Finite Group Databasebeta

← C151×#18875 #50323800

Subgroups — C151×#18875 #50323800

Order 38656.

986 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3508×C151C211yesmaximal, characteristic
2#3249×C151C211yesmaximal, characteristic, complemented
3#3208×C151C211yesmaximal, characteristic, complemented
4#3076×C151C211yesmaximal, characteristic, complemented
5#1641×C151C212yesmaximal, complemented
6#1640×C151C212yesmaximal, complemented
7#1631×C151C212yesmaximal, complemented
8#1630×C151C212yesmaximal, complemented
9C2×#331×C151C211yesmaximal, characteristic, complemented
10#1481×C151C211yesmaximal, characteristic, complemented
11#1477×C151C211yesmaximal, characteristic, complemented
12#559×C151—21no—
13#559×C151C2212yes—
14#550×C151—21no—
15#550×C151C2212yes—
16#546×C151—22no—
17#546×C151—22no—
18#540×C151—22no—
19#535×C151—21no—
20#533×C151C2211yescharacteristic
21#533×C151C2211yescharacteristic
22#532×C151—21no—
23#532×C151—21no—
24C2×#127×C151C2211yescharacteristic
25#518×C151—21no—
26C4×#41×C151C2211yescharacteristic
27#492×C151C2212yes—
28#486×C151C2212yes—
29#479×C151—21no—
30#478×C151—21no—
31#477×C151—21no—
32#477×C151C2211yescharacteristic, complemented
33#477×C151C2212yescomplemented
34#476×C151C2211yescharacteristic, complemented
35#475×C151C2211yescharacteristic, complemented
36#474×C151C2211yescharacteristic, complemented
37#467×C151—22no—
38#462×C151—21no—
39#462×C151C2212yescomplemented
40#460×C151—22no—
41#450×C151—22no—
42#431×C151C2211yescharacteristic
43#427×C151—21no—
44C2×C4⋊C8×C151C2211yescharacteristic
45C2×C4⋊C8×C151C2211yescharacteristic
46#416×C151—21no—
47#331×C151C2212yes—
48#331×C151C2212yes—
49Q8⋊C8×C151C2212yescomplemented
50Q8⋊C8×C151C2212yescomplemented
51Q8⋊C8×C151C2212yescomplemented
52D4⋊C8×C151C2212yescomplemented
53C2×#41×C151—22no—
54C2×#41×C151—22no—
55C2×#41×C151D411yescharacteristic
56C22×D4×C151—21no—
57C22×D4×C151—21no—
58C22×D4×C151D412yes—
59C2×Q16×C151—22no—
60C2×SD16×C151—24no—
61C22×C8×C151—21no—
62C22×C8×C151—21no—
63C4⋊Q8×C151D411yescharacteristic
64C4⋊Q8×C151D411yescharacteristic
65C4⋊Q8×C151C2311yescharacteristic
66C4⋊Q8×C151C2311yescharacteristic
67#127×C151—22no—
68#127×C151—21no—
69#127×C151—21no—
70#127×C151D412yes—
71#127×C151C2312yes—
72#125×C151—21no—
73#125×C151—21no—
74#124×C151—22no—
75#124×C151—22no—
76#124×C151—22no—
77#123×C151—24no—
78#123×C151—24no—
79C22⋊Q8×C151—21no—
80C22⋊Q8×C151—21no—
81C22⋊Q8×C151—21no—
82C22⋊Q8×C151D412yes—
83#121×C151—22no—
84#121×C151—22no—
85#121×C151—22no—
86#121×C151—24no—
87#121×C151—24no—
88#121×C151—22no—
89#120×C151—24no—
90C4×Q8×C151—21no—
91C4×Q8×C151—21no—
92C4×Q8×C151—21no—
93C4×Q8×C151D411yescharacteristic
94C4×Q8×C151C2312yes—
95C4×Q8×C151C2311yescharacteristic
96C4×D4×C151—22no—
97C4×D4×C151—22no—
98C4×D4×C151—22no—
