Finite Group Databasebeta

← C151×#21503 #50326428

Subgroups — C151×#21503 #50326428

Order 38656.

1260 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#459×C151C211yesmaximal, characteristic
2C2×#442×C151C212yesmaximal, complemented
3C4×#127×C151C211yesmaximal, characteristic, complemented
4C42×D4×C151C211yesmaximal, characteristic, complemented
5#1982×C151C211yesmaximal, characteristic, complemented
6#1948×C151C211yesmaximal, characteristic, complemented
7#1926×C151C212yesmaximal, complemented
8#1879×C151C212yesmaximal, complemented
9#1802×C151C211yesmaximal, characteristic, complemented
10#1767×C151C211yesmaximal, characteristic, complemented
11#1762×C151C212yesmaximal, complemented
12C22×D8×C151C2212yes—
13C2×#127×C151C2211yescharacteristic
14C2×C4×D4×C151—21no—
15C2×C4×D4×C151—21no—
16C2×C4×D4×C151C2211yescharacteristic
17C2×C4×D4×C151C2212yes—
18C2×C4×D4×C151C2211yescharacteristic
19C2×C4×D4×C151C2211yescharacteristic
20C22×C42×C151—21no—
21#459×C151C418yescomplemented
22#442×C151—28no—
23C2×#106×C151C2212yes—
24C2×C4⋊C8×C151C2211yescharacteristic
25C2×C4⋊C8×C151C2211yescharacteristic
26C2×#102×C151C2212yes—
27C2×#102×C151C2212yes—
28C2×#102×C151C2212yes—
29C2×#102×C151C2212yes—
30C2×#97×C151C2212yes—
31#390×C151—21no—
32#390×C151—21no—
33#390×C151C2212yes—
34#384×C151C2211yescharacteristic
35C4×C4⋊C4×C151C2211yescharacteristic
36C4×C4⋊C4×C151C2211yescharacteristic
37C4×C4⋊C4×C151C2211yescharacteristic
38C4×C22⋊C4×C151—21no—
39C4×C22⋊C4×C151—21no—
40C4×C22⋊C4×C151—21no—
41C43×C151C2211yescharacteristic
42#340×C151C2212yes—
43#340×C151C2212yes—
44#336×C151C2212yes—
45C22×D4×C151—21no—
46C22×D4×C151—21no—
47C22×D4×C151C2312yes—
48C22×D4×C151C2311yescharacteristic
49C23×C4×C151—21no—
50C23×C4×C151—21no—
51C23×C4×C151—21no—
52C23×C4×C151—22no—
53C2×D8×C151—24no—
54C2×D8×C151—24no—
55C2×D8×C151C2×C418yescomplemented
56C22×C8×C151C2312yes—
57C22×C8×C151C2312yes—
58#127×C151—22no—
59#127×C151C2×C414yescomplemented
60C4×D4×C151—22no—
61C4×D4×C151—22no—
62C4×D4×C151—24no—
63C4×D4×C151—22no—
64C4×D4×C151—22no—
65C4×D4×C151—24no—
66C4×D4×C151—44no—
67C4×D4×C151—44no—
68C4×D4×C151C2×C414yescomplemented
69C4×D4×C151D414yescomplemented
70C2×C4⋊C4×C151—21no—
71C2×C4⋊C4×C151—21no—
72C2×C4⋊C4×C151C2312yes—
73C2×C4⋊C4×C151C2311yescharacteristic
74C2×C4⋊C4×C151C2311yescharacteristic
75C2×C4⋊C4×C151C2311yescharacteristic
76C2×C22⋊C4×C151—21no—
77C2×C22⋊C4×C151—21no—
78C2×C22⋊C4×C151—21no—
79C2×C22⋊C4×C151—22no—
80C2×C22⋊C4×C151—22no—
81C2×C22⋊C4×C151—22no—
82C2×C42×C151—21no—
83C2×C42×C151—21no—
84C2×C42×C151—21no—
85C2×C42×C151—44no—
86C2×C42×C151D411yescharacteristic
87C2×C42×C151D411yescharacteristic
88C2×C42×C151D411yescharacteristic
89C2×C42×C151D411yescharacteristic
90C2×C42×C151C2311yescharacteristic
91C2×C42×C151C2311yescharacteristic
92C2×C42×C151C2311yescharacteristic
