Finite Group Databasebeta

← C151×#23950 #50328875

Subgroups — C151×#23950 #50328875

Order 38656.

1242 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3493×C151C211yesmaximal, characteristic, complemented
2#3005×C151C212yesmaximal, complemented
3#2297×C151C211yesmaximal, characteristic, complemented
4#2028×C151C211yesmaximal, characteristic, complemented
5#2027×C151C211yesmaximal, characteristic, complemented
6#2007×C151C211yesmaximal, characteristic, complemented
7#2006×C151C211yesmaximal, characteristic, complemented
8#1896×C151C212yesmaximal, complemented
9#1883×C151C212yesmaximal, complemented
10#1882×C151C212yesmaximal, complemented
11#1822×C151C211yesmaximal, characteristic, complemented
12C22×#41×C151C2211yescharacteristic
13C2×#137×C151C2212yes—
14C2×AGL(1,ℤ/8)×C151C2212yes—
15#556×C151—22no—
16#555×C151—22no—
17#553×C151—22no—
18#552×C151—22no—
19#544×C151C2211yescharacteristic
20#538×C151—22no—
21#537×C151C2211yescharacteristic
22#529×C151—22no—
23#525×C151C2212yes—
24#520×C151C2212yes—
25C2×#117×C151C2211yescharacteristic
26C22×C42×C151C2211yescharacteristic
27#452×C151—22no—
28#452×C151—22no—
29#449×C151—22no—
30#449×C151—22no—
31C2×#104×C151C2212yes—
32C2×#104×C151C2212yes—
33#419×C151C2212yes—
34#418×C151C2212yes—
35#407×C151C2212yes—
36#391×C151C2211yescharacteristic
37#391×C151C2211yescharacteristic
38#390×C151C2211yescharacteristic
39#390×C151C2211yescharacteristic
40#386×C151—22no—
41C4×C22⋊C4×C151—22no—
42#343×C151C2212yes—
43#343×C151C2211yescharacteristic
44#343×C151C2211yescharacteristic
45#339×C151C2212yes—
46#339×C151C2212yes—
47C2×#41×C151—22no—
48C2×#41×C151—22no—
49C2×#41×C151—22no—
50C2×#41×C151—22no—
51C2×#41×C151—42no—
52C2×#41×C151C2312yes—
53C22×Q8×C151C2311yescharacteristic
54C22×D4×C151—21no—
55C22×D4×C151C2311yescharacteristic
56C23×C4×C151—21no—
57C23×C4×C151—21no—
58C23×C4×C151C2311yescharacteristic
59#137×C151—24no—
60#137×C151—42no—
61AGL(1,ℤ/8)×C151—24no—
62AGL(1,ℤ/8)×C151—42no—
63C2×Q16×C151—22no—
64C2×SD16×C151—22no—
65C2×SD16×C151—22no—
66C2×D8×C151—22no—
67C2×M16×C151C2312yes—
68C2×M16×C151C2312yes—
69C4⋊Q8×C151—22no—
70#126×C151—24no—
71#126×C151—24no—
72#125×C151—22no—
73#125×C151—22no—
74#124×C151—22no—
75#124×C151—24no—
76#123×C151—24no—
77#123×C151—22no—
78C22⋊Q8×C151—22no—
79C22⋊Q8×C151—22no—
80C22⋊Q8×C151D412yes—
81C22⋊Q8×C151D412yes—
82#121×C151—24no—
83#121×C151—22no—
84#121×C151—22no—
85#121×C151D412yes—
86#121×C151D412yes—
87#120×C151—22no—
88C4×Q8×C151—22no—
89C4×Q8×C151—22no—
90C4×D4×C151—22no—
91C4×D4×C151—22no—
92C4×D4×C151—24no—
93C4×D4×C151—22no—
94C4×D4×C151—22no—
95#117×C151—24no—
96#117×C151—22no—
97#117×C151—22no—
98#117×C151—22no—
99#117×C151C2312yes—
100C2×C4⋊C4×C151—22no—
101C2×C4⋊C4×C151C2311yescharacteristic
102C2×C4⋊C4×C151C2311yescharacteristic
103C2×C22⋊C4×C151—22no—
104C2×C22⋊C4×C151—22no—
105C2×C22⋊C4×C151D412yes—
106C2×C42×C151—21no—
107C2×C42×C151—42no—
108C2×C42×C151D412yes—
109C2×C42×C151C2311yescharacteristic
110C2×C42×C151C2311yescharacteristic
111#104×C151—42no—
112#104×C151—42no—
113#103×C151—22no—
114#103×C151—22no—
115#102×C151—22no—
116#102×C151—22no—
117C22⋊C8×C151—22no—
118C22⋊C8×C151—22no—
119#95×C151—22no—
120#95×C151—22no—
