Finite Group Databasebeta

← #26107871

Subgroups — #26107871

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243332C212yesmaximal, complemented
2C4×#13093939C211yesmaximal, characteristic, complemented
3#20241904C212yesmaximal, complemented
4#20241726C211yesmaximal, characteristic, complemented
5C2×C47×#550C211yesmaximal, characteristic
6C2×#13093951—24no—
7C2×#13093939—21no—
8#15689087C2212yes—
9C4×C22⋊Dic47C2212yes—
10#15688890—21no—
11#15688857—24no—
12#15688849—21no—
13#15688847—21no—
14C22×#41×C47C411yescharacteristic, complemented
15C22×#41×C47C2211yescharacteristic
16C47×#550—24no—
17C47×#550—24no—
18C2×#127×C47—21no—
19C2×#127×C47C411yescharacteristic, complemented
20C2×#121×C47—21no—
21C2×#121×C47—21no—
22C2×#121×C47C411yescharacteristic, complemented
23C2×#121×C47C2211yescharacteristic
24C2×C4×Q8×C47C411yescharacteristic, complemented
25C2×C4×D4×C47—21no—
26C2×C4×D4×C47C2211yescharacteristic
27C2×#10625329—22no—
28C2×C22⋊Dic47—22no—
29C2×C4×Dic47—22no—
30#13093951—48no—
31#13093939—24no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C47⋊#95—22no—
35C2×#41×C47—22no—
36C2×#41×C47—22no—
37C2×#41×C47—22no—
38C2×#41×C47—44no—
39C2×#41×C47—44no—
40C2×#41×C47D414yes—
41C22×Q8×C47C2×C411yescharacteristic
42C22×D4×C47—21no—
43C22×D4×C47—21no—
44C22×D4×C47—21no—
45C22×D4×C47—21no—
46C22×D4×C47—21no—
47C22×D4×C47—21no—
48C22×D4×C47C2×C411yescharacteristic
49C22×D4×C47C2×C411yescharacteristic
50C22×D4×C47D411yescharacteristic
51C23×C4×C47—21no—
52C23×C4×C47—21no—
53C23×C4×C47C2×C411yescharacteristic
54C23×C4×C47C2311yescharacteristic
55#127×C47—28no—
56#127×C47—22no—
57#127×C47—22no—
58#121×C47—28no—
59#121×C47—28no—
60#121×C47—22no—
61#121×C47—22no—
62#121×C47—22no—
63#121×C47D414yes—
64C4×Q8×C47—22no—
65C4×Q8×C47—22no—
66C4×D4×C47—28no—
67C4×D4×C47—22no—
68C4×D4×C47—22no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47C2×C411yescharacteristic
71C2×C4⋊C4×C47D411yescharacteristic
72C2×C22⋊C4×C47—21no—
73C2×C22⋊C4×C47—21no—
74C2×C22⋊C4×C47D411yescharacteristic
75C2×C22⋊C4×C47D411yescharacteristic
76C2×C42×C47—21no—
77C2×C42×C47C2×C411yescharacteristic
78C24×C47—21no—
79C24×C47—21no—
80C24×C47C22⋊C411yescharacteristic
81C24×C47C2×D411yescharacteristic
82#41×C47—28no—
83#41×C47—44no—
84#41×C47—48no—
85#41×C47—48no—
86#41×C47—44no—
87#41×C47—44no—
88C2×Q8×C47—24no—
89C2×Q8×C47—21no—
90C2×Q8×C47C22⋊C412yes—
91C2×D4×C47—24no—
92C2×D4×C47—24no—
93C2×D4×C47—22no—
94C2×D4×C47—24no—
95C2×D4×C47—24no—
96C2×D4×C47—24no—
97C2×D4×C47—24no—
98C2×D4×C47—22no—
99C2×D4×C47—21no—
100C2×D4×C47—21no—
101C2×D4×C47—42no—
102C2×D4×C47—41no—
103C2×D4×C47—42no—
104C2×D4×C47—44no—
105C2×D4×C47—44no—
106C2×D4×C47—42no—
107C2×D4×C47—42no—
108C2×D4×C47—82no—
109C2×D4×C47—82no—
110C2×D4×C47—82no—
111C2×D4×C47C22⋊C412yes—
112C2×D4×C47C22⋊C412yes—
113C22×C4×C47—21no—
114C22×C4×C47—24no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—24no—
119C22×C4×C47—42no—
120C22×C4×C47—44no—
121C22×C4×C47—44no—
122C22×C4×C47—42no—
123C22×C4×C47C22⋊C411yescharacteristic
124C22×C4×C47C22⋊C411yescharacteristic
125C22×C4×C47C22⋊C411yescharacteristic
126C22×C4×C47C22×C411yescharacteristic
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×Dic47—42no—
131C4⋊C4×C47—24no—
132C4⋊C4×C47—21no—
133C4⋊C4×C47—22no—
134C4⋊C4×C47C22⋊C412yes—
135C22⋊C4×C47—24no—
136C22⋊C4×C47—22no—
137C22⋊C4×C47—24no—
138C22⋊C4×C47—22no—
139C42×C47—24no—
140C42×C47—21no—
141C42×C47—21no—
142C22⋊Dic47—24no—
