Finite Group Databasebeta

← C151×#24176 #50329101

Subgroups — C151×#24176 #50329101

Order 38656.

1238 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506×C151C211yesmaximal, characteristic, complemented
2#3077×C151C211yesmaximal, characteristic, complemented
3#3036×C151C211yesmaximal, characteristic, complemented
4C4×#127×C151C211yesmaximal, characteristic, complemented
5#2037×C151C212yesmaximal, complemented
6#2002×C151C212yesmaximal, complemented
7#1994×C151C211yesmaximal, characteristic, complemented
8#1962×C151C211yesmaximal, characteristic, complemented
9#1883×C151C212yesmaximal, complemented
10#1859×C151C212yesmaximal, complemented
11#1800×C151C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C151C2212yes—
13#559×C151—22no—
14#556×C151—21no—
15#555×C151—21no—
16#553×C151—22no—
17#550×C151—21no—
18#547×C151—21no—
19#538×C151—22no—
20#535×C151C2212yes—
21#533×C151C2211yescharacteristic
22#532×C151C2211yescharacteristic
23C2×#127×C151C2211yescharacteristic
24#525×C151—22no—
25#520×C151C2212yes—
26C2×C4×D4×C151—22no—
27C2×C4×D4×C151C2211yescharacteristic
28C2×C4×D4×C151C2211yescharacteristic
29#478×C151—22no—
30#476×C151—21no—
31#474×C151—21no—
32#463×C151—22no—
33#461×C151—21no—
34#459×C151—21no—
35#428×C151C2212yes—
36#423×C151C2211yescharacteristic
37#423×C151C2211yescharacteristic
38C2×#104×C151C2212yes—
39C2×#104×C151C2212yes—
40#418×C151C2212yes—
41#417×C151C2212yes—
42C2×#100×C151C2212yes—
43#390×C151—22no—
44#390×C151C2211yescharacteristic
45#390×C151C2211yescharacteristic
46#384×C151C2211yescharacteristic
47C43×C151C2211yescharacteristic
48#343×C151C2212yes—
49#339×C151C2212yes—
50#339×C151C2212yes—
51C2×#41×C151—22no—
52C2×#41×C151C2312yes—
53C22×D4×C151—22no—
54C22×D4×C151C2311yescharacteristic
55C22×D4×C151C2311yescharacteristic
56C23×C4×C151—21no—
57C23×C4×C151D412yes—
58AGL(1,ℤ/8)×C151—24no—
59AGL(1,ℤ/8)×C151—42no—
60C2×SD16×C151—22no—
61C2×D8×C151—22no—
62C2×M16×C151C2312yes—
63C2×M16×C151C2312yes—
64C4⋊Q8×C151D411yescharacteristic, complemented
65C4⋊Q8×C151D411yescharacteristic, complemented
66C4⋊Q8×C151D411yescharacteristic, complemented
67#127×C151—22no—
68#127×C151—22no—
69#127×C151—22no—
70#127×C151D411yescharacteristic, complemented
71#126×C151—24no—
72#125×C151D412yescomplemented
73#124×C151—24no—
74#124×C151—22no—
75#124×C151D412yescomplemented
76#123×C151—24no—
77#123×C151—24no—
78C22⋊Q8×C151—22no—
79C22⋊Q8×C151—21no—
80C22⋊Q8×C151—21no—
81#121×C151—22no—
82#121×C151—24no—
83#121×C151—21no—
84#121×C151—21no—
85#121×C151—22no—
86#121×C151—22no—
87#120×C151—24no—
88C4×Q8×C151—21no—
89C4×Q8×C151—21no—
90C4×Q8×C151—22no—
91C4×D4×C151—24no—
92C4×D4×C151—22no—
93C4×D4×C151—24no—
94C4×D4×C151—21no—
95C4×D4×C151—21no—
96C4×D4×C151—21no—
97C4×D4×C151—21no—
98C4×D4×C151—21no—
99C4×D4×C151—21no—
100C4×D4×C151—21no—
101C4×D4×C151—21no—
102C4×D4×C151—21no—
103C4×D4×C151—21no—
104C4×D4×C151—48no—
105#117×C151—22no—
106#117×C151—22no—
107#117×C151C2312yes—
108C2×C4⋊C4×C151—22no—
