Finite Group Databasebeta

← C151×#24171 #50329096

Subgroups — C151×#24171 #50329096

Order 38656.

1106 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511×C151C211yesmaximal, characteristic, complemented
2#3036×C151C211yesmaximal, characteristic, complemented
3C4×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic, complemented
4#2563×C151C211yesmaximal, characteristic, complemented
5#2040×C151C212yesmaximal, complemented
6#2006×C151C212yesmaximal, complemented
7#1997×C151C211yesmaximal, characteristic, complemented
8#1980×C151C211yesmaximal, characteristic, complemented
9#1883×C151C212yesmaximal, complemented
10#1860×C151C212yesmaximal, complemented
11#1806×C151C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C151C2211yescharacteristic
13C2×AGL(1,ℤ/8)×C151C2212yes—
14#556×C151—21no—
15#555×C151—21no—
16#554×C151—21no—
17#553×C151—22no—
18#548×C151—22no—
19#547×C151—21no—
20#542×C151—22no—
21#537×C151C2212yes—
22#533×C151—21no—
23#532×C151—21no—
24#532×C151C2211yescharacteristic
25C2×C4⋊Q8×C151C2211yescharacteristic
26#520×C151C2212yes—
27C4×#41×C151C2211yescharacteristic
28C2×C4×D4×C151C2211yescharacteristic
29#463×C151—22no—
30#461×C151—21no—
31#459×C151—21no—
32#442×C151—21no—
33#440×C151—21no—
34C4×SD16×C151—21no—
35C4×D8×C151—21no—
36#428×C151C2211yescharacteristic
37#423×C151C2211yescharacteristic
38C2×#104×C151C2212yes—
39C2×#104×C151C2212yes—
40C2×#104×C151C2212yes—
41#418×C151C2211yescharacteristic
42#417×C151C2212yes—
43#417×C151C2211yescharacteristic
44C4×M16×C151C2211yescharacteristic
45#393×C151—22no—
46#390×C151C2212yes—
47#382×C151C2212yes—
48#380×C151—22no—
49C4×C4⋊C4×C151C2211yescharacteristic
50#343×C151C2212yes—
51#343×C151C2212yes—
52#339×C151C2212yes—
53C2×#41×C151—22no—
54C2×#41×C151C2311yescharacteristic
55C2×#41×C151C2312yes—
56C22×Q8×C151D412yes—
57C22×D4×C151C2311yescharacteristic
58C23×C4×C151—21no—
59AGL(1,ℤ/8)×C151—22no—
60AGL(1,ℤ/8)×C151—24no—
61AGL(1,ℤ/8)×C151—22no—
62AGL(1,ℤ/8)×C151—42no—
63C2×SD16×C151—21no—
64C2×SD16×C151—22no—
65C2×D8×C151—21no—
66C2×D8×C151—22no—
67C2×M16×C151C2312yes—
68C2×M16×C151C2311yescharacteristic
69C2×M16×C151C2311yescharacteristic
70C4⋊Q8×C151—21no—
71C4⋊Q8×C151—21no—
72C4⋊Q8×C151—21no—
73C4⋊Q8×C151—21no—
74C4⋊Q8×C151D411yescharacteristic
75#127×C151D411yescharacteristic
76#126×C151—24no—
77#125×C151—22no—
78#124×C151—22no—
79#124×C151—24no—
80#124×C151D412yes—
81#123×C151—24no—
82C22⋊Q8×C151—22no—
83C22⋊Q8×C151—24no—
84C22⋊Q8×C151—22no—
85C22⋊Q8×C151—22no—
86#121×C151—22no—
87#121×C151—22no—
88#121×C151—22no—
89#121×C151—21no—
90#121×C151—21no—
91#121×C151—24no—
92C4×Q8×C151—22no—
93C4×Q8×C151—22no—
94C4×Q8×C151D411yescharacteristic
95C4×Q8×C151D411yescharacteristic
96C4×D4×C151—22no—
97C4×D4×C151—22no—
98C4×D4×C151—21no—
99C4×D4×C151—21no—
100C4×D4×C151—24no—
101C4×D4×C151—21no—
102C4×D4×C151—21no—
103C4×D4×C151—21no—
104C4×D4×C151—21no—
105C4×D4×C151—21no—
106C4×D4×C151—21no—
107C4×D4×C151D411yescharacteristic
108C4×D4×C151D411yescharacteristic
109#117×C151—22no—
110#117×C151—22no—
