Finite Group Databasebeta

← C151×#19043 #50323968

Subgroups — C151×#19043 #50323968

Order 38656.

950 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3558×C151C211yesmaximal, characteristic
2#3215×C151C211yesmaximal, characteristic, complemented
3#3181×C151C211yesmaximal, characteristic, complemented
4#3135×C151C211yesmaximal, characteristic, complemented
5#1662×C151C212yesmaximal, complemented
6#1661×C151C212yesmaximal, complemented
7#1659×C151C212yesmaximal, complemented
8#1659×C151C212yesmaximal, complemented
9#1532×C151C211yesmaximal, characteristic, complemented
10#1530×C151C211yesmaximal, characteristic, complemented
11C2×#329×C151C211yesmaximal, characteristic, complemented
12#562×C151C2212yes—
13H(3,𝔽4)×C151C2212yes—
14#559×C151—21no—
15#548×C151—21no—
16#546×C151—22no—
17#546×C151—21no—
18D42×C151—21no—
19#542×C151—21no—
20#538×C151—22no—
21#538×C151—21no—
22#535×C151—21no—
23#534×C151—21no—
24#533×C151C2211yescharacteristic, complemented
25#532×C151C2211yescharacteristic
26#529×C151C2211yescharacteristic
27C2×#124×C151C2211yescharacteristic
28#501×C151C2211yescharacteristic, complemented
29#497×C151C2211yescharacteristic, complemented
30#497×C151C2212yescomplemented
31#491×C151—21no—
32#491×C151C2211yescharacteristic, complemented
33#490×C151—21no—
34#489×C151C2212yescomplemented
35#489×C151C2212yescomplemented
36#489×C151C2211yescharacteristic, complemented
37#489×C151C2212yescomplemented
38#467×C151—21no—
39#465×C151—21no—
40#464×C151—22no—
41#451×C151—21no—
42#450×C151—21no—
43#441×C151—21no—
44#440×C151—21no—
45#432×C151C2211yescharacteristic, complemented
46#416×C151—22no—
47C2×#97×C151C2211yescharacteristic, complemented
48C2×#97×C151C2211yescharacteristic, complemented
49#330×C151C2212yescomplemented
50#329×C151C2212yescomplemented
51#329×C151C2212yescomplemented
52#329×C151C2212yescomplemented
53#329×C151C2212yescomplemented
54#329×C151C2212yescomplemented
55C2×#41×C151—21no—
56C2×#41×C151D412yes—
57C22×Q8×C151D411yescharacteristic
58C22×D4×C151—22no—
59C22×D4×C151—21no—
60C22×D4×C151D411yescharacteristic
61C2×Q16×C151—21no—
62C2×Q16×C151—22no—
63C2×SD16×C151—21no—
64C2×SD16×C151—22no—
65C22×C8×C151—21no—
66C22×C8×C151—21no—
67C4⋊Q8×C151D411yescharacteristic
68C4⋊Q8×C151C2311yescharacteristic
69C4⋊Q8×C151C2311yescharacteristic
70#127×C151—21no—
71#127×C151D411yescharacteristic
72#126×C151—22no—
73#126×C151—21no—
74#125×C151—21no—
75#125×C151C2311yescharacteristic
76#124×C151—21no—
77#124×C151—21no—
78#124×C151—21no—
79#124×C151—22no—
80#124×C151—22no—
81#124×C151—22no—
82#124×C151D412yes—
83#124×C151C2312yes—
84#124×C151C2312yes—
85#124×C151C2311yescharacteristic
86#123×C151—22no—
87#123×C151—22no—
88#123×C151—21no—
89#123×C151—22no—
90#123×C151—22no—
91C22⋊Q8×C151—21no—
92C22⋊Q8×C151—21no—
93C22⋊Q8×C151—21no—
94C22⋊Q8×C151—22no—
95C22⋊Q8×C151—22no—
96C22⋊Q8×C151D411yescharacteristic
97#121×C151—22no—
98#121×C151—22no—
99#121×C151—22no—
100#121×C151—21no—
101#121×C151—22no—
102#121×C151—22no—
103#121×C151—21no—
104#121×C151—21no—
105#121×C151D411yescharacteristic
