Finite Group Databasebeta

← C47×#19279 #26058977

Subgroups — C47×#19279 #26058977

Order 12032.

712 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3167C211yesmaximal, characteristic, complemented
2C47×#3077C211yesmaximal, characteristic, complemented
3C47×#3036C211yesmaximal, characteristic
4C2×C47×#436C211yesmaximal, characteristic, complemented
5C47×#1699C212yesmaximal, complemented
6C47×#1697C212yesmaximal, complemented
7C47×#1691C212yesmaximal, complemented
8C47×#1685C212yesmaximal, complemented
9C47×#1567C211yesmaximal, characteristic, complemented
10C47×#1550C211yesmaximal, characteristic, complemented
11C47×#1547C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C47—21no—
13C22×M16×C47—21no—
14C47×#533C2211yescharacteristic
15C47×#532C2211yescharacteristic
16C2×C4×D4×C47C2211yescharacteristic
17C47×#498C2211yescharacteristic, complemented
18C47×#492C2211yescharacteristic, complemented
19C47×#487C2211yescharacteristic, complemented
20C47×#486C2211yescharacteristic, complemented
21C47×#478—22no—
22C47×#476C2212yes—
23C47×#474C2212yes—
24C47×#465—21no—
25C47×#465—21no—
26C47×#463—22no—
27C47×#461C2212yes—
28C47×#459C2212yes—
29C4×SD16×C47—21no—
30C4×SD16×C47—21no—
31C47×#436—21no—
32C47×#436—21no—
33C47×#436C2212yes—
34C47×#436C2212yes—
35C47×#428—21no—
36C47×#423C2211yescharacteristic
37C47×#423C2211yescharacteristic
38C2×C4⋊C8×C47C2211yescharacteristic
39C47×#418—21no—
40C47×#417—21no—
41C2×C22⋊C8×C47—21no—
42C2×C4×C8×C47C2211yescharacteristic
43C47×#335C2212yescomplemented
44C47×#335C2212yescomplemented
45C47×#333C2212yescomplemented
46C47×#332C2212yescomplemented
47C47×#332C2212yescomplemented
48C47×#331C2212yescomplemented
49C2×#41×C47—21no—
50C22×D4×C47D411yescharacteristic
51C23×C4×C47—21no—
52AGL(1,ℤ/8)×C47—44no—
53C2×SD16×C47—22no—
54C2×SD16×C47—21no—
55C2×SD16×C47—21no—
56C2×D8×C47—22no—
57C2×M16×C47—21no—
58C2×M16×C47—21no—
59C2×M16×C47—22no—
60C2×M16×C47—22no—
61C2×M16×C47—44no—
62C22×C8×C47—21no—
63C22×C8×C47D411yescharacteristic
64C22×C8×C47D411yescharacteristic
65C4⋊Q8×C47C2311yescharacteristic
66C4⋊Q8×C47C2311yescharacteristic
67C4⋊Q8×C47C2311yescharacteristic
68#127×C47C2311yescharacteristic
69#125×C47—21no—
70#124×C47—21no—
71C22⋊Q8×C47—21no—
72C22⋊Q8×C47—21no—
73#121×C47—21no—
74#121×C47—21no—
75C4×Q8×C47—21no—
76C4×Q8×C47—21no—
77C4×D4×C47—21no—
78C4×D4×C47—21no—
79C4×D4×C47—21no—
80C4×D4×C47—21no—
81C4×D4×C47D412yes—
82C4×D4×C47C2312yes—
83#117×C47—21no—
84C2×C4⋊C4×C47D411yescharacteristic
85C2×C22⋊C4×C47—21no—
86C2×C42×C47C2311yescharacteristic
87#107×C47—22no—
88#106×C47—21no—
89#106×C47—21no—
90#106×C47—22no—
91C4⋊C8×C47—21no—
92C4⋊C8×C47—21no—
93C4⋊C8×C47—21no—
94C4⋊C8×C47C2312yes—
95C4⋊C8×C47C2312yes—
96C4⋊C8×C47C2312yes—
97#103×C47—21no—
98#103×C47—21no—
99#103×C47—22no—
100#102×C47—22no—
101#102×C47—22no—
102#102×C47—21no—
103#102×C47—21no—
104#102×C47—22no—
105C22⋊C8×C47—22no—
106C22⋊C8×C47—22no—
107C4×C8×C47D411yescharacteristic
108C4×C8×C47D411yescharacteristic
109C4×C8×C47C2311yescharacteristic
