Finite Group Databasebeta

← C151×#24409 #50329334

Subgroups — C151×#24409 #50329334

Order 38656.

1336 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3585×C151C211yesmaximal, characteristic
2#3042×C151C212yesmaximal, complemented
3#3028×C151C211yesmaximal, characteristic, complemented
4#3009×C151C211yesmaximal, characteristic, complemented
5#2883×C151C211yesmaximal, characteristic, complemented
6#2053×C151C212yesmaximal, complemented
7#2009×C151C212yesmaximal, complemented
8#1909×C151C211yesmaximal, characteristic, complemented
9#1904×C151C212yesmaximal, complemented
10#1868×C151C211yesmaximal, characteristic, complemented
11#1794×C151C211yesmaximal, characteristic, complemented
12#585×C151C2211yescharacteristic
13C2×H(5,𝔽2)×C151C2211yescharacteristic
14#577×C151—24no—
15#576×C151—24no—
16#575×C151—22no—
17#575×C151C2212yes—
18C2×#137×C151C2211yescharacteristic, complemented
19C2×AGL(1,ℤ/8)×C151—22no—
20C2×AGL(1,ℤ/8)×C151C2212yes—
21C22×SD16×C151C2211yescharacteristic, complemented
22#568×C151C2211yescharacteristic
23#536×C151C2211yescharacteristic
24#536×C151C2212yescomplemented
25#518×C151C2212yescomplemented
26#463×C151—24no—
27#461×C151—24no—
28#458×C151—24no—
29#452×C151—22no—
30#450×C151—22no—
31#424×C151C2212yescomplemented
32#421×C151C2212yescomplemented
33#419×C151C2212yescomplemented
34#418×C151C2211yescharacteristic, complemented
35#416×C151C2211yescharacteristic, complemented
36#416×C151C2212yescomplemented
37#410×C151C2211yescharacteristic
38#410×C151C2211yescharacteristic
39#410×C151C2212yescomplemented
40#410×C151C2212yescomplemented
41C2×#99×C151—24no—
42#408×C151C2211yescharacteristic
43#344×C151C2212yescomplemented
44H(3,ℤ/4)×C151C2212yescomplemented
452−1+4×C151—22no—
462−1+4×C151—21no—
47H(5,𝔽2)×C151—22no—
48H(5,𝔽2)×C151—24no—
49H(5,𝔽2)×C151—21no—
50H(5,𝔽2)×C151—22no—
51H(5,𝔽2)×C151—42no—
52C2×#41×C151—22no—
53C2×#41×C151—21no—
54C2×#41×C151—42no—
55C2×#41×C151D412yes—
56C2×#41×C151C2311yescharacteristic
57C2×#41×C151C2312yes—
58C2×#41×C151C2311yescharacteristic
59C22×Q8×C151C2311yescharacteristic
60C22×D4×C151—22no—
61C22×D4×C151—21no—
62C22×D4×C151D412yes—
63C22×D4×C151C2311yescharacteristic
64#137×C151—24no—
65#137×C151—22no—
66#137×C151—22no—
67#137×C151—42no—
68AGL(1,ℤ/8)×C151—28no—
69AGL(1,ℤ/8)×C151—24no—
70AGL(1,ℤ/8)×C151—22no—
71AGL(1,ℤ/8)×C151—24no—
72AGL(1,ℤ/8)×C151—24no—
73AGL(1,ℤ/8)×C151—22no—
74#135×C151—28no—
75#135×C151—24no—
76C2×Q16×C151—21no—
77C2×SD16×C151—24no—
78C2×SD16×C151—21no—
79C2×SD16×C151—22no—
80C2×SD16×C151—24no—
81C2×SD16×C151D414yes—
82C2×D8×C151—24no—
83C2×D8×C151D414yes—
84#131×C151—24no—
85C2×M16×C151—22no—
86C2×M16×C151C2311yescharacteristic
87C2×M16×C151C2312yes—
88C22×C8×C151C2311yescharacteristic
89C4⋊Q8×C151—22no—
90C4⋊Q8×C151—21no—
91#124×C151—22no—
92#124×C151—21no—
