Finite Group Databasebeta

← C151×#24419 #50329344

Subgroups — C151×#24419 #50329344

Order 38656.

1360 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3486×C151C211yesmaximal, characteristic
2#3056×C151C212yesmaximal, complemented
3C2×#456×C151C211yesmaximal, characteristic, complemented
4C22×#101×C151C211yesmaximal, characteristic, complemented
5#2434×C151C211yesmaximal, characteristic, complemented
6#2038×C151C212yesmaximal, complemented
7#2005×C151C212yesmaximal, complemented
8#1886×C151C211yesmaximal, characteristic, complemented
9#1871×C151C212yesmaximal, complemented
10#1857×C151C211yesmaximal, characteristic, complemented
11#1793×C151C211yesmaximal, characteristic, complemented
12C2×2−1+4×C151C2211yescharacteristic
13C23×Q8×C151C2211yescharacteristic
14C2×#137×C151C2212yes—
15C22×M16×C151C2212yes—
16D4×Q8×C151—22no—
17D4×Q8×C151—22no—
18#548×C151—22no—
19#548×C151—22no—
20#536×C151—22no—
21#536×C151C2212yes—
22C2×C4⋊Q8×C151C2211yescharacteristic
23C2×C4⋊Q8×C151C2211yescharacteristic
24#525×C151—22no—
25#525×C151C2212yes—
26C2×C22⋊Q8×C151C2211yescharacteristic
27#519×C151C2211yescharacteristic
28#470×C151—22no—
29#468×C151—22no—
30#467×C151—22no—
31#465×C151—22no—
32#456×C151—24no—
33#428×C151C2212yes—
34C2×#104×C151C2212yes—
35C2×#104×C151C2212yes—
36#419×C151C2212yes—
37#417×C151C2212yes—
38C2×#101×C151—24no—
39C2×#101×C151C2211yescharacteristic
40C2×#101×C151C2212yes—
41C2×#101×C151C2211yescharacteristic
42#395×C151—22no—
43#391×C151C2211yescharacteristic
44#391×C151C2211yescharacteristic
45#385×C151—22no—
46C4×C22⋊C4×C151C2211yescharacteristic
47#339×C151C2212yes—
48#339×C151C2212yes—
492−1+4×C151—22no—
502−1+4×C151—22no—
512−1+4×C151—42no—
52C2×#41×C151—22no—
53C2×#41×C151—22no—
54C2×#41×C151C2312yes—
55C22×Q8×C151—22no—
56C22×Q8×C151—22no—
57C22×Q8×C151—22no—
58C22×Q8×C151—22no—
59C22×Q8×C151—22no—
60C22×Q8×C151—42no—
61C22×Q8×C151D412yes—
62C22×Q8×C151C2311yescharacteristic
63C22×Q8×C151C2311yescharacteristic
64C22×Q8×C151C2311yescharacteristic
65C23×C4×C151D412yes—
66C23×C4×C151C2311yescharacteristic
67#137×C151—24no—
68#137×C151—42no—
69C2×Q16×C151—22no—
70C2×SD16×C151—22no—
71C2×M16×C151—22no—
72C2×M16×C151—24no—
73C2×M16×C151—42no—
74C2×M16×C151C2312yes—
75C2×M16×C151C2312yes—
76C22×C8×C151—22no—
77C4⋊Q8×C151—24no—
78C4⋊Q8×C151—22no—
79C4⋊Q8×C151—22no—
80C4⋊Q8×C151—22no—
81C4⋊Q8×C151—42no—
82#124×C151—24no—
83#124×C151—22no—
84#123×C151—24no—
85#123×C151—22no—
86C22⋊Q8×C151—22no—
87C22⋊Q8×C151—22no—
88C22⋊Q8×C151—24no—
89C22⋊Q8×C151—24no—
90C22⋊Q8×C151—22no—
91C22⋊Q8×C151—22no—
92C22⋊Q8×C151—22no—
93C22⋊Q8×C151—22no—
94C22⋊Q8×C151D412yes—
95C22⋊Q8×C151D412yes—
96#121×C151—22no—
97#121×C151—22no—
98#121×C151D412yes—
99#121×C151D412yes—
100#120×C151—22no—
101C4×Q8×C151—22no—
