Finite Group Databasebeta

← C151×#24423 #50329348

Subgroups — C151×#24423 #50329348

Order 38656.

1284 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3586×C151C211yesmaximal, characteristic
2#3069×C151C212yesmaximal, complemented
3#3022×C151C211yesmaximal, characteristic, complemented
4#3011×C151C211yesmaximal, characteristic, complemented
5#2889×C151C211yesmaximal, characteristic, complemented
6#2049×C151C212yesmaximal, complemented
7#2020×C151C212yesmaximal, complemented
8#1915×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×#137×C151C2212yes—
21C2×#137×C151C2211yescharacteristic, complemented
22C22×Q16×C151C2211yescharacteristic, complemented
23#568×C151C2211yescharacteristic
24#568×C151C2212yescomplemented
25#536×C151C2211yescharacteristic
26#473×C151—24no—
27#471×C151—24no—
28#456×C151—22no—
29#455×C151—22no—
30#452×C151—22no—
31#451×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×#101×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×Q8×C151C2311yescharacteristic
66#137×C151—28no—
67#137×C151—24no—
68#137×C151—24no—
69#137×C151—24no—
70#137×C151—24no—
71#137×C151—22no—
72#137×C151—24no—
73#137×C151—42no—
74#137×C151D414yes—
75AGL(1,ℤ/8)×C151—24no—
76AGL(1,ℤ/8)×C151—22no—
77AGL(1,ℤ/8)×C151D414yes—
78#135×C151—28no—
79#135×C151—24no—
80C2×Q16×C151—24no—
81C2×Q16×C151—22no—
82C2×Q16×C151—21no—
83C2×Q16×C151—22no—
84C2×Q16×C151—22no—
85C2×Q16×C151—42no—
86C2×SD16×C151—24no—
87C2×SD16×C151—22no—
88C2×SD16×C151—21no—
89C2×SD16×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
100C4⋊Q8×C151—21no—
101#124×C151—21no—
102#123×C151—22no—
103C22⋊Q8×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—24no—
112#101×C151—21no—
113#101×C151—22no—
114#101×C151D412yes—
115#100×C151—21no—
116#100×C151—22no—
117#100×C151D412yes—
118C22⋊C8×C151—22no—
119C22⋊C8×C151—42no—
120#41×C151—22no—
121#41×C151—22no—
122#41×C151—22no—
123#41×C151—24no—
124#41×C151—24no—
125#41×C151—22no—
126#41×C151—22no—
127#41×C151—22no—
128#41×C151—22no—
129#41×C151—42no—
130#41×C151—44no—
131#41×C151—44no—
132#41×C151—42no—
133#41×C151—48no—
134#41×C151—42no—
135#41×C151—42no—
136#41×C151—42no—
137#41×C151C2×D414yes—
138C2×Q8×C151—22no—
139C2×Q8×C151—21no—
140C2×Q8×C151—22no—
141C2×Q8×C151—24no—
142C2×Q8×C151—22no—
143C2×Q8×C151—22no—
144C2×Q8×C151—21no—
145C2×Q8×C151—21no—
146C2×Q8×C151—22no—
147C2×Q8×C151—22no—
148C2×Q8×C151—42no—
149C2×Q8×C151—44no—
150C2×Q8×C151—42no—
151C2×Q8×C151C2×D412yes—
152C2×Q8×C151C2×D412yes—
153C2×Q8×C151C2×D411yescharacteristic
154C2×Q8×C151C2×D411yescharacteristic
155C2×D4×C151—21no—
156C2×D4×C151—22no—
157C2×D4×C151—22no—
158C2×D4×C151—22no—
159C2×D4×C151—42no—
160C2×D4×C151—44no—
161C2×D4×C151—42no—
162C2×D4×C151C2×D412yes—
163C2×D4×C151C2×D411yescharacteristic
164C2×D4×C151C2×D411yescharacteristic
165C2×D4×C151#4112yes—
166C22×C4×C151—21no—
167C22×C4×C151—22no—
168C22×C4×C151—21no—
169C22×C4×C151—22no—
170C22×C4×C151—22no—
171C22×C4×C151—21no—
