Finite Group Databasebeta

← C47×#27026 #26066724

Subgroups — C47×#27026 #26066724

Order 12032.

726 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#2452C211yesmaximal, characteristic
2C47×#2370C211yesmaximal, characteristic, complemented
3C47×#2065C211yesmaximal, characteristic
4C47×#2059C211yesmaximal, characteristic, complemented
5C47×#2058C211yesmaximal, characteristic, complemented
6C47×#2025C211yesmaximal, characteristic
7C47×#1988C211yesmaximal, characteristic, complemented
8C47×#1943C211yesmaximal, characteristic, complemented
9C47×#1926C211yesmaximal, characteristic
10C47×#1925C211yesmaximal, characteristic
11C47×#1920C211yesmaximal, characteristic, complemented
12C47×#1920C211yesmaximal, characteristic, complemented
13C47×#1917C211yesmaximal, characteristic, complemented
14C47×#1864C211yesmaximal, characteristic
15C47×#1762C211yesmaximal, characteristic
16C2×C4⋊Q8×C47C2211yescharacteristic
17C2×#125×C47C2211yescharacteristic
18C2×C22⋊Q8×C47—21no—
19C2×C4×D4×C47C2211yescharacteristic
20C2×#107×C47C2211yescharacteristic
21C2×#106×C47C2211yescharacteristic
22C2×#106×C47C2211yescharacteristic
23C2×#106×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C2×#102×C47C2211yescharacteristic
26C2×#102×C47C2211yescharacteristic
27C2×#102×C47C2211yescharacteristic
28C2×#102×C47C2211yescharacteristic
29C2×#97×C47C2211yescharacteristic
30C47×#396—21no—
31C47×#393—21no—
32C47×#390C2211yescharacteristic
33C47×#390C2211yescharacteristic
34C47×#389—21no—
35C47×#389C2211yescharacteristic
36C47×#389C2211yescharacteristic
37C47×#389C2211yescharacteristic
38C47×#389C2211yescharacteristic
39C47×#387—21no—
40C47×#387—21no—
41C47×#385—21no—
42C4×C4⋊C4×C47—21no—
43C4×C4⋊C4×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2211yescharacteristic
45C4×C4⋊C4×C47C2211yescharacteristic
46C4×C4⋊C4×C47C2211yescharacteristic
47C47×#340C2211yescharacteristic
48C47×#340C2211yescharacteristic
49C47×#340C2211yescharacteristic
50C47×#340C2211yescharacteristic
51C47×#340C2211yescharacteristic
52C47×#340C2211yescharacteristic
53C47×#340C2211yescharacteristic
54C47×#340C2211yescharacteristic
55C47×#336C2211yescharacteristic
56C47×#336C2211yescharacteristic
57C47×#336C2211yescharacteristic
58C47×#336C2211yescharacteristic
59C22×Q8×C47—21no—
60C22×D4×C47C2311yescharacteristic
61C23×C4×C47—21no—
62C22×C8×C47C2311yescharacteristic
63C22×C8×C47C2311yescharacteristic
64C22×C8×C47C2311yescharacteristic
65C22×C8×C47C2311yescharacteristic
66C4⋊Q8×C47—22no—
67C4⋊Q8×C47—41no—
68#125×C47—22no—
69#125×C47—22no—
70C22⋊Q8×C47—44no—
71C4×D4×C47—22no—
72C4×D4×C47—41no—
73C2×C4⋊C4×C47—21no—
74C2×C4⋊C4×C47—21no—
75C2×C4⋊C4×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—21no—
78C2×C4⋊C4×C47—21no—
79C2×C4⋊C4×C47—21no—
80C2×C4⋊C4×C47D411yescharacteristic
81C2×C4⋊C4×C47D411yescharacteristic
82C2×C4⋊C4×C47C2311yescharacteristic
83C2×C4⋊C4×C47C2311yescharacteristic
84C2×C4⋊C4×C47C2311yescharacteristic
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47C2311yescharacteristic
87C2×C4⋊C4×C47C2311yescharacteristic
