Finite Group Databasebeta

← C151×#23976 #50328901

Subgroups — C151×#23976 #50328901

Order 38656.

1482 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555×C151C211yesmaximal, characteristic, complemented
2C2×#447×C151C211yesmaximal, characteristic, complemented
3C2×#418×C151C211yesmaximal, characteristic, complemented
4#2408×C151C211yesmaximal, characteristic, complemented
5#2023×C151C212yesmaximal, complemented
6#1999×C151C212yesmaximal, complemented
7#1895×C151C211yesmaximal, characteristic, complemented
8#1879×C151C212yesmaximal, complemented
9#1875×C151C212yesmaximal, complemented
10#1870×C151C211yesmaximal, characteristic, complemented
11#1825×C151C211yesmaximal, characteristic, complemented
12C23×D4×C151C2211yescharacteristic
13C22×D8×C151C2212yes—
14C22×M16×C151C2211yescharacteristic
15#555×C151—24no—
16#555×C151—24no—
17C2×C4⋊Q8×C151C2211yescharacteristic, complemented
18C2×#124×C151—21no—
19C2×#124×C151—21no—
20C2×#121×C151—21no—
21C2×#121×C151C2212yes—
22C2×#121×C151C2211yescharacteristic
23C2×#121×C151C2211yescharacteristic, complemented
24C2×#120×C151—21no—
25C2×C4×D4×C151C2211yescharacteristic
26C2×C4×D4×C151C2211yescharacteristic, complemented
27C2×#117×C151—21no—
28C2×#117×C151C2211yescharacteristic
29C2×#117×C151C2211yescharacteristic, complemented
30#447×C151—24no—
31#418×C151—24no—
32C2×#102×C151C2212yes—
33C2×#102×C151C2212yes—
34C2×#102×C151C2212yes—
35C2×#102×C151C2212yescomplemented
36C2×C22⋊C8×C151C2211yescharacteristic
37C2×C22⋊C8×C151C2212yescomplemented
38#394×C151—21no—
39#392×C151—21no—
40#390×C151C2212yes—
41#380×C151C2211yescharacteristic
42#379×C151—21no—
43#340×C151C2212yescomplemented
44#340×C151C2212yescomplemented
45#340×C151C2212yescomplemented
46#336×C151C2212yescomplemented
47C25×C151—21no—
48C22×Q8×C151—21no—
49C22×D4×C151—21no—
50C22×D4×C151—21no—
51C22×D4×C151—21no—
52C22×D4×C151—22no—
53C22×D4×C151—24no—
54C22×D4×C151—22no—
55C22×D4×C151—22no—
56C22×D4×C151—42no—
57C22×D4×C151D411yescharacteristic
58C22×D4×C151C2311yescharacteristic
59C22×D4×C151C2312yes—
60C23×C4×C151—21no—
61C23×C4×C151C2311yescharacteristic
62C23×C4×C151C2311yescharacteristic
63C2×D8×C151—24no—
64C2×D8×C151—24no—
65C2×D8×C151—42no—
66C2×M16×C151—22no—
67C2×M16×C151—42no—
68C22×C8×C151C2312yes—
69C22×C8×C151C2312yes—
70C4⋊Q8×C151—22no—
71C4⋊Q8×C151—22no—
72#124×C151—28no—
73#124×C151—28no—
74#121×C151—28no—
75#121×C151—22no—
76#121×C151—22no—
77#121×C151—22no—
78#121×C151—24no—
79#121×C151—42no—
80#121×C151D414yescomplemented
81#120×C151—44no—
82C4×D4×C151—22no—
83C4×D4×C151—22no—
84C4×D4×C151—22no—
85C4×D4×C151—22no—
86#117×C151—28no—
87#117×C151—22no—
88#117×C151—22no—
89#117×C151—22no—
90#117×C151D414yescomplemented
91C2×C4⋊C4×C151—21no—
92C2×C4⋊C4×C151D411yescharacteristic
93C2×C4⋊C4×C151C2312yes—
94C2×C4⋊C4×C151C2311yescharacteristic
95C2×C4⋊C4×C151C2311yescharacteristic
96C2×C4⋊C4×C151C2311yescharacteristic
97C2×C22⋊C4×C151—21no—
98C2×C22⋊C4×C151—21no—
99C2×C22⋊C4×C151—21no—
