Finite Group Databasebeta

← C22×#16810911 #28251553

Subgroups — C22×#16810911 #28251553

Order 13568.

2790 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133216C211yesmaximal, characteristic, complemented
2C22×D4×Dic53C211yesmaximal, characteristic
3C2×#16810911C2124yesmaximal, direct product factor
4C22×#13133149C211yesmaximal, characteristic, complemented
5C22×#13133148C211yesmaximal, characteristic, complemented
6C22×#13133146C211yesmaximal, characteristic, complemented
7C22×#13133138C211yesmaximal, characteristic
8C22×#102×C53C211yesmaximal, characteristic
9C23×Dic106—21no—
10C24×Dic53—21no—
11C23×#88322C2211yescharacteristic
12C2×#13133216—26no—
13C2×#13133216C22112yescomplemented
14C2×D4×Dic53—26no—
15C2×D4×Dic53C22112yes—
16C22×C4⋊Dic53C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C22×#11719919—21no—
19C22×C22⋊Dic53—21no—
20C22×C4×Dic53C2211yescharacteristic
21#16810911C22164yesdirect product factor
22C2×#13133149C22112yescomplemented
23C2×#13133148C22112yescomplemented
24C2×#13133146C22112yescomplemented
25C2×#13133138C22112yes—
26C23×D4×C53C2211yescharacteristic
27C23×C8×C53C2211yescharacteristic
28C22×C4⋊C4×C53C2211yescharacteristic
29C2×#102×C53C22112yes—
30C22×Dic106—212no—
31C22×Dic106—48no—
32C23×Dic53—21no—
33C23×Dic53—21no—
34C23×Dic53—212no—
35C23×Dic53—48no—
36C23×Dic53D411yescharacteristic
37C23×Dic53D411yescharacteristic
38C22×#88322—24no—
39C22×#88322C2316yes—
40#13133216—224no—
41#13133216C23116yescomplemented
42D4×Dic53—224no—
43D4×Dic53C23116yes—
44C2×C4⋊Dic53—23no—
45C2×C4⋊Dic53—23no—
46C2×C4⋊Dic53C2316yes—
47C2×C4⋊Dic53C2316yes—
48C2×#11719919—212no—
49C2×C22⋊Dic53—212no—
50C2×C4×Dic53D416yes—
51C2×C4×Dic53C2316yes—
52#13133149C23116yescomplemented
53#13133148C23116yescomplemented
54#13133146C23116yescomplemented
55#13133138C23116yes—
56C25×C53—21no—
57C22×D4×C53—23no—
58C22×D4×C53—24no—
59C22×D4×C53Q818yescomplemented
60C22×D4×C53C2316yes—
61C23×C4×C53—21no—
62C23×C4×C53C2311yescharacteristic
63C22×C8×C53—24no—
64C22×C8×C53C2316yes—
65C2×C4⋊C4×C53—23no—
66C2×C4⋊C4×C53C2316yes—
67#102×C53C23116yes—
68C24×C53—28no—
69C24×C53—26no—
70C24×C53—44no—
71C24×C53—44no—
72C24×C53C2×D411yescharacteristic
73C2×D4×C53—224no—
74C2×D4×C53—26no—
75C2×D4×C53—212no—
76C2×D4×C53C2×Q8124yescomplemented
77C2×D4×C53C2414yes—
78C22×C4×C53—26no—
79C22×C4×C53—44no—
80C22×C4×C53C2×D413yes—
81C22×C4×C53C2×Q814yes—
82C22×C4×C53#4114yes—
83C22×C4×C53C2413yes—
84C22×Dic53—24no—
85C22×Dic53—26no—
86C22×Dic53—26no—
87C22×Dic53—216no—
88C22×Dic53—44no—
89C22×Dic53—44no—
90C22×Dic53—448no—
91C22×Dic53SD1618yes—
92C22×Dic53C2×D416yes—
93C22×Dic53C2×D416yes—
94C2×Dic106—216no—
95C2×Dic106—448no—
96C2×C8×C53—212no—
97C2×C8×C53C2414yes—
98C4⋊C4×C53—26no—
99C4⋊C4×C53C2414yes—
100C22⋊Dic53—216no—
101C4⋊Dic53—26no—
102C4⋊Dic53—26no—
103C4⋊Dic53C2414yes—
104C4⋊Dic53C2414yes—
105#11719919—216no—
106C4×Dic53C2×D4112yes—