93#106×C151—24no—
94C4⋊C8×C151—22no—
95C4⋊C8×C151C2×C414yescomplemented
96#102×C151—24no—
97#102×C151—24no—
98#102×C151—24no—
99#102×C151C2×C418yescomplemented
100#97×C151—24no—
101#95×C151—21no—
102#95×C151—21no—
103#95×C151—21no—
104C24×C151—21no—
105C24×C151—22no—
106C2×D4×C151—24no—
107C2×D4×C151—24no—
108C2×D4×C151—22no—
109C2×D4×C151—21no—
110C2×D4×C151—21no—
111C2×D4×C151—21no—
112C2×D4×C151—22no—
113C2×D4×C151—22no—
114C2×D4×C151—22no—
115C2×D4×C151—22no—
116C2×D4×C151—42no—
117C2×D4×C151—42no—
118C2×D4×C151—42no—
119C2×D4×C151—42no—
120C2×D4×C151C22×C414yes—
121C2×D4×C151C22×C412yes—
122C2×D4×C151C2×D412yes—
123C2×D4×C151#4112yes—
124C22×C4×C151—24no—
125C22×C4×C151—21no—
126C22×C4×C151—21no—
127C22×C4×C151—21no—
128C22×C4×C151—24no—
129C22×C4×C151—24no—
130C22×C4×C151—22no—
131C22×C4×C151—42no—
132C22×C4×C151—42no—
133C22×C4×C151—42no—
134C22×C4×C151—44no—
135C22×C4×C151—44no—
136C22×C4×C151—42no—
137C22×C4×C151—42no—
138C22×C4×C151—42no—
139C22×C4×C151—44no—
140C22×C4×C151C2×D411yescharacteristic
141C22×C4×C151C2×D411yescharacteristic
142C22×C4×C151C2×D411yescharacteristic
143C22×C4×C151C2×D411yescharacteristic
144C22×C4×C151C2×D411yescharacteristic
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2411yescharacteristic
147D8×C151—216no—
148C2×C8×C151—22no—
149C2×C8×C151—22no—
150C2×C8×C151—22no—
151C2×C8×C151—22no—
152C2×C8×C151—22no—
153C2×C8×C151C22×C414yes—
154C4⋊C4×C151—21no—
155C4⋊C4×C151—21no—
156C4⋊C4×C151—21no—
157C4⋊C4×C151—22no—
158C4⋊C4×C151—22no—
159C4⋊C4×C151—42no—
160C4⋊C4×C151—42no—
161C4⋊C4×C151C22×C412yes—
162C4⋊C4×C151C2×D412yes—
163C4⋊C4×C151#4112yes—
164C22⋊C4×C151—24no—
165C22⋊C4×C151—24no—
166C22⋊C4×C151—24no—
167C22⋊C4×C151—44no—
168C22⋊C4×C151—44no—
169C22⋊C4×C151—44no—
170C42×C151—22no—
171C42×C151—22no—
172C42×C151—22no—
173C42×C151—21no—
174C42×C151—21no—
175C42×C151—41no—
176C42×C151—42no—
177C42×C151—42no—
178C42×C151—42no—
179C42×C151C22×C412yes—
180C42×C151C2×D412yes—
181C42×C151#4112yes—
182C42×C151#4112yes—
183C23×C151—22no—
184C23×C151—22no—
185C23×C151—22no—
186C23×C151—24no—
187C23×C151—41no—
188C23×C151—41no—
189C23×C151—41no—
190C23×C151—41no—
191C23×C151—42no—
192C23×C151—42no—
193C23×C151—42no—
194C23×C151—42no—
195C23×C151—42no—
196C23×C151—42no—
197C23×C151C22×D411yescharacteristic
198D4×C151—24no—
199D4×C151—22no—
200D4×C151—24no—
201D4×C151—22no—
202D4×C151—22no—
203D4×C151—48no—
204D4×C151—42no—
205D4×C151—42no—
206D4×C151—48no—
207D4×C151C4×D414yescomplemented
208C2×C4×C151—21no—
209C2×C4×C151—21no—
210C2×C4×C151—21no—
211C2×C4×C151—21no—
212C2×C4×C151—21no—
213C2×C4×C151—21no—
214C2×C4×C151—21no—
215C2×C4×C151—21no—
216C2×C4×C151—22no—
217C2×C4×C151—22no—
218C2×C4×C151—21no—
219C2×C4×C151—22no—
220C2×C4×C151—21no—
221C2×C4×C151—21no—
222C2×C4×C151—48no—
223C2×C4×C151—48no—
224C2×C4×C151—41no—
