Finite Group Databasebeta

← #25110508

Subgroups — #25110508

Order 11008.

796 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145065C211yesmaximal, characteristic
2#19143511C211yesmaximal, characteristic, complemented
3#19143455C211yesmaximal, characteristic, complemented
4#19135493C211yesmaximal, characteristic, complemented
5#19135130C211yesmaximal, characteristic, complemented
6#19135125C211yesmaximal, characteristic, complemented
7#19135119C211yesmaximal, characteristic, complemented
8#19135114C211yesmaximal, characteristic, complemented
9#19135087C211yesmaximal, characteristic, complemented
10#19135080C211yesmaximal, characteristic, complemented
11#19135078C211yesmaximal, characteristic, complemented
12#19135071C211yesmaximal, characteristic, complemented
13#19134720C211yesmaximal, characteristic, complemented
14#19134350C211yesmaximal, characteristic, complemented
15C43×#1977C211yesmaximal, characteristic, complemented
16C2×#13063356C2211yescharacteristic
17C2×#13063351C2211yescharacteristic
18C2×#13063347C2211yescharacteristic
19#14583665C2211yescharacteristic
20#14583665C2211yescharacteristic
21#14583473C2211yescharacteristic
22#14583466C2211yescharacteristic
23#14583377C2211yescharacteristic
24#14583374C2211yescharacteristic
25#14583290C412yescomplemented
26#14583285C412yescomplemented
27#14583280C412yescomplemented
28#14583270C412yescomplemented
29#14583020—22no—
30#14583015—22no—
31#14583000—22no—
32SD16×Dic43—22no—
33#14582937C2211yescharacteristic
34#14582915C2211yescharacteristic
35#14582915C2211yescharacteristic
36#14582915C2211yescharacteristic
37#14582915C2211yescharacteristic
38#14582908C2211yescharacteristic
39#14582904C2211yescharacteristic
40#14582872C2211yescharacteristic
41#14582872C2211yescharacteristic
42#14582854C2211yescharacteristic
43#14582854C2211yescharacteristic
44#14582671C2211yescharacteristic
45#14582671C2211yescharacteristic
46C43⋊#338C2211yescharacteristic
47#14582656C2211yescharacteristic
48#14582656C2211yescharacteristic
49#14582656C2211yescharacteristic
50#14582655C2211yescharacteristic
51#14582655C2211yescharacteristic
52C22×SD16×C43C2211yescharacteristic
53C2×#108×C43C2211yescharacteristic
54C43×#419C2211yescharacteristic
55C43×#418C2211yescharacteristic
56C2×#101×C43C2211yescharacteristic
57C2×#100×C43C2211yescharacteristic
58C43×#405C2211yescharacteristic
59C2×#10621898C2311yescharacteristic
60C2×#10621898C2311yescharacteristic
61#13063356—41no—
62#13063356—41no—
63#13063351—41no—
64#13063351—41no—
65#13063347—21no—
66#13063347—21no—
67#13063347—21no—
68#13063347—21no—
69C2×Dic172C2×C411yescharacteristic, complemented
70C2×#10621885C2×C412yescomplemented
71C2×#10621873C2×C411yescharacteristic, complemented
72C2×#10621873C2×C411yescharacteristic, complemented
73C2×#10621883C2×C412yes—
74C2×#10621884C2×C412yescomplemented
75C2×D344C2×C411yescharacteristic, complemented
76C2×#10621882C2×C412yes—
77#13063333—22no—
78#13063333—22no—
79#13063331—41no—
80#13063331—41no—
81C2×#10621877C2311yescharacteristic
82C2×#10621877C2311yescharacteristic
83M16×D43—41no—
84M16×D43—41no—
85#13063323C2×C412yescomplemented
86#13063321C2×C412yes—
87#13063314C2×C412yescomplemented
88#13063312C2×C412yes—
89#13063308—41no—
90#13063301—41no—
91#13063296—41no—
92#13063294—41no—
93Q8×Dic43—22no—
94Q8×Dic43—41no—
95D4×Dic43—22no—
96D4×Dic43—41no—
97#13063284—41no—
98#13063284C2311yescharacteristic
99#13063284C2311yescharacteristic
100#13063284C2311yescharacteristic
101#13063284C2311yescharacteristic
102C2×C22⋊Dic43—41no—
103#13063259—21no—