110C4×C8×C47C2311yescharacteristic
111C24×C47—21no—
112#41×C47—42no—
113#41×C47—42no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—22no—
119C2×D4×C47—22no—
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—41no—
124C2×D4×C47C2×D412yes—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47—41no—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47#4111yescharacteristic
134SD16×C47—42no—
135SD16×C47—44no—
136SD16×C47—42no—
137D8×C47—44no—
138M16×C47—44no—
139M16×C47—42no—
140M16×C47—42no—
141M16×C47—44no—
142C2×C8×C47—22no—
143C2×C8×C47—22no—
144C2×C8×C47—22no—
145C2×C8×C47—22no—
146C2×C8×C47—41no—
147C2×C8×C47—42no—
148C2×C8×C47C2×D411yescharacteristic
149C2×C8×C47C2×D411yescharacteristic
150C2×C8×C47C2×D411yescharacteristic
151C2×C8×C47C2×D411yescharacteristic
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—41no—
160C4⋊C4×C47—41no—
161C4⋊C4×C47C2×D412yes—
162C22⋊C4×C47—22no—
163C22⋊C4×C47—22no—
164C22⋊C4×C47—41no—
165C22⋊C4×C47—41no—
166C22⋊C4×C47—41no—
167C42×C47—41no—
168C42×C47C2×D411yescharacteristic
169C42×C47C2×D411yescharacteristic
170C42×C47#4111yescharacteristic
171C42×C47C2411yescharacteristic
172C23×C47—22no—
173C23×C47—42no—
174C23×C47—42no—
175C23×C47—42no—
176C23×C47—41no—
177C23×C47#12111yescharacteristic
178Q8×C47—41no—
179Q8×C47—41no—
180Q8×C47—41no—
181Q8×C47—41no—
182Q8×C47—81no—
183D4×C47—22no—
184D4×C47—22no—
185D4×C47—41no—
186D4×C47—41no—
187D4×C47—41no—
188D4×C47—41no—
189D4×C47—42no—
190D4×C47—81no—
191D4×C47—82no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—22no—
195C2×C4×C47—41no—
196C2×C4×C47—44no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—82no—
204C2×C4×C47#12111yescharacteristic
205C2×C4×C47#12111yescharacteristic
206C2×C4×C47#12111yescharacteristic
207C2×C4×C47AGL(1,ℤ/8)12yes—
208C2×C4×C47#13712yes—
209C2×C4×C47C22×D411yescharacteristic
210C2×C4×C47C22×D411yescharacteristic
211C2×C4×C47C2×#4111yescharacteristic
212C8×C47—21no—
213C8×C47—21no—
214C8×C47—21no—
215C8×C47—42no—
216C8×C47—42no—
217C8×C47—42no—
218C8×C47AGL(1,ℤ/8)12yescomplemented
219#19279C4711yesmaximal, characteristic, direct product factor, Sylow
220C22×C47—21no—
221C22×C47—21no—
222C22×C47—42no—
223C22×C47—42no—
224C22×C47—42no—
225C22×C47—84no—
226C22×C47—81no—
227C22×C47—81no—
228C22×C47—81no—
229C22×C47#46812yes—
230C22×C47C2×#12111yescharacteristic
231C4×C47—21no—
232C4×C47—21no—
233C4×C47—21no—
234C4×C47—42no—
235C4×C47—81no—
236C4×C47—81no—
237C4×C47—81no—
238C4×C47#46812yes—
239C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C47C2×#13711yescharacteristic
242C4×C47#57511yescharacteristic
243#3167C2×C4711yescharacteristic, complemented
244#3077C2×C4711yescharacteristic, complemented
245#3036C2×C4711yescharacteristic
246C2×#436C2×C4711yescharacteristic, complemented
247#1699C2×C4712yescomplemented
248#1697C2×C4712yescomplemented
249#1691C2×C4712yescomplemented
250#1685C2×C4712yescomplemented
251#1567C2×C4711yescharacteristic, complemented
252#1550C2×C4711yescharacteristic, complemented