225C2×C4×C151—41no—
226C2×C4×C151—41no—
227C2×C4×C151—41no—
228C2×C4×C151—41no—
229C2×C4×C151—41no—
230C2×C4×C151—48no—
231C2×C4×C151—42no—
232C2×C4×C151—42no—
233C2×C4×C151—42no—
234C2×C4×C151—88no—
235C2×C4×C151C4×D412yes—
236C2×C4×C151C4×D412yes—
237C2×C4×C151#12112yes—
238C2×C4×C151#12112yes—
239C2×C4×C151AGL(1,ℤ/8)12yes—
240C2×C4×C151AGL(1,ℤ/8)12yes—
241C2×C4×C151AGL(1,ℤ/8)12yes—
242C2×C4×C151C23×C411yescharacteristic
243C2×C4×C151C22×D411yescharacteristic
244C2×C4×C151C2×#4111yescharacteristic
245C2×C4×C151C2×#4111yescharacteristic
246C2×C4×C151C2×#4111yescharacteristic
247C2×C4×C151C2×#4111yescharacteristic
248C8×C151—24no—
249C8×C151—42no—
250C22×C151—24no—
251C22×C151—42no—
252C22×C151—42no—
253C22×C151—42no—
254C22×C151—42no—
255C22×C151—42no—
256C22×C151—42no—
257C22×C151—44no—
258C22×C151—44no—
259C22×C151—44no—
260C22×C151—82no—
261C22×C151—82no—
262C22×C151—82no—
263C22×C151—82no—
264C22×C151C2×C4×D411yescharacteristic
265C22×C151C2×#12111yescharacteristic
266C22×C151#52511yescharacteristic
267C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C151#57511yescharacteristic
271C4×C151—22no—
272C4×C151—22no—
273C4×C151—22no—
274C4×C151—22no—
275C4×C151—42no—
276C4×C151—41no—
277C4×C151—42no—
278C4×C151—42no—
279C4×C151—82no—
280C4×C151#44214yescomplemented
281C4×C151C2×C4×D411yescharacteristic
282C4×C151C4×#4111yescharacteristic
283C4×C151#52911yescharacteristic
284C4×C151#53211yescharacteristic
285C2×C151—44no—
286C2×C151—84no—
287C2×C151C4×#12111yescharacteristic
288C2×C151C2×#44211yescharacteristic
289C2×C151C4×AGL(1,ℤ/8)11yescharacteristic
290C2×C151#303611yescharacteristic
291C2×C151#303811yescharacteristic
292C2×C151#310911yescharacteristic
293C2×C151#311511yescharacteristic
294#21503C15111yesmaximal, characteristic, direct product factor, Sylow
295C151#2150311yescharacteristic, direct product factor, Sylow
296C2×#459C2×C15111yescharacteristic
297C2×#442C2×C15112yescomplemented
298C4×#127C2×C15111yescharacteristic, complemented
299C42×D4C2×C15111yescharacteristic, complemented
300#1982C2×C15111yescharacteristic, complemented
301#1948C2×C15111yescharacteristic, complemented
302#1926C2×C15112yescomplemented
303#1879C2×C15112yescomplemented
304#1802C2×C15111yescharacteristic, complemented
305#1767C2×C15111yescharacteristic, complemented
306#1762C2×C15112yescomplemented
307C22×D8C22×C15112yes—
308C2×#127C22×C15111yescharacteristic
309C2×C4×D4—21no—
310C2×C4×D4—21no—
311C2×C4×D4C22×C15112yes—
312C2×C4×D4C22×C15111yescharacteristic
313C2×C4×D4C22×C15111yescharacteristic
314C2×C4×D4C22×C15111yescharacteristic
315C22×C42—21no—
316#459C4×C15118yescomplemented
317#442—28no—
318C2×#106C22×C15112yes—
319C2×C4⋊C8C22×C15111yescharacteristic
320C2×C4⋊C8C22×C15111yescharacteristic
321C2×#102C22×C15112yes—
322C2×#102C22×C15112yes—
323C2×#102C22×C15112yes—