104#13063257—21no—
105#13063251—22no—
106#13063251—22no—
107#13063251—41no—
108#13063251—41no—
109#13063249—22no—
110#13063246—22no—
111#13063239—41no—
112#13063239—41no—
113#13063235C2×C412yescomplemented
114#13063235D412yescomplemented
115C8×Dic43C2×C412yescomplemented
116C8×Dic43D412yescomplemented
117C22×Q8×C43C2311yescharacteristic
118C22×D4×C43C2311yescharacteristic
119C2×SD16×C43—21no—
120C2×SD16×C43—21no—
121C2×SD16×C43—21no—
122C2×SD16×C43—21no—
123C2×SD16×C43C2×C412yes—
124C2×SD16×C43C2×C412yes—
125C2×M16×C43C2311yescharacteristic
126C2×M16×C43C2311yescharacteristic
127C2×M16×C43C2311yescharacteristic
128C22×C8×C43C2311yescharacteristic
129#117×C43C2311yescharacteristic
130#108×C43—21no—
131#108×C43—21no—
132#103×C43—41no—
133#102×C43—41no—
134#101×C43—41no—
135#100×C43—41no—
136#97×C43—21no—
137C4×C8×C43—21no—
138C24×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43C22×C412yes—
145C2×D4×C43—21no—
146C2×D4×C43—21no—
147C2×D4×C43—21no—
148C2×D4×C43—21no—
149C2×D4×C43—21no—
150C2×D4×C43C22×C412yes—
151C22×C4×C43—21no—
152C22×C4×C43C2411yescharacteristic
153C2×C22⋊D43—21no—
154C2×C22⋊D43—21no—
155C2×C22⋊D43—42no—
156C22×Dic43—42no—
157#10621898—21no—
158#10621898—21no—
159#10621898—21no—
160#10621898—21no—
161#10621898—21no—
162#10621898—21no—
163#10621898—21no—
164#10621898—21no—
165C2×D172C22×C411yescharacteristic
166C2×D172C22×C411yescharacteristic
167C2×C4×D43—21no—
168C2×C4×D43—21no—
169C2×Dic86C22×C411yescharacteristic
170C2×Dic86C22×C411yescharacteristic
171SD16×C43—22no—
172SD16×C43—22no—
173SD16×C43—22no—
174SD16×C43—22no—
175M16×C43—21no—
176M16×C43—21no—
177M16×C43—21no—
178M16×C43—21no—
179M16×C43—21no—
180M16×C43—21no—
181C2×C8×C43—21no—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43C22×C411yescharacteristic
185C2×C8×C43C22×C411yescharacteristic
186C2×C8×C43C2×D411yescharacteristic
187C2×C8×C43C2×D411yescharacteristic
188C2×C8×C43#4111yescharacteristic
189C2×C8×C43#4111yescharacteristic
190C4⋊C4×C43—21no—
191C22⋊C4×C43—21no—
192C42×C43—21no—
193C22⋊Dic43—21no—
194C22⋊Dic43—21no—
195C22⋊Dic43—21no—
196C22⋊Dic43—21no—
197C22⋊Dic43—42no—
198C22⋊Dic43—42no—
199#10621885—42no—
200#10621884—42no—
201#10621883—42no—
202#10621882—42no—
203(C2×D43)⋊C4—42no—
204C4⋊Dic43—21no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—42no—
209Dic43⋊C4—42no—
210C4×Dic43—21no—
211C4×Dic43—21no—
212C4×Dic43—42no—
213C4×Dic43C22×C412yes—
214C4×Dic43C2×D412yes—
215#10621877—21no—
216#10621877—21no—
217#10621877—21no—
218#10621877—21no—
219C2×C43⋊C8C22×C412yes—
220C2×C43⋊C8C2×D412yes—
221Dic172—21no—
222Dic172—21no—
223D344—21no—
224D344—21no—
225#10621873—21no—
226#10621873—21no—
227#10621873—21no—
228#10621873—21no—
229#10621872—42no—
230#10621872—42no—
231C8×D43—42no—
232C8×D43—42no—
233C23×C43—22no—
234C23×C43—41no—
235C23×C43—41no—
236C23×C43—161no—
237C23×C43H(5,𝔽2)11yescharacteristic
238C22×D43—21no—
239C22×D43—21no—
240Q8×C43—22no—
241Q8×C43—22no—
242Q8×C43—41no—
243Q8×C43—41no—
244D4×C43—22no—
245D4×C43—22no—
246D4×C43—41no—
247D4×C43—41no—
248C2×C4×C43—22no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43C23×C411yescharacteristic
253C2×C4×C43C22×D411yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×C4×C43H(5,𝔽2)11yescharacteristic
256C2×C4×C43H(5,𝔽2)11yescharacteristic
257C2×C4×C43H(5,𝔽2)11yescharacteristic
258C22⋊D43—41no—
259C22⋊D43—41no—
260C22⋊D43—41no—
261C22⋊D43—41no—
262C22⋊D43—162no—
263C2×Dic43—22no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—22no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—162no—
