Finite Group Databasebeta

← C47×#19167 #26058865

Subgroups — C47×#19167 #26058865

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3139C211yesmaximal, characteristic, complemented
2C47×#3101C211yesmaximal, characteristic
3C2×C47×#474C211yesmaximal, characteristic
4D4×M16×C47C211yesmaximal, characteristic
5C47×#1703C212yesmaximal, complemented
6C47×#1699C212yesmaximal, complemented
7C47×#1683C212yesmaximal, complemented
8C47×#1679C212yesmaximal, complemented
9C47×#1565C211yesmaximal, characteristic, complemented
10C47×#1550C211yesmaximal, characteristic, complemented
11C47×#1546C211yesmaximal, characteristic, complemented
12C22×M16×C47—21no—
13C47×#533C2211yescharacteristic
14C2×C4⋊Q8×C47C2211yescharacteristic
15C2×C4×D4×C47C2211yescharacteristic
16C47×#491—22no—
17C47×#487C2212yescomplemented
18C47×#487C2212yescomplemented
19C47×#483—21no—
20C47×#476C2211yescharacteristic
21C47×#476C2211yescharacteristic
22C47×#474—21no—
23C47×#474—21no—
24C47×#474C2212yes—
25C47×#474C2212yes—
26C47×#461C2211yescharacteristic
27C47×#461C2211yescharacteristic
28C47×#450—21no—
29C4×SD16×C47—21no—
30C4×SD16×C47—21no—
31C47×#436C2212yes—
32C47×#435—22no—
33C8×D4×C47C2212yes—
34C47×#431C2211yescharacteristic
35C2×#107×C47—21no—
36C47×#423C2211yescharacteristic
37C2×C4⋊C8×C47C2211yescharacteristic
38C47×#419—21no—
39C47×#419—21no—
40C2×#102×C47—21no—
41C47×#407—21no—
42C4×M16×C47C2211yescharacteristic
43C47×#335C2212yescomplemented
44C47×#334C2212yescomplemented
45C47×#333C2212yescomplemented
46C47×#332C2212yescomplemented
47C47×#332C2212yescomplemented
48C47×#332C2212yescomplemented
49C22×Q8×C47—21no—
50C22×D4×C47D411yescharacteristic
51C23×C4×C47—21no—
52C2×SD16×C47—21no—
53C2×SD16×C47—21no—
54C2×M16×C47—21no—
55C2×M16×C47—22no—
56C2×M16×C47—44no—
57C2×M16×C47D411yescharacteristic
58C2×M16×C47D411yescharacteristic
59C22×C8×C47—21no—
60C22×C8×C47—22no—
61C4⋊Q8×C47—21no—
62C4⋊Q8×C47—21no—
63C4⋊Q8×C47—21no—
64C4⋊Q8×C47C2311yescharacteristic
65C4⋊Q8×C47C2312yes—
66C4⋊Q8×C47C2311yescharacteristic
67#125×C47—21no—
68C22⋊Q8×C47—21no—
69C22⋊Q8×C47—21no—
70C4×Q8×C47—21no—
71C4×Q8×C47—21no—
72C4×D4×C47—21no—
73C4×D4×C47—21no—
74C4×D4×C47D411yescharacteristic
75C4×D4×C47D411yescharacteristic
76C4×D4×C47C2311yescharacteristic
77C4×D4×C47C2311yescharacteristic
78#117×C47—21no—
79C2×C4⋊C4×C47—21no—
80C2×C4⋊C4×C47—21no—
81C2×C4⋊C4×C47D411yescharacteristic
82C2×C22⋊C4×C47—21no—
83C2×C42×C47C2311yescharacteristic
84#107×C47—22no—
85#107×C47—22no—
86#106×C47—21no—
87#106×C47—21no—
88C4⋊C8×C47—21no—
89C4⋊C8×C47—21no—
90C4⋊C8×C47C2311yescharacteristic
91C4⋊C8×C47C2311yescharacteristic
92C4⋊C8×C47C2312yes—
93C4⋊C8×C47C2312yes—
94#103×C47—22no—
95#103×C47—22no—
96#103×C47—22no—
97#103×C47—22no—
98#103×C47—21no—
99#103×C47—21no—
100#102×C47—21no—
101#102×C47—21no—
102#102×C47—22no—
103#102×C47—22no—
104C22⋊C8×C47—21no—
105C22⋊C8×C47—22no—
106C22⋊C8×C47—22no—
107#97×C47D412yes—
