Finite Group Databasebeta

← C151×#24170 #50329095

Subgroups — C151×#24170 #50329095

Order 38656.

1162 conjugacy classes of subgroups.

The subgroup lattice has 674 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#2542×C151C211yesmaximal, characteristic, complemented
5#2039×C151C212yesmaximal, complemented
6#2006×C151C212yesmaximal, complemented
7#1997×C151C211yesmaximal, characteristic, complemented
8#1980×C151C211yesmaximal, characteristic, complemented
9#1883×C151C212yesmaximal, complemented
10#1859×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
26C2×C22⋊Q8×C151—22no—
27#520×C151C2212yes—
28C4×#41×C151C2211yescharacteristic
29C2×C4×D4×C151C2211yescharacteristic
30#463×C151—22no—
31#461×C151—21no—
32#459×C151—21no—
33#442×C151—21no—
34#440×C151—21no—
35C4×SD16×C151—21no—
36C4×D8×C151—21no—
37#428×C151C2211yescharacteristic
38#423×C151C2211yescharacteristic
39C2×#104×C151C2212yes—
40C2×#104×C151C2212yes—
41C2×#104×C151C2212yes—
42#418×C151C2211yescharacteristic
43#417×C151C2212yes—
44#417×C151C2211yescharacteristic
45C4×M16×C151C2211yescharacteristic
46#399×C151—22no—
47#390×C151C2212yes—
48#384×C151C2212yes—
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—
86C22⋊Q8×C151—48no—
87#121×C151—22no—
88#121×C151—22no—
89#121×C151—22no—
90#121×C151—21no—
91#121×C151—21no—
92#121×C151—24no—
93C4×Q8×C151—22no—
94C4×Q8×C151—22no—
95C4×Q8×C151D411yescharacteristic
96C4×Q8×C151D411yescharacteristic
97C4×D4×C151—22no—
98C4×D4×C151—22no—
99C4×D4×C151—21no—
100C4×D4×C151—21no—
101C4×D4×C151—24no—
102C4×D4×C151—21no—
103C4×D4×C151—21no—
104C4×D4×C151—21no—
105C4×D4×C151—21no—
106C4×D4×C151—21no—
107C4×D4×C151—21no—
108C4×D4×C151D411yescharacteristic
109C4×D4×C151D411yescharacteristic
110#117×C151—22no—
111#117×C151—22no—
112#117×C151—21no—
113#117×C151C2311yescharacteristic
114C2×C4⋊C4×C151—21no—
115C2×C4⋊C4×C151—22no—
116C2×C4⋊C4×C151—24no—
117C2×C4⋊C4×C151D412yes—
118C2×C4⋊C4×C151C2311yescharacteristic
119C2×C4⋊C4×C151C2312yes—
120C2×C22⋊C4×C151—24no—
121C2×C22⋊C4×C151—21no—
122C2×C42×C151—21no—
123C2×C42×C151C2312yes—
124C2×C42×C151C2311yescharacteristic
125#107×C151—21no—
126#106×C151—21no—
127C4⋊C8×C151—21no—
128C4⋊C8×C151—21no—
129#104×C151—24no—
130#104×C151—42no—
131#104×C151—42no—
132#103×C151—21no—
133#103×C151—22no—
134#102×C151—21no—
135#102×C151—22no—
136#102×C151—21no—
137#102×C151—21no—
138#97×C151—21no—
139C4×C8×C151—21no—
140#95×C151—21no—
141C24×C151—21no—
142#41×C151—22no—
143#41×C151—22no—
144#41×C151—21no—
145#41×C151—21no—
146#41×C151—21no—
147#41×C151—22no—
148#41×C151—22no—
149#41×C151—21no—
150#41×C151—84no—
151C2×Q8×C151—22no—
152C2×Q8×C151—22no—
153C2×Q8×C151—44no—
154C2×Q8×C151C2×D412yes—
155C2×Q8×C151C2×D411yescharacteristic
156C2×D4×C151—21no—
