Finite Group Databasebeta

← C151×#24420 #50329345

Subgroups — C151×#24420 #50329345

Order 38656.

1326 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3585×C151C211yesmaximal, characteristic, complemented
2#3068×C151C212yesmaximal, complemented
3#3021×C151C211yesmaximal, characteristic, complemented
4#3008×C151C211yesmaximal, characteristic, complemented
5#2888×C151C211yesmaximal, characteristic, complemented
6#2048×C151C212yesmaximal, complemented
7#2018×C151C212yesmaximal, complemented
8#1914×C151C211yesmaximal, characteristic, complemented
9#1881×C151C212yesmaximal, complemented
10#1858×C151C211yesmaximal, characteristic, complemented
11#1791×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×AGL(1,ℤ/8)×C151—22no—
19C2×AGL(1,ℤ/8)×C151C2212yes—
20C2×AGL(1,ℤ/8)×C151C2211yescharacteristic, complemented
21C2×AGL(1,ℤ/8)×C151C2212yescomplemented
22C22×D8×C151C2211yescharacteristic, complemented
23#568×C151C2211yescharacteristic
24#568×C151C2212yescomplemented
25#535×C151C2211yescharacteristic
26#472×C151—24no—
27#471×C151—24no—
28#454×C151—22no—
29#453×C151—22no—
30#449×C151—22no—
31#447×C151—22no—
32#430×C151C2212yescomplemented
33#421×C151C2212yescomplemented
34#418×C151C2211yescharacteristic, complemented
35#416×C151C2211yescharacteristic, complemented
36#413×C151—22no—
37#413×C151C2211yescharacteristic
38#413×C151C2211yescharacteristic
39#413×C151C2212yescomplemented
40C2×#100×C151—22no—
41C2×#100×C151C2212yescomplemented
42C2×#100×C151C2212yescomplemented
43#408×C151C2211yescharacteristic
44#408×C151C2212yescomplemented
45#344×C151C2212yescomplemented
46#341×C151C2212yescomplemented
472−1+4×C151—22no—
482−1+4×C151—21no—
49H(5,𝔽2)×C151—22no—
50H(5,𝔽2)×C151—24no—
51H(5,𝔽2)×C151—21no—
52H(5,𝔽2)×C151—22no—
53H(5,𝔽2)×C151—42no—
54C2×#41×C151—22no—
55C2×#41×C151—21no—
56C2×#41×C151—42no—
57C2×#41×C151D412yes—
58C2×#41×C151C2312yes—
59C2×#41×C151C2311yescharacteristic
60C2×#41×C151C2311yescharacteristic
61C22×D4×C151—21no—
62C22×D4×C151—22no—
63C22×D4×C151D412yes—
64C22×D4×C151C2311yescharacteristic
65C22×D4×C151C2311yescharacteristic
66#137×C151—24no—
67#137×C151—22no—
68#137×C151D414yescomplemented
69AGL(1,ℤ/8)×C151—28no—
70AGL(1,ℤ/8)×C151—24no—
71AGL(1,ℤ/8)×C151—22no—
72AGL(1,ℤ/8)×C151—24no—
73AGL(1,ℤ/8)×C151—24no—
74AGL(1,ℤ/8)×C151—24no—
75AGL(1,ℤ/8)×C151—24no—
76AGL(1,ℤ/8)×C151—42no—
77AGL(1,ℤ/8)×C151D414yescomplemented
78#135×C151—28no—
79#135×C151—24no—
80C2×SD16×C151—24no—
81C2×SD16×C151—22no—
82C2×SD16×C151—21no—
83C2×SD16×C151—22no—
84C2×D8×C151—24no—
85C2×D8×C151—22no—
86C2×D8×C151—21no—
87C2×D8×C151—22no—
88C2×D8×C151—22no—
89C2×D8×C151—42no—
90#131×C151—24no—
91#131×C151—24no—
92#131×C151—42no—
93C2×M16×C151—22no—
94C2×M16×C151—24no—
95C2×M16×C151C2312yes—
96C2×M16×C151C2311yescharacteristic
97C2×M16×C151C2312yes—
98C2×M16×C151C2312yes—
99C22×C8×C151C2311yescharacteristic