88C2×C4⋊C4×C47C2311yescharacteristic
89C2×C22⋊C4×C47—21no—
90C2×C22⋊C4×C47—21no—
91C2×C22⋊C4×C47—21no—
92C2×C22⋊C4×C47—21no—
93C2×C22⋊C4×C47—21no—
94C2×C42×C47—21no—
95C2×C42×C47—21no—
96C2×C42×C47—21no—
97C2×C42×C47C2311yescharacteristic
98C2×C42×C47C2311yescharacteristic
99C2×C42×C47C2311yescharacteristic
100#107×C47—22no—
101#106×C47—22no—
102#106×C47—22no—
103#106×C47—41no—
104C4⋊C8×C47—41no—
105#102×C47—22no—
106#102×C47—22no—
107#102×C47—41no—
108#102×C47Q814yescomplemented
109#97×C47—41no—
110#95×C47—21no—
111#95×C47—21no—
112#95×C47—21no—
113#95×C47—21no—
114#95×C47—21no—
115#95×C47—21no—
116#95×C47—21no—
117#95×C47D411yescharacteristic
118#95×C47D411yescharacteristic
119C24×C47—21no—
120C2×Q8×C47—42no—
121C2×Q8×C47—42no—
122C2×Q8×C47—81no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47C2×Q812yes—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—42no—
137C22×C4×C47—42no—
138C22×C4×C47—81no—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2411yescharacteristic
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47C2×Q812yes—
158C4⋊C4×C47—24no—
159C4⋊C4×C47—24no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—41no—
172C4⋊C4×C47—42no—
173C4⋊C4×C47—42no—
174C4⋊C4×C47—42no—
175C4⋊C4×C47—42no—
176C4⋊C4×C47—81no—
177C4⋊C4×C47C2×Q812yes—
178C4⋊C4×C47#4112yes—
179C4⋊C4×C47#4112yes—
180C4⋊C4×C47#4112yes—
181C22⋊C4×C47—42no—
182C22⋊C4×C47—42no—
183C22⋊C4×C47—42no—
184C22⋊C4×C47—42no—
185C22⋊C4×C47—81no—
186C42×C47—21no—
187C42×C47—21no—
188C42×C47—21no—
189C42×C47—21no—
190C42×C47—21no—
191C42×C47—21no—
192C42×C47—42no—
193C42×C47—81no—
194C42×C47—81no—
195C23×C47—22no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47C22×D411yescharacteristic
203Q8×C47—84no—
204D4×C47—22no—
205D4×C47—22no—
206D4×C47—41no—
207D4×C47—41no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—84no—
247C2×C4×C47AGL(1,ℤ/8)12yes—
248C2×C4×C47C22×Q811yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47H(5,𝔽2)11yescharacteristic
253C2×C4×C472−1+411yescharacteristic
254C8×C47—22no—
255C8×C47—41no—
256C8×C47—41no—
257C8×C47—41no—
258#27026C4711yesmaximal, characteristic, direct product factor, Sylow
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—81no—
263C22×C47—81no—
264C22×C47—81no—
265C22×C47—81no—
266C22×C47#52511yescharacteristic
267C22×C47#54611yescharacteristic
268C22×C47D4×Q811yescharacteristic
269C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C47#57511yescharacteristic
271C22×C47#57711yescharacteristic
272C22×C47#57711yescharacteristic
273C4×C47—22no—
274C4×C47—41no—
275C4×C47—42no—
276C4×C47—41no—
277C4×C47—42no—
278C4×C47—42no—
279C4×C47—41no—
280C4×C47—42no—
281C4×C47—41no—
282C4×C47—41no—
283C4×C47—81no—
284C4×C47—81no—
285C4×C47—81no—
286C4×C47#53311yescharacteristic
287C4×C47#54211yescharacteristic
288C4×C47#55411yescharacteristic
289C4×C47#55611yescharacteristic
290#2452C2×C4711yescharacteristic
291#2370C2×C4711yescharacteristic, complemented
292#2065C2×C4711yescharacteristic
293#2059C2×C4711yescharacteristic, complemented
294#2058C2×C4711yescharacteristic, complemented
