Finite Group Databasebeta

← C151×#19305 #50324230

Subgroups — C151×#19305 #50324230

Order 38656.

678 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3182×C151C211yesmaximal, characteristic, complemented
2#3109×C151C211yesmaximal, characteristic
3#3077×C151C211yesmaximal, characteristic, complemented
4D4×M16×C151C211yesmaximal, characteristic, complemented
5#1703×C151C211yesmaximal, characteristic, complemented
6#1697×C151C211yesmaximal, characteristic, complemented
7#1695×C151C211yesmaximal, characteristic, complemented
8#1685×C151C211yesmaximal, characteristic, complemented
9#1683×C151C211yesmaximal, characteristic, complemented
10#1677×C151C211yesmaximal, characteristic, complemented
11#1675×C151C211yesmaximal, characteristic, complemented
12#1665×C151C211yesmaximal, characteristic, complemented
13#1569×C151C211yesmaximal, characteristic, complemented
14#1560×C151C211yesmaximal, characteristic, complemented
15#1559×C151C211yesmaximal, characteristic, complemented
16C22×M16×C151—21no—
17#533×C151C2211yescharacteristic
18#532×C151C2211yescharacteristic
19C2×C4×D4×C151C2211yescharacteristic
20#500×C151C2211yescharacteristic, complemented
21#492×C151C2211yescharacteristic, complemented
22#487×C151C2211yescharacteristic, complemented
23#486×C151C2211yescharacteristic, complemented
24#481×C151—21no—
25#478×C151—21no—
26#478×C151—21no—
27#476×C151C2211yescharacteristic
28#476×C151C2211yescharacteristic
29#476×C151C2211yescharacteristic
30#476×C151C2211yescharacteristic
31#474×C151C2211yescharacteristic
32#474×C151C2211yescharacteristic
33#468×C151—21no—
34#468×C151—21no—
35#459×C151C2211yescharacteristic
36#459×C151C2211yescharacteristic
37#447×C151—21no—
38#442×C151—21no—
39#442×C151—21no—
40#440×C151—21no—
41#440×C151—21no—
42#436×C151C2211yescharacteristic
43#436×C151C2211yescharacteristic
44#435×C151—21no—
45#435×C151—21no—
46C8×D4×C151C2211yescharacteristic
47C8×D4×C151C2211yescharacteristic
48#432×C151C2211yescharacteristic
49#428×C151—21no—
50#423×C151C2211yescharacteristic
51#423×C151C2211yescharacteristic
52#418×C151—21no—
53#407×C151—21no—
54C4×M16×C151C2211yescharacteristic
55#335×C151C2211yescharacteristic, complemented
56#335×C151C2211yescharacteristic, complemented
57#334×C151C2211yescharacteristic, complemented
58#334×C151C2211yescharacteristic, complemented
59#333×C151C2211yescharacteristic, complemented
60#333×C151C2211yescharacteristic, complemented
61#333×C151C2211yescharacteristic, complemented
62#333×C151C2211yescharacteristic, complemented
63#332×C151C2211yescharacteristic, complemented
64#332×C151C2211yescharacteristic, complemented
65#331×C151C2211yescharacteristic, complemented
66#331×C151C2211yescharacteristic, complemented
67C2×#41×C151—21no—
68C22×D4×C151D411yescharacteristic
69C23×C4×C151—21no—
70C2×SD16×C151—21no—
71C2×SD16×C151—21no—
72C2×D8×C151—21no—
73C2×D8×C151—21no—
74C2×M16×C151—21no—
75C2×M16×C151—21no—
76C2×M16×C151—21no—
77C2×M16×C151—21no—
78C2×M16×C151—44no—
79C2×M16×C151D411yescharacteristic
80C2×M16×C151D411yescharacteristic
81C22×C8×C151—21no—
82C22×C8×C151—21no—
83C4⋊Q8×C151C2311yescharacteristic
84C4⋊Q8×C151C2311yescharacteristic
85C4⋊Q8×C151C2311yescharacteristic
86#127×C151C2311yescharacteristic
87#125×C151—21no—
88#124×C151—21no—
89C22⋊Q8×C151—21no—
90C22⋊Q8×C151—21no—
91#121×C151—21no—
