Finite Group Databasebeta

← C22×#16810382 #27381473

Subgroups — C22×#16810382 #27381473

Order 13568.

2830 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719917C211yesmaximal, characteristic
2C23×C4×Dic53C211yesmaximal, characteristic
3C2×#16810382C2112yesmaximal, direct product factor
4C22×#117×C53C211yesmaximal, characteristic
5C24×Dic53—21no—
6C22×#11719917—24no—
7C22×#11719917—23no—
8C22×#11719917C418yescomplemented
9C22×#11719917C2216yes—
10C23×#88322—21no—
11C22×C4×Dic53—23no—
12C22×C4×Dic53—24no—
13C22×C4×Dic53C418yescomplemented
14C22×C4×Dic53C2216yes—
15#16810382C22116yesdirect product factor
16C24×C4×C53C2211yescharacteristic
17C2×#117×C53—23no—
18C2×#117×C53—23no—
19C2×#117×C53C416yescomplemented
20C2×#117×C53C2216yes—
21C22×C4⋊C4×C53C412yescomplemented
22C22×C22⋊C4×C53—21no—
23C22×C42×C53—21no—
24C23×Dic53—216no—
25C23×Dic53—28no—
26C23×Dic53—26no—
27C2×#11719917—224no—
28C2×#11719917—212no—
29C2×#11719917—26no—
30C2×#11719917C2×C4124yescomplemented
31C2×#11719917C2314yes—
32C22×#88322—28no—
33C22×#88322—26no—
34C2×C4×Dic53—224no—
35C2×C4×Dic53—212no—
36C2×C4×Dic53—26no—
37C2×C4×Dic53C2×C4124yescomplemented
38C2×C4×Dic53C2314yes—
39C25×C53D411yescharacteristic
40C23×C4×C53—22no—
41C23×C4×C53C2×C414yes—
42C23×C4×C53C2×C414yes—
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53C2×C413yes—
45C23×C4×C53D413yes—
46C23×C4×C53D414yes—
47C23×C4×C53D413yes—
48C23×C4×C53D411yescharacteristic
49C23×C4×C53Q814yes—
50C23×C4×C53C2313yes—
51#117×C53—212no—
52#117×C53—26no—
53#117×C53—26no—
54#117×C53C2×C4112yescomplemented
55#117×C53C2314yes—
56C2×C4⋊C4×C53—26no—
57C2×C4⋊C4×C53C2×C4112yescomplemented
58C2×C22⋊C4×C53—26no—
59C2×C22⋊C4×C53—26no—
60C2×C42×C53—26no—
61C2×C42×C53—26no—
62C24×C53—28no—
63C24×C53C22⋊C414yes—
64C24×C53C22⋊C413yes—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C4⋊C414yes—
67C24×C53C2×D413yes—
68C22×C4×C53—212no—
69C22×C4×C53—48no—
70C22×C4×C53C42112yes—
71C22×C4×C53C4214yes—
72C22×C4×C53C22⋊C4112yes—
73C22×C4×C53C22⋊C413yes—
74C22×C4×C53C22⋊C413yes—
75C22×C4×C53C22⋊C416yes—
76C22×C4×C53C22⋊C414yes—
77C22×C4×C53C22⋊C4112yes—
78C22×C4×C53C4⋊C4112yes—
79C22×C4×C53C4⋊C4112yes—
80C22×C4×C53C4⋊C414yes—
81C22×C4×C53C4⋊C4112yes—
82C22×C4×C53C4⋊C414yes—
83C22×C4×C53C22×C416yes—
84C22×C4×C53C22×C416yes—
85C22×C4×C53C22×C413yes—
86C22×C4×C53C22×C413yes—
87C22×C4×C53C2×D413yes—
88C22×C4×C53C2×D416yes—
89C22×C4×C53C2×D413yes—
90C22×C4×C53C2×D413yes—