100#127×C151—21no—
101#124×C151—21no—
102#121×C151—22no—
103#120×C151—22no—
104#117×C151C2311yescharacteristic
105#108×C151—24no—
106#104×C151—24no—
107#103×C151—21no—
108#102×C151—21no—
109#102×C151—22no—
110#101×C151—21no—
111#101×C151—22no—
112#101×C151D412yes—
113#100×C151—21no—
114#100×C151—22no—
115#100×C151—24no—
116#100×C151—24no—
117#100×C151D412yes—
118C22⋊C8×C151—22no—
119C22⋊C8×C151—42no—
120C24×C151—22no—
121C24×C151—21no—
122C24×C151—21no—
123#41×C151—22no—
124#41×C151—22no—
125#41×C151—22no—
126#41×C151—24no—
127#41×C151—24no—
128#41×C151—22no—
129#41×C151—22no—
130#41×C151—22no—
131#41×C151—22no—
132#41×C151—42no—
133#41×C151—44no—
134#41×C151—44no—
135#41×C151—42no—
136#41×C151—42no—
137#41×C151—42no—
138#41×C151—42no—
139#41×C151C2×D414yes—
140C2×Q8×C151—22no—
141C2×Q8×C151—42no—
142C2×Q8×C151C2×D412yes—
143C2×Q8×C151C2×D411yescharacteristic
144C2×Q8×C151C2×D411yescharacteristic
145C2×D4×C151—21no—
146C2×D4×C151—22no—
147C2×D4×C151—24no—
148C2×D4×C151—22no—
149C2×D4×C151—22no—
150C2×D4×C151—22no—
151C2×D4×C151—22no—
152C2×D4×C151—21no—
153C2×D4×C151—22no—
154C2×D4×C151—21no—
155C2×D4×C151—21no—
156C2×D4×C151—22no—
157C2×D4×C151—22no—
158C2×D4×C151—42no—
159C2×D4×C151—44no—
160C2×D4×C151—48no—
161C2×D4×C151—42no—
162C2×D4×C151—44no—
163C2×D4×C151—42no—
164C2×D4×C151—42no—
165C2×D4×C151C2×D412yes—
166C2×D4×C151C2×D412yes—
167C2×D4×C151C2×D411yescharacteristic
168C2×D4×C151C2×D411yescharacteristic
169C2×D4×C151#4112yes—
170C22×C4×C151—21no—
171C22×C4×C151—22no—
172C22×C4×C151—22no—
173C22×C4×C151—42no—
174C22×C4×C151C2×D412yes—
175C22×C4×C151C2411yescharacteristic
176Q16×C151—24no—
177SD16×C151—28no—
178SD16×C151—24no—
179SD16×C151—24no—
180SD16×C151—44no—
181SD16×C151—42no—
182D8×C151—28no—
183D8×C151—24no—
184D8×C151—44no—
185D8×C151—44no—
186D8×C151—42no—
187M16×C151—24no—
188M16×C151—22no—
189M16×C151—22no—
190M16×C151—22no—
191M16×C151—22no—
192M16×C151—22no—
193M16×C151—44no—
194M16×C151C2×D414yes—
195M16×C151C2×D414yes—
196C2×C8×C151—24no—
197C2×C8×C151—22no—
198C2×C8×C151—22no—
199C2×C8×C151—21no—
200C2×C8×C151—21no—
201C2×C8×C151—22no—
202C2×C8×C151—22no—
203C2×C8×C151—44no—
204C4⋊C4×C151—21no—
205C22⋊C4×C151—21no—
206C22⋊C4×C151—42no—
207C42×C151—21no—
208C23×C151—21no—
209C23×C151—22no—
210C23×C151—22no—
211C23×C151—44no—
212C23×C151—42no—
213C23×C151—42no—
214C23×C151—41no—
215C23×C151—42no—
216C23×C151—161no—
217C23×C151—161no—
218C23×C151—162no—
219C23×C151#12112yes—
220C23×C151C2×#4111yescharacteristic
221Q8×C151—21no—
222Q8×C151—22no—
223Q8×C151—22no—
224Q8×C151—22no—
225Q8×C151—42no—
226Q8×C151#12012yes—
227D4×C151—24no—
228D4×C151—24no—
229D4×C151—22no—
230D4×C151—22no—
231D4×C151—21no—
232D4×C151—22no—
233D4×C151—22no—
234D4×C151—22no—
235D4×C151—44no—
236D4×C151—44no—
237D4×C151—42no—
238D4×C151—44no—
239D4×C151—42no—
240D4×C151—42no—
241D4×C151—41no—
242D4×C151—41no—