107C4×Dic53C2414yes—
108C2×#88322—212no—
109C2×#88322C2414yes—
110C23×C53—224no—
111C23×C53—24no—
112C23×C53—412no—
113C23×C53—412no—
114C23×C53—432no—
115C23×C53C22⋊Q814yes—
116C23×C53C2×SD1614yes—
117C23×C53AGL(1,ℤ/8)14yes—
118C23×C53C22×D413yes—
119D4×C53—224no—
120D4×C53—24no—
121D4×C53C22×Q818yescomplemented
122C2×C4×C53—24no—
123C2×C4×C53—412no—
124C2×C4×C53C22⋊Q8112yes—
125C2×C4×C53C22×D413yes—
126C2×C4×C53C22×Q816yes—
127C2×C4×C53C2×#4116yes—
128C2×C4×C53C2511yescharacteristic
129C2×Dic53—212no—
130C2×Dic53—24no—
131C2×Dic53—24no—
132C2×Dic53—412no—
133C2×Dic53—412no—
134C2×Dic53—432no—
135C2×Dic53C2×SD16124yes—
136C2×Dic53C22×D414yes—
137C2×Dic53C22×D414yes—
138Dic106—432no—
139C8×C53—24no—
140#88322—24no—
141C22×#476—531nomaximal, Sylow
142C22×C53—28no—
143C22×C53—44no—
144C22×C53—44no—
145C22×C53—448no—
146C22×C53#476116yescomplemented
147C22×C53C2×C22⋊Q816yes—
148C22×C53C22×SD1616yes—
149C22×C53C2×AGL(1,ℤ/8)16yes—
150C22×C53C23×D411yescharacteristic
151C4×C53—44no—
152C4×C53C2×C22⋊Q816yes—
153C4×C53C23×Q811yescharacteristic
154C4×C53C22×#4111yescharacteristic
155Dic53—24no—
156Dic53—44no—
157Dic53—44no—
158Dic53C22×SD1618yes—
159C22×C4⋊Q8—531no—
160C22×C4×D4—531no—
161C2×#476—5324no—
162C22×#106—531no—
163C22×#106—531no—
164C22×C4⋊C8—531no—
165C22×#102—531no—
166C22×#102D5311yescharacteristic
167C2×C53—48no—
168C2×C53C2×#476112yescomplemented
169C2×C53C22×C22⋊Q811yescharacteristic
170C2×C53C23×SD1611yescharacteristic
171C2×C53C22×AGL(1,ℤ/8)11yescharacteristic
172C23×Q8—1061no—
173C23×D4C2×D5311yescharacteristic
174C24×C4—1061no—
175C23×C8—531no—
176C23×C8C2×D5311yescharacteristic
177C2×C4⋊Q8—5312no—
178C2×C4⋊Q8—1066no—
179C2×C4×D4—5312no—
180C2×C4×D4—1066no—
181C22×C4⋊C4—531no—
182C22×C4⋊C4—531no—
183C22×C4⋊C4—1061no—
184C22×C4⋊C4C2×D5311yescharacteristic
185C22×C22⋊C4—1061no—
186C22×C42—531no—
187#476—5364no—
188C2×#106—5312no—
189C2×#106—5312no—
190C2×C4⋊C8—5312no—
191C2×#102—5312no—
192C2×#102C2×D53112yes—
193C53C22×#47611yescharacteristic, complemented, Sylow
194C25—21no—
195C22×Q8—10612no—
196C22×Q8—2128no—
197C22×D4—23no—
198C22×D4—24no—
199C22×D4Dic10618yescomplemented
200C22×D4C22×D5316yes—
201C23×C4—21no—
202C23×C4—531no—
203C23×C4—531no—
204C23×C4—10612no—
205C23×C4—1061no—
206C23×C4—1061no—
207C23×C4—2128no—
208C23×C4C22×D5311yescharacteristic
209C22×C8—24no—
210C22×C8—536no—
211C22×C8—1064no—
212C22×C8C22×D5316yes—
213C4⋊Q8—5316no—
214C4⋊Q8—10624no—
215C4×D4—5316no—
216C4×D4—10624no—
217C2×C4⋊C4—23no—
218C2×C4⋊C4—536no—
219C2×C4⋊C4—536no—
220C2×C4⋊C4—10612no—
221C2×C4⋊C4—1063no—
222C2×C4⋊C4—1063no—
223C2×C4⋊C4C22×D5316yes—
224C2×C22⋊C4—10612no—
225C2×C42—536no—
226C2×C42—536no—
227#106—5316no—
228#106—5316no—
229C4⋊C8—5316no—
230#102—5316no—
231#102C22×D53116yes—
232C24—28no—
233C24—26no—
234C24—44no—
235C24—44no—
236C24D4×D5311yescharacteristic
237C2×Q8—10616no—
238C2×Q8—21248no—
239C2×D4—224no—
