Finite Group Databasebeta

← C151×#18574 #50323499

Subgroups — C151×#18574 #50323499

Order 38656.

1238 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3498×C151C211yesmaximal, characteristic
2#3254×C151C211yesmaximal, characteristic, complemented
3#3138×C151C211yesmaximal, characteristic, complemented
4#3107×C151C211yesmaximal, characteristic, complemented
5#1642×C151C212yesmaximal, complemented
6#1641×C151C212yesmaximal, complemented
7#1639×C151C212yesmaximal, complemented
8#1639×C151C212yesmaximal, complemented
9#1545×C151C211yesmaximal, characteristic, complemented
10#1489×C151C211yesmaximal, characteristic, complemented
11#1484×C151C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)×C151—21no—
13#550×C151—21no—
14#550×C151C2212yes—
15D42×C151—22no—
16D42×C151—22no—
17D42×C151—21no—
18D42×C151C2212yes—
19#535×C151—22no—
20#535×C151—21no—
21#533×C151C2211yescharacteristic, complemented
22#532×C151—21no—
23#532×C151C2211yescharacteristic
24C2×#127×C151—21no—
25C2×#127×C151C2211yescharacteristic
26C4×#41×C151C2211yescharacteristic
27#490×C151—22no—
28#486×C151C2212yescomplemented
29#486×C151C2212yescomplemented
30#485×C151—21no—
31#477×C151—21no—
32#475×C151C2211yescharacteristic, complemented
33#474×C151C2211yescharacteristic, complemented
34#474×C151C2212yescomplemented
35#468×C151—22no—
36#462×C151C2211yescharacteristic, complemented
37#461×C151C2211yescharacteristic, complemented
38#461×C151C2212yescomplemented
39#460×C151—21no—
40#452×C151—21no—
41C4×Q16×C151—21no—
42C4×SD16×C151—21no—
43#431×C151C2211yescharacteristic, complemented
44#423×C151C2211yescharacteristic, complemented
45#418×C151—22no—
46C4×M16×C151C2211yescharacteristic, complemented
47#331×C151C2212yescomplemented
48#331×C151C2212yescomplemented
49Q8⋊C8×C151C2212yescomplemented
50D4⋊C8×C151C2212yescomplemented
51D4⋊C8×C151C2212yescomplemented
52D4⋊C8×C151C2212yescomplemented
53H(5,𝔽2)×C151—44no—
54H(5,𝔽2)×C151—44no—
55C2×#41×C151—22no—
56C2×#41×C151—21no—
57C2×#41×C151—22no—
58C2×#41×C151D411yescharacteristic
59C22×D4×C151—22no—
60C22×D4×C151—24no—
61C22×D4×C151—22no—
62C22×D4×C151—22no—
63C22×D4×C151—22no—
64C22×D4×C151—21no—
65C22×D4×C151—21no—
66C22×D4×C151D412yes—
67C2×Q16×C151—21no—
68C2×SD16×C151—21no—
69C2×SD16×C151—22no—
70C2×M16×C151—21no—
71C2×M16×C151—21no—
72C4⋊Q8×C151D411yescharacteristic
73C4⋊Q8×C151C2311yescharacteristic
74C4⋊Q8×C151C2311yescharacteristic
75#127×C151—22no—
76#127×C151—22no—
77#127×C151—22no—
78#127×C151—21no—
79#127×C151—21no—
80#127×C151D411yescharacteristic
81#127×C151D412yes—
82#127×C151C2312yes—
83#125×C151—21no—
84#124×C151—22no—
85#124×C151—21no—
86C22⋊Q8×C151—21no—
87C22⋊Q8×C151—21no—
88#121×C151—22no—
89#121×C151—24no—
90#121×C151—24no—
91#121×C151—21no—
92#121×C151—22no—
93#121×C151—22no—
94#121×C151—22no—
95#121×C151D412yes—
96#120×C151—24no—
97#120×C151—24no—
98#120×C151—24no—
99C4×Q8×C151—21no—
100C4×Q8×C151—21no—
101C4×Q8×C151D411yescharacteristic
102C4×Q8×C151C2311yescharacteristic
103C4×D4×C151—22no—
104C4×D4×C151—21no—
105C4×D4×C151—21no—
106C4×D4×C151—22no—
107C4×D4×C151—21no—
108C4×D4×C151D411yescharacteristic
