Finite Group Databasebeta

← #26190256

Subgroups — #26190256

Order 12032.

1192 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254310C212yesmaximal, complemented
2#20249209C211yesmaximal, characteristic, complemented
3#20242357C214yesmaximal, complemented
4#20242340C214yesmaximal, complemented
5#20241948C212yesmaximal, complemented
6#20241702C211yesmaximal, characteristic, complemented
7C47×#3449C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic, complemented
9#15689495—22no—
10#15689475—22no—
11#15689328—24no—
12#15689261C2218yescomplemented
13#15689176C2212yescomplemented
14#15689062—21no—
15#15689059—21no—
16#15689043C2212yescomplemented
17#15689035C2212yes—
18#15689021—21no—
19#15688989C2211yescharacteristic, complemented
20#15688983—21no—
21#15688857C2218yescomplemented
22#15688849C2212yescomplemented
23#15688847C2212yes—
24C22×#41×C47—21no—
25C47×#537—21no—
26C47×#537C2212yes—
27C47×#536—22no—
28C47×#535—22no—
29C47×#534—21no—
30C47×#534C2212yes—
31C47×#532—21no—
32C47×#532C2212yes—
33C47×#525—24no—
34C2×#117×C47C2211yescharacteristic
35C22×Dic94C2312yes—
36C2×#10625335—28no—
37#13094024—22no—
38#13094024—22no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C2×Dic47⋊C4—22no—
43C2×C22⋊Dic47—22no—
44C4⋊(C47⋊C8)—22no—
45#13093947—28no—
46C22⋊(C47⋊C8)C2314yes—
47C4×C47⋊C8—22no—
48C47⋊#95—22no—
49C47⋊#95—21no—
50C47⋊#95—22no—
51C47⋊#95C2312yes—
52C2×#41×C47—22no—
53C2×#41×C47—22no—
54C2×#41×C47—22no—
55C2×#41×C47—22no—
56C2×#41×C47—44no—
57C2×#41×C47—44no—
58C22×Q8×C47—21no—
59C22×D4×C47—21no—
60C22×D4×C47—22no—
61C23×C4×C47—22no—
62C23×C4×C47D411yescharacteristic
63C4⋊Q8×C47—21no—
64C4⋊Q8×C47D412yes—
65#127×C47—21no—
66#127×C47D412yes—
67#124×C47—22no—
68#124×C47—22no—
69#123×C47—24no—
70#123×C47—24no—
71C22⋊Q8×C47—24no—
72C22⋊Q8×C47—24no—
73#121×C47—24no—
74#121×C47—24no—
75#121×C47—22no—
76#121×C47—22no—
77#121×C47—22no—
78#121×C47C2314yes—
79#120×C47—24no—
80#120×C47—24no—
81C4×D4×C47—22no—
82C4×D4×C47—22no—
83C4×D4×C47—22no—
84C4×D4×C47D414yes—
85#117×C47—22no—
86#117×C47—22no—
87C2×C4⋊C4×C47D411yescharacteristic
88C2×C4⋊C4×C47C2311yescharacteristic
89C2×C22⋊C4×C47D411yescharacteristic
90C2×C22⋊C4×C47C2311yescharacteristic
91C2×C42×C47D411yescharacteristic
92C2×C42×C47C2311yescharacteristic
93C24×C47—22no—
94C24×C47C2×D411yescharacteristic
95#41×C47—44no—
96#41×C47—48no—
97#41×C47—48no—
98#41×C47—44no—
99#41×C47—44no—
100#41×C47—44no—
101C2×Q8×C47—22no—
102C2×Q8×C47—22no—
103C2×Q8×C47—44no—
104C2×D4×C47—24no—
105C2×D4×C47—22no—
106C2×D4×C47—24no—
107C2×D4×C47—22no—
108C2×D4×C47—22no—
109C2×D4×C47—22no—
110C2×D4×C47—22no—
111C2×D4×C47—44no—
112C2×D4×C47—44no—
113C2×D4×C47—44no—
114C2×D4×C47C2×D414yes—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—22no—
118C22×C4×C47—24no—
119C22×C4×C47—22no—
120C22×C4×C47—24no—
121C22×C4×C47—48no—
122C22×C4×C47SD1614yescomplemented
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2411yescharacteristic
129C22×Dic47—22no—
130C22×Dic47—22no—
131C2×Dic94—24no—
132C2×Dic94—82no—
133C4⋊C4×C47—22no—
134C4⋊C4×C47—22no—
135C4⋊C4×C47—22no—
136C4⋊C4×C47—22no—
137C4⋊C4×C47—22no—
138C4⋊C4×C47C2×D414yes—
139C22⋊C4×C47—24no—
140C22⋊C4×C47—24no—
141C22⋊C4×C47—22no—
142C22⋊C4×C47—24no—
143C22⋊C4×C47—22no—
144C22⋊C4×C47—24no—
145C42×C47—21no—
146C42×C47—21no—
147C42×C47C2×D412yes—