253#1547C2×C4711yescharacteristic, complemented
254C2×C47—22no—
255C2×C47—82no—
256C2×C47—161no—
257C2×C47#303611yescharacteristic
258C2×C47C2×#46811yescharacteristic
259C2×C47#306711yescharacteristic
260C2×AGL(1,ℤ/8)—21no—
261C22×M16—21no—
262#533C22×C4711yescharacteristic
263#532C22×C4711yescharacteristic
264C2×C4×D4C22×C4711yescharacteristic
265#498C22×C4711yescharacteristic, complemented
266#492C22×C4711yescharacteristic, complemented
267#487C22×C4711yescharacteristic, complemented
268#486C22×C4711yescharacteristic, complemented
269#478—22no—
270#476C22×C4712yes—
271#474C22×C4712yes—
272#465—21no—
273#465—21no—
274#463—22no—
275#461C22×C4712yes—
276#459C22×C4712yes—
277C4×SD16—21no—
278C4×SD16—21no—
279#436—21no—
280#436—21no—
281#436C22×C4712yes—
282#436C22×C4712yes—
283#428—21no—
284#423C22×C4711yescharacteristic
285#423C22×C4711yescharacteristic
286C2×C4⋊C8C22×C4711yescharacteristic
287#418—21no—
288#417—21no—
289C2×C22⋊C8—21no—
290C2×C4×C8C22×C4711yescharacteristic
291#335C22×C4712yescomplemented
292#335C22×C4712yescomplemented
293#333C22×C4712yescomplemented
294#332C22×C4712yescomplemented
295#332C22×C4712yescomplemented
296#331C22×C4712yescomplemented
297C47#1927911yescharacteristic, direct product factor, Sylow
298C2×#41—21no—
299C22×D4D4×C4711yescharacteristic
300C23×C4—21no—
301AGL(1,ℤ/8)—44no—
302C2×SD16—22no—
303C2×SD16—21no—
304C2×SD16—21no—
305C2×D8—22no—
306C2×M16—21no—
307C2×M16—21no—
308C2×M16—22no—
309C2×M16—22no—
310C2×M16—44no—
311C22×C8—21no—
312C22×C8D4×C4711yescharacteristic
313C22×C8D4×C4711yescharacteristic
314C4⋊Q8C23×C4711yescharacteristic
315C4⋊Q8C23×C4711yescharacteristic
316C4⋊Q8C23×C4711yescharacteristic
317#127C23×C4711yescharacteristic
318#125—21no—
319#124—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322#121—21no—
323#121—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4D4×C4712yes—
331C4×D4C23×C4712yes—
332#117—21no—
333C2×C4⋊C4D4×C4711yescharacteristic
334C2×C22⋊C4—21no—
335C2×C42C23×C4711yescharacteristic
336#107—22no—
337#106—21no—
338#106—21no—
339#106—22no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8C23×C4712yes—
344C4⋊C8C23×C4712yes—
345C4⋊C8C23×C4712yes—
346#103—21no—
347#103—21no—
348#103—22no—
349#102—22no—
350#102—22no—
351#102—21no—
352#102—21no—
353#102—22no—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C4×C8D4×C4711yescharacteristic
357C4×C8D4×C4711yescharacteristic
358C4×C8C23×C4711yescharacteristic
359C4×C8C23×C4711yescharacteristic
360C24—21no—
361#41—42no—
362#41—42no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4C2×D4×C4712yes—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4C2×D4×C4711yescharacteristic
381C22×C4C2×D4×C4711yescharacteristic
382C22×C4#41×C4711yescharacteristic
383SD16—42no—
384SD16—44no—
385SD16—42no—
386D8—44no—
387M16—44no—
388M16—42no—
389M16—42no—
390M16—44no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8C2×D4×C4711yescharacteristic
398C2×C8C2×D4×C4711yescharacteristic
399C2×C8C2×D4×C4711yescharacteristic
400C2×C8C2×D4×C4711yescharacteristic