111#117×C151—21no—
112#117×C151C2311yescharacteristic
113C2×C4⋊C4×C151—22no—
114C2×C4⋊C4×C151—21no—
115C2×C4⋊C4×C151D412yes—
116C2×C4⋊C4×C151C2311yescharacteristic
117C2×C4⋊C4×C151C2312yes—
118C2×C22⋊C4×C151—21no—
119C2×C22⋊C4×C151—24no—
120C2×C42×C151—21no—
121C2×C42×C151C2312yes—
122C2×C42×C151C2311yescharacteristic
123#107×C151—21no—
124#106×C151—21no—
125C4⋊C8×C151—21no—
126C4⋊C8×C151—21no—
127#104×C151—24no—
128#104×C151—42no—
129#104×C151—42no—
130#103×C151—21no—
131#103×C151—22no—
132#102×C151—21no—
133#102×C151—22no—
134#102×C151—21no—
135#102×C151—21no—
136#97×C151—21no—
137C4×C8×C151—21no—
138#95×C151—21no—
139#95×C151—24no—
140C24×C151—21no—
141#41×C151—22no—
142#41×C151—22no—
143#41×C151—21no—
144#41×C151—21no—
145#41×C151—21no—
146#41×C151—22no—
147#41×C151—22no—
148#41×C151—21no—
149#41×C151—84no—
150C2×Q8×C151—22no—
151C2×Q8×C151—22no—
152C2×Q8×C151—82no—
153C2×Q8×C151C2×D412yes—
154C2×Q8×C151C2×D411yescharacteristic
155C2×D4×C151—21no—
156C2×D4×C151—22no—
157C2×D4×C151—24no—
158C2×D4×C151—21no—
159C2×D4×C151—21no—
160C2×D4×C151—21no—
161C2×D4×C151—21no—
162C2×D4×C151—21no—
163C2×D4×C151—21no—
164C2×D4×C151—21no—
165C2×D4×C151C2×D412yes—
166C2×D4×C151C2×D411yescharacteristic
167C2×D4×C151#4112yescomplemented
168C22×C4×C151—24no—
169C22×C4×C151—21no—
170C22×C4×C151—21no—
171C22×C4×C151—21no—
172C22×C4×C151—22no—
173C22×C4×C151—21no—
174C22×C4×C151—22no—
175C22×C4×C151—44no—
176C22×C4×C151—41no—
177C22×C4×C151—41no—
178C22×C4×C151C2×D411yescharacteristic
179C22×C4×C151C2×D414yes—
180C22×C4×C151C2×D411yescharacteristic
181C22×C4×C151C2411yescharacteristic
182SD16×C151—24no—
183SD16×C151—44no—
184D8×C151—24no—
185D8×C151—44no—
186M16×C151—21no—
187M16×C151—22no—
188M16×C151—21no—
189M16×C151—22no—
190M16×C151—21no—
191M16×C151#4112yescomplemented
192C2×C8×C151—21no—
193C2×C8×C151—21no—
194C2×C8×C151—22no—
195C4⋊C4×C151—24no—
196C4⋊C4×C151—24no—
197C4⋊C4×C151—22no—
198C4⋊C4×C151—22no—
199C4⋊C4×C151—21no—
200C4⋊C4×C151—21no—
201C4⋊C4×C151—21no—
202C4⋊C4×C151—21no—
203C4⋊C4×C151—22no—
204C4⋊C4×C151—22no—
205C4⋊C4×C151—21no—
206C4⋊C4×C151—21no—
207C4⋊C4×C151—42no—
208C4⋊C4×C151C2×D411yescharacteristic
209C4⋊C4×C151C2×D411yescharacteristic
210C22⋊C4×C151—24no—
211C22⋊C4×C151—24no—
212C22⋊C4×C151—24no—
213C22⋊C4×C151—21no—
214C22⋊C4×C151—21no—
215C22⋊C4×C151—21no—
216C22⋊C4×C151—22no—
217C22⋊C4×C151—22no—
218C22⋊C4×C151—41no—
219C22⋊C4×C151—41no—
220C22⋊C4×C151—84no—
221C42×C151—24no—
222C42×C151—21no—
223C42×C151—21no—
224C42×C151—21no—
225C42×C151—22no—
226C42×C151—22no—
227C42×C151C2×D411yescharacteristic
228C42×C151C2×D411yescharacteristic
229C42×C151C2×D411yescharacteristic
230C42×C151C2×D411yescharacteristic
231C23×C151—21no—
232C23×C151—22no—
233C23×C151—22no—
234C23×C151—42no—
235C23×C151—42no—
236C23×C151—41no—
237C23×C151—41no—
238C23×C151C22×D411yescharacteristic
239Q8×C151—21no—
240Q8×C151—21no—
241Q8×C151—22no—
242Q8×C151—22no—
243Q8×C151—44no—
244Q8×C151—82no—
245D4×C151—21no—
246D4×C151—22no—
247D4×C151—21no—
248D4×C151—22no—
