Finite Group Databasebeta

← C151×#18631 #50323556

Subgroups — C151×#18631 #50323556

Order 38656.

1262 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#3498×C151C211yesmaximal, characteristic
2#3206×C151C211yesmaximal, characteristic, complemented
3C2×#486×C151C211yesmaximal, characteristic, complemented
4#3079×C151C211yesmaximal, characteristic, complemented
5#1642×C151C214yesmaximal, complemented
6#1639×C151C214yesmaximal, complemented
7#1545×C151C211yesmaximal, characteristic, complemented
8#1492×C151C211yesmaximal, characteristic, complemented
9#1476×C151C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C151—21no—
11#550×C151—21no—
12#550×C151C2212yes—
13D42×C151—22no—
14D42×C151—22no—
15D42×C151—21no—
16D42×C151C2212yes—
17#535×C151—22no—
18#535×C151—21no—
19#532×C151—21no—
20#532×C151C2211yescharacteristic
21C2×C4⋊Q8×C151C2211yescharacteristic, complemented
22C2×#127×C151—21no—
23C2×#127×C151C2211yescharacteristic
24C4×#41×C151C2211yescharacteristic
25#492×C151—21no—
26#487×C151—21no—
27#486×C151—22no—
28#486×C151C2214yescomplemented
29#477×C151—21no—
30#476×C151—21no—
31#475×C151C2212yescomplemented
32#474×C151C2212yescomplemented
33#461×C151C2214yescomplemented
34#448×C151—22no—
35#431×C151C2211yescharacteristic, complemented
36#428×C151—21no—
37#423×C151C2211yescharacteristic, complemented
38#417×C151—21no—
39C2×#102×C151—22no—
40C2×C4×C8×C151C2211yescharacteristic, complemented
41#331×C151C2214yescomplemented
42Q8⋊C8×C151C2212yescomplemented
43D4⋊C8×C151C2212yescomplemented
44D4⋊C8×C151C2214yescomplemented
45H(5,𝔽2)×C151—44no—
46H(5,𝔽2)×C151—44no—
47C2×#41×C151—22no—
48C2×#41×C151—21no—
49C2×#41×C151—22no—
50C2×#41×C151D411yescharacteristic
51C22×Q8×C151—21no—
52C22×D4×C151—22no—
53C22×D4×C151—24no—
54C22×D4×C151—22no—
55C22×D4×C151—22no—
56C22×D4×C151—22no—
57C22×D4×C151—21no—
58C22×D4×C151—21no—
59C22×D4×C151D412yes—
60C2×SD16×C151—24no—
61C2×M16×C151—21no—
62C22×C8×C151—21no—
63C4⋊Q8×C151—21no—
64C4⋊Q8×C151—21no—
65C4⋊Q8×C151—21no—
66C4⋊Q8×C151D411yescharacteristic
67C4⋊Q8×C151C2312yes—
68#127×C151—22no—
69#127×C151—22no—
70#127×C151—22no—
71#127×C151—21no—
72#127×C151—21no—
73#127×C151D412yes—
74#127×C151D411yescharacteristic
75#127×C151C2312yes—
76#124×C151—22no—
77#124×C151—21no—
78#121×C151—22no—
79#121×C151—24no—
80#121×C151—24no—
81#121×C151—21no—
82#121×C151—22no—
83#121×C151—22no—
84#121×C151—22no—
85#121×C151D412yes—
86#120×C151—24no—
87#120×C151—24no—
88#120×C151—24no—
89C4×Q8×C151D411yescharacteristic
90C4×Q8×C151C2311yescharacteristic
91C4×D4×C151—22no—
92C4×D4×C151—21no—
93C4×D4×C151—21no—
94C4×D4×C151—22no—
95C4×D4×C151—21no—
96C4×D4×C151D411yescharacteristic
97C4×D4×C151C2312yes—
98C4×D4×C151C2311yescharacteristic
99#117×C151—21no—
100#117×C151D411yescharacteristic
101C2×C4⋊C4×C151—21no—
102C2×C4⋊C4×C151—21no—
103C2×C42×C151—21no—
104C2×C42×C151C2311yescharacteristic
105#107×C151—22no—
106#106×C151—22no—
107C4⋊C8×C151—21no—
108C4⋊C8×C151C2312yes—
109C4⋊C8×C151C2312yes—
110#103×C151—24no—
111#103×C151—22no—
112#102×C151—28no—
113#102×C151—22no—
114C22⋊C8×C151—21no—
115C4×C8×C151—21no—
116C4×C8×C151—21no—
117C4×C8×C151C2312yes—
118C24×C151—24no—
119C24×C151—22no—
120#41×C151—22no—
121#41×C151—48no—
122#41×C151—44no—
123#41×C151—44no—
124#41×C151—42no—
125#41×C151—42no—
126#41×C151SD1614yescomplemented
127C2×Q8×C151—21no—
128C2×Q8×C151—22no—
129C2×Q8×C151—41no—
130C2×Q8×C151—82no—
131C2×Q8×C151C2×D411yescharacteristic
132C2×D4×C151—21no—
