Finite Group Databasebeta

← #26383443

Subgroups — #26383443

Order 12032.

1918 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20249424C214yesmaximal, complemented
2#20249379C214yesmaximal, complemented
3#20249368C211yesmaximal, characteristic, complemented
4#20249351C212yesmaximal, complemented
5#20249334C211yesmaximal, characteristic, complemented
6#20249179C212yesmaximal, complemented
7#20249138C211yesmaximal, characteristic
8#20246652C214yesmaximal, complemented
9#20246633C214yesmaximal, complemented
10#20245950C211yesmaximal, characteristic, complemented
11#125×Dic47C211yesmaximal, characteristic, complemented
12#20245894C212yesmaximal, complemented
13#20245846C212yesmaximal, complemented
14#117×Dic47C211yesmaximal, characteristic, complemented
15C47×#2629C211yesmaximal, characteristic, complemented
16C2×#13094021C2211yescharacteristic
17C2×#13093998C2212yes—
18C2×#13093986C2212yes—
19C2×#13093986C2214yes—
20C2×(C22⋊D47)⋊C4C2214yes—
21C2×#13093983C2211yescharacteristic
22C2×#13093982C2214yes—
23C22×C4×Dic47C2211yescharacteristic
24#15689062C2214yes—
25#15689060C2214yes—
26#15689056C2214yes—
27#15689052C2214yes—
28#15689050C2214yes—
29#15689050C2214yes—
30#15689049C2214yes—
31#15689045C2214yes—
32#15689040C2214yes—
33#15689040C2214yes—
34#15689040C2214yes—
35#15689039C2214yes—
36#15689039C2214yes—
37#15689032C2214yes—
38#15689025C2212yes—
39#15689024C2212yes—
40#15689024C2212yes—
41#15689024C2214yes—
42#15689024C2212yes—
43#15689022C2212yes—
44#15689022C2212yes—
45#15689019C2212yes—
46#15689015C2214yes—
47#15689013C2214yes—
48#15689012C2214yes—
49#15689009C2214yes—
50#15689000C2214yes—
51#15688989C2211yescharacteristic
52C4⋊C4×Dic47C2212yes—
53C4⋊C4×Dic47C2212yes—
54C4⋊C4×Dic47C2214yes—
55C4⋊C4×Dic47C2212yes—
56C22⋊C4×Dic47C2212yes—
57C22⋊C4×Dic47C2212yes—
58#15688970C2214yes—
59#15688970C2214yes—
60#15688969C2211yescharacteristic
61#15688969C2211yescharacteristic
62C42×Dic47C2211yescharacteristic
63C42×Dic47C2211yescharacteristic
64C2×#125×C47C2211yescharacteristic
65C2×#117×C47C2211yescharacteristic
66C47×#398C2214yes—
67C47×#396C2214yes—
68C47×#389C2212yes—
69C47×#384C2211yescharacteristic
70C47×#380C2212yes—
71C22×D188C2312yes—
72C22×D188C2314yes—
73C22×C22⋊D47C2314yes—
74C23×Dic47C2311yescharacteristic
75C23×Dic47C2311yescharacteristic
76C22×C4×D47C2314yes—
77#13094021—24no—
78#13093998—28no—
79#13093986—28no—
80#13093986—48no—
81(C22⋊D47)⋊C4—48no—
82#13093983—24no—
83#13093982—48no—
84C2×C4⋊Dic47C2312yes—
85C2×C4⋊Dic47C2312yes—
86C2×C4⋊Dic47C2314yes—
87C2×C4⋊Dic47C2314yes—
88C2×Dic47⋊C4C2314yes—
89C2×C22⋊Dic47C2312yes—
90C2×(C2×D47)⋊C4C2314yes—
91C2×(C2×D47)⋊C4C2314yes—
92C2×(C2×D47)⋊C4C2314yes—
93C2×(C2×D47)⋊C4C2314yes—
94C2×(C2×D47)⋊C4C2314yes—
95C2×(C2×D47)⋊C4C2314yes—
96C2×(C2×D47)⋊C4C2314yes—
97C2×(C2×D47)⋊C4C2314yes—
98C2×(C2×D47)⋊C4C2314yes—
99C2×(C2×D47)⋊C4C2314yes—
100C2×C4×Dic47—24no—
101C2×C4×Dic47C2311yescharacteristic
102C2×C4×Dic47C2311yescharacteristic
103C2×C4×Dic47C2312yes—
104C2×C4×Dic47C2312yes—
105C2×C4×Dic47C2311yescharacteristic
106C2×C4×Dic47C2311yescharacteristic
107C2×C4×Dic47C2314yes—
108C2×C4×Dic47C2311yescharacteristic
109C2×C4×Dic47C2312yes—
110C2×C4×Dic47C2312yes—
111C2×C4×Dic47C2311yescharacteristic
112C47⋊#95C2312yes—
113C47⋊#95C2312yes—
