Finite Group Databasebeta

← C2×#20245289 #26369722

Subgroups — C2×#20245289 #26369722

Order 12032.

710 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689559C211yesmaximal, characteristic
2C2×#15689244C211yesmaximal, characteristic, complemented
3C2×#15689235C211yesmaximal, characteristic, complemented
4#20245289C218yesmaximal, direct product factor
5C2×#15688947C211yesmaximal, characteristic, complemented
6C2×#15688934C211yesmaximal, characteristic, complemented
7C2×#15688900C211yesmaximal, characteristic, complemented
8C2×C47×#361C211yesmaximal, characteristic, complemented
9C2×#13094046C2211yescharacteristic
10C2×#13094032—21no—
11C2×D4×Dic47—21no—
12#15689559C414yescomplemented
13#15689559C2214yes—
14C2×#13093968C412yescomplemented
15C2×#13093965C412yescomplemented
16C2×#13093964C2211yescharacteristic
17#15689244—22no—
18#15689244C2214yescomplemented
19#15689235—22no—
20#15689235C2214yescomplemented
21C2×#13093956C2211yescharacteristic
22C2×#13093950—21no—
23C2×#13093945—21no—
24C2×#13093934C2211yescharacteristic
25#15688947C2214yescomplemented
26#15688934C2214yescomplemented
27#15688900C2214yescomplemented
28C22×D8×C47C2211yescharacteristic
29C2×M32×C47C2211yescharacteristic
30C2×#108×C47C2211yescharacteristic
31C47×#361C2214yescomplemented
32C2×#10625350—21no—
33C23×Dic47—41no—
34#13094046—21no—
35#13094046—21no—
36#13094046C2×C412yescomplemented
37#13094046C2312yes—
38C2×Dic188C2×C411yescharacteristic, complemented
39C2×#10625325—21no—
40C2×D376C2×C411yescharacteristic, complemented
41#13094032—24no—
42#13094032—42no—
43C2×#10625324—41no—
44C22×C47⋊C8D411yescharacteristic
45C2×C8×D47—41no—
46D4×Dic47—24no—
47D4×Dic47—42no—
48C2×C4⋊Dic47—21no—
49C2×C22⋊Dic47—41no—
50C2×C4×Dic47D411yescharacteristic
51#13093968C2×C418yescomplemented
52#13093965C2×C418yescomplemented
53#13093964C2×C412yescomplemented
54#13093964C2312yes—
55C2×C47⋊C16C2×C412yescomplemented
56#13093956—21no—
57#13093956C2312yes—
58#13093950—24no—
59#13093945—24no—
60#13093934D412yes—
61#13093934C2312yes—
62C22×D4×C47—21no—
63C2×D8×C47—21no—
64C2×D8×C47—21no—
65C2×D8×C47—21no—
66C2×D8×C47C2×C412yes—
67C2×D8×C47C2×C412yes—
68C2×D8×C47C2312yes—
69C2×M16×C47—21no—
70C22×C8×C47C2311yescharacteristic
71M32×C47—21no—
72M32×C47C2312yes—
73C2×C16×C47—21no—
74#108×C47—21no—
75#108×C47C2312yes—
76C24×C47—41no—
77C2×D4×C47—22no—
78C2×D4×C47—22no—
79C2×D4×C47—22no—
80C2×D4×C47—41no—
81C2×D4×C47—41no—
82C2×D4×C47—41no—
83C22×C4×C47C2×D411yescharacteristic
84C2×C22⋊D47—41no—
85C22×Dic47—21no—
86C22×Dic47—44no—
87C22×Dic47—41no—
88C22×Dic47—84no—
89#10625350—22no—
90#10625350—22no—
91#10625350—41no—
92#10625350—41no—
93C2×D188—21no—
94C2×C4×D47—41no—
95C2×Dic94—21no—
96D8×C47—22no—
97D8×C47—22no—
98D8×C47—22no—
99D8×C47C22×C414yes—
100M16×C47—22no—
101M16×C47—41no—
102C2×C8×C47—41no—
103C2×C8×C47C22×C411yescharacteristic
104C2×C8×C47C22×C411yescharacteristic
105C2×C8×C47C2×D411yescharacteristic
106C2×C8×C47#4111yescharacteristic
107C2×C8×C47#4111yescharacteristic
108C2×C8×C47C2411yescharacteristic
109C22⋊Dic47—44no—
110C4⋊Dic47—22no—
111C4⋊Dic47—41no—
112C4×Dic47—21no—
113C4×Dic47C2×D412yes—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8C2×D412yes—
117Dic188—21no—
118Dic188C22×C412yescomplemented
119D376—21no—
120D376C22×C412yescomplemented
121#10625325—22no—
122#10625325—22no—
123#10625324—44no—
124C8×D47—44no—
125C16×C47—22no—
126C47⋊C16C22×C414yescomplemented
127C23×C47—42no—
128C23×C47—42no—
129C23×C47—42no—
130C23×C47—82no—
131C23×C47—82no—
132C23×C47AGL(1,ℤ/8)11yescharacteristic
133C22×D47—41no—
134D4×C47—24no—
135D4×C47—42no—
136D4×C47—42no—
137D4×C47—42no—
138C2×C4×C47C4×D411yescharacteristic
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C47C22×D411yescharacteristic
144C22⋊D47—42no—
145C22⋊D47—42no—
146C2×Dic47—22no—
147C2×Dic47—42no—
148C2×Dic47—41no—
149C2×Dic47—42no—
150C2×Dic47—88no—
151C2×Dic47—82no—
152D188—22no—
153D188—41no—
154C4×D47—42no—
155C4×D47—42no—
156Dic94—22no—
157Dic94—41no—
158C8×C47—42no—
159C8×C47C4×D411yescharacteristic
160C8×C47C4×D411yescharacteristic
161C8×C47C23×C411yescharacteristic
162C8×C47C2×#4111yescharacteristic
163C47⋊C8—22no—
164C2×AGL(1,ℤ/16)—471nomaximal, Sylow
165C22×C47—22no—
166C22×C47—44no—
167C22×C47—82no—
168C22×C47—82no—
169C22×C47—84no—
170C22×C47—84no—
171C22×C47#44211yescharacteristic
172C22×C47#44211yescharacteristic
