Finite Group Databasebeta

← C22×#16810384 #27381475

Subgroups — C22×#16810384 #27381475

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
1C22×#13133176C211yesmaximal, characteristic
2C23×C4×Dic53C211yesmaximal, characteristic
3C2×#16810384C2112yesmaximal, direct product factor
4C23×M16×C53C211yesmaximal, characteristic
5C24×Dic53—21no—
6C2×#13133176—23no—
7C2×#13133176—23no—
8C2×#13133176C416yescomplemented
9C2×#13133176C2216yes—
10C22×C4⋊Dic53C412yescomplemented
11C22×C22⋊Dic53—21no—
12C22×C4×Dic53—23no—
13C22×C4×Dic53—24no—
14C22×C4×Dic53—21no—
15C22×C4×Dic53C418yescomplemented
16C22×C4×Dic53C2216yes—
17#16810384C22116yesdirect product factor
18C24×C4×C53C2211yescharacteristic
19C22×M16×C53—24no—
20C22×M16×C53—23no—
21C22×M16×C53C418yescomplemented
22C22×M16×C53C2216yes—
23C23×C8×C53—21no—
24C23×Dic53—216no—
25C23×Dic53—28no—
26C23×Dic53—26no—
27C23×Dic53—22no—
28#13133176—212no—
29#13133176—26no—
30#13133176—26no—
31#13133176C2×C4112yescomplemented
32#13133176C2314yes—
33C2×C4⋊Dic53—26no—
34C2×C4⋊Dic53C2×C4112yescomplemented
35C2×C22⋊Dic53—26no—
36C2×C22⋊Dic53—26no—
37C2×C4×Dic53—224no—
38C2×C4×Dic53—26no—
39C2×C4×Dic53—212no—
40C2×C4×Dic53—26no—
41C2×C4×Dic53—26no—
42C2×C4×Dic53C2×C4124yescomplemented
43C2×C4×Dic53C2314yes—
44C25×C53D411yescharacteristic
45C23×C4×C53C2×C414yes—
46C23×C4×C53C2×C414yes—
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53C2×C413yes—
49C23×C4×C53D413yes—
50C23×C4×C53D414yes—
51C23×C4×C53D413yes—
52C23×C4×C53D411yescharacteristic
53C23×C4×C53Q814yes—
54C23×C4×C53C2313yes—
55C2×M16×C53—224no—
56C2×M16×C53—212no—
57C2×M16×C53—26no—
58C2×M16×C53C2×C4124yescomplemented
59C2×M16×C53C2314yes—
60C22×C8×C53—28no—
61C22×C8×C53—26no—
62C24×C53—28no—
63C24×C53C22⋊C413yes—
64C24×C53C22⋊C414yes—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C4⋊C414yes—
67C24×C53C2×D413yes—
68C22×C4×C53C42112yes—
69C22×C4×C53C4214yes—
70C22×C4×C53C22⋊C4112yes—
71C22×C4×C53C22⋊C413yes—
72C22×C4×C53C22⋊C413yes—
73C22×C4×C53C22⋊C416yes—
74C22×C4×C53C22⋊C414yes—
75C22×C4×C53C22⋊C4112yes—
76C22×C4×C53C4⋊C4112yes—
77C22×C4×C53C4⋊C4112yes—
78C22×C4×C53C4⋊C4112yes—
79C22×C4×C53C4⋊C414yes—
80C22×C4×C53C4⋊C414yes—
81C22×C4×C53C22×C416yes—
82C22×C4×C53C22×C416yes—
83C22×C4×C53C22×C413yes—
84C22×C4×C53C22×C413yes—
85C22×C4×C53C2×D413yes—
86C22×C4×C53C2×D413yes—
87C22×C4×C53C2×D416yes—
88C22×C4×C53C2×D413yes—
89C22×C4×C53C2×Q816yes—
90C22×C4×C53C2411yescharacteristic
91C22×Dic53—264no—
92C22×Dic53—248no—
93C22×Dic53—224no—
94C22×Dic53—24no—
95C22×Dic53—212no—
96C22×Dic53—48no—
97M16×C53—224no—
98M16×C53—24no—
99M16×C53C22×C418yescomplemented
100C2×C8×C53—224no—
101C2×C8×C53—24no—
102C22⋊Dic53—212no—
103C22⋊Dic53—24no—
104C4⋊Dic53—212no—
105C4⋊Dic53C22×C418yescomplemented
106C4×Dic53—224no—
107C4×Dic53—212no—
108C4×Dic53—24no—
109C4×Dic53—24no—
110C4×Dic53C22×C418yescomplemented
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#95124yes—
124C2×C4×C53#95112yes—
125C2×C4×C53#95112yes—