93#123×C151—22no—
94#123×C151—24no—
95C22⋊Q8×C151—22no—
96C22⋊Q8×C151—42no—
97C4×D4×C151—24no—
98C4×D4×C151—42no—
99#117×C151C2311yescharacteristic
100#117×C151C2312yes—
101#117×C151C2312yes—
102C2×C22⋊C4×C151—24no—
103C4⋊C8×C151—24no—
104#104×C151—42no—
105#103×C151—24no—
106#103×C151—21no—
107#103×C151—22no—
108#102×C151—21no—
109#102×C151—22no—
110#102×C151—22no—
111#99×C151—22no—
112#99×C151—24no—
113#99×C151—24no—
114#99×C151D414yes—
115C22⋊C8×C151—22no—
116C24×C151—21no—
117C24×C151—22no—
118#41×C151—22no—
119#41×C151—22no—
120#41×C151—24no—
121#41×C151—24no—
122#41×C151—22no—
123#41×C151—22no—
124#41×C151—22no—
125#41×C151—22no—
126#41×C151—22no—
127#41×C151—22no—
128#41×C151—42no—
129#41×C151—44no—
130#41×C151—42no—
131#41×C151—44no—
132#41×C151—42no—
133#41×C151—42no—
134#41×C151—42no—
135C2×Q8×C151—22no—
136C2×Q8×C151—22no—
137C2×Q8×C151—21no—
138C2×Q8×C151—21no—
139C2×Q8×C151—22no—
140C2×Q8×C151—42no—
141C2×Q8×C151C2×D412yes—
142C2×Q8×C151C2×D411yescharacteristic
143C2×Q8×C151#4111yescharacteristic
144C2×D4×C151—22no—
145C2×D4×C151—24no—
146C2×D4×C151—22no—
147C2×D4×C151—21no—
148C2×D4×C151—22no—
149C2×D4×C151—22no—
150C2×D4×C151—21no—
151C2×D4×C151—22no—
152C2×D4×C151—42no—
153C2×D4×C151—44no—
154C2×D4×C151—42no—
155C2×D4×C151—48no—
156C2×D4×C151—44no—
157C2×D4×C151—42no—
158C2×D4×C151C2×D412yes—
159C2×D4×C151C2×D412yes—
160C2×D4×C151C2×D411yescharacteristic
161C2×D4×C151C2×D412yes—
162C2×D4×C151C2×D414yes—
163C2×D4×C151#4111yescharacteristic
164C22×C4×C151—22no—
165C22×C4×C151—21no—
166C22×C4×C151—22no—
167C22×C4×C151—21no—
168C22×C4×C151—42no—
169C22×C4×C151—44no—
170C22×C4×C151C2×D412yes—
171C22×C4×C151C2411yescharacteristic
172Q16×C151—24no—
173Q16×C151—42no—
174SD16×C151—24no—
175SD16×C151—24no—
176SD16×C151—28no—
177SD16×C151—42no—
178SD16×C151—44no—
179D8×C151—24no—
180D8×C151—28no—
181M16×C151—24no—
182M16×C151—22no—
183M16×C151—22no—
184M16×C151—22no—
185C2×C8×C151—24no—
186C2×C8×C151—21no—
187C2×C8×C151—21no—
188C2×C8×C151—22no—
189C2×C8×C151C2×D414yes—
190C4⋊C4×C151—22no—
191C4⋊C4×C151—21no—
192C4⋊C4×C151—22no—
193C4⋊C4×C151—44no—
194C4⋊C4×C151—42no—
195C22⋊C4×C151—22no—
196C22⋊C4×C151—21no—
197C22⋊C4×C151—44no—
198C22⋊C4×C151—44no—
199C22⋊C4×C151—42no—
200C22⋊C4×C151C2×D414yes—
201C42×C151—22no—
202C42×C151—21no—
203C42×C151—22no—
204C23×C151—22no—
205C23×C151—24no—
206C23×C151—21no—
207C23×C151—22no—
208C23×C151—44no—
209C23×C151—42no—
210C23×C151—161no—
211C23×C151—162no—
212C23×C151#12112yes—
213C23×C151C22×D411yescharacteristic
214Q8×C151—22no—
215Q8×C151—22no—
216Q8×C151—21no—
217Q8×C151—21no—
218Q8×C151—22no—
219Q8×C151—42no—
220Q8×C151—41no—
221Q8×C151—41no—
222Q8×C151—42no—
223D4×C151—24no—
224D4×C151—22no—
225D4×C151—21no—
226D4×C151—22no—
227D4×C151—21no—
228D4×C151—22no—
229D4×C151—24no—
230D4×C151—22no—
231D4×C151—44no—
