Finite Group Databasebeta

← C151×#22431 #50327356

Subgroups — C151×#22431 #50327356

Order 38656.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547×C151C211yesmaximal, characteristic
2C2×#436×C151C211yesmaximal, characteristic, complemented
3C2×#417×C151C212yesmaximal, complemented
4D4×C4⋊C4×C151C211yesmaximal, characteristic, complemented
5#1981×C151C211yesmaximal, characteristic, complemented
6#1956×C151C211yesmaximal, characteristic, complemented
7#1894×C151C212yesmaximal, complemented
8#1872×C151C212yesmaximal, complemented
9#1802×C151C211yesmaximal, characteristic, complemented
10#1788×C151C212yesmaximal, complemented
11C4×#102×C151C211yesmaