Finite Group Databasebeta

← C47×#12705 #26052403

Subgroups — C47×#12705 #26052403

Order 12032.

1186 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3484C211yesmaximal, characteristic
2C47×#2121C214yesmaximal, complemented
3C47×#1783C211yesmaximal, characteristic, complemented
4C47×#1781C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C47C411yescharacteristic, complemented
6C23×D4×C47C2211yescharacteristic
7C47×#546—24no—
8D42×C47—24no—
9C47×#535—21no—
10C47×#534—21no—
11C47×#534C2212yes—
12C47×#525—21no—
13C2×#120×C47—21no—
14C2×#120×C47C412yescomplemented
15C47×#518C411yescharacteristic, complemented
16C2×#100×C47—21no—
17C2×#100×C47C2212yescomplemented
18C2×#99×C47—21no—
19C2×#99×C47C2212yescomplemented
20C47×#407—21no—
21C47×#379—21no—
22C47×#352—24no—
23C47×AΣL(1,𝔽16)—24no—
24C25×C47C2×C411yescharacteristic
25C25×C47D411yescharacteristic
26H(5,𝔽2)×C47—42no—
27H(5,𝔽2)×C47—42no—
28C2×#41×C47—21no—
29C2×#41×C47—22no—
30C22×D4×C47—21no—
31C22×D4×C47—22no—
32C22×D4×C47—22no—
33C22×D4×C47—22no—
34C22×D4×C47—24no—
35C22×D4×C47—42no—
36C22×D4×C47—42no—
37C22×D4×C47C2×C411yescharacteristic
38C22×D4×C47C2×C412yes—
39C22×D4×C47D411yescharacteristic
40C22×D4×C47D411yescharacteristic
41C22×D4×C47C2311yescharacteristic
42C23×C4×C47D411yescharacteristic
43C2×M16×C47—22no—
44#127×C47—22no—
45#124×C47—22no—
46#123×C47—22no—
47#123×C47—22no—
48#123×C47D414yes—
49C22⋊Q8×C47—22no—
50#121×C47—22no—
51#121×C47—24no—
52#121×C47—24no—
53#121×C47—24no—
54#120×C47—24no—
55#120×C47—22no—
56#120×C47—22no—
57#120×C47—22no—
58#120×C47—22no—
59#120×C47—24no—
60#120×C47D414yes—
61C4×D4×C47—24no—
62C4×D4×C47—24no—
63#117×C47—21no—
64C2×C22⋊C4×C47—22no—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—22no—
67C2×C22⋊C4×C47—22no—
68C2×C22⋊C4×C47—42no—
69C2×C22⋊C4×C47C2×C412yes—
70#100×C47—24no—
71#100×C47—42no—
72#99×C47—24no—
73#99×C47—42no—
74C22⋊C8×C47—42no—
75C24×C47—21no—
76C24×C47—22no—
77C24×C47—22no—
78C24×C47—22no—
79C24×C47—41no—
80C24×C47—41no—
81C24×C47—41no—
82C24×C47—82no—
83C24×C47—82no—
84C24×C47C22⋊C411yescharacteristic
85C24×C47C22⋊C411yescharacteristic
86C24×C47C22⋊C411yescharacteristic
87C24×C47C22⋊C412yescomplemented
88C24×C47C22×C411yescharacteristic
89C24×C47C2×D411yescharacteristic
90#41×C47—42no—
91#41×C47—48no—
92#41×C47—42no—
93C2×Q8×C47—21no—
94C2×D4×C47—22no—
95C2×D4×C47—22no—
96C2×D4×C47—24no—
97C2×D4×C47—22no—
98C2×D4×C47—21no—
99C2×D4×C47—22no—
100C2×D4×C47—22no—
101C2×D4×C47—22no—
102C2×D4×C47—22no—
103C2×D4×C47—24no—
104C2×D4×C47—24no—
105C2×D4×C47—22no—
106C2×D4×C47—22no—
107C2×D4×C47—22no—
108C2×D4×C47—41no—
109C2×D4×C47—41no—
110C2×D4×C47—41no—
111C2×D4×C47—41no—
112C2×D4×C47—416no—
113C2×D4×C47—48no—
114C2×D4×C47—42no—
115C2×D4×C47—42no—
116C2×D4×C47—42no—
117C2×D4×C47—42no—
118C2×D4×C47—84no—
119C2×D4×C47—84no—
120C2×D4×C47C2×D414yes—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—41no—
128C22×C4×C47—42no—
129C22×C4×C47C22⋊C411yescharacteristic
130C22×C4×C47C22⋊C412yescomplemented
131C22×C4×C47C2×D411yescharacteristic
132M16×C47—42no—
133M16×C47—42no—
134C2×C8×C47—42no—
135C4⋊C4×C47—22no—
136C4⋊C4×C47—24no—
137C22⋊C4×C47—24no—
138C22⋊C4×C47—22no—
139C22⋊C4×C47—24no—