109C2×C4⋊C4×C151—22no—
110C2×C4⋊C4×C151C2311yescharacteristic
111C2×C4⋊C4×C151C2311yescharacteristic
112C2×C22⋊C4×C151—21no—
113C2×C22⋊C4×C151—22no—
114C2×C22⋊C4×C151—22no—
115C2×C22⋊C4×C151D412yes—
116C2×C42×C151—21no—
117C2×C42×C151—21no—
118C2×C42×C151C2311yescharacteristic
119C2×C42×C151C2311yescharacteristic
120C2×C42×C151C2311yescharacteristic
121#107×C151—22no—
122#106×C151—22no—
123C4⋊C8×C151—21no—
124C4⋊C8×C151—21no—
125C4⋊C8×C151—21no—
126C4⋊C8×C151—21no—
127#104×C151—24no—
128#104×C151—42no—
129#103×C151—22no—
130#102×C151—22no—
131#102×C151—22no—
132#102×C151—22no—
133#100×C151—42no—
134C24×C151—21no—
135C24×C151C2×D412yes—
136#41×C151—22no—
137#41×C151—22no—
138#41×C151—22no—
139#41×C151—22no—
140#41×C151—84no—
141C2×Q8×C151—21no—
142C2×Q8×C151—21no—
143C2×Q8×C151C2×D412yes—
144C2×D4×C151—28no—
145C2×D4×C151—22no—
146C2×D4×C151—24no—
147C2×D4×C151—24no—
148C2×D4×C151—21no—
149C2×D4×C151—21no—
150C2×D4×C151—21no—
151C2×D4×C151—21no—
152C2×D4×C151—21no—
153C2×D4×C151—21no—
154C2×D4×C151—21no—
155C2×D4×C151—21no—
156C2×D4×C151—21no—
157C2×D4×C151—21no—
158C2×D4×C151—21no—
159C2×D4×C151—21no—
160C2×D4×C151—21no—
161C2×D4×C151—44no—
162C2×D4×C151—44no—
163C2×D4×C151C2×D412yes—
164C2×D4×C151#4112yescomplemented
165C22×C4×C151—21no—
166C22×C4×C151—21no—
167C22×C4×C151—24no—
168C22×C4×C151—22no—
169C22×C4×C151—22no—
170C22×C4×C151—22no—
171C22×C4×C151—21no—
172C22×C4×C151—44no—
173C22×C4×C151—44no—
174C22×C4×C151—41no—
175C22×C4×C151—41no—
176C22×C4×C151C2×D411yescharacteristic
177C22×C4×C151C2×D412yes—
178C22×C4×C151C2×D411yescharacteristic
179C22×C4×C151C2411yescharacteristic
180SD16×C151—44no—
181D8×C151—44no—
182M16×C151—22no—
183M16×C151—22no—
184M16×C151—22no—
185M16×C151—22no—
186C2×C8×C151—22no—
187C2×C8×C151—22no—
188C4⋊C4×C151—24no—
189C4⋊C4×C151—21no—
190C4⋊C4×C151—21no—
191C4⋊C4×C151—22no—
192C4⋊C4×C151—21no—
193C4⋊C4×C151—22no—
194C4⋊C4×C151—21no—
195C4⋊C4×C151—21no—
196C4⋊C4×C151—21no—
197C4⋊C4×C151—21no—
198C4⋊C4×C151—21no—
199C4⋊C4×C151—44no—
200C4⋊C4×C151C2×D412yes—
201C4⋊C4×C151C2×D412yes—
202C22⋊C4×C151—24no—
203C22⋊C4×C151—24no—
204C22⋊C4×C151—24no—
205C22⋊C4×C151—22no—
206C22⋊C4×C151—22no—
207C22⋊C4×C151—22no—
208C22⋊C4×C151—22no—
209C22⋊C4×C151—22no—
210C22⋊C4×C151—44no—
211C22⋊C4×C151—44no—
212C22⋊C4×C151—41no—
213C22⋊C4×C151—41no—
214C42×C151—24no—
215C42×C151—24no—
216C42×C151—22no—
217C42×C151—21no—
218C42×C151—21no—
219C42×C151—21no—
220C42×C151—42no—
221C42×C151C2×D411yescharacteristic
222C42×C151C2×D411yescharacteristic
223C42×C151C2×D411yescharacteristic
224C42×C151C2×D411yescharacteristic
225C42×C151#4112yescomplemented
226C23×C151—22no—
227C23×C151—22no—
228C23×C151—22no—
229C23×C151—22no—
230C23×C151—22no—
231C23×C151—42no—
232C23×C151—42no—
233C23×C151—42no—
234C23×C151—42no—
235C23×C151—41no—
236C23×C151—41no—
237C23×C151C22×D411yescharacteristic
238Q8×C151—22no—
239Q8×C151—22no—
240Q8×C151—42no—
241Q8×C151—81no—
242D4×C151—22no—
243D4×C151—22no—