143C22⋊Dic47—42no—
144C4×Dic47—44no—
145#10625329—44no—
146C2×C47⋊C8—42no—
147C23×C47—21no—
148C23×C47—21no—
149C23×C47—21no—
150C23×C47—22no—
151C23×C47—22no—
152C23×C47—21no—
153C23×C47—21no—
154C23×C47—41no—
155C23×C47—41no—
156C23×C47—42no—
157C23×C47—42no—
158C23×C47—42no—
159C23×C47—42no—
160C23×C47—41no—
161C23×C47—41no—
162C23×C47—41no—
163C23×C47—41no—
164C23×C47—41no—
165C23×C47—41no—
166C23×C47C2×C22⋊C411yescharacteristic
167C23×C47#12012yes—
168Q8×C47—24no—
169Q8×C47—22no—
170Q8×C47#9914yescomplemented
171D4×C47—24no—
172D4×C47—44no—
173D4×C47—48no—
174D4×C47—42no—
175D4×C47—48no—
176D4×C47—48no—
177D4×C47—42no—
178D4×C47—42no—
179D4×C47—84no—
180D4×C47—84no—
181D4×C47—84no—
182C2×C4×C47—24no—
183C2×C4×C47—21no—
184C2×C4×C47—22no—
185C2×C4×C47—21no—
186C2×C4×C47—24no—
187C2×C4×C47—22no—
188C2×C4×C47—21no—
189C2×C4×C47—22no—
190C2×C4×C47—21no—
191C2×C4×C47—41no—
192C2×C4×C47—42no—
193C2×C4×C47—48no—
194C2×C4×C47—48no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47#10412yescomplemented
201C2×C4×C47C2×C22⋊C411yescharacteristic
202C2×C4×C47C2×C22⋊C411yescharacteristic
203C2×C4×C47#12012yes—
204C2×Dic47—44no—
205C2×Dic47—82no—
206C2×Dic47—82no—
207C47⋊C8—82no—
208#8361—471nomaximal, Sylow
209C22×C47—21no—
210C22×C47—21no—
211C22×C47—22no—
212C22×C47—22no—
213C22×C47—22no—
214C22×C47—41no—
215C22×C47—42no—
216C22×C47—42no—
217C22×C47—42no—
218C22×C47—41no—
219C22×C47—42no—
220C22×C47—41no—
221C22×C47—42no—
222C22×C47—42no—
223C22×C47—81no—
224C22×C47—81no—
225C22×C47—81no—
226C22×C47—81no—
227C22×C47—82no—
228C22×C47—82no—
229C22×C47—82no—
230C22×C47—82no—
231C22×C47#37911yescharacteristic
232C22×C47C2×#10411yescharacteristic
233C22×C47#42111yescharacteristic
234C4×C47—22no—
235C4×C47—24no—
236C4×C47—21no—
237C4×C47—22no—
238C4×C47—42no—
239C4×C47—41no—
240C4×C47—41no—
241C4×C47C2×#9912yes—
242Dic47—84no—
243C2×#550D4711yescharacteristic
244#1789—472no—
245C4×#99—471no—
246#1507—472no—
247#1465—471no—
248C2×C47—21no—
249C2×C47—21no—
250C2×C47—42no—
251C2×C47—81no—
252C2×C47—82no—
253C2×C47—82no—
254C2×C47#178111yescharacteristic
255C2×C47#178911yescharacteristic
256C2×C47#179511yescharacteristic
257C22×#41Dic4711yescharacteristic, complemented
258C22×#41C2×D4711yescharacteristic
259#550—24no—
260#550—24no—
261C2×#127—21no—
262C2×#127Dic4711yescharacteristic, complemented
263C2×#121—21no—
264C2×#121—21no—
265C2×#121Dic4711yescharacteristic, complemented
266C2×#121C2×D4711yescharacteristic
267C2×C4×Q8Dic4711yescharacteristic, complemented
268C2×C4×D4—21no—
269C2×C4×D4C2×D4711yescharacteristic
270C2×#104—944no—
271C2×#99—941no—
272#407—472no—
273C4×C22⋊C4—472no—
274#342—941no—
275#327—944no—
276#324—941no—
277#323—941no—
278C47#836111yescharacteristic, complemented, Sylow
279C2×#41—22no—
280C2×#41—22no—
281C2×#41—22no—
282C2×#41—44no—
283C2×#41—44no—
284C2×#41C22⋊D4714yes—
285C22×Q8C2×Dic4711yescharacteristic
286C22×D4—21no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—21no—
290C22×D4—21no—
291C22×D4—21no—
292C22×D4C2×Dic4711yescharacteristic
293C22×D4C2×Dic4711yescharacteristic
294C22×D4C22⋊D4711yescharacteristic
295C23×C4—21no—
296C23×C4—21no—
297C23×C4C2×Dic4711yescharacteristic
298C23×C4C22×D4711yescharacteristic
299C2×M16—942no—
300#127—28no—
301#127—22no—
302#127—22no—
303#121—28no—
304#121—28no—
305#121—22no—
306#121—22no—
307#121—22no—
308#121C22⋊D4714yes—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×D4—28no—
312C4×D4—22no—