140C22⋊C4×C47—22no—
141C22⋊C4×C47—24no—
142C22⋊C4×C47—24no—
143C22⋊C4×C47—42no—
144C22⋊C4×C47—42no—
145C22⋊C4×C47—84no—
146C22⋊C4×C47C22⋊C414yescomplemented
147C42×C47—22no—
148C23×C47—21no—
149C23×C47—21no—
150C23×C47—21no—
151C23×C47—21no—
152C23×C47—21no—
153C23×C47—21no—
154C23×C47—21no—
155C23×C47—21no—
156C23×C47—21no—
157C23×C47—21no—
158C23×C47—22no—
159C23×C47—21no—
160C23×C47—21no—
161C23×C47—42no—
162C23×C47—44no—
163C23×C47—44no—
164C23×C47—44no—
165C23×C47—44no—
166C23×C47—44no—
167C23×C47—42no—
168C23×C47—42no—
169C23×C47—82no—
170C23×C47—82no—
171C23×C47—82no—
172C23×C47—81no—
173C23×C47—81no—
174C23×C47—81no—
175C23×C47—81no—
176C23×C47—82no—
177C23×C47—82no—
178C23×C47—82no—
179C23×C47—82no—
180C23×C47—82no—
181C23×C47—82no—
182C23×C47—82no—
183C23×C47—82no—
184C23×C47#9912yes—
185C23×C47C2×C22⋊C412yes—
186C23×C47C2×C22⋊C411yescharacteristic
187C23×C47#12012yes—
188Q8×C47—42no—
189D4×C47—28no—
190D4×C47—44no—
191D4×C47—44no—
192D4×C47—44no—
193D4×C47—42no—
194D4×C47—44no—
195D4×C47—44no—
196D4×C47—48no—
197D4×C47—42no—
198D4×C47—42no—
199D4×C47—84no—
200D4×C47—84no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—24no—
206C2×C4×C47—22no—
207C2×C4×C47—44no—
208C2×C4×C47—44no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—82no—
212C2×C4×C47—82no—
213C2×C4×C47—82no—
214C2×C4×C47—84no—
215C2×C4×C47#9912yes—
216C2×C4×C47#12012yes—
217C8×C47—82no—
218#12705C4711yesmaximal, characteristic, direct product factor, Sylow
219C22×C47—22no—
220C22×C47—22no—
221C22×C47—22no—
222C22×C47—22no—
223C22×C47—22no—
224C22×C47—22no—
225C22×C47—41no—
226C22×C47—42no—
227C22×C47—42no—
228C22×C47—42no—
229C22×C47—42no—
230C22×C47—42no—
231C22×C47—41no—
232C22×C47—41no—
233C22×C47—41no—
234C22×C47—44no—
235C22×C47—42no—
236C22×C47—41no—
237C22×C47—42no—
238C22×C47—41no—
239C22×C47—81no—
240C22×C47—81no—
241C22×C47—81no—
242C22×C47—81no—
243C22×C47—81no—
244C22×C47—81no—
245C22×C47—88no—
246C22×C47—88no—
247C22×C47—162no—
248C22×C47—162no—
249C22×C47#37911yescharacteristic
250C22×C47C2×#9911yescharacteristic
251C22×C47C2×#9911yescharacteristic
252C4×C47—22no—
253C4×C47—22no—
254C4×C47—44no—
255C4×C47—42no—
256C4×C47—162no—
257#3484C2×C4711yescharacteristic
258#2121C2×C4714yescomplemented
259#1783C2×C4711yescharacteristic, complemented
260#1781C2×C4711yescharacteristic, complemented
261C2×C47—21no—
262C2×C47—21no—
263C2×C47—21no—
264C2×C47—42no—
265C2×C47—42no—
266C2×C47—42no—
267C2×C47—42no—
268C2×C47—81no—
269C2×C47—82no—
270C2×C47#178111yescharacteristic
271C2×H(5,𝔽2)C4×C4711yescharacteristic, complemented
272C23×D4C22×C4711yescharacteristic
273#546—24no—
274D42—24no—
275#535—21no—
276#534—21no—
277#534C22×C4712yes—
278#525—21no—
279C2×#120—21no—
280C2×#120C4×C4712yescomplemented
281#518C4×C4711yescharacteristic, complemented
282C2×#100—21no—
283C2×#100C22×C4712yescomplemented
284C2×#99—21no—
285C2×#99C22×C4712yescomplemented
286#407—21no—
287#379—21no—
288#352—24no—
289AΣL(1,𝔽16)—24no—
290C47#1270511yescharacteristic, direct product factor, Sylow
291C25C2×C4×C4711yescharacteristic
292C25D4×C4711yescharacteristic
293H(5,𝔽2)—42no—
294H(5,𝔽2)—42no—
295C2×#41—21no—
296C2×#41—22no—
