Finite Group Databasebeta

← C151×#24422 #50329347

Subgroups — C151×#24422 #50329347

Order 38656.

1300 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#3586×C151C211yesmaximal, characteristic, complemented
2#3068×C151C212yesmaximal, complemented
3#3021×C151C211yesmaximal, characteristic, complemented
4#3010×C151C211yesmaximal, characteristic, complemented
5#2889×C151C211yesmaximal, characteristic, complemented
6#2049×C151C212yesmaximal, complemented
7#2019×C151C212yesmaximal, complemented
8#1914×C151C211yesmaximal, characteristic, complemented
9#1881×C151C212yesmaximal, complemented
10#1858×C151C211yesmaximal, characteristic, complemented
11#1791×C151C211yesmaximal, characteristic, complemented
12#585×C151C2211yescharacteristic
13C2×2−1+4×C151C2211yescharacteristic
14#578×C151—24no—
15#577×C151—24no—
16#575×C151—22no—
17#575×C151C2212yes—
18C2×#137×C151—22no—
19C2×#137×C151C2212yes—
20C2×AGL(1,ℤ/8)×C151C2212yes—
21C2×AGL(1,ℤ/8)×C151C2211yescharacteristic, complemented
22C22×SD16×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#448×C151—22no—
32#430×C151C2212yescomplemented
33#421×C151C2212yes—
34#419×C151C2211yescharacteristic, complemented
35#416×C151C2211yescharacteristic, complemented
36#413×C151—22no—
37#413×C151C2211yescharacteristic
38#413×C151C2212yes—
39#413×C151C2211yescharacteristic
40C2×#101×C151—22no—
41C2×#101×C151C2212yescomplemented
42C2×#100×C151C2212yescomplemented
43#408×C151C2211yescharacteristic
44#408×C151C2212yes—
45#344×C151C2212yescomplemented
46#341×C151C2212yescomplemented
472−1+4×C151—24no—
482−1+4×C151—22no—
492−1+4×C151—21no—
502−1+4×C151—42no—
51H(5,𝔽2)×C151—22no—
52H(5,𝔽2)×C151—21no—
53H(5,𝔽2)×C151—22no—
54C2×#41×C151—22no—
55C2×#41×C151—21no—
56C2×#41×C151—22no—
57C2×#41×C151—42no—
58C2×#41×C151D412yes—
59C2×#41×C151C2312yes—
60C2×#41×C151C2311yescharacteristic
61C2×#41×C151C2311yescharacteristic
62C22×Q8×C151—21no—
63C22×Q8×C151D412yes—
64C22×Q8×C151C2311yescharacteristic
65C22×D4×C151C2311yescharacteristic
66#137×C151—28no—
67#137×C151—24no—
68#137×C151—24no—
69#137×C151—22no—
70#137×C151—24no—
71#137×C151D414yescomplemented
72AGL(1,ℤ/8)×C151—24no—
73AGL(1,ℤ/8)×C151—22no—
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×Q16×C151—24no—
81C2×Q16×C151—22no—
82C2×SD16×C151—24no—
83C2×SD16×C151—22no—
84C2×SD16×C151—21no—
85C2×SD16×C151—22no—
86C2×SD16×C151—22no—
87C2×SD16×C151—42no—
88C2×D8×C151—21no—
89C2×D8×C151—22no—
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#103×C151—22no—
109#102×C151—21no—
110#101×C151—24no—
111#101×C151—21no—
112#101×C151—22no—
113#101×C151D412yes—
114#100×C151—21no—
115#100×C151—22no—
116#100×C151—24no—
117#100×C151D412yes—
118C22⋊C8×C151—22no—
119C22⋊C8×C151—42no—
120C24×C151—21no—
121#41×C151—22no—
122#41×C151—22no—
123#41×C151—22no—
124#41×C151—24no—
125#41×C151—24no—
126#41×C151—22no—
127#41×C151—22no—
128#41×C151—22no—
129#41×C151—22no—
130#41×C151—42no—
131#41×C151—44no—
132#41×C151—44no—
133#41×C151—42no—
134#41×C151—48no—
135#41×C151—42no—
136#41×C151—42no—
137#41×C151—42no—
138#41×C151C2×D414yes—
139C2×Q8×C151—22no—
140C2×Q8×C151—21no—
141C2×Q8×C151—22no—
142C2×Q8×C151—24no—
