Finite Group Databasebeta

← C151×#24527 #50329452

Subgroups — C151×#24527 #50329452

Order 38656.

1182 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#450×C151C211yesmaximal, characteristic
2C22×#102×C151C211yesmaximal, characteristic
3#2503×C151C211yesmaximal, characteristic, complemented
4D4×C22⋊C4×C151C211yesmaximal, characteristic
5#2073×C151C212yesmaximal, complemented
6#2055×C151C212yesmaximal, complemented
7#1892×C151C211yesmaximal, characteristic, complemented
8#1877×C151C212yesmaximal, complemented
9#1850×C151C211yesmaximal, characteristic, complemented
10#1823×C151C211yesmaximal, characteristic, complemented
11#1804×C151C212yesmaximal, complemented
12C23×D4×C151C2211yescharacteristic
13C22×SD16×C151C2211yescharacteristic
14C22×SD16×C151C2211yescharacteristic
15C23×C8×C151C2211yescharacteristic
16C2×C22⋊Q8×C151C2211yescharacteristic
17C2×C4×D4×C151C2212yes—
18C22×C4⋊C4×C151C2211yescharacteristic
19C22×C22⋊C4×C151—21no—
20#450×C151—24no—
21C2×C4⋊C8×C151C2212yescomplemented
22C2×#103×C151C2212yescomplemented
23C2×#102×C151—24no—
24C2×#102×C151C2212yes—
25C2×#102×C151C2212yes—
26C2×#102×C151C2211yescharacteristic
27C2×#102×C151C2211yescharacteristic
28C2×C22⋊C8×C151C2211yescharacteristic
29#400×C151—22no—
30#391×C151C2212yescomplemented
31#390×C151C2212yes—
32#386×C151—22no—
33#382×C151C2212yescomplemented
34#381×C151—21no—
35#381×C151—21no—
36#380×C151C2211yescharacteristic
37#379×C151—21no—
38C4×C22⋊C4×C151C2211yescharacteristic
39#340×C151C2212yescomplemented
40#340×C151C2212yescomplemented
41#340×C151C2212yescomplemented
42#336×C151C2212yescomplemented
43C25×C151—21no—
44C22×Q8×C151C2311yescharacteristic
45C22×D4×C151—21no—
46C22×D4×C151—22no—
47C22×D4×C151—24no—
48C22×D4×C151—42no—
49C22×D4×C151D414yes—
50C22×D4×C151C2311yescharacteristic
51C22×D4×C151C2311yescharacteristic
52C23×C4×C151—22no—
53C23×C4×C151—21no—
54C23×C4×C151C2311yescharacteristic
55C2×SD16×C151—22no—
56C2×SD16×C151—22no—
57C2×SD16×C151—24no—
58C2×SD16×C151—42no—
59C22×C8×C151—22no—
60C22×C8×C151—42no—
61C22×C8×C151C2311yescharacteristic
62C22×C8×C151C2312yes—
63C22×C8×C151C2311yescharacteristic
64C22⋊Q8×C151—22no—
65C22⋊Q8×C151—22no—
66C4×D4×C151—24no—
67C4×D4×C151—42no—
68C2×C4⋊C4×C151—22no—
69C2×C4⋊C4×C151—22no—
70C2×C4⋊C4×C151—21no—
71C2×C4⋊C4×C151—21no—
72C2×C4⋊C4×C151C2312yes—
73C2×C4⋊C4×C151C2311yescharacteristic
74C2×C4⋊C4×C151C2312yes—
75C2×C22⋊C4×C151—22no—
76C2×C22⋊C4×C151—22no—
77C2×C22⋊C4×C151—22no—
78C2×C22⋊C4×C151—21no—
79C2×C22⋊C4×C151—21no—
80C2×C22⋊C4×C151—44no—
81C2×C22⋊C4×C151D411yescharacteristic
82C2×C22⋊C4×C151D411yescharacteristic
83C2×C42×C151C2312yes—
84C4⋊C8×C151—42no—
85#103×C151—42no—
86#102×C151—24no—
87#102×C151—24no—
88#102×C151—22no—
89#102×C151—22no—
90C22⋊C8×C151—22no—
91#95×C151—21no—
92#95×C151—22no—
93#95×C151—22no—
94#95×C151—22no—
95#95×C151—22no—
96#95×C151D412yes—
97C24×C151—21no—
98C24×C151—21no—
99C24×C151—24no—
100C24×C151—44no—
101C24×C151—42no—
102C24×C151—42no—
103C24×C151—82no—
104C24×C151C2×D411yescharacteristic
105C2×Q8×C151—21no—
106C2×Q8×C151—21no—
107C2×Q8×C151—22no—
108C2×Q8×C151—22no—
109C2×D4×C151—28no—