102C4×Q8×C151—22no—
103C4×D4×C151—24no—
104C4×D4×C151—24no—
105C4×D4×C151—22no—
106C4×D4×C151—22no—
107#117×C151—22no—
108#117×C151C2312yes—
109C2×C4⋊C4×C151—22no—
110C2×C4⋊C4×C151—22no—
111C2×C4⋊C4×C151—21no—
112C2×C4⋊C4×C151C2311yescharacteristic
113C2×C22⋊C4×C151—21no—
114C2×C22⋊C4×C151—21no—
115C2×C42×C151C2311yescharacteristic
116C2×C42×C151C2311yescharacteristic
117#107×C151—22no—
118#106×C151—22no—
119#104×C151—24no—
120#104×C151—42no—
121#103×C151—22no—
122#103×C151—22no—
123#103×C151—22no—
124#102×C151—22no—
125#101×C151—24no—
126#101×C151—22no—
127#101×C151D414yescomplemented
128#95×C151—21no—
129#95×C151—22no—
130#95×C151—22no—
131C24×C151C2×D411yescharacteristic
132#41×C151—22no—
133#41×C151—22no—
134#41×C151—22no—
135#41×C151—22no—
136#41×C151—44no—
137#41×C151—44no—
138#41×C151—44no—
139#41×C151—44no—
140C2×Q8×C151—22no—
141C2×Q8×C151—24no—
142C2×Q8×C151—24no—
143C2×Q8×C151—22no—
144C2×Q8×C151—21no—
145C2×Q8×C151—22no—
146C2×Q8×C151—21no—
147C2×Q8×C151—22no—
148C2×Q8×C151—22no—
149C2×Q8×C151—22no—
150C2×Q8×C151—21no—
151C2×Q8×C151—21no—
152C2×Q8×C151—22no—
153C2×Q8×C151—42no—
154C2×Q8×C151—44no—
155C2×Q8×C151—44no—
156C2×Q8×C151—44no—
157C2×Q8×C151—44no—
158C2×Q8×C151—44no—
159C2×Q8×C151—44no—
160C2×Q8×C151C2×D412yes—
161C2×Q8×C151C2×D412yes—
162C2×Q8×C151#4112yes—
163C2×D4×C151—22no—
164C2×D4×C151—22no—
165C2×D4×C151—22no—
166C2×D4×C151—22no—
167C2×D4×C151C2×D412yes—
168C22×C4×C151—21no—
169C22×C4×C151—21no—
170C22×C4×C151—22no—
171C22×C4×C151—24no—
172C22×C4×C151—22no—
173C22×C4×C151—22no—
174C22×C4×C151—21no—
175C22×C4×C151—21no—
176C22×C4×C151—24no—
177C22×C4×C151—22no—
178C22×C4×C151—21no—
179C22×C4×C151—21no—
180C22×C4×C151—42no—
181C22×C4×C151C2×D412yes—
182C22×C4×C151C2×D411yescharacteristic
183C22×C4×C151C2×D412yes—
184C22×C4×C151C2×D412yes—
185C22×C4×C151C2×D412yes—
186C22×C4×C151C2411yescharacteristic
187Q16×C151—44no—
188SD16×C151—44no—
189M16×C151—22no—
190M16×C151—22no—
191M16×C151—22no—
192M16×C151—44no—
193M16×C151C2×D414yes—
194C2×C8×C151—22no—
195C2×C8×C151—22no—
196C2×C8×C151—44no—
197C4⋊C4×C151—22no—
198C4⋊C4×C151—22no—
199C4⋊C4×C151—24no—
200C4⋊C4×C151—24no—
201C4⋊C4×C151—22no—
202C4⋊C4×C151—22no—
203C4⋊C4×C151—22no—
204C4⋊C4×C151—44no—
205C4⋊C4×C151—42no—
206C4⋊C4×C151—44no—
207C4⋊C4×C151C2×D412yes—
208C4⋊C4×C151C2×D412yes—
209C22⋊C4×C151—24no—
210C22⋊C4×C151—24no—
211C22⋊C4×C151—24no—
212C22⋊C4×C151—22no—
213C22⋊C4×C151—22no—
214C22⋊C4×C151—22no—
215C22⋊C4×C151—22no—
216C22⋊C4×C151—22no—
217C22⋊C4×C151—41no—
218C22⋊C4×C151—41no—
219C42×C151—24no—
220C42×C151—22no—
221C42×C151—21no—
222C42×C151—21no—
223C42×C151—22no—
224C23×C151—21no—
225C23×C151—21no—
226C23×C151—22no—
227C23×C151—22no—
228C23×C151—41no—
229C23×C151#12011yescharacteristic
230C23×C151#12011yescharacteristic