92#121×C151—21no—
93C4×Q8×C151—21no—
94C4×Q8×C151—21no—
95C4×D4×C151—21no—
96C4×D4×C151—21no—
97C4×D4×C151—21no—
98C4×D4×C151—21no—
99C4×D4×C151D411yescharacteristic
100C4×D4×C151D411yescharacteristic
101C4×D4×C151C2311yescharacteristic
102C4×D4×C151C2311yescharacteristic
103#117×C151—21no—
104C2×C4⋊C4×C151D411yescharacteristic
105C2×C22⋊C4×C151—21no—
106C2×C42×C151C2311yescharacteristic
107#107×C151—21no—
108#107×C151—21no—
109#107×C151—21no—
110#107×C151—21no—
111#106×C151—21no—
112#106×C151—21no—
113#106×C151—21no—
114#106×C151—21no—
115C4⋊C8×C151—21no—
116C4⋊C8×C151—21no—
117C4⋊C8×C151C2311yescharacteristic
118C4⋊C8×C151C2311yescharacteristic
119C4⋊C8×C151C2311yescharacteristic
120C4⋊C8×C151C2311yescharacteristic
121C4⋊C8×C151C2311yescharacteristic
122C4⋊C8×C151C2311yescharacteristic
123#103×C151—21no—
124#103×C151—21no—
125#102×C151—21no—
126#102×C151—21no—
127#102×C151—21no—
128#102×C151—21no—
129#102×C151—21no—
130#102×C151—21no—
131#102×C151—21no—
132#102×C151—21no—
133#102×C151—21no—
134#102×C151—21no—
135C22⋊C8×C151—21no—
136C22⋊C8×C151—21no—
137C22⋊C8×C151—21no—
138C22⋊C8×C151—21no—
139C22⋊C8×C151—21no—
140#97×C151—21no—
141#97×C151D411yescharacteristic
142#97×C151D411yescharacteristic
143C4×C8×C151C2311yescharacteristic
144C4×C8×C151C2311yescharacteristic
145C24×C151—21no—
146#41×C151—82no—
147C2×Q8×C151—21no—
148C2×Q8×C151—21no—
149C2×Q8×C151—21no—
150C2×D4×C151—22no—
151C2×D4×C151—21no—
152C2×D4×C151—21no—
153C2×D4×C151—21no—
154C2×D4×C151—21no—
155C2×D4×C151—41no—
156C2×D4×C151—41no—
157C2×D4×C151C2×D411yescharacteristic
158C2×D4×C151C2×D411yescharacteristic
159C22×C4×C151—21no—
160C22×C4×C151—21no—
161C22×C4×C151—21no—
162C22×C4×C151—21no—
163C22×C4×C151—21no—
164C22×C4×C151—42no—
165C22×C4×C151—42no—
166C22×C4×C151—41no—
167C22×C4×C151C2×D411yescharacteristic
168C22×C4×C151C2×D411yescharacteristic
169C22×C4×C151#4111yescharacteristic
170SD16×C151—42no—
171SD16×C151—42no—
172D8×C151—42no—
173D8×C151—42no—
174M16×C151—22no—
175M16×C151—41no—
176M16×C151—42no—
177M16×C151—42no—
178M16×C151—42no—
179M16×C151—42no—
180C2×C8×C151—21no—
181C2×C8×C151—21no—
182C2×C8×C151—21no—
183C2×C8×C151—21no—
184C2×C8×C151—21no—
185C2×C8×C151—21no—
186C2×C8×C151—42no—
187C2×C8×C151—42no—
188C2×C8×C151C2×D411yescharacteristic
189C2×C8×C151C2×D411yescharacteristic
190C2×C8×C151C2×D411yescharacteristic
191C2×C8×C151C2×D411yescharacteristic
192C4⋊C4×C151—21no—
193C4⋊C4×C151—21no—
194C4⋊C4×C151—21no—
195C4⋊C4×C151—21no—
196C4⋊C4×C151—21no—
197C4⋊C4×C151—21no—
198C4⋊C4×C151—21no—
199C4⋊C4×C151—41no—
200C4⋊C4×C151—41no—
201C4⋊C4×C151C2×D411yescharacteristic
202C4⋊C4×C151C2×D411yescharacteristic
203C22⋊C4×C151—21no—
204C22⋊C4×C151—21no—
205C22⋊C4×C151—21no—
206C22⋊C4×C151—21no—
207C22⋊C4×C151—41no—
208C22⋊C4×C151—41no—
209C22⋊C4×C151—41no—
210C42×C151—41no—
211C42×C151C2×D411yescharacteristic
212C42×C151C2×D411yescharacteristic
213C42×C151#4111yescharacteristic
214C42×C151C2411yescharacteristic
215C23×C151—21no—
216C23×C151—21no—
217C23×C151—42no—
218C23×C151—42no—
219C23×C151—41no—
220C23×C151—81no—
221C23×C151#12111yescharacteristic
222Q8×C151—41no—
223Q8×C151—41no—
224Q8×C151—81no—
225Q8×C151—81no—
226D4×C151—21no—
227D4×C151—21no—
228D4×C151—21no—
229D4×C151—42no—
230D4×C151—41no—
231D4×C151—41no—
232D4×C151—41no—
233D4×C151—41no—
234D4×C151—81no—