121C24×C151—21no—
122C24×C151C2×D411yescharacteristic
123#41×C151—22no—
124#41×C151—22no—
125#41×C151—22no—
126#41×C151—22no—
127#41×C151—44no—
128#41×C151—44no—
129#41×C151—44no—
130#41×C151—44no—
131C2×Q8×C151—22no—
132C2×Q8×C151—21no—
133C2×Q8×C151—22no—
134C2×Q8×C151—22no—
135C2×Q8×C151C2×D412yes—
136C2×Q8×C151#4112yes—
137C2×D4×C151—24no—
138C2×D4×C151—22no—
139C2×D4×C151—22no—
140C2×D4×C151—22no—
141C2×D4×C151—22no—
142C2×D4×C151—22no—
143C2×D4×C151—21no—
144C2×D4×C151—22no—
145C2×D4×C151—42no—
146C2×D4×C151—42no—
147C2×D4×C151C2×D412yes—
148C2×D4×C151#4112yes—
149C22×C4×C151—21no—
150C22×C4×C151—21no—
151C22×C4×C151—22no—
152C22×C4×C151—24no—
153C22×C4×C151—21no—
154C22×C4×C151—21no—
155C22×C4×C151—21no—
156C22×C4×C151—24no—
157C22×C4×C151—22no—
158C22×C4×C151—42no—
159C22×C4×C151—42no—
160C22×C4×C151—42no—
161C22×C4×C151—42no—
162C22×C4×C151—41no—
163C22×C4×C151—41no—
164C22×C4×C151C2×D411yescharacteristic
165C22×C4×C151C2×D412yes—
166C22×C4×C151C2×D412yes—
167C22×C4×C151C2×D412yes—
168C22×C4×C151C2×D412yes—
169C22×C4×C151C2411yescharacteristic
170Q16×C151—44no—
171SD16×C151—44no—
172SD16×C151—44no—
173D8×C151—44no—
174M16×C151—22no—
175M16×C151—22no—
176M16×C151—22no—
177M16×C151—22no—
178C2×C8×C151—22no—
179C2×C8×C151—22no—
180C4⋊C4×C151—24no—
181C4⋊C4×C151—24no—
182C4⋊C4×C151—22no—
183C4⋊C4×C151—22no—
184C4⋊C4×C151—22no—
185C4⋊C4×C151—22no—
186C4⋊C4×C151—22no—
187C4⋊C4×C151—22no—
188C4⋊C4×C151C2×D412yes—
189C4⋊C4×C151C2×D412yes—
190C22⋊C4×C151—24no—
191C22⋊C4×C151—24no—
192C22⋊C4×C151—24no—
193C22⋊C4×C151—22no—
194C22⋊C4×C151—22no—
195C22⋊C4×C151—22no—
196C22⋊C4×C151—22no—
197C22⋊C4×C151—22no—
198C22⋊C4×C151—44no—
199C22⋊C4×C151—44no—
200C42×C151—24no—
201C42×C151—22no—
202C42×C151—22no—
203C42×C151—22no—
204C42×C151—22no—
205C42×C151—22no—
206C42×C151—42no—
207C23×C151—21no—
208C23×C151—21no—
209C23×C151—22no—
210C23×C151—22no—
211C23×C151—41no—
212C23×C151—41no—
213C23×C151—42no—
214C23×C151—42no—
215C23×C151—41no—
216C23×C151—41no—
217C23×C151#12012yes—
218C23×C151C22×D411yescharacteristic
219Q8×C151—24no—
220Q8×C151—22no—
221Q8×C151—22no—
222Q8×C151—42no—
223Q8×C151—82no—
224D4×C151—22no—
225D4×C151—22no—
226D4×C151—24no—
227D4×C151—48no—
228D4×C151—42no—
229D4×C151—42no—
230D4×C151—42no—
231D4×C151—82no—
232D4×C151—82no—
233D4×C151—82no—
234C2×C4×C151—21no—
235C2×C4×C151—21no—
236C2×C4×C151—22no—
237C2×C4×C151—22no—
238C2×C4×C151—22no—
239C2×C4×C151—22no—
240C2×C4×C151—22no—
241C2×C4×C151—22no—
242C2×C4×C151—22no—
243C2×C4×C151—22no—
244C2×C4×C151—22no—
245C2×C4×C151—22no—
246C2×C4×C151—22no—
247C2×C4×C151—22no—
248C2×C4×C151—44no—
249C2×C4×C151—44no—
250C2×C4×C151—41no—
251C2×C4×C151—41no—
252C2×C4×C151—48no—
253C2×C4×C151—41no—
254C2×C4×C151—41no—
255C2×C4×C151—41no—
256C2×C4×C151—41no—
257C2×C4×C151—42no—
258C2×C4×C151—42no—
259C2×C4×C151—42no—
260C2×C4×C151—42no—
261C2×C4×C151—84no—
262C2×C4×C151#12011yescharacteristic
263C2×C4×C151#12011yescharacteristic
264C2×C4×C151C22×D412yes—
265C2×C4×C151C2×#4111yescharacteristic
266C2×C4×C151C2×#4111yescharacteristic
267C2×C4×C151H(5,𝔽2)12yes—
268C8×C151—42no—
269C8×C151—42no—
270C22×C151—21no—
271C22×C151—21no—
272C22×C151—22no—
273C22×C151—22no—
274C22×C151—22no—
275C22×C151—42no—