231C23×C151C22×D411yescharacteristic
232Q8×C151—24no—
233Q8×C151—24no—
234Q8×C151—22no—
235Q8×C151—22no—
236Q8×C151—48no—
237Q8×C151—44no—
238Q8×C151—42no—
239Q8×C151—42no—
240Q8×C151—42no—
241Q8×C151—42no—
242D4×C151—22no—
243D4×C151—22no—
244D4×C151—44no—
245D4×C151—44no—
246D4×C151—42no—
247D4×C151—42no—
248D4×C151—44no—
249C2×C4×C151—22no—
250C2×C4×C151—21no—
251C2×C4×C151—22no—
252C2×C4×C151—21no—
253C2×C4×C151—22no—
254C2×C4×C151—22no—
255C2×C4×C151—22no—
256C2×C4×C151—22no—
257C2×C4×C151—22no—
258C2×C4×C151—22no—
259C2×C4×C151—22no—
260C2×C4×C151—22no—
261C2×C4×C151—22no—
262C2×C4×C151—22no—
263C2×C4×C151—22no—
264C2×C4×C151—22no—
265C2×C4×C151—44no—
266C2×C4×C151—44no—
267C2×C4×C151—48no—
268C2×C4×C151—41no—
269C2×C4×C151—41no—
270C2×C4×C151—41no—
271C2×C4×C151—41no—
272C2×C4×C151—42no—
273C2×C4×C151—42no—
274C2×C4×C151—42no—
275C2×C4×C151—41no—
276C2×C4×C151—41no—
277C2×C4×C151—41no—
278C2×C4×C151—41no—
279C2×C4×C151#12012yes—
280C2×C4×C151#12114yes—
281C2×C4×C151C22×D411yescharacteristic
282C2×C4×C151C22×D412yes—
283C2×C4×C151C2×#4111yescharacteristic
284C2×C4×C151H(5,𝔽2)12yes—
285C8×C151—24no—
286C8×C151—42no—
287C22×C151—21no—
288C22×C151—21no—
289C22×C151—21no—
290C22×C151—21no—
291C22×C151—21no—
292C22×C151—21no—
293C22×C151—21no—
294C22×C151—42no—
295C22×C151—42no—
296C22×C151—42no—
297C22×C151—42no—
298C22×C151—42no—
299C22×C151#45612yescomplemented
300C22×C151C2×#12011yescharacteristic
301C22×C151C2×#12111yescharacteristic
302C22×C151#53411yescharacteristic
303C4×C151—22no—
304C4×C151—24no—
305C4×C151—42no—
306C4×C151—42no—
307C4×C151—42no—
308C4×C151—42no—
309C4×C151—42no—
310C4×C151—42no—
311C4×C151—81no—
312C4×C151—81no—
313C4×C151D4212yes—
314C4×C151#54612yes—
315C2×C151—21no—
316C2×C151—21no—
317C2×C151—22no—
318C2×C151—41no—
319C2×C151—82no—
320C2×C151#241011yescharacteristic
321C2×C151C2×#45611yescharacteristic
322C2×C151#302011yescharacteristic
323#24419C15111yesmaximal, characteristic, direct product factor, Sylow
324C151#2441911yescharacteristic, direct product factor, Sylow
325#3486C2×C15111yescharacteristic
326#3056C2×C15112yescomplemented
327C2×#456C2×C15111yescharacteristic, complemented
328C22×#101C2×C15111yescharacteristic, complemented
329#2434C2×C15111yescharacteristic, complemented
330#2038C2×C15112yescomplemented
331#2005C2×C15112yescomplemented
332#1886C2×C15111yescharacteristic, complemented
333#1871C2×C15112yescomplemented
334#1857C2×C15111yescharacteristic, complemented
335#1793C2×C15111yescharacteristic, complemented
336C2×2−1+4C22×C15111yescharacteristic
337C23×Q8C22×C15111yescharacteristic
338C2×#137C22×C15112yes—
339C22×M16C22×C15112yes—
340D4×Q8—22no—
341D4×Q8—22no—
342#548—22no—
343#548—22no—
344#536—22no—
345#536C22×C15112yes—