100C2×C22⋊C4×C151—22no—
101C2×C22⋊C4×C151—22no—
102C2×C22⋊C4×C151—22no—
103C2×C22⋊C4×C151—21no—
104C2×C22⋊C4×C151—21no—
105C2×C22⋊C4×C151—21no—
106C2×C22⋊C4×C151D411yescharacteristic
107C2×C22⋊C4×C151D411yescharacteristic
108C2×C42×C151—21no—
109C2×C42×C151—21no—
110C2×C42×C151C2311yescharacteristic
111#102×C151—24no—
112#102×C151—24no—
113#102×C151—24no—
114#102×C151—42no—
115C22⋊C8×C151—22no—
116C22⋊C8×C151—42no—
117#95×C151—21no—
118C24×C151—21no—
119C24×C151—22no—
120C24×C151—44no—
121C24×C151—42no—
122C24×C151—42no—
123C24×C151—42no—
124C24×C151—82no—
125C24×C151C2×D411yescharacteristic
126C24×C151C2×D411yescharacteristic
127C2×Q8×C151—24no—
128C2×Q8×C151—82no—
129C2×D4×C151—28no—
130C2×D4×C151—22no—
131C2×D4×C151—24no—
132C2×D4×C151—24no—
133C2×D4×C151—22no—
134C2×D4×C151—22no—
135C2×D4×C151—22no—
136C2×D4×C151—21no—
137C2×D4×C151—22no—
138C2×D4×C151—22no—
139C2×D4×C151—22no—
140C2×D4×C151—22no—
141C2×D4×C151—22no—
142C2×D4×C151—48no—
143C2×D4×C151—41no—
144C2×D4×C151—44no—
145C2×D4×C151—44no—
146C2×D4×C151—44no—
147C2×D4×C151—44no—
148C2×D4×C151—42no—
149C2×D4×C151—82no—
150C2×D4×C151—82no—
151C2×D4×C151—82no—
152C2×D4×C151C2×D412yes—
153C2×D4×C151#4114yes—
154C22×C4×C151—22no—
155C22×C4×C151—22no—
156C22×C4×C151—24no—
157C22×C4×C151—21no—
158C22×C4×C151—21no—
159C22×C4×C151—21no—
160C22×C4×C151—21no—
161C22×C4×C151—21no—
162C22×C4×C151—22no—
163C22×C4×C151—22no—
164C22×C4×C151—22no—
165C22×C4×C151—42no—
166C22×C4×C151—42no—
167C22×C4×C151C2×D411yescharacteristic
168C22×C4×C151C2×D411yescharacteristic
169C22×C4×C151C2×D411yescharacteristic
170C22×C4×C151C2×D412yes—
171C22×C4×C151C2×D412yes—
172C22×C4×C151C2×D411yescharacteristic
173C22×C4×C151C2411yescharacteristic
174D8×C151—48no—
175M16×C151—44no—
176C2×C8×C151—22no—
177C2×C8×C151—22no—
178C2×C8×C151—22no—
179C2×C8×C151—22no—
180C2×C8×C151—22no—
181C2×C8×C151—22no—
182C4⋊C4×C151—24no—
183C4⋊C4×C151—22no—
184C4⋊C4×C151—21no—
185C4⋊C4×C151—21no—
186C4⋊C4×C151—21no—
187C4⋊C4×C151—22no—
188C4⋊C4×C151—22no—
189C4⋊C4×C151C2×D412yes—
190C4⋊C4×C151C2×D412yes—
191C4⋊C4×C151C2×D412yes—
192C22⋊C4×C151—24no—
193C22⋊C4×C151—24no—
194C22⋊C4×C151—24no—
195C22⋊C4×C151—24no—
196C22⋊C4×C151—22no—
197C22⋊C4×C151—24no—
198C22⋊C4×C151—22no—
199C22⋊C4×C151—44no—
200C22⋊C4×C151—42no—
201C22⋊C4×C151—44no—
202C22⋊C4×C151—82no—
203C42×C151—24no—
204C42×C151—24no—
205C42×C151—21no—
206C42×C151—21no—
207C23×C151—22no—
208C23×C151—21no—
209C23×C151—21no—
210C23×C151—22no—
211C23×C151—22no—
212C23×C151—21no—
213C23×C151—24no—
214C23×C151—22no—
215C23×C151—21no—
216C23×C151—21no—
217C23×C151—22no—
218C23×C151—48no—
219C23×C151—42no—
220C23×C151—42no—
221C23×C151—42no—
222C23×C151—42no—
223C23×C151—41no—
224C23×C151—42no—
225C23×C151—41no—
226C23×C151—42no—
227C23×C151—42no—
228C23×C151—42no—
229C23×C151—84no—
230C23×C151—84no—
231C23×C151—88no—
232C23×C151—88no—
233C23×C151#12012yes—
234C23×C151C22×D411yescharacteristic
235Q8×C151—84no—
236D4×C151—24no—
