Finite Group Databasebeta

← #26274352

Subgroups — #26274352

Order 12032.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20247113C211yesmaximal, characteristic
2#20246275C211yesmaximal, characteristic
3#20244960C211yesmaximal, characteristic, complemented
4#20244891C211yesmaximal, characteristic, complemented
5#20244887C211yesmaximal, characteristic, complemented
6#20244724C211yesmaximal, characteristic, complemented
7#20244344C211yesmaximal, characteristic, complemented
8#20244153C211yesmaximal, characteristic, complemented
9#20243968C211yesmaximal, characteristic, complemented
10#20243964C211yesmaximal, characteristic, complemented
11#20243580C211yesmaximal, characteristic
12#20243571C211yesmaximal, characteristic
13#20243480C211yesmaximal, characteristic
14#20243108C211yesmaximal, characteristic
15C47×#1809C211yesmaximal, characteristic
16C2×#13094015C2211yescharacteristic
17C2×#13093983C2211yescharacteristic
18C22×C4⋊Dic47C2211yescharacteristic
19C2×#13093955C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×Dic47⋊C8C2211yescharacteristic
22C2×Dic47⋊C8C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24#15689065—21no—
25#15689055—21no—
26#15689022C2211yescharacteristic
27#15689019C2211yescharacteristic
28#15689019C2211yescharacteristic
29#15689019C2211yescharacteristic
30#15689017—21no—
31#15689017—21no—
32#15688998—21no—
33#15688988C2211yescharacteristic
34#15688987—21no—
35#15688981—21no—
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4×C22⋊Dic47C2211yescharacteristic
39#15688968—21no—
40#15688886C2211yescharacteristic
41#15688886C2211yescharacteristic
42#15688886C2211yescharacteristic
43#15688886C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52C22×M16×C47C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C47×#380C2211yescharacteristic
56C47×#380C2211yescharacteristic
57C47×#340C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C23×Dic47—21no—
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62#13094015—22no—
63#13094015—41no—
64#13093983—22no—
65#13093983Q814yes—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—41no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47D411yescharacteristic
81C2×C22⋊Dic47D411yescharacteristic
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47C2311yescharacteristic
86C2×C4×Dic47C2311yescharacteristic
87#13093955—22no—
88#13093955—41no—
89Dic47⋊C8—22no—
90Dic47⋊C8—41no—
91C22⋊(C47⋊C8)—22no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95D411yescharacteristic
101C47⋊#95D411yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C2×M16×C47—22no—
104C2×M16×C47—22no—
105C2×M16×C47—41no—
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114#107×C47—41no—
115#106×C47—22no—
116#95×C47—21no—
117#95×C47—21no—
118C24×C47C2×D411yescharacteristic
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×Q812yes—
129C22×C4×C47#4112yes—
130C22×C4×C47C2411yescharacteristic
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—42no—
136C22×Dic47—42no—
137C22×Dic47—81no—
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142M16×C47—44no—
143C2×C8×C47—21no—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47#4112yes—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C22⋊C4×C47—42no—
158C22⋊C4×C47—81no—
159C22⋊Dic47—24no—
160C22⋊Dic47—41no—
161C22⋊Dic47—41no—
162C4⋊Dic47—21no—
163C4⋊Dic47—21no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—42no—
168C4⋊Dic47—42no—
169C4⋊Dic47—81no—
170C4⋊Dic47C2×Q812yes—
171C4⋊Dic47C2×Q812yes—
