Finite Group Databasebeta

← #26226680

Subgroups — #26226680

Order 12032.

786 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689145C211yesmaximal, characteristic
2#20245784C211yesmaximal, characteristic
3#20245781C211yesmaximal, characteristic
4#20244159C211yesmaximal, characteristic, complemented
5#20244158C211yesmaximal, characteristic, complemented
6#20244157C211yesmaximal, characteristic, complemented
7#20244156C211yesmaximal, characteristic, complemented
8#20244155C211yesmaximal, characteristic, complemented
9#20244154C211yesmaximal, characteristic, complemented
10#20244153C211yesmaximal, characteristic, complemented
11#20243422C211yesmaximal, characteristic
12#20243420C211yesmaximal, characteristic
13#20243420C211yesmaximal, characteristic
14#20243418C211yesmaximal, characteristic
15C47×#1949C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C22×#10625324C2211yescharacteristic
18C2×C42×D47C2211yescharacteristic
19C2×#13093958C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25#15689145C418yescomplemented
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×Dic47⋊C8C2211yescharacteristic
28C2×Dic47⋊C8C2211yescharacteristic
29#15688990—21no—
30#15688990—21no—
31#15688989C2211yescharacteristic
32#15688988C2211yescharacteristic
33#15688988C2211yescharacteristic
34#15688985—21no—
35#15688985—21no—
36#15688980C2211yescharacteristic
37#15688980C2211yescharacteristic
38#15688980C2211yescharacteristic
39#15688980C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43C4×(C2×D47)⋊C4—21no—
44#15688970—21no—
45#15688969—21no—
46#15688968—21no—
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×C4⋊C8×C47C2211yescharacteristic
56C47×#382C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C2×#10625324—41no—
64C2×#10625324—41no—
65C2×#10625324—41no—
66C2×#10625324—41no—
67C2×#10625324C2×C414yes—
68C2×#10625324C2×C414yes—
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C42×D47C2×C414yes—
86C42×D47Q814yes—
87#13093958—41no—
88#13093957—41no—
89#13093957—41no—
90#13093956—41no—
91#13093955—41no—
92#13093955—41no—
93C4⋊(C47⋊C8)C2×C414yes—
94Dic47⋊C8C2×C414yes—
95Dic47⋊C8C2×C414yes—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C42×C47—21no—
116C2×C42×C47C2311yescharacteristic
117#107×C47—41no—
118#106×C47—41no—
119C4⋊C8×C47C2×C414yes—
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123C23×D47C2×D411yescharacteristic
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2411yescharacteristic
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47C2×D411yescharacteristic
136C22×Dic47C2×D411yescharacteristic
137C22×Dic47C2×D411yescharacteristic
138C2×C4×D47—21no—
139C2×C4×D47—21no—
140C2×C4×D47—21no—
141C2×C4×D47—21no—
142C2×C4×D47—41no—
143C2×C4×D47—41no—
144C2×C4×D47—41no—
145C2×C4×D47—41no—
146C2×C4×D47C4⋊C414yes—
147C2×C4×D47C4⋊C414yes—
148C2×C4×D47C22×C412yes—
149C2×C4×D47C2×Q812yes—
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47C22×C412yes—
155C2×C8×C47C22×C412yes—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C42×C47—42no—
165C42×C47C22×C412yes—
166C42×C47C2×Q812yes—
167(C2×D47)⋊C4—42no—
168(C2×D47)⋊C4—42no—
169(C2×D47)⋊C4—42no—
170(C2×D47)⋊C4—42no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4×Dic47—42no—
180C4×Dic47—42no—
181C4×Dic47—42no—
182C4×Dic47C4⋊C414yes—
183C4×Dic47C4⋊C414yes—
184C4×Dic47C22×C412yes—
185C4×Dic47C2×Q812yes—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8C22×C412yes—
191C2×C47⋊C8C22×C412yes—
192#10625324—44no—
193#10625324—44no—
194C23×C47C22×D411yescharacteristic
