Finite Group Databasebeta

← C151×#18258 #50323183

Subgroups — C151×#18258 #50323183

Order 38656.

1276 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3538×C151C211yesmaximal, characteristic, complemented
2#3003×C151C211yesmaximal, characteristic
3C2×#450×C151C211yesmaximal, characteristic, complemented
4#2435×C151C211yesmaximal, characteristic, complemented
5#1599×C151C214yesmaximal, complemented
6#1597×C151C214yesmaximal, complemented
7#1508×C151C211yesmaximal, characteristic, complemented
8#1501×C151C211yesmaximal, characteristic, complemented
9#1472×C151C211yesmaximal, characteristic, complemented
10C23×D4×C151C2211yescharacteristic
11C2×AGL(1,ℤ/8)×C151—21no—
12C22×SD16×C151—21no—
13#563×C151—22no—
14#552×C151—21no—
15#552×C151—21no—
16#551×C151—22no—
17D4×Q8×C151—21no—
18D4×Q8×C151—21no—
19#544×C151—21no—
20#544×C151C2212yescomplemented
21#541×C151—21no—
22#539×C151—21no—
23#539×C151C2212yescomplemented
24#536×C151—21no—
25C2×#124×C151—21no—
26#525×C151C2211yescharacteristic
27C2×C22⋊Q8×C151C2211yescharacteristic
28C2×C22⋊Q8×C151C2211yescharacteristic
29#450×C151—22no—
30#450×C151—21no—
31#450×C151C2214yes—
32#450×C151C2212yes—
33#447×C151—21no—
34#447×C151C2212yes—
35#418×C151—21no—
36C2×#102×C151—21no—
37#407×C151C2211yescharacteristic
38C2×C22⋊C8×C151C2211yescharacteristic
39#394×C151C2214yes—
40#390×C151—21no—
41#379×C151—22no—
42C4×C22⋊C4×C151C2211yescharacteristic
43#328×C151C2214yescomplemented
44#328×C151C2212yescomplemented
45#327×C151C2212yescomplemented
46#324×C151C2214yescomplemented
47C25×C151—21no—
48C2×#41×C151D411yescharacteristic
49C22×Q8×C151—21no—
50C22×Q8×C151D411yescharacteristic
51C22×Q8×C151D411yescharacteristic
52C22×D4×C151—21no—
53C22×D4×C151—21no—
54C22×D4×C151—21no—
55C22×D4×C151—21no—
56C22×D4×C151—21no—
57C22×D4×C151—44no—
58C22×D4×C151C2312yes—
59C23×C4×C151C2311yescharacteristic
60AGL(1,ℤ/8)×C151—44no—
61C2×SD16×C151—24no—
62C2×SD16×C151—22no—
63C2×SD16×C151—44no—
64C2×D8×C151—22no—
65C2×M16×C151—21no—
66C22×C8×C151—21no—
67C4⋊Q8×C151—21no—
68C4⋊Q8×C151—22no—
69C4⋊Q8×C151—22no—
70C4⋊Q8×C151—22no—
71#126×C151—22no—
72#126×C151—22no—
73#126×C151—22no—
74#126×C151—22no—
75#125×C151—21no—
76#125×C151—22no—
77#125×C151—22no—
78#125×C151—22no—
79#125×C151—22no—
80#124×C151—22no—
81#124×C151—44no—
82#123×C151—22no—
83#123×C151—22no—
84#123×C151—22no—
85#123×C151—21no—
86#123×C151—22no—
87C22⋊Q8×C151—21no—
88C22⋊Q8×C151—21no—
89C22⋊Q8×C151—22no—
90C22⋊Q8×C151—22no—
91C22⋊Q8×C151—21no—
92C22⋊Q8×C151—21no—
93C22⋊Q8×C151—21no—
94C22⋊Q8×C151—21no—
95C22⋊Q8×C151—21no—
96C22⋊Q8×C151—21no—
97C22⋊Q8×C151—22no—
98C22⋊Q8×C151—22no—
99C22⋊Q8×C151D412yes—
100C22⋊Q8×C151D411yescharacteristic
101C22⋊Q8×C151C2312yes—
102C22⋊Q8×C151C2311yescharacteristic
103#121×C151D411yescharacteristic
104#121×C151C2311yescharacteristic
105#120×C151—21no—
106C4×Q8×C151—21no—