235D4×C151—82no—
236D4×C151#13712yes—
237C2×C4×C151—21no—
238C2×C4×C151—21no—
239C2×C4×C151—21no—
240C2×C4×C151—21no—
241C2×C4×C151—21no—
242C2×C4×C151—21no—
243C2×C4×C151—41no—
244C2×C4×C151—42no—
245C2×C4×C151—42no—
246C2×C4×C151—42no—
247C2×C4×C151—41no—
248C2×C4×C151—41no—
249C2×C4×C151—41no—
250C2×C4×C151—41no—
251C2×C4×C151—41no—
252C2×C4×C151—41no—
253C2×C4×C151—41no—
254C2×C4×C151—82no—
255C2×C4×C151#12111yescharacteristic
256C2×C4×C151#12111yescharacteristic
257C2×C4×C151#12111yescharacteristic
258C2×C4×C151C22×D411yescharacteristic
259C2×C4×C151C22×D411yescharacteristic
260C2×C4×C151C2×#4111yescharacteristic
261C8×C151—21no—
262C8×C151—21no—
263C8×C151—21no—
264C8×C151—41no—
265C8×C151—41no—
266C8×C151—41no—
267C8×C151—41no—
268C8×C151—41no—
269C8×C151—41no—
270C8×C151AGL(1,ℤ/8)12yes—
271C22×C151—21no—
272C22×C151—21no—
273C22×C151—22no—
274C22×C151—21no—
275C22×C151—41no—
276C22×C151—41no—
277C22×C151—41no—
278C22×C151—41no—
279C22×C151—41no—
280C22×C151—84no—
281C22×C151—81no—
282C22×C151—81no—
283C22×C151—81no—
284C22×C151C2×#12111yescharacteristic
285C4×C151—21no—
286C4×C151—21no—
287C4×C151—21no—
288C4×C151—21no—
289C4×C151—41no—
290C4×C151—41no—
291C4×C151—81no—
292C4×C151—81no—
293C4×C151—81no—
294C4×C151C2×AGL(1,ℤ/8)11yescharacteristic
295C4×C151C2×#13711yescharacteristic
296C4×C151#57511yescharacteristic
297C4×C151#57511yescharacteristic
298C2×C151—41no—
299C2×C151—42no—
300C2×C151—81no—
301C2×C151—161no—
302C2×C151#303611yescharacteristic
303C2×C151#305611yescharacteristic
304C2×C151#306811yescharacteristic
305#19305C15111yesmaximal, characteristic, direct product factor, Sylow
306C151#1930511yescharacteristic, direct product factor, Sylow
307#3182C2×C15111yescharacteristic, complemented
308#3109C2×C15111yescharacteristic
309#3077C2×C15111yescharacteristic, complemented
310D4×M16C2×C15111yescharacteristic, complemented
311#1703C2×C15111yescharacteristic, complemented
312#1697C2×C15111yescharacteristic, complemented
313#1695C2×C15111yescharacteristic, complemented
314#1685C2×C15111yescharacteristic, complemented
315#1683C2×C15111yescharacteristic, complemented
316#1677C2×C15111yescharacteristic, complemented
317#1675C2×C15111yescharacteristic, complemented
318#1665C2×C15111yescharacteristic, complemented
319#1569C2×C15111yescharacteristic, complemented
320#1560C2×C15111yescharacteristic, complemented
321#1559C2×C15111yescharacteristic, complemented
322C22×M16—21no—
323#533C22×C15111yescharacteristic
324#532C22×C15111yescharacteristic
325C2×C4×D4C22×C15111yescharacteristic
326#500C22×C15111yescharacteristic, complemented
327#492C22×C15111yescharacteristic, complemented
328#487C22×C15111yescharacteristic, complemented
329#486C22×C15111yescharacteristic, complemented
330#481—21no—
331#478—21no—
332#478—21no—
333#476C22×C15111yescharacteristic
334#476C22×C15111yescharacteristic
335#476C22×C15111yescharacteristic
336#476C22×C15111yescharacteristic
337#474C22×C15111yescharacteristic
338#474C22×C15111yescharacteristic
339#468—21no—
340#468—21no—
341#459C22×C15111yescharacteristic
342#459C22×C15111yescharacteristic
343#447—21no—
344#442—21no—
345#442—21no—
346#440—21no—
347#440—21no—
348#436C22×C15111yescharacteristic
349#436C22×C15111yescharacteristic
350#435—21no—
351#435—21no—
352C8×D4C22×C15111yescharacteristic
353C8×D4C22×C15111yescharacteristic