244D4×C151—22no—
245D4×C151—22no—
246D4×C151—416no—
247D4×C151—42no—
248D4×C151—42no—
249D4×C151—41no—
250D4×C151—41no—
251D4×C151—41no—
252D4×C151—41no—
253D4×C151—41no—
254D4×C151—41no—
255D4×C151—42no—
256D4×C151—81no—
257D4×C151—84no—
258D4×C151—84no—
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—22no—
266C2×C4×C151—22no—
267C2×C4×C151—22no—
268C2×C4×C151—41no—
269C2×C4×C151—41no—
270C2×C4×C151—41no—
271C2×C4×C151—42no—
272C2×C4×C151—42no—
273C2×C4×C151—44no—
274C2×C4×C151—42no—
275C2×C4×C151—42no—
276C2×C4×C151—41no—
277C2×C4×C151—82no—
278C2×C4×C151—82no—
279C2×C4×C151—84no—
280C2×C4×C151—82no—
281C2×C4×C151#12011yescharacteristic
282C2×C4×C151#12011yescharacteristic
283C2×C4×C151#12011yescharacteristic
284C2×C4×C151#12011yescharacteristic
285C2×C4×C151C22×D411yescharacteristic
286C2×C4×C151C22×D411yescharacteristic
287C2×C4×C151C2×#4111yescharacteristic
288C2×C4×C151C2×#4111yescharacteristic
289C2×C4×C151H(5,𝔽2)11yescharacteristic
290C2×C4×C1512−1+411yescharacteristic
291C8×C151—42no—
292C8×C151—42no—
293C22×C151—21no—
294C22×C151—21no—
295C22×C151—44no—
296C22×C151—44no—
297C22×C151—42no—
298C22×C151—42no—
299C22×C151—42no—
300C22×C151—42no—
301C22×C151—42no—
302C22×C151—42no—
303C22×C151—42no—
304C22×C151—82no—
305C22×C151—82no—
306C22×C151—81no—
307C22×C151—81no—
308C22×C151C2×#12011yescharacteristic
309C22×C151#52511yescharacteristic
310C22×C151#53711yescharacteristic
311C4×C151—24no—
312C4×C151—21no—
313C4×C151—21no—
314C4×C151—21no—
315C4×C151—42no—
316C4×C151—42no—
317C4×C151—42no—
318C4×C151—42no—
319C4×C151—42no—
320C4×C151—81no—
321C4×C151#45312yes—
322C4×C151#54611yescharacteristic
323C4×C151#54711yescharacteristic
324C4×C151#54811yescharacteristic
325C4×C151#55011yescharacteristic
326C2×C151—21no—
327C2×C151—21no—
328C2×C151—82no—
329C2×C151—82no—
330C2×C151—82no—
331C2×C151#240711yescharacteristic
332C2×C151C2×#45311yescharacteristic
333C2×C151#301811yescharacteristic
334#24176C15111yesmaximal, characteristic, direct product factor, Sylow
335C151#2417611yescharacteristic, direct product factor, Sylow
336#3506C2×C15111yescharacteristic, complemented
337#3077C2×C15111yescharacteristic, complemented
338#3036C2×C15111yescharacteristic, complemented
339C4×#127C2×C15111yescharacteristic, complemented
340#2037C2×C15112yescomplemented
341#2002C2×C15112yescomplemented
342#1994C2×C15111yescharacteristic, complemented
343#1962C2×C15111yescharacteristic, complemented
344#1883C2×C15112yescomplemented
345#1859C2×C15112yescomplemented
346#1800C2×C15111yescharacteristic, complemented
347C2×AGL(1,ℤ/8)C22×C15112yes—
348#559—22no—
349#556—21no—
350#555—21no—
351#553—22no—
352#550—21no—
353#547—21no—
354#538—22no—
355#535C22×C15112yes—
356#533C22×C15111yescharacteristic
357#532C22×C15111yescharacteristic
358C2×#127C22×C15111yescharacteristic
359#525—22no—
360#520C22×C15112yes—
361C2×C4×D4—22no—
362C2×C4×D4C22×C15111yescharacteristic
363C2×C4×D4C22×C15111yescharacteristic
364#478—22no—