270D172—21no—
271D172—21no—
272D172—21no—
273D172—21no—
274C4×D43—41no—
275C4×D43—41no—
276C4×D43—41no—
277C4×D43—41no—
278Dic86—21no—
279Dic86—21no—
280Dic86—21no—
281Dic86—21no—
282C8×C43—41no—
283C8×C43—41no—
284C8×C43—41no—
285C8×C43C4×D411yescharacteristic
286C8×C43C4×D411yescharacteristic
287C8×C43C4×D411yescharacteristic
288C8×C43C4×D411yescharacteristic
289C43⋊C8—22no—
290C43⋊C8—22no—
291#22284—431nomaximal, Sylow
292C22×C43—41no—
293C22×C43—42no—
294C22×C43—82no—
295C22×C43—82no—
296C22×C43—161no—
297C22×C43#51811yescharacteristic
298C22×C43#53511yescharacteristic
299C22×C43#54011yescharacteristic
300C2×D43—41no—
301C2×D43—41no—
302C2×D43—81no—
303C2×D43—81no—
304C4×C43—42no—
305C4×C43—81no—
306C4×C43C2×C4×D411yescharacteristic
307C4×C43#51811yescharacteristic
308C4×C43#53511yescharacteristic
309C4×C43#54011yescharacteristic
310Dic43—42no—
311Dic43—42no—
312Dic43—41no—
313Dic43—41no—
314Dic43—81no—
315Dic43—81no—
316#3154—431no—
317#2957—431no—
318#2949—431no—
319#1977D4311yescharacteristic, complemented
320#1976—431no—
321#1912—431no—
322#1911—431no—
323#1910—431no—
324#1909—431no—
325#1905—431no—
326#1904—431no—
327#1903—431no—
328#1902—431no—
329#1833—431no—
330#1747—431no—
331C2×C43—21no—
332C2×C43—21no—
333C2×C43—21no—
334C2×C43—82no—
335C2×C43#232811yescharacteristic
336D43—81no—
337D43—81no—
338C2×#137—431no—
339C2×AGL(1,ℤ/8)—431no—
340C2×#135—431no—
341C22×SD16C2×D4311yescharacteristic
342#568—431no—
343#568—431no—
344#536—431no—
345#535—431no—
346#519—431no—
347#518—431no—
348#497—432no—
349#496—432no—
350#495—432no—
351#492—432no—
352#444—862no—
353#443—862no—
354#440—862no—
355C4×SD16—862no—
356C2×#108C2×D4311yescharacteristic
357#427—431no—
358#421—431no—
359#421—431no—
360#421—431no—
361#421—431no—
362#419—431no—
363#419C2×D4311yescharacteristic
364#418—431no—
365#418C2×D4311yescharacteristic
366C2×#101C2×D4311yescharacteristic
367C2×#100C2×D4311yescharacteristic
368#410—431no—
369#410—431no—
370#405—431no—
371#405—431no—
372#405C2×D4311yescharacteristic
373#344—431no—
374#344—431no—
375#338—431no—
376H(3,ℤ/4)—431no—
377H(3,ℤ/4)—431no—
378H(3,ℤ/4)—431no—
379H(3,ℤ/4)—431no—
380H(3,ℤ/4)—431no—
381C43#2228411yescharacteristic, complemented, Sylow
382C2×#41—431no—
383C2×#41—431no—
384C22×Q8C22×D4311yescharacteristic
385C22×D4C22×D4311yescharacteristic
386#137—1721no—
387#137—1721no—
388AGL(1,ℤ/8)—1721no—
389AGL(1,ℤ/8)—1721no—
390#135—861no—
391#135—861no—
392#135—861no—
393#135—861no—
394C2×Q16—432no—
395C2×Q16—431no—
396C2×SD16—21no—
397C2×SD16—21no—
398C2×SD16—21no—
399C2×SD16—21no—
400C2×SD16—432no—
401C2×SD16—432no—
402C2×SD16—431no—
403C2×SD16—431no—
404C2×SD16C4×D4312yes—
405C2×SD16C4×D4312yes—
406C2×D8—432no—
407C2×D8—431no—
408#131—862no—
409#131—862no—
410#131—1721no—
411#131—1721no—
412C2×M16—431no—
413C2×M16—431no—
414C2×M16—1721no—
415C2×M16—1721no—
416C2×M16C22×D4311yescharacteristic
417C2×M16C22×D4311yescharacteristic
418C2×M16C22×D4311yescharacteristic
419C22×C8C22×D4311yescharacteristic
420C4⋊Q8—432no—
421#127—432no—
422#124—432no—
423#124—432no—
424#123—1721no—
425C22⋊Q8—1721no—
426#121—1721no—
427#120—1721no—
428C4×Q8—862no—
429C4×Q8—1721no—
430C4×D4—862no—
431C4×D4—1721no—
432#117—431no—
433#117—431no—
434#117—431no—
435#117—431no—
436#117—1721no—
437#117C22×D4311yescharacteristic
438C2×C22⋊C4—1721no—
439#108—21no—
440#108—21no—
441#107—861no—
442#106—861no—
443#104—862no—
444#104—862no—
445#104—1721no—
446#104—1721no—
447#103—41no—
448#103—862no—
449#102—41no—
450#102—862no—
451#101—41no—
452#100—41no—
453#99—1721no—