126C2×C4×C53C2×C4216yes—
127C2×C4×C53C2×C4216yes—
128C2×C4×C53C2×C22⋊C416yes—
129C2×C4×C53C2×C22⋊C413yes—
130C2×C4×C53C2×C22⋊C413yes—
131C2×C4×C53C2×C22⋊C416yes—
132C2×C4×C53C2×C22⋊C416yes—
133C2×C4×C53C2×C4⋊C416yes—
134C2×C4×C53C2×C4⋊C416yes—
135C2×C4×C53C2×C4⋊C416yes—
136C2×C4×C53C2×C4⋊C416yes—
137C2×C4×C53C2×C4⋊C416yes—
138C2×C4×C53C23×C411yescharacteristic
139C2×C4×C53C23×C411yescharacteristic
140C2×C4×C53C23×C411yescharacteristic
141C2×C4×C53C22×D411yescharacteristic
142C2×C4×C53C22×D411yescharacteristic
143C2×C4×C53C22×Q811yescharacteristic
144C2×Dic53—216no—
145C2×Dic53—296no—
146C2×Dic53—28no—
147C2×Dic53—28no—
148C2×Dic53—424no—
149C8×C53—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×C53C2×#9516yes—
164C4×C53C2×#9516yes—
165C4×C53C22×C4211yescharacteristic
166C4×C53C22×C22⋊C411yescharacteristic
167C4×C53C22×C4⋊C411yescharacteristic
168C4×C53C22×C4⋊C411yescharacteristic
169Dic53—216no—
170Dic53—48no—
171C23×M16D5311yescharacteristic
172C22×#117—531no—
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—23no—
184C22×M16—24no—
185C22×M16Dic5318yescomplemented
186C22×M16C2×D5316yes—
187C23×C8—21no—
188C2×#117—536no—
189C2×#117—536no—
190C2×#117—1063no—
191C2×#117—1063no—
192C22×C4⋊C4—532no—
193C22×C22⋊C4—1061no—
194C22×C42—538no—
195C22×C42—536no—
196C22×C42—1063no—
197C22×C42—1064no—
198C22×C42—1061no—
199#339—5316no—
200C53C22×#33911yescharacteristic, complemented, Sylow
201C25D21211yescharacteristic
202C23×C4—10616no—
203C23×C4—1066no—
204C23×C4—1068no—
205C23×C4—1062no—
206C23×C4Dic10614yes—
207C23×C4C4×D5311yescharacteristic
208C23×C4C4×D5313yes—
209C23×C4C4×D5314yes—
210C23×C4D21213yes—
211C23×C4C2×Dic5314yes—
212C23×C4C22⋊D5313yes—
213C23×C4C22⋊D5314yes—
214C23×C4C22⋊D5311yescharacteristic
215C23×C4C22×D5313yes—
216C2×M16—212no—
217C2×M16—26no—
218C2×M16—224no—
219C2×M16C2×Dic53124yescomplemented
220C2×M16C22×D5314yes—
221C22×C8—26no—
222C22×C8—28no—
223#117—5312no—
224#117—534no—
225#117—1066no—
226#117—1066no—
227#117—10612no—
228C2×C4⋊C4—5312no—
229C2×C4⋊C4—1066no—
230C2×C22⋊C4—1066no—
231C2×C22⋊C4—1066no—
232C2×C42—5324no—
233C2×C42—534no—
234C2×C42—1066no—
235C2×C42—1066no—
236C2×C42—10612no—
237C2×C42—1066no—
238C2×C42—10624no—
239C24—28no—
240C24C4⋊Dic5314yes—
241C24#1171992113yes—
242C24#1171992114yes—
243C24#1171992111yescharacteristic
244C24C2×D21213yes—
245C22×C4—10648no—
246C22×C4—10664no—
247C22×C4—10624no—
248C22×C4—1064no—
249C22×C4—10612no—
250C22×C4—2128no—
251C22×C4C4×Dic53112yes—
252C22×C4C4×Dic5314yes—
253C22×C4#11719919112yes—
254C22×C4#11719919112yes—
255C22×C4#1171991914yes—
256C22×C4#1171991914yes—
257C22×C4C4⋊Dic53112yes—
258C22×C4#11719921112yes—
259C22×C4#1171992113yes—
260C22×C4#1171992113yes—
261C22×C4#1171992116yes—
262C22×C4C22⋊Dic5314yes—
263C22×C4C22⋊Dic53112yes—
264C22×C4C2×Dic10616yes—
265C22×C4C2×C4×D5316yes—
266C22×C4C2×C4×D5313yes—
267C22×C4C2×C4×D5313yes—
268C22×C4C2×D21213yes—
269C22×C4C22×Dic5316yes—
270C22×C4C2×C22⋊D5313yes—