109C4×D4×C151C2311yescharacteristic
110C4×D4×C151C2312yes—
111#117×C151—21no—
112#117×C151—21no—
113#117×C151D411yescharacteristic
114C2×C42×C151—21no—
115C2×C42×C151C2311yescharacteristic
116#107×C151—21no—
117#107×C151—22no—
118#106×C151—21no—
119C4⋊C8×C151—21no—
120C4⋊C8×C151C2312yes—
121C4⋊C8×C151C2311yescharacteristic
122C4⋊C8×C151C2311yescharacteristic
123#103×C151—21no—
124#103×C151—21no—
125#103×C151—22no—
126#103×C151—21no—
127#102×C151—24no—
128#102×C151—24no—
129#102×C151—21no—
130#102×C151—22no—
131C22⋊C8×C151—21no—
132#97×C151—21no—
133C4×C8×C151—21no—
134C4×C8×C151C2312yes—
135C24×C151—24no—
136C24×C151—22no—
137#41×C151—22no—
138#41×C151—22no—
139#41×C151—48no—
140#41×C151—42no—
141#41×C151—42no—
142#41×C151—44no—
143#41×C151—44no—
144C2×Q8×C151—21no—
145C2×Q8×C151—21no—
146C2×Q8×C151—21no—
147C2×Q8×C151C2×D411yescharacteristic
148C2×D4×C151—21no—
149C2×D4×C151—24no—
150C2×D4×C151—24no—
151C2×D4×C151—22no—
152C2×D4×C151—24no—
153C2×D4×C151—24no—
154C2×D4×C151—22no—
155C2×D4×C151—21no—
156C2×D4×C151—22no—
157C2×D4×C151—21no—
158C2×D4×C151—22no—
159C2×D4×C151—22no—
160C2×D4×C151—22no—
161C2×D4×C151—48no—
162C2×D4×C151—42no—
163C2×D4×C151—42no—
164C2×D4×C151—44no—
165C2×D4×C151—416no—
166C2×D4×C151—44no—
167C2×D4×C151—44no—
168C2×D4×C151—81no—
169C2×D4×C151—81no—
170C2×D4×C151—82no—
171C2×D4×C151—82no—
172C2×D4×C151—84no—
173C2×D4×C151C2×D412yes—
174C2×D4×C151C2×D411yescharacteristic
175C2×D4×C151C2×D414yes—
176C22×C4×C151—21no—
177C22×C4×C151—24no—
178C22×C4×C151—21no—
179C22×C4×C151—21no—
180C22×C4×C151—22no—
181C22×C4×C151C2×D411yescharacteristic
182C22×C4×C151C2×D411yescharacteristic
183C22×C4×C151C2×D411yescharacteristic
184Q16×C151—42no—
185SD16×C151—42no—
186SD16×C151—82no—
187M16×C151—42no—
188M16×C151—81no—
189C2×C8×C151—21no—
190C2×C8×C151—21no—
191C2×C8×C151—22no—
192C2×C8×C151—22no—
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—21no—
200C4⋊C4×C151—41no—
201C4⋊C4×C151—41no—
202C4⋊C4×C151C2×D411yescharacteristic
203C4⋊C4×C151C2×D411yescharacteristic
204C4⋊C4×C151C2×D412yes—
205C22⋊C4×C151—24no—
206C22⋊C4×C151—24no—
207C22⋊C4×C151—21no—
208C22⋊C4×C151—21no—
209C22⋊C4×C151—22no—
210C22⋊C4×C151—21no—
211C22⋊C4×C151—22no—
212C22⋊C4×C151—41no—
213C42×C151—21no—
214C42×C151—21no—
215C42×C151—21no—
216C42×C151—41no—
217C42×C151C2×D411yescharacteristic
218C42×C151C2×D411yescharacteristic
219C42×C151C2×D411yescharacteristic
220C42×C151C2411yescharacteristic
221C23×C151—21no—
222C23×C151—22no—
223C23×C151—24no—
224C23×C151—44no—
225C23×C151—48no—
226C23×C151—82no—
227C23×C151—82no—
228C23×C151—84no—
229C23×C151—84no—
230C23×C151#12011yescharacteristic
231Q8×C151—21no—
232Q8×C151—42no—
233Q8×C151—41no—
234Q8×C151—41no—
235Q8×C151—41no—
236Q8×C151—41no—
237Q8×C151AGL(1,ℤ/8)12yes—
238D4×C151—24no—
239D4×C151—21no—
240D4×C151—22no—
241D4×C151—22no—
242D4×C151—44no—
243D4×C151—44no—
244D4×C151—42no—
245D4×C151—48no—
246D4×C151—44no—
247D4×C151—48no—
248D4×C151—42no—
249D4×C151—42no—
250D4×C151—42no—
251D4×C151—42no—
252D4×C151—44no—
253D4×C151—42no—
254D4×C151—82no—
255D4×C151—84no—
256D4×C151—84no—
257D4×C151AGL(1,ℤ/8)12yes—