249D4×C151—22no—
250D4×C151—22no—
251D4×C151—42no—
252D4×C151—42no—
253D4×C151—41no—
254D4×C151—42no—
255D4×C151—41no—
256D4×C151—41no—
257D4×C151—41no—
258D4×C151—42no—
259D4×C151—84no—
260C2×C4×C151—22no—
261C2×C4×C151—24no—
262C2×C4×C151—22no—
263C2×C4×C151—21no—
264C2×C4×C151—21no—
265C2×C4×C151—21no—
266C2×C4×C151—41no—
267C2×C4×C151—42no—
268C2×C4×C151—42no—
269C2×C4×C151—42no—
270C2×C4×C151—41no—
271C2×C4×C151—41no—
272C2×C4×C151—42no—
273C2×C4×C151—44no—
274C2×C4×C151—42no—
275C2×C4×C151—42no—
276C2×C4×C151—42no—
277C2×C4×C151—41no—
278C2×C4×C151—42no—
279C2×C4×C151—82no—
280C2×C4×C151—84no—
281C2×C4×C151—82no—
282C2×C4×C151#12011yescharacteristic
283C2×C4×C151#12011yescharacteristic
284C2×C4×C151#12011yescharacteristic
285C2×C4×C151#12011yescharacteristic
286C2×C4×C151C22×D411yescharacteristic
287C2×C4×C151C22×D411yescharacteristic
288C2×C4×C151C2×#4111yescharacteristic
289C2×C4×C151C2×#4111yescharacteristic
290C2×C4×C151H(5,𝔽2)11yescharacteristic
291C2×C4×C1512−1+411yescharacteristic
292C8×C151—22no—
293C8×C151—41no—
294C8×C151—42no—
295C22×C151—21no—
296C22×C151—21no—
297C22×C151—44no—
298C22×C151—42no—
299C22×C151—41no—
300C22×C151—42no—
301C22×C151—41no—
302C22×C151—42no—
303C22×C151—41no—
304C22×C151—42no—
305C22×C151—82no—
306C22×C151—81no—
307C22×C151—81no—
308C22×C151C2×#12011yescharacteristic
309C22×C151#54811yescharacteristic
310C22×C151#55011yescharacteristic
311C4×C151—21no—
312C4×C151—21no—
313C4×C151—21no—
314C4×C151—42no—
315C4×C151—41no—
316C4×C151—41no—
317C4×C151—41no—
318C4×C151—44no—
319C4×C151—82no—
320C4×C151—81no—
321C4×C151#45412yes—
322C4×C151#52511yescharacteristic
323C4×C151#53711yescharacteristic
324C4×C151#54611yescharacteristic
325C4×C151#54711yescharacteristic
326C2×C151—21no—
327C2×C151—21no—
328C2×C151—81no—
329C2×C151—82no—
330C2×C151—82no—
331C2×C151#240711yescharacteristic
332C2×C151C2×#45411yescharacteristic
333C2×C151#301911yescharacteristic
334#24171C15111yesmaximal, characteristic, direct product factor, Sylow
335C151#2417111yescharacteristic, direct product factor, Sylow
336#3511C2×C15111yescharacteristic, complemented
337#3036C2×C15111yescharacteristic, complemented
338C4×AGL(1,ℤ/8)C2×C15111yescharacteristic, complemented
339#2563C2×C15111yescharacteristic, complemented
340#2040C2×C15112yescomplemented
341#2006C2×C15112yescomplemented
342#1997C2×C15111yescharacteristic, complemented
343#1980C2×C15111yescharacteristic, complemented
344#1883C2×C15112yescomplemented
345#1860C2×C15112yescomplemented
346#1806C2×C15111yescharacteristic, complemented
347C2×AGL(1,ℤ/8)C22×C15111yescharacteristic
348C2×AGL(1,ℤ/8)C22×C15112yes—
349#556—21no—
350#555—21no—
351#554—21no—
352#553—22no—
353#548—22no—
354#547—21no—
355#542—22no—
356#537C22×C15112yes—
357#533—21no—
358#532—21no—
359#532C22×C15111yescharacteristic
360C2×C4⋊Q8C22×C15111yescharacteristic
361#520C22×C15112yes—
362C4×#41C22×C15111yescharacteristic
363C2×C4×D4C22×C15111yescharacteristic
364#463—22no—
365#461—21no—
366#459—21no—
367#442—21no—
368#440—21no—