173C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
174C2×D47—42no—
175C2×D47—82no—
176C4×C47#44211yescharacteristic
177C4×C47#44211yescharacteristic
178C4×C47C2×C4×D411yescharacteristic
179C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
180Dic47—42no—
181Dic47—42no—
182Dic47—82no—
183C2×#509—471no—
184C2×#444—471no—
185C2×#442—471no—
186AGL(1,ℤ/16)—478no—
187C2×#365—471no—
188C2×#361D4711yescharacteristic, complemented
189C2×#360—471no—
190C2×#347—471no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—84no—
194C2×C47AGL(1,ℤ/16)12yescomplemented
195C2×C47C2×#44211yescharacteristic
196D47—82no—
197C2×#135—471no—
198C22×D8C2×D4711yescharacteristic
199C2×#131—941no—
200C2×C4×D4—941no—
201#509—474no—
202#509—474no—
203C2×SD32—472no—
204C2×D16—472no—
205C2×M32—471no—
206C2×M32C2×D4711yescharacteristic
207#444—474no—
208#444—942no—
209#442—474no—
210#442—942no—
211C2×#108C2×D4711yescharacteristic
212C2×#106—471no—
213C2×#104—941no—
214C2×#102—941no—
215C2×#97—471no—
216#365—474no—
217#361C2×D4714yescomplemented
218#360—474no—
219#347—474no—
220C47C2×AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
221C2×#41—941no—
222C22×D4—21no—
223C23×C4—1881no—
224#135—472no—
225#135—472no—
226#135—941no—
227#135—941no—
228C2×Q16—471no—
229C2×SD16—941no—
230C2×D8—21no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—471no—
234C2×D8C4×D4712yes—
235C2×D8C4×D4712yes—
236C2×D8C22×D4712yes—
237#131—944no—
238#131—1882no—
239C2×M16—21no—
240C2×M16—1881no—
241C22×C8—471no—
242C22×C8—1881no—
243C22×C8C22×D4711yescharacteristic
244C4×D4—944no—
245C4×D4—1882no—
246C2×C4⋊C4—941no—
247C2×C22⋊C4—1881no—
248C2×C42—471no—
249SD32—478no—
250D16—478no—
251M32—21no—
252M32—472no—
253M32—472no—
254M32C22×D4712yes—
255C2×C16—21no—
256C2×C16—472no—
257#108—21no—
258#108C22×D4712yes—
259#106—472no—
260#106—941no—
261#104—944no—
262#102—944no—
263#97—472no—
264#97—472no—
265C24—41no—
266#41—942no—
267#41—942no—
268#41—1881no—
269#41—1881no—
270C2×Q8—941no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—941no—
278C2×D4—1881no—
279C22×C4—941no—
280C22×C4—3764no—
281C22×C4—1884no—
282C22×C4—1881no—
283C22×C4—1881no—
284C22×C4D4×D4711yescharacteristic
285Q16—472no—
286Q16—941no—
287SD16—942no—
288SD16—942no—
289D8—22no—
290D8—22no—
291D8—22no—
292D8—472no—
293D8—941no—
294D8C2×C4×D4714yes—
295M16—22no—
296M16—41no—
297M16—1884no—
298C2×C8—41no—
299C2×C8—472no—
300C2×C8—941no—
301C2×C8—941no—
302C2×C8—1884no—
303C2×C8C2×C4×D4711yescharacteristic
304C2×C8C2×C4×D4711yescharacteristic
305C2×C8D4×D4711yescharacteristic
306C2×C8#1062535211yescharacteristic
307C2×C8#1062535211yescharacteristic
308C2×C8C23×D4711yescharacteristic
309C4⋊C4—942no—
310C4⋊C4—1881no—
311C22⋊C4—1884no—
312C42—472no—
313C42—941no—
314C16—22no—
315C16—474no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—82no—
320C23—82no—
321C23—1881no—
322C23#1309405111yescharacteristic
323Q8—942no—
324Q8—1881no—
325D4—24no—
326D4—42no—
327D4—42no—
328D4—42no—
329D4—942no—
330D4—1882no—
331D4—1882no—
332D4—1881no—
333C2×C4—942no—
334C2×C4—3762no—
335C2×C4—3768no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4—1881no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4(C22⋊D47)⋊C411yescharacteristic
342C2×C4(C22⋊D47)⋊C411yescharacteristic
343C2×C4#1309404811yescharacteristic
344C2×C4#1309404811yescharacteristic
345C2×C4#1309405111yescharacteristic
346C2×C4C2×D4×D4711yescharacteristic
347C8—42no—
348C8—942no—
349C8(C22⋊D47)⋊C411yescharacteristic
350C8(C22⋊D47)⋊C411yescharacteristic
351C8C22×C4×D4711yescharacteristic
352C8C2×#1062535211yescharacteristic
353C22—22no—
354C22—44no—
355C22—82no—
356C22—82no—
357C22—84no—
358C22—84no—
359C22—3762no—
360C22—1882no—
361C22#1568923211yescharacteristic
362C22#1568923211yescharacteristic
363C22C2×#1309405111yescharacteristic
364C4—3762no—
365C4—1882no—
366C4—1882no—
367C4#1568923211yescharacteristic
368C4#1568923211yescharacteristic
369C4C2×(C22⋊D47)⋊C411yescharacteristic
370C4C2×#1309404811yescharacteristic
371C2—21no—
372C2—21no—
373C2—84no—
374C2—3762no—
375C2#2024528912yesdirect product factor
376C2C2×#1568923211yescharacteristic
subgroupG/H
centerC22#15689232
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC2×C47×#361C2
socleC22×C47#442
prime radicalC2×#15689559C2
cosocleC2×C8×C47C24
hypercenterC2×#361D47
hypocenterC47C2×AGL(1,ℤ/16)
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#361