195C22×D47—21no—
196C22×D47—21no—
197C22×D47—21no—
198C22×D47—21no—
199C22×D47—21no—
200C22×D47—21no—
201C22×D47C2×C4⋊C412yes—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47C2×C4⋊C412yes—
219C2×C4×C47C2×C4⋊C412yes—
220C2×C4×C47C23×C411yescharacteristic
221C2×C4×C47C22×Q811yescharacteristic
222C2×C4×C47H(5,𝔽2)11yescharacteristic
223C2×C4×C47H(5,𝔽2)11yescharacteristic
224C2×C4×C472−1+411yescharacteristic
225C2×C4×C472−1+411yescharacteristic
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47C2×C4⋊C412yes—
245C2×Dic47C2×C4⋊C412yes—
246C2×Dic47C2×C4⋊C412yes—
247C4×D47—22no—
248C4×D47—22no—
249C4×D47—22no—
250C4×D47—22no—
251C4×D47—44no—
252C4×D47—44no—
253C8×C47—22no—
254C8×C47—22no—
255C47⋊C8—22no—
256C47⋊C8—22no—
257#21965—471nomaximal, Sylow
258C22×C47C22×C4⋊C411yescharacteristic
259C22×C47#53711yescharacteristic
260C22×C47#53711yescharacteristic
261C22×C47#57511yescharacteristic
262C22×C47#57511yescharacteristic
263C22×C47#57511yescharacteristic
264C22×C47#57511yescharacteristic
265C2×D47—22no—
266C2×D47—22no—
267C2×D47—22no—
268C2×D47—41no—
269C2×D47—41no—
270C2×D47—41no—
271C2×D47—41no—
272C4×C47—22no—
273C4×C47—22no—
274C4×C47—81no—
275C4×C47—81no—
276C4×C47—81no—
277C4×C47—81no—
278C4×C47#51811yescharacteristic
279C4×C47#51911yescharacteristic
280C4×C47#54411yescharacteristic
281C4×C47#54411yescharacteristic
282Dic47—22no—
283Dic47—22no—
284Dic47—22no—
285Dic47—81no—
286Dic47—81no—
287Dic47—81no—
288Dic47—81no—
289C2×#423—471no—
290#2283—471no—
291#2283—471no—
292#1949—471no—
293#1949—471no—
294#1949—471no—
295#1949—471no—
296#1949—471no—
297#1949—471no—
298#1949—471no—
299#1949D4711yescharacteristic, complemented
300#1809—471no—
301#1809—471no—
302#1809—471no—
303#1809—471no—
304C2×C47#231611yescharacteristic
305C2×C47#291111yescharacteristic
306C2×C47#291111yescharacteristic
307C2×C47#325611yescharacteristic
308C2×C47#325611yescharacteristic
309C2×C47#325611yescharacteristic
310C2×C47#325611yescharacteristic
311D47—42no—
312C22×M16—471no—
313C22×M16—471no—
314C22×C42—471no—
315C2×#107—471no—
316C2×#107—471no—
317C2×#107—471no—
318C2×#107C2×D4711yescharacteristic
319C2×#106—471no—
320C2×#106—471no—
321C2×#106—471no—
322C2×#106C2×D4711yescharacteristic
323#423—478no—
324C2×C4⋊C8—471no—
325C2×C4⋊C8—471no—
326C2×C4⋊C8—471no—
327C2×C4⋊C8C2×D4711yescharacteristic
328#383—941no—
329#383—941no—
330#382—471no—
331#382—471no—
332#382—471no—
333#382C2×D4711yescharacteristic
334#381—941no—
335#381—941no—
336#380—471no—
337#380—471no—
338#380—471no—
339#380—471no—
340C4×C4⋊C4—471no—
341C4×C4⋊C4—471no—
342C4×C4⋊C4—471no—
343C4×C4⋊C4C2×D4711yescharacteristic
344C4×C22⋊C4—941no—
345C4×C22⋊C4—941no—
346#376—941no—
347#376—941no—
348#340—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#340C2×D4711yescharacteristic
355#340C2×D4711yescharacteristic
356C47#2196511yescharacteristic, complemented, Sylow
357C23×C4—471no—
358C23×C4—471no—
359C23×C4—471no—
360C2×M16—474no—
361C2×M16—474no—
362C2×M16—1881no—
363C2×M16—1881no—
364C2×M16—1881no—
365C2×M16—1881no—
366C22×C8—471no—
367C22×C8—471no—
368C22×C8C22×D4711yescharacteristic
369C22×C8C22×D4711yescharacteristic
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C4⋊C4C22×D4711yescharacteristic
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C22⋊C4—941no—
379C2×C22⋊C4—941no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—941no—
382C2×C42—21no—
383C2×C42—471no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—474no—
387C2×C42—474no—
388C2×C42—941no—
389C2×C42—941no—
390C2×C42—941no—