369C4×SD16—21no—
370C4×D8—21no—
371#428C22×C15111yescharacteristic
372#423C22×C15111yescharacteristic
373C2×#104C22×C15112yes—
374C2×#104C22×C15112yes—
375C2×#104C22×C15112yes—
376#418C22×C15111yescharacteristic
377#417C22×C15112yes—
378#417C22×C15111yescharacteristic
379C4×M16C22×C15111yescharacteristic
380#393—22no—
381#390C22×C15112yes—
382#382C22×C15112yes—
383#380—22no—
384C4×C4⋊C4C22×C15111yescharacteristic
385#343C22×C15112yes—
386#343C22×C15112yes—
387#339C22×C15112yes—
388C2×#41—22no—
389C2×#41C23×C15111yescharacteristic
390C2×#41C23×C15112yes—
391C22×Q8D4×C15112yes—
392C22×D4C23×C15111yescharacteristic
393C23×C4—21no—
394AGL(1,ℤ/8)—22no—
395AGL(1,ℤ/8)—24no—
396AGL(1,ℤ/8)—22no—
397AGL(1,ℤ/8)—42no—
398C2×SD16—21no—
399C2×SD16—22no—
400C2×D8—21no—
401C2×D8—22no—
402C2×M16C23×C15112yes—
403C2×M16C23×C15111yescharacteristic
404C2×M16C23×C15111yescharacteristic
405C4⋊Q8—21no—
406C4⋊Q8—21no—
407C4⋊Q8—21no—
408C4⋊Q8—21no—
409C4⋊Q8D4×C15111yescharacteristic
410#127D4×C15111yescharacteristic
411#126—24no—
412#125—22no—
413#124—22no—
414#124—24no—
415#124D4×C15112yes—
416#123—24no—
417C22⋊Q8—22no—
418C22⋊Q8—24no—
419C22⋊Q8—22no—
420C22⋊Q8—22no—
421#121—22no—
422#121—22no—
423#121—22no—
424#121—21no—
425#121—21no—
426#121—24no—
427C4×Q8—22no—
428C4×Q8—22no—
429C4×Q8D4×C15111yescharacteristic
430C4×Q8D4×C15111yescharacteristic
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—24no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4D4×C15111yescharacteristic
443C4×D4D4×C15111yescharacteristic
444#117—22no—
445#117—22no—
446#117—21no—
447#117C23×C15111yescharacteristic
448C2×C4⋊C4—22no—
449C2×C4⋊C4—21no—
450C2×C4⋊C4D4×C15112yes—
451C2×C4⋊C4C23×C15111yescharacteristic
452C2×C4⋊C4C23×C15112yes—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—24no—
455C2×C42—21no—
456C2×C42C23×C15112yes—
457C2×C42C23×C15111yescharacteristic
458#107—21no—
459#106—21no—
460C4⋊C8—21no—
461C4⋊C8—21no—
462#104—24no—
463#104—42no—
464#104—42no—
465#103—21no—
466#103—22no—
467#102—21no—
468#102—22no—
469#102—21no—
470#102—21no—
471#97—21no—
472C4×C8—21no—
473#95—21no—
474#95—24no—
475C24—21no—
476#41—22no—
477#41—22no—
478#41—21no—
479#41—21no—
480#41—21no—
481#41—22no—
482#41—22no—
483#41—21no—
484#41—84no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—82no—
488C2×Q8C2×D4×C15112yes—
489C2×Q8C2×D4×C15111yescharacteristic
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—24no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4C2×D4×C15112yes—
501C2×D4C2×D4×C15111yescharacteristic
502C2×D4#41×C15112yescomplemented
503C22×C4—24no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—44no—
511C22×C4—41no—
512C22×C4—41no—
513C22×C4C2×D4×C15111yescharacteristic
514C22×C4C2×D4×C15114yes—
515C22×C4C2×D4×C15111yescharacteristic
516C22×C4C24×C15111yescharacteristic
517SD16—24no—
518SD16—44no—
519D8—24no—
520D8—44no—
521M16—21no—
522M16—22no—
523M16—21no—
524M16—22no—
525M16—21no—
526M16#41×C15112yescomplemented
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—42no—