99C4×D4×C151—21no—
100C4×D4×C151—21no—
101C4×D4×C151—22no—
102C4×D4×C151—22no—
103C4×D4×C151—22no—
104C4×D4×C151D411yescharacteristic
105C4×D4×C151C2311yescharacteristic
106#117×C151—21no—
107#117×C151—21no—
108#117×C151—21no—
109#117×C151D411yescharacteristic
110C2×C22⋊C4×C151—24no—
111C2×C42×C151—21no—
112C2×C42×C151C2311yescharacteristic
113#107×C151—21no—
114#107×C151—21no—
115#107×C151—22no—
116#106×C151—21no—
117#106×C151—21no—
118C4⋊C8×C151—21no—
119C4⋊C8×C151—21no—
120C4⋊C8×C151—21no—
121C4⋊C8×C151C2312yes—
122C4⋊C8×C151C2311yescharacteristic
123C4⋊C8×C151C2311yescharacteristic
124#103×C151—22no—
125#103×C151—21no—
126#103×C151—21no—
127#103×C151—22no—
128#102×C151—21no—
129#102×C151—21no—
130#102×C151—24no—
131C22⋊C8×C151—21no—
132C4×C8×C151C2312yes—
133C24×C151—22no—
134#41×C151—22no—
135#41×C151—22no—
136#41×C151—82no—
137#41×C151—82no—
138#41×C151—84no—
139C2×Q8×C151—21no—
140C2×Q8×C151—21no—
141C2×Q8×C151—21no—
142C2×Q8×C151C2×D411yescharacteristic
143C2×Q8×C151C2×D412yes—
144C2×D4×C151—21no—
145C2×D4×C151—24no—
146C2×D4×C151—24no—
147C2×D4×C151—22no—
148C2×D4×C151—22no—
149C2×D4×C151—22no—
150C2×D4×C151—22no—
151C2×D4×C151—22no—
152C2×D4×C151—42no—
153C2×D4×C151—42no—
154C2×D4×C151—81no—
155C2×D4×C151—81no—
156C2×D4×C151—82no—
157C2×D4×C151C2×D411yescharacteristic
158C2×D4×C151C2×D414yes—
159C22×C4×C151—24no—
160C22×C4×C151—24no—
161C22×C4×C151—21no—
162C22×C4×C151—21no—
163C22×C4×C151—21no—
164C22×C4×C151C2×D411yescharacteristic
165C22×C4×C151C2×D411yescharacteristic
166C22×C4×C151C2×D411yescharacteristic
167Q16×C151—82no—
168SD16×C151—84no—
169C2×C8×C151—22no—
170C2×C8×C151—21no—
171C2×C8×C151—21no—
172C2×C8×C151—22no—
173C2×C8×C151—42no—
174C2×C8×C151—81no—
175C4⋊C4×C151—21no—
176C4⋊C4×C151—21no—
177C4⋊C4×C151—21no—
178C4⋊C4×C151—21no—
179C4⋊C4×C151—22no—
180C4⋊C4×C151—21no—
181C4⋊C4×C151—22no—
182C4⋊C4×C151—21no—
183C4⋊C4×C151—21no—
184C4⋊C4×C151—21no—
185C4⋊C4×C151—21no—
186C4⋊C4×C151—41no—
187C4⋊C4×C151—41no—
188C4⋊C4×C151C2×D412yes—
189C4⋊C4×C151C2×D411yescharacteristic
190C4⋊C4×C151C2×D411yescharacteristic
191C22⋊C4×C151—24no—
192C22⋊C4×C151—24no—
193C22⋊C4×C151—24no—
194C22⋊C4×C151—24no—
195C22⋊C4×C151—21no—
196C22⋊C4×C151—22no—
197C22⋊C4×C151—21no—
198C22⋊C4×C151—21no—
199C22⋊C4×C151—41no—
200C42×C151—21no—
201C42×C151—21no—
202C42×C151—21no—
203C42×C151—22no—
204C42×C151—41no—
205C42×C151C2×D411yescharacteristic
206C42×C151C2×D411yescharacteristic
207C42×C151C2×D411yescharacteristic
208C42×C151C2411yescharacteristic
209C23×C151—24no—
210C23×C151—21no—
211C23×C151—44no—
212C23×C151—82no—
213C23×C151—82no—
214C23×C151#12011yescharacteristic
215Q8×C151—21no—
216Q8×C151—21no—
217Q8×C151—22no—
218Q8×C151—22no—
219Q8×C151—42no—
220Q8×C151—42no—
221Q8×C151—41no—
222Q8×C151—41no—
223D4×C151—21no—
224D4×C151—21no—
225D4×C151—44no—