106#120×C151—22no—
107#120×C151—22no—
108#120×C151—22no—
109#120×C151—22no—
110#120×C151—22no—
111C4×Q8×C151—21no—
112C4×Q8×C151—21no—
113C4×Q8×C151—21no—
114C4×Q8×C151D411yescharacteristic
115C4×D4×C151—22no—
116C4×D4×C151—22no—
117C4×D4×C151—21no—
118C4×D4×C151—21no—
119C4×D4×C151—21no—
120C4×D4×C151—21no—
121C4×D4×C151D411yescharacteristic
122#117×C151—21no—
123#117×C151—21no—
124C2×C22⋊C4×C151—22no—
125C2×C22⋊C4×C151—21no—
126C2×C22⋊C4×C151—21no—
127C2×C42×C151C2311yescharacteristic
128#107×C151—21no—
129#106×C151—21no—
130C4⋊C8×C151—21no—
131#103×C151—21no—
132#103×C151—21no—
133#103×C151—22no—
134#103×C151—22no—
135#103×C151—21no—
136#103×C151—22no—
137#102×C151—22no—
138#102×C151—22no—
139#102×C151—21no—
140#102×C151—22no—
141C22⋊C8×C151—21no—
142#97×C151—21no—
143#97×C151—21no—
144#97×C151C2312yes—
145#97×C151C2311yescharacteristic
146#97×C151C2311yescharacteristic
147#97×C151C2312yes—
148C24×C151—22no—
149C24×C151—21no—
150#41×C151—44no—
151#41×C151—82no—
152C2×Q8×C151—21no—
153C2×Q8×C151—21no—
154C2×Q8×C151—21no—
155C2×Q8×C151—81no—
156C2×Q8×C151C2×D412yes—
157C2×Q8×C151C2×D411yescharacteristic
158C2×Q8×C151C2×D412yes—
159C2×D4×C151—22no—
160C2×D4×C151—21no—
161C2×D4×C151—22no—
162C2×D4×C151—22no—
163C2×D4×C151—22no—
164C2×D4×C151—22no—
165C2×D4×C151—22no—
166C2×D4×C151—21no—
167C2×D4×C151—21no—
168C2×D4×C151—21no—
169C2×D4×C151—81no—
170C2×D4×C151—82no—
171C2×D4×C151—84no—
172C2×D4×C151C2×D412yes—
173C2×D4×C151C2×D411yescharacteristic
174C2×D4×C151C2×D412yes—
175C22×C4×C151—22no—
176C22×C4×C151—22no—
177C22×C4×C151—21no—
178C22×C4×C151—21no—
179C22×C4×C151—22no—
180C22×C4×C151C2×D411yescharacteristic
181C22×C4×C151C2×D411yescharacteristic
182C22×C4×C151C2×D411yescharacteristic
183Q16×C151—42no—
184Q16×C151—82no—
185SD16×C151—42no—
186SD16×C151—82no—
187C2×C8×C151—21no—
188C2×C8×C151—22no—
189C2×C8×C151—21no—
190C2×C8×C151—22no—
191C2×C8×C151—42no—
192C2×C8×C151—42no—
193C4⋊C4×C151—22no—
194C4⋊C4×C151—22no—
195C4⋊C4×C151—21no—
196C4⋊C4×C151—21no—
197C4⋊C4×C151—21no—
198C4⋊C4×C151—21no—
199C4⋊C4×C151—21no—
200C4⋊C4×C151—21no—
201C4⋊C4×C151—21no—
202C4⋊C4×C151—21no—
203C4⋊C4×C151—21no—
204C4⋊C4×C151—21no—
205C4⋊C4×C151—41no—
206C4⋊C4×C151—41no—
207C4⋊C4×C151C2×D411yescharacteristic
208C4⋊C4×C151C2×D411yescharacteristic
209C22⋊C4×C151—22no—
210C22⋊C4×C151—22no—
211C22⋊C4×C151—22no—
212C22⋊C4×C151—22no—
213C22⋊C4×C151—22no—
214C22⋊C4×C151—21no—
215C22⋊C4×C151—22no—
216C22⋊C4×C151—21no—
217C22⋊C4×C151—22no—
218C22⋊C4×C151—21no—
219C22⋊C4×C151—22no—
220C22⋊C4×C151—22no—
221C22⋊C4×C151—21no—
222C22⋊C4×C151—21no—
223C22⋊C4×C151—22no—
224C22⋊C4×C151—22no—
225C22⋊C4×C151—41no—
226C42×C151—22no—
227C42×C151—21no—
228C42×C151—41no—
229C42×C151C2×D411yescharacteristic
230C42×C151C2×D411yescharacteristic
231C42×C151C2×D411yescharacteristic
232C42×C151C2411yescharacteristic
233C23×C151—22no—
234C23×C151—21no—
235C23×C151—22no—
236C23×C151—42no—
237C23×C151—81no—
238C23×C151—81no—
239C23×C151—82no—
240C23×C151—82no—
241C23×C151—82no—
242C23×C151#12011yescharacteristic
