Finite Group Databasebeta

← C22×#16810846 #28250901

Subgroups — C22×#16810846 #28250901

Order 13568.

2918 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic212C211yesmaximal, characteristic, complemented
2C23×#11719913C211yesmaximal, characteristic
3C22×#13133205C211yesmaximal, characteristic
4C22×C4×Dic106C211yesmaximal, characteristic, complemented
5C2×#16810846C2124yesmaximal, direct product factor
6C22×#13133142C211yesmaximal, characteristic, complemented
7C22×#13133139C211yesmaximal, characteristic
8C22×C4⋊C8×C53C211yesmaximal, characteristic, complemented
9C23×Dic106C2211yescharacteristic
10C23×Dic106C2211yescharacteristic
11C23×D212C2211yescharacteristic
12C22×Dic212—28no—
13C22×Dic212C22112yescomplemented
14C22×#11719913—28no—
15C22×#11719913C22112yes—
16C2×#13133205—26no—
17C2×#13133205C22112yes—
18C2×C4×Dic106—26no—
19C2×C4×Dic106C22112yescomplemented
20C22×C4⋊Dic53C2211yescharacteristic
21C22×#11719919—21no—
22C22×#11719921—21no—
23C22×#11719921—21no—
24C22×C4×Dic53—21no—
25#16810846C22164yesdirect product factor
26C2×#13133142C22112yescomplemented
27C2×#13133139C22112yes—
28C23×C8×C53C2211yescharacteristic
29C23×C8×C53C2211yescharacteristic
30C22×C42×C53C2211yescharacteristic
31C2×C4⋊C8×C53C22112yescomplemented
32C24×D53—21no—
33C22×Dic106—23no—
34C22×Dic106—24no—
35C22×Dic106—24no—
36C22×Dic106—24no—
37C22×Dic106—23no—
38C22×Dic106D418yes—
39C22×Dic106C2316yes—
40C22×Dic106C2316yes—
41C22×D212—24no—
42C22×D212—24no—
43C22×D212—23no—
44C22×D212C2316yes—
45C23×Dic53—21no—
46C23×Dic53—21no—
47C23×Dic53—21no—
48C2×Dic212—248no—
49C2×Dic212C23116yescomplemented
50C2×#11719913—248no—
51C2×#11719913C23116yes—
52#13133205—224no—
53#13133205C23116yes—
54C4×Dic106—224no—
55C4×Dic106C23116yescomplemented
56C2×C4⋊Dic53—23no—
57C2×C4⋊Dic53C2316yes—
58C2×#11719919—212no—
59C2×#11719921—212no—
60C2×#11719921—212no—
61C2×C4×Dic53—212no—
62#13133142C23116yescomplemented
63#13133139C23116yes—
64C23×C4×C53D411yescharacteristic
65C23×C4×C53D411yescharacteristic
66C23×C4×C53C2311yescharacteristic
67C22×C8×C53—24no—
68C22×C8×C53—24no—
69C22×C8×C53C2316yes—
70C22×C8×C53C2316yes—
71C2×C42×C53D416yes—
72C2×C42×C53C2316yes—
73C4⋊C8×C53C23116yescomplemented
74C24×C53C2×D411yescharacteristic
75C23×D53—26no—
76C23×D53—44no—
77C23×D53—44no—
78C23×D53—44no—
79C22×C4×C53—24no—
80C22×C4×C53—24no—
81C22×C4×C53C2×D416yes—
82C22×C4×C53C2×D416yes—
83C22×C4×C53C2×D413yes—
84C22×C4×C53C2×D414yes—
85C22×C4×C53#4114yes—
86C22×C4×C53C2413yes—
87C22×Dic53—28no—
88C22×Dic53—26no—
89C22×Dic53—26no—
90C22×Dic53—26no—
91C22×Dic53—44no—
92C22×Dic53—44no—
93C2×D212—212no—
94C2×D212—212no—
95C2×D212—26no—
96C2×D212—412no—
97C2×D212C2414yes—
98C2×Dic106—224no—
99C2×Dic106—26no—
100C2×Dic106—212no—
101C2×Dic106—212no—
102C2×Dic106—212no—
103C2×Dic106—26no—
104C2×Dic106—412no—
105C2×Dic106C2×D4124yes—
106C2×Dic106C2414yes—
107C2×Dic106C2414yes—
108C2×C8×C53—212no—
109C2×C8×C53—212no—
110C2×C8×C53C2414yes—
111C2×C8×C53C2414yes—
112C42×C53C2×D4112yes—
113C42×C53C2414yes—
114#11719921—216no—
115#11719921—216no—
116C4⋊Dic53—26no—