226D4×C151—44no—
227D4×C151—42no—
228D4×C151—42no—
229D4×C151—42no—
230D4×C151—42no—
231D4×C151—42no—
232D4×C151—44no—
233D4×C151—82no—
234D4×C151—84no—
235D4×C151—84no—
236C2×C4×C151—21no—
237C2×C4×C151—21no—
238C2×C4×C151—22no—
239C2×C4×C151—21no—
240C2×C4×C151—22no—
241C2×C4×C151—21no—
242C2×C4×C151—21no—
243C2×C4×C151—41no—
244C2×C4×C151—42no—
245C2×C4×C151—41no—
246C2×C4×C151—41no—
247C2×C4×C151—42no—
248C2×C4×C151—41no—
249C2×C4×C151—41no—
250C2×C4×C151—84no—
251C2×C4×C151—84no—
252C2×C4×C151#12011yescharacteristic
253C2×C4×C151#12011yescharacteristic
254C2×C4×C151#12011yescharacteristic
255C2×C4×C151#13512yes—
256C2×C4×C151#13712yes—
257C2×C4×C151C22×D411yescharacteristic
258C2×C4×C151C22×D411yescharacteristic
259C2×C4×C151C22×D411yescharacteristic
260C8×C151—42no—
261C8×C151—41no—
262C8×C151—41no—
263C8×C151—42no—
264C22×C151—21no—
265C22×C151—21no—
266C22×C151—41no—
267C22×C151—44no—
268C22×C151—41no—
269C22×C151—44no—
270C22×C151—41no—
271C22×C151—44no—
272C22×C151—164no—
273C22×C151#45212yes—
274C22×C151C2×#12011yescharacteristic
275C4×C151—21no—
276C4×C151—21no—
277C4×C151—21no—
278C4×C151—41no—
279C4×C151—41no—
280C4×C151—42no—
281C4×C151—41no—
282C4×C151—42no—
283C4×C151—81no—
284C4×C151—81no—
285C4×C151#45212yes—
286C4×C151C2×#13511yescharacteristic
287C4×C151C2×#13711yescharacteristic
288C4×C151#57611yescharacteristic
289C4×C151#57711yescharacteristic
290C2×C151—22no—
291C2×C151—81no—
292C2×C151—84no—
293C2×C151C2×#45211yescharacteristic
294C2×C151#301211yescharacteristic
295C2×C151#301911yescharacteristic
296#18875C15111yesmaximal, characteristic, direct product factor, Sylow
297C151#1887511yescharacteristic, direct product factor, Sylow
298#3508C2×C15111yescharacteristic
299#3249C2×C15111yescharacteristic, complemented
300#3208C2×C15111yescharacteristic, complemented
301#3076C2×C15111yescharacteristic, complemented
302#1641C2×C15112yescomplemented
303#1640C2×C15112yescomplemented
304#1631C2×C15112yescomplemented
305#1630C2×C15112yescomplemented
306C2×#331C2×C15111yescharacteristic, complemented
307#1481C2×C15111yescharacteristic, complemented
308#1477C2×C15111yescharacteristic, complemented
309#559—21no—
310#559C22×C15112yes—
311#550—21no—
312#550C22×C15112yes—
313#546—22no—
314#546—22no—
315#540—22no—
316#535—21no—
317#533C22×C15111yescharacteristic
318#533C22×C15111yescharacteristic
319#532—21no—
320#532—21no—
321C2×#127C22×C15111yescharacteristic
322#518—21no—
323C4×#41C22×C15111yescharacteristic
324#492C22×C15112yes—
325#486C22×C15112yes—
326#479—21no—
327#478—21no—
328#477—21no—
329#477C22×C15111yescharacteristic, complemented
330#477C22×C15112yescomplemented
331#476C22×C15111yescharacteristic, complemented
332#475C22×C15111yescharacteristic, complemented
333#474C22×C15111yescharacteristic, complemented