107C4×Q8×C151—21no—
108C4×D4×C151—22no—
109C4×D4×C151—21no—
110C4×D4×C151—21no—
111C4×D4×C151—22no—
112#117×C151—21no—
113#117×C151—22no—
114#117×C151—22no—
115#117×C151D411yescharacteristic
116C2×C4⋊C4×C151—21no—
117C2×C4⋊C4×C151D411yescharacteristic
118C2×C4⋊C4×C151D411yescharacteristic
119C2×C4⋊C4×C151C2312yes—
120C2×C22⋊C4×C151—24no—
121C2×C22⋊C4×C151—24no—
122C2×C22⋊C4×C151—21no—
123C2×C22⋊C4×C151—21no—
124C2×C22⋊C4×C151D412yes—
125C2×C42×C151—21no—
126#102×C151—24no—
127#102×C151—22no—
128#102×C151—22no—
129C22⋊C8×C151—21no—
130C22⋊C8×C151—21no—
131C22⋊C8×C151C2312yes—
132C22⋊C8×C151C2312yes—
133#95×C151C2312yes—
134C24×C151—22no—
135C24×C151—44no—
136C24×C151—44no—
137C24×C151—41no—
138C24×C151—41no—
139C24×C151—82no—
140C24×C151C2×D411yescharacteristic
141#41×C151—42no—
142C2×Q8×C151—22no—
143C2×Q8×C151—21no—
144C2×Q8×C151—21no—
145C2×Q8×C151—21no—
146C2×Q8×C151—42no—
147C2×Q8×C151—42no—
148C2×Q8×C151—82no—
149C2×Q8×C151C2×D412yes—
150C2×Q8×C151C2×D411yescharacteristic
151C2×D4×C151—24no—
152C2×D4×C151—24no—
153C2×D4×C151—24no—
154C2×D4×C151—21no—
155C2×D4×C151—21no—
156C2×D4×C151—21no—
157C2×D4×C151—22no—
158C2×D4×C151—22no—
159C2×D4×C151—42no—
160C2×D4×C151—42no—
161C2×D4×C151—42no—
162C2×D4×C151—44no—
163C2×D4×C151—44no—
164C2×D4×C151—44no—
165C2×D4×C151—41no—
166C2×D4×C151—41no—
167C2×D4×C151C2×D411yescharacteristic
168C22×C4×C151—21no—
169C22×C4×C151—22no—
170C22×C4×C151—21no—
171C22×C4×C151—21no—
172C22×C4×C151—22no—
173C22×C4×C151—42no—
174C22×C4×C151C2×D412yes—
175C22×C4×C151C2×D411yescharacteristic
176C22×C4×C151C2×D411yescharacteristic
177C22×C4×C151C2×D411yescharacteristic
178C22×C4×C151C2×D412yes—
179C22×C4×C151C2×D411yescharacteristic
180C22×C4×C151C2×D411yescharacteristic
181C22×C4×C151C2411yescharacteristic
182SD16×C151—48no—
183SD16×C151—44no—
184D8×C151—44no—
185M16×C151—42no—
186C2×C8×C151—22no—
187C2×C8×C151—22no—
188C2×C8×C151—42no—
189C4⋊C4×C151—22no—
190C4⋊C4×C151—22no—
191C4⋊C4×C151—22no—
192C4⋊C4×C151—22no—
193C4⋊C4×C151—22no—
194C4⋊C4×C151—22no—
195C4⋊C4×C151—22no—
196C4⋊C4×C151—22no—
197C4⋊C4×C151—22no—
198C4⋊C4×C151—21no—
199C4⋊C4×C151—21no—
200C4⋊C4×C151—21no—
201C4⋊C4×C151—21no—
202C4⋊C4×C151—22no—
203C4⋊C4×C151—22no—
204C4⋊C4×C151—22no—
205C4⋊C4×C151—21no—
206C4⋊C4×C151C2×D412yes—
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—22no—
215C22⋊C4×C151—21no—
216C22⋊C4×C151—21no—
217C22⋊C4×C151—22no—
218C22⋊C4×C151—21no—
219C22⋊C4×C151—21no—
220C22⋊C4×C151—21no—
221C22⋊C4×C151—42no—
222C22⋊C4×C151—48no—
223C22⋊C4×C151—84no—
224C42×C151—22no—
225C42×C151—22no—
226C42×C151—22no—
227C42×C151—21no—
228C42×C151—42no—
229C23×C151—22no—
230C23×C151—22no—
231C23×C151—24no—
232C23×C151—21no—
233C23×C151—41no—
234C23×C151—44no—
235C23×C151—42no—
236C23×C151—44no—
237C23×C151—44no—
238C23×C151—42no—
239C23×C151—44no—
240C23×C151—82no—
241C23×C151—82no—