237D4×C151—24no—
238D4×C151—24no—
239D4×C151—44no—
240D4×C151—48no—
241D4×C151—41no—
242D4×C151—41no—
243D4×C151—42no—
244D4×C151—42no—
245D4×C151—84no—
246D4×C151—84no—
247D4×C151—88no—
248C2×C4×C151—21no—
249C2×C4×C151—22no—
250C2×C4×C151—21no—
251C2×C4×C151—22no—
252C2×C4×C151—22no—
253C2×C4×C151—24no—
254C2×C4×C151—21no—
255C2×C4×C151—22no—
256C2×C4×C151—22no—
257C2×C4×C151—21no—
258C2×C4×C151—22no—
259C2×C4×C151—22no—
260C2×C4×C151—22no—
261C2×C4×C151—48no—
262C2×C4×C151—41no—
263C2×C4×C151—41no—
264C2×C4×C151—42no—
265C2×C4×C151—42no—
266C2×C4×C151—41no—
267C2×C4×C151—41no—
268C2×C4×C151—42no—
269C2×C4×C151—42no—
270C2×C4×C151—42no—
271C2×C4×C151—42no—
272C2×C4×C151#12012yes—
273C2×C4×C151AGL(1,ℤ/8)12yes—
274C2×C4×C151C22×D411yescharacteristic
275C2×C4×C151C22×D411yescharacteristic
276C2×C4×C151C2×#4112yes—
277C2×C4×C151H(5,𝔽2)12yes—
278C8×C151—42no—
279C8×C151—42no—
280C22×C151—22no—
281C22×C151—22no—
282C22×C151—22no—
283C22×C151—42no—
284C22×C151—42no—
285C22×C151—41no—
286C22×C151—44no—
287C22×C151—42no—
288C22×C151—42no—
289C22×C151—41no—
290C22×C151—41no—
291C22×C151—42no—
292C22×C151—42no—
293C22×C151—44no—
294C22×C151—42no—
295C22×C151—44no—
296C22×C151—816no—
297C22×C151—82no—
298C22×C151—82no—
299C22×C151—82no—
300C22×C151—82no—
301C22×C151—82no—
302C22×C151—82no—
303C22×C151C2×#12011yescharacteristic
304C22×C151#54612yes—
305C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C151#57511yescharacteristic
307C22×C151#57612yes—
308C4×C151—22no—
309C4×C151—22no—
310C4×C151—41no—
311C4×C151—42no—
312C4×C151—41no—
313C4×C151—42no—
314C4×C151—42no—
315C4×C151—82no—
316C4×C151#52511yescharacteristic
317C4×C151#53511yescharacteristic
318C4×C151#54612yes—
319C2×C151—42no—
320C2×C151—81no—
321C2×C151—82no—
322C2×C151—84no—
323C2×C151#240411yescharacteristic
324C2×C151#300611yescharacteristic
325C2×C151#300811yescharacteristic
326C2×C151#331912yes—
327C2×C151#332512yes—
328#23976C15111yesmaximal, characteristic, direct product factor, Sylow
329C151#2397611yescharacteristic, direct product factor, Sylow
330C2×#555C2×C15111yescharacteristic, complemented
331C2×#447C2×C15111yescharacteristic, complemented
332C2×#418C2×C15111yescharacteristic, complemented
333#2408C2×C15111yescharacteristic, complemented
334#2023C2×C15112yescomplemented
335#1999C2×C15112yescomplemented
336#1895C2×C15111yescharacteristic, complemented
337#1879C2×C15112yescomplemented
338#1875C2×C15112yescomplemented
339#1870C2×C15111yescharacteristic, complemented
340#1825C2×C15111yescharacteristic, complemented
341C23×D4C22×C15111yescharacteristic
342C22×D8C22×C15112yes—
343C22×M16C22×C15111yescharacteristic
344#555—24no—
345#555—24no—
346C2×C4⋊Q8C22×C15111yescharacteristic, complemented
347C2×#124—21no—
348C2×#124—21no—
349C2×#121—21no—
350C2×#121C22×C15112yes—
351C2×#121C22×C15111yescharacteristic
352C2×#121C22×C15111yescharacteristic, complemented
353C2×#120—21no—
354C2×C4×D4C22×C15111yescharacteristic
