Finite Group Databasebeta

← C47×#24857 #26064555

Subgroups — C47×#24857 #26064555

Order 12032.

720 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3079C211yesmaximal, characteristic
2C2×C47×#477C211yesmaximal, characteristic, complemented
3C4×#137×C47C211yesmaximal, characteristic
4C47×#2906C211yesmaximal, characteristic
5C47×#2089C212yesmaximal, complemented
6C47×#2064C212yesmaximal, complemented
7C47×#1997C211yesmaximal, characteristic, complemented
8C47×#1950C211yesmaximal, characteristic, complemented
9C47×#1908C212yesmaximal
10C47×#1816C212yesmaximal
11C47×#1801C211yesmaximal, characteristic, complemented
12C2×#137×C47C2211yescharacteristic
13C2×#131×C47C2212yes—
14C2×C4⋊Q8×C47C2211yescharacteristic
15C4×#41×C47C2211yescharacteristic
16C2×C4×Q8×C47C2211yescharacteristic
17C47×#477—21no—
18C47×#477—21no—
19C47×#477—21no—
20C47×#477—22no—
21C47×#476—21no—
22C47×#475—21no—
23C47×#474—21no—
24C47×#441—21no—
25C47×#440—21no—
26C4×Q16×C47—21no—
27C4×SD16×C47—21no—
28C2×#108×C47C2212yes—
29C47×#428C2212yes—
30C47×#428C2211yescharacteristic
31C2×#107×C47C2212yes—
32C47×#424—22no—
33C47×#424C2212yes—
34C47×#423—21no—
35C47×#423C2211yescharacteristic
36C47×#423C2211yescharacteristic
37C2×C4⋊C8×C47—21no—
38C2×C4⋊C8×C47C2211yescharacteristic
39C47×#419C2211yescharacteristic
40C47×#417C2212yes—
41C47×#417C2211yescharacteristic
42C2×#103×C47C2212yes—
43C2×#101×C47C2212yes—
44C4×M16×C47C2211yescharacteristic
45C47×#343C2212yes—
46C47×#343C2212yes—
47C47×#343C2212yes—
48C47×#341C2212yes—
49C2×#41×C47C2311yescharacteristic
50C22×Q8×C47—21no—
51C22×Q8×C47C2311yescharacteristic
52#137×C47—22no—
53#137×C47Q814yescomplemented
54C2×Q16×C47—21no—
55C2×SD16×C47—21no—
56#131×C47—24no—
57#131×C47—42no—
58C2×M16×C47—22no—
59C2×M16×C47C2312yes—
60C2×M16×C47C2312yes—
61C2×M16×C47C2311yescharacteristic
62C2×M16×C47C2311yescharacteristic
63C22×C8×C47—22no—
64C22×C8×C47C2312yes—
65C4⋊Q8×C47—21no—
66C4⋊Q8×C47—21no—
67C4⋊Q8×C47—21no—
68C4⋊Q8×C47—21no—
69C4×Q8×C47—21no—
70C4×Q8×C47—21no—
71C4×Q8×C47—21no—
72C4×Q8×C47—21no—
73C4×Q8×C47D411yescharacteristic
74C4×Q8×C47D411yescharacteristic
75C4×D4×C47—21no—
76C4×D4×C47—21no—
77C4×D4×C47D411yescharacteristic
78C4×D4×C47D411yescharacteristic
79#117×C47—21no—
80#117×C47C2311yescharacteristic
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47—21no—
83C2×C4⋊C4×C47C2311yescharacteristic
84C2×C4⋊C4×C47C2312yes—
85C2×C42×C47—21no—
86C2×C42×C47—21no—
87C2×C42×C47C2311yescharacteristic
88#108×C47—42no—
89#107×C47—22no—
90#107×C47—22no—
91#107×C47—22no—
92#107×C47—21no—
93#106×C47—22no—
94#106×C47—21no—
95C4⋊C8×C47—22no—
96C4⋊C8×C47—22no—
97C4⋊C8×C47—21no—
98C4⋊C8×C47—21no—
99C4⋊C8×C47—21no—
100C4⋊C8×C47—21no—
101C4⋊C8×C47D411yescharacteristic
102C4⋊C8×C47D411yescharacteristic
103C4⋊C8×C47D412yes—
104#103×C47—22no—
105#103×C47—21no—
106#103×C47—22no—
107#103×C47—22no—