242C23×C151—82no—
243C23×C151—816no—
244C23×C151—82no—
245C23×C151#12011yescharacteristic
246C23×C151#12011yescharacteristic
247C23×C151C22×D411yescharacteristic
248Q8×C151—42no—
249Q8×C151—41no—
250Q8×C151—42no—
251Q8×C151—42no—
252Q8×C151—81no—
253Q8×C151—82no—
254D4×C151—24no—
255D4×C151—24no—
256D4×C151—41no—
257D4×C151—42no—
258D4×C151—42no—
259D4×C151—44no—
260D4×C151—42no—
261D4×C151—42no—
262D4×C151—44no—
263D4×C151—81no—
264C2×C4×C151—22no—
265C2×C4×C151—22no—
266C2×C4×C151—22no—
267C2×C4×C151—22no—
268C2×C4×C151—21no—
269C2×C4×C151—22no—
270C2×C4×C151—21no—
271C2×C4×C151—21no—
272C2×C4×C151—22no—
273C2×C4×C151—22no—
274C2×C4×C151—24no—
275C2×C4×C151—41no—
276C2×C4×C151—41no—
277C2×C4×C151—41no—
278C2×C4×C151—42no—
279C2×C4×C151—42no—
280C2×C4×C151—42no—
281C2×C4×C151—42no—
282C2×C4×C151—42no—
283C2×C4×C151—42no—
284C2×C4×C151—81no—
285C2×C4×C151—82no—
286C2×C4×C151#12011yescharacteristic
287C2×C4×C151#12011yescharacteristic
288C2×C4×C151C22×D411yescharacteristic
289C2×C4×C151C22×D411yescharacteristic
290C8×C151—42no—
291C8×C151—42no—
292C22×C151—21no—
293C22×C151—21no—
294C22×C151—21no—
295C22×C151—21no—
296C22×C151—21no—
297C22×C151—21no—
298C22×C151—21no—
299C22×C151—44no—
300C22×C151—48no—
301C22×C151—41no—
302C22×C151—41no—
303C22×C151—41no—
304C22×C151—44no—
305C22×C151—84no—
306C22×C151—88no—
307C22×C151—88no—
308C22×C151—82no—
309C22×C151—82no—
310C22×C151#45412yes—
311C22×C151C2×#12011yescharacteristic
312C22×C151#57712yes—
313C4×C151—21no—
314C4×C151—21no—
315C4×C151—21no—
316C4×C151—42no—
317C4×C151—41no—
318C4×C151—42no—
319C4×C151—41no—
320C4×C151—41no—
321C4×C151—42no—
322C4×C151—42no—
323C4×C151—82no—
324C4×C151B(4,𝔽2)12yes—
325C2×C151—22no—
326C2×C151—22no—
327C2×C151—41no—
328C2×C151—81no—
329C2×C151—84no—
330C2×C151#301011yescharacteristic
331C2×C151C2×#45411yescharacteristic
332C2×C151C2×B(4,𝔽2)11yescharacteristic
333#18258C15111yesmaximal, characteristic, direct product factor, Sylow
334C151#1825811yescharacteristic, direct product factor, Sylow
335#3538C2×C15111yescharacteristic, complemented
336#3003C2×C15111yescharacteristic
337C2×#450C2×C15111yescharacteristic, complemented
338#2435C2×C15111yescharacteristic, complemented
339#1599C2×C15114yescomplemented
340#1597C2×C15114yescomplemented
341#1508C2×C15111yescharacteristic, complemented
342#1501C2×C15111yescharacteristic, complemented
343#1472C2×C15111yescharacteristic, complemented
344C23×D4C22×C15111yescharacteristic
345C2×AGL(1,ℤ/8)—21no—
346C22×SD16—21no—
347#563—22no—
348#552—21no—
349#552—21no—
350#551—22no—
351D4×Q8—21no—
352D4×Q8—21no—
353#544—21no—
354#544C22×C15112yescomplemented
355#541—21no—
356#539—21no—
357#539C22×C15112yescomplemented
358#536—21no—
359C2×#124—21no—
360#525C22×C15111yescharacteristic
361C2×C22⋊Q8C22×C15111yescharacteristic
362C2×C22⋊Q8C22×C15111yescharacteristic
363#450—22no—