117C4⋊Dic53C2414yes—
118#11719919—216no—
119C4×Dic53—216no—
120Dic212—232no—
121#11719913—232no—
122C23×C53#12114yes—
123C23×C53#13514yes—
124C23×C53#13714yes—
125C23×C53C22×D413yes—
126C22×D53—24no—
127C22×D53—412no—
128C22×D53—412no—
129C22×D53—412no—
130C22×D53—816no—
131C2×C4×C53—212no—
132C2×C4×C53—212no—
133C2×C4×C53#121112yes—
134C2×C4×C53C22×D416yes—
135C2×C4×C53C22×D414yes—
136C2×C4×C53C22×D414yes—
137C2×C4×C53C22×D413yes—
138C2×C4×C53C2×#4116yes—
139C2×C4×C53C2511yescharacteristic
140C2×Dic53—224no—
141C2×Dic53—24no—
142C2×Dic53—24no—
143C2×Dic53—24no—
144C2×Dic53—412no—
145C2×Dic53—412no—
146D212—24no—
147D212—24no—
148D212—412no—
149Dic106—224no—
150Dic106—24no—
151Dic106—24no—
152Dic106—24no—
153Dic106—412no—
154Dic106C22×D418yes—
155C8×C53—24no—
156C8×C53—24no—
157C22×#464—531nomaximal, Sylow
158C22×C53#464116yescomplemented
159C22×C53C2×#12116yes—
160C22×C53C2×#13516yes—
161C22×C53C2×#13716yes—
162C22×C53C23×D411yescharacteristic
163C2×D53—44no—
164C2×D53—44no—
165C2×D53—44no—
166C2×D53—824no—
167C4×C53—24no—
168C4×C53—24no—
169C4×C53C2×#12116yes—
170C4×C53C23×D411yescharacteristic
171C4×C53C22×#4111yescharacteristic
172Dic53—28no—
173Dic53—44no—
174Dic53—44no—
175C23×Q16—531no—
176C23×SD16—531no—
177C22×#124—531no—
178C22×C4×Q8—531no—
179C2×#464—5324no—
180C22×C4⋊C8D5311yescharacteristic, complemented
181C22×#103—531no—
182C22×#102—531no—
183C2×C53C2×#464112yescomplemented
184C2×C53C22×#12111yescharacteristic
185C2×C53C22×#13511yescharacteristic
186C2×C53C22×#13711yescharacteristic
187D53—84no—
188C23×Q8—531no—
189C23×Q8—531no—
190C23×D4—531no—
191C22×Q16—5312no—
192C22×Q16—1068no—
193C22×SD16—5312no—
194C22×SD16—1068no—
195C23×C8C2×D5311yescharacteristic
196C23×C8C2×D5311yescharacteristic
197C2×#124—5312no—
198C2×#124—1066no—
199C2×C4×Q8—5312no—
200C2×C4×Q8—1066no—
201C22×C4⋊C4—531no—
202C22×C4⋊C4—1061no—
203C22×C22⋊C4—1061no—
204C22×C22⋊C4—1061no—
205C22×C42—1061no—
206C22×C42C2×D5311yescharacteristic
207#464—5364no—
208C2×C4⋊C8C2×D53112yescomplemented
209C2×#103—5312no—
210C2×#102—5312no—
211C53C22×#46411yescharacteristic, complemented, Sylow
212C25—1061no—
213C22×Q8—538no—
214C22×Q8—536no—
215C22×Q8—536no—
216C22×Q8—1063no—
217C22×Q8—1063no—
218C22×Q8—1064no—
219C22×Q8—1064no—
220C22×Q8—1064no—
221C22×D4—536no—
222C22×D4—1063no—
223C22×D4—1064no—
224C22×D4—1064no—
225C23×C4—1061no—
226C23×C4—1061no—
227C23×C4—1061no—
228C23×C4D21211yescharacteristic
229C23×C4D21211yescharacteristic
230C23×C4C22×D5311yescharacteristic
231C2×Q16—5316no—
232C2×Q16—10648no—
233C2×SD16—5316no—
234C2×SD16—10648no—
235C22×C8—24no—
236C22×C8—24no—
237C22×C8C22×D5316yes—
238C22×C8C22×D5316yes—
239#124—5316no—
240#124—10624no—
241C4×Q8—5316no—
242C4×Q8—10624no—
243C2×C4⋊C4—536no—
244C2×C4⋊C4—1063no—
245C2×C4⋊C4—10612no—
246C2×C22⋊C4—10612no—
247C2×C22⋊C4—10612no—
248C2×C42—10612no—
249C2×C42D21216yes—
250C2×C42C22×D5316yes—
251C4⋊C8C22×D53116yescomplemented
252#103—5316no—
253#102—5316no—
254C24—1066no—
255C24—2124no—
256C24—2124no—
257C24—2124no—
258C24C2×D21211yescharacteristic
259C2×Q8—5324no—