110C2×D4×C151—24no—
111C2×D4×C151—21no—
112C2×D4×C151—22no—
113C2×D4×C151—21no—
114C2×D4×C151—22no—
115C2×D4×C151—22no—
116C2×D4×C151—44no—
117C2×D4×C151—48no—
118C2×D4×C151—44no—
119C2×D4×C151C2×D412yes—
120C2×D4×C151C2×D412yes—
121C2×D4×C151#4114yes—
122C22×C4×C151—22no—
123C22×C4×C151—22no—
124C22×C4×C151—21no—
125C22×C4×C151—21no—
126C22×C4×C151—22no—
127C22×C4×C151—22no—
128C22×C4×C151—44no—
129C22×C4×C151—44no—
130C22×C4×C151—42no—
131C22×C4×C151—42no—
132C22×C4×C151—42no—
133C22×C4×C151—82no—
134C22×C4×C151C2×D412yes—
135C22×C4×C151C2×D412yes—
136C22×C4×C151C2×D411yescharacteristic
137C22×C4×C151C2×D412yes—
138C22×C4×C151#4112yes—
139C22×C4×C151C2411yescharacteristic
140SD16×C151—44no—
141SD16×C151—44no—
142C2×C8×C151—21no—
143C2×C8×C151—21no—
144C2×C8×C151—22no—
145C2×C8×C151—22no—
146C2×C8×C151—22no—
147C2×C8×C151—22no—
148C2×C8×C151—22no—
149C2×C8×C151—44no—
150C4⋊C4×C151—22no—
151C4⋊C4×C151—21no—
152C4⋊C4×C151—22no—
153C4⋊C4×C151—22no—
154C4⋊C4×C151—21no—
155C4⋊C4×C151—22no—
156C4⋊C4×C151—42no—
157C4⋊C4×C151—42no—
158C22⋊C4×C151—22no—
159C22⋊C4×C151—22no—
160C22⋊C4×C151—44no—
161C22⋊C4×C151—44no—
162C22⋊C4×C151—42no—
163C22⋊C4×C151—42no—
164C22⋊C4×C151—82no—
165C42×C151—22no—
166C42×C151—22no—
167C23×C151—22no—
168C23×C151—22no—
169C23×C151—22no—
170C23×C151—24no—
171C23×C151—22no—
172C23×C151—48no—
173C23×C151—41no—
174C23×C151—41no—
175C23×C151—42no—
176C23×C151—42no—
177C23×C151—41no—
178C23×C151—41no—
179C23×C151—42no—
180C23×C151—48no—
181C23×C151—84no—
182C23×C151—88no—
183C23×C151—88no—
184C23×C151#12112yes—
185C23×C151#13512yes—
186C23×C151AGL(1,ℤ/8)12yes—
187C23×C151C22×D411yescharacteristic
188Q8×C151—41no—
189Q8×C151—41no—
190Q8×C151—42no—
191D4×C151—22no—
192D4×C151—22no—
193D4×C151—22no—
194D4×C151—24no—
195D4×C151—48no—
196D4×C151—42no—
197D4×C151#12314yes—
198C2×C4×C151—22no—
199C2×C4×C151—22no—
200C2×C4×C151—22no—
201C2×C4×C151—24no—
202C2×C4×C151—22no—
203C2×C4×C151—22no—
204C2×C4×C151—22no—
205C2×C4×C151—22no—
206C2×C4×C151—22no—
207C2×C4×C151—22no—
208C2×C4×C151—41no—
209C2×C4×C151—41no—
210C2×C4×C151—42no—
211C2×C4×C151—42no—
212C2×C4×C151—42no—
213C2×C4×C151—42no—
214C2×C4×C151—42no—
215C2×C4×C151—42no—
216C2×C4×C151—42no—
217C2×C4×C151—42no—
218C2×C4×C151—88no—
219C2×C4×C151—84no—
220C2×C4×C151#12112yes—
221C2×C4×C151C22×D411yescharacteristic
222C2×C4×C151C2×#4112yes—
223C2×C4×C151C2×#4111yescharacteristic
224C2×C4×C151H(5,𝔽2)12yes—
225C8×C151—41no—
226C8×C151—42no—
227C8×C151—41no—
228C22×C151—22no—
229C22×C151—24no—
230C22×C151—22no—
231C22×C151—24no—
232C22×C151—24no—
233C22×C151—42no—
234C22×C151—44no—
235C22×C151—42no—
236C22×C151—44no—
237C22×C151—42no—
238C22×C151—42no—
239C22×C151—42no—
240C22×C151—816no—
241C22×C151—82no—
242C22×C151—82no—
243C22×C151#46314yes—
244C22×C151C2×#12111yescharacteristic
245C22×C151#54612yes—
246C22×C151C2×#13511yescharacteristic
247C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C151#57712yes—
249C4×C151—22no—
250C4×C151—42no—
251C4×C151—42no—
252C4×C151—81no—
253C4×C151—82no—
254C4×C151—81no—
255C4×C151—82no—
256C4×C151C2×#12311yescharacteristic