108C4×C8×C47—21no—
109C4×C8×C47C2312yes—
110C24×C47—21no—
111C2×Q8×C47—22no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—41no—
115C2×Q8×C47—42no—
116C2×Q8×C47—42no—
117C2×D4×C47—22no—
118C2×D4×C47—21no—
119C2×D4×C47—41no—
120C2×D4×C47C2×D411yescharacteristic
121C2×D4×C47C2×D411yescharacteristic
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—42no—
128C22×C4×C47—42no—
129C22×C4×C47—41no—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47#4111yescharacteristic
133SD16×C47—42no—
134SD16×C47—42no—
135M16×C47—22no—
136M16×C47—22no—
137M16×C47—44no—
138M16×C47—42no—
139C2×C8×C47—21no—
140C2×C8×C47—21no—
141C2×C8×C47—22no—
142C2×C8×C47—22no—
143C2×C8×C47—44no—
144C2×C8×C47—42no—
145C2×C8×C47C2×D412yes—
146C2×C8×C47C2×D412yes—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—41no—
155C4⋊C4×C47—41no—
156C4⋊C4×C47—41no—
157C4⋊C4×C47—41no—
158C4⋊C4×C47C2×D411yescharacteristic
159C4⋊C4×C47C2×D411yescharacteristic
160C22⋊C4×C47—21no—
161C22⋊C4×C47—21no—
162C22⋊C4×C47—21no—
163C22⋊C4×C47—21no—
164C22⋊C4×C47—41no—
165C42×C47—41no—
166C42×C47C2×D411yescharacteristic
167C42×C47C2×D411yescharacteristic
168C42×C47#4111yescharacteristic
169C42×C47C2411yescharacteristic
170C23×C47—21no—
171C23×C47—21no—
172C23×C47—42no—
173C23×C47—42no—
174C23×C47—81no—
175C23×C47#12111yescharacteristic
176Q8×C47—42no—
177Q8×C47—42no—
178Q8×C47—41no—
179Q8×C47—41no—
180Q8×C47—81no—
181Q8×C47—82no—
182D4×C47—21no—
183D4×C47—21no—
184D4×C47—21no—
185D4×C47—42no—
186D4×C47#13712yes—
187C2×C4×C47—21no—
188C2×C4×C47—21no—
189C2×C4×C47—21no—
190C2×C4×C47—21no—
191C2×C4×C47—41no—
192C2×C4×C47—42no—
193C2×C4×C47—41no—
194C2×C4×C47—41no—
195C2×C4×C47—41no—
196C2×C4×C47—41no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—42no—
202C2×C4×C47—82no—
203C2×C4×C47#12111yescharacteristic
204C2×C4×C47#12111yescharacteristic
205C2×C4×C47#12111yescharacteristic
206C2×C4×C47#13512yes—
207C2×C4×C47AGL(1,ℤ/8)12yes—
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47C22×D411yescharacteristic
210C2×C4×C47C2×#4111yescharacteristic
211C8×C47—22no—
212C8×C47—22no—
213C8×C47—42no—
214C8×C47—41no—
215C8×C47—42no—
216C8×C47—41no—
217#19167C4711yesmaximal, characteristic, direct product factor, Sylow
218C22×C47—21no—
219C22×C47—21no—
220C22×C47—22no—
221C22×C47—41no—
222C22×C47—41no—
223C22×C47—41no—
224C22×C47—41no—
225C22×C47—41no—
226C22×C47—84no—
227C22×C47#46312yes—
228C22×C47C2×#12111yescharacteristic
229C4×C47—21no—
230C4×C47—21no—
231C4×C47—21no—
232C4×C47—41no—
233C4×C47—41no—
234C4×C47—82no—
235C4×C47—81no—
236C4×C47—81no—
237C4×C47#46312yes—
238C4×C47C2×#13511yescharacteristic
239C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C47C2×#13711yescharacteristic
241C4×C47#57511yescharacteristic
242#3139C2×C4711yescharacteristic, complemented
243#3101C2×C4711yescharacteristic
244C2×#474C2×C4711yescharacteristic
245D4×M16C2×C4711yescharacteristic