232D4×C151—42no—
233D4×C151—44no—
234D4×C151—44no—
235D4×C151—42no—
236D4×C151—42no—
237D4×C151—42no—
238D4×C151#12014yes—
239C2×C4×C151—22no—
240C2×C4×C151—22no—
241C2×C4×C151—21no—
242C2×C4×C151—22no—
243C2×C4×C151—24no—
244C2×C4×C151—44no—
245C2×C4×C151—44no—
246C2×C4×C151—42no—
247C2×C4×C151—42no—
248C2×C4×C151—42no—
249C2×C4×C151—42no—
250C2×C4×C151—41no—
251C2×C4×C151—41no—
252C2×C4×C151—42no—
253C2×C4×C151—42no—
254C2×C4×C151—164no—
255C2×C4×C151#12112yes—
256C2×C4×C151C22×D411yescharacteristic
257C2×C4×C151C22×D412yes—
258C2×C4×C151C2×#4111yescharacteristic
259C2×C4×C151H(5,𝔽2)12yes—
260C8×C151—24no—
261C8×C151—41no—
262C8×C151—41no—
263C22×C151—22no—
264C22×C151—24no—
265C22×C151—41no—
266C22×C151—42no—
267C22×C151—42no—
268C22×C151—41no—
269C22×C151—84no—
270C22×C151—82no—
271C22×C151—82no—
272C22×C151—81no—
273C22×C151—164no—
274C22×C151—162no—
275C22×C151C2×#12111yescharacteristic
276C22×C151D4212yes—
277C4×C151—22no—
278C4×C151—44no—
279C4×C151—41no—
280C4×C151—42no—
281C4×C151—81no—
282C4×C151—82no—
283C4×C151—81no—
284C4×C151C2×#12011yescharacteristic
285C4×C151#53411yescharacteristic
286C4×C151#54612yes—
287C2×C151—22no—
288C2×C151—21no—
289C2×C151—42no—
290C2×C151—44no—
291C2×C151—82no—
292C2×C151—81no—
293C2×C151#241011yescharacteristic
294#24409C15111yesmaximal, characteristic, direct product factor, Sylow
295C151#2440911yescharacteristic, direct product factor, Sylow
296#3585C2×C15111yescharacteristic
297#3042C2×C15112yescomplemented
298#3028C2×C15111yescharacteristic, complemented
299#3009C2×C15111yescharacteristic, complemented
300#2883C2×C15111yescharacteristic, complemented
301#2053C2×C15112yescomplemented
302#2009C2×C15112yescomplemented
303#1909C2×C15111yescharacteristic, complemented
304#1904C2×C15112yescomplemented
305#1868C2×C15111yescharacteristic, complemented
306#1794C2×C15111yescharacteristic, complemented
307#585C22×C15111yescharacteristic
308C2×H(5,𝔽2)C22×C15111yescharacteristic
309#577—24no—
310#576—24no—
311#575—22no—
312#575C22×C15112yes—
313C2×#137C22×C15111yescharacteristic, complemented
314C2×AGL(1,ℤ/8)—22no—
315C2×AGL(1,ℤ/8)C22×C15112yes—
316C22×SD16C22×C15111yescharacteristic, complemented
317#568C22×C15111yescharacteristic
318#536C22×C15111yescharacteristic
319#536C22×C15112yescomplemented
320#518C22×C15112yescomplemented
321#463—24no—
322#461—24no—
323#458—24no—
324#452—22no—
325#450—22no—
326#424C22×C15112yescomplemented
327#421C22×C15112yescomplemented
328#419C22×C15112yescomplemented
329#418C22×C15111yescharacteristic, complemented
330#416C22×C15112yescomplemented
331#416C22×C15111yescharacteristic, complemented
332#410C22×C15111yescharacteristic
333#410C22×C15111yescharacteristic
334#410C22×C15112yescomplemented
335#410C22×C15112yescomplemented
336C2×#99—24no—
337#408C22×C15111yescharacteristic