243Q8×C151—42no—
244Q8×C151—42no—
245Q8×C151—41no—
246Q8×C151—42no—
247Q8×C151—41no—
248Q8×C151—42no—
249Q8×C151—41no—
250D4×C151—42no—
251D4×C151—41no—
252D4×C151—42no—
253D4×C151—42no—
254D4×C151—42no—
255D4×C151—44no—
256D4×C151—42no—
257D4×C151—81no—
258D4×C151—82no—
259D4×C151—82no—
260D4×C151—82no—
261D4×C151—82no—
262D4×C151—82no—
263C2×C4×C151—22no—
264C2×C4×C151—22no—
265C2×C4×C151—21no—
266C2×C4×C151—22no—
267C2×C4×C151—21no—
268C2×C4×C151—21no—
269C2×C4×C151—41no—
270C2×C4×C151—41no—
271C2×C4×C151—42no—
272C2×C4×C151—41no—
273C2×C4×C151—41no—
274C2×C4×C151—82no—
275C2×C4×C151—81no—
276C2×C4×C151—82no—
277C2×C4×C151—82no—
278C2×C4×C151#12011yescharacteristic
279C2×C4×C151#12011yescharacteristic
280C2×C4×C151#12011yescharacteristic
281C2×C4×C151C22×D411yescharacteristic
282C2×C4×C151C22×D411yescharacteristic
283C2×C4×C151C22×D411yescharacteristic
284C8×C151—22no—
285C8×C151—42no—
286C8×C151—41no—
287C8×C151—42no—
288C22×C151—21no—
289C22×C151—21no—
290C22×C151—42no—
291C22×C151—42no—
292C22×C151—41no—
293C22×C151—41no—
294C22×C151—42no—
295C22×C151—42no—
296C22×C151—41no—
297C22×C151—42no—
298C22×C151—42no—
299C22×C151—162no—
300C22×C151—164no—
301C22×C151#45512yes—
302C22×C151C2×#12011yescharacteristic
303C4×C151—21no—
304C4×C151—21no—
305C4×C151—21no—
306C4×C151—21no—
307C4×C151—21no—
308C4×C151—21no—
309C4×C151—41no—
310C4×C151—42no—
311C4×C151—41no—
312C4×C151—41no—
313C4×C151—42no—
314C4×C151—81no—
315C4×C151—81no—
316C2×C151—22no—
317C2×C151—82no—
318C2×C151—82no—
319C2×C151—81no—
320C2×C151C2×#45511yescharacteristic
321C2×C151#301811yescharacteristic
322C2×C151#301911yescharacteristic
323#19043C15111yesmaximal, characteristic, direct product factor, Sylow
324C151#1904311yescharacteristic, direct product factor, Sylow
325#3558C2×C15111yescharacteristic
326#3215C2×C15111yescharacteristic, complemented
327#3181C2×C15111yescharacteristic, complemented
328#3135C2×C15111yescharacteristic, complemented
329#1662C2×C15112yescomplemented
330#1661C2×C15112yescomplemented
331#1659C2×C15112yescomplemented
332#1659C2×C15112yescomplemented
333#1532C2×C15111yescharacteristic, complemented
334#1530C2×C15111yescharacteristic, complemented
335C2×#329C2×C15111yescharacteristic, complemented
336#562C22×C15112yes—
337H(3,𝔽4)C22×C15112yes—
338#559—21no—
339#548—21no—
340#546—22no—
341#546—21no—
342D42—21no—
343#542—21no—
344#538—22no—
345#538—21no—
346#535—21no—
347#534—21no—
348#533C22×C15111yescharacteristic, complemented
349#532C22×C15111yescharacteristic
350#529C22×C15111yescharacteristic
351C2×#124C22×C15111yescharacteristic
352#501C22×C15111yescharacteristic, complemented
353#497C22×C15111yescharacteristic, complemented
354#497C22×C15112yescomplemented
355#491—21no—
356#491C22×C15111yescharacteristic, complemented
357#490—21no—
358#489C22×C15112yescomplemented
359#489C22×C15112yescomplemented
360#489C22×C15111yescharacteristic, complemented
361#489C22×C15112yescomplemented
362#467—21no—
363#465—21no—
364#464—22no—
365#451—21no—
366#450—21no—
367#441—21no—
368#440—21no—