258C2×C4×C151—24no—
259C2×C4×C151—21no—
260C2×C4×C151—21no—
261C2×C4×C151—21no—
262C2×C4×C151—21no—
263C2×C4×C151—21no—
264C2×C4×C151—22no—
265C2×C4×C151—22no—
266C2×C4×C151—21no—
267C2×C4×C151—21no—
268C2×C4×C151—48no—
269C2×C4×C151—42no—
270C2×C4×C151—42no—
271C2×C4×C151—41no—
272C2×C4×C151—41no—
273C2×C4×C151—42no—
274C2×C4×C151—41no—
275C2×C4×C151—41no—
276C2×C4×C151#12011yescharacteristic
277C2×C4×C151#12011yescharacteristic
278C2×C4×C151#12011yescharacteristic
279C2×C4×C151C22×D411yescharacteristic
280C2×C4×C151C22×D411yescharacteristic
281C2×C4×C151C22×D411yescharacteristic
282C8×C151—42no—
283C8×C151—41no—
284C8×C151—41no—
285C8×C151—42no—
286C22×C151—21no—
287C22×C151—22no—
288C22×C151—21no—
289C22×C151—21no—
290C22×C151—41no—
291C22×C151—41no—
292C22×C151—44no—
293C22×C151—42no—
294C22×C151—42no—
295C22×C151—44no—
296C22×C151—44no—
297C22×C151—42no—
298C22×C151—164no—
299C22×C151—168no—
300C22×C151C2×#12011yescharacteristic
301C4×C151—21no—
302C4×C151—22no—
303C4×C151—21no—
304C4×C151—21no—
305C4×C151—42no—
306C4×C151—41no—
307C4×C151—41no—
308C4×C151—81no—
309C4×C151—81no—
310C4×C151#45512yes—
311C4×C151C2×AGL(1,ℤ/8)11yescharacteristic
312C4×C151C2×AGL(1,ℤ/8)11yescharacteristic
313C4×C151#57611yescharacteristic
314C4×C151#57711yescharacteristic
315C2×C151—41no—
316C2×C151—42no—
317C2×C151—82no—
318C2×C151—84no—
319C2×C151#300311yescharacteristic
320C2×C151#301211yescharacteristic
321C2×C151C2×#45511yescharacteristic
322#18574C15111yesmaximal, characteristic, direct product factor, Sylow
323C151#1857411yescharacteristic, direct product factor, Sylow
324#3498C2×C15111yescharacteristic
325#3254C2×C15111yescharacteristic, complemented
326#3138C2×C15111yescharacteristic, complemented
327#3107C2×C15111yescharacteristic, complemented
328#1642C2×C15112yescomplemented
329#1641C2×C15112yescomplemented
330#1639C2×C15112yescomplemented
331#1639C2×C15112yescomplemented
332#1545C2×C15111yescharacteristic, complemented
333#1489C2×C15111yescharacteristic, complemented
334#1484C2×C15111yescharacteristic, complemented
335C2×H(5,𝔽2)—21no—
336#550—21no—
337#550C22×C15112yes—
338D42—22no—
339D42—22no—
340D42—21no—
341D42C22×C15112yes—
342#535—22no—
343#535—21no—
344#533C22×C15111yescharacteristic, complemented
345#532—21no—
346#532C22×C15111yescharacteristic
347C2×#127—21no—
348C2×#127C22×C15111yescharacteristic
349C4×#41C22×C15111yescharacteristic
350#490—22no—
351#486C22×C15112yescomplemented
352#486C22×C15112yescomplemented
353#485—21no—
354#477—21no—
355#475C22×C15111yescharacteristic, complemented
356#474C22×C15111yescharacteristic, complemented
357#474C22×C15112yescomplemented
358#468—22no—
359#462C22×C15111yescharacteristic, complemented
360#461C22×C15111yescharacteristic, complemented
361#461C22×C15112yescomplemented
362#460—21no—
363#452—21no—
364C4×Q16—21no—
365C4×SD16—21no—
366#431C22×C15111yescharacteristic, complemented
367#423C22×C15111yescharacteristic, complemented
368#418—22no—
369C4×M16C22×C15111yescharacteristic, complemented
370#331C22×C15112yescomplemented
371#331C22×C15112yescomplemented
372Q8⋊C8C22×C15112yescomplemented
373D4⋊C8C22×C15112yescomplemented