157C2×D4×C151—22no—
158C2×D4×C151—24no—
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×C151—21no—
166C2×D4×C151C2×D412yes—
167C2×D4×C151C2×D411yescharacteristic
168C2×D4×C151#4112yescomplemented
169C22×C4×C151—22no—
170C22×C4×C151—24no—
171C22×C4×C151—21no—
172C22×C4×C151—21no—
173C22×C4×C151—21no—
174C22×C4×C151—22no—
175C22×C4×C151—21no—
176C22×C4×C151—44no—
177C22×C4×C151—41no—
178C22×C4×C151—41no—
179C22×C4×C151C2×D411yescharacteristic
180C22×C4×C151C2×D414yes—
181C22×C4×C151C2×D411yescharacteristic
182C22×C4×C151C2411yescharacteristic
183SD16×C151—24no—
184SD16×C151—44no—
185D8×C151—24no—
186D8×C151—44no—
187M16×C151—21no—
188M16×C151—22no—
189M16×C151—21no—
190M16×C151—22no—
191M16×C151—21no—
192M16×C151#4112yescomplemented
193C2×C8×C151—21no—
194C2×C8×C151—21no—
195C2×C8×C151—22no—
196C4⋊C4×C151—24no—
197C4⋊C4×C151—24no—
198C4⋊C4×C151—24no—
199C4⋊C4×C151—22no—
200C4⋊C4×C151—22no—
201C4⋊C4×C151—21no—
202C4⋊C4×C151—21no—
203C4⋊C4×C151—21no—
204C4⋊C4×C151—21no—
205C4⋊C4×C151—22no—
206C4⋊C4×C151—22no—
207C4⋊C4×C151—21no—
208C4⋊C4×C151—21no—
209C4⋊C4×C151—48no—
210C4⋊C4×C151C2×D411yescharacteristic
211C4⋊C4×C151C2×D411yescharacteristic
212C22⋊C4×C151—24no—
213C22⋊C4×C151—24no—
214C22⋊C4×C151—24no—
215C22⋊C4×C151—21no—
216C22⋊C4×C151—21no—
217C22⋊C4×C151—21no—
218C22⋊C4×C151—22no—
219C22⋊C4×C151—22no—
220C22⋊C4×C151—48no—
221C22⋊C4×C151—41no—
222C22⋊C4×C151—41no—
223C42×C151—24no—
224C42×C151—21no—
225C42×C151—21no—
226C42×C151—21no—
227C42×C151—22no—
228C42×C151—22no—
229C42×C151C2×D411yescharacteristic
230C42×C151C2×D411yescharacteristic
231C42×C151C2×D411yescharacteristic
232C42×C151C2×D411yescharacteristic
233C23×C151—21no—
234C23×C151—22no—
235C23×C151—22no—
236C23×C151—42no—
237C23×C151—42no—
238C23×C151—41no—
239C23×C151—41no—
240C23×C151C22×D411yescharacteristic
241Q8×C151—21no—
242Q8×C151—21no—
243Q8×C151—22no—
244Q8×C151—22no—
245Q8×C151—44no—
246Q8×C151—82no—
247D4×C151—21no—
248D4×C151—22no—
249D4×C151—21no—
250D4×C151—22no—
251D4×C151—22no—
252D4×C151—22no—
253D4×C151—42no—
254D4×C151—42no—
255D4×C151—41no—
256D4×C151—42no—
257D4×C151—41no—
258D4×C151—41no—
259D4×C151—41no—
260D4×C151—42no—
261D4×C151—84no—
262C2×C4×C151—22no—
263C2×C4×C151—24no—
264C2×C4×C151—24no—
265C2×C4×C151—22no—
266C2×C4×C151—21no—
267C2×C4×C151—21no—
268C2×C4×C151—21no—
269C2×C4×C151—41no—
270C2×C4×C151—42no—
271C2×C4×C151—42no—
272C2×C4×C151—42no—
273C2×C4×C151—42no—
274C2×C4×C151—42no—
275C2×C4×C151—41no—
276C2×C4×C151—41no—
277C2×C4×C151—42no—
278C2×C4×C151—44no—
279C2×C4×C151—42no—
280C2×C4×C151—41no—
281C2×C4×C151—82no—
282C2×C4×C151—84no—
283C2×C4×C151—82no—
284C2×C4×C151#12011yescharacteristic
285C2×C4×C151#12011yescharacteristic
286C2×C4×C151#12011yescharacteristic
287C2×C4×C151#12011yescharacteristic