365#476—21no—
366#474—21no—
367#463—22no—
368#461—21no—
369#459—21no—
370#428C22×C15112yes—
371#423C22×C15111yescharacteristic
372#423C22×C15111yescharacteristic
373C2×#104C22×C15112yes—
374C2×#104C22×C15112yes—
375#418C22×C15112yes—
376#417C22×C15112yes—
377C2×#100C22×C15112yes—
378#390—22no—
379#390C22×C15111yescharacteristic
380#390C22×C15111yescharacteristic
381#384C22×C15111yescharacteristic
382C43C22×C15111yescharacteristic
383#343C22×C15112yes—
384#339C22×C15112yes—
385#339C22×C15112yes—
386C2×#41—22no—
387C2×#41C23×C15112yes—
388C22×D4—22no—
389C22×D4C23×C15111yescharacteristic
390C22×D4C23×C15111yescharacteristic
391C23×C4—21no—
392C23×C4D4×C15112yes—
393AGL(1,ℤ/8)—24no—
394AGL(1,ℤ/8)—42no—
395C2×SD16—22no—
396C2×D8—22no—
397C2×M16C23×C15112yes—
398C2×M16C23×C15112yes—
399C4⋊Q8D4×C15111yescharacteristic, complemented
400C4⋊Q8D4×C15111yescharacteristic, complemented
401C4⋊Q8D4×C15111yescharacteristic, complemented
402#127—22no—
403#127—22no—
404#127—22no—
405#127D4×C15111yescharacteristic, complemented
406#126—24no—
407#125D4×C15112yescomplemented
408#124—24no—
409#124—22no—
410#124D4×C15112yescomplemented
411#123—24no—
412#123—24no—
413C22⋊Q8—22no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416#121—22no—
417#121—24no—
418#121—21no—
419#121—21no—
420#121—22no—
421#121—22no—
422#120—24no—
423C4×Q8—22no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×D4—24no—
427C4×D4—22no—
428C4×D4—24no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—48no—
440#117—22no—
441#117—22no—
442#117C23×C15112yes—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—22no—
445C2×C4⋊C4C23×C15111yescharacteristic
446C2×C4⋊C4C23×C15111yescharacteristic
447C2×C22⋊C4—22no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4D4×C15112yes—
451C2×C42—21no—
452C2×C42—21no—
453C2×C42C23×C15111yescharacteristic
454C2×C42C23×C15111yescharacteristic
455C2×C42C23×C15111yescharacteristic
456#107—22no—
457#106—22no—
458C4⋊C8—21no—
459C4⋊C8—21no—
460C4⋊C8—21no—
461C4⋊C8—21no—
462#104—24no—
463#104—42no—
464#103—22no—
465#102—22no—
466#102—22no—
467#102—22no—
468#100—42no—
469C24—21no—
470C24C2×D4×C15112yes—
471#41—22no—
472#41—22no—
473#41—22no—
474#41—22no—
475#41—84no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8C2×D4×C15112yes—
479C2×D4—28no—
480C2×D4—22no—
481C2×D4—24no—
482C2×D4—24no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—44no—
497C2×D4—44no—
498C2×D4C2×D4×C15112yes—
499C2×D4#41×C15112yescomplemented
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—41no—
510C22×C4—41no—
511C22×C4C2×D4×C15111yescharacteristic
512C22×C4C2×D4×C15112yes—
513C22×C4C2×D4×C15111yescharacteristic
514C22×C4C24×C15111yescharacteristic
515SD16—44no—
516D8—44no—
517M16—22no—
518M16—22no—
519M16—22no—
520M16—22no—
521C2×C8—22no—
522C2×C8—22no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—44no—
535C4⋊C4C2×D4×C15112yes—
536C4⋊C4C2×D4×C15112yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C42—24no—
