Finite Group Databasebeta

← C151×#24265 #50329190

Subgroups — C151×#24265 #50329190

Order 38656.

1478 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#135×C151C211yesmaximal, characteristic, complemented
2C2×#465×C151C212yesmaximal, complemented
3C2×C8×D4×C151C211yesmaximal, characteristic, complemented
4#2407×C151C212yesmaximal, complemented
5#2056×C151C211yesmaximal, characteristic, complemented
6#2054×C151C211yesmaximal, characteristic, complemented
7#2001×C151C211yesmaximal, characteristic, complemented
8#1999×C151C211yesmaximal, characteristic, complemented
9#1968×C151C212yesmaximal, complemented
10#1807×C151C211yesmaximal, characteristic, complemented
11#1788×C151C212yesmaximal, complemented
12C22×#41×C151C2212yes—
13C2×#135×C151—28no—
14C2×#135×C151—24no—
15C22×Q16×C151C2211yescharacteristic
16C22×SD16×C151C2212yes—
17C22×D8×C151C2211yescharacteristic
18C23×C8×C151C2211yescharacteristic
19C23×C8×C151C2211yescharacteristic
20C2×C22⋊Q8×C151C2212yes—
21C2×#121×C151C2212yes—
22C2×C4×D4×C151C2211yescharacteristic
23#465×C151—28no—
24C8×D4×C151—24no—
25C2×#106×C151C2212yes—
26C2×C4⋊C8×C151C2211yescharacteristic
27C2×#103×C151C2212yes—
28C2×#103×C151C2212yes—
29C2×#102×C151C2212yes—
30C2×#102×C151C2212yes—
31C2×C22⋊C8×C151C2211yescharacteristic
32C2×C22⋊C8×C151C2211yescharacteristic
33C2×C4×C8×C151C2211yescharacteristic
34#394×C151—22no—
35#392×C151—22no—
36#391×C151C2212yes—
37#390×C151C2212yes—
38#386×C151—22no—
39#385×C151—22no—
40#380×C151C2212yes—
41#340×C151C2211yescharacteristic
42#340×C151C2211yescharacteristic
43C2×#41×C151—24no—
44C2×#41×C151—24no—
45C2×#41×C151—24no—
46C2×#41×C151—24no—
47C2×#41×C151—42no—
48C2×#41×C151—42no—
49C22×Q8×C151C2312yes—
50C22×D4×C151—22no—
51C22×D4×C151—22no—
52C22×D4×C151D411yescharacteristic
53C22×D4×C151C2312yes—
54C23×C4×C151—22no—
55C23×C4×C151C2311yescharacteristic
56C23×C4×C151C2311yescharacteristic
57#135×C151—232no—
58C2×Q16×C151—22no—
59C2×Q16×C151—24no—
60C2×SD16×C151—24no—
61C2×SD16×C151—24no—
62C2×SD16×C151D418yescomplemented
63C2×D8×C151—24no—
64C2×D8×C151—22no—
65C22×C8×C151—24no—
66C22×C8×C151—22no—
67C22×C8×C151—24no—
68C22×C8×C151D414yescomplemented
69C22×C8×C151C2311yescharacteristic
70C22×C8×C151C2311yescharacteristic
71C22×C8×C151C2311yescharacteristic
72C22×C8×C151C2311yescharacteristic
73C22⋊Q8×C151—24no—
74C22⋊Q8×C151—42no—
75#121×C151—24no—
76#121×C151—42no—
77C4×D4×C151—22no—
78C4×D4×C151—22no—
79C2×C4⋊C4×C151—22no—
80C2×C4⋊C4×C151D411yescharacteristic
81C2×C4⋊C4×C151C2312yes—
82C2×C4⋊C4×C151C2312yes—
83C2×C22⋊C4×C151—22no—
84C2×C22⋊C4×C151—22no—
85C2×C22⋊C4×C151—22no—
86C2×C22⋊C4×C151—22no—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151D411yescharacteristic
89C2×C22⋊C4×C151D411yescharacteristic
90C2×C42×C151C2311yescharacteristic
91#106×C151—24no—
92C4⋊C8×C151—22no—
93#103×C151—24no—
94#103×C151—42no—
95#102×C151—24no—
96#102×C151—42no—
97C22⋊C8×C151—22no—
98C22⋊C8×C151—22no—
99C4×C8×C151—22no—
100#95×C151—22no—
101#95×C151—22no—
102#95×C151—22no—
103C24×C151—22no—
104C24×C151C2×D411yescharacteristic
105C24×C151C2×D411yescharacteristic
106#41×C151—216no—
107#41×C151—48no—
108#41×C151—48no—
109#41×C151—48no—
110C2×Q8×C151—22no—
111C2×Q8×C151—22no—