354#432C22×C15111yescharacteristic
355#428—21no—
356#423C22×C15111yescharacteristic
357#423C22×C15111yescharacteristic
358#418—21no—
359#407—21no—
360C4×M16C22×C15111yescharacteristic
361#335C22×C15111yescharacteristic, complemented
362#335C22×C15111yescharacteristic, complemented
363#334C22×C15111yescharacteristic, complemented
364#334C22×C15111yescharacteristic, complemented
365#333C22×C15111yescharacteristic, complemented
366#333C22×C15111yescharacteristic, complemented
367#333C22×C15111yescharacteristic, complemented
368#333C22×C15111yescharacteristic, complemented
369#332C22×C15111yescharacteristic, complemented
370#332C22×C15111yescharacteristic, complemented
371#331C22×C15111yescharacteristic, complemented
372#331C22×C15111yescharacteristic, complemented
373C2×#41—21no—
374C22×D4D4×C15111yescharacteristic
375C23×C4—21no—
376C2×SD16—21no—
377C2×SD16—21no—
378C2×D8—21no—
379C2×D8—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—44no—
385C2×M16D4×C15111yescharacteristic
386C2×M16D4×C15111yescharacteristic
387C22×C8—21no—
388C22×C8—21no—
389C4⋊Q8C23×C15111yescharacteristic
390C4⋊Q8C23×C15111yescharacteristic
391C4⋊Q8C23×C15111yescharacteristic
392#127C23×C15111yescharacteristic
393#125—21no—
394#124—21no—
395C22⋊Q8—21no—
396C22⋊Q8—21no—
397#121—21no—
398#121—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4D4×C15111yescharacteristic
406C4×D4D4×C15111yescharacteristic
407C4×D4C23×C15111yescharacteristic
408C4×D4C23×C15111yescharacteristic
409#117—21no—
410C2×C4⋊C4D4×C15111yescharacteristic
411C2×C22⋊C4—21no—
412C2×C42C23×C15111yescharacteristic
413#107—21no—
414#107—21no—
415#107—21no—
416#107—21no—
417#106—21no—
418#106—21no—
419#106—21no—
420#106—21no—
421C4⋊C8—21no—
422C4⋊C8—21no—
423C4⋊C8C23×C15111yescharacteristic
424C4⋊C8C23×C15111yescharacteristic
425C4⋊C8C23×C15111yescharacteristic
426C4⋊C8C23×C15111yescharacteristic
427C4⋊C8C23×C15111yescharacteristic
428C4⋊C8C23×C15111yescharacteristic
429#103—21no—
430#103—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—21no—
435#102—21no—
436#102—21no—
437#102—21no—
438#102—21no—
439#102—21no—
440#102—21no—
441C22⋊C8—21no—
442C22⋊C8—21no—
443C22⋊C8—21no—
444C22⋊C8—21no—
445C22⋊C8—21no—
446#97—21no—
447#97D4×C15111yescharacteristic
448#97D4×C15111yescharacteristic
449C4×C8C23×C15111yescharacteristic
450C4×C8C23×C15111yescharacteristic
451C24—21no—
452#41—82no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4C2×D4×C15111yescharacteristic
464C2×D4C2×D4×C15111yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4C2×D4×C15111yescharacteristic
474C22×C4C2×D4×C15111yescharacteristic
475C22×C4#41×C15111yescharacteristic
476SD16—42no—
477SD16—42no—
478D8—42no—
479D8—42no—
480M16—22no—
481M16—41no—
482M16—42no—
483M16—42no—
484M16—42no—
485M16—42no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8C2×D4×C15111yescharacteristic
495C2×C8C2×D4×C15111yescharacteristic
496C2×C8C2×D4×C15111yescharacteristic
497C2×C8C2×D4×C15111yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4C2×D4×C15111yescharacteristic
508C4⋊C4C2×D4×C15111yescharacteristic
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C42—41no—
517C42C2×D4×C15111yescharacteristic
518C42C2×D4×C15111yescharacteristic
519C42#41×C15111yescharacteristic
520C42C24×C15111yescharacteristic
