Finite Group Databasebeta

← C2×#20247103 #26384770

Subgroups — C2×#20247103 #26384770

Order 12032.

2726 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094016C211yesmaximal, characteristic, complemented
2C22×#13093982C211yesmaximal, characteristic
3#20247103C2116yesmaximal, direct product factor
4C2×#15689058C212yesmaximal
5C2×#15689047C212yesmaximal
6C2×#15689019C211yesmaximal, characteristic, complemented
7C2×#15689018C212yesmaximal, complemented
8C2×#15689005C212yesmaximal, complemented
9C2×#15688992C211yesmaximal, characteristic, complemented
10C2×#15688980C211yesmaximal, characteristic
11C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
12C2×C47×#389C211yesmaximal, characteristic, complemented
13C23×C4×D47C2211yescharacteristic
14C2×#13094016—28no—
15C2×#13094016C2218yescomplemented
16C2×#13093982—28no—
17C2×#13093982C2218yes—
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20C22×Dic47⋊C4C2212yes—
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×(C2×D47)⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2211yescharacteristic
27C22×(C2×D47)⋊C4C2211yescharacteristic
28C22×(C2×D47)⋊C4C2212yes—
29C22×C4×Dic47C2211yescharacteristic
30C22×C4×Dic47C2212yes—
31#15689058C22116yes—
32#15689047C22116yes—
33#15689019C2218yescomplemented
34#15689018C22116yescomplemented
35#15689005C22116yescomplemented
36#15688992C2218yescomplemented
37#15688980C2218yes—
38C4×(C2×D47)⋊C4C2218yes—
39C2×C47⋊#95C2211yescharacteristic
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2211yescharacteristic
43C22×C4⋊C4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2212yes—
46C22×C42×C47C2211yescharacteristic
47C47×#389C2218yescomplemented
48C2×#95×C47C2212yes—
49C24×D47C2311yescharacteristic
50C23×Dic47C2312yes—
51C23×Dic47C2312yes—
52C23×Dic47C2311yescharacteristic
53C23×Dic47C2311yescharacteristic
54C23×Dic47C2311yescharacteristic
55C22×C4×D47—24no—
56C22×C4×D47—24no—
57C22×C4×D47Q818yes—
58C22×C4×D47C2314yes—
59#13094016—232no—
60#13093982—232no—
61C2×C4⋊Dic47—24no—
62C2×C4⋊Dic47D418yes—
63C2×C4⋊Dic47C2314yes—
64C2×C4⋊Dic47C2314yes—
65C2×Dic47⋊C4—28no—
66C2×Dic47⋊C4—28no—
67C2×Dic47⋊C4—28no—
68C2×Dic47⋊C4—28no—
69C2×Dic47⋊C4—44no—
70C2×Dic47⋊C4C2318yes—
71C2×Dic47⋊C4C2318yes—
72C2×Dic47⋊C4C2318yes—
73C2×Dic47⋊C4C2318yes—
74C2×Dic47⋊C4C2318yes—
75C2×(C2×D47)⋊C4—28no—
76C2×(C2×D47)⋊C4—24no—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—44no—
79C2×(C2×D47)⋊C4C2314yes—
80C2×(C2×D47)⋊C4C2314yes—
81C2×(C2×D47)⋊C4C2318yes—
82C2×(C2×D47)⋊C4C2318yes—
83C2×C4×Dic47—24no—
84C2×C4×Dic47—28no—
85C2×C4×Dic47C2314yes—
86C2×C4×Dic47C2318yes—
87C47⋊#95C2314yes—
88C47⋊#95C2314yes—
89C47⋊#95C2318yes—
90C47⋊#95C2318yes—
91C23×C4×C47C2312yes—
92C23×C4×C47C2311yescharacteristic
93C23×C4×C47C2311yescharacteristic
94C23×C4×C47C2312yes—
95C23×C4×C47C2311yescharacteristic
96C2×C4⋊C4×C47—24no—
97C2×C4⋊C4×C47—24no—
98C2×C4⋊C4×C47—44no—
99C2×C4⋊C4×C47C2314yes—
100C2×C4⋊C4×C47C2314yes—
101C2×C4⋊C4×C47C2318yes—
102C2×C42×C47—24no—
103C2×C42×C47C2314yes—
104#95×C47C2318yes—
105C24×C47C2411yescharacteristic
106C23×D47—22no—
107C23×D47—22no—
108C23×D47—22no—
109C23×D47—22no—
110C23×D47—22no—