114C47⋊#95C2312yes—
115C47⋊#95C2312yes—
116C47⋊#95C2314yes—
117C47⋊#95C2314yes—
118C47⋊#95C2314yes—
119C47⋊#95C2314yes—
120C47⋊#95C2314yes—
121C23×C4×C47C2311yescharacteristic
122#125×C47—24no—
123#117×C47—24no—
124C2×C4⋊C4×C47C2312yes—
125C2×C4⋊C4×C47C2314yes—
126C2×C4⋊C4×C47C2312yes—
127C2×C4⋊C4×C47C2312yes—
128C2×C4⋊C4×C47C2314yes—
129C2×C4⋊C4×C47C2314yes—
130C2×C22⋊C4×C47C2312yes—
131C2×C22⋊C4×C47C2314yes—
132C2×C42×C47C2311yescharacteristic
133C2×C42×C47C2311yescharacteristic
134#95×C47C2314yes—
135#95×C47C2314yes—
136C24×C47C2411yescharacteristic
137C23×D47C2414yes—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47#4114yes—
141C22×C4×C47C2412yes—
142C22×C4×C47C2414yes—
143C22×C4×C47C2414yes—
144C22×C4×C47C2412yes—
145C22×C4×C47C2411yescharacteristic
146C22×C4×C47C2411yescharacteristic
147C2×C22⋊D47—28no—
148C2×C22⋊D47—44no—
149C2×C22⋊D47—44no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—24no—
153C22×Dic47—42no—
154C22×Dic47C2411yescharacteristic
155C22×Dic47C2414yes—
156C22×Dic47C2412yes—
157C22×Dic47C2412yes—
158C22×Dic47C2411yescharacteristic
159C22×Dic47C2411yescharacteristic
160C22×Dic47C2411yescharacteristic
161C2×D188—24no—
162C2×D188—24no—
163C2×D188—28no—
164C2×D188—24no—
165C2×D188—44no—
166C2×D188—44no—
167C2×C4×D47—28no—
168C2×C4×D47—28no—
169C2×C4×D47—28no—
170C4⋊C4×C47—24no—
171C4⋊C4×C47—28no—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—44no—
175C4⋊C4×C47—44no—
176C22⋊C4×C47—24no—
177C22⋊C4×C47—44no—
178C42×C47—22no—
179C42×C47#4114yes—
180C22⋊Dic47—24no—
181(C2×D47)⋊C4—28no—
182(C2×D47)⋊C4—28no—
183(C2×D47)⋊C4—44no—
184(C2×D47)⋊C4—44no—
185(C2×D47)⋊C4—44no—
186(C2×D47)⋊C4—44no—
187(C2×D47)⋊C4—44no—
188(C2×D47)⋊C4—44no—
189(C2×D47)⋊C4—44no—
190(C2×D47)⋊C4—44no—
191C4⋊Dic47—24no—
192C4⋊Dic47—24no—
193C4⋊Dic47—44no—
194C4⋊Dic47—44no—
195Dic47⋊C4—44no—
196C4×Dic47—22no—
197C4×Dic47—22no—
198C4×Dic47—22no—
199C4×Dic47—22no—
200C4×Dic47—22no—
201C4×Dic47—24no—
202C4×Dic47—28no—
203C4×Dic47—24no—
204C4×Dic47—42no—
205C4×Dic47—42no—
206C4×Dic47#4114yes—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—22no—
210C23×C47—22no—
211C23×C47C2×#4112yes—
212C23×C47C2511yescharacteristic
213C22×D47—24no—
214C22×D47—24no—
215C22×D47—24no—
216C22×D47—24no—
217C22×D47—24no—
218C22×D47—24no—
219C22×D47—24no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—28no—
225C2×C4×C47—24no—
226C2×C4×C47—22no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—22no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47C2×#4112yes—
235C2×C4×C47C2×#4114yes—
236C2×C4×C47C2×#4112yes—
237C2×C4×C47C2×#4112yes—
238C2×C4×C47C2×#4112yes—
239C22⋊D47—416no—
240C2×Dic47—22no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—24no—
253C2×Dic47—22no—
254C2×Dic47—44no—
255C2×Dic47—44no—
256C2×Dic47C2×#4112yes—
257C2×Dic47C2×#4112yes—
258C2×Dic47H(5,𝔽2)14yes—
259D188—416no—
260D188—48no—
261C4×D47—416no—
262#34621—471nomaximal, Sylow
263C22×C47—22no—
264C22×C47—24no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—42no—
268C22×C47C22×#4111yescharacteristic
269C22×C47C22×#4111yescharacteristic
270C22×C47C22×#4111yescharacteristic
271C22×C47C2×H(5,𝔽2)12yes—
272C22×C47#58512yes—
273C2×D47—44no—
274C2×D47—44no—
275C2×D47—44no—
276C2×D47—44no—
277C2×D47—44no—
278C2×D47—44no—
279C2×D47—44no—