324C2×#102C22×C15112yes—
325C2×#97C22×C15112yes—
326#390—21no—
327#390—21no—
328#390C22×C15112yes—
329#384C22×C15111yescharacteristic
330C4×C4⋊C4C22×C15111yescharacteristic
331C4×C4⋊C4C22×C15111yescharacteristic
332C4×C4⋊C4C22×C15111yescharacteristic
333C4×C22⋊C4—21no—
334C4×C22⋊C4—21no—
335C4×C22⋊C4—21no—
336C43C22×C15111yescharacteristic
337#340C22×C15112yes—
338#340C22×C15112yes—
339#336C22×C15112yes—
340C22×D4—21no—
341C22×D4—21no—
342C22×D4C23×C15112yes—
343C22×D4C23×C15111yescharacteristic
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—22no—
348C2×D8—24no—
349C2×D8—24no—
350C2×D8C2×C4×C15118yescomplemented
351C22×C8C23×C15112yes—
352C22×C8C23×C15112yes—
353#127—22no—
354#127C2×C4×C15114yescomplemented
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—24no—
360C4×D4—24no—
361C4×D4—44no—
362C4×D4—44no—
363C4×D4C2×C4×C15114yescomplemented
364C4×D4D4×C15114yescomplemented
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4C23×C15112yes—
368C2×C4⋊C4C23×C15111yescharacteristic
369C2×C4⋊C4C23×C15111yescharacteristic
370C2×C4⋊C4C23×C15111yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—44no—
381C2×C42D4×C15111yescharacteristic
382C2×C42D4×C15111yescharacteristic
383C2×C42D4×C15111yescharacteristic
384C2×C42D4×C15111yescharacteristic
385C2×C42C23×C15111yescharacteristic
386C2×C42C23×C15111yescharacteristic
387C2×C42C23×C15111yescharacteristic
388#106—24no—
389C4⋊C8—22no—
390C4⋊C8C2×C4×C15114yescomplemented
391#102—24no—
392#102—24no—
393#102—24no—
394#102C2×C4×C15118yescomplemented
395#97—24no—
396#95—21no—
397#95—21no—
398#95—21no—
399C24—21no—
400C24—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4C22×C4×C15114yes—
416C2×D4C22×C4×C15112yes—
417C2×D4C2×D4×C15112yes—
418C2×D4#41×C15112yes—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4C2×D4×C15111yescharacteristic
436C22×C4C2×D4×C15111yescharacteristic
437C22×C4C2×D4×C15111yescharacteristic
438C22×C4C2×D4×C15111yescharacteristic
439C22×C4C2×D4×C15111yescharacteristic
440C22×C4C2×D4×C15111yescharacteristic
441C22×C4C24×C15111yescharacteristic
442D8—216no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8C22×C4×C15114yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4C22×C4×C15112yes—
457C4⋊C4C2×D4×C15112yes—
458C4⋊C4#41×C15112yes—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—41no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42C22×C4×C15112yes—
475C42C2×D4×C15112yes—
476C42#41×C15112yes—
477C42#41×C15112yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C22×D4×C15111yescharacteristic
493D4—24no—
494D4—22no—
495D4—24no—
496D4—22no—
497D4—22no—
498D4—48no—
499D4—48no—
500D4—42no—
501D4—42no—
502D4C4×D4×C15114yescomplemented
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—88no—
530C2×C4C4×D4×C15112yes—
531C2×C4C4×D4×C15112yes—
532C2×C4#121×C15112yes—