243D4×C151—42no—
244D4×C151—42no—
245D4×C151—162no—
246D4×C151#12012yes—
247C2×C4×C151—21no—
248C2×C4×C151—22no—
249C2×C4×C151—22no—
250C2×C4×C151—24no—
251C2×C4×C151—22no—
252C2×C4×C151—44no—
253C2×C4×C151—44no—
254C2×C4×C151—42no—
255C2×C4×C151—42no—
256C2×C4×C151—42no—
257C2×C4×C151—41no—
258C2×C4×C151#12112yes—
259C2×C4×C151C22×D411yescharacteristic
260C2×C4×C151C22×D412yes—
261C2×C4×C151C22×D411yescharacteristic
262C2×C4×C151H(5,𝔽2)12yes—
263C8×C151—24no—
264C8×C151—24no—
265C8×C151—41no—
266C8×C151—41no—
267C8×C151—42no—
268C22×C151—22no—
269C22×C151—22no—
270C22×C151—41no—
271C22×C151—42no—
272C22×C151—42no—
273C22×C151—42no—
274C22×C151—81no—
275C22×C151—82no—
276C22×C151—82no—
277C22×C151—82no—
278C22×C151—81no—
279C22×C151—162no—
280C22×C151—162no—
281C22×C151—164no—
282C22×C151—162no—
283C22×C151C2×#12111yescharacteristic
284C22×C151#54612yes—
285C4×C151—22no—
286C4×C151—22no—
287C4×C151—42no—
288C4×C151—81no—
289C4×C151C2×#12011yescharacteristic
290C4×C151#53411yescharacteristic
291C4×C151D4212yes—
292C2×C151—21no—
293C2×C151—22no—
294C2×C151—42no—
295C2×C151—42no—
296C2×C151—82no—
297C2×C151—82no—
298C2×C151—161no—
299C2×C151#241011yescharacteristic
300#24420C15111yesmaximal, characteristic, direct product factor, Sylow
301C151#2442011yescharacteristic, direct product factor, Sylow
302#3585C2×C15111yescharacteristic, complemented
303#3068C2×C15112yescomplemented
304#3021C2×C15111yescharacteristic, complemented
305#3008C2×C15111yescharacteristic, complemented
306#2888C2×C15111yescharacteristic, complemented
307#2048C2×C15112yescomplemented
308#2018C2×C15112yescomplemented
309#1914C2×C15111yescharacteristic, complemented
310#1881C2×C15112yescomplemented
311#1858C2×C15111yescharacteristic, complemented
312#1791C2×C15111yescharacteristic, complemented
313#585C22×C15111yescharacteristic
314C2×H(5,𝔽2)C22×C15111yescharacteristic
315#577—24no—
316#576—24no—
317#575—22no—
318#575C22×C15112yes—
319C2×AGL(1,ℤ/8)—22no—
320C2×AGL(1,ℤ/8)C22×C15112yes—
321C2×AGL(1,ℤ/8)C22×C15111yescharacteristic, complemented
322C2×AGL(1,ℤ/8)C22×C15112yescomplemented
323C22×D8C22×C15111yescharacteristic, complemented
324#568C22×C15111yescharacteristic
325#568C22×C15112yescomplemented
326#535C22×C15111yescharacteristic
327#472—24no—
328#471—24no—
329#454—22no—
330#453—22no—
331#449—22no—
332#447—22no—
333#430C22×C15112yescomplemented
334#421C22×C15112yescomplemented
335#418C22×C15111yescharacteristic, complemented
336#416C22×C15111yescharacteristic, complemented
337#413—22no—
338#413C22×C15111yescharacteristic
339#413C22×C15111yescharacteristic
340#413C22×C15112yescomplemented
341C2×#100—22no—
342C2×#100C22×C15112yescomplemented
343C2×#100C22×C15112yescomplemented
344#408C22×C15111yescharacteristic
345#408C22×C15112yescomplemented
346#344C22×C15112yescomplemented
347#341C22×C15112yescomplemented
3482−1+4—22no—