276C22×C151—42no—
277C22×C151—44no—
278C22×C151—42no—
279C22×C151—42no—
280C22×C151—42no—
281C22×C151—42no—
282C22×C151—82no—
283C22×C151—81no—
284C22×C151—81no—
285C22×C151C2×#12011yescharacteristic
286C22×C151#52511yescharacteristic
287C22×C151#53511yescharacteristic
288C4×C151—21no—
289C4×C151—21no—
290C4×C151—42no—
291C4×C151—42no—
292C4×C151—42no—
293C4×C151—42no—
294C4×C151—42no—
295C4×C151—42no—
296C4×C151—42no—
297C4×C151—42no—
298C4×C151#54612yes—
299C4×C151#54612yes—
300C2×C151—21no—
301C2×C151—21no—
302C2×C151—42no—
303C2×C151—82no—
304C2×C151—82no—
305C2×C151#240411yescharacteristic
306C2×C151#301811yescharacteristic
307C2×C151#301811yescharacteristic
308#23950C15111yesmaximal, characteristic, direct product factor, Sylow
309C151#2395011yescharacteristic, direct product factor, Sylow
310#3493C2×C15111yescharacteristic, complemented
311#3005C2×C15112yescomplemented
312#2297C2×C15111yescharacteristic, complemented
313#2028C2×C15111yescharacteristic, complemented
314#2027C2×C15111yescharacteristic, complemented
315#2007C2×C15111yescharacteristic, complemented
316#2006C2×C15111yescharacteristic, complemented
317#1896C2×C15112yescomplemented
318#1883C2×C15112yescomplemented
319#1882C2×C15112yescomplemented
320#1822C2×C15111yescharacteristic, complemented
321C22×#41C22×C15111yescharacteristic
322C2×#137C22×C15112yes—
323C2×AGL(1,ℤ/8)C22×C15112yes—
324#556—22no—
325#555—22no—
326#553—22no—
327#552—22no—
328#544C22×C15111yescharacteristic
329#538—22no—
330#537C22×C15111yescharacteristic
331#529—22no—
332#525C22×C15112yes—
333#520C22×C15112yes—
334C2×#117C22×C15111yescharacteristic
335C22×C42C22×C15111yescharacteristic
336#452—22no—
337#452—22no—
338#449—22no—
339#449—22no—
340C2×#104C22×C15112yes—
341C2×#104C22×C15112yes—
342#419C22×C15112yes—
343#418C22×C15112yes—
344#407C22×C15112yes—
345#391C22×C15111yescharacteristic
346#391C22×C15111yescharacteristic
347#390C22×C15111yescharacteristic
348#390C22×C15111yescharacteristic
349#386—22no—
350C4×C22⋊C4—22no—
351#343C22×C15112yes—
352#343C22×C15111yescharacteristic
353#343C22×C15111yescharacteristic
354#339C22×C15112yes—
355#339C22×C15112yes—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41—42no—
361C2×#41C23×C15112yes—
362C22×Q8C23×C15111yescharacteristic
363C22×D4—21no—
364C22×D4C23×C15111yescharacteristic
365C23×C4—21no—
366C23×C4—21no—
367C23×C4C23×C15111yescharacteristic
368#137—24no—
369#137—42no—
370AGL(1,ℤ/8)—24no—
371AGL(1,ℤ/8)—42no—
372C2×Q16—22no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×D8—22no—
376C2×M16C23×C15112yes—
377C2×M16C23×C15112yes—
378C4⋊Q8—22no—
379#126—24no—
380#126—24no—
381#125—22no—
382#125—22no—
383#124—22no—
384#124—24no—
385#123—24no—
386#123—22no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8D4×C15112yes—
390C22⋊Q8D4×C15112yes—
391#121—24no—
392#121—22no—
393#121—22no—
394#121D4×C15112yes—
395#121D4×C15112yes—
396#120—22no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—24no—
402C4×D4—22no—
403C4×D4—22no—
404#117—24no—
405#117—22no—
406#117—22no—
407#117—22no—
408#117C23×C15112yes—
409C2×C4⋊C4—22no—
410C2×C4⋊C4C23×C15111yescharacteristic
411C2×C4⋊C4C23×C15111yescharacteristic
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4D4×C15112yes—