240C2×D4—26no—
241C2×D4—212no—
242C2×D4C2×Dic106124yescomplemented
243C2×D4C23×D5314yes—
244C22×C4—26no—
245C22×C4—44no—
246C22×C4—538no—
247C22×C4—536no—
248C22×C4—536no—
249C22×C4—10616no—
250C22×C4—1064no—
251C22×C4—1066no—
252C22×C4—1066no—
253C22×C4—21248no—
254C22×C4—2124no—
255C22×C4—2124no—
256C22×C4C2×Dic10614yes—
257C22×C4D4×D5313yes—
258C22×C4#1171994714yes—
259C22×C4C23×D5313yes—
260C2×C8—212no—
261C2×C8—534no—
262C2×C8—10612no—
263C2×C8C23×D5314yes—
264C4⋊C4—26no—
265C4⋊C4—534no—
266C4⋊C4—534no—
267C4⋊C4—10616no—
268C4⋊C4—1066no—
269C4⋊C4—1066no—
270C4⋊C4C23×D5314yes—
271C22⋊C4—10616no—
272C42—5312no—
273C42—534no—
274C23—224no—
275C23—24no—
276C23—412no—
277C23—412no—
278C23—432no—
279C23#1313319414yes—
280C23SD16×D5314yes—
281C23#1313324114yes—
282C23C2×D4×D5313yes—
283Q8—21232no—
284D4—224no—
285D4—24no—
286D4C22×Dic10618yescomplemented
287C2×C4—24no—
288C2×C4—412no—
289C2×C4—5324no—
290C2×C4—534no—
291C2×C4—534no—
292C2×C4—10612no—
293C2×C4—1064no—
294C2×C4—1064no—
295C2×C4—21232no—
296C2×C4—21212no—
297C2×C4—21212no—
298C2×C4#13133194112yes—
299C2×C4C2×D4×D5313yes—
300C2×C4C22×Dic10616yes—
301C2×C4C2×#1171994716yes—
302C2×C4C24×D5311yescharacteristic
303C8—24no—
304C8—1064no—
305C22—28no—
306C22—448no—
307C22—44no—
308C22—44no—
309C22#16810911116yesdirect product factor
310C22C2×#1313319416yes—
311C22C2×SD16×D5316yes—
312C22C2×#1313324116yes—
313C22C22×D4×D5311yescharacteristic
314C4—44no—
315C4—538no—
316C4—1064no—
317C4—2124no—
318C4—2124no—
319C4C2×#1313319416yes—
320C4C23×Dic10611yescharacteristic
321C4C22×#1171994711yescharacteristic
322C2—48no—
323C2C2×#16810911112yesdirect product factor
324C2C22×#1313319411yescharacteristic
325C2C22×SD16×D5311yescharacteristic
326C2C22×#1313324111yescharacteristic
subgroupG/H
centerC24D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC22×#102×C53C2
socleC24×C53C2×D4
prime radicalC23×D4×C53C22
hypercenterC22×#102D53
hypocenterC53C22×#476
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
1. C22×#13133216C2semidirect
3. C2×#16810911C2direct (N × H)
4. C22×#13133149C2semidirect
5. C22×#13133148C2semidirect
6. C22×#13133146C2semidirect
13. C2×#13133216C22semidirect
21. #16810911C22direct (N × H)
22. C2×#13133149C22semidirect
23. C2×#13133148C22semidirect
24. C2×#13133146C22semidirect
41. #13133216C23semidirect
52. #13133149C23semidirect
53. #13133148C23semidirect
54. #13133146C23semidirect
59. C22×D4×C53Q8semidirect
76. C2×D4×C53C2×Q8semidirect
121. D4×C53C22×Q8semidirect
146. C22×C53#476semidirect
168. C2×C53C2×#476semidirect
193. C53C22×#476semidirect
199. C22×D4Dic106semidirect
242. C2×D4C2×Dic106semidirect
286. D4C22×Dic106semidirect
309. C22#16810911direct (N × H)
323. C2C2×#16810911direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332162
C22×D4×Dic532
C2×#168109112
C22×#131331492
C22×#131331482
C22×#131331462
C22×#131331382
C22×#102×C532
C22×#47653