334#467—22no—
335#462—21no—
336#462C22×C15112yescomplemented
337#460—22no—
338#450—22no—
339#431C22×C15111yescharacteristic
340#427—21no—
341C2×C4⋊C8C22×C15111yescharacteristic
342C2×C4⋊C8C22×C15111yescharacteristic
343#416—21no—
344#331C22×C15112yes—
345#331C22×C15112yes—
346Q8⋊C8C22×C15112yescomplemented
347Q8⋊C8C22×C15112yescomplemented
348Q8⋊C8C22×C15112yescomplemented
349D4⋊C8C22×C15112yescomplemented
350C2×#41—22no—
351C2×#41—22no—
352C2×#41D4×C15111yescharacteristic
353C22×D4—21no—
354C22×D4—21no—
355C22×D4D4×C15112yes—
356C2×Q16—22no—
357C2×SD16—24no—
358C22×C8—21no—
359C22×C8—21no—
360C4⋊Q8D4×C15111yescharacteristic
361C4⋊Q8D4×C15111yescharacteristic
362C4⋊Q8C23×C15111yescharacteristic
363C4⋊Q8C23×C15111yescharacteristic
364#127—22no—
365#127—21no—
366#127—21no—
367#127D4×C15112yes—
368#127C23×C15112yes—
369#125—21no—
370#125—21no—
371#124—22no—
372#124—22no—
373#124—22no—
374#123—24no—
375#123—24no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8D4×C15112yes—
380#121—22no—
381#121—22no—
382#121—22no—
383#121—24no—
384#121—24no—
385#121—22no—
386#120—24no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8D4×C15111yescharacteristic
391C4×Q8C23×C15111yescharacteristic
392C4×Q8C23×C15112yes—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4D4×C15111yescharacteristic
402C4×D4C23×C15111yescharacteristic
403#117—21no—
404#117—21no—
405#117—21no—
406#117D4×C15111yescharacteristic
407C2×C22⋊C4—24no—
408C2×C42—21no—
409C2×C42C23×C15111yescharacteristic
410#107—21no—
411#107—21no—
412#107—22no—
413#106—21no—
414#106—21no—
415C4⋊C8—21no—
416C4⋊C8—21no—
417C4⋊C8—21no—
418C4⋊C8C23×C15112yes—
419C4⋊C8C23×C15111yescharacteristic
420C4⋊C8C23×C15111yescharacteristic
421#103—21no—
422#103—21no—
423#103—22no—
424#103—22no—
425#102—21no—
426#102—21no—
427#102—24no—
428C22⋊C8—21no—
429C4×C8C23×C15112yes—
430C24—22no—
431#41—22no—
432#41—22no—
433#41—82no—
434#41—82no—
435#41—84no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C2×D4×C15111yescharacteristic
440C2×Q8C2×D4×C15112yes—
441C2×D4—21no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—81no—
452C2×D4—81no—
453C2×D4—82no—
454C2×D4C2×D4×C15111yescharacteristic
455C2×D4C2×D4×C15114yes—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4C2×D4×C15111yescharacteristic
462C22×C4C2×D4×C15111yescharacteristic
463C22×C4C2×D4×C15111yescharacteristic
464Q16—82no—
465SD16—84no—
466C2×C8—22no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—42no—
471C2×C8—81no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4C2×D4×C15112yes—
486C4⋊C4C2×D4×C15111yescharacteristic
487C4⋊C4C2×D4×C15111yescharacteristic
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—41no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—22no—
501C42—41no—
502C42C2×D4×C15111yescharacteristic