346C2×C4⋊Q8C22×C15111yescharacteristic
347C2×C4⋊Q8C22×C15111yescharacteristic
348#525—22no—
349#525C22×C15112yes—
350C2×C22⋊Q8C22×C15111yescharacteristic
351#519C22×C15111yescharacteristic
352#470—22no—
353#468—22no—
354#467—22no—
355#465—22no—
356#456—24no—
357#428C22×C15112yes—
358C2×#104C22×C15112yes—
359C2×#104C22×C15112yes—
360#419C22×C15112yes—
361#417C22×C15112yes—
362C2×#101—24no—
363C2×#101C22×C15111yescharacteristic
364C2×#101C22×C15112yes—
365C2×#101C22×C15111yescharacteristic
366#395—22no—
367#391C22×C15111yescharacteristic
368#391C22×C15111yescharacteristic
369#385—22no—
370C4×C22⋊C4C22×C15111yescharacteristic
371#339C22×C15112yes—
372#339C22×C15112yes—
3732−1+4—22no—
3742−1+4—22no—
3752−1+4—42no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41C23×C15112yes—
379C22×Q8—22no—
380C22×Q8—22no—
381C22×Q8—22no—
382C22×Q8—22no—
383C22×Q8—22no—
384C22×Q8—42no—
385C22×Q8D4×C15112yes—
386C22×Q8C23×C15111yescharacteristic
387C22×Q8C23×C15111yescharacteristic
388C22×Q8C23×C15111yescharacteristic
389C23×C4D4×C15112yes—
390C23×C4C23×C15111yescharacteristic
391#137—24no—
392#137—42no—
393C2×Q16—22no—
394C2×SD16—22no—
395C2×M16—22no—
396C2×M16—24no—
397C2×M16—42no—
398C2×M16C23×C15112yes—
399C2×M16C23×C15112yes—
400C22×C8—22no—
401C4⋊Q8—24no—
402C4⋊Q8—22no—
403C4⋊Q8—22no—
404C4⋊Q8—22no—
405C4⋊Q8—42no—
406#124—24no—
407#124—22no—
408#123—24no—
409#123—22no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8—24no—
413C22⋊Q8—24no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8D4×C15112yes—
419C22⋊Q8D4×C15112yes—
420#121—22no—
421#121—22no—
422#121D4×C15112yes—
423#121D4×C15112yes—
424#120—22no—
425C4×Q8—22no—
426C4×Q8—22no—
427C4×D4—24no—
428C4×D4—24no—
429C4×D4—22no—
430C4×D4—22no—
431#117—22no—
432#117C23×C15112yes—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—22no—
435C2×C4⋊C4—22no—
436C2×C4⋊C4C23×C15111yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C42C23×C15111yescharacteristic
440C2×C42C23×C15111yescharacteristic
441#107—22no—
442#106—22no—
443#104—24no—
444#104—42no—
445#103—22no—
446#103—22no—
447#103—22no—
448#102—22no—
449#101—22no—
450#101—24no—
451#101D4×C15114yescomplemented
452#95—21no—
453#95—22no—
454#95—22no—
455C24C2×D4×C15111yescharacteristic
456#41—22no—
457#41—22no—
458#41—22no—
459#41—22no—
460#41—44no—
461#41—44no—
462#41—44no—
463#41—44no—
464C2×Q8—22no—
465C2×Q8—24no—
466C2×Q8—24no—
467C2×Q8—22no—
468C2×Q8—21no—
469C2×Q8—22no—
470C2×Q8—21no—
471C2×Q8—22no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—22no—
477C2×Q8—42no—
478C2×Q8—44no—
479C2×Q8—44no—
480C2×Q8—44no—
481C2×Q8—44no—
482C2×Q8—44no—
483C2×Q8—44no—
484C2×Q8C2×D4×C15112yes—
485C2×Q8C2×D4×C15112yes—
486C2×Q8#41×C15112yes—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4C2×D4×C15112yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—42no—