91C22×C4×C53C2×Q816yes—
92C22×C4×C53C2411yescharacteristic
93C22×Dic53—264no—
94C22×Dic53—248no—
95C22×Dic53—224no—
96C22×Dic53—24no—
97C4⋊C4×C53—212no—
98C4⋊C4×C53C22×C418yescomplemented
99C22⋊C4×C53—212no—
100C22⋊C4×C53—24no—
101C42×C53—212no—
102C42×C53—24no—
103C4×Dic53—224no—
104C4×Dic53—24no—
105C4×Dic53C22×C418yescomplemented
106#11719917—224no—
107#11719917—24no—
108#11719917C22×C418yescomplemented
109C2×#88322—224no—
110C2×#88322—24no—
111C23×C53—216no—
112C23×C53—216no—
113C23×C53—212no—
114C23×C53—212no—
115C23×C53#95112yes—
116C23×C53#9514yes—
117C23×C53#10418yes—
118C23×C53C2×C22⋊C416yes—
119C23×C53C2×C22⋊C413yes—
120C23×C53C2×C22⋊C413yes—
121C23×C53C2×C4⋊C416yes—
122C23×C53C22×D411yescharacteristic
123C2×C4×C53—28no—
124C2×C4×C53—424no—
125C2×C4×C53#95124yes—
126C2×C4×C53#95112yes—
127C2×C4×C53#95112yes—
128C2×C4×C53C2×C4216yes—
129C2×C4×C53C2×C4216yes—
130C2×C4×C53C2×C22⋊C416yes—
131C2×C4×C53C2×C22⋊C416yes—
132C2×C4×C53C2×C22⋊C416yes—
133C2×C4×C53C2×C22⋊C413yes—
134C2×C4×C53C2×C22⋊C413yes—
135C2×C4×C53C2×C4⋊C416yes—
136C2×C4×C53C2×C4⋊C416yes—
137C2×C4×C53C2×C4⋊C416yes—
138C2×C4×C53C2×C4⋊C416yes—
139C2×C4×C53C2×C4⋊C416yes—
140C2×C4×C53C23×C411yescharacteristic
141C2×C4×C53C23×C411yescharacteristic
142C2×C4×C53C23×C411yescharacteristic
143C2×C4×C53C22×D411yescharacteristic
144C2×C4×C53C22×D411yescharacteristic
145C2×C4×C53C22×Q811yescharacteristic
146C2×Dic53—216no—
147C2×Dic53—296no—
148C2×Dic53—28no—
149#88322—28no—
150C22×#339—531nomaximal, Sylow
151C22×C53—22no—
152C22×C53—26no—
153C22×C53—212no—
154C22×C53—224no—
155C22×C53—212no—
156C22×C53#339116yescomplemented
157C22×C53C2×#9516yes—
158C22×C53C2×#9516yes—
159C22×C53C2×#104112yes—
160C22×C53C22×C22⋊C411yescharacteristic
161C22×C53C22×C22⋊C411yescharacteristic
162C22×C53C22×C4⋊C411yescharacteristic
163C4×C53—48no—
164C4×C53C2×#9516yes—
165C4×C53C2×#9516yes—
166C4×C53C22×C4211yescharacteristic
167C4×C53C22×C22⋊C411yescharacteristic
168C4×C53C22×C4⋊C411yescharacteristic
169C4×C53C22×C4⋊C411yescharacteristic
170Dic53—216no—
171C23×M16—531no—
172C22×#117D5311yescharacteristic
173C23×C42—531no—
174C2×#339—5312no—
175C2×C53—21no—
176C2×C53—21no—
177C2×C53—26no—
178C2×C53C2×#339112yescomplemented
179C2×C53C22×#9511yescharacteristic
180C2×C53C22×#10412yes—
181C24×C4—1061no—
182C24×C4C2×D5311yescharacteristic
183C22×M16—538no—
184C22×M16—536no—
185C22×M16—1064no—
186C22×M16—1063no—
187C23×C8—1061no—
188C2×#117—23no—