550C42—24no—
551C42—22no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42C2×D4×C15111yescharacteristic
557C42C2×D4×C15111yescharacteristic
558C42C2×D4×C15111yescharacteristic
559C42C2×D4×C15111yescharacteristic
560C42#41×C15112yescomplemented
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23C22×D4×C15111yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—42no—
576Q8—81no—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—416no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—81no—
592D4—84no—
593D4—84no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—84no—
615C2×C4—82no—
616C2×C4#120×C15111yescharacteristic
617C2×C4#120×C15111yescharacteristic
618C2×C4#120×C15111yescharacteristic
619C2×C4#120×C15111yescharacteristic
620C2×C4C22×D4×C15111yescharacteristic
621C2×C4C22×D4×C15111yescharacteristic
622C2×C4C2×#41×C15111yescharacteristic
623C2×C4C2×#41×C15111yescharacteristic
624C2×C4H(5,𝔽2)×C15111yescharacteristic
625C2×C42−1+4×C15111yescharacteristic
626C8—42no—
627C8—42no—
628C22—21no—
629C22—21no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22C2×#120×C15111yescharacteristic
644C22#525×C15111yescharacteristic
645C22#537×C15111yescharacteristic
646C4—24no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—81no—
656C4#453×C15112yes—
657C4#546×C15111yescharacteristic
658C4#547×C15111yescharacteristic
659C4#548×C15111yescharacteristic
660C4#550×C15111yescharacteristic
661C2—21no—
662C2—21no—
663C2—82no—
664C2—82no—
665C2—82no—
666C2#2407×C15111yescharacteristic
667C2C2×#453×C15111yescharacteristic
668C2#3018×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3506×C151C2semidirect
2. #3077×C151C2semidirect
3. #3036×C151C2semidirect
4. C4×#127×C151C2semidirect
5. #2037×C151C2semidirect
6. #2002×C151C2semidirect
7. #1994×C151C2semidirect
8. #1962×C151C2semidirect
9. #1883×C151C2semidirect
10. #1859×C151C2semidirect
11. #1800×C151C2semidirect
64. C4⋊Q8×C151D4semidirect
65. C4⋊Q8×C151D4semidirect
66. C4⋊Q8×C151D4semidirect
70. #127×C151D4semidirect
72. #125×C151D4semidirect
75. #124×C151D4semidirect
164. C2×D4×C151#41semidirect
225. C42×C151#41semidirect
334. #24176C151direct (N × H)
335. C151#24176direct (N × H)
336. #3506C2×C151semidirect
337. #3077C2×C151semidirect
338. #3036C2×C151semidirect
339. C4×#127C2×C151semidirect
340. #2037C2×C151semidirect
341. #2002C2×C151semidirect
342. #1994C2×C151semidirect
343. #1962C2×C151semidirect
344. #1883C2×C151semidirect
345. #1859C2×C151semidirect
346. #1800C2×C151semidirect
399. C4⋊Q8D4×C151semidirect
400. C4⋊Q8D4×C151semidirect
401. C4⋊Q8D4×C151semidirect
405. #127D4×C151semidirect
407. #125D4×C151semidirect
410. #124D4×C151semidirect
499. C2×D4#41×C151semidirect
560. C42#41×C151semidirect

Maximal subgroups

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

subgroupindex
#3506×C1512
#3077×C1512
#3036×C1512
C4×#127×C1512
#2037×C1512
#2002×C1512
#1994×C1512
#1962×C1512
#1883×C1512
#1859×C1512
#1800×C1512
#24176151