288C2×C4×C151C22×D411yescharacteristic
289C2×C4×C151C22×D411yescharacteristic
290C2×C4×C151C2×#4111yescharacteristic
291C2×C4×C151C2×#4111yescharacteristic
292C2×C4×C151H(5,𝔽2)11yescharacteristic
293C2×C4×C1512−1+411yescharacteristic
294C8×C151—22no—
295C8×C151—41no—
296C8×C151—42no—
297C22×C151—21no—
298C22×C151—21no—
299C22×C151—44no—
300C22×C151—42no—
301C22×C151—41no—
302C22×C151—42no—
303C22×C151—41no—
304C22×C151—42no—
305C22×C151—41no—
306C22×C151—42no—
307C22×C151—82no—
308C22×C151—81no—
309C22×C151—81no—
310C22×C151C2×#12011yescharacteristic
311C22×C151#54811yescharacteristic
312C22×C151#55011yescharacteristic
313C4×C151—21no—
314C4×C151—21no—
315C4×C151—21no—
316C4×C151—44no—
317C4×C151—42no—
318C4×C151—41no—
319C4×C151—41no—
320C4×C151—44no—
321C4×C151—41no—
322C4×C151—81no—
323C4×C151#45312yes—
324C4×C151#52511yescharacteristic
325C4×C151#53711yescharacteristic
326C4×C151#54611yescharacteristic
327C4×C151#54711yescharacteristic
328C2×C151—21no—
329C2×C151—21no—
330C2×C151—81no—
331C2×C151—82no—
332C2×C151—82no—
333C2×C151#240711yescharacteristic
334C2×C151C2×#45311yescharacteristic
335C2×C151#302011yescharacteristic
336#24170C15111yesmaximal, characteristic, direct product factor, Sylow
337C151#2417011yescharacteristic, direct product factor, Sylow
338#3511C2×C15111yescharacteristic, complemented
339#3036C2×C15111yescharacteristic, complemented
340C4×AGL(1,ℤ/8)C2×C15111yescharacteristic, complemented
341#2542C2×C15111yescharacteristic, complemented
342#2039C2×C15112yescomplemented
343#2006C2×C15112yescomplemented
344#1997C2×C15111yescharacteristic, complemented
345#1980C2×C15111yescharacteristic, complemented
346#1883C2×C15112yescomplemented
347#1859C2×C15112yescomplemented
348#1806C2×C15111yescharacteristic, complemented
349C2×AGL(1,ℤ/8)C22×C15111yescharacteristic
350C2×AGL(1,ℤ/8)C22×C15112yes—
351#556—21no—
352#555—21no—
353#554—21no—
354#553—22no—
355#548—22no—
356#547—21no—
357#542—22no—
358#537C22×C15112yes—
359#533—21no—
360#532—21no—
361#532C22×C15111yescharacteristic
362C2×C4⋊Q8C22×C15111yescharacteristic
363C2×C22⋊Q8—22no—
364#520C22×C15112yes—
365C4×#41C22×C15111yescharacteristic
366C2×C4×D4C22×C15111yescharacteristic
367#463—22no—
368#461—21no—
369#459—21no—
370#442—21no—
371#440—21no—
372C4×SD16—21no—
373C4×D8—21no—
374#428C22×C15111yescharacteristic
375#423C22×C15111yescharacteristic
376C2×#104C22×C15112yes—
377C2×#104C22×C15112yes—
378C2×#104C22×C15112yes—
379#418C22×C15111yescharacteristic
380#417C22×C15112yes—
381#417C22×C15111yescharacteristic
382C4×M16C22×C15111yescharacteristic
383#399—22no—
384#390C22×C15112yes—
385#384C22×C15112yes—
386C4×C4⋊C4C22×C15111yescharacteristic
387#343C22×C15112yes—
388#343C22×C15112yes—
389#339C22×C15112yes—
390C2×#41—22no—