3492−1+4—21no—
350H(5,𝔽2)—24no—
351H(5,𝔽2)—22no—
352H(5,𝔽2)—22no—
353H(5,𝔽2)—21no—
354H(5,𝔽2)—42no—
355C2×#41—21no—
356C2×#41—22no—
357C2×#41—42no—
358C2×#41D4×C15112yes—
359C2×#41C23×C15112yes—
360C2×#41C23×C15111yescharacteristic
361C2×#41C23×C15111yescharacteristic
362C22×D4—22no—
363C22×D4—21no—
364C22×D4D4×C15112yes—
365C22×D4C23×C15111yescharacteristic
366C22×D4C23×C15111yescharacteristic
367#137—24no—
368#137—22no—
369#137D4×C15114yescomplemented
370AGL(1,ℤ/8)—28no—
371AGL(1,ℤ/8)—24no—
372AGL(1,ℤ/8)—22no—
373AGL(1,ℤ/8)—24no—
374AGL(1,ℤ/8)—24no—
375AGL(1,ℤ/8)—24no—
376AGL(1,ℤ/8)—24no—
377AGL(1,ℤ/8)—42no—
378AGL(1,ℤ/8)D4×C15114yescomplemented
379#135—28no—
380#135—24no—
381C2×SD16—24no—
382C2×SD16—22no—
383C2×SD16—21no—
384C2×SD16—22no—
385C2×D8—24no—
386C2×D8—22no—
387C2×D8—22no—
388C2×D8—21no—
389C2×D8—22no—
390C2×D8—42no—
391#131—24no—
392#131—24no—
393#131—42no—
394C2×M16—22no—
395C2×M16—24no—
396C2×M16C23×C15112yes—
397C2×M16C23×C15111yescharacteristic
398C2×M16C23×C15112yes—
399C2×M16C23×C15112yes—
400C22×C8C23×C15111yescharacteristic
401#127—21no—
402#124—21no—
403#121—22no—
404#120—22no—
405#117C23×C15111yescharacteristic
406#108—24no—
407#104—24no—
408#103—21no—
409#102—21no—
410#102—22no—
411#101—21no—
412#101—22no—
413#101D4×C15112yes—
414#100—21no—
415#100—22no—
416#100—24no—
417#100—24no—
418#100D4×C15112yes—
419C22⋊C8—22no—
420C22⋊C8—42no—
421C24—22no—
422C24—21no—
423C24—21no—
424#41—22no—
425#41—22no—
426#41—22no—
427#41—24no—
428#41—24no—
429#41—22no—
430#41—22no—
431#41—22no—
432#41—22no—
433#41—42no—
434#41—44no—
435#41—44no—
436#41—42no—
437#41—42no—
438#41—42no—
439#41—42no—
440#41C2×D4×C15114yes—
441C2×Q8—22no—
442C2×Q8—42no—
443C2×Q8C2×D4×C15112yes—
444C2×Q8C2×D4×C15111yescharacteristic
445C2×Q8C2×D4×C15111yescharacteristic
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—48no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×D4×C15112yes—
467C2×D4C2×D4×C15112yes—
468C2×D4C2×D4×C15111yescharacteristic
469C2×D4C2×D4×C15111yescharacteristic
470C2×D4#41×C15112yes—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4C2×D4×C15112yes—
476C22×C4C24×C15111yescharacteristic
477Q16—24no—
478SD16—28no—
479SD16—24no—
480SD16—24no—
481SD16—44no—
482SD16—42no—
483D8—28no—
484D8—24no—
485D8—44no—
486D8—44no—
487D8—42no—
488M16—24no—
489M16—22no—
490M16—22no—
491M16—22no—
492M16—22no—
493M16—22no—
494M16—44no—
495M16C2×D4×C15114yes—
496M16C2×D4×C15114yes—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—44no—
505C4⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—42no—
508C42—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23—161no—
518C23—161no—
519C23—162no—
520C23#121×C15112yes—
521C23C2×#41×C15111yescharacteristic
522Q8—21no—
523Q8—22no—
524Q8—22no—
525Q8—22no—
526Q8—42no—
527Q8#120×C15112yes—
528D4—24no—
529D4—24no—
530D4—22no—
531D4—22no—
532D4—21no—