189C2×#117—23no—
190C2×#117Dic5316yescomplemented
191C2×#117C2×D5316yes—
192C22×C4⋊C4Dic5312yescomplemented
193C22×C22⋊C4—21no—
194C22×C42—21no—
195C22×C42—538no—
196C22×C42—536no—
197C22×C42—1064no—
198C22×C42—1063no—
199#339—5316no—
200C53C22×#33911yescharacteristic, complemented, Sylow
201C25C22⋊D5311yescharacteristic
202C23×C4—22no—
203C23×C4—10616no—
204C23×C4—1066no—
205C23×C4—1068no—
206C23×C4Dic10614yes—
207C23×C4C4×D5314yes—
208C23×C4C4×D5314yes—
209C23×C4D21214yes—
210C23×C4C2×Dic5311yescharacteristic
211C23×C4C2×Dic5313yes—
212C23×C4C22⋊D5313yes—
213C23×C4C22⋊D5313yes—
214C23×C4C22⋊D5311yescharacteristic
215C23×C4C22×D5313yes—
216C2×M16—5324no—
217C2×M16—534no—
218C2×M16—10612no—
219C2×M16—1066no—
220C2×M16—10624no—
221C22×C8—1066no—
222C22×C8—1068no—
223#117—26no—
224#117—26no—
225#117—212no—
226#117C2×Dic53112yescomplemented
227#117C22×D5314yes—
228C2×C4⋊C4—26no—
229C2×C4⋊C4C2×Dic53112yescomplemented
230C2×C22⋊C4—26no—
231C2×C22⋊C4—26no—
232C2×C42—26no—
233C2×C42—26no—
234C2×C42—5324no—
235C2×C42—534no—
236C2×C42—10612no—
237C2×C42—1066no—
238C2×C42—10624no—
239C24—28no—
240C24#1171991914yes—
241C24#1171992114yes—
242C24C22⋊Dic5313yes—
243C24C22⋊Dic5311yescharacteristic
244C24C2×C22⋊D5313yes—
245C22×C4—212no—
246C22×C4—48no—
247C22×C4—10648no—
248C22×C4—10664no—
249C22×C4—10624no—
250C22×C4—1064no—
251C22×C4C4×Dic53112yes—
252C22×C4C4×Dic5314yes—
253C22×C4#11719919112yes—
254C22×C4#11719919112yes—
255C22×C4#1171991914yes—
256C22×C4C4⋊Dic53112yes—
257C22×C4C4⋊Dic5314yes—
258C22×C4#11719921112yes—
259C22×C4#1171992114yes—
260C22×C4#11719921112yes—
261C22×C4C22⋊Dic5313yes—
262C22×C4C22⋊Dic5313yes—
263C22×C4C22⋊Dic5316yes—
264C22×C4C2×Dic10616yes—
265C22×C4C2×C4×D5316yes—
266C22×C4C2×C4×D5316yes—
267C22×C4C2×D21216yes—
268C22×C4C22×Dic5313yes—
269C22×C4C22×Dic5313yes—
270C22×C4C2×C22⋊D5313yes—
271C22×C4C2×C22⋊D5313yes—
272C22×C4C2×C22⋊D5313yes—
273C22×C4C23×D5311yescharacteristic
274M16—538no—
275M16—10624no—
276M16—1064no—
277C2×C8—10624no—
278C2×C8—1064no—
279C4⋊C4—212no—
280C4⋊C4C22×Dic5318yescomplemented
281C22⋊C4—212no—
282C22⋊C4—24no—
283C42—212no—
284C42—24no—
285C42—538no—
286C42—10624no—
287C42—1064no—
288C23—212no—
289C23—212no—
290C23—216no—
291C23—216no—
292C23#13133124112yes—
293C23#1313312414yes—
294C23#1313314418yes—
295C23C2×#1171992116yes—
296C23C2×C22⋊Dic5313yes—
297C23C2×C22⋊Dic5313yes—
298C23C2×#1171991916yes—
299C23C22×C22⋊D5311yescharacteristic
300C2×C4—28no—