374D4⋊C8C22×C15112yescomplemented
375D4⋊C8C22×C15112yescomplemented
376H(5,𝔽2)—44no—
377H(5,𝔽2)—44no—
378C2×#41—22no—
379C2×#41—22no—
380C2×#41—21no—
381C2×#41D4×C15111yescharacteristic
382C22×D4—22no—
383C22×D4—24no—
384C22×D4—22no—
385C22×D4—22no—
386C22×D4—22no—
387C22×D4—21no—
388C22×D4—21no—
389C22×D4D4×C15112yes—
390C2×Q16—21no—
391C2×SD16—21no—
392C2×SD16—22no—
393C2×M16—21no—
394C2×M16—21no—
395C4⋊Q8D4×C15111yescharacteristic
396C4⋊Q8C23×C15111yescharacteristic
397C4⋊Q8C23×C15111yescharacteristic
398#127—22no—
399#127—22no—
400#127—22no—
401#127—21no—
402#127—21no—
403#127D4×C15111yescharacteristic
404#127D4×C15112yes—
405#127C23×C15112yes—
406#125—21no—
407#124—22no—
408#124—21no—
409C22⋊Q8—21no—
410C22⋊Q8—21no—
411#121—22no—
412#121—24no—
413#121—24no—
414#121—21no—
415#121—22no—
416#121—22no—
417#121—22no—
418#121D4×C15112yes—
419#120—24no—
420#120—24no—
421#120—24no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8D4×C15111yescharacteristic
425C4×Q8C23×C15111yescharacteristic
426C4×D4—22no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—22no—
430C4×D4—21no—
431C4×D4D4×C15111yescharacteristic
432C4×D4C23×C15111yescharacteristic
433C4×D4C23×C15112yes—
434#117—21no—
435#117—21no—
436#117D4×C15111yescharacteristic
437C2×C42—21no—
438C2×C42C23×C15111yescharacteristic
439#107—21no—
440#107—22no—
441#106—21no—
442C4⋊C8—21no—
443C4⋊C8C23×C15111yescharacteristic
444C4⋊C8C23×C15111yescharacteristic
445C4⋊C8C23×C15112yes—
446#103—21no—
447#103—21no—
448#103—21no—
449#103—22no—
450#102—24no—
451#102—24no—
452#102—21no—
453#102—22no—
454C22⋊C8—21no—
455#97—21no—
456C4×C8—21no—
457C4×C8C23×C15112yes—
458C24—24no—
459C24—22no—
460#41—22no—
461#41—22no—
462#41—48no—
463#41—42no—
464#41—42no—
465#41—44no—
466#41—44no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8C2×D4×C15111yescharacteristic
471C2×D4—21no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—48no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—44no—
488C2×D4—416no—
489C2×D4—44no—
490C2×D4—44no—
491C2×D4—81no—
492C2×D4—81no—
493C2×D4—82no—
494C2×D4—82no—
495C2×D4—84no—
496C2×D4C2×D4×C15112yes—
497C2×D4C2×D4×C15111yescharacteristic
498C2×D4C2×D4×C15114yes—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4C2×D4×C15111yescharacteristic
505C22×C4C2×D4×C15111yescharacteristic
506C22×C4C2×D4×C15111yescharacteristic
507Q16—42no—
508SD16—42no—
509SD16—82no—
510M16—42no—
511M16—81no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—22no—
515C2×C8—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4C2×D4×C15111yescharacteristic
526C4⋊C4C2×D4×C15111yescharacteristic
527C4⋊C4C2×D4×C15112yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—22no—
535C22⋊C4—41no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—41no—
540C42C2×D4×C15111yescharacteristic
541C42C2×D4×C15111yescharacteristic
542C42C2×D4×C15111yescharacteristic
543C42C24×C15111yescharacteristic
544C23—21no—
545C23—22no—
546C23—24no—
547C23—44no—
548C23—48no—
549C23—82no—
550C23—82no—
551C23—84no—
552C23—84no—
553C23#120×C15111yescharacteristic
554Q8—21no—
555Q8—42no—