108#103×C47—21no—
109#103×C47—21no—
110#102×C47—22no—
111#102×C47—21no—
112#101×C47—42no—
113#97×C47—21no—
114C4×C8×C47—21no—
115#41×C47—21no—
116#41×C47—21no—
117#41×C47C2×Q812yes—
118#41×C47#4112yes—
119C2×Q8×C47—21no—
120C2×Q8×C47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—41no—
125C2×Q8×C47—41no—
126C2×Q8×C47—82no—
127C2×Q8×C47C2×D411yescharacteristic
128C2×Q8×C47C2×Q812yes—
129C2×D4×C47—21no—
130C2×D4×C47C2×D411yescharacteristic
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—22no—
137C22×C4×C47#4111yescharacteristic
138C22×C4×C47#4111yescharacteristic
139C22×C4×C47C2411yescharacteristic
140Q16×C47—24no—
141SD16×C47—24no—
142M16×C47—21no—
143M16×C47—22no—
144M16×C47—21no—
145M16×C47—22no—
146M16×C47—22no—
147M16×C47—22no—
148M16×C47—21no—
149M16×C47—44no—
150M16×C47C2×Q812yes—
151C2×C8×C47—21no—
152C2×C8×C47—22no—
153C2×C8×C47—22no—
154C2×C8×C47—21no—
155C2×C8×C47—22no—
156C2×C8×C47—22no—
157C2×C8×C47—44no—
158C2×C8×C47C2×D412yes—
159C2×C8×C47C2×D412yes—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—41no—
167C4⋊C4×C47—41no—
168C4⋊C4×C47—41no—
169C4⋊C4×C47—41no—
170C4⋊C4×C47C2×D411yescharacteristic
171C4⋊C4×C47C2×D411yescharacteristic
172C22⋊C4×C47—21no—
173C22⋊C4×C47—21no—
174C22⋊C4×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—41no—
179C42×C47—41no—
180C42×C47C2×D411yescharacteristic
181C42×C47C2×D411yescharacteristic
182C42×C47C2×D411yescharacteristic
183C42×C47C2×D411yescharacteristic
184C23×C47—21no—
185C23×C47C2×#4111yescharacteristic
186Q8×C47—21no—
187Q8×C47—22no—
188Q8×C47—22no—
189Q8×C47—41no—
190Q8×C47—41no—
191Q8×C47—82no—
192Q8×C47—82no—
193Q8×C47C22⋊Q812yes—
194D4×C47—22no—
195D4×C47—22no—
196D4×C47—21no—
197D4×C47C22⋊Q812yes—
198C2×C4×C47—21no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—42no—
209C2×C4×C47—41no—
210C2×C4×C47—42no—
211C2×C4×C47—81no—
212C2×C4×C47—81no—
213C2×C4×C47—82no—
214C2×C4×C47#12111yescharacteristic
215C2×C4×C47#12111yescharacteristic
216C2×C4×C47#12111yescharacteristic
217C2×C4×C47#12111yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×Q811yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47H(5,𝔽2)11yescharacteristic
223C2×C4×C472−1+411yescharacteristic
224C8×C47—22no—
225C8×C47—22no—
226C8×C47—22no—
227C8×C47—41no—
228C8×C47—42no—
229C8×C47—42no—
230#24857C4711yesmaximal, characteristic, direct product factor, Sylow
231C22×C47—21no—
232C22×C47—21no—
233C22×C47—22no—
234C22×C47—41no—
235C22×C47—41no—
236C22×C47C2×#12111yescharacteristic
237C22×C47#55211yescharacteristic
238C22×C47#55411yescharacteristic
239C4×C47—22no—
240C4×C47—21no—
241C4×C47—21no—
242C4×C47—21no—
243C4×C47—42no—
244C4×C47—41no—
245C4×C47—41no—
246C4×C47—81no—
247C4×C47—82no—
248C4×C47#47312yes—
249C4×C47C2×C22⋊Q811yescharacteristic
250C4×C47#53711yescharacteristic
251C4×C47#54611yescharacteristic
252C4×C47D4×Q811yescharacteristic