338#344C22×C15112yescomplemented
339H(3,ℤ/4)C22×C15112yescomplemented
3402−1+4—22no—
3412−1+4—21no—
342H(5,𝔽2)—22no—
343H(5,𝔽2)—24no—
344H(5,𝔽2)—22no—
345H(5,𝔽2)—21no—
346H(5,𝔽2)—42no—
347C2×#41—21no—
348C2×#41—22no—
349C2×#41—42no—
350C2×#41D4×C15112yes—
351C2×#41C23×C15112yes—
352C2×#41C23×C15111yescharacteristic
353C2×#41C23×C15111yescharacteristic
354C22×Q8C23×C15111yescharacteristic
355C22×D4—21no—
356C22×D4—22no—
357C22×D4D4×C15112yes—
358C22×D4C23×C15111yescharacteristic
359#137—24no—
360#137—22no—
361#137—22no—
362#137—42no—
363AGL(1,ℤ/8)—28no—
364AGL(1,ℤ/8)—24no—
365AGL(1,ℤ/8)—22no—
366AGL(1,ℤ/8)—24no—
367AGL(1,ℤ/8)—24no—
368AGL(1,ℤ/8)—22no—
369#135—28no—
370#135—24no—
371C2×Q16—21no—
372C2×SD16—24no—
373C2×SD16—22no—
374C2×SD16—24no—
375C2×SD16—21no—
376C2×SD16D4×C15114yes—
377C2×D8—24no—
378C2×D8D4×C15114yes—
379#131—24no—
380C2×M16—22no—
381C2×M16C23×C15111yescharacteristic
382C2×M16C23×C15112yes—
383C22×C8C23×C15111yescharacteristic
384C4⋊Q8—21no—
385C4⋊Q8—22no—
386#124—21no—
387#124—22no—
388#123—22no—
389#123—24no—
390C22⋊Q8—22no—
391C22⋊Q8—42no—
392C4×D4—24no—
393C4×D4—42no—
394#117C23×C15112yes—
395#117C23×C15111yescharacteristic
396#117C23×C15112yes—
397C2×C22⋊C4—24no—
398C4⋊C8—24no—
399#104—42no—
400#103—21no—
401#103—22no—
402#103—24no—
403#102—21no—
404#102—22no—
405#102—22no—
406#99—22no—
407#99—24no—
408#99—24no—
409#99D4×C15114yes—
410C22⋊C8—22no—
411C24—21no—
412C24—22no—
413#41—22no—
414#41—22no—
415#41—24no—
416#41—24no—
417#41—22no—
418#41—22no—
419#41—22no—
420#41—22no—
421#41—22no—
422#41—22no—
423#41—42no—
424#41—44no—
425#41—42no—
426#41—44no—
427#41—42no—
428#41—42no—
429#41—42no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—42no—
436C2×Q8C2×D4×C15112yes—
437C2×Q8C2×D4×C15111yescharacteristic
438C2×Q8#41×C15111yescharacteristic
439C2×D4—22no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—42no—
450C2×D4—48no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4C2×D4×C15112yes—
454C2×D4C2×D4×C15112yes—
455C2×D4C2×D4×C15111yescharacteristic
456C2×D4C2×D4×C15112yes—
457C2×D4C2×D4×C15114yes—
458C2×D4#41×C15111yescharacteristic
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4C2×D4×C15112yes—
466C22×C4C24×C15111yescharacteristic
467Q16—24no—
468Q16—42no—
469SD16—24no—
470SD16—24no—
471SD16—28no—
472SD16—42no—
473SD16—44no—
474D8—24no—
475D8—28no—
476M16—24no—
477M16—22no—
478M16—22no—
479M16—22no—
480C2×C8—24no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—22no—
484C2×C8C2×D4×C15114yes—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4C2×D4×C15114yes—
496C42—22no—
497C42—21no—
498C42—22no—
499C23—22no—
500C23—24no—
501C23—21no—
502C23—22no—
503C23—44no—
504C23—42no—
505C23—161no—
506C23—162no—
507C23#121×C15112yes—
508C23C22×D4×C15111yescharacteristic
509Q8—22no—
510Q8—22no—
511Q8—21no—