172C4⋊Dic47#4112yes—
173Dic47⋊C4—42no—
174Dic47⋊C4—81no—
175Dic47⋊C4—81no—
176Dic47⋊C4—81no—
177C4×Dic47—24no—
178C4×Dic47—21no—
179C4×Dic47—21no—
180C4×Dic47—21no—
181C4×Dic47—21no—
182C4×Dic47—42no—
183C4×Dic47—42no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47C22⋊Q812yes—
197C23×C47C22×D411yescharacteristic
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47C22⋊Q812yes—
217C2×C4×C47C22×Q811yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47H(5,𝔽2)11yescharacteristic
222C2×C4×C472−1+411yescharacteristic
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—22no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—84no—
248C8×C47—22no—
249C8×C47—41no—
250C47⋊C8—41no—
251C47⋊C8—41no—
252#26019—471nomaximal, Sylow
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—41no—
257C22×C47—41no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47C2×C22⋊Q811yescharacteristic
261C22×C47#54611yescharacteristic
262C22×C47#54711yescharacteristic
263C22×C47#57511yescharacteristic
264C22×C47#57511yescharacteristic
265C22×C47#57711yescharacteristic
266C22×C47#57711yescharacteristic
267C4×C47—22no—
268C4×C47—81no—
269C4×C47—81no—
270C4×C47—81no—
271C4×C47—81no—
272C4×C47#53311yescharacteristic
273C4×C47#54411yescharacteristic
274C4×C47#55111yescharacteristic
275C4×C47#55211yescharacteristic
276Dic47—42no—
277Dic47—42no—
278Dic47—41no—
279Dic47—41no—
280Dic47—41no—
281Dic47—41no—
282Dic47—42no—
283Dic47—81no—
284#2442—471no—
285#2346—471no—
286#2101—471no—
287#2088—471no—
288#2088—471no—
289#2059—471no—
290#1989—471no—
291#1949—471no—
292#1919—471no—
293#1919—471no—
294#1853—471no—
295#1851—471no—
296#1825—471no—
297#1809D4711yescharacteristic
298#1736—471no—
299C2×C47—42no—
300C2×C47#243811yescharacteristic
301C2×C47#308011yescharacteristic
302C2×C47#308411yescharacteristic
303C2×C47#332111yescharacteristic
304C2×C47#332311yescharacteristic
305C2×C47#334811yescharacteristic
306C2×C47#335111yescharacteristic
307C22×M16C2×D4711yescharacteristic
308C2×#125—471no—
309C2×#117—471no—
310C22×C4⋊C4—471no—
311C2×#107C2×D4711yescharacteristic
312C2×#106—471no—
313C2×#106—471no—
314C2×#106C2×D4711yescharacteristic
315C2×C4⋊C8—471no—
316C2×C4⋊C8—471no—
317C2×C22⋊C8—471no—
318#400—941no—
319#398—941no—
320#389—471no—
321#389—471no—
322#389—471no—
323#389—471no—
324#388—941no—
325#388—941no—
326#385—941no—
327#382—471no—
328#382—941no—
329#380—941no—
330#380C2×D4711yescharacteristic
331#380C2×D4711yescharacteristic
332C4×C4⋊C4—471no—
333C4×C4⋊C4—471no—
334C4×C22⋊C4—471no—
335#376—941no—
336#340—471no—
337#340—471no—
338#340—471no—
339#340—471no—
340#340—471no—
341#340—471no—
342#340—471no—
343#340—471no—
344#340—471no—
345#340—471no—
346#340C2×D4711yescharacteristic
347#340C2×D4711yescharacteristic
348#336—471no—
349#336—471no—
350C47#2601911yescharacteristic, complemented, Sylow
351C23×C4—941no—
352C23×C4C22×D4711yescharacteristic
353C2×M16—22no—
354C2×M16—22no—
355C2×M16—41no—
356C22×C8—471no—
357C22×C8—471no—
358C22×C8C22×D4711yescharacteristic
359C22×C8C22×D4711yescharacteristic
360#125—942no—
361#125—1881no—
362#117—474no—
363#117—942no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—942no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—1881no—
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—471no—
384C2×C22⋊C4—471no—
385C2×C22⋊C4—941no—
386C2×C42—471no—
387C2×C42—471no—
388C2×C42—941no—
389C2×C42—941no—
390C2×C42—941no—
391#107—41no—
392#106—22no—
393#106—942no—
394#106—1881no—
395C4⋊C8—942no—