253#3079C2×C4711yescharacteristic
254C2×#477C2×C4711yescharacteristic, complemented
255C4×#137C2×C4711yescharacteristic
256#2906C2×C4711yescharacteristic
257#2089C2×C4712yescomplemented
258#2064C2×C4712yescomplemented
259#1997C2×C4711yescharacteristic, complemented
260#1950C2×C4711yescharacteristic, complemented
261#1908C2×C4712yes—
262#1816C2×C4712yes—
263#1801C2×C4711yescharacteristic, complemented
264C2×C47—21no—
265C2×C47—21no—
266C2×C47—42no—
267C2×C47#241711yescharacteristic
268C2×C47C2×#47311yescharacteristic
269C2×C47#306811yescharacteristic
270C2×#137C22×C4711yescharacteristic
271C2×#131C22×C4712yes—
272C2×C4⋊Q8C22×C4711yescharacteristic
273C4×#41C22×C4711yescharacteristic
274C2×C4×Q8C22×C4711yescharacteristic
275#477—21no—
276#477—21no—
277#477—21no—
278#477—22no—
279#476—21no—
280#475—21no—
281#474—21no—
282#441—21no—
283#440—21no—
284C4×Q16—21no—
285C4×SD16—21no—
286C2×#108C22×C4712yes—
287#428C22×C4712yes—
288#428C22×C4711yescharacteristic
289C2×#107C22×C4712yes—
290#424—22no—
291#424C22×C4712yes—
292#423—21no—
293#423C22×C4711yescharacteristic
294#423C22×C4711yescharacteristic
295C2×C4⋊C8—21no—
296C2×C4⋊C8C22×C4711yescharacteristic
297#419C22×C4711yescharacteristic
298#417C22×C4712yes—
299#417C22×C4711yescharacteristic
300C2×#103C22×C4712yes—
301C2×#101C22×C4712yes—
302C4×M16C22×C4711yescharacteristic
303#343C22×C4712yes—
304#343C22×C4712yes—
305#343C22×C4712yes—
306#341C22×C4712yes—
307C47#2485711yescharacteristic, direct product factor, Sylow
308C2×#41C23×C4711yescharacteristic
309C22×Q8—21no—
310C22×Q8C23×C4711yescharacteristic
311#137—22no—
312#137Q8×C4714yescomplemented
313C2×Q16—21no—
314C2×SD16—21no—
315#131—24no—
316#131—42no—
317C2×M16—22no—
318C2×M16C23×C4712yes—
319C2×M16C23×C4712yes—
320C2×M16C23×C4711yescharacteristic
321C2×M16C23×C4711yescharacteristic
322C22×C8—22no—
323C22×C8C23×C4712yes—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×Q8D4×C4711yescharacteristic
333C4×Q8D4×C4711yescharacteristic
334C4×D4—21no—
335C4×D4—21no—
336C4×D4D4×C4711yescharacteristic
337C4×D4D4×C4711yescharacteristic
338#117—21no—
339#117C23×C4711yescharacteristic
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4C23×C4711yescharacteristic
343C2×C4⋊C4C23×C4712yes—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42C23×C4711yescharacteristic
347#108—42no—
348#107—22no—
349#107—22no—
350#107—21no—
351#107—22no—
352#106—21no—
353#106—22no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8D4×C4711yescharacteristic
361C4⋊C8D4×C4711yescharacteristic
362C4⋊C8D4×C4712yes—
363#103—21no—
364#103—22no—
365#103—22no—
366#103—22no—
367#103—21no—
368#103—21no—
369#102—21no—
370#102—22no—
371#101—42no—
372#97—21no—
373C4×C8—21no—
374#41—21no—
375#41—21no—
376#41C2×Q8×C4712yes—
377#41#41×C4712yes—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—82no—
386C2×Q8C2×D4×C4711yescharacteristic