112C2×Q8×C151—22no—
113C2×Q8×C151—22no—
114C2×Q8×C151—22no—
115C2×Q8×C151C2×D414yes—
116C2×D4×C151—22no—
117C2×D4×C151—22no—
118C2×D4×C151—22no—
119C2×D4×C151—22no—
120C2×D4×C151—22no—
121C2×D4×C151—22no—
122C2×D4×C151—22no—
123C2×D4×C151—22no—
124C2×D4×C151—44no—
125C2×D4×C151—44no—
126C2×D4×C151—44no—
127C2×D4×C151—44no—
128C2×D4×C151—44no—
129C2×D4×C151—82no—
130C2×D4×C151C2×D414yes—
131C22×C4×C151—22no—
132C22×C4×C151—22no—
133C22×C4×C151—22no—
134C22×C4×C151—21no—
135C22×C4×C151—22no—
136C22×C4×C151—21no—
137C22×C4×C151—22no—
138C22×C4×C151—22no—
139C22×C4×C151—22no—
140C22×C4×C151—44no—
141C22×C4×C151—44no—
142C22×C4×C151—44no—
143C22×C4×C151C2×D411yescharacteristic
144C22×C4×C151C2×D411yescharacteristic
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2×D412yes—
147C22×C4×C151C2×D411yescharacteristic
148C22×C4×C151#4112yes—
149C22×C4×C151C2411yescharacteristic
150Q16×C151—28no—
151SD16×C151—216no—
152D8×C151—28no—
153C2×C8×C151—22no—
154C2×C8×C151—28no—
155C2×C8×C151—21no—
156C2×C8×C151—21no—
157C2×C8×C151—22no—
158C2×C8×C151—22no—
159C2×C8×C151—44no—
160C2×C8×C151C2×D412yes—
161C2×C8×C151C2×D412yes—
162C2×C8×C151C2×D414yes—
163C4⋊C4×C151—22no—
164C4⋊C4×C151—22no—
165C4⋊C4×C151—22no—
166C4⋊C4×C151—22no—
167C4⋊C4×C151—22no—
168C4⋊C4×C151—44no—
169C22⋊C4×C151—22no—
170C22⋊C4×C151—22no—
171C22⋊C4×C151—44no—
172C22⋊C4×C151—44no—
173C22⋊C4×C151—44no—
174C22⋊C4×C151—82no—
175C22⋊C4×C151—82no—
176C42×C151—21no—
177C42×C151—21no—
178C23×C151—21no—
179C23×C151—22no—
180C23×C151—22no—
181C23×C151—21no—
182C23×C151—21no—
183C23×C151—21no—
184C23×C151—22no—
185C23×C151—22no—
186C23×C151—24no—
187C23×C151—42no—
188C23×C151—42no—
189C23×C151—42no—
190C23×C151—42no—
191C23×C151—42no—
192C23×C151—42no—
193C23×C151#12112yes—
194C23×C151#13512yescomplemented
195C23×C151C22×D411yescharacteristic
196Q8×C151—24no—
197Q8×C151—24no—
198Q8×C151—42no—
199Q8×C151—42no—
200D4×C151—28no—
201D4×C151—24no—
202D4×C151—24no—
203D4×C151—416no—
204D4×C151—42no—
205D4×C151—42no—
206D4×C151—88no—
207C2×C4×C151—21no—
208C2×C4×C151—22no—
209C2×C4×C151—21no—
210C2×C4×C151—22no—
211C2×C4×C151—22no—
212C2×C4×C151—24no—
213C2×C4×C151—21no—
214C2×C4×C151—21no—
215C2×C4×C151—24no—
216C2×C4×C151—22no—
217C2×C4×C151—22no—
218C2×C4×C151—416no—
219C2×C4×C151—42no—
220C2×C4×C151—42no—
221C2×C4×C151—41no—
222C2×C4×C151—41no—
223C2×C4×C151—42no—
224C2×C4×C151—42no—
225C2×C4×C151—42no—
226C2×C4×C151—42no—
227C2×C4×C151—42no—
228C2×C4×C151—42no—
229C2×C4×C151—42no—
230C2×C4×C151#12112yes—
231C2×C4×C151#13512yescomplemented
232C2×C4×C151C22×D411yescharacteristic
233C2×C4×C151C22×D412yes—
234C2×C4×C151C2×#4111yescharacteristic
235C2×C4×C151H(5,𝔽2)12yes—
236C8×C151—22no—
237C8×C151—41no—
238C8×C151—41no—
239C8×C151#12014yes—
240C22×C151—22no—
241C22×C151—24no—
242C22×C151—22no—
243C22×C151—22no—
244C22×C151—22no—
245C22×C151—41no—
246C22×C151—42no—
247C22×C151—42no—
248C22×C151—42no—
249C22×C151—41no—
250C22×C151—44no—
251C22×C151—44no—
252C22×C151—44no—
253C22×C151—82no—
254C22×C151—82no—
255C22×C151—82no—
256C22×C151—82no—
257C22×C151C2×#12111yescharacteristic
258C22×C151D4212yes—
259C22×C151C2×#13511yescharacteristic
260C22×C151C2×#13511yescharacteristic