415C2×C42—21no—
416C2×C42—42no—
417C2×C42D4×C15112yes—
418C2×C42C23×C15111yescharacteristic
419C2×C42C23×C15111yescharacteristic
420#104—42no—
421#104—42no—
422#103—22no—
423#103—22no—
424#102—22no—
425#102—22no—
426C22⋊C8—22no—
427C22⋊C8—22no—
428#95—22no—
429#95—22no—
430C24—21no—
431C24C2×D4×C15111yescharacteristic
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—44no—
437#41—44no—
438#41—44no—
439#41—44no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8C2×D4×C15112yes—
445C2×Q8#41×C15112yes—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4C2×D4×C15112yes—
457C2×D4#41×C15112yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4C2×D4×C15111yescharacteristic
474C22×C4C2×D4×C15112yes—
475C22×C4C2×D4×C15112yes—
476C22×C4C2×D4×C15112yes—
477C22×C4C2×D4×C15112yes—
478C22×C4C24×C15111yescharacteristic
479Q16—44no—
480SD16—44no—
481SD16—44no—
482D8—44no—
483M16—22no—
484M16—22no—
485M16—22no—
486M16—22no—
487C2×C8—22no—
488C2×C8—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4C2×D4×C15112yes—
498C4⋊C4C2×D4×C15112yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—42no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23#120×C15112yes—
527C23C22×D4×C15111yescharacteristic
528Q8—22no—
529Q8—24no—
530Q8—22no—
531Q8—42no—
532Q8—82no—
533D4—22no—
534D4—24no—
535D4—22no—
536D4—48no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—82no—
541D4—82no—
542D4—82no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—48no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—84no—
571C2×C4#120×C15111yescharacteristic
572C2×C4#120×C15111yescharacteristic
573C2×C4C22×D4×C15112yes—
574C2×C4C2×#41×C15111yescharacteristic
575C2×C4C2×#41×C15111yescharacteristic
576C2×C4H(5,𝔽2)×C15112yes—
577C8—42no—
578C8—42no—
579C22—21no—
580C22—21no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—82no—
592C22—81no—
593C22—81no—
594C22C2×#120×C15111yescharacteristic
595C22#525×C15111yescharacteristic
596C22#535×C15111yescharacteristic
597C4—21no—
598C4—21no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4#546×C15112yes—
608C4#546×C15112yes—
609C2—21no—
610C2—21no—
611C2—42no—
612C2—82no—
613C2—82no—
614C2#2404×C15111yescharacteristic
615C2#3018×C15111yescharacteristic
616C2#3018×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3493×C151C2semidirect
2. #3005×C151C2semidirect
3. #2297×C151C2semidirect
4. #2028×C151C2semidirect
5. #2027×C151C2semidirect
6. #2007×C151C2semidirect
7. #2006×C151C2semidirect
8. #1896×C151C2semidirect
9. #1883×C151C2semidirect
10. #1882×C151C2semidirect
11. #1822×C151C2semidirect
308. #23950C151direct (N × H)
309. C151#23950direct (N × H)
310. #3493C2×C151semidirect
311. #3005C2×C151semidirect
312. #2297C2×C151semidirect
313. #2028C2×C151semidirect
314. #2027C2×C151semidirect
315. #2007C2×C151semidirect
316. #2006C2×C151semidirect
317. #1896C2×C151semidirect
318. #1883C2×C151semidirect
319. #1882C2×C151semidirect
320. #1822C2×C151semidirect

Maximal subgroups

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

subgroupindex
#3493×C1512
#3005×C1512
#2297×C1512
#2028×C1512
#2027×C1512
#2007×C1512
#2006×C1512
#1896×C1512
#1883×C1512
#1882×C1512
#1822×C1512
#23950151