148C42×C47C2×D412yes—
149C22⋊Dic47—82no—
150#10625335—88no—
151C4⋊Dic47—42no—
152C4⋊Dic47—41no—
153C4⋊Dic47—41no—
154C4⋊Dic47—42no—
155Dic47⋊C4—82no—
156C2×C47⋊C8—24no—
157C23×C47—21no—
158C23×C47—21no—
159C23×C47—24no—
160C23×C47—41no—
161C23×C47—41no—
162C23×C47—44no—
163C23×C47—82no—
164C23×C47—82no—
165C23×C47C2×SD1612yes—
166C23×C47C22×D411yescharacteristic
167Q8×C47—44no—
168Q8×C47—44no—
169D4×C47—44no—
170D4×C47—44no—
171D4×C47—48no—
172D4×C47—44no—
173D4×C47—44no—
174D4×C47—44no—
175D4×C47—82no—
176D4×C47—82no—
177D4×C47—82no—
178C2×C4×C47—21no—
179C2×C4×C47—21no—
180C2×C4×C47—24no—
181C2×C4×C47—24no—
182C2×C4×C47—21no—
183C2×C4×C47—21no—
184C2×C4×C47—24no—
185C2×C4×C47—41no—
186C2×C4×C47—41no—
187C2×C4×C47—48no—
188C2×C4×C47—44no—
189C2×C4×C47—48no—
190C2×C4×C47—41no—
191C2×C4×C47—41no—
192C2×C4×C47#12012yes—
193C2×C4×C47#12011yescharacteristic
194C2×C4×C47#12011yescharacteristic
195C2×C4×C47C2×SD1612yes—
196C2×C4×C47C2×SD1612yes—
197C2×C4×C47AGL(1,ℤ/8)12yes—
198C2×C4×C47C22×D412yes—
199C2×Dic47—42no—
200C2×Dic47—42no—
201C2×Dic47—42no—
202C2×Dic47—42no—
203C2×Dic47—42no—
204C2×Dic47—42no—
205Dic94—48no—
206C47⋊C8—44no—
207#18301—471nomaximal, Sylow
208C22×C47—21no—
209C22×C47—21no—
210C22×C47—22no—
211C22×C47—42no—
212C22×C47—41no—
213C22×C47—41no—
214C22×C47—44no—
215C22×C47—44no—
216C22×C47—44no—
217C22×C47—81no—
218C22×C47—88no—
219C22×C47C2×#12011yescharacteristic
220C22×C47C22×SD1611yescharacteristic
221C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
222C4×C47—22no—
223C4×C47—22no—
224C4×C47—22no—
225C4×C47—44no—
226C4×C47—41no—
227C4×C47—41no—
228C4×C47#45012yes—
229C4×C47#45012yes—
230Dic47—82no—
231Dic47—82no—
232#3449D4711yescharacteristic
233#3206—472no—
234#2628—471no—
235#1604—474no—
236#1601—474no—
237#1517—472no—
238#1458—471no—
239C2×C47—21no—
240C2×C47—21no—
241C2×C47—42no—
242C2×C47—84no—
243C2×C47C2×#45011yescharacteristic
244C2×C47#302211yescharacteristic
245C2×C47#302711yescharacteristic
246C22×#41—21no—
247#537—21no—
248#537C2×D4712yes—
249#536—22no—
250#535—22no—
251#534—21no—
252#534C2×D4712yes—
253#532—21no—
254#532C2×D4712yes—
255C2×C4⋊Q8—471no—
256#525—24no—
257C2×#117C2×D4711yescharacteristic
258#492—942no—
259#487—942no—
260#461—944no—
261#448—478no—
262#431—472no—
263#400—941no—
264#399—941no—
265#395—472no—
266#393—472no—
267#389—941no—
268#382—471no—
269#380—941no—
270#327—478no—
271#324—472no—
272#323—472no—
273C47#1830111yescharacteristic, complemented, Sylow
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—22no—
278C2×#41—44no—
279C2×#41—44no—
280C22×Q8—21no—
281C22×Q8—472no—
282C22×D4—21no—
283C22×D4—22no—
284C23×C4—22no—
285C23×C4C22⋊D4711yescharacteristic
286C2×SD16—948no—
287C4⋊Q8—21no—
288C4⋊Q8—942no—
289C4⋊Q8—942no—
290C4⋊Q8C22⋊D4712yes—
291#127—21no—
292#127C22⋊D4712yes—
293#124—22no—
294#124—22no—
295#123—24no—
296#123—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299#121—24no—
300#121—24no—
301#121—22no—
302#121—22no—
303#121—22no—
304#121C22×D4714yes—
305#120—24no—
306#120—24no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4C22⋊D4714yes—
311#117—22no—
312#117—22no—
313C2×C4⋊C4—942no—
314C2×C4⋊C4—941no—
315C2×C4⋊C4—941no—
316C2×C4⋊C4—941no—
317C2×C4⋊C4C22⋊D4711yescharacteristic
318C2×C4⋊C4C22×D4711yescharacteristic
319C2×C22⋊C4—942no—
320C2×C22⋊C4C22⋊D4711yescharacteristic
