Finite Group Databasebeta

← #26220699

Subgroups — #26220699

Order 12032.

730 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252186C211yesmaximal, characteristic
2#20252033C211yesmaximal, characteristic, complemented
3#20252032C211yesmaximal, characteristic, complemented
4#20244278C211yesmaximal, characteristic
5#20244273C211yesmaximal, characteristic
6#20244140C211yesmaximal, characteristic, complemented
7#20244135C211yesmaximal, characteristic, complemented
8#20244100C211yesmaximal, characteristic, complemented
9#20244096C211yesmaximal, characteristic, complemented
10#20244093C211yesmaximal, characteristic, complemented
11#20244089C211yesmaximal, characteristic, complemented
12#20243295C211yesmaximal, characteristic, complemented
13#20243199C211yesmaximal, characteristic, complemented
14#20243075C211yesmaximal, characteristic, complemented
15C47×#1759C211yesmaximal, characteristic, complemented
16C2×#13094044C2211yescharacteristic
17C4×#10625350C2211yescharacteristic
18#15689233C412yescomplemented
19#15689228C412yescomplemented
20#15689223C412yescomplemented
21#15689221C412yescomplemented
22#15689185—22no—
23#15689184—22no—
24#15689176—22no—
25#15689174—22no—
26C2×#13093959C2211yescharacteristic
27C2×#13093959C2211yescharacteristic
28#15689159C2211yescharacteristic
29#15689151C2211yescharacteristic
30#15689151C2211yescharacteristic
31#15689150C2211yescharacteristic
32#15689150C2211yescharacteristic
33C2×#13093952C2211yescharacteristic
34C2×#13093952C2211yescharacteristic
35#15689119C2211yescharacteristic
36#15689117C2211yescharacteristic
37#15689091C2211yescharacteristic
38#15689091C2211yescharacteristic
39#15689090C2211yescharacteristic
40#15689090C2211yescharacteristic
41C4×#10625329C2211yescharacteristic
42C2×#13093935C2211yescharacteristic
43C2×#13093935C2211yescharacteristic
44C2×#13093935C2211yescharacteristic
45C2×C4×C47⋊C8C2211yescharacteristic
46C2×C4×C47⋊C8C2211yescharacteristic
47C2×C4×C47⋊C8C2211yescharacteristic
48#15688887C2211yescharacteristic
49#15688887C2211yescharacteristic
50#15688883C2211yescharacteristic
51#15688883C2211yescharacteristic
52C4×#41×C47C2211yescharacteristic
53C2×#104×C47C2211yescharacteristic
54C2×#104×C47C2211yescharacteristic
55C4×M16×C47C2211yescharacteristic
56C47×#376C2211yescharacteristic
57C47×#339C2211yescharacteristic
58C47×#339C2211yescharacteristic
59C2×#10625350C2311yescharacteristic
60#13094044—22no—
61#13094044—22no—
62C2×#10625337C2×C411yescharacteristic
63C2×#10625335C2×C411yescharacteristic
64C2×#10625336C2×C411yescharacteristic
65C2×#10625334C2×C411yescharacteristic
66C2×#10625329C2311yescharacteristic
67C2×#10625329C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72C4×Dic94C2×C412yes—
73C4×D188C2×C412yes—
74C4×C22⋊D47—22no—
75#13093983C2311yescharacteristic
76#13093981—21no—
77C42×D47—21no—
78#13093959—22no—
79#13093959—22no—
80#13093957C2×C412yes—
81#13093955C2×C412yes—
82C4⋊(C47⋊C8)—22no—
83C4⋊(C47⋊C8)—22no—
84Dic47⋊C8—22no—
85Dic47⋊C8—22no—
86#13093952—22no—
87#13093952—22no—
88#13093948C2×C412yes—
89#13093947C2×C412yes—
90#13093945C2×C412yes—