503C42C2×D4×C15111yescharacteristic
504C42C2×D4×C15111yescharacteristic
505C42C24×C15111yescharacteristic
506C23—24no—
507C23—21no—
508C23—44no—
509C23—82no—
510C23—82no—
511C23#120×C15111yescharacteristic
512Q8—21no—
513Q8—21no—
514Q8—22no—
515Q8—22no—
516Q8—42no—
517Q8—42no—
518Q8—41no—
519Q8—41no—
520D4—21no—
521D4—21no—
522D4—44no—
523D4—44no—
524D4—42no—
525D4—44no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—82no—
531D4—84no—
532D4—84no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4#120×C15111yescharacteristic
550C2×C4#120×C15111yescharacteristic
551C2×C4#120×C15111yescharacteristic
552C2×C4#135×C15112yes—
553C2×C4#137×C15112yes—
554C2×C4C22×D4×C15111yescharacteristic
555C2×C4C22×D4×C15111yescharacteristic
556C2×C4C22×D4×C15111yescharacteristic
557C8—42no—
558C8—41no—
559C8—41no—
560C8—42no—
561C22—21no—
562C22—21no—
563C22—41no—
564C22—44no—
565C22—41no—
566C22—44no—
567C22—41no—
568C22—44no—
569C22—164no—
570C22#452×C15112yes—
571C22C2×#120×C15111yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4#452×C15112yes—
583C4C2×#135×C15111yescharacteristic
584C4C2×#137×C15111yescharacteristic
585C4#576×C15111yescharacteristic
586C4#577×C15111yescharacteristic
587C2—22no—
588C2—84no—
589C2—81no—
590C2C2×#452×C15111yescharacteristic
591C2#3012×C15111yescharacteristic
592C2#3019×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC42C24×C151
prime radical#3508×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. #3249×C151C2semidirect
3. #3208×C151C2semidirect
4. #3076×C151C2semidirect
5. #1641×C151C2semidirect
6. #1640×C151C2semidirect
7. #1631×C151C2semidirect
8. #1630×C151C2semidirect
9. C2×#331×C151C2semidirect
10. #1481×C151C2semidirect
11. #1477×C151C2semidirect
32. #477×C151C22semidirect
33. #477×C151C22semidirect
34. #476×C151C22semidirect
35. #475×C151C22semidirect
36. #474×C151C22semidirect
39. #462×C151C22semidirect
49. Q8⋊C8×C151C22semidirect
50. Q8⋊C8×C151C22semidirect
51. Q8⋊C8×C151C22semidirect
52. D4⋊C8×C151C22semidirect
296. #18875C151direct (N × H)
297. C151#18875direct (N × H)
299. #3249C2×C151semidirect
300. #3208C2×C151semidirect
301. #3076C2×C151semidirect
302. #1641C2×C151semidirect
303. #1640C2×C151semidirect
304. #1631C2×C151semidirect
305. #1630C2×C151semidirect
306. C2×#331C2×C151semidirect
307. #1481C2×C151semidirect
308. #1477C2×C151semidirect
329. #477C22×C151semidirect
330. #477C22×C151semidirect
331. #476C22×C151semidirect
332. #475C22×C151semidirect
333. #474C22×C151semidirect
336. #462C22×C151semidirect
346. Q8⋊C8C22×C151semidirect
347. Q8⋊C8C22×C151semidirect
348. Q8⋊C8C22×C151semidirect
349. D4⋊C8C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3508×C1512
#3249×C1512
#3208×C1512
#3076×C1512
#1641×C1512
#1640×C1512
#1631×C1512
#1630×C1512
C2×#331×C1512
#1481×C1512
#1477×C1512
#18875151