111C23×D47—22no—
112C23×D47C2×Q814yes—
113C23×D47C2412yes—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—24no—
117C22×C4×C47—24no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—24no—
121C22×C4×C47—24no—
122C22×C4×C47—22no—
123C22×C4×C47—24no—
124C22×C4×C47—24no—
125C22×C4×C47—22no—
126C22×C4×C47C2×D414yes—
127C22×C4×C47C2×Q814yes—
128C22×C4×C47#4114yes—
129C22×C4×C47C2412yes—
130C22×C4×C47C2412yes—
131C22×C4×C47C2414yes—
132C22×C4×C47C2414yes—
133C22×C4×C47C2412yes—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—24no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—24no—
141C22×Dic47—22no—
142C22×Dic47—24no—
143C22×Dic47—24no—
144C22×Dic47—24no—
145C22×Dic47C2×D418yes—
146C22×Dic47C2×Q814yes—
147C22×Dic47#4114yes—
148C22×Dic47#4114yes—
149C22×Dic47C2414yes—
150C22×Dic47C2414yes—
151C22×Dic47C2412yes—
152C22×Dic47C2412yes—
153C22×Dic47C2412yes—
154C2×C4×D47—232no—
155C2×C4×D47—28no—
156C2×C4×D47—28no—
157C2×C4×D47C2×Q8116yes—
158C4⋊C4×C47—28no—
159C4⋊C4×C47—28no—
160C4⋊C4×C47—48no—
161C42×C47—28no—
162(C2×D47)⋊C4—216no—
163(C2×D47)⋊C4—28no—
164(C2×D47)⋊C4—28no—
165(C2×D47)⋊C4—48no—
166C4⋊Dic47—28no—
167C4⋊Dic47C2×D4116yes—
168Dic47⋊C4—216no—
169Dic47⋊C4—216no—
170Dic47⋊C4—216no—
171Dic47⋊C4—216no—
172Dic47⋊C4—48no—
173C4×Dic47—28no—
174C4×Dic47—216no—
175C23×C47C22×D412yes—
176C23×C47C22×Q812yes—
177C23×C47C2×#4112yes—
178C23×C47C2×#4112yes—
179C23×C47C2×#4112yes—
180C23×C47H(5,𝔽2)12yes—
181C23×C472−1+412yes—
182C23×C47C2511yescharacteristic
183C22×D47—28no—
184C22×D47—28no—
185C22×D47—28no—
186C22×D47—22no—
187C22×D47—22no—
188C22×D47—22no—
189C22×D47—22no—
190C22×D47—22no—
191C22×D47—22no—
192C22×D47—44no—
193C22×D47—44no—
194C22×D47—44no—
195C22×D47—44no—
196C22×D47C22×Q814yes—
197C2×C4×C47—24no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—24no—
201C2×C4×C47—24no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—44no—
210C2×C4×C47—44no—
211C2×C4×C47—48no—
212C2×C4×C47—48no—
213C2×C4×C47C22⋊Q8116yes—
214C2×C4×C47C22×D414yes—
215C2×C4×C47C22×Q814yes—
216C2×C4×C47C2×#4114yes—
217C2×Dic47—28no—
218C2×Dic47—216no—
219C2×Dic47—28no—
220C2×Dic47—28no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—24no—
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—24no—
228C2×Dic47—22no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—48no—
233C2×Dic47C22×D418yes—
234C2×Dic47C22×Q814yes—
235C2×Dic47C2×#4114yes—
236C2×Dic47C2×#4114yes—
237C4×D47—232no—
238C2×#2441—471nomaximal, Sylow
239C22×C47C2×C22⋊Q814yes—
240C22×C47#53314yes—
241C22×C47#54414yes—
242C22×C47#54714yes—
243C22×C47#55014yes—
244C22×C47#55214yes—
245C22×C47#55314yes—
246C22×C47C23×D411yescharacteristic
247C22×C47C23×Q811yescharacteristic
248C22×C47C22×#4111yescharacteristic
249C22×C47C22×#4111yescharacteristic
250C22×C47C22×#4111yescharacteristic
251C22×C47C2×H(5,𝔽2)11yescharacteristic
252C22×C47C2×2−1+411yescharacteristic
253C2×D47—24no—
254C2×D47—24no—
255C2×D47—24no—
256C2×D47—42no—
257C2×D47—416no—
258C2×D47—42no—
259C2×D47—42no—
260C2×D47—42no—
261C4×C47—42no—
262C4×C47—44no—
263C4×C47—42no—
264C4×C47—44no—
265C4×C47C2×C22⋊Q818yes—
266Dic47—24no—
267Dic47—28no—