313C4×D4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4C2×Dic4711yescharacteristic
316C2×C4⋊C4C22⋊D4711yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—942no—
320C2×C22⋊C4C22⋊D4711yescharacteristic
321C2×C22⋊C4C22⋊D4711yescharacteristic
322C2×C42—21no—
323C2×C42—942no—
324C2×C42C2×Dic4711yescharacteristic
325#104—1888no—
326#99—944no—
327C22⋊C8—942no—
328C22⋊C8—942no—
329#95—942no—
330C24—21no—
331C24—21no—
332C24C22⋊Dic4711yescharacteristic
333C24C2×C22⋊D4711yescharacteristic
334#41—28no—
335#41—44no—
336#41—48no—
337#41—48no—
338#41—44no—
339#41—44no—
340C2×Q8—24no—
341C2×Q8—21no—
342C2×Q8C22⋊Dic4712yes—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—42no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—82no—
363C2×D4C22⋊Dic4712yes—
364C2×D4C22⋊Dic4712yes—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—1882no—
376C22×C4C22⋊Dic4711yescharacteristic
377C22×C4C22⋊Dic4711yescharacteristic
378C22×C4C22⋊Dic4711yescharacteristic
379C22×C4C22×Dic4711yescharacteristic
380C22×C4C2×C22⋊D4712yes—
381C22×C4C2×C22⋊D4712yes—
382C22×C4C2×C22⋊D4711yescharacteristic
383M16—1884no—
384C2×C8—1882no—
385C4⋊C4—24no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4C22⋊Dic4712yes—
389C22⋊C4—24no—
390C22⋊C4—22no—
391C22⋊C4—24no—
392C22⋊C4—22no—
393C22⋊C4—944no—
394C22⋊C4—1882no—
395C42—24no—
396C42—21no—
397C42—21no—
398C42—1884no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—22no—
403C23—22no—
404C23—21no—
405C23—21no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23C2×C22⋊Dic4711yescharacteristic
419C23#1309399312yes—
420Q8—24no—
421Q8—22no—
422Q8#1309393914yescomplemented
423D4—24no—
424D4—44no—
425D4—48no—
426D4—42no—
427D4—48no—
428D4—48no—
429D4—42no—
430D4—42no—
431D4—84no—
432D4—84no—
433D4—84no—
434C2×C4—24no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—3762no—
453C2×C4—3762no—
454C2×C4—1884no—
455C2×C4#1309395112yescomplemented
456C2×C4C2×C22⋊Dic4711yescharacteristic
457C2×C4C2×C22⋊Dic4711yescharacteristic
458C2×C4#1309399312yes—
459C8—3762no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—42no—
466C22—41no—
467C22—42no—
468C22—41no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—41no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22#1568897911yescharacteristic
483C22C2×#1309395111yescharacteristic
484C22#1568914011yescharacteristic
485C4—22no—
486C4—24no—
487C4—21no—
488C4—22no—
489C4—41no—
490C4—42no—
491C4—41no—
492C4—3764no—
493C4C2×#1309393912yes—
494C2—21no—
495C2—21no—
496C2—42no—
497C2—82no—
498C2—82no—
499C2—81no—
500C2#2024330311yescharacteristic
501C2#2024333211yescharacteristic
502C2#2024335811yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC2×C47×#550C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenterC2×#550D47
hypocenterC47#8361
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #20243332C2semidirect
2. C4×#13093939C2semidirect
3. #20241904C2semidirect
4. #20241726C2semidirect
14. C22×#41×C47C4semidirect
19. C2×#127×C47C4semidirect
22. C2×#121×C47C4semidirect
24. C2×C4×Q8×C47C4semidirect
170. Q8×C47#99semidirect
200. C2×C4×C47#104semidirect
257. C22×#41Dic47semidirect
262. C2×#127Dic47semidirect
265. C2×#121Dic47semidirect
267. C2×C4×Q8Dic47semidirect
278. C47#8361semidirect
422. Q8#13093939semidirect
455. C2×C4#13093951semidirect

Maximal subgroups

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

subgroupindex
#202433322
C4×#130939392
#202419042
#202417262
C2×C47×#5502
#836147