261C22×C151#57712yes—
262C4×C151—22no—
263C4×C151—22no—
264C4×C151—41no—
265C4×C151—44no—
266C4×C151—41no—
267C4×C151—82no—
268C4×C151—82no—
269C4×C151C2×#12011yescharacteristic
270C4×C151#53511yescharacteristic
271C4×C151#54612yes—
272C2×C151—42no—
273C2×C151—42no—
274C2×C151—84no—
275C2×C151#240811yescharacteristic
276C2×C151#305011yescharacteristic
277C2×C151#306011yescharacteristic
278C2×C151#328212yes—
279C2×C151#328412yes—
280#24265C15111yesmaximal, characteristic, direct product factor, Sylow
281C151#2426511yescharacteristic, direct product factor, Sylow
282C22×#135C2×C15111yescharacteristic, complemented
283C2×#465C2×C15112yescomplemented
284C2×C8×D4C2×C15111yescharacteristic, complemented
285#2407C2×C15112yescomplemented
286#2056C2×C15111yescharacteristic, complemented
287#2054C2×C15111yescharacteristic, complemented
288#2001C2×C15111yescharacteristic, complemented
289#1999C2×C15111yescharacteristic, complemented
290#1968C2×C15112yescomplemented
291#1807C2×C15111yescharacteristic, complemented
292#1788C2×C15112yescomplemented
293C22×#41C22×C15112yes—
294C2×#135—28no—
295C2×#135—24no—
296C22×Q16C22×C15111yescharacteristic
297C22×SD16C22×C15112yes—
298C22×D8C22×C15111yescharacteristic
299C23×C8C22×C15111yescharacteristic
300C23×C8C22×C15111yescharacteristic
301C2×C22⋊Q8C22×C15112yes—
302C2×#121C22×C15112yes—
303C2×C4×D4C22×C15111yescharacteristic
304#465—28no—
305C8×D4—24no—
306C2×#106C22×C15112yes—
307C2×C4⋊C8C22×C15111yescharacteristic
308C2×#103C22×C15112yes—
309C2×#103C22×C15112yes—
310C2×#102C22×C15112yes—
311C2×#102C22×C15112yes—
312C2×C22⋊C8C22×C15111yescharacteristic
313C2×C22⋊C8C22×C15111yescharacteristic
314C2×C4×C8C22×C15111yescharacteristic
315#394—22no—
316#392—22no—
317#391C22×C15112yes—
318#390C22×C15112yes—
319#386—22no—
320#385—22no—
321#380C22×C15112yes—
322#340C22×C15111yescharacteristic
323#340C22×C15111yescharacteristic
324C2×#41—24no—
325C2×#41—24no—
326C2×#41—24no—
327C2×#41—24no—
328C2×#41—42no—
329C2×#41—42no—
330C22×Q8C23×C15112yes—
331C22×D4—22no—
332C22×D4—22no—
333C22×D4D4×C15111yescharacteristic
334C22×D4C23×C15112yes—
335C23×C4—22no—
336C23×C4C23×C15111yescharacteristic
337C23×C4C23×C15111yescharacteristic
338#135—232no—
339C2×Q16—24no—
340C2×Q16—22no—
341C2×SD16—24no—
342C2×SD16—24no—
343C2×SD16D4×C15118yescomplemented
344C2×D8—24no—
345C2×D8—22no—
346C22×C8—24no—
347C22×C8—22no—
348C22×C8—24no—
349C22×C8D4×C15114yescomplemented
350C22×C8C23×C15111yescharacteristic
351C22×C8C23×C15111yescharacteristic
352C22×C8C23×C15111yescharacteristic
353C22×C8C23×C15111yescharacteristic
354C22⋊Q8—24no—
355C22⋊Q8—42no—
356#121—24no—
357#121—42no—
358C4×D4—22no—
359C4×D4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4D4×C15111yescharacteristic
362C2×C4⋊C4C23×C15112yes—
363C2×C4⋊C4C23×C15112yes—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4D4×C15111yescharacteristic
370C2×C22⋊C4D4×C15111yescharacteristic
371C2×C42C23×C15111yescharacteristic
372#106—24no—
373C4⋊C8—22no—
374#103—24no—
375#103—42no—
376#102—24no—
377#102—42no—
378C22⋊C8—22no—
379C22⋊C8—22no—
380C4×C8—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384C24—22no—
385C24C2×D4×C15111yescharacteristic
386C24C2×D4×C15111yescharacteristic