257C4×C151#53411yescharacteristic
258C4×C151#55112yes—
259C2×C151—24no—
260C2×C151—44no—
261C2×C151—82no—
262C2×C151#241211yescharacteristic
263C2×C151C2×#46311yescharacteristic
264C2×C151#304211yescharacteristic
265C2×C151#329612yes—
266C2×C151#331012yes—
267#24527C15111yesmaximal, characteristic, direct product factor, Sylow
268C151#2452711yescharacteristic, direct product factor, Sylow
269C2×#450C2×C15111yescharacteristic
270C22×#102C2×C15111yescharacteristic
271#2503C2×C15111yescharacteristic, complemented
272D4×C22⋊C4C2×C15111yescharacteristic
273#2073C2×C15112yescomplemented
274#2055C2×C15112yescomplemented
275#1892C2×C15111yescharacteristic, complemented
276#1877C2×C15112yescomplemented
277#1850C2×C15111yescharacteristic, complemented
278#1823C2×C15111yescharacteristic, complemented
279#1804C2×C15112yescomplemented
280C23×D4C22×C15111yescharacteristic
281C22×SD16C22×C15111yescharacteristic
282C22×SD16C22×C15111yescharacteristic
283C23×C8C22×C15111yescharacteristic
284C2×C22⋊Q8C22×C15111yescharacteristic
285C2×C4×D4C22×C15112yes—
286C22×C4⋊C4C22×C15111yescharacteristic
287C22×C22⋊C4—21no—
288#450—24no—
289C2×C4⋊C8C22×C15112yescomplemented
290C2×#103C22×C15112yescomplemented
291C2×#102—24no—
292C2×#102C22×C15112yes—
293C2×#102C22×C15111yescharacteristic
294C2×#102C22×C15112yes—
295C2×#102C22×C15111yescharacteristic
296C2×C22⋊C8C22×C15111yescharacteristic
297#400—22no—
298#391C22×C15112yescomplemented
299#390C22×C15112yes—
300#386—22no—
301#382C22×C15112yescomplemented
302#381—21no—
303#381—21no—
304#380C22×C15111yescharacteristic
305#379—21no—
306C4×C22⋊C4C22×C15111yescharacteristic
307#340C22×C15112yescomplemented
308#340C22×C15112yescomplemented
309#340C22×C15112yescomplemented
310#336C22×C15112yescomplemented
311C25—21no—
312C22×Q8C23×C15111yescharacteristic
313C22×D4—24no—
314C22×D4—22no—
315C22×D4—21no—
316C22×D4—42no—
317C22×D4D4×C15114yes—
318C22×D4C23×C15111yescharacteristic
319C22×D4C23×C15111yescharacteristic
320C23×C4—21no—
321C23×C4—22no—
322C23×C4C23×C15111yescharacteristic
323C2×SD16—24no—
324C2×SD16—22no—
325C2×SD16—22no—
326C2×SD16—42no—
327C22×C8—22no—
328C22×C8—42no—
329C22×C8C23×C15112yes—
330C22×C8C23×C15111yescharacteristic
331C22×C8C23×C15111yescharacteristic
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C4×D4—24no—
335C4×D4—42no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4C23×C15112yes—
341C2×C4⋊C4C23×C15112yes—
342C2×C4⋊C4C23×C15111yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4D4×C15111yescharacteristic
350C2×C22⋊C4D4×C15111yescharacteristic
351C2×C42C23×C15112yes—
352C4⋊C8—42no—
353#103—42no—
354#102—24no—
355#102—24no—
356#102—22no—
357#102—22no—
358C22⋊C8—22no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95D4×C15112yes—
365C24—21no—
366C24—21no—
367C24—24no—
368C24—44no—
369C24—42no—
370C24—42no—
371C24—82no—
372C24C2×D4×C15111yescharacteristic
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×D4—28no—
378C2×D4—24no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4C2×D4×C15112yes—
388C2×D4C2×D4×C15112yes—
389C2×D4#41×C15114yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4C2×D4×C15112yes—