280C4×C47—22no—
281C4×C47—24no—
282C4×C47—44no—
283C4×C47—42no—
284C4×C47—44no—
285C4×C47#52914yes—
286Dic47—22no—
287Dic47—41no—
288Dic47—41no—
289Dic47—42no—
290Dic47—42no—
291Dic47—44no—
292Dic47#54014yes—
293#2651—474no—
294#2648—474no—
295#2647—471no—
296#2644—472no—
297#2641—471no—
298#2629D4711yescharacteristic, complemented
299#2624—472no—
300#2619—471no—
301#2395—474no—
302#2392—474no—
303C4×#127—471no—
304C4×#125—471no—
305C4×#121—472no—
306#2293—472no—
307C4×#117—471no—
308C2×C47—42no—
309C2×C47C2×#52911yescharacteristic
310C2×C47C2×#54011yescharacteristic
311C2×C47#345511yescharacteristic
312C2×C47#346012yes—
313C2×C47#346012yes—
314D47—84no—
315C2×#127—471no—
316C2×#125C2×D4711yescharacteristic
317C2×#121—472no—
318C2×C4×D4—472no—
319C2×C4×D4—474no—
320C2×C4×D4—474no—
321C2×#117—471no—
322C2×#117—474no—
323C2×#117C2×D4711yescharacteristic
324C22×C42—471no—
325#400—474no—
326#399—474no—
327#398—474no—
328#398C2×D4714yes—
329#397—474no—
330#397—474no—
331#397—474no—
332#397—474no—
333#396—474no—
334#396C2×D4714yes—
335#394—474no—
336#394—474no—
337#394—474no—
338#394—474no—
339#394—474no—
340#392—474no—
341#390—472no—
342#390—474no—
343#390—472no—
344#390—472no—
345#390—472no—
346#389—472no—
347#389—472no—
348#389—472no—
349#389C2×D4712yes—
350#388—474no—
351#388—474no—
352#388—474no—
353#387—474no—
354#386—474no—
355#384C2×D4711yescharacteristic
356#382—471no—
357#380C2×D4712yes—
358C4×C4⋊C4—474no—
359C4×C4⋊C4—472no—
360C4×C4⋊C4—472no—
361C4×C4⋊C4—472no—
362C4×C22⋊C4—472no—
363C4×C22⋊C4—472no—
364C4×C22⋊C4—474no—
365C4×C22⋊C4—474no—
366#376—471no—
367#376—471no—
368C43—471no—
369C43—471no—
370C47#3462111yescharacteristic, complemented, Sylow
371C22×D4—474no—
372C22×D4—472no—
373C22×D4—474no—
374C23×C4—471no—
375C23×C4—471no—
376C23×C4—474no—
377C23×C4C22×D4711yescharacteristic
378#127—944no—
379#125—24no—
380#121—948no—
381C4×D4—948no—
382C4×D4—1888no—
383C4×D4—1888no—
384#117—24no—
385#117—944no—
386#117—1888no—
387C2×C4⋊C4—472no—
388C2×C4⋊C4—472no—
389C2×C4⋊C4—474no—
390C2×C4⋊C4—474no—
391C2×C4⋊C4—474no—
392C2×C4⋊C4C22×D4712yes—
393C2×C4⋊C4C22×D4712yes—
394C2×C4⋊C4C22×D4712yes—
395C2×C4⋊C4C22×D4714yes—
396C2×C4⋊C4C22×D4714yes—
397C2×C4⋊C4C22×D4714yes—
398C2×C22⋊C4—472no—
399C2×C22⋊C4—474no—
400C2×C22⋊C4—474no—
401C2×C22⋊C4—474no—
402C2×C22⋊C4—474no—
403C2×C22⋊C4—474no—
404C2×C22⋊C4—474no—
405C2×C22⋊C4—474no—
406C2×C22⋊C4—474no—
407C2×C22⋊C4—474no—
408C2×C22⋊C4—474no—
409C2×C22⋊C4C22×D4712yes—
410C2×C22⋊C4C22×D4714yes—
411C2×C42—471no—
412C2×C42—472no—
413C2×C42—474no—
414C2×C42—471no—
415C2×C42—471no—
416C2×C42—472no—
417C2×C42—471no—
418C2×C42—471no—
419C2×C42—471no—
420C2×C42—472no—
421C2×C42—472no—
422C2×C42—944no—
423C2×C42C22×D4711yescharacteristic
424C2×C42C22×D4711yescharacteristic
425#95—472no—
426#95—472no—
427#95—472no—
428#95—472no—
429#95—474no—
430#95—474no—
431#95—474no—
432#95—474no—
433#95—474no—
434#95C22×D4714yes—
435#95C22×D4714yes—
436C24—474no—
437C24C23×D4711yescharacteristic
438C2×D4—948no—
439C2×D4—944no—
440C2×D4—948no—
441C2×D4—944no—
442C2×D4—944no—
443C2×D4—1884no—
444C2×D4—1884no—
445C2×D4—1884no—
446C2×D4—1884no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—471no—
450C22×C4—471no—
451C22×C4—471no—
452C22×C4—474no—
453C22×C4—472no—
454C22×C4—472no—