391C2×#41C23×C15111yescharacteristic
392C2×#41C23×C15112yes—
393C22×Q8D4×C15112yes—
394C22×D4C23×C15111yescharacteristic
395C23×C4—21no—
396AGL(1,ℤ/8)—22no—
397AGL(1,ℤ/8)—24no—
398AGL(1,ℤ/8)—22no—
399AGL(1,ℤ/8)—42no—
400C2×SD16—21no—
401C2×SD16—22no—
402C2×D8—21no—
403C2×D8—22no—
404C2×M16C23×C15112yes—
405C2×M16C23×C15111yescharacteristic
406C2×M16C23×C15111yescharacteristic
407C4⋊Q8—21no—
408C4⋊Q8—21no—
409C4⋊Q8—21no—
410C4⋊Q8—21no—
411C4⋊Q8D4×C15111yescharacteristic
412#127D4×C15111yescharacteristic
413#126—24no—
414#125—22no—
415#124—22no—
416#124—24no—
417#124D4×C15112yes—
418#123—24no—
419C22⋊Q8—22no—
420C22⋊Q8—24no—
421C22⋊Q8—22no—
422C22⋊Q8—22no—
423C22⋊Q8—48no—
424#121—22no—
425#121—22no—
426#121—22no—
427#121—21no—
428#121—21no—
429#121—24no—
430C4×Q8—22no—
431C4×Q8—22no—
432C4×Q8D4×C15111yescharacteristic
433C4×Q8D4×C15111yescharacteristic
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—24no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4D4×C15111yescharacteristic
446C4×D4D4×C15111yescharacteristic
447#117—22no—
448#117—22no—
449#117—21no—
450#117C23×C15111yescharacteristic
451C2×C4⋊C4—21no—
452C2×C4⋊C4—22no—
453C2×C4⋊C4—24no—
454C2×C4⋊C4D4×C15112yes—
455C2×C4⋊C4C23×C15111yescharacteristic
456C2×C4⋊C4C23×C15112yes—
457C2×C22⋊C4—21no—
458C2×C22⋊C4—24no—
459C2×C42—21no—
460C2×C42C23×C15112yes—
461C2×C42C23×C15111yescharacteristic
462#107—21no—
463#106—21no—
464C4⋊C8—21no—
465C4⋊C8—21no—
466#104—24no—
467#104—42no—
468#104—42no—
469#103—21no—
470#103—22no—
471#102—21no—
472#102—22no—
473#102—21no—
474#102—21no—
475#97—21no—
476C4×C8—21no—
477#95—21no—
478C24—21no—
479#41—22no—
480#41—22no—
481#41—21no—
482#41—21no—
483#41—21no—
484#41—22no—
485#41—22no—
486#41—21no—
487#41—84no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—44no—
491C2×Q8C2×D4×C15112yes—
492C2×Q8C2×D4×C15111yescharacteristic
493C2×D4—21no—
494C2×D4—22no—
495C2×D4—24no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4C2×D4×C15112yes—
504C2×D4C2×D4×C15111yescharacteristic
505C2×D4#41×C15112yescomplemented
506C22×C4—22no—
507C22×C4—24no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—21no—
513C22×C4—44no—
514C22×C4—41no—
515C22×C4—41no—
516C22×C4C2×D4×C15111yescharacteristic
517C22×C4C2×D4×C15114yes—
518C22×C4C2×D4×C15111yescharacteristic
519C22×C4C24×C15111yescharacteristic
520SD16—24no—
521SD16—44no—
522D8—24no—
523D8—44no—
524M16—21no—
525M16—22no—
526M16—21no—
527M16—22no—
528M16—21no—
529M16#41×C15112yescomplemented
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—48no—
547C4⋊C4C2×D4×C15111yescharacteristic
548C4⋊C4C2×D4×C15111yescharacteristic
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—48no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C42—24no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—22no—
