16—23no—
200C2×Q16—136no—
201C2×Q16—263no—
202C2×Q16C22×D1316yes—
203C2×SD16—212no—
204C2×SD16—2612no—
205C2×D8—23no—
206C2×D8—136no—
207C2×D8—263no—
208C2×D8C22×D1316yes—
209C22×C8C22⋊D1311yescharacteristic
210C22×C8C22⋊D1316yes—
211C22×C8C22×D1311yescharacteristic
212C22×C8C22×D1316yes—
213Q32—1316no—
214SD32—1316no—
215SD32—1316no—
216D16—1316no—
217C2×C16—136no—
218C2×C16—136no—
219C2×C16—1316no—
220C2×C13—24no—
221C2×C13—84no—
222C2×C13C2×#50816yescomplemented
223C2×C13C23×D811yescharacteristic
224D13—84no—
225C24—41no—
226C24—521no—
227C24#610411yescharacteristic
228#41—216no—
229#41—424no—
230#41—2616no—
231#41—5224no—
232C2×Q8—26no—
233C2×Q8—43no—
234C2×Q8—266no—
235C2×Q8—523no—
236C2×D4—26no—
237C2×D4—412no—
238C2×D4—43no—
239C2×D4—266no—
240C2×D4—5212no—
241C2×D4—523no—
242C22×C4—41no—
243C22×C4—412no—
244C22×C4—521no—
245C22×C4—5212no—
246C22×C4#610416yes—
247C22×C4#610411yescharacteristic
248C22×C4C2×C22⋊D1311yescharacteristic
249C22×C4C2×C22⋊D1316yes—
250Q16—26no—
251Q16—134no—
252Q16—266no—
253Q16C23×D1314yes—
254SD16—216no—
255SD16—2616no—
256D8—26no—
257D8—134no—
258D8—266no—
259D8C23×D1314yes—
260C2×C8C2×C22⋊D1313yes—
261C2×C8C2×C22⋊D1316yes—
262C2×C8C2×C22⋊D13112yes—
263C2×C8C23×D1313yes—
264C2×C8C23×D1314yes—
265C16—134no—
266C16—134no—
267C13C22×#50811yescharacteristic, complemented, Sylow
268C23—24no—
269C23—46no—
270C23—84no—
271C23—526no—
272C23—1044no—
273C23C2×#610416yes—
274C23C2×#610411yescharacteristic
275Q8—24no—
276Q8—46no—
277Q8—264no—
278Q8—526no—
279D4—24no—
280D4—416no—
281D4—46no—
282D4—264no—
283D4—5216no—
284D4—526no—
285C2×C4—46no—
286C2×C4—416no—
287C2×C4—526no—
288C2×C4—5216no—
289C2×C4C2×#610416yes—
290C2×C4C2×#610413yes—
291C2×C4C2×#6104112yes—
292C2×C4C22×C22⋊D1313yes—
293C2×C4C22×C22⋊D1314yes—
294C8C22×C22⋊D1313yes—
295C8C22×C22⋊D1314yes—
296C8C24×D1311yescharacteristic
297C22—212no—
298C22—44no—
299C22—812no—
300C22—524no—
301C22—10412no—
302C22#1171885514yesdirect product factor
303C22C22×#610414yes—
304C22C22×#610413yes—
305C4—44no—
306C4—524no—
307C4C22×#610414yes—
308C4C22×#610413yes—
309C4C23×C22⋊D1311yescharacteristic
310C2—24no—
311C2—84no—
312C2—1044no—
313C2C2×#1171885516yesdirect product factor
314C2C23×#610411yescharacteristic
subgroupG/H
centerC22×C4#6104
commutator, cosocleC8×C13C25
FrattiniC8C24×D13
FittingC22×C13×#135C2
socleC23×C13C2×D8
hypercenterC22×#135D13
hypocenterC13C22×#508
LCSG ▷ C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C23×C8 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
1. C22×#88158C2semidirect
2. C2×#11718855C2direct (N × H)
3. C22×#88069C2semidirect
4. C22×#88067C2semidirect
5. C22×#88068C2semidirect
6. C22×#88066C2semidirect
7. C23×#6090C2semidirect
8. C22×C13×#135C2semidirect
12. C2×C13×#135C22semidirect
19. C2×#88158C22semidirect
22. #11718855C22direct (N × H)
23. C2×#88069C22semidirect
24. C2×#88068C22semidirect
25. C2×#88067C22semidirect
26. C2×#88066C22semidirect
29. C22×#6090C22semidirect
44. C13×#135C23semidirect
57. #88158C23semidirect
61. #88069C23semidirect
62. #88068C23semidirect
63. #88067C23semidirect
64. #88066C23semidirect
67. C2×#6090C23semidirect
112. C22×#135D13semidirect
153. C2×#135C2×D13semidirect
172. C22×C13#508semidirect
198. #135C22×D13semidirect
222. C2×C13C2×#508semidirect
267. C13C22×#508semidirect
302. C22#11718855direct (N × H)
313. C2C2×#11718855direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#881582
C2×#117188552
C22×#880692
C22×#880672
C22×#880682
C22×#880662
C23×#60902
C22×C13×#1352
C22×#50813