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4C2×D4×C4712yes—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C42—41no—
417C42C2×D4×C4711yescharacteristic
418C42C2×D4×C4711yescharacteristic
419C42#41×C4711yescharacteristic
420C42C24×C4711yescharacteristic
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23#121×C4711yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432D4—22no—
433D4—22no—
434D4—42no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—81no—
440D4—82no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—82no—
453C2×C4#121×C4711yescharacteristic
454C2×C4#121×C4711yescharacteristic
455C2×C4#121×C4711yescharacteristic
456C2×C4AGL(1,ℤ/8)×C4712yes—
457C2×C4#137×C4712yes—
458C2×C4C22×D4×C4711yescharacteristic
459C2×C4C22×D4×C4711yescharacteristic
460C2×C4C2×#41×C4711yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—42no—
465C8—42no—
466C8—42no—
467C8AGL(1,ℤ/8)×C4712yescomplemented
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—84no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22C47×#46812yes—
478C22C2×#121×C4711yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4C47×#46812yes—
487C4C2×AGL(1,ℤ/8)×C4711yescharacteristic
488C4C2×AGL(1,ℤ/8)×C4711yescharacteristic
489C4C2×#137×C4711yescharacteristic
490C4C47×#57511yescharacteristic
491C2—22no—
492C2—82no—
493C2—161no—
494C2C47×#303611yescharacteristic
495C2C2×C47×#46811yescharacteristic
496C2C47×#306711yescharacteristic
subgroupG/H
center, socleC22×C47C2×#121
commutator, Frattini, cosocleC42C24×C47
prime radicalC47×#3036C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C47 ◁ C2×C42×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C47×#3167C2semidirect
2. C47×#3077C2semidirect
4. C2×C47×#436C2semidirect
5. C47×#1699C2semidirect
6. C47×#1697C2semidirect
7. C47×#1691C2semidirect
8. C47×#1685C2semidirect
9. C47×#1567C2semidirect
10. C47×#1550C2semidirect
11. C47×#1547C2semidirect
17. C47×#498C22semidirect
18. C47×#492C22semidirect
19. C47×#487C22semidirect
20. C47×#486C22semidirect
43. C47×#335C22semidirect
44. C47×#335C22semidirect
45. C47×#333C22semidirect
46. C47×#332C22semidirect
47. C47×#332C22semidirect
48. C47×#331C22semidirect
218. C8×C47AGL(1,ℤ/8)semidirect
219. #19279C47direct (N × H)
243. #3167C2×C47semidirect
244. #3077C2×C47semidirect
246. C2×#436C2×C47semidirect
247. #1699C2×C47semidirect
248. #1697C2×C47semidirect
249. #1691C2×C47semidirect
250. #1685C2×C47semidirect
251. #1567C2×C47semidirect
252. #1550C2×C47semidirect
253. #1547C2×C47semidirect
265. #498C22×C47semidirect
266. #492C22×C47semidirect
267. #487C22×C47semidirect
268. #486C22×C47semidirect
291. #335C22×C47semidirect
292. #335C22×C47semidirect
293. #333C22×C47semidirect
294. #332C22×C47semidirect
295. #332C22×C47semidirect
296. #331C22×C47semidirect
297. C47#19279direct (N × H)
467. C8AGL(1,ℤ/8)×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#31672
C47×#30772
C47×#30362
C2×C47×#4362
C47×#16992
C47×#16972
C47×#16912
C47×#16852
C47×#15672
C47×#15502
C47×#15472
#1927947