505C22×C4C2×D4×C15112yes—
506C22×C4C2×D4×C15111yescharacteristic
507C22×C4C2×D4×C15112yes—
508C22×C4C2×D4×C15112yes—
509C22×C4C2×D4×C15112yes—
510C22×C4C24×C15111yescharacteristic
511Q16—44no—
512SD16—44no—
513M16—22no—
514M16—22no—
515M16—22no—
516M16—44no—
517M16C2×D4×C15114yes—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—44no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4C2×D4×C15112yes—
532C4⋊C4C2×D4×C15112yes—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C42—24no—
544C42—22no—
545C42—21no—
546C42—21no—
547C42—22no—
548C23—22no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—41no—
553C23#120×C15111yescharacteristic
554C23#120×C15111yescharacteristic
555C23C22×D4×C15111yescharacteristic
556Q8—24no—
557Q8—24no—
558Q8—22no—
559Q8—22no—
560Q8—48no—
561Q8—44no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566D4—22no—
567D4—22no—
568D4—44no—
569D4—44no—
570D4—44no—
571D4—42no—
572D4—42no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4#120×C15112yes—
604C2×C4#121×C15114yes—
605C2×C4C22×D4×C15111yescharacteristic
606C2×C4C22×D4×C15112yes—
607C2×C4C2×#41×C15111yescharacteristic
608C2×C4H(5,𝔽2)×C15112yes—
609C8—24no—
610C8—42no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22#456×C15112yescomplemented
624C22C2×#120×C15111yescharacteristic
625C22C2×#121×C15111yescharacteristic
626C22#534×C15111yescharacteristic
627C4—22no—
628C4—24no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4D42×C15112yes—
638C4#546×C15112yes—
639C2—21no—
640C2—21no—
641C2—22no—
642C2—41no—
643C2—82no—
644C2#2410×C15111yescharacteristic
645C2C2×#456×C15111yescharacteristic
646C2#3020×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
prime radical#3486×C151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3056×C151C2semidirect
3. C2×#456×C151C2semidirect
4. C22×#101×C151C2semidirect
5. #2434×C151C2semidirect
6. #2038×C151C2semidirect
7. #2005×C151C2semidirect
8. #1886×C151C2semidirect
9. #1871×C151C2semidirect
10. #1857×C151C2semidirect
11. #1793×C151C2semidirect
127. #101×C151D4semidirect
299. C22×C151#456semidirect
323. #24419C151direct (N × H)
324. C151#24419direct (N × H)
326. #3056C2×C151semidirect
327. C2×#456C2×C151semidirect
328. C22×#101C2×C151semidirect
329. #2434C2×C151semidirect
330. #2038C2×C151semidirect
331. #2005C2×C151semidirect
332. #1886C2×C151semidirect
333. #1871C2×C151semidirect
334. #1857C2×C151semidirect
335. #1793C2×C151semidirect
451. #101D4×C151semidirect
623. C22#456×C151semidirect

Maximal subgroups

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

subgroupindex
#3486×C1512
#3056×C1512
C2×#456×C1512
C22×#101×C1512
#2434×C1512
#2038×C1512
#2005×C1512
#1886×C1512
#1871×C1512
#1857×C1512
#1793×C1512
#24419151