512Q8—21no—
513Q8—22no—
514Q8—42no—
515Q8—41no—
516Q8—41no—
517Q8—42no—
518D4—24no—
519D4—22no—
520D4—21no—
521D4—22no—
522D4—21no—
523D4—24no—
524D4—22no—
525D4—22no—
526D4—44no—
527D4—42no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4#120×C15114yes—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—164no—
550C2×C4#121×C15112yes—
551C2×C4C22×D4×C15111yescharacteristic
552C2×C4C22×D4×C15112yes—
553C2×C4C2×#41×C15111yescharacteristic
554C2×C4H(5,𝔽2)×C15112yes—
555C8—24no—
556C8—41no—
557C8—41no—
558C22—22no—
559C22—24no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—84no—
565C22—82no—
566C22—81no—
567C22—82no—
568C22—164no—
569C22—162no—
570C22C2×#121×C15111yescharacteristic
571C22D42×C15112yes—
572C4—22no—
573C4—44no—
574C4—41no—
575C4—42no—
576C4—82no—
577C4—81no—
578C4—81no—
579C4C2×#120×C15111yescharacteristic
580C4#534×C15111yescharacteristic
581C4#546×C15112yes—
582C2—22no—
583C2—21no—
584C2—42no—
585C2—44no—
586C2—81no—
587C2—82no—
588C2#2410×C15111yescharacteristic
subgroupG/H
center, socleC2×C151#2410
commutator, Frattini, cosocleC22×C4C24×C151
prime radical#3585×C151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C151 ◁ C22×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3042×C151C2semidirect
3. #3028×C151C2semidirect
4. #3009×C151C2semidirect
5. #2883×C151C2semidirect
6. #2053×C151C2semidirect
7. #2009×C151C2semidirect
8. #1909×C151C2semidirect
9. #1904×C151C2semidirect
10. #1868×C151C2semidirect
11. #1794×C151C2semidirect
18. C2×#137×C151C22semidirect
21. C22×SD16×C151C22semidirect
24. #536×C151C22semidirect
25. #518×C151C22semidirect
31. #424×C151C22semidirect
32. #421×C151C22semidirect
33. #419×C151C22semidirect
34. #418×C151C22semidirect
35. #416×C151C22semidirect
36. #416×C151C22semidirect
39. #410×C151C22semidirect
40. #410×C151C22semidirect
43. #344×C151C22semidirect
44. H(3,ℤ/4)×C151C22semidirect
294. #24409C151direct (N × H)
295. C151#24409direct (N × H)
297. #3042C2×C151semidirect
298. #3028C2×C151semidirect
299. #3009C2×C151semidirect
300. #2883C2×C151semidirect
301. #2053C2×C151semidirect
302. #2009C2×C151semidirect
303. #1909C2×C151semidirect
304. #1904C2×C151semidirect
305. #1868C2×C151semidirect
306. #1794C2×C151semidirect
313. C2×#137C22×C151semidirect
316. C22×SD16C22×C151semidirect
319. #536C22×C151semidirect
320. #518C22×C151semidirect
326. #424C22×C151semidirect
327. #421C22×C151semidirect
328. #419C22×C151semidirect
329. #418C22×C151semidirect
330. #416C22×C151semidirect
331. #416C22×C151semidirect
334. #410C22×C151semidirect
335. #410C22×C151semidirect
338. #344C22×C151semidirect
339. H(3,ℤ/4)C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3585×C1512
#3042×C1512
#3028×C1512
#3009×C1512
#2883×C1512
#2053×C1512
#2009×C1512
#1909×C1512
#1904×C1512
#1868×C1512
#1794×C1512
#24409151