271C22×C4C2×C22⋊D5316yes—
272C22×C4C2×C22⋊D5313yes—
273C22×C4C23×D5311yescharacteristic
274M16—224no—
275M16—24no—
276M16C22×Dic5318yescomplemented
277C2×C8—224no—
278C2×C8—24no—
279C4⋊C4—538no—
280C4⋊C4—10612no—
281C22⋊C4—10612no—
282C22⋊C4—1064no—
283C42—538no—
284C42—10624no—
285C42—10612no—
286C42—1064no—
287C42—1064no—
288C23—212no—
289C23—212no—
290C23—216no—
291C23—216no—
292C23#13133124112yes—
293C23#1313312414yes—
294C23#1313314318yes—
295C23C2×#1171992113yes—
296C23C2×#1171992116yes—
297C23C2×#1171992113yes—
298C23C2×C4⋊Dic5316yes—
299C23C22×D21211yescharacteristic
300C2×C4—10616no—
301C2×C4—10696no—
302C2×C4—1068no—
303C2×C4—1068no—
304C2×C4—21224no—
305C2×C4#13133124124yes—
306C2×C4#13133124112yes—
307C2×C4#13133124112yes—
308C2×C4C2×C4×Dic5316yes—
309C2×C4C2×C4×Dic5316yes—
310C2×C4C2×#1171992116yes—
311C2×C4C2×#1171992113yes—
312C2×C4C2×#1171992113yes—
313C2×C4C2×C22⋊Dic5316yes—
314C2×C4C2×C22⋊Dic5316yes—
315C2×C4C2×#1171991916yes—
316C2×C4C2×#1171991916yes—
317C2×C4C2×#1171991916yes—
318C2×C4C2×#1171991916yes—
319C2×C4C2×C4⋊Dic5316yes—
320C2×C4C22×C4×D5311yescharacteristic
321C2×C4C22×C4×D5311yescharacteristic
322C2×C4C23×Dic5311yescharacteristic
323C2×C4C22×C22⋊D5311yescharacteristic
324C2×C4C22×C22⋊D5311yescharacteristic
325C2×C4C22×Dic10611yescharacteristic
326C8—28no—
327C22—22no—
328C22—26no—
329C22—212no—
330C22—224no—
331C22—212no—
332C22#16810384116yesdirect product factor
333C22C2×#1313312416yes—
334C22C2×#1313312416yes—
335C22C2×#13133143112yes—
336C22C22×#1171992111yescharacteristic
337C22C22×#1171992111yescharacteristic
338C22C22×C4⋊Dic5311yescharacteristic
339C4—10616no—
340C4—2128no—
341C4C2×#1313312416yes—
342C4C2×#1313312416yes—
343C4C22×C4×Dic5311yescharacteristic
344C4C22×C22⋊Dic5311yescharacteristic
345C4C22×#1171991911yescharacteristic
346C4C22×#1171991911yescharacteristic
347C2—21no—
348C2—21no—
349C2—26no—
350C2C2×#16810384112yesdirect product factor
351C2C22×#1313312411yescharacteristic
352C2C22×#1313314312yes—
subgroupG/H
centerC23×C4C22⋊D53
commutatorC4×C53C22×C42
FrattiniC22×C4C23×D53
FittingC23×M16×C53C2
socleC24×C53C22⋊C4
prime radicalC25×C53D4
cosocleC22×C4×C53C24
hypercenterC23×M16D53
hypocenterC53C22×#339
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16810384C2direct (N × H)
8. C2×#13133176C4semidirect
10. C22×C4⋊Dic53C4semidirect
15. C22×C4×Dic53C4semidirect
17. #16810384C22direct (N × H)
21. C22×M16×C53C4semidirect
31. #13133176C2×C4semidirect
34. C2×C4⋊Dic53C2×C4semidirect
42. C2×C4×Dic53C2×C4semidirect
58. C2×M16×C53C2×C4semidirect
99. M16×C53C22×C4semidirect
105. C4⋊Dic53C22×C4semidirect
110. C4×Dic53C22×C4semidirect
156. C22×C53#339semidirect
178. C2×C53C2×#339semidirect
185. C22×M16Dic53semidirect
200. C53C22×#339semidirect
219. C2×M16C2×Dic53semidirect
276. M16C22×Dic53semidirect
332. C22#16810384direct (N × H)
350. C2C2×#16810384direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331762
C23×C4×Dic532
C2×#168103842
C23×M16×C532
C22×#33953