369#432C22×C15111yescharacteristic, complemented
370#416—22no—
371C2×#97C22×C15111yescharacteristic, complemented
372C2×#97C22×C15111yescharacteristic, complemented
373#330C22×C15112yescomplemented
374#329C22×C15112yescomplemented
375#329C22×C15112yescomplemented
376#329C22×C15112yescomplemented
377#329C22×C15112yescomplemented
378#329C22×C15112yescomplemented
379C2×#41—21no—
380C2×#41D4×C15112yes—
381C22×Q8D4×C15111yescharacteristic
382C22×D4—22no—
383C22×D4—21no—
384C22×D4D4×C15111yescharacteristic
385C2×Q16—21no—
386C2×Q16—22no—
387C2×SD16—21no—
388C2×SD16—22no—
389C22×C8—21no—
390C22×C8—21no—
391C4⋊Q8D4×C15111yescharacteristic
392C4⋊Q8C23×C15111yescharacteristic
393C4⋊Q8C23×C15111yescharacteristic
394#127—21no—
395#127D4×C15111yescharacteristic
396#126—22no—
397#126—21no—
398#125—21no—
399#125C23×C15111yescharacteristic
400#124—21no—
401#124—22no—
402#124—22no—
403#124—22no—
404#124—21no—
405#124—21no—
406#124D4×C15112yes—
407#124C23×C15112yes—
408#124C23×C15111yescharacteristic
409#124C23×C15112yes—
410#123—22no—
411#123—22no—
412#123—21no—
413#123—22no—
414#123—22no—
415C22⋊Q8—21no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8—21no—
419C22⋊Q8—21no—
420C22⋊Q8D4×C15111yescharacteristic
421#121—22no—
422#121—22no—
423#121—22no—
424#121—21no—
425#121—22no—
426#121—22no—
427#121—21no—
428#121—21no—
429#121D4×C15111yescharacteristic
430#120—22no—
431#120—22no—
432#120—22no—
433#120—22no—
434#120—22no—
435C4×Q8—21no—
436C4×Q8—21no—
437C4×Q8—21no—
438C4×Q8D4×C15111yescharacteristic
439C4×D4—22no—
440C4×D4—22no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4D4×C15111yescharacteristic
446#117—21no—
447#117—21no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C42C23×C15111yescharacteristic
452#107—21no—
453#106—21no—
454C4⋊C8—21no—
455#103—22no—
456#103—22no—
457#103—21no—
458#103—22no—
459#103—21no—
460#103—21no—
461#102—22no—
462#102—22no—
463#102—21no—
464#102—22no—
465C22⋊C8—21no—
466#97—21no—
467#97—21no—
468#97C23×C15111yescharacteristic
469#97C23×C15111yescharacteristic
470#97C23×C15112yes—
471#97C23×C15112yes—
472C24—22no—
473C24—21no—
474#41—44no—
475#41—82no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—81no—
480C2×Q8C2×D4×C15112yes—
481C2×Q8C2×D4×C15111yescharacteristic
482C2×Q8C2×D4×C15112yes—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—81no—
494C2×D4—82no—
495C2×D4—84no—
496C2×D4C2×D4×C15112yes—
497C2×D4C2×D4×C15111yescharacteristic
498C2×D4C2×D4×C15112yes—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4C2×D4×C15111yescharacteristic
505C22×C4C2×D4×C15111yescharacteristic
506C22×C4C2×D4×C15111yescharacteristic
507Q16—42no—
508Q16—82no—
509SD16—42no—
510SD16—82no—
511C2×C8—21no—
512C2×C8—22no—
513C2×C8—21no—
514C2×C8—22no—
515C2×C8—42no—
516C2×C8—42no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4C2×D4×C15111yescharacteristic
532C4⋊C4C2×D4×C15111yescharacteristic
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—41no—
550C42—22no—
551C42—21no—
552C42—41no—
553C42C2×D4×C15111yescharacteristic
554C42C2×D4×C15111yescharacteristic