91#13093944C2×C412yes—
92C22⋊(C47⋊C8)—22no—
93C22⋊(C47⋊C8)—22no—
94(C2×D47)⋊C8—22no—
95(C2×D47)⋊C8—22no—
96#13093935—21no—
97#13093935—22no—
98#13093935—22no—
99#13093935C2×C412yes—
100#13093935C2×C412yes—
101C4×C47⋊C8—21no—
102C4×C47⋊C8—22no—
103C4×C47⋊C8—22no—
104C4×C47⋊C8D412yes—
105C4×C47⋊C8D412yes—
106C2×#41×C47C2311yescharacteristic
107C2×M16×C47C2311yescharacteristic
108C2×M16×C47C2311yescharacteristic
109C4×Q8×C47C2×C412yes—
110C4×D4×C47—22no—
111C4×D4×C47C2×C412yes—
112#117×C47—21no—
113#117×C47C2311yescharacteristic
114C2×C42×C47—21no—
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117C2×C42×C47C2311yescharacteristic
118#104×C47—22no—
119#104×C47—22no—
120#97×C47—21no—
121C4×C8×C47—21no—
122#95×C47—22no—
123#41×C47—21no—
124#41×C47—21no—
125#41×C47—21no—
126#41×C47—21no—
127C2×Q8×C47C22×C411yescharacteristic
128C2×D4×C47—21no—
129C2×D4×C47C22×C411yescharacteristic
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—22no—
133C22×C4×C47—21no—
134C22×C4×C47#4112yes—
135C22×C4×C47C2411yescharacteristic
136C2×C22⋊D47—21no—
137#10625350—21no—
138#10625350—21no—
139#10625350—21no—
140#10625350—21no—
141C2×D188C22×C411yescharacteristic
142C2×C4×D47—22no—
143C2×C4×D47—21no—
144C2×Dic94C22×C411yescharacteristic
145M16×C47—21no—
146M16×C47—21no—
147M16×C47—21no—
148M16×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47—21no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47C22×C412yes—
153C22⋊C4×C47—21no—
154C22⋊C4×C47—22no—
155C42×C47—22no—
156C42×C47—21no—
157C42×C47—21no—
158C42×C47—21no—
159C42×C47C22×C412yes—
160C42×C47C2×D411yescharacteristic
161C42×C47C2×D411yescharacteristic
162C42×C47#4112yes—
163C42×C47#4112yes—
164C22⋊Dic47—21no—
165#10625337—22no—
166#10625336—22no—
167#10625335—22no—
168#10625334—22no—
169(C2×D47)⋊C4—22no—
170C4⋊Dic47C22×C412yes—
171Dic47⋊C4—22no—
172C4×Dic47—22no—
173C4×Dic47—21no—
174#10625329—21no—
175#10625329—21no—
176#10625329—21no—
177#10625329—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8C22×C412yes—
187C2×C47⋊C8C22×C411yescharacteristic
188C2×C47⋊C8C22×C411yescharacteristic
189C2×C47⋊C8C2×D412yes—
190C2×C47⋊C8C2×D412yes—
191C2×C47⋊C8#4112yes—
192C2×C47⋊C8#4112yes—
193C23×C47—21no—
194C23×C47C2×#4111yescharacteristic
195C22×D47—21no—
196Q8×C47—21no—
197Q8×C47—21no—
198D4×C47—21no—
199D4×C47—21no—
200D4×C47—41no—
201D4×C47—41no—
202C2×C4×C47—22no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—42no—
207C2×C4×C47—44no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47C4×D412yes—
213C2×C4×C47C4×D412yes—
214C2×C4×C47C23×C411yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C22⋊D47—41no—
221C22⋊D47—41no—
222C2×Dic47—21no—
223C2×Dic47—22no—
224D188—21no—
225D188—21no—
226C4×D47—44no—
227C4×D47—41no—
228C4×D47—41no—
229Dic94—21no—
230Dic94—21no—
231C8×C47—41no—
232C8×C47—41no—
233C47⋊C8—22no—
234C47⋊C8—22no—
235C47⋊C8—22no—
236C47⋊C8—41no—
237C47⋊C8—41no—
238C47⋊C8#11712yes—
239C47⋊C8#11712yes—
240#21296—471nomaximal, Sylow
241C22×C47—22no—
242C22×C47—42no—
243C22×C47—41no—
244C22×C47C2×C4×D411yescharacteristic
245C22×C47#52911yescharacteristic