533D4—22no—
534D4—22no—
535D4—22no—
536D4—44no—
537D4—44no—
538D4—42no—
539D4—44no—
540D4—42no—
541D4—42no—
542D4—41no—
543D4—41no—
544D4—42no—
545D4—42no—
546D4—162no—
547D4#120×C15112yes—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4#121×C15112yes—
560C2×C4C22×D4×C15111yescharacteristic
561C2×C4C22×D4×C15112yes—
562C2×C4C22×D4×C15111yescharacteristic
563C2×C4H(5,𝔽2)×C15112yes—
564C8—24no—
565C8—24no—
566C8—41no—
567C8—41no—
568C8—42no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—81no—
580C22—162no—
581C22—162no—
582C22—164no—
583C22—162no—
584C22C2×#121×C15111yescharacteristic
585C22#546×C15112yes—
586C4—22no—
587C4—22no—
588C4—42no—
589C4—81no—
590C4C2×#120×C15111yescharacteristic
591C4#534×C15111yescharacteristic
592C4D42×C15112yes—
593C2—21no—
594C2—22no—
595C2—42no—
596C2—42no—
597C2—82no—
598C2—82no—
599C2—161no—
600C2#2410×C15111yescharacteristic
subgroupG/H
center, socleC2×C151#2410
commutator, Frattini, cosocleC22×C4C24×C151
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
1. #3585×C151C2semidirect
2. #3068×C151C2semidirect
3. #3021×C151C2semidirect
4. #3008×C151C2semidirect
5. #2888×C151C2semidirect
6. #2048×C151C2semidirect
7. #2018×C151C2semidirect
8. #1914×C151C2semidirect
9. #1881×C151C2semidirect
10. #1858×C151C2semidirect
11. #1791×C151C2semidirect
20. C2×AGL(1,ℤ/8)×C151C22semidirect
21. C2×AGL(1,ℤ/8)×C151C22semidirect
22. C22×D8×C151C22semidirect
24. #568×C151C22semidirect
32. #430×C151C22semidirect
33. #421×C151C22semidirect
34. #418×C151C22semidirect
35. #416×C151C22semidirect
39. #413×C151C22semidirect
41. C2×#100×C151C22semidirect
42. C2×#100×C151C22semidirect
44. #408×C151C22semidirect
45. #344×C151C22semidirect
46. #341×C151C22semidirect
68. #137×C151D4semidirect
77. AGL(1,ℤ/8)×C151D4semidirect
300. #24420C151direct (N × H)
301. C151#24420direct (N × H)
302. #3585C2×C151semidirect
303. #3068C2×C151semidirect
304. #3021C2×C151semidirect
305. #3008C2×C151semidirect
306. #2888C2×C151semidirect
307. #2048C2×C151semidirect
308. #2018C2×C151semidirect
309. #1914C2×C151semidirect
310. #1881C2×C151semidirect
311. #1858C2×C151semidirect
312. #1791C2×C151semidirect
321. C2×AGL(1,ℤ/8)C22×C151semidirect
322. C2×AGL(1,ℤ/8)C22×C151semidirect
323. C22×D8C22×C151semidirect
325. #568C22×C151semidirect
333. #430C22×C151semidirect
334. #421C22×C151semidirect
335. #418C22×C151semidirect
336. #416C22×C151semidirect
340. #413C22×C151semidirect
342. C2×#100C22×C151semidirect
343. C2×#100C22×C151semidirect
345. #408C22×C151semidirect
346. #344C22×C151semidirect
347. #341C22×C151semidirect
369. #137D4×C151semidirect
378. AGL(1,ℤ/8)D4×C151semidirect

Maximal subgroups

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

subgroupindex
#3585×C1512
#3068×C1512
#3021×C1512
#3008×C1512
#2888×C1512
#2048×C1512
#2018×C1512
#1914×C1512
#1881×C1512
#1858×C1512
#1791×C1512
#24420151