555C42C2×D4×C15111yescharacteristic
556C42C24×C15111yescharacteristic
557C23—22no—
558C23—21no—
559C23—22no—
560C23—42no—
561C23—81no—
562C23—81no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23#120×C15111yescharacteristic
567Q8—41no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—41no—
572Q8—42no—
573Q8—41no—
574D4—41no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—81no—
582D4—82no—
583D4—82no—
584D4—82no—
585D4—82no—
586D4—82no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4#120×C15111yescharacteristic
603C2×C4#120×C15111yescharacteristic
604C2×C4#120×C15111yescharacteristic
605C2×C4C22×D4×C15111yescharacteristic
606C2×C4C22×D4×C15111yescharacteristic
607C2×C4C22×D4×C15111yescharacteristic
608C8—22no—
609C8—42no—
610C8—41no—
611C8—42no—
612C22—21no—
613C22—21no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—162no—
624C22—164no—
625C22#455×C15112yes—
626C22C2×#120×C15111yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—42no—
638C4—81no—
639C4—81no—
640C2—22no—
641C2—82no—
642C2—81no—
643C2—82no—
644C2C2×#455×C15111yescharacteristic
645C2#3018×C15111yescharacteristic
646C2#3019×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC42C24×C151
prime radical#3558×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. #3215×C151C2semidirect
3. #3181×C151C2semidirect
4. #3135×C151C2semidirect
5. #1662×C151C2semidirect
6. #1661×C151C2semidirect
7. #1659×C151C2semidirect
8. #1659×C151C2semidirect
9. #1532×C151C2semidirect
10. #1530×C151C2semidirect
11. C2×#329×C151C2semidirect
24. #533×C151C22semidirect
28. #501×C151C22semidirect
29. #497×C151C22semidirect
30. #497×C151C22semidirect
32. #491×C151C22semidirect
34. #489×C151C22semidirect
35. #489×C151C22semidirect
36. #489×C151C22semidirect
37. #489×C151C22semidirect
45. #432×C151C22semidirect
47. C2×#97×C151C22semidirect
48. C2×#97×C151C22semidirect
49. #330×C151C22semidirect
50. #329×C151C22semidirect
51. #329×C151C22semidirect
52. #329×C151C22semidirect
53. #329×C151C22semidirect
54. #329×C151C22semidirect
323. #19043C151direct (N × H)
324. C151#19043direct (N × H)
326. #3215C2×C151semidirect
327. #3181C2×C151semidirect
328. #3135C2×C151semidirect
329. #1662C2×C151semidirect
330. #1661C2×C151semidirect
331. #1659C2×C151semidirect
332. #1659C2×C151semidirect
333. #1532C2×C151semidirect
334. #1530C2×C151semidirect
335. C2×#329C2×C151semidirect
348. #533C22×C151semidirect
352. #501C22×C151semidirect
353. #497C22×C151semidirect
354. #497C22×C151semidirect
356. #491C22×C151semidirect
358. #489C22×C151semidirect
359. #489C22×C151semidirect
360. #489C22×C151semidirect
361. #489C22×C151semidirect
369. #432C22×C151semidirect
371. C2×#97C22×C151semidirect
372. C2×#97C22×C151semidirect
373. #330C22×C151semidirect
374. #329C22×C151semidirect
375. #329C22×C151semidirect
376. #329C22×C151semidirect
377. #329C22×C151semidirect
378. #329C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3558×C1512
#3215×C1512
#3181×C1512
#3135×C1512
#1662×C1512
#1661×C1512
#1659×C1512
#1532×C1512
#1530×C1512
C2×#329×C1512
#19043151