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—81no—
527C23#121×C15111yescharacteristic
528Q8—41no—
529Q8—41no—
530Q8—81no—
531Q8—81no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—42no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—81no—
541D4—82no—
542D4#137×C15112yes—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—82no—
561C2×C4#121×C15111yescharacteristic
562C2×C4#121×C15111yescharacteristic
563C2×C4#121×C15111yescharacteristic
564C2×C4C22×D4×C15111yescharacteristic
565C2×C4C22×D4×C15111yescharacteristic
566C2×C4C2×#41×C15111yescharacteristic
567C8—21no—
568C8—21no—
569C8—21no—
570C8—41no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—41no—
575C8—41no—
576C8AGL(1,ℤ/8)×C15112yes—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—21no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—84no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22C2×#121×C15111yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4C2×AGL(1,ℤ/8)×C15111yescharacteristic
601C4C2×#137×C15111yescharacteristic
602C4#575×C15111yescharacteristic
603C4#575×C15111yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—161no—
608C2#3036×C15111yescharacteristic
609C2#3056×C15111yescharacteristic
610C2#3068×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#121
commutator, Frattini, cosocleC42C24×C151
prime radical#3109×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
1. #3182×C151C2semidirect
3. #3077×C151C2semidirect
4. D4×M16×C151C2semidirect
5. #1703×C151C2semidirect
6. #1697×C151C2semidirect
7. #1695×C151C2semidirect
8. #1685×C151C2semidirect
9. #1683×C151C2semidirect
10. #1677×C151C2semidirect
11. #1675×C151C2semidirect
12. #1665×C151C2semidirect
13. #1569×C151C2semidirect
14. #1560×C151C2semidirect
15. #1559×C151C2semidirect
20. #500×C151C22semidirect
21. #492×C151C22semidirect
22. #487×C151C22semidirect
23. #486×C151C22semidirect
55. #335×C151C22semidirect
56. #335×C151C22semidirect
57. #334×C151C22semidirect
58. #334×C151C22semidirect
59. #333×C151C22semidirect
60. #333×C151C22semidirect
61. #333×C151C22semidirect
62. #333×C151C22semidirect
63. #332×C151C22semidirect
64. #332×C151C22semidirect
65. #331×C151C22semidirect
66. #331×C151C22semidirect
305. #19305C151direct (N × H)
306. C151#19305direct (N × H)
307. #3182C2×C151semidirect
309. #3077C2×C151semidirect
310. D4×M16C2×C151semidirect
311. #1703C2×C151semidirect
312. #1697C2×C151semidirect
313. #1695C2×C151semidirect
314. #1685C2×C151semidirect
315. #1683C2×C151semidirect
316. #1677C2×C151semidirect
317. #1675C2×C151semidirect
318. #1665C2×C151semidirect
319. #1569C2×C151semidirect
320. #1560C2×C151semidirect
321. #1559C2×C151semidirect
326. #500C22×C151semidirect
327. #492C22×C151semidirect
328. #487C22×C151semidirect
329. #486C22×C151semidirect
361. #335C22×C151semidirect
362. #335C22×C151semidirect
363. #334C22×C151semidirect
364. #334C22×C151semidirect
365. #333C22×C151semidirect
366. #333C22×C151semidirect
367. #333C22×C151semidirect
368. #333C22×C151semidirect
369. #332C22×C151semidirect
370. #332C22×C151semidirect
371. #331C22×C151semidirect
372. #331C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3182×C1512
#3109×C1512
#3077×C1512
D4×M16×C1512
#1703×C1512
#1697×C1512
#1695×C1512
#1685×C1512
#1683×C1512
#1677×C1512
#1675×C1512
#1665×C1512
#1569×C1512
#1560×C1512
#1559×C1512
#19305151