387#41—216no—
388#41—48no—
389#41—48no—
390#41—48no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8C2×D4×C15114yes—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—82no—
411C2×D4C2×D4×C15114yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4C2×D4×C15111yescharacteristic
425C22×C4C2×D4×C15111yescharacteristic
426C22×C4C2×D4×C15111yescharacteristic
427C22×C4C2×D4×C15112yes—
428C22×C4C2×D4×C15111yescharacteristic
429C22×C4#41×C15112yes—
430C22×C4C24×C15111yescharacteristic
431Q16—28no—
432SD16—216no—
433D8—28no—
434C2×C8—22no—
435C2×C8—28no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—44no—
441C2×C8C2×D4×C15112yes—
442C2×C8C2×D4×C15112yes—
443C2×C8C2×D4×C15114yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C42—21no—
458C42—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23#121×C15112yes—
475C23#135×C15112yescomplemented
476C23C22×D4×C15111yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—42no—
480Q8—42no—
481D4—28no—
482D4—24no—
483D4—24no—
484D4—416no—
485D4—42no—
486D4—42no—
487D4—88no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4#121×C15112yes—
512C2×C4#135×C15112yescomplemented
513C2×C4C22×D4×C15111yescharacteristic
514C2×C4C22×D4×C15112yes—
515C2×C4C2×#41×C15111yescharacteristic
516C2×C4H(5,𝔽2)×C15112yes—
517C8—22no—
518C8—41no—
519C8—41no—
520C8#120×C15114yes—
521C22—22no—
522C22—24no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—41no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22C2×#121×C15111yescharacteristic
539C22D42×C15112yes—
540C22C2×#135×C15111yescharacteristic
541C22C2×#135×C15111yescharacteristic
542C22#577×C15112yes—
543C4—22no—
544C4—22no—
545C4—41no—
546C4—44no—
547C4—41no—
548C4—82no—
549C4—82no—
550C4C2×#120×C15111yescharacteristic
551C4#535×C15111yescharacteristic
552C4#546×C15112yes—
553C2—42no—
554C2—42no—
555C2—84no—
556C2#2408×C15111yescharacteristic
557C2#3050×C15111yescharacteristic
558C2#3060×C15111yescharacteristic
559C2#3282×C15112yes—
560C2#3284×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. C22×#135×C151C2semidirect
2. C2×#465×C151C2semidirect
3. C2×C8×D4×C151C2semidirect
4. #2407×C151C2semidirect
5. #2056×C151C2semidirect
6. #2054×C151C2semidirect
7. #2001×C151C2semidirect
8. #1999×C151C2semidirect
9. #1968×C151C2semidirect
10. #1807×C151C2semidirect
11. #1788×C151C2semidirect
62. C2×SD16×C151D4semidirect
68. C22×C8×C151D4semidirect
194. C23×C151#135semidirect
231. C2×C4×C151#135semidirect
280. #24265C151direct (N × H)
281. C151#24265direct (N × H)
282. C22×#135C2×C151semidirect
283. C2×#465C2×C151semidirect
284. C2×C8×D4C2×C151semidirect
285. #2407C2×C151semidirect
286. #2056C2×C151semidirect
287. #2054C2×C151semidirect
288. #2001C2×C151semidirect
289. #1999C2×C151semidirect
290. #1968C2×C151semidirect
291. #1807C2×C151semidirect
292. #1788C2×C151semidirect
343. C2×SD16D4×C151semidirect
349. C22×C8D4×C151semidirect
475. C23#135×C151semidirect
512. C2×C4#135×C151semidirect

Maximal subgroups

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

subgroupindex
C22×#135×C1512
C2×#465×C1512
C2×C8×D4×C1512
#2407×C1512
#2056×C1512
#2054×C1512
#2001×C1512
#1999×C1512
#1968×C1512
#1807×C1512
#1788×C1512
#24265151