133C2×D4×C151—24no—
134C2×D4×C151—24no—
135C2×D4×C151—22no—
136C2×D4×C151—24no—
137C2×D4×C151—24no—
138C2×D4×C151—22no—
139C2×D4×C151—21no—
140C2×D4×C151—22no—
141C2×D4×C151—21no—
142C2×D4×C151—22no—
143C2×D4×C151—22no—
144C2×D4×C151—22no—
145C2×D4×C151—48no—
146C2×D4×C151—44no—
147C2×D4×C151—416no—
148C2×D4×C151—44no—
149C2×D4×C151—44no—
150C2×D4×C151—42no—
151C2×D4×C151—42no—
152C2×D4×C151—82no—
153C2×D4×C151—84no—
154C2×D4×C151—84no—
155C2×D4×C151C2×D412yes—
156C2×D4×C151C2×D411yescharacteristic
157C2×D4×C151C2×D414yes—
158C22×C4×C151—21no—
159C22×C4×C151—24no—
160C22×C4×C151—21no—
161C22×C4×C151—21no—
162C22×C4×C151—22no—
163C22×C4×C151—41no—
164C22×C4×C151C2×D411yescharacteristic
165C22×C4×C151C2×D411yescharacteristic
166C22×C4×C151C2×D411yescharacteristic
167SD16×C151—84no—
168M16×C151—42no—
169C2×C8×C151—22no—
170C2×C8×C151—22no—
171C2×C8×C151—22no—
172C2×C8×C151—42no—
173C4⋊C4×C151—21no—
174C4⋊C4×C151—21no—
175C4⋊C4×C151—21no—
176C4⋊C4×C151—22no—
177C4⋊C4×C151—22no—
178C4⋊C4×C151—41no—
179C4⋊C4×C151—41no—
180C4⋊C4×C151C2×D412yes—
181C4⋊C4×C151C2×D411yescharacteristic
182C4⋊C4×C151C2×D411yescharacteristic
183C22⋊C4×C151—24no—
184C22⋊C4×C151—24no—
185C22⋊C4×C151—21no—
186C22⋊C4×C151—21no—
187C22⋊C4×C151—22no—
188C22⋊C4×C151—21no—
189C22⋊C4×C151—22no—
190C42×C151—21no—
191C42×C151—21no—
192C42×C151—21no—
193C42×C151C2×D411yescharacteristic
194C42×C151C2×D411yescharacteristic
195C42×C151C2×D411yescharacteristic
196C42×C151C2411yescharacteristic
197C23×C151—21no—
198C23×C151—22no—
199C23×C151—24no—
200C23×C151—48no—
201C23×C151—44no—
202C23×C151—82no—
203C23×C151—82no—
204C23×C151—84no—
205C23×C151—84no—
206C23×C151#12011yescharacteristic
207Q8×C151—21no—
208Q8×C151—42no—
209Q8×C151—44no—
210Q8×C151—82no—
211Q8×C151C2×SD1612yes—
212D4×C151—24no—
213D4×C151—21no—
214D4×C151—24no—
215D4×C151—44no—
216D4×C151—48no—
217D4×C151—48no—
218D4×C151—44no—
219D4×C151—42no—
220D4×C151—42no—
221D4×C151—44no—
222D4×C151—44no—
223D4×C151—42no—
224D4×C151—44no—
225D4×C151—42no—
226D4×C151—82no—
227D4×C151—84no—
228D4×C151—84no—
229D4×C151C2×SD1612yes—
230C2×C4×C151—21no—
231C2×C4×C151—22no—
232C2×C4×C151—24no—
233C2×C4×C151—22no—
234C2×C4×C151—22no—
235C2×C4×C151—21no—
236C2×C4×C151—21no—
237C2×C4×C151—48no—
238C2×C4×C151—42no—
239C2×C4×C151—42no—
240C2×C4×C151—42no—
241C2×C4×C151—42no—
242C2×C4×C151—42no—
243C2×C4×C151—82no—
244C2×C4×C151#12011yescharacteristic
245C2×C4×C151#12011yescharacteristic
246C2×C4×C151#12011yescharacteristic
247C2×C4×C151C2×SD1612yes—
248C2×C4×C151AGL(1,ℤ/8)12yes—
249C2×C4×C151C22×D411yescharacteristic
250C2×C4×C151C22×D411yescharacteristic
251C2×C4×C151C22×D411yescharacteristic
252C8×C151—42no—
253C8×C151—42no—
254C8×C151—42no—
255C22×C151—21no—
256C22×C151—21no—
257C22×C151—22no—
258C22×C151—41no—
259C22×C151—41no—
260C22×C151—44no—
261C22×C151—42no—
262C22×C151—42no—
263C22×C151—44no—
264C22×C151—44no—
265C22×C151—42no—
266C22×C151—164no—
267C22×C151—168no—
268C22×C151#45012yes—
269C22×C151C2×#12011yescharacteristic
270C4×C151—22no—
271C4×C151—21no—
272C4×C151—21no—
273C4×C151—42no—
274C4×C151—41no—
275C4×C151—41no—
276C4×C151—82no—
277C4×C151#45012yes—
278C4×C151#45512yes—
279C4×C151C22×SD1611yescharacteristic
280C4×C151C2×AGL(1,ℤ/8)11yescharacteristic
281C4×C151#57612yes—
282C2×C151—22no—
283C2×C151—42no—
284C2×C151—82no—
285C2×C151—84no—
286C2×C151C2×#45011yescharacteristic
287C2×C151#300811yescharacteristic
288C2×C151C2×#45511yescharacteristic