556Q8—41no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8AGL(1,ℤ/8)×C15112yes—
561D4—24no—
562D4—21no—
563D4—22no—
564D4—22no—
565D4—44no—
566D4—44no—
567D4—42no—
568D4—48no—
569D4—44no—
570D4—48no—
571D4—44no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—82no—
578D4—84no—
579D4—84no—
580D4AGL(1,ℤ/8)×C15112yes—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4#120×C15111yescharacteristic
600C2×C4#120×C15111yescharacteristic
601C2×C4#120×C15111yescharacteristic
602C2×C4C22×D4×C15111yescharacteristic
603C2×C4C22×D4×C15111yescharacteristic
604C2×C4C22×D4×C15111yescharacteristic
605C8—42no—
606C8—41no—
607C8—41no—
608C8—42no—
609C22—21no—
610C22—22no—
611C22—21no—
612C22—21no—
613C22—41no—
614C22—41no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—164no—
622C22—168no—
623C22C2×#120×C15111yescharacteristic
624C4—21no—
625C4—22no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4#455×C15112yes—
634C4C2×AGL(1,ℤ/8)×C15111yescharacteristic
635C4C2×AGL(1,ℤ/8)×C15111yescharacteristic
636C4#576×C15111yescharacteristic
637C4#577×C15111yescharacteristic
638C2—41no—
639C2—42no—
640C2—82no—
641C2—84no—
642C2#3003×C15111yescharacteristic
643C2#3012×C15111yescharacteristic
644C2C2×#455×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC42C24×C151
prime radical#3498×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
2. #3254×C151C2semidirect
3. #3138×C151C2semidirect
4. #3107×C151C2semidirect
5. #1642×C151C2semidirect
6. #1641×C151C2semidirect
7. #1639×C151C2semidirect
8. #1639×C151C2semidirect
9. #1545×C151C2semidirect
10. #1489×C151C2semidirect
11. #1484×C151C2semidirect
21. #533×C151C22semidirect
28. #486×C151C22semidirect
29. #486×C151C22semidirect
32. #475×C151C22semidirect
33. #474×C151C22semidirect
34. #474×C151C22semidirect
36. #462×C151C22semidirect
37. #461×C151C22semidirect
38. #461×C151C22semidirect
43. #431×C151C22semidirect
44. #423×C151C22semidirect
46. C4×M16×C151C22semidirect
47. #331×C151C22semidirect
48. #331×C151C22semidirect
49. Q8⋊C8×C151C22semidirect
50. D4⋊C8×C151C22semidirect
51. D4⋊C8×C151C22semidirect
52. D4⋊C8×C151C22semidirect
322. #18574C151direct (N × H)
323. C151#18574direct (N × H)
325. #3254C2×C151semidirect
326. #3138C2×C151semidirect
327. #3107C2×C151semidirect
328. #1642C2×C151semidirect
329. #1641C2×C151semidirect
330. #1639C2×C151semidirect
331. #1639C2×C151semidirect
332. #1545C2×C151semidirect
333. #1489C2×C151semidirect
334. #1484C2×C151semidirect
344. #533C22×C151semidirect
351. #486C22×C151semidirect
352. #486C22×C151semidirect
355. #475C22×C151semidirect
356. #474C22×C151semidirect
357. #474C22×C151semidirect
359. #462C22×C151semidirect
360. #461C22×C151semidirect
361. #461C22×C151semidirect
366. #431C22×C151semidirect
367. #423C22×C151semidirect
369. C4×M16C22×C151semidirect
370. #331C22×C151semidirect
371. #331C22×C151semidirect
372. Q8⋊C8C22×C151semidirect
373. D4⋊C8C22×C151semidirect
374. D4⋊C8C22×C151semidirect
375. D4⋊C8C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3498×C1512
#3254×C1512
#3138×C1512
#3107×C1512
#1642×C1512
#1641×C1512
#1639×C1512
#1545×C1512
#1489×C1512
#1484×C1512
#18574151