396C4⋊C8—1881no—
397C22⋊C8—942no—
398#95—21no—
399#95—21no—
400#95—471no—
401#95—471no—
402#95—941no—
403#95—941no—
404#95—941no—
405#95—941no—
406#95—941no—
407#95—941no—
408#95—941no—
409#95—941no—
410C24D4×D4711yescharacteristic
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—471no—
420C22×C4—471no—
421C22×C4—471no—
422C22×C4—471no—
423C22×C4—941no—
424C22×C4—941no—
425C22×C4—941no—
426C22×C4—941no—
427C22×C4—3761no—
428C22×C4—1882no—
429C22×C4—1882no—
430C22×C4#1062535012yes—
431C22×C4D4×D4711yescharacteristic
432C22×C4Q8×D4712yes—
433C22×C4C23×D4711yescharacteristic
434M16—44no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—941no—
441C2×C8—941no—
442C2×C8—941no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8#1062535212yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—472no—
456C4⋊C4—472no—
457C4⋊C4—472no—
458C4⋊C4—941no—
459C4⋊C4—941no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—3761no—
464C4⋊C4—3761no—
465C4⋊C4—3761no—
466C4⋊C4—3761no—
467C4⋊C4—1882no—
468C4⋊C4—1882no—
469C4⋊C4—1882no—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C22⋊C4—944no—
473C22⋊C4—1881no—
474C22⋊C4—1881no—
475C42—944no—
476C42—941no—
477C42—941no—
478C42—941no—
479C42—941no—
480C42—1882no—
481C42—1882no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1309400212yes—
489C23C2×D4×D4711yescharacteristic
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—941no—
509C2×C4—941no—
510C2×C4—942no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—942no—
514C2×C4—942no—
515C2×C4—941no—
516C2×C4—941no—
517C2×C4—941no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—3764no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4#1309400212yes—
534C2×C4C2×Q8×D4711yescharacteristic
535C2×C4#41×D4711yescharacteristic
536C2×C4C2×#1062535211yescharacteristic
537C2×C4C2×#1062535011yescharacteristic
538C2×C4#1309406911yescharacteristic
539C2×C4#1309407111yescharacteristic
540C8—22no—
541C8—41no—
542C8—1881no—
543C8—1881no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1309400211yescharacteristic
552C22#1568976211yescharacteristic
553C22#1568977311yescharacteristic
554C22#1568991911yescharacteristic
555C22#1568992211yescharacteristic
556C22#1568993211yescharacteristic
557C22#1568993311yescharacteristic
558C4—22no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—3761no—
564C4—1882no—
565C4—1882no—
566C4—1881no—
567C4—1881no—
568C4—1882no—
569C4—1881no—
570C4—1881no—
571C4#1568967511yescharacteristic
572C4#1568975211yescharacteristic
573C4#1568979911yescharacteristic
574C4#1568980611yescharacteristic
575C2—42no—
576C2#2024707011yescharacteristic
577C2#2025320911yescharacteristic
578C2#2025324911yescharacteristic
579C2#2025556911yescharacteristic
580C2#2025561011yescharacteristic
581C2#2025589811yescharacteristic
582C2#2025594311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1809C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#1809D47
hypocenterC47#26019
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
3. #20244960C2semidirect
4. #20244891C2semidirect
5. #20244887C2semidirect
6. #20244724C2semidirect
7. #20244344C2semidirect
8. #20244153C2semidirect
9. #20243968C2semidirect
10. #20243964C2semidirect
350. C47#26019semidirect

Maximal subgroups

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

subgroupindex
#202471132
#202462752
#202449602
#202448912
#202448872
#202447242
#202443442
#202441532
#202439682
#202439642
#202435802
#202435712
#202434802
#202431082
C47×#18092
#2601947