268Dic47—24no—
269Dic47—24no—
270Dic47—44no—
271C22×#125—471no—
272C22×#117—471no—
273#2441—4716no—
274C2×#399—472no—
275C2×#396—472no—
276C2×#389—472no—
277C2×#389—471no—
278C2×#389D4711yescharacteristic, complemented
279C2×#387—472no—
280C2×#384—471no—
281C2×#380—471no—
282C2×C4×C22⋊C4—471no—
283C2×C47#244118yescomplemented
284C2×C47C22×C22⋊Q811yescharacteristic
285C2×C47C2×#53311yescharacteristic
286C2×C47C2×#54411yescharacteristic
287C2×C47C2×#54711yescharacteristic
288C2×C47C2×#55011yescharacteristic
289C2×C47C2×#55211yescharacteristic
290C2×C47C2×#55311yescharacteristic
291D47—44no—
292C24×C4—471no—
293C2×#125—478no—
294C2×#125—948no—
295C2×#117—478no—
296C2×#117—948no—
297C22×C4⋊C4—471no—
298C22×C4⋊C4—472no—
299C22×C4⋊C4—472no—
300C22×C4⋊C4—472no—
301C22×C4⋊C4—472no—
302C22×C4⋊C4—471no—
303C22×C4⋊C4—472no—
304C22×C4⋊C4C2×D4711yescharacteristic
305C22×C4⋊C4C2×D4711yescharacteristic
306C22×C4⋊C4C2×D4712yes—
307C22×C22⋊C4—471no—
308C22×C22⋊C4—472no—
309C22×C22⋊C4—471no—
310C22×C22⋊C4—472no—
311C22×C42—472no—
312C22×C42—471no—
313C22×C42C2×D4711yescharacteristic
314#399—4716no—
315#396—4716no—
316#389—4716no—
317#389—478no—
318#389C2×D4718yescomplemented
319#387—4716no—
320#384—478no—
321#380—478no—
322C4×C22⋊C4—478no—
323C2×#95—472no—
324C2×#95—471no—
325C2×#95—471no—
326C2×#95—472no—
327C2×#95C2×D4712yes—
328C47C2×#244111yescharacteristic, complemented, Sylow
329C25—471no—
330C23×C4—472no—
331C23×C4—471no—
332C23×C4—471no—
333C23×C4—471no—
334C23×C4—478no—
335C23×C4—474no—
336C23×C4—472no—
337C23×C4—944no—
338C23×C4—944no—
339C23×C4C22×D4711yescharacteristic
340C23×C4C22×D4712yes—
341C23×C4C22×D4711yescharacteristic
342C23×C4C22×D4712yes—
343C23×C4C22×D4711yescharacteristic
344#125—9432no—
345#117—9432no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—478no—
350C2×C4⋊C4—474no—
351C2×C4⋊C4—478no—
352C2×C4⋊C4—478no—
353C2×C4⋊C4—474no—
354C2×C4⋊C4—478no—
355C2×C4⋊C4—478no—
356C2×C4⋊C4—478no—
357C2×C4⋊C4—948no—
358C2×C4⋊C4—948no—
359C2×C4⋊C4—948no—
360C2×C4⋊C4—948no—
361C2×C4⋊C4—944no—
362C2×C4⋊C4—1884no—
363C2×C4⋊C4C22×D4714yes—
364C2×C4⋊C4C22×D4714yes—
365C2×C4⋊C4C22×D4718yes—
366C2×C22⋊C4—478no—
367C2×C22⋊C4—474no—
368C2×C22⋊C4—474no—
369C2×C22⋊C4—478no—
370C2×C22⋊C4—948no—
371C2×C22⋊C4—944no—
372C2×C22⋊C4—944no—
373C2×C22⋊C4—1884no—
374C2×C42—24no—
375C2×C42—478no—
376C2×C42—474no—
377C2×C42—948no—
378C2×C42—944no—
379C2×C42C22×D4714yes—
380#95—474no—
381#95—474no—
382#95—478no—
383#95—478no—
384#95C22×D4718yes—
385C24—474no—
386C24—472no—
387C24—942no—
388C24—942no—
389C24—942no—
390C24—942no—
391C24—942no—
392C24—942no—
393C24C23×D4711yescharacteristic
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—478no—
407C22×C4—474no—
408C22×C4—474no—
409C22×C4—474no—
410C22×C4—472no—
411C22×C4—472no—
412C22×C4—474no—
413C22×C4—472no—
414C22×C4—4716no—
415C22×C4—474no—
416C22×C4—942no—
417C22×C4—944no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—944no—
423C22×C4—942no—
424C22×C4—9432no—
425C22×C4—948no—
426C22×C4—948no—
427C22×C4—944no—
428C22×C4—944no—
429C22×C4—944no—
430C22×C4#1062535014yes—
431C22×C4D4×D4714yes—
432C22×C4Q8×D4714yes—
433C22×C4C23×D4714yes—