143C2×Q8×C151—22no—
144C2×Q8×C151—22no—
145C2×Q8×C151—42no—
146C2×Q8×C151—44no—
147C2×Q8×C151—42no—
148C2×Q8×C151C2×D412yes—
149C2×Q8×C151C2×D412yes—
150C2×Q8×C151C2×D411yescharacteristic
151C2×Q8×C151C2×D411yescharacteristic
152C2×D4×C151—21no—
153C2×D4×C151—22no—
154C2×D4×C151—22no—
155C2×D4×C151—22no—
156C2×D4×C151—22no—
157C2×D4×C151—21no—
158C2×D4×C151—21no—
159C2×D4×C151—22no—
160C2×D4×C151—42no—
161C2×D4×C151—44no—
162C2×D4×C151—42no—
163C2×D4×C151—42no—
164C2×D4×C151C2×D412yes—
165C2×D4×C151C2×D411yescharacteristic
166C2×D4×C151C2×D411yescharacteristic
167C2×D4×C151#4112yes—
168C22×C4×C151—21no—
169C22×C4×C151—22no—
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—28no—
177Q16×C151—24no—
178SD16×C151—28no—
179SD16×C151—24no—
180SD16×C151—24no—
181SD16×C151—44no—
182SD16×C151—44no—
183SD16×C151—42no—
184D8×C151—24no—
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—42no—
211C23×C151—41no—
212C23×C151—42no—
213C23×C151—161no—
214C23×C151#12112yes—
215C23×C151C2×#4111yescharacteristic
216Q8×C151—21no—
217Q8×C151—22no—
218Q8×C151—22no—
219Q8×C151—22no—
220Q8×C151—22no—
221Q8×C151—22no—
222Q8×C151—44no—
223Q8×C151—42no—
224Q8×C151—42no—
225Q8×C151#12012yes—
226D4×C151—24no—
227D4×C151—24no—
228D4×C151—22no—
229D4×C151—21no—
230D4×C151—22no—
231D4×C151—22no—
232D4×C151—42no—
233D4×C151—44no—
234D4×C151—44no—
235D4×C151—42no—
236D4×C151—42no—
237D4×C151—41no—
238D4×C151—41no—
239D4×C151—42no—
240D4×C151—162no—
241D4×C151#12012yes—
242C2×C4×C151—21no—
243C2×C4×C151—22no—
244C2×C4×C151—22no—
245C2×C4×C151—24no—
246C2×C4×C151—22no—
247C2×C4×C151—22no—
248C2×C4×C151—44no—
249C2×C4×C151—44no—
250C2×C4×C151—44no—
251C2×C4×C151—42no—
252C2×C4×C151—42no—
253C2×C4×C151—42no—
254C2×C4×C151—42no—
255C2×C4×C151—41no—
256C2×C4×C151#12112yes—
257C2×C4×C151C22×D411yescharacteristic
258C2×C4×C151C22×D412yes—
259C2×C4×C151C22×D411yescharacteristic
260C2×C4×C151H(5,𝔽2)12yes—
261C8×C151—24no—
262C8×C151—24no—
263C8×C151—41no—
264C8×C151—41no—
265C8×C151—42no—
266C22×C151—22no—
267C22×C151—22no—
268C22×C151—41no—
269C22×C151—42no—
270C22×C151—42no—
271C22×C151—81no—
272C22×C151—82no—
273C22×C151—82no—
274C22×C151—81no—
275C22×C151—162no—
276C22×C151—162no—
277C22×C151C2×#12111yescharacteristic
278C22×C151#54612yes—
279C4×C151—22no—
280C4×C151—22no—
281C4×C151—42no—
282C4×C151—42no—
283C4×C151—81no—
284C4×C151C2×#12011yescharacteristic
285C4×C151#53411yescharacteristic
286C4×C151D4212yes—
287C2×C151—21no—
288C2×C151—22no—
289C2×C151—42no—
290C2×C151—42no—
291C2×C151—82no—
292C2×C151—161no—
293C2×C151#241011yescharacteristic
294#24422C15111yesmaximal, characteristic, direct product factor, Sylow
295C151#2442211yescharacteristic, direct product factor, Sylow
296#3586C2×C15111yescharacteristic, complemented
297#3068C2×C15112yescomplemented
298#3021C2×C15111yescharacteristic, complemented
299#3010C2×C15111yescharacteristic, complemented
300#2889C2×C15111yescharacteristic, complemented
301#2049C2×C15112yescomplemented
302#2019C2×C15112yescomplemented
303#1914C2×C15111yescharacteristic, complemented
304#1881C2×C15112yescomplemented
305#1858C2×C15111yescharacteristic, complemented