403C22×C4C2×D4×C15112yes—
404C22×C4C2×D4×C15111yescharacteristic
405C22×C4C2×D4×C15112yes—
406C22×C4#41×C15112yes—
407C22×C4C24×C15111yescharacteristic
408SD16—44no—
409SD16—44no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—44no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—82no—
433C42—22no—
434C42—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—48no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—48no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—84no—
450C23—88no—
451C23—88no—
452C23#121×C15112yes—
453C23#135×C15112yes—
454C23AGL(1,ℤ/8)×C15112yes—
455C23C22×D4×C15111yescharacteristic
456Q8—41no—
457Q8—41no—
458Q8—42no—
459D4—22no—
460D4—22no—
461D4—22no—
462D4—24no—
463D4—48no—
464D4—42no—
465D4#123×C15114yes—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—88no—
487C2×C4—84no—
488C2×C4#121×C15112yes—
489C2×C4C22×D4×C15111yescharacteristic
490C2×C4C2×#41×C15112yes—
491C2×C4C2×#41×C15111yescharacteristic
492C2×C4H(5,𝔽2)×C15112yes—
493C8—41no—
494C8—42no—
495C8—41no—
496C22—22no—
497C22—24no—
498C22—22no—
499C22—24no—
500C22—24no—
501C22—42no—
502C22—44no—
503C22—42no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—816no—
509C22—82no—
510C22—82no—
511C22#463×C15114yes—
512C22C2×#121×C15111yescharacteristic
513C22#546×C15112yes—
514C22C2×#135×C15111yescharacteristic
515C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
516C22#577×C15112yes—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—81no—
521C4—82no—
522C4—81no—
523C4—82no—
524C4C2×#123×C15111yescharacteristic
525C4#534×C15111yescharacteristic
526C4#551×C15112yes—
527C2—24no—
528C2—44no—
529C2—82no—
530C2#2412×C15111yescharacteristic
531C2C2×#463×C15111yescharacteristic
532C2#3042×C15111yescharacteristic
533C2#3296×C15112yes—
534C2#3310×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC23×D4×C151C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C4×C22⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
3. #2503×C151C2semidirect
5. #2073×C151C2semidirect
6. #2055×C151C2semidirect
7. #1892×C151C2semidirect
8. #1877×C151C2semidirect
9. #1850×C151C2semidirect
10. #1823×C151C2semidirect
11. #1804×C151C2semidirect
21. C2×C4⋊C8×C151C22semidirect
22. C2×#103×C151C22semidirect
30. #391×C151C22semidirect
33. #382×C151C22semidirect
39. #340×C151C22semidirect
40. #340×C151C22semidirect
41. #340×C151C22semidirect
42. #336×C151C22semidirect
267. #24527C151direct (N × H)
268. C151#24527direct (N × H)
271. #2503C2×C151semidirect
273. #2073C2×C151semidirect
274. #2055C2×C151semidirect
275. #1892C2×C151semidirect
276. #1877C2×C151semidirect
277. #1850C2×C151semidirect
278. #1823C2×C151semidirect
279. #1804C2×C151semidirect
289. C2×C4⋊C8C22×C151semidirect
290. C2×#103C22×C151semidirect
298. #391C22×C151semidirect
301. #382C22×C151semidirect
307. #340C22×C151semidirect
308. #340C22×C151semidirect
309. #340C22×C151semidirect
310. #336C22×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#450×C1512
C22×#102×C1512
#2503×C1512
D4×C22⋊C4×C1512
#2073×C1512
#2055×C1512
#1892×C1512
#1877×C1512
#1850×C1512
#1823×C1512
#1804×C1512
#24527151