391C2×C42C22×D4711yescharacteristic
392#107—41no—
393#107—1881no—
394#107—1881no—
395#107—1881no—
396#106—41no—
397#106—1881no—
398#106—1881no—
399#106—1881no—
400C4⋊C8—474no—
401C4⋊C8—474no—
402C4⋊C8—474no—
403C4⋊C8C4×D4714yes—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420C24—471no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—471no—
426C22×C4—471no—
427C22×C4—471no—
428C22×C4—472no—
429C22×C4—472no—
430C22×C4—474no—
431C22×C4—474no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4—941no—
440C22×C4—1881no—
441C22×C4—1881no—
442C22×C4—1881no—
443C22×C4—1881no—
444C22×C4D4×D4711yescharacteristic
445C22×C4D4×D4711yescharacteristic
446C22×C4C23×D4711yescharacteristic
447M16—1884no—
448M16—1884no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—472no—
454C2×C8—472no—
455C2×C8—941no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8C2×C4×D4712yes—
460C2×C8C2×C4×D4712yes—
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—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C22⋊C4—1882no—
478C22⋊C4—1882no—
479C22⋊C4—1882no—
480C22⋊C4—1882no—
481C42—42no—
482C42—474no—
483C42—474no—
484C42—472no—
485C42—472no—
486C42—1882no—
487C42—1882no—
488C42—1882no—
489C42C2×C4×D4712yes—
490C42Q8×D4712yes—
491C23—472no—
492C23—941no—
493C23—941no—
494C23—941no—
495C23—941no—
496C23—941no—
497C23—941no—
498C23C2×D4×D4711yescharacteristic
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—472no—
516C2×C4—472no—
517C2×C4—472no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—1884no—
529C2×C4—1884no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4C4⋊C4×D4712yes—
543C2×C4C4⋊C4×D4712yes—
544C2×C4C22×C4×D4711yescharacteristic
545C2×C4C2×Q8×D4711yescharacteristic
546C2×C4#1309406911yescharacteristic
547C2×C4#1309407011yescharacteristic
548C2×C4#1309407111yescharacteristic
549C2×C4#1309407211yescharacteristic
550C8—22no—
551C8—22no—
552C8—942no—
553C8—942no—
554C22—942no—
555C22—942no—
556C22—942no—
557C22—1881no—
558C22—1881no—
559C22—1881no—
560C22—1881no—
561C22C2×C4⋊C4×D4711yescharacteristic
562C22#1568970211yescharacteristic
563C22#1568970311yescharacteristic
564C22#1568991911yescharacteristic
565C22#1568992011yescharacteristic
566C22#1568992011yescharacteristic
567C22#1568992211yescharacteristic
568C4—22no—
569C4—22no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—942no—
575C4—942no—
576C4—942no—
577C4—3761no—
578C4—3761no—
579C4—3761no—
580C4—3761no—
581C4#1568959511yescharacteristic
582C4#1568959911yescharacteristic
583C4#1568975011yescharacteristic
584C4#1568975311yescharacteristic
585C2—1882no—
586C2#2024600311yescharacteristic
587C2#2025197511yescharacteristic
588C2#2025197611yescharacteristic
589C2#2025478611yescharacteristic
590C2#2025478711yescharacteristic
591C2#2025478811yescharacteristic
592C2#2025479111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1949C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1949D47
hypocenterC47#21965
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #20244159C2semidirect
5. #20244158C2semidirect
6. #20244157C2semidirect
7. #20244156C2semidirect
8. #20244155C2semidirect
9. #20244154C2semidirect
10. #20244153C2semidirect
15. C47×#1949C2semidirect
25. #15689145C4semidirect
299. #1949D47semidirect
356. C47#21965semidirect

Maximal subgroups

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

subgroupindex
C2×#156891452
#202457842
#202457812
#202441592
#202441582
#202441572
#202441562
#202441552
#202441542
#202441532
#202434222
#202434202
#202434182
C47×#19492
#2196547