455C22×C4—471no—
456C22×C4—948no—
457C22×C4—948no—
458C22×C4—948no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—944no—
462C22×C4—1882no—
463C22×C4#1062535414yes—
464C22×C4C23×D4712yes—
465C22×C4C23×D4714yes—
466C22×C4C23×D4714yes—
467C22×C4C23×D4712yes—
468C22×C4C23×D4711yescharacteristic
469C22×C4C23×D4711yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—944no—
477C4⋊C4—944no—
478C4⋊C4—1884no—
479C4⋊C4—1884no—
480C4⋊C4—1884no—
481C22⋊C4—24no—
482C22⋊C4—44no—
483C22⋊C4—948no—
484C22⋊C4—944no—
485C22⋊C4—948no—
486C22⋊C4—1884no—
487C22⋊C4—1884no—
488C22⋊C4—1884no—
489C22⋊C4—1884no—
490C22⋊C4—1884no—
491C22⋊C4—1884no—
492C22⋊C4—1884no—
493C22⋊C4—1884no—
494C42—22no—
495C42—474no—
496C42—942no—
497C42—942no—
498C42—944no—
499C42—948no—
500C42—944no—
501C42—942no—
502C42—942no—
503C42—942no—
504C42—1882no—
505C42—1882no—
506C42#1062535214yes—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—944no—
512C23—944no—
513C23—944no—
514C23—944no—
515C23—944no—
516C23—944no—
517C23—944no—
518C23C2×#1062535412yes—
519C23C24×D4711yescharacteristic
520D4—18816no—
521D4—18816no—
522D4—1888no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—472no—
538C2×C4—474no—
539C2×C4—472no—
540C2×C4—942no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—944no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—944no—
549C2×C4—942no—
550C2×C4—942no—
551C2×C4—942no—
552C2×C4—942no—
553C2×C4—944no—
554C2×C4—18816no—
555C2×C4—1884no—
556C2×C4—1884no—
557C2×C4#41×D4712yes—
558C2×C4C2×#1062535214yes—
559C2×C4C2×#1062535212yes—
560C2×C4C2×#1062535412yes—
561C2×C4C2×#1062535412yes—
562C22—24no—
563C22—22no—
564C22—41no—
565C22—42no—
566C22—41no—
567C22—1884no—
568C22—1884no—
569C22—1884no—
570C22—1884no—
571C22—1884no—
572C22—1884no—
573C22—1884no—
574C22C2×#41×D4711yescharacteristic
575C22C22×#1062535211yescharacteristic
576C22C22×#1062535411yescharacteristic
577C22H(5,𝔽2)×D4712yes—
578C22#1568996012yes—
579C4—24no—
580C4—22no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—474no—
585C4—942no—
586C4—1881no—
587C4—1884no—
588C4—1882no—
589C4—1882no—
590C4—1881no—
591C4#1568965914yes—
592C2—42no—
593C2—3764no—
594C2C2×#1568965911yescharacteristic
595C2#540×D4711yescharacteristic
596C2#2025677911yescharacteristic
597C2#2025680512yes—
598C2#2025681112yes—
subgroupG/H
center, FrattiniC23C24×D47
commutator, socle, cosocleC23×C47C25
FittingC47×#2629C2
prime radical#20249138C2
hypercenter#2629D47
hypocenterC47#34621
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2629

Complements

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

NH ≅ G/Ntype
1. #20249424C2semidirect
2. #20249379C2semidirect
3. #20249368C2semidirect
4. #20249351C2semidirect
5. #20249334C2semidirect
6. #20249179C2semidirect
8. #20246652C2semidirect
9. #20246633C2semidirect
10. #20245950C2semidirect
11. #125×Dic47C2semidirect
12. #20245894C2semidirect
13. #20245846C2semidirect
14. #117×Dic47C2semidirect
15. C47×#2629C2semidirect
298. #2629D47semidirect
370. C47#34621semidirect

Maximal subgroups

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

subgroupindex
#202494242
#202493792
#202493682
#202493512
#202493342
#202491792
#202491382
#202466522
#202466332
#202459502
#125×Dic472
#202458942
#202458462
#117×Dic472
C47×#26292
#3462147