297C22×D4—21no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—24no—
302C22×D4—42no—
303C22×D4—42no—
304C22×D4C2×C4×C4711yescharacteristic
305C22×D4C2×C4×C4712yes—
306C22×D4D4×C4711yescharacteristic
307C22×D4D4×C4711yescharacteristic
308C22×D4C23×C4711yescharacteristic
309C23×C4D4×C4711yescharacteristic
310C2×M16—22no—
311#127—22no—
312#124—22no—
313#123—22no—
314#123—22no—
315#123D4×C4714yes—
316C22⋊Q8—22no—
317#121—22no—
318#121—24no—
319#121—24no—
320#121—24no—
321#120—24no—
322#120—22no—
323#120—22no—
324#120—22no—
325#120—22no—
326#120—24no—
327#120D4×C4714yes—
328C4×D4—24no—
329C4×D4—24no—
330#117—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4C2×C4×C4712yes—
337#100—24no—
338#100—42no—
339#99—24no—
340#99—42no—
341C22⋊C8—42no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—82no—
350C24—82no—
351C24C22⋊C4×C4711yescharacteristic
352C24C22⋊C4×C4711yescharacteristic
353C24C22⋊C4×C4711yescharacteristic
354C24C22⋊C4×C4712yescomplemented
355C24C22×C4×C4711yescharacteristic
356C24C2×D4×C4711yescharacteristic
357#41—42no—
358#41—48no—
359#41—42no—
360C2×Q8—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4C2×D4×C4714yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4C22⋊C4×C4711yescharacteristic
397C22×C4C22⋊C4×C4712yescomplemented
398C22×C4C2×D4×C4711yescharacteristic
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—84no—
413C22⋊C4C22⋊C4×C4714yescomplemented
414C42—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23#99×C4712yes—
452C23C2×C22⋊C4×C4712yes—
453C23C2×C22⋊C4×C4711yescharacteristic
454C23#120×C4712yes—
455Q8—42no—
456D4—28no—
457D4—44no—
458D4—44no—
459D4—44no—
460D4—42no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—42no—
465D4—42no—
466D4—84no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4#99×C4712yes—
483C2×C4#120×C4712yes—
484C8—82no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—88no—
512C22—88no—
513C22—162no—
514C22—162no—
515C22C47×#37911yescharacteristic
516C22C2×#99×C4711yescharacteristic
517C22C2×#99×C4711yescharacteristic
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—162no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—82no—
532C2C47×#178111yescharacteristic
subgroupG/H
center, socleC2×C47#1781
commutatorC24C22×C4×C47
Frattini, cosocleC22×D4C23×C47
prime radicalC47×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C47 ◁ C23×C47 ◁ C23×D4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C47×#2121C2semidirect
3. C47×#1783C2semidirect
4. C47×#1781C2semidirect
5. C2×H(5,𝔽2)×C47C4semidirect
14. C2×#120×C47C4semidirect
15. C47×#518C4semidirect
17. C2×#100×C47C22semidirect
19. C2×#99×C47C22semidirect
87. C24×C47C22⋊C4semidirect
130. C22×C4×C47C22⋊C4semidirect
146. C22⋊C4×C47C22⋊C4semidirect
218. #12705C47direct (N × H)
258. #2121C2×C47semidirect
259. #1783C2×C47semidirect
260. #1781C2×C47semidirect
271. C2×H(5,𝔽2)C4×C47semidirect
280. C2×#120C4×C47semidirect
281. #518C4×C47semidirect
283. C2×#100C22×C47semidirect
285. C2×#99C22×C47semidirect
290. C47#12705direct (N × H)
354. C24C22⋊C4×C47semidirect
397. C22×C4C22⋊C4×C47semidirect
413. C22⋊C4C22⋊C4×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#34842
C47×#21212
C47×#17832
C47×#17812
#1270547