306#1791C2×C15111yescharacteristic, complemented
307#585C22×C15111yescharacteristic
308C2×2−1+4C22×C15111yescharacteristic
309#578—24no—
310#577—24no—
311#575—22no—
312#575C22×C15112yes—
313C2×#137—22no—
314C2×#137C22×C15112yes—
315C2×AGL(1,ℤ/8)C22×C15112yes—
316C2×AGL(1,ℤ/8)C22×C15111yescharacteristic, complemented
317C22×SD16C22×C15111yescharacteristic, complemented
318#568C22×C15111yescharacteristic
319#568C22×C15112yescomplemented
320#535C22×C15111yescharacteristic
321#472—24no—
322#471—24no—
323#454—22no—
324#453—22no—
325#449—22no—
326#448—22no—
327#430C22×C15112yescomplemented
328#421C22×C15112yes—
329#419C22×C15111yescharacteristic, complemented
330#416C22×C15111yescharacteristic, complemented
331#413—22no—
332#413C22×C15112yes—
333#413C22×C15111yescharacteristic
334#413C22×C15111yescharacteristic
335C2×#101—22no—
336C2×#101C22×C15112yescomplemented
337C2×#100C22×C15112yescomplemented
338#408C22×C15111yescharacteristic
339#408C22×C15112yes—
340#344C22×C15112yescomplemented
341#341C22×C15112yescomplemented
3422−1+4—24no—
3432−1+4—22no—
3442−1+4—21no—
3452−1+4—42no—
346H(5,𝔽2)—22no—
347H(5,𝔽2)—22no—
348H(5,𝔽2)—21no—
349C2×#41—22no—
350C2×#41—21no—
351C2×#41—22no—
352C2×#41—42no—
353C2×#41D4×C15112yes—
354C2×#41C23×C15112yes—
355C2×#41C23×C15111yescharacteristic
356C2×#41C23×C15111yescharacteristic
357C22×Q8—21no—
358C22×Q8D4×C15112yes—
359C22×Q8C23×C15111yescharacteristic
360C22×D4C23×C15111yescharacteristic
361#137—28no—
362#137—24no—
363#137—22no—
364#137—24no—
365#137—24no—
366#137D4×C15114yescomplemented
367AGL(1,ℤ/8)—24no—
368AGL(1,ℤ/8)—22no—
369AGL(1,ℤ/8)—24no—
370AGL(1,ℤ/8)—24no—
371AGL(1,ℤ/8)—42no—
372AGL(1,ℤ/8)D4×C15114yescomplemented
373#135—28no—
374#135—24no—
375C2×Q16—24no—
376C2×Q16—22no—
377C2×SD16—24no—
378C2×SD16—22no—
379C2×SD16—22no—
380C2×SD16—22no—
381C2×SD16—21no—
382C2×SD16—42no—
383C2×D8—22no—
384C2×D8—21no—
385#131—24no—
386#131—24no—
387#131—42no—
388C2×M16—22no—
389C2×M16—24no—
390C2×M16C23×C15112yes—
391C2×M16C23×C15111yescharacteristic
392C2×M16C23×C15112yes—
393C2×M16C23×C15112yes—
394C22×C8C23×C15111yescharacteristic
395#127—21no—
396#124—21no—
397#121—22no—
398#120—22no—
399#117C23×C15111yescharacteristic
400#108—24no—
401#104—24no—
402#103—21no—
403#103—22no—
404#102—21no—
405#101—21no—
406#101—22no—
407#101—24no—
408#101D4×C15112yes—
409#100—21no—
410#100—22no—
411#100—24no—
412#100D4×C15112yes—
413C22⋊C8—22no—
414C22⋊C8—42no—
415C24—21no—
416#41—22no—
417#41—22no—
418#41—22no—
419#41—24no—
420#41—24no—
421#41—22no—
422#41—22no—
423#41—22no—
424#41—22no—
425#41—42no—
426#41—44no—
427#41—44no—
428#41—42no—
429#41—48no—
430#41—42no—
431#41—42no—
432#41—42no—
433#41C2×D4×C15114yes—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8C2×D4×C15112yes—
444C2×Q8C2×D4×C15112yes—
445C2×Q8C2×D4×C15111yescharacteristic
446C2×Q8C2×D4×C15111yescharacteristic
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4C2×D4×C15112yes—
460C2×D4C2×D4×C15111yescharacteristic
461C2×D4C2×D4×C15111yescharacteristic
462C2×D4#41×C15112yes—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4C2×D4×C15112yes—
470C22×C4C24×C15111yescharacteristic