289#18631C15111yesmaximal, characteristic, direct product factor, Sylow
290C151#1863111yescharacteristic, direct product factor, Sylow
291#3498C2×C15111yescharacteristic
292#3206C2×C15111yescharacteristic, complemented
293C2×#486C2×C15111yescharacteristic, complemented
294#3079C2×C15111yescharacteristic, complemented
295#1642C2×C15114yescomplemented
296#1639C2×C15114yescomplemented
297#1545C2×C15111yescharacteristic, complemented
298#1492C2×C15111yescharacteristic, complemented
299#1476C2×C15111yescharacteristic, complemented
300C2×H(5,𝔽2)—21no—
301#550—21no—
302#550C22×C15112yes—
303D42—22no—
304D42—22no—
305D42—21no—
306D42C22×C15112yes—
307#535—22no—
308#535—21no—
309#532—21no—
310#532C22×C15111yescharacteristic
311C2×C4⋊Q8C22×C15111yescharacteristic, complemented
312C2×#127—21no—
313C2×#127C22×C15111yescharacteristic
314C4×#41C22×C15111yescharacteristic
315#492—21no—
316#487—21no—
317#486—22no—
318#486C22×C15114yescomplemented
319#477—21no—
320#476—21no—
321#475C22×C15112yescomplemented
322#474C22×C15112yescomplemented
323#461C22×C15114yescomplemented
324#448—22no—
325#431C22×C15111yescharacteristic, complemented
326#428—21no—
327#423C22×C15111yescharacteristic, complemented
328#417—21no—
329C2×#102—22no—
330C2×C4×C8C22×C15111yescharacteristic, complemented
331#331C22×C15114yescomplemented
332Q8⋊C8C22×C15112yescomplemented
333D4⋊C8C22×C15112yescomplemented
334D4⋊C8C22×C15114yescomplemented
335H(5,𝔽2)—44no—
336H(5,𝔽2)—44no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—21no—
340C2×#41D4×C15111yescharacteristic
341C22×Q8—21no—
342C22×D4—22no—
343C22×D4—24no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4D4×C15112yes—
350C2×SD16—24no—
351C2×M16—21no—
352C22×C8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8D4×C15111yescharacteristic
357C4⋊Q8C23×C15112yes—
358#127—22no—
359#127—22no—
360#127—22no—
361#127—21no—
362#127—21no—
363#127D4×C15111yescharacteristic
364#127D4×C15112yes—
365#127C23×C15112yes—
366#124—22no—
367#124—21no—
368#121—22no—
369#121—24no—
370#121—24no—
371#121—21no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121D4×C15112yes—
376#120—24no—
377#120—24no—
378#120—24no—
379C4×Q8D4×C15111yescharacteristic
380C4×Q8C23×C15111yescharacteristic
381C4×D4—22no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—22no—
385C4×D4—21no—
386C4×D4D4×C15111yescharacteristic
387C4×D4C23×C15111yescharacteristic
388C4×D4C23×C15112yes—
389#117—21no—
390#117D4×C15111yescharacteristic
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C42—21no—
394C2×C42C23×C15111yescharacteristic
395#107—22no—
396#106—22no—
397C4⋊C8—21no—
398C4⋊C8C23×C15112yes—
399C4⋊C8C23×C15112yes—
400#103—22no—
401#103—24no—
402#102—28no—
403#102—22no—
404C22⋊C8—21no—
405C4×C8—21no—
406C4×C8—21no—
407C4×C8C23×C15112yes—
408C24—24no—
409C24—22no—
410#41—22no—
411#41—48no—
412#41—44no—
413#41—44no—
414#41—42no—
415#41—42no—
416#41SD16×C15114yescomplemented
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—41no—
420C2×Q8—82no—
421C2×Q8C2×D4×C15111yescharacteristic
422C2×D4—21no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—48no—
436C2×D4—44no—
437C2×D4—416no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—82no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4C2×D4×C15112yes—
446C2×D4C2×D4×C15111yescharacteristic
447C2×D4C2×D4×C15114yes—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4C2×D4×C15111yescharacteristic
455C22×C4C2×D4×C15111yescharacteristic
456C22×C4C2×D4×C15111yescharacteristic