543C4⋊C4C2×D4×C15111yescharacteristic
544C4⋊C4C2×D4×C15111yescharacteristic
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—84no—
556C42—24no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—22no—
561C42—22no—
562C42C2×D4×C15111yescharacteristic
563C42C2×D4×C15111yescharacteristic
564C42C2×D4×C15111yescharacteristic
565C42C2×D4×C15111yescharacteristic
566C23—21no—
567C23—22no—
568C23—22no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—41no—
573C23C22×D4×C15111yescharacteristic
574Q8—21no—
575Q8—21no—
576Q8—22no—
577Q8—22no—
578Q8—44no—
579Q8—82no—
580D4—21no—
581D4—22no—
582D4—21no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—42no—
587D4—42no—
588D4—41no—
589D4—42no—
590D4—41no—
591D4—42no—
592D4—41no—
593D4—41no—
594D4—84no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—82no—
615C2×C4—84no—
616C2×C4—82no—
617C2×C4#120×C15111yescharacteristic
618C2×C4#120×C15111yescharacteristic
619C2×C4#120×C15111yescharacteristic
620C2×C4#120×C15111yescharacteristic
621C2×C4C22×D4×C15111yescharacteristic
622C2×C4C22×D4×C15111yescharacteristic
623C2×C4C2×#41×C15111yescharacteristic
624C2×C4C2×#41×C15111yescharacteristic
625C2×C4H(5,𝔽2)×C15111yescharacteristic
626C2×C42−1+4×C15111yescharacteristic
627C8—22no—
628C8—41no—
629C8—42no—
630C22—21no—
631C22—21no—
632C22—44no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22—41no—
639C22—42no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22C2×#120×C15111yescharacteristic
644C22#548×C15111yescharacteristic
645C22#550×C15111yescharacteristic
646C4—21no—
647C4—21no—
648C4—21no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—44no—
654C4—82no—
655C4—81no—
656C4#454×C15112yes—
657C4#525×C15111yescharacteristic
658C4#537×C15111yescharacteristic
659C4#546×C15111yescharacteristic
660C4#547×C15111yescharacteristic
661C2—21no—
662C2—21no—
663C2—82no—
664C2—81no—
665C2—82no—
666C2#2407×C15111yescharacteristic
667C2C2×#454×C15111yescharacteristic
668C2#3019×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3511×C151C2semidirect
2. #3036×C151C2semidirect
3. C4×AGL(1,ℤ/8)×C151C2semidirect
4. #2563×C151C2semidirect
5. #2040×C151C2semidirect
6. #2006×C151C2semidirect
7. #1997×C151C2semidirect
8. #1980×C151C2semidirect
9. #1883×C151C2semidirect
10. #1860×C151C2semidirect
11. #1806×C151C2semidirect
167. C2×D4×C151#41semidirect
191. M16×C151#41semidirect
334. #24171C151direct (N × H)
335. C151#24171direct (N × H)
336. #3511C2×C151semidirect
337. #3036C2×C151semidirect
338. C4×AGL(1,ℤ/8)C2×C151semidirect
339. #2563C2×C151semidirect
340. #2040C2×C151semidirect
341. #2006C2×C151semidirect
342. #1997C2×C151semidirect
343. #1980C2×C151semidirect
344. #1883C2×C151semidirect
345. #1860C2×C151semidirect
346. #1806C2×C151semidirect
502. C2×D4#41×C151semidirect
526. M16#41×C151semidirect

Maximal subgroups

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

subgroupindex
#3511×C1512
#3036×C1512
C4×AGL(1,ℤ/8)×C1512
#2563×C1512
#2040×C1512
#2006×C1512
#1997×C1512
#1980×C1512
#1883×C1512
#1860×C1512
#1806×C1512
#24171151