246C22×C47#52911yescharacteristic
247C2×D47—42no—
248C2×D47—41no—
249C4×C47—22no—
250C4×C47—22no—
251C4×C47—42no—
252C4×C47—42no—
253C4×C47—42no—
254C4×C47—41no—
255C4×C47C2×#11711yescharacteristic
256C4×C47C4×#4111yescharacteristic
257C4×C47#53211yescharacteristic
258C4×C47#53211yescharacteristic
259Dic47—42no—
260Dic47—41no—
261#2943—471no—
262#2922—471no—
263#2922—471no—
264#1975—471no—
265#1974—471no—
266#1946—471no—
267#1945—471no—
268#1938—471no—
269#1938—471no—
270#1937—471no—
271#1937—471no—
272#1778—471no—
273#1759—471no—
274#1759D4711yescharacteristic, complemented
275#1726—471no—
276C2×C47—21no—
277C2×C47—21no—
278C2×C47—81no—
279C2×C47#229311yescharacteristic
280C2×C47#295512yes—
281D47—81no—
282C2×#135—471no—
283C4×#41—471no—
284C4×#41C2×D4711yescharacteristic
285#442—472no—
286#441—472no—
287#440—472no—
288#440—472no—
289#433—942no—
290#433—942no—
291#431—942no—
292#431—942no—
293C2×#108—471no—
294C2×#108—471no—
295#427—471no—
296#424—471no—
297#424—471no—
298#424—471no—
299#424—471no—
300C2×#104—471no—
301C2×#104—471no—
302C2×#104C2×D4711yescharacteristic
303C2×#104C2×D4711yescharacteristic
304#416—471no—
305#416—471no—
306#408—471no—
307#408—471no—
308#408—471no—
309#408—471no—
310C4×M16—471no—
311C4×M16C2×D4711yescharacteristic
312C2×#97—471no—
313C2×#97—471no—
314C2×#97—471no—
315C2×C4×C8—471no—
316C2×C4×C8—471no—
317C2×C4×C8—471no—
318#376C2×D4711yescharacteristic
319#341—471no—
320#341—471no—
321#339—471no—
322#339—471no—
323#339C2×D4711yescharacteristic
324#339C2×D4711yescharacteristic
325C47#2129611yescharacteristic, complemented, Sylow
326C2×#41—471no—
327C2×#41C22×D4711yescharacteristic
328#135—942no—
329#135—942no—
330C2×Q16—471no—
331C2×SD16—471no—
332C2×SD16—471no—
333C2×D8—471no—
334C2×M16—471no—
335C2×M16—471no—
336C2×M16C22×D4711yescharacteristic
337C2×M16C22×D4711yescharacteristic
338C22×C8—471no—
339C22×C8—471no—
340C22×C8—471no—
341C22×C8—471no—
342C4×Q8—472no—
343C4×Q8C4×D4712yes—
344C4×D4—22no—
345C4×D4—472no—
346C4×D4—942no—
347C4×D4C4×D4712yes—
348#117—21no—
349#117—471no—
350#117—941no—
351#117C22×D4711yescharacteristic
352C2×C42—21no—
353C2×C42—941no—
354C2×C42C22×D4711yescharacteristic
355C2×C42C22×D4711yescharacteristic
356C2×C42C22×D4711yescharacteristic
357#108—942no—
358#108—942no—
359#107—472no—
360#106—472no—
361C4⋊C8—942no—
362C4⋊C8—942no—
363C4⋊C8—942no—
364C4⋊C8—942no—
365#104—22no—
366#104—22no—
367#104—942no—
368#104—942no—
369#103—472no—
370#103—472no—
371#102—472no—
372#102—472no—
373C22⋊C8—942no—
374C22⋊C8—942no—
375C22⋊C8—942no—
376C22⋊C8—942no—
377#97—21no—
378#97—472no—
379#97—472no—
380#97—942no—
381#97—942no—
382#97—941no—
383C4×C8—21no—
384C4×C8—472no—
385C4×C8—472no—
386C4×C8—942no—
387C4×C8—942no—
388C4×C8—941no—
389#95—22no—
390#41—21no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—941no—
395#41—941no—
396#41—941no—
397#41—941no—
398C2×Q8—471no—
399C2×Q8C2×C4×D4711yescharacteristic
400C2×D4—21no—
401C2×D4—471no—
402C2×D4—941no—
403C2×D4C2×C4×D4711yescharacteristic
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—942no—
409C22×C4—941no—