246#1703C2×C4712yescomplemented
247#1699C2×C4712yescomplemented
248#1683C2×C4712yescomplemented
249#1679C2×C4712yescomplemented
250#1565C2×C4711yescharacteristic, complemented
251#1550C2×C4711yescharacteristic, complemented
252#1546C2×C4711yescharacteristic, complemented
253C2×C47—22no—
254C2×C47—42no—
255C2×C47—81no—
256C2×C47C2×#46311yescharacteristic
257C2×C47#305611yescharacteristic
258C2×C47#306911yescharacteristic
259C22×M16—21no—
260#533C22×C4711yescharacteristic
261C2×C4⋊Q8C22×C4711yescharacteristic
262C2×C4×D4C22×C4711yescharacteristic
263#491—22no—
264#487C22×C4712yescomplemented
265#487C22×C4712yescomplemented
266#483—21no—
267#476C22×C4711yescharacteristic
268#476C22×C4711yescharacteristic
269#474—21no—
270#474—21no—
271#474C22×C4712yes—
272#474C22×C4712yes—
273#461C22×C4711yescharacteristic
274#461C22×C4711yescharacteristic
275#450—21no—
276C4×SD16—21no—
277C4×SD16—21no—
278#436C22×C4712yes—
279#435—22no—
280C8×D4C22×C4712yes—
281#431C22×C4711yescharacteristic
282C2×#107—21no—
283#423C22×C4711yescharacteristic
284C2×C4⋊C8C22×C4711yescharacteristic
285#419—21no—
286#419—21no—
287C2×#102—21no—
288#407—21no—
289C4×M16C22×C4711yescharacteristic
290#335C22×C4712yescomplemented
291#334C22×C4712yescomplemented
292#333C22×C4712yescomplemented
293#332C22×C4712yescomplemented
294#332C22×C4712yescomplemented
295#332C22×C4712yescomplemented
296C47#1916711yescharacteristic, direct product factor, Sylow
297C22×Q8—21no—
298C22×D4D4×C4711yescharacteristic
299C23×C4—21no—
300C2×SD16—21no—
301C2×SD16—21no—
302C2×M16—21no—
303C2×M16—22no—
304C2×M16—44no—
305C2×M16D4×C4711yescharacteristic
306C2×M16D4×C4711yescharacteristic
307C22×C8—21no—
308C22×C8—22no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8C23×C4712yes—
313C4⋊Q8C23×C4711yescharacteristic
314C4⋊Q8C23×C4711yescharacteristic
315#125—21no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4D4×C4711yescharacteristic
323C4×D4D4×C4711yescharacteristic
324C4×D4C23×C4711yescharacteristic
325C4×D4C23×C4711yescharacteristic
326#117—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4D4×C4711yescharacteristic
330C2×C22⋊C4—21no—
331C2×C42C23×C4711yescharacteristic
332#107—22no—
333#107—22no—
334#106—21no—
335#106—21no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8C23×C4712yes—
339C4⋊C8C23×C4711yescharacteristic
340C4⋊C8C23×C4711yescharacteristic
341C4⋊C8C23×C4712yes—
342#103—22no—
343#103—22no—
344#103—22no—
345#103—22no—
346#103—21no—
347#103—21no—
348#102—22no—
349#102—22no—
350#102—21no—
351#102—21no—
352C22⋊C8—21no—
353C22⋊C8—22no—
354C22⋊C8—22no—
355#97D4×C4712yes—
356C4×C8—21no—
357C4×C8C23×C4712yes—
358C24—21no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4C2×D4×C4711yescharacteristic
369C2×D4C2×D4×C4711yescharacteristic
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4C2×D4×C4711yescharacteristic
379C22×C4C2×D4×C4711yescharacteristic
380C22×C4#41×C4711yescharacteristic