295#2025C2×C4711yescharacteristic
296#1988C2×C4711yescharacteristic, complemented
297#1943C2×C4711yescharacteristic, complemented
298#1926C2×C4711yescharacteristic
299#1925C2×C4711yescharacteristic
300#1920C2×C4711yescharacteristic, complemented
301#1920C2×C4711yescharacteristic, complemented
302#1917C2×C4711yescharacteristic, complemented
303#1864C2×C4711yescharacteristic
304#1762C2×C4711yescharacteristic
305C2×C47—82no—
306C2×C47#246511yescharacteristic
307C2×C47#311511yescharacteristic
308C2×C47#312511yescharacteristic
309C2×C47#331011yescharacteristic
310C2×C47#332611yescharacteristic
311C2×C47#338411yescharacteristic
312C2×C47#338811yescharacteristic
313C2×C4⋊Q8C22×C4711yescharacteristic
314C2×#125C22×C4711yescharacteristic
315C2×C22⋊Q8—21no—
316C2×C4×D4C22×C4711yescharacteristic
317C2×#107C22×C4711yescharacteristic
318C2×#106C22×C4711yescharacteristic
319C2×#106C22×C4711yescharacteristic
320C2×#106C22×C4711yescharacteristic
321C2×C4⋊C8C22×C4711yescharacteristic
322C2×#102C22×C4711yescharacteristic
323C2×#102C22×C4711yescharacteristic
324C2×#102C22×C4711yescharacteristic
325C2×#102C22×C4711yescharacteristic
326C2×#97C22×C4711yescharacteristic
327#396—21no—
328#393—21no—
329#390C22×C4711yescharacteristic
330#390C22×C4711yescharacteristic
331#389—21no—
332#389C22×C4711yescharacteristic
333#389C22×C4711yescharacteristic
334#389C22×C4711yescharacteristic
335#389C22×C4711yescharacteristic
336#387—21no—
337#387—21no—
338#385—21no—
339C4×C4⋊C4—21no—
340C4×C4⋊C4C22×C4711yescharacteristic
341C4×C4⋊C4C22×C4711yescharacteristic
342C4×C4⋊C4C22×C4711yescharacteristic
343C4×C4⋊C4C22×C4711yescharacteristic
344#340C22×C4711yescharacteristic
345#340C22×C4711yescharacteristic
346#340C22×C4711yescharacteristic
347#340C22×C4711yescharacteristic
348#340C22×C4711yescharacteristic
349#340C22×C4711yescharacteristic
350#340C22×C4711yescharacteristic
351#340C22×C4711yescharacteristic
352#336C22×C4711yescharacteristic
353#336C22×C4711yescharacteristic
354#336C22×C4711yescharacteristic
355#336C22×C4711yescharacteristic
356C47#2702611yescharacteristic, direct product factor, Sylow
357C22×Q8—21no—
358C22×D4C23×C4711yescharacteristic
359C23×C4—21no—
360C22×C8C23×C4711yescharacteristic
361C22×C8C23×C4711yescharacteristic
362C22×C8C23×C4711yescharacteristic
363C22×C8C23×C4711yescharacteristic
364C4⋊Q8—22no—
365C4⋊Q8—41no—
366#125—22no—
367#125—22no—
368C22⋊Q8—44no—
369C4×D4—22no—
370C4×D4—41no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4D4×C4711yescharacteristic
379C2×C4⋊C4D4×C4711yescharacteristic
380C2×C4⋊C4C23×C4711yescharacteristic
381C2×C4⋊C4C23×C4711yescharacteristic
382C2×C4⋊C4C23×C4711yescharacteristic
383C2×C4⋊C4C23×C4711yescharacteristic
384C2×C4⋊C4C23×C4711yescharacteristic
385C2×C4⋊C4C23×C4711yescharacteristic
386C2×C4⋊C4C23×C4711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42C23×C4711yescharacteristic
396C2×C42C23×C4711yescharacteristic
397C2×C42C23×C4711yescharacteristic
398#107—22no—
399#106—22no—
400#106—22no—
401#106—41no—
402C4⋊C8—41no—
403#102—22no—
404#102—22no—
405#102—41no—