410C22×C4#1062535212yes—
411C22×C4C23×D4711yescharacteristic
412Q16—942no—
413SD16—942no—
414SD16—942no—
415D8—942no—
416M16—21no—
417M16—21no—
418M16—21no—
419M16—21no—
420M16—941no—
421M16—941no—
422M16—941no—
423M16—941no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—471no—
427C2×C8—471no—
428C2×C8—472no—
429C2×C8—472no—
430C2×C8—472no—
431C2×C8—472no—
432C2×C8—472no—
433C2×C8—941no—
434C2×C8—941no—
435C2×C8—941no—
436C2×C8—941no—
437C2×C8—941no—
438C2×C8—941no—
439C2×C8—941no—
440C2×C8—941no—
441C4⋊C4—22no—
442C4⋊C4—472no—
443C4⋊C4—942no—
444C4⋊C4C2×C4×D4712yes—
445C22⋊C4—21no—
446C22⋊C4—22no—
447C22⋊C4—941no—
448C22⋊C4—942no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—942no—
454C42—941no—
455C42C2×C4×D4712yes—
456C42#1062535012yes—
457C42#1062535012yes—
458C42D4×D4711yescharacteristic
459C42D4×D4711yescharacteristic
460C23—21no—
461C23—941no—
462C23C2×#1062535211yescharacteristic
463Q8—21no—
464Q8—21no—
465Q8—941no—
466Q8—941no—
467D4—21no—
468D4—21no—
469D4—41no—
470D4—41no—
471D4—941no—
472D4—941no—
473D4—1881no—
474D4—1881no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—942no—
486C2×C4—941no—
487C2×C4—1884no—
488C2×C4—1881no—
489C2×C4—1881no—
490C2×C4(C22⋊D47)⋊C412yes—
491C2×C4(C22⋊D47)⋊C412yes—
492C2×C4C22×C4×D4711yescharacteristic
493C2×C4C2×D4×D4711yescharacteristic
494C2×C4#41×D4711yescharacteristic
495C2×C4#41×D4711yescharacteristic
496C2×C4C2×#1062535011yescharacteristic
497C2×C4C2×#1062535011yescharacteristic
498C8—41no—
499C8—41no—
500C8—472no—
501C8—472no—
502C8—942no—
503C8—942no—
504C8—942no—
505C8—1881no—
506C8—1881no—
507C22—22no—
508C22—41no—
509C22—42no—
510C22—1881no—
511C22—1882no—
512C22C2×(C22⋊D47)⋊C411yescharacteristic
513C22#1568965311yescharacteristic
514C22#1568965311yescharacteristic
515C4—22no—
516C4—22no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—42no—
521C4—1881no—
522C4—1882no—
523C4#117×D4711yescharacteristic
524C4C4×#1062535011yescharacteristic
525C4#1568967011yescharacteristic
526C4#1568967011yescharacteristic
527C2—21no—
528C2—21no—
529C2—81no—
530C2—3761no—
531C2#2024585111yescharacteristic
532C2#2025228712yes—
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1759C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094044C22
cosocleC22×C4×C47C24
hypercenter#1759D47
hypocenterC47#21296
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
2. #20252033C2semidirect
3. #20252032C2semidirect
6. #20244140C2semidirect
7. #20244135C2semidirect
8. #20244100C2semidirect
9. #20244096C2semidirect
10. #20244093C2semidirect
11. #20244089C2semidirect
12. #20243295C2semidirect
13. #20243199C2semidirect
14. #20243075C2semidirect
15. C47×#1759C2semidirect
18. #15689233C4semidirect
19. #15689228C4semidirect
20. #15689223C4semidirect
21. #15689221C4semidirect
274. #1759D47semidirect
325. C47#21296semidirect

Maximal subgroups

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

subgroupindex
#202521862
#202520332
#202520322
#202442782
#202442732
#202441402
#202441352
#202441002
#202440962
#202440932
#202440892
#202432952
#202431992
#202430752
C47×#17592
#2129647