172C22×C4×C151—42no—
173C22×C4×C151C2×D412yes—
174C22×C4×C151C2411yescharacteristic
175Q16×C151—28no—
176Q16×C151—24no—
177Q16×C151—44no—
178Q16×C151—44no—
179Q16×C151—42no—
180SD16×C151—28no—
181SD16×C151—24no—
182SD16×C151—24no—
183SD16×C151—44no—
184SD16×C151—42no—
185D8×C151—24no—
186M16×C151—24no—
187M16×C151—22no—
188M16×C151—22no—
189M16×C151—22no—
190M16×C151—22no—
191M16×C151—22no—
192M16×C151—44no—
193M16×C151C2×D414yes—
194M16×C151C2×D414yes—
195C2×C8×C151—24no—
196C2×C8×C151—22no—
197C2×C8×C151—22no—
198C2×C8×C151—21no—
199C2×C8×C151—21no—
200C2×C8×C151—22no—
201C2×C8×C151—22no—
202C2×C8×C151—44no—
203C4⋊C4×C151—21no—
204C4⋊C4×C151—42no—
205C22⋊C4×C151—21no—
206C22⋊C4×C151—42no—
207C42×C151—21no—
208C23×C151—21no—
209C23×C151—22no—
210C23×C151—42no—
211C23×C151#12112yes—
212C23×C151C2×#4111yescharacteristic
213Q8×C151—21no—
214Q8×C151—22no—
215Q8×C151—22no—
216Q8×C151—22no—
217Q8×C151—22no—
218Q8×C151—22no—
219Q8×C151—44no—
220Q8×C151—42no—
221Q8×C151—42no—
222Q8×C151—41no—
223Q8×C151—41no—
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#12012yes—
238C2×C4×C151—21no—
239C2×C4×C151—22no—
240C2×C4×C151—22no—
241C2×C4×C151—24no—
242C2×C4×C151—22no—
243C2×C4×C151—22no—
244C2×C4×C151—44no—
245C2×C4×C151—44no—
246C2×C4×C151—44no—
247C2×C4×C151—42no—
248C2×C4×C151—42no—
249C2×C4×C151—42no—
250C2×C4×C151—42no—
251C2×C4×C151—41no—
252C2×C4×C151—41no—
253C2×C4×C151—42no—
254C2×C4×C151#12112yes—
255C2×C4×C151C22×D411yescharacteristic
256C2×C4×C151C22×D412yes—
257C2×C4×C151C22×D411yescharacteristic
258C2×C4×C151H(5,𝔽2)12yes—
259C8×C151—24no—
260C8×C151—24no—
261C8×C151—41no—
262C8×C151—41no—
263C8×C151—42no—
264C22×C151—22no—
265C22×C151—22no—
266C22×C151—41no—
267C22×C151—42no—
268C22×C151—42no—
269C22×C151—81no—
270C22×C151—82no—
271C22×C151—162no—
272C22×C151C2×#12111yescharacteristic
273C22×C151#54612yes—
274C4×C151—22no—
275C4×C151—22no—
276C4×C151—42no—
277C4×C151—42no—
278C4×C151—81no—
279C4×C151—81no—
280C4×C151C2×#12011yescharacteristic
281C4×C151#53411yescharacteristic
282C4×C151D4212yes—
283C2×C151—21no—
284C2×C151—22no—
285C2×C151—42no—
286C2×C151—42no—
287C2×C151—82no—
288C2×C151#241011yescharacteristic
289#24423C15111yesmaximal, characteristic, direct product factor, Sylow
290C151#2442311yescharacteristic, direct product factor, Sylow
291#3586C2×C15111yescharacteristic
292#3069C2×C15112yescomplemented
293#3022C2×C15111yescharacteristic, complemented
294#3011C2×C15111yescharacteristic, complemented
295#2889C2×C15111yescharacteristic, complemented
296#2049C2×C15112yescomplemented
297#2020C2×C15112yescomplemented
298#1915C2×C15111yescharacteristic, complemented
299#1881C2×C15112yescomplemented
300#1858C2×C15111yescharacteristic, complemented
301#1791C2×C15111yescharacteristic, complemented
302#585C22×C15111yescharacteristic
303C2×2−1+4C22×C15111yescharacteristic
304#578—24no—
305#577—24no—
306#575—22no—
307#575C22×C15112yes—
308C2×#137—22no—
309C2×#137C22×C15112yes—