364#450—21no—
365#450C22×C15114yes—
366#450C22×C15112yes—
367#447—21no—
368#447C22×C15112yes—
369#418—21no—
370C2×#102—21no—
371#407C22×C15111yescharacteristic
372C2×C22⋊C8C22×C15111yescharacteristic
373#394C22×C15114yes—
374#390—21no—
375#379—22no—
376C4×C22⋊C4C22×C15111yescharacteristic
377#328C22×C15114yescomplemented
378#328C22×C15112yescomplemented
379#327C22×C15112yescomplemented
380#324C22×C15114yescomplemented
381C25—21no—
382C2×#41D4×C15111yescharacteristic
383C22×Q8—21no—
384C22×Q8D4×C15111yescharacteristic
385C22×Q8D4×C15111yescharacteristic
386C22×D4—21no—
387C22×D4—21no—
388C22×D4—21no—
389C22×D4—21no—
390C22×D4—21no—
391C22×D4—44no—
392C22×D4C23×C15112yes—
393C23×C4C23×C15111yescharacteristic
394AGL(1,ℤ/8)—44no—
395C2×SD16—24no—
396C2×SD16—22no—
397C2×SD16—44no—
398C2×D8—22no—
399C2×M16—21no—
400C22×C8—21no—
401C4⋊Q8—21no—
402C4⋊Q8—22no—
403C4⋊Q8—22no—
404C4⋊Q8—22no—
405#126—22no—
406#126—22no—
407#126—22no—
408#126—22no—
409#125—21no—
410#125—22no—
411#125—22no—
412#125—22no—
413#125—22no—
414#124—22no—
415#124—44no—
416#123—22no—
417#123—22no—
418#123—22no—
419#123—21no—
420#123—22no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8—22no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C22⋊Q8—21no—
430C22⋊Q8—21no—
431C22⋊Q8—21no—
432C22⋊Q8—22no—
433C22⋊Q8D4×C15112yes—
434C22⋊Q8D4×C15111yescharacteristic
435C22⋊Q8C23×C15112yes—
436C22⋊Q8C23×C15111yescharacteristic
437#121D4×C15111yescharacteristic
438#121C23×C15111yescharacteristic
439#120—21no—
440C4×Q8—21no—
441C4×Q8—21no—
442C4×D4—22no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4—22no—
446#117—21no—
447#117—22no—
448#117—22no—
449#117D4×C15111yescharacteristic
450C2×C4⋊C4—21no—
451C2×C4⋊C4D4×C15111yescharacteristic
452C2×C4⋊C4D4×C15111yescharacteristic
453C2×C4⋊C4C23×C15112yes—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—24no—
457C2×C22⋊C4—24no—
458C2×C22⋊C4D4×C15112yes—
459C2×C42—21no—
460#102—24no—
461#102—22no—
462#102—22no—
463C22⋊C8—21no—
464C22⋊C8—21no—
465C22⋊C8C23×C15112yes—
466C22⋊C8C23×C15112yes—
467#95C23×C15112yes—
468C24—22no—
469C24—44no—
470C24—44no—
471C24—41no—
472C24—41no—
473C24—82no—
474C24C2×D4×C15111yescharacteristic
475#41—42no—
476C2×Q8—22no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—82no—
483C2×Q8C2×D4×C15112yes—
484C2×Q8C2×D4×C15111yescharacteristic
485C2×D4—24no—
486C2×D4—24no—
487C2×D4—24no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—44no—
497C2×D4—44no—
498C2×D4—44no—
499C2×D4—41no—
500C2×D4—41no—
501C2×D4C2×D4×C15111yescharacteristic
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—42no—
508C22×C4C2×D4×C15112yes—
509C22×C4C2×D4×C15111yescharacteristic
510C22×C4C2×D4×C15111yescharacteristic
511C22×C4C2×D4×C15111yescharacteristic
512C22×C4C2×D4×C15112yes—
513C22×C4C2×D4×C15111yescharacteristic
514C22×C4C2×D4×C15111yescharacteristic