321C2×C22⋊C4C22×D4711yescharacteristic
322C2×C42C22⋊D4711yescharacteristic
323C2×C42C22×D4711yescharacteristic
324C4⋊C8—942no—
325#103—948no—
326C22⋊C8—474no—
327C4×C8—942no—
328#95—472no—
329#95—942no—
330#95—941no—
331#95—942no—
332C24—22no—
333C24C2×C22⋊D4711yescharacteristic
334#41—44no—
335#41—48no—
336#41—48no—
337#41—44no—
338#41—44no—
339#41—44no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—44no—
343C2×Q8—944no—
344C2×Q8—3762no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4C2×C22⋊D4714yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—48no—
363C22×C4—942no—
364C22×C4—942no—
365C22×C4#1062533514yescomplemented
366C22×C4C2×C22⋊D4711yescharacteristic
367C22×C4C2×C22⋊D4711yescharacteristic
368C22×C4C2×C22⋊D4711yescharacteristic
369C22×C4C2×C22⋊D4711yescharacteristic
370C22×C4C2×C22⋊D4711yescharacteristic
371C22×C4C23×D4711yescharacteristic
372SD16—3768no—
373C2×C8—944no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—3762no—
380C4⋊C4—1882no—
381C4⋊C4—1881no—
382C4⋊C4—1881no—
383C4⋊C4—1882no—
384C4⋊C4C2×C22⋊D4714yes—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—22no—
390C22⋊C4—24no—
391C22⋊C4—3762no—
392C42—21no—
393C42—21no—
394C42C2×C22⋊D4712yes—
395C42C2×C22⋊D4712yes—
396C23—21no—
397C23—21no—
398C23—24no—
399C23—41no—
400C23—41no—
401C23—44no—
402C23—82no—
403C23—82no—
404C23C2×#1062533512yes—
405C23C22×C22⋊D4711yescharacteristic
406Q8—44no—
407Q8—44no—
408Q8—1888no—
409D4—44no—
410D4—44no—
411D4—48no—
412D4—44no—
413D4—44no—
414D4—44no—
415D4—82no—
416D4—82no—
417D4—82no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—24no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—1882no—
433C2×C4—1882no—
434C2×C4—1882no—
435C2×C4—1882no—
436C2×C4—1882no—
437C2×C4—1882no—
438C2×C4#1309399312yes—
439C2×C4#1309399311yescharacteristic
440C2×C4#1309399311yescharacteristic
441C2×C4C2×#1062533512yes—
442C2×C4C2×#1062533512yes—
443C2×C4#1309405012yes—
444C2×C4C22×C22⋊D4712yes—
445C8—1884no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—42no—
450C22—41no—
451C22—41no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—81no—
456C22—88no—
457C22C2×#1309399311yescharacteristic
458C22C22×#1062533511yescharacteristic
459C22C2×#1309405011yescharacteristic
460C4—22no—
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—44no—
466C4—3762no—
467C4—3762no—
468C4#1568927412yes—
469C4#1568927412yes—
470C2—21no—
471C2—21no—
472C2—42no—
473C2—84no—
474C2C2×#1568927411yescharacteristic
475C2#2025276711yescharacteristic
476C2#2025279011yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#3449C2
socleC22×C47C2×#120
hypercenter#3449D47
hypocenterC47#18301
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #20254310C2semidirect
2. #20249209C2semidirect
3. #20242357C2semidirect
4. #20242340C2semidirect
5. #20241948C2semidirect
6. #20241702C2semidirect
8. C2×#13094024C22semidirect
12. #15689261C22semidirect
13. #15689176C22semidirect
16. #15689043C22semidirect
19. #15688989C22semidirect
21. #15688857C22semidirect
22. #15688849C22semidirect
122. C22×C4×C47SD16semidirect
273. C47#18301semidirect
365. C22×C4#10625335semidirect

Maximal subgroups

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

subgroupindex
#202543102
#202492092
#202423572
#202423402
#202419482
#202417022
C47×#34492
#1830147