533C2×C4#121×C15112yes—
534C2×C4AGL(1,ℤ/8)×C15112yes—
535C2×C4AGL(1,ℤ/8)×C15112yes—
536C2×C4AGL(1,ℤ/8)×C15112yes—
537C2×C4C23×C4×C15111yescharacteristic
538C2×C4C22×D4×C15111yescharacteristic
539C2×C4C2×#41×C15111yescharacteristic
540C2×C4C2×#41×C15111yescharacteristic
541C2×C4C2×#41×C15111yescharacteristic
542C2×C4C2×#41×C15111yescharacteristic
543C8—24no—
544C8—42no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22C2×C4×D4×C15111yescharacteristic
560C22C2×#121×C15111yescharacteristic
561C22#525×C15111yescharacteristic
562C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
563C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
564C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
565C22#575×C15111yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—42no—
571C4—41no—
572C4—42no—
573C4—42no—
574C4—82no—
575C4#442×C15114yescomplemented
576C4C2×C4×D4×C15111yescharacteristic
577C4C4×#41×C15111yescharacteristic
578C4#529×C15111yescharacteristic
579C4#532×C15111yescharacteristic
580C2—44no—
581C2—84no—
582C2C4×#121×C15111yescharacteristic
583C2C2×#442×C15111yescharacteristic
584C2C4×AGL(1,ℤ/8)×C15111yescharacteristic
585C2#3036×C15111yescharacteristic
586C2#3038×C15111yescharacteristic
587C2#3109×C15111yescharacteristic
588C2#3115×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutatorC2×C4C23×C4×C151
Frattini, cosocleC22×C4C24×C151
prime radicalC2×#459×C151C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C43×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#442×C151C2semidirect
3. C4×#127×C151C2semidirect
4. C42×D4×C151C2semidirect
5. #1982×C151C2semidirect
6. #1948×C151C2semidirect
7. #1926×C151C2semidirect
8. #1879×C151C2semidirect
9. #1802×C151C2semidirect
10. #1767×C151C2semidirect
11. #1762×C151C2semidirect
21. #459×C151C4semidirect
55. C2×D8×C151C2×C4semidirect
59. #127×C151C2×C4semidirect
68. C4×D4×C151C2×C4semidirect
69. C4×D4×C151D4semidirect
95. C4⋊C8×C151C2×C4semidirect
99. #102×C151C2×C4semidirect
207. D4×C151C4×D4semidirect
280. C4×C151#442semidirect
294. #21503C151direct (N × H)
295. C151#21503direct (N × H)
297. C2×#442C2×C151semidirect
298. C4×#127C2×C151semidirect
299. C42×D4C2×C151semidirect
300. #1982C2×C151semidirect
301. #1948C2×C151semidirect
302. #1926C2×C151semidirect
303. #1879C2×C151semidirect
304. #1802C2×C151semidirect
305. #1767C2×C151semidirect
306. #1762C2×C151semidirect
316. #459C4×C151semidirect
350. C2×D8C2×C4×C151semidirect
354. #127C2×C4×C151semidirect
363. C4×D4C2×C4×C151semidirect
364. C4×D4D4×C151semidirect
390. C4⋊C8C2×C4×C151semidirect
394. #102C2×C4×C151semidirect
502. D4C4×D4×C151semidirect
575. C4#442×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#459×C1512
C2×#442×C1512
C4×#127×C1512
C42×D4×C1512
#1982×C1512
#1948×C1512
#1926×C1512
#1879×C1512
#1802×C1512
#1767×C1512
#1762×C1512
#21503151