515C22×C4C24×C15111yescharacteristic
516SD16—48no—
517SD16—44no—
518D8—44no—
519M16—42no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—42no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4C2×D4×C15112yes—
541C4⋊C4C2×D4×C15111yescharacteristic
542C4⋊C4C2×D4×C15111yescharacteristic
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—22no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—48no—
556C22⋊C4—42no—
557C22⋊C4—84no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—21no—
562C42—42no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—21no—
567C23—41no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—82no—
575C23—82no—
576C23—82no—
577C23—816no—
578C23—82no—
579C23#120×C15111yescharacteristic
580C23#120×C15111yescharacteristic
581C23C22×D4×C15111yescharacteristic
582Q8—42no—
583Q8—41no—
584Q8—42no—
585Q8—42no—
586Q8—81no—
587Q8—82no—
588D4—24no—
589D4—24no—
590D4—41no—
591D4—42no—
592D4—42no—
593D4—44no—
594D4—42no—
595D4—42no—
596D4—44no—
597D4—81no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—81no—
619C2×C4—82no—
620C2×C4#120×C15111yescharacteristic
621C2×C4#120×C15111yescharacteristic
622C2×C4C22×D4×C15111yescharacteristic
623C2×C4C22×D4×C15111yescharacteristic
624C8—42no—
625C8—42no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—44no—
634C22—48no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—44no—
639C22—84no—
640C22—88no—
641C22—88no—
642C22—82no—
643C22—82no—
644C22#454×C15112yes—
645C22C2×#120×C15111yescharacteristic
646C22#577×C15112yes—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4B(4,𝔽2)×C15112yes—
659C2—22no—
660C2—22no—
661C2—41no—
662C2—84no—
663C2—81no—
664C2#3010×C15111yescharacteristic
665C2C2×#454×C15111yescharacteristic
666C2C2×B(4,𝔽2)×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
prime radical#3003×C151C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3538×C151C2semidirect
3. C2×#450×C151C2semidirect
4. #2435×C151C2semidirect
5. #1599×C151C2semidirect
6. #1597×C151C2semidirect
7. #1508×C151C2semidirect
8. #1501×C151C2semidirect
9. #1472×C151C2semidirect
20. #544×C151C22semidirect
23. #539×C151C22semidirect
43. #328×C151C22semidirect
44. #328×C151C22semidirect
45. #327×C151C22semidirect
46. #324×C151C22semidirect
333. #18258C151direct (N × H)
334. C151#18258direct (N × H)
335. #3538C2×C151semidirect
337. C2×#450C2×C151semidirect
338. #2435C2×C151semidirect
339. #1599C2×C151semidirect
340. #1597C2×C151semidirect
341. #1508C2×C151semidirect
342. #1501C2×C151semidirect
343. #1472C2×C151semidirect
354. #544C22×C151semidirect
357. #539C22×C151semidirect
377. #328C22×C151semidirect
378. #328C22×C151semidirect
379. #327C22×C151semidirect
380. #324C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3538×C1512
#3003×C1512
C2×#450×C1512
#2435×C1512
#1599×C1512
#1597×C1512
#1508×C1512
#1501×C1512
#1472×C1512
#18258151