355C2×C4×D4C22×C15111yescharacteristic, complemented
356C2×#117—21no—
357C2×#117C22×C15111yescharacteristic
358C2×#117C22×C15111yescharacteristic, complemented
359#447—24no—
360#418—24no—
361C2×#102C22×C15112yes—
362C2×#102C22×C15112yes—
363C2×#102C22×C15112yes—
364C2×#102C22×C15112yescomplemented
365C2×C22⋊C8C22×C15111yescharacteristic
366C2×C22⋊C8C22×C15112yescomplemented
367#394—21no—
368#392—21no—
369#390C22×C15112yes—
370#380C22×C15111yescharacteristic
371#379—21no—
372#340C22×C15112yescomplemented
373#340C22×C15112yescomplemented
374#340C22×C15112yescomplemented
375#336C22×C15112yescomplemented
376C25—21no—
377C22×Q8—21no—
378C22×D4—24no—
379C22×D4—21no—
380C22×D4—21no—
381C22×D4—22no—
382C22×D4—22no—
383C22×D4—22no—
384C22×D4—21no—
385C22×D4—42no—
386C22×D4D4×C15111yescharacteristic
387C22×D4C23×C15112yes—
388C22×D4C23×C15111yescharacteristic
389C23×C4—21no—
390C23×C4C23×C15111yescharacteristic
391C23×C4C23×C15111yescharacteristic
392C2×D8—24no—
393C2×D8—24no—
394C2×D8—42no—
395C2×M16—22no—
396C2×M16—42no—
397C22×C8C23×C15112yes—
398C22×C8C23×C15112yes—
399C4⋊Q8—22no—
400C4⋊Q8—22no—
401#124—28no—
402#124—28no—
403#121—28no—
404#121—22no—
405#121—22no—
406#121—22no—
407#121—24no—
408#121—42no—
409#121D4×C15114yescomplemented
410#120—44no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415#117—28no—
416#117—22no—
417#117—22no—
418#117—22no—
419#117D4×C15114yescomplemented
420C2×C4⋊C4—21no—
421C2×C4⋊C4D4×C15111yescharacteristic
422C2×C4⋊C4C23×C15112yes—
423C2×C4⋊C4C23×C15111yescharacteristic
424C2×C4⋊C4C23×C15111yescharacteristic
425C2×C4⋊C4C23×C15111yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—22no—
434C2×C22⋊C4—22no—
435C2×C22⋊C4D4×C15111yescharacteristic
436C2×C22⋊C4D4×C15111yescharacteristic
437C2×C42—21no—
438C2×C42—21no—
439C2×C42C23×C15111yescharacteristic
440#102—24no—
441#102—24no—
442#102—24no—
443#102—42no—
444C22⋊C8—22no—
445C22⋊C8—42no—
446#95—21no—
447C24—21no—
448C24—22no—
449C24—44no—
450C24—42no—
451C24—42no—
452C24—42no—
453C24—82no—
454C24C2×D4×C15111yescharacteristic
455C24C2×D4×C15111yescharacteristic
456C2×Q8—24no—
457C2×Q8—82no—
458C2×D4—28no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—48no—
472C2×D4—41no—
473C2×D4—44no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—42no—
478C2×D4—82no—
479C2×D4—82no—
480C2×D4—82no—
481C2×D4C2×D4×C15112yes—
482C2×D4#41×C15114yes—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4C2×D4×C15111yescharacteristic
497C22×C4C2×D4×C15111yescharacteristic
498C22×C4C2×D4×C15111yescharacteristic
499C22×C4C2×D4×C15112yes—
500C22×C4C2×D4×C15112yes—
501C22×C4C2×D4×C15111yescharacteristic
502C22×C4C24×C15111yescharacteristic
503D8—48no—
504M16—44no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4C2×D4×C15112yes—
519C4⋊C4C2×D4×C15112yes—
520C4⋊C4C2×D4×C15112yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—82no—
532C42—24no—
533C42—24no—
534C42—21no—
535C42—21no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—48no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—84no—