381SD16—42no—
382SD16—42no—
383M16—22no—
384M16—22no—
385M16—44no—
386M16—42no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—44no—
392C2×C8—42no—
393C2×C8C2×D4×C4712yes—
394C2×C8C2×D4×C4712yes—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4C2×D4×C4711yescharacteristic
407C4⋊C4C2×D4×C4711yescharacteristic
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C42—41no—
414C42C2×D4×C4711yescharacteristic
415C42C2×D4×C4711yescharacteristic
416C42#41×C4711yescharacteristic
417C42C24×C4711yescharacteristic
418C23—21no—
419C23—21no—
420C23—42no—
421C23—42no—
422C23—81no—
423C23#121×C4711yescharacteristic
424Q8—42no—
425Q8—42no—
426Q8—41no—
427Q8—41no—
428Q8—81no—
429Q8—82no—
430D4—21no—
431D4—21no—
432D4—21no—
433D4—42no—
434D4#137×C4712yes—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4#121×C4711yescharacteristic
452C2×C4#121×C4711yescharacteristic
453C2×C4#121×C4711yescharacteristic
454C2×C4#135×C4712yes—
455C2×C4AGL(1,ℤ/8)×C4712yes—
456C2×C4C22×D4×C4711yescharacteristic
457C2×C4C22×D4×C4711yescharacteristic
458C2×C4C2×#41×C4711yescharacteristic
459C8—22no—
460C8—22no—
461C8—42no—
462C8—41no—
463C8—42no—
464C8—41no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—84no—
474C22C47×#46312yes—
475C22C2×#121×C4711yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—41no—
481C4—82no—
482C4—81no—
483C4—81no—
484C4C47×#46312yes—
485C4C2×#135×C4711yescharacteristic
486C4C2×AGL(1,ℤ/8)×C4711yescharacteristic
487C4C2×#137×C4711yescharacteristic
488C4C47×#57511yescharacteristic
489C2—22no—
490C2—42no—
491C2—81no—
492C2C2×C47×#46311yescharacteristic
493C2C47×#305611yescharacteristic
494C2C47×#306911yescharacteristic
subgroupG/H
center, socleC22×C47C2×#121
commutator, Frattini, cosocleC42C24×C47
prime radicalC22×D4×C47D4
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×#3139C2semidirect
5. C47×#1703C2semidirect
6. C47×#1699C2semidirect
7. C47×#1683C2semidirect
8. C47×#1679C2semidirect
9. C47×#1565C2semidirect
10. C47×#1550C2semidirect
11. C47×#1546C2semidirect
17. C47×#487C22semidirect
18. C47×#487C22semidirect
43. C47×#335C22semidirect
44. C47×#334C22semidirect
45. C47×#333C22semidirect
46. C47×#332C22semidirect
47. C47×#332C22semidirect
48. C47×#332C22semidirect
217. #19167C47direct (N × H)
242. #3139C2×C47semidirect
246. #1703C2×C47semidirect
247. #1699C2×C47semidirect
248. #1683C2×C47semidirect
249. #1679C2×C47semidirect
250. #1565C2×C47semidirect
251. #1550C2×C47semidirect
252. #1546C2×C47semidirect
264. #487C22×C47semidirect
265. #487C22×C47semidirect
290. #335C22×C47semidirect
291. #334C22×C47semidirect
292. #333C22×C47semidirect
293. #332C22×C47semidirect
294. #332C22×C47semidirect
295. #332C22×C47semidirect
296. C47#19167direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#31392
C47×#31012
C2×C47×#4742
D4×M16×C472
C47×#17032
C47×#16992
C47×#16832
C47×#16792
C47×#15652
C47×#15502
C47×#15462
#1916747