301C2×C4—424no—
302C2×C4—10616no—
303C2×C4—10696no—
304C2×C4—1068no—
305C2×C4#13133124124yes—
306C2×C4#13133124112yes—
307C2×C4#13133124112yes—
308C2×C4C2×C4×Dic5316yes—
309C2×C4C2×C4×Dic5316yes—
310C2×C4C2×#1171992116yes—
311C2×C4C2×#1171992116yes—
312C2×C4C2×#1171992116yes—
313C2×C4C2×C22⋊Dic5313yes—
314C2×C4C2×C22⋊Dic5313yes—
315C2×C4C2×#1171991916yes—
316C2×C4C2×#1171991916yes—
317C2×C4C2×#1171991916yes—
318C2×C4C2×C4⋊Dic5316yes—
319C2×C4C2×C4⋊Dic5316yes—
320C2×C4C22×C4×D5311yescharacteristic
321C2×C4C22×C4×D5311yescharacteristic
322C2×C4C23×Dic5311yescharacteristic
323C2×C4C22×C22⋊D5311yescharacteristic
324C2×C4C22×D21211yescharacteristic
325C2×C4C22×Dic10611yescharacteristic
326C8—1068no—
327C22—22no—
328C22—26no—
329C22—212no—
330C22—224no—
331C22—212no—
332C22#16810382116yesdirect product factor
333C22C2×#1313312416yes—
334C22C2×#1313312416yes—
335C22C2×#13133144112yes—
336C22C22×#1171992111yescharacteristic
337C22C22×C22⋊Dic5311yescharacteristic
338C22C22×#1171991911yescharacteristic
339C4—48no—
340C4—10616no—
341C4C2×#1313312416yes—
342C4C2×#1313312416yes—
343C4C22×C4×Dic5311yescharacteristic
344C4C22×#1171992111yescharacteristic
345C4C22×#1171991911yescharacteristic
346C4C22×C4⋊Dic5311yescharacteristic
347C2—21no—
348C2—21no—
349C2—26no—
350C2C2×#16810382112yesdirect product factor
351C2C22×#1313312411yescharacteristic
352C2C22×#1313314412yes—
subgroupG/H
centerC23×C4C22⋊D53
commutatorC4×C53C22×C42
FrattiniC22×C4C23×D53
FittingC22×#117×C53C2
socleC24×C53C22⋊C4
prime radicalC25×C53D4
cosocleC22×C4×C53C24
hypercenterC22×#117D53
hypocenterC53C22×#339
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#16810382C2direct (N × H)
8. C22×#11719917C4semidirect
13. C22×C4×Dic53C4semidirect
15. #16810382C22direct (N × H)
19. C2×#117×C53C4semidirect
21. C22×C4⋊C4×C53C4semidirect
30. C2×#11719917C2×C4semidirect
37. C2×C4×Dic53C2×C4semidirect
54. #117×C53C2×C4semidirect
57. C2×C4⋊C4×C53C2×C4semidirect
98. C4⋊C4×C53C22×C4semidirect
105. C4×Dic53C22×C4semidirect
108. #11719917C22×C4semidirect
156. C22×C53#339semidirect
178. C2×C53C2×#339semidirect
190. C2×#117Dic53semidirect
192. C22×C4⋊C4Dic53semidirect
200. C53C22×#339semidirect
226. #117C2×Dic53semidirect
229. C2×C4⋊C4C2×Dic53semidirect
280. C4⋊C4C22×Dic53semidirect
332. C22#16810382direct (N × H)
350. C2C2×#16810382direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199172
C23×C4×Dic532
C2×#168103822
C22×#117×C532
C22×#33953