565C42—22no—
566C42C2×D4×C15111yescharacteristic
567C42C2×D4×C15111yescharacteristic
568C42C2×D4×C15111yescharacteristic
569C42C2×D4×C15111yescharacteristic
570C23—21no—
571C23—22no—
572C23—22no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—41no—
577C23C22×D4×C15111yescharacteristic
578Q8—21no—
579Q8—21no—
580Q8—22no—
581Q8—22no—
582Q8—44no—
583Q8—82no—
584D4—21no—
585D4—22no—
586D4—21no—
587D4—22no—
588D4—22no—
589D4—22no—
590D4—42no—
591D4—42no—
592D4—41no—
593D4—42no—
594D4—41no—
595D4—42no—
596D4—41no—
597D4—41no—
598D4—84no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—24no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—82no—
619C2×C4—84no—
620C2×C4—82no—
621C2×C4#120×C15111yescharacteristic
622C2×C4#120×C15111yescharacteristic
623C2×C4#120×C15111yescharacteristic
624C2×C4#120×C15111yescharacteristic
625C2×C4C22×D4×C15111yescharacteristic
626C2×C4C22×D4×C15111yescharacteristic
627C2×C4C2×#41×C15111yescharacteristic
628C2×C4C2×#41×C15111yescharacteristic
629C2×C4H(5,𝔽2)×C15111yescharacteristic
630C2×C42−1+4×C15111yescharacteristic
631C8—22no—
632C8—41no—
633C8—42no—
634C22—21no—
635C22—21no—
636C22—44no—
637C22—42no—
638C22—41no—
639C22—42no—
640C22—41no—
641C22—42no—
642C22—41no—
643C22—42no—
644C22—82no—
645C22—81no—
646C22—81no—
647C22C2×#120×C15111yescharacteristic
648C22#548×C15111yescharacteristic
649C22#550×C15111yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—44no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—44no—
658C4—41no—
659C4—81no—
660C4#453×C15112yes—
661C4#525×C15111yescharacteristic
662C4#537×C15111yescharacteristic
663C4#546×C15111yescharacteristic
664C4#547×C15111yescharacteristic
665C2—21no—
666C2—21no—
667C2—82no—
668C2—81no—
669C2—82no—
670C2#2407×C15111yescharacteristic
671C2C2×#453×C15111yescharacteristic
672C2#3020×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. #2542×C151C2semidirect
5. #2039×C151C2semidirect
6. #2006×C151C2semidirect
7. #1997×C151C2semidirect
8. #1980×C151C2semidirect
9. #1883×C151C2semidirect
10. #1859×C151C2semidirect
11. #1806×C151C2semidirect
168. C2×D4×C151#41semidirect
192. M16×C151#41semidirect
336. #24170C151direct (N × H)
337. C151#24170direct (N × H)
338. #3511C2×C151semidirect
339. #3036C2×C151semidirect
340. C4×AGL(1,ℤ/8)C2×C151semidirect
341. #2542C2×C151semidirect
342. #2039C2×C151semidirect
343. #2006C2×C151semidirect
344. #1997C2×C151semidirect
345. #1980C2×C151semidirect
346. #1883C2×C151semidirect
347. #1859C2×C151semidirect
348. #1806C2×C151semidirect
505. C2×D4#41×C151semidirect
529. 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
#2542×C1512
#2039×C1512
#2006×C1512
#1997×C1512
#1980×C1512
#1883×C1512
#1859×C1512
#1806×C1512
#24170151