310C2×#137C22×C15112yes—
311C2×#137C22×C15111yescharacteristic, complemented
312C22×Q16C22×C15111yescharacteristic, complemented
313#568C22×C15111yescharacteristic
314#568C22×C15112yescomplemented
315#536C22×C15111yescharacteristic
316#473—24no—
317#471—24no—
318#456—22no—
319#455—22no—
320#452—22no—
321#451—22no—
322#430C22×C15112yescomplemented
323#421C22×C15112yes—
324#419C22×C15111yescharacteristic, complemented
325#416C22×C15111yescharacteristic, complemented
326#413—22no—
327#413C22×C15112yes—
328#413C22×C15111yescharacteristic
329#413C22×C15111yescharacteristic
330C2×#101—22no—
331C2×#101C22×C15112yescomplemented
332C2×#101C22×C15112yescomplemented
333#408C22×C15111yescharacteristic
334#408C22×C15112yes—
335#344C22×C15112yescomplemented
336#341C22×C15112yescomplemented
3372−1+4—24no—
3382−1+4—22no—
3392−1+4—21no—
3402−1+4—42no—
341H(5,𝔽2)—22no—
342H(5,𝔽2)—22no—
343H(5,𝔽2)—21no—
344C2×#41—22no—
345C2×#41—21no—
346C2×#41—22no—
347C2×#41—42no—
348C2×#41D4×C15112yes—
349C2×#41C23×C15112yes—
350C2×#41C23×C15111yescharacteristic
351C2×#41C23×C15111yescharacteristic
352C22×Q8—21no—
353C22×Q8D4×C15112yes—
354C22×Q8C23×C15111yescharacteristic
355C22×Q8C23×C15111yescharacteristic
356#137—28no—
357#137—24no—
358#137—22no—
359#137—24no—
360#137—24no—
361#137—24no—
362#137—24no—
363#137—42no—
364#137D4×C15114yes—
365AGL(1,ℤ/8)—24no—
366AGL(1,ℤ/8)—22no—
367AGL(1,ℤ/8)D4×C15114yes—
368#135—28no—
369#135—24no—
370C2×Q16—24no—
371C2×Q16—22no—
372C2×Q16—22no—
373C2×Q16—22no—
374C2×Q16—21no—
375C2×Q16—42no—
376C2×SD16—24no—
377C2×SD16—22no—
378C2×SD16—22no—
379C2×SD16—21no—
380#131—24no—
381#131—24no—
382#131—42no—
383C2×M16—22no—
384C2×M16—24no—
385C2×M16C23×C15112yes—
386C2×M16C23×C15111yescharacteristic
387C2×M16C23×C15112yes—
388C2×M16C23×C15112yes—
389C22×C8C23×C15111yescharacteristic
390C4⋊Q8—21no—
391#124—21no—
392#123—22no—
393C22⋊Q8—22no—
394#117C23×C15111yescharacteristic
395#108—24no—
396#104—24no—
397#103—21no—
398#103—22no—
399#102—21no—
400#101—21no—
401#101—22no—
402#101—24no—
403#101—24no—
404#101D4×C15112yes—
405#100—21no—
406#100—22no—
407#100D4×C15112yes—
408C22⋊C8—22no—
409C22⋊C8—42no—
410#41—22no—
411#41—22no—
412#41—22no—
413#41—24no—
414#41—24no—
415#41—22no—
416#41—22no—
417#41—22no—
418#41—22no—
419#41—42no—
420#41—44no—
421#41—44no—
422#41—42no—
423#41—48no—
424#41—42no—
425#41—42no—
426#41—42no—
427#41C2×D4×C15114yes—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8C2×D4×C15112yes—
442C2×Q8C2×D4×C15112yes—
443C2×Q8C2×D4×C15111yescharacteristic
444C2×Q8C2×D4×C15111yescharacteristic
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4C2×D4×C15112yes—
453C2×D4C2×D4×C15111yescharacteristic
454C2×D4C2×D4×C15111yescharacteristic
455C2×D4#41×C15112yes—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4C2×D4×C15112yes—
464C22×C4C24×C15111yescharacteristic
465Q16—28no—
466Q16—24no—
467Q16—44no—
468Q16—44no—
469Q16—42no—
470SD16—28no—
471SD16—24no—
472SD16—24no—