434C22×C4C23×D4712yes—
435C22×C4C23×D4712yes—
436C22×C4C23×D4714yes—
437C22×C4C23×D4712yes—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—48no—
441C4⋊C4—4716no—
442C4⋊C4—9416no—
443C4⋊C4—9416no—
444C4⋊C4—9416no—
445C4⋊C4—9416no—
446C4⋊C4—948no—
447C4⋊C4—1888no—
448C22⋊C4—948no—
449C22⋊C4—948no—
450C22⋊C4—9416no—
451C22⋊C4—1888no—
452C42—28no—
453C42—9416no—
454C42—948no—
455C23—474no—
456C23—948no—
457C23—948no—
458C23—948no—
459C23—942no—
460C23—942no—
461C23—942no—
462C23—942no—
463C23—942no—
464C23—942no—
465C23—1884no—
466C23—1884no—
467C23—1884no—
468C23—1884no—
469C23C2×D4×D4712yes—
470C23C2×Q8×D4712yes—
471C23#41×D4712yes—
472C23#41×D4712yes—
473C23C2×#1062535012yes—
474C23#1309406912yes—
475C23#1309407212yes—
476C23C24×D4711yescharacteristic
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—478no—
494C2×C4—474no—
495C2×C4—474no—
496C2×C4—474no—
497C2×C4—9416no—
498C2×C4—948no—
499C2×C4—948no—
500C2×C4—948no—
501C2×C4—942no—
502C2×C4—944no—
503C2×C4—942no—
504C2×C4—942no—
505C2×C4—942no—
506C2×C4—942no—
507C2×C4—944no—
508C2×C4—942no—
509C2×C4—9432no—
510C2×C4—944no—
511C2×C4—944no—
512C2×C4—944no—
513C2×C4—1888no—
514C2×C4#13094002116yes—
515C2×C4C2×D4×D4714yes—
516C2×C4C2×Q8×D4714yes—
517C2×C4C2×#1062535014yes—
518C22—944no—
519C22—944no—
520C22—944no—
521C22—18816no—
522C22—1882no—
523C22—1882no—
524C22—1882no—
525C22—1882no—
526C22C2×#1309400214yes—
527C22#1568967914yes—
528C22#1568975014yes—
529C22#1568977514yes—
530C22#1568979114yes—
531C22#1568981414yes—
532C22#1568982314yes—
533C22C22×D4×D4711yescharacteristic
534C22C22×Q8×D4711yescharacteristic
535C22C2×#41×D4711yescharacteristic
536C22C2×#41×D4711yescharacteristic
537C22C22×#1062535011yescharacteristic
538C22C2×#1309406911yescharacteristic
539C22C2×#1309407211yescharacteristic
540C4—42no—
541C4—42no—
542C4—44no—
543C4—44no—
544C4—948no—
545C4—944no—
546C4—944no—
547C4—944no—
548C4—1884no—
549C4C2×#1309400218yes—
550C2—1884no—
551C2#2024710318yesdirect product factor
552C2C22×#1309400211yescharacteristic
553C2C2×#1568967911yescharacteristic
554C2C2×#1568975011yescharacteristic
555C2C2×#1568977511yescharacteristic
556C2C2×#1568979111yescharacteristic
557C2C2×#1568981411yescharacteristic
558C2C2×#1568982311yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#389C2
socleC24×C47C24
prime radicalC24×D47C23
hypercenterC2×#389D47
hypocenterC47C2×#2441
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#13094016C2semidirect
3. #20247103C2direct (N × H)
6. C2×#15689019C2semidirect
7. C2×#15689018C2semidirect
8. C2×#15689005C2semidirect
9. C2×#15688992C2semidirect
12. C2×C47×#389C2semidirect
15. C2×#13094016C22semidirect
33. #15689019C22semidirect
34. #15689018C22semidirect
35. #15689005C22semidirect
36. #15688992C22semidirect
47. C47×#389C22semidirect
278. C2×#389D47semidirect
283. C2×C47#2441semidirect
318. #389C2×D47semidirect
328. C47C2×#2441semidirect
551. C2#20247103direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940162
C22×#130939822
#202471032
C2×#156890582
C2×#156890472
C2×#156890192
C2×#156890182
C2×#156890052
C2×#156889922
C2×#156889802
C2×C4×(C2×D47)⋊C42
C2×C47×#3892
C2×#244147