Complements

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

NH ≅ G/Ntype
2. C2×#15689244C2semidirect
3. C2×#15689235C2semidirect
4. #20245289C2direct (N × H)
5. C2×#15688947C2semidirect
6. C2×#15688934C2semidirect
7. C2×#15688900C2semidirect
8. C2×C47×#361C2semidirect
12. #15689559C4semidirect
14. C2×#13093968C4semidirect
15. C2×#13093965C4semidirect
18. #15689244C22semidirect
20. #15689235C22semidirect
25. #15688947C22semidirect
26. #15688934C22semidirect
27. #15688900C22semidirect
31. C47×#361C22semidirect
36. #13094046C2×C4semidirect
38. C2×Dic188C2×C4semidirect
40. C2×D376C2×C4semidirect
51. #13093968C2×C4semidirect
52. #13093965C2×C4semidirect
53. #13093964C2×C4semidirect
55. C2×C47⋊C16C2×C4semidirect
118. Dic188C22×C4semidirect
120. D376C22×C4semidirect
126. C47⋊C16C22×C4semidirect
188. C2×#361D47semidirect
194. C2×C47AGL(1,ℤ/16)semidirect
217. #361C2×D47semidirect
220. C47C2×AGL(1,ℤ/16)semidirect
375. C2#20245289direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156895592
C2×#156892442
C2×#156892352
#202452892
C2×#156889472
C2×#156889342
C2×#156889002
C2×C47×#3612
C2×AGL(1,ℤ/16)47