260C2×Q8—534no—
261C2×Q8—534no—
262C2×Q8—10624no—
263C2×Q8—1066no—
264C2×Q8—1066no—
265C2×Q8—10612no—
266C2×Q8—10612no—
267C2×Q8—10612no—
268C2×Q8—21212no—
269C2×D4—534no—
270C2×D4—10612no—
271C2×D4—1066no—
272C2×D4—10612no—
273C2×D4—21212no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—1068no—
277C22×C4—1066no—
278C22×C4—1066no—
279C22×C4—1066no—
280C22×C4—2124no—
281C22×C4—2124no—
282C22×C4C2×D21216yes—
283C22×C4C2×D21216yes—
284C22×C4C2×D21213yes—
285C22×C4D4×D5314yes—
286C22×C4#1171994914yes—
287C22×C4C23×D5313yes—
288Q16—10632no—
289SD16—10632no—
290C2×C8—212no—
291C2×C8—212no—
292C2×C8C23×D5314yes—
293C2×C8C23×D5314yes—
294C4⋊C4—534no—
295C4⋊C4—1066no—
296C4⋊C4—10616no—
297C22⋊C4—10616no—
298C22⋊C4—10616no—
299C42—10616no—
300C42C2×D212112yes—
301C42C23×D5314yes—
302C23—1064no—
303C23—42416no—
304C23—21212no—
305C23—21212no—
306C23—21212no—
307C23#1313319214yes—
308C23#1313323914yes—
309C23#1313325014yes—
310C23C22×D21213yes—
311Q8—538no—
312Q8—10624no—
313Q8—1064no—
314Q8—1064no—
315Q8—1064no—
316Q8—21212no—
317D4—1064no—
318D4—1064no—
319D4—21212no—
320C2×C4—212no—
321C2×C4—212no—
322C2×C4—10624no—
323C2×C4—1064no—
324C2×C4—1064no—
325C2×C4—1064no—
326C2×C4—21212no—
327C2×C4—21212no—
328C2×C4#13133192112yes—
329C2×C4C2×D4×D5316yes—
330C2×C4C22×D21214yes—
331C2×C4C22×D21214yes—
332C2×C4C22×D21213yes—
333C2×C4C2×#1171994916yes—
334C2×C4C24×D5311yescharacteristic
335C8—24no—
336C8—24no—
337C22—42424no—
338C22—2124no—
339C22—2124no—
340C22—2124no—
341C22#16810846116yesdirect product factor
342C22C2×#1313319216yes—
343C22C2×#1313323916yes—
344C22C2×#1313325016yes—
345C22C23×D21211yescharacteristic
346C4—24no—
347C4—24no—
348C4—1068no—
349C4—2124no—
350C4—2124no—
351C4C2×#1313319216yes—
352C4C22×D4×D5311yescharacteristic
353C4C22×#1171994911yescharacteristic
354C2—4244no—
355C2C2×#16810846112yesdirect product factor
356C2C22×#1313319211yescharacteristic
357C2C22×#1313323911yescharacteristic
358C2C22×#1313325011yescharacteristic
subgroupG/H
centerC24C2×D212
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC22×C4⋊C8×C53C2
socleC24×C53C2×D4
prime radicalC23×D212C22
hypercenterC22×C4⋊C8D53
hypocenterC53C22×#464
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C23×Dic212C2semidirect
4. C22×C4×Dic106C2semidirect
5. C2×#16810846C2direct (N × H)
6. C22×#13133142C2semidirect
8. C22×C4⋊C8×C53C2semidirect
13. C22×Dic212C22semidirect
19. C2×C4×Dic106C22semidirect
25. #16810846C22direct (N × H)
26. C2×#13133142C22semidirect
31. C2×C4⋊C8×C53C22semidirect
49. C2×Dic212C23semidirect
55. C4×Dic106C23semidirect
62. #13133142C23semidirect
73. C4⋊C8×C53C23semidirect
158. C22×C53#464semidirect
180. C22×C4⋊C8D53semidirect
183. C2×C53C2×#464semidirect
208. C2×C4⋊C8C2×D53semidirect
211. C53C22×#464semidirect
251. C4⋊C8C22×D53semidirect
341. C22#16810846direct (N × H)
355. C2C2×#16810846direct (N × H)

Maximal subgroups

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

subgroupindex
C23×Dic2122
C23×#117199132
C22×#131332052
C22×C4×Dic1062
C2×#168108462
C22×#131331422
C22×#131331392
C22×C4⋊C8×C532
C22×#46453