559C23—84no—
560C23—88no—
561C23—88no—
562C23#120×C15112yes—
563C23C22×D4×C15111yescharacteristic
564Q8—84no—
565D4—24no—
566D4—24no—
567D4—24no—
568D4—44no—
569D4—48no—
570D4—41no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—84no—
575D4—84no—
576D4—88no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—48no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4#120×C15112yes—
602C2×C4AGL(1,ℤ/8)×C15112yes—
603C2×C4C22×D4×C15111yescharacteristic
604C2×C4C22×D4×C15111yescharacteristic
605C2×C4C2×#41×C15112yes—
606C2×C4H(5,𝔽2)×C15112yes—
607C8—42no—
608C8—42no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—41no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—816no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22C2×#120×C15111yescharacteristic
633C22#546×C15112yes—
634C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
635C22#575×C15111yescharacteristic
636C22#576×C15112yes—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—82no—
645C4#525×C15111yescharacteristic
646C4#535×C15111yescharacteristic
647C4#546×C15112yes—
648C2—42no—
649C2—81no—
650C2—84no—
651C2—82no—
652C2#2404×C15111yescharacteristic
653C2#3006×C15111yescharacteristic
654C2#3008×C15111yescharacteristic
655C2#3319×C15112yes—
656C2#3325×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C4×D4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#555×C151C2semidirect
2. C2×#447×C151C2semidirect
3. C2×#418×C151C2semidirect
4. #2408×C151C2semidirect
5. #2023×C151C2semidirect
6. #1999×C151C2semidirect
7. #1895×C151C2semidirect
8. #1879×C151C2semidirect
9. #1875×C151C2semidirect
10. #1870×C151C2semidirect
11. #1825×C151C2semidirect
17. C2×C4⋊Q8×C151C22semidirect
23. C2×#121×C151C22semidirect
26. C2×C4×D4×C151C22semidirect
29. C2×#117×C151C22semidirect
35. C2×#102×C151C22semidirect
37. C2×C22⋊C8×C151C22semidirect
43. #340×C151C22semidirect
44. #340×C151C22semidirect
45. #340×C151C22semidirect
46. #336×C151C22semidirect
80. #121×C151D4semidirect
90. #117×C151D4semidirect
328. #23976C151direct (N × H)
329. C151#23976direct (N × H)
330. C2×#555C2×C151semidirect
331. C2×#447C2×C151semidirect
332. C2×#418C2×C151semidirect
333. #2408C2×C151semidirect
334. #2023C2×C151semidirect
335. #1999C2×C151semidirect
336. #1895C2×C151semidirect
337. #1879C2×C151semidirect
338. #1875C2×C151semidirect
339. #1870C2×C151semidirect
340. #1825C2×C151semidirect
346. C2×C4⋊Q8C22×C151semidirect
352. C2×#121C22×C151semidirect
355. C2×C4×D4C22×C151semidirect
358. C2×#117C22×C151semidirect
364. C2×#102C22×C151semidirect
366. C2×C22⋊C8C22×C151semidirect
372. #340C22×C151semidirect
373. #340C22×C151semidirect
374. #340C22×C151semidirect
375. #336C22×C151semidirect
409. #121D4×C151semidirect
419. #117D4×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#555×C1512
C2×#447×C1512
C2×#418×C1512
#2408×C1512
#2023×C1512
#1999×C1512
#1895×C1512
#1879×C1512
#1875×C1512
#1870×C1512
#1825×C1512
#23976151