454#99—1721no—
455#97—21no—
456#97—432no—
457#97—432no—
458C4×C8—21no—
459C4×C8—432no—
460C4×C8—432no—
461C24—21no—
462#41—861no—
463#41—861no—
464#41—861no—
465#41—861no—
466#41—861no—
467#41—861no—
468#41—861no—
469#41—861no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—431no—
476C2×Q8—431no—
477C2×Q8C2×C4×D4312yes—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—431no—
484C2×D4—431no—
485C2×D4—861no—
486C2×D4—861no—
487C2×D4—1722no—
488C2×D4C2×C4×D4312yes—
489C22×C4—21no—
490C22×C4—861no—
491C22×C4—861no—
492C22×C4—1722no—
493C22×C4C23×D4311yescharacteristic
494Q16—861no—
495Q16—861no—
496Q16—1722no—
497SD16—22no—
498SD16—22no—
499SD16—22no—
500SD16—22no—
501SD16—861no—
502SD16—861no—
503SD16—861no—
504SD16—861no—
505SD16—1722no—
506SD16—1722no—
507D8—861no—
508D8—861no—
509D8—1722no—
510M16—21no—
511M16—21no—
512M16—21no—
513M16—21no—
514M16—21no—
515M16—21no—
516M16—861no—
517M16—861no—
518M16—861no—
519M16—861no—
520M16—1722no—
521M16—1722no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—432no—
526C2×C8—432no—
527C2×C8—1722no—
528C2×C8—1722no—
529C2×C8C2×C4×D4311yescharacteristic
530C2×C8C2×C4×D4311yescharacteristic
531C2×C8D4×D4311yescharacteristic
532C2×C8D4×D4311yescharacteristic
533C2×C8#1062190011yescharacteristic
534C2×C8#1062190011yescharacteristic
535C4⋊C4—21no—
536C4⋊C4—861no—
537C4⋊C4—861no—
538C4⋊C4—861no—
539C4⋊C4—861no—
540C4⋊C4—1722no—
541C4⋊C4—1722no—
542C22⋊C4—21no—
543C22⋊C4—861no—
544C22⋊C4—861no—
545C22⋊C4—861no—
546C22⋊C4—861no—
547C22⋊C4—1722no—
548C22⋊C4—1722no—
549C22⋊C4—1722no—
550C42—21no—
551C42—432no—
552C42—432no—
553C42—861no—
554C42—861no—
555C42—1722no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—161no—
560C23—861no—
561C23—861no—
562C23#1306337111yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—861no—
568Q8—861no—
569Q8—861no—
570Q8—861no—
571D4—22no—
572D4—22no—
573D4—41no—
574D4—41no—
575D4—861no—
576D4—861no—
577D4—861no—
578D4—861no—
579D4—1721no—
580D4—1721no—
581D4—1721no—
582D4—1721no—
583D4—6882no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—862no—
589C2×C4—861no—
590C2×C4—861no—
591C2×C4—862no—
592C2×C4—1721no—
593C2×C4—1721no—
594C2×C4—1721no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4—1721no—
598C2×C4—6882no—
599C2×C4C22×C4×D4311yescharacteristic
600C2×C4C2×D4×D4311yescharacteristic
601C2×C4C2×#1062190011yescharacteristic
602C2×C4#1306337011yescharacteristic
603C2×C4#1306337011yescharacteristic
604C2×C4#1306337111yescharacteristic
605C8—41no—
606C8—41no—
607C8—41no—
608C8—862no—
609C8—862no—
610C8(C22⋊D43)⋊C411yescharacteristic
611C8(C22⋊D43)⋊C411yescharacteristic
612C8(C22⋊D43)⋊C411yescharacteristic
613C8(C22⋊D43)⋊C411yescharacteristic
614C22—41no—
615C22—42no—
616C22—82no—
617C22—82no—
618C22—161no—
619C22—3441no—
620C22—3441no—
621C22—1721no—
622C22—1721no—
623C22#1458337111yescharacteristic
624C22#1458346711yescharacteristic
625C22#1458350211yescharacteristic
626C4—42no—
627C4—81no—
628C4—3441no—
629C4—3441no—
630C4—1722no—
631C4—1722no—
632C4—1721no—
633C4—1721no—
634C4C2×(C22⋊D43)⋊C411yescharacteristic
635C4#1458337111yescharacteristic
636C4#1458346211yescharacteristic
637C4#1458350211yescharacteristic
638C2—21no—
639C2—21no—
640C2—21no—
641C2—82no—
642C2—3441no—
643C2—3441no—
644C2#1913731911yescharacteristic
subgroupG/H
centerC2#19137319
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1977C2
socleC2×C43#2328
prime radical#19145065C2
cosocleC22×C4×C43C24
hypercenter#1977D43
hypocenterC43#22284
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1977