473SD16—44no—
474SD16—42no—
475D8—24no—
476M16—24no—
477M16—22no—
478M16—22no—
479M16—22no—
480M16—22no—
481M16—22no—
482M16—44no—
483M16C2×D4×C15114yes—
484M16C2×D4×C15114yes—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—44no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C22⋊C4—21no—
496C22⋊C4—42no—
497C42—21no—
498C23—21no—
499C23—22no—
500C23—42no—
501C23#121×C15112yes—
502C23C2×#41×C15111yescharacteristic
503Q8—21no—
504Q8—22no—
505Q8—22no—
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—44no—
510Q8—42no—
511Q8—42no—
512Q8—41no—
513Q8—41no—
514Q8—42no—
515Q8#120×C15112yes—
516D4—24no—
517D4—24no—
518D4—22no—
519D4—21no—
520D4—22no—
521D4—22no—
522D4—42no—
523D4—44no—
524D4—44no—
525D4—42no—
526D4—42no—
527D4#120×C15112yes—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4#121×C15112yes—
545C2×C4C22×D4×C15111yescharacteristic
546C2×C4C22×D4×C15112yes—
547C2×C4C22×D4×C15111yescharacteristic
548C2×C4H(5,𝔽2)×C15112yes—
549C8—24no—
550C8—24no—
551C8—41no—
552C8—41no—
553C8—42no—
554C22—22no—
555C22—22no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—82no—
561C22—162no—
562C22C2×#121×C15111yescharacteristic
563C22#546×C15112yes—
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—81no—
569C4—81no—
570C4C2×#120×C15111yescharacteristic
571C4#534×C15111yescharacteristic
572C4D42×C15112yes—
573C2—21no—
574C2—22no—
575C2—42no—
576C2—42no—
577C2—82no—
578C2#2410×C15111yescharacteristic
subgroupG/H
center, socleC2×C151#2410
commutator, Frattini, cosocleC22×C4C24×C151
prime radical#3586×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. #3069×C151C2semidirect
3. #3022×C151C2semidirect
4. #3011×C151C2semidirect
5. #2889×C151C2semidirect
6. #2049×C151C2semidirect
7. #2020×C151C2semidirect
8. #1915×C151C2semidirect
9. #1881×C151C2semidirect
10. #1858×C151C2semidirect
11. #1791×C151C2semidirect
21. C2×#137×C151C22semidirect
22. C22×Q16×C151C22semidirect
24. #568×C151C22semidirect
32. #430×C151C22semidirect
34. #419×C151C22semidirect
35. #416×C151C22semidirect
41. C2×#101×C151C22semidirect
42. C2×#101×C151C22semidirect
45. #344×C151C22semidirect
46. #341×C151C22semidirect
289. #24423C151direct (N × H)
290. C151#24423direct (N × H)
292. #3069C2×C151semidirect
293. #3022C2×C151semidirect
294. #3011C2×C151semidirect
295. #2889C2×C151semidirect
296. #2049C2×C151semidirect
297. #2020C2×C151semidirect
298. #1915C2×C151semidirect
299. #1881C2×C151semidirect
300. #1858C2×C151semidirect
301. #1791C2×C151semidirect
311. C2×#137C22×C151semidirect
312. C22×Q16C22×C151semidirect
314. #568C22×C151semidirect
322. #430C22×C151semidirect
324. #419C22×C151semidirect
325. #416C22×C151semidirect
331. C2×#101C22×C151semidirect
332. C2×#101C22×C151semidirect
335. #344C22×C151semidirect
336. #341C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3586×C1512
#3069×C1512
#3022×C1512
#3011×C1512
#2889×C1512
#2049×C1512
#2020×C1512
#1915×C1512
#1881×C1512
#1858×C1512
#1791×C1512
#24423151