387C2×Q8C2×Q8×C4712yes—
388C2×D4—21no—
389C2×D4C2×D4×C4711yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4#41×C4711yescharacteristic
397C22×C4#41×C4711yescharacteristic
398C22×C4C24×C4711yescharacteristic
399Q16—24no—
400SD16—24no—
401M16—21no—
402M16—22no—
403M16—22no—
404M16—22no—
405M16—21no—
406M16—22no—
407M16—21no—
408M16—44no—
409M16C2×Q8×C4712yes—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—44no—
417C2×C8C2×D4×C4712yes—
418C2×C8C2×D4×C4712yes—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4C2×D4×C4711yescharacteristic
430C4⋊C4C2×D4×C4711yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—41no—
438C42—41no—
439C42C2×D4×C4711yescharacteristic
440C42C2×D4×C4711yescharacteristic
441C42C2×D4×C4711yescharacteristic
442C42C2×D4×C4711yescharacteristic
443C23—21no—
444C23C2×#41×C4711yescharacteristic
445Q8—21no—
446Q8—22no—
447Q8—22no—
448Q8—41no—
449Q8—41no—
450Q8—82no—
451Q8—82no—
452Q8C22⋊Q8×C4712yes—
453D4—22no—
454D4—22no—
455D4—21no—
456D4C22⋊Q8×C4712yes—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4#121×C4711yescharacteristic
474C2×C4#121×C4711yescharacteristic
475C2×C4#121×C4711yescharacteristic
476C2×C4#121×C4711yescharacteristic
477C2×C4C22×D4×C4711yescharacteristic
478C2×C4C22×D4×C4711yescharacteristic
479C2×C4C22×Q8×C4711yescharacteristic
480C2×C4C2×#41×C4711yescharacteristic
481C2×C4H(5,𝔽2)×C4711yescharacteristic
482C2×C42−1+4×C4711yescharacteristic
483C8—22no—
484C8—22no—
485C8—22no—
486C8—41no—
487C8—42no—
488C8—42no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22C2×#121×C4711yescharacteristic
495C22C47×#55211yescharacteristic
496C22C47×#55411yescharacteristic
497C4—21no—
498C4—21no—
499C4—21no—
500C4—22no—
501C4—41no—
502C4—41no—
503C4—42no—
504C4—82no—
505C4—81no—
506C4C47×#47312yes—
507C4C2×C22⋊Q8×C4711yescharacteristic
508C4C47×#53711yescharacteristic
509C4C47×#54611yescharacteristic
510C4D4×Q8×C4711yescharacteristic
511C2—21no—
512C2—21no—
513C2—42no—
514C2C47×#241711yescharacteristic
515C2C2×C47×#47311yescharacteristic
516C2C47×#306811yescharacteristic
subgroupG/H
center, socleC22×C47C2×#121
commutator, Frattini, cosocleC22×C4C24×C47
prime radicalC2×#41×C47C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ C2×C42×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C47×#477C2semidirect
5. C47×#2089C2semidirect
6. C47×#2064C2semidirect
7. C47×#1997C2semidirect
8. C47×#1950C2semidirect
11. C47×#1801C2semidirect
53. #137×C47Q8semidirect
230. #24857C47direct (N × H)
254. C2×#477C2×C47semidirect
257. #2089C2×C47semidirect
258. #2064C2×C47semidirect
259. #1997C2×C47semidirect
260. #1950C2×C47semidirect
263. #1801C2×C47semidirect
307. C47#24857direct (N × H)
312. #137Q8×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#30792
C2×C47×#4772
C4×#137×C472
C47×#29062
C47×#20892
C47×#20642
C47×#19972
C47×#19502
C47×#19082
C47×#18162
C47×#18012
#2485747