457SD16—84no—
458M16—42no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—42no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4C2×D4×C15112yes—
471C4⋊C4C2×D4×C15111yescharacteristic
472C4⋊C4C2×D4×C15111yescharacteristic
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42C2×D4×C15111yescharacteristic
484C42C2×D4×C15111yescharacteristic
485C42C2×D4×C15111yescharacteristic
486C42C24×C15111yescharacteristic
487C23—21no—
488C23—22no—
489C23—24no—
490C23—48no—
491C23—44no—
492C23—82no—
493C23—82no—
494C23—84no—
495C23—84no—
496C23#120×C15111yescharacteristic
497Q8—21no—
498Q8—42no—
499Q8—44no—
500Q8—82no—
501Q8C2×SD16×C15112yes—
502D4—24no—
503D4—21no—
504D4—24no—
505D4—44no—
506D4—48no—
507D4—48no—
508D4—44no—
509D4—42no—
510D4—44no—
511D4—44no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—44no—
516D4—82no—
517D4—84no—
518D4—84no—
519D4C2×SD16×C15112yes—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—82no—
534C2×C4#120×C15111yescharacteristic
535C2×C4#120×C15111yescharacteristic
536C2×C4#120×C15111yescharacteristic
537C2×C4C2×SD16×C15112yes—
538C2×C4AGL(1,ℤ/8)×C15112yes—
539C2×C4C22×D4×C15111yescharacteristic
540C2×C4C22×D4×C15111yescharacteristic
541C2×C4C22×D4×C15111yescharacteristic
542C8—42no—
543C8—42no—
544C8—42no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—41no—
549C22—41no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—164no—
557C22—168no—
558C22#450×C15112yes—
559C22C2×#120×C15111yescharacteristic
560C4—22no—
561C4—21no—
562C4—21no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—82no—
567C4#450×C15112yes—
568C4#455×C15112yes—
569C4C22×SD16×C15111yescharacteristic
570C4C2×AGL(1,ℤ/8)×C15111yescharacteristic
571C4#576×C15112yes—
572C2—22no—
573C2—42no—
574C2—82no—
575C2—84no—
576C2C2×#450×C15111yescharacteristic
577C2#3008×C15111yescharacteristic
578C2C2×#455×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC42C24×C151
prime radical#3498×C151C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3206×C151C2semidirect
3. C2×#486×C151C2semidirect
4. #3079×C151C2semidirect
5. #1642×C151C2semidirect
6. #1639×C151C2semidirect
7. #1545×C151C2semidirect
8. #1492×C151C2semidirect
9. #1476×C151C2semidirect
21. C2×C4⋊Q8×C151C22semidirect
28. #486×C151C22semidirect
31. #475×C151C22semidirect
32. #474×C151C22semidirect
33. #461×C151C22semidirect
35. #431×C151C22semidirect
37. #423×C151C22semidirect
40. C2×C4×C8×C151C22semidirect
41. #331×C151C22semidirect
42. Q8⋊C8×C151C22semidirect
43. D4⋊C8×C151C22semidirect
44. D4⋊C8×C151C22semidirect
126. #41×C151SD16semidirect
289. #18631C151direct (N × H)
290. C151#18631direct (N × H)
292. #3206C2×C151semidirect
293. C2×#486C2×C151semidirect
294. #3079C2×C151semidirect
295. #1642C2×C151semidirect
296. #1639C2×C151semidirect
297. #1545C2×C151semidirect
298. #1492C2×C151semidirect
299. #1476C2×C151semidirect
311. C2×C4⋊Q8C22×C151semidirect
318. #486C22×C151semidirect
321. #475C22×C151semidirect
322. #474C22×C151semidirect
323. #461C22×C151semidirect
325. #431C22×C151semidirect
327. #423C22×C151semidirect
330. C2×C4×C8C22×C151semidirect
331. #331C22×C151semidirect
332. Q8⋊C8C22×C151semidirect
333. D4⋊C8C22×C151semidirect
334. D4⋊C8C22×C151semidirect
416. #41SD16×C151semidirect

Maximal subgroups

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

subgroupindex
#3498×C1512
#3206×C1512
C2×#486×C1512
#3079×C1512
#1642×C1512
#1639×C1512
#1545×C1512
#1492×C1512
#1476×C1512
#18631151