406#102Q8×C4714yescomplemented
407#97—41no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95D4×C4711yescharacteristic
416#95D4×C4711yescharacteristic
417C24—21no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—81no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C2×Q8×C4712yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4C2×D4×C4711yescharacteristic
438C22×C4C2×D4×C4711yescharacteristic
439C22×C4C2×D4×C4711yescharacteristic
440C22×C4C2×D4×C4711yescharacteristic
441C22×C4C2×D4×C4711yescharacteristic
442C22×C4C2×D4×C4711yescharacteristic
443C22×C4C24×C4711yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8C2×Q8×C4712yes—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4C2×Q8×C4712yes—
476C4⋊C4#41×C4712yes—
477C4⋊C4#41×C4712yes—
478C4⋊C4#41×C4712yes—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—81no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42—81no—
492C42—81no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C22×D4×C4711yescharacteristic
501Q8—84no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—84no—
545C2×C4AGL(1,ℤ/8)×C4712yes—
546C2×C4C22×Q8×C4711yescharacteristic
547C2×C4C2×#41×C4711yescharacteristic
548C2×C4C2×#41×C4711yescharacteristic
549C2×C4C2×#41×C4711yescharacteristic
550C2×C4H(5,𝔽2)×C4711yescharacteristic
551C2×C42−1+4×C4711yescharacteristic
552C8—22no—
553C8—41no—
554C8—41no—
555C8—41no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22C47×#52511yescharacteristic
564C22C47×#54611yescharacteristic
565C22D4×Q8×C4711yescharacteristic
566C22C2×AGL(1,ℤ/8)×C4711yescharacteristic
567C22C47×#57511yescharacteristic
568C22C47×#57711yescharacteristic
569C22C47×#57711yescharacteristic
570C4—22no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4C47×#53311yescharacteristic
584C4C47×#54211yescharacteristic
585C4C47×#55411yescharacteristic
586C4C47×#55611yescharacteristic
587C2—82no—
588C2C47×#246511yescharacteristic
589C2C47×#311511yescharacteristic
590C2C47×#312511yescharacteristic
591C2C47×#331011yescharacteristic
592C2C47×#332611yescharacteristic
593C2C47×#338411yescharacteristic
594C2C47×#338811yescharacteristic
subgroupG/H
center, socleC23×C47C22×D4
commutator, Frattini, cosocleC22×C4C24×C47
prime radicalC22×D4×C47C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C47 ◁ C4×C4⋊C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C47×#2370C2semidirect
4. C47×#2059C2semidirect
5. C47×#2058C2semidirect
7. C47×#1988C2semidirect
8. C47×#1943C2semidirect
11. C47×#1920C2semidirect
12. C47×#1920C2semidirect
13. C47×#1917C2semidirect
108. #102×C47Q8semidirect
258. #27026C47direct (N × H)
291. #2370C2×C47semidirect
293. #2059C2×C47semidirect
294. #2058C2×C47semidirect
296. #1988C2×C47semidirect
297. #1943C2×C47semidirect
300. #1920C2×C47semidirect
301. #1920C2×C47semidirect
302. #1917C2×C47semidirect
356. C47#27026direct (N × H)
406. #102Q8×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#24522
C47×#23702
C47×#20652
C47×#20592
C47×#20582
C47×#20252
C47×#19882
C47×#19432
C47×#19262
C47×#19252
C47×#19202
C47×#19172
C47×#18642
C47×#17622
#2702647