Complements

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

NH ≅ G/Ntype
2. #19143511C2semidirect
3. #19143455C2semidirect
4. #19135493C2semidirect
5. #19135130C2semidirect
6. #19135125C2semidirect
7. #19135119C2semidirect
8. #19135114C2semidirect
9. #19135087C2semidirect
10. #19135080C2semidirect
11. #19135078C2semidirect
12. #19135071C2semidirect
13. #19134720C2semidirect
14. #19134350C2semidirect
15. C43×#1977C2semidirect
25. #14583290C4semidirect
26. #14583285C4semidirect
27. #14583280C4semidirect
28. #14583270C4semidirect
69. C2×Dic172C2×C4semidirect
70. C2×#10621885C2×C4semidirect
71. C2×#10621873C2×C4semidirect
72. C2×#10621873C2×C4semidirect
74. C2×#10621884C2×C4semidirect
75. C2×D344C2×C4semidirect
85. #13063323C2×C4semidirect
87. #13063314C2×C4semidirect
113. #13063235C2×C4semidirect
114. #13063235D4semidirect
115. C8×Dic43C2×C4semidirect
116. C8×Dic43D4semidirect
319. #1977D43semidirect
381. C43#22284semidirect

Maximal subgroups

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

subgroupindex
#191450652
#191435112
#191434552
#191354932
#191351302
#191351252
#191351192
#191351142
#191350872
#191350802
#191350782
#191350712
#191347202
#191343502
C43×#19772
#2228443