471Q16—28no—
472Q16—24no—
473SD16—28no—
474SD16—24no—
475SD16—24no—
476SD16—44no—
477SD16—44no—
478SD16—42no—
479D8—24no—
480D8—44no—
481D8—42no—
482M16—24no—
483M16—22no—
484M16—22no—
485M16—22no—
486M16—22no—
487M16—22no—
488M16—44no—
489M16C2×D4×C15114yes—
490M16C2×D4×C15114yes—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—44no—
499C4⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—42no—
502C42—21no—
503C23—21no—
504C23—22no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—161no—
509C23#121×C15112yes—
510C23C2×#41×C15111yescharacteristic
511Q8—21no—
512Q8—22no—
513Q8—22no—
514Q8—22no—
515Q8—22no—
516Q8—22no—
517Q8—44no—
518Q8—42no—
519Q8—42no—
520Q8#120×C15112yes—
521D4—24no—
522D4—24no—
523D4—22no—
524D4—21no—
525D4—22no—
526D4—22no—
527D4—42no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—41no—
533D4—41no—
534D4—42no—
535D4—162no—
536D4#120×C15112yes—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4#121×C15112yes—
552C2×C4C22×D4×C15111yescharacteristic
553C2×C4C22×D4×C15112yes—
554C2×C4C22×D4×C15111yescharacteristic
555C2×C4H(5,𝔽2)×C15112yes—
556C8—24no—
557C8—24no—
558C8—41no—
559C8—41no—
560C8—42no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—81no—
570C22—162no—
571C22—162no—
572C22C2×#121×C15111yescharacteristic
573C22#546×C15112yes—
574C4—22no—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—81no—
579C4C2×#120×C15111yescharacteristic
580C4#534×C15111yescharacteristic
581C4D42×C15112yes—
582C2—21no—
583C2—22no—
584C2—42no—
585C2—42no—
586C2—82no—
587C2—161no—
588C2#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. #3586×C151C2semidirect
2. #3068×C151C2semidirect
3. #3021×C151C2semidirect
4. #3010×C151C2semidirect
5. #2889×C151C2semidirect
6. #2049×C151C2semidirect
7. #2019×C151C2semidirect
8. #1914×C151C2semidirect
9. #1881×C151C2semidirect
10. #1858×C151C2semidirect
11. #1791×C151C2semidirect
21. C2×AGL(1,ℤ/8)×C151C22semidirect
22. C22×SD16×C151C22semidirect
24. #568×C151C22semidirect
32. #430×C151C22semidirect
34. #419×C151C22semidirect
35. #416×C151C22semidirect
41. C2×#101×C151C22semidirect
42. C2×#100×C151C22semidirect
45. #344×C151C22semidirect
46. #341×C151C22semidirect
71. #137×C151D4semidirect
77. AGL(1,ℤ/8)×C151D4semidirect
294. #24422C151direct (N × H)
295. C151#24422direct (N × H)
296. #3586C2×C151semidirect
297. #3068C2×C151semidirect
298. #3021C2×C151semidirect
299. #3010C2×C151semidirect
300. #2889C2×C151semidirect
301. #2049C2×C151semidirect
302. #2019C2×C151semidirect
303. #1914C2×C151semidirect
304. #1881C2×C151semidirect
305. #1858C2×C151semidirect
306. #1791C2×C151semidirect
316. C2×AGL(1,ℤ/8)C22×C151semidirect
317. C22×SD16C22×C151semidirect
319. #568C22×C151semidirect
327. #430C22×C151semidirect
329. #419C22×C151semidirect
330. #416C22×C151semidirect
336. C2×#101C22×C151semidirect
337. C2×#100C22×C151semidirect
340. #344C22×C151semidirect
341. #341C22×C151semidirect
366. #137D4×C151semidirect
372. 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
#3586×C1512
#3068×C1512
#3021×C1512
#3010×C1512
#2889×C1512
#2049×C1512
#2019×C1512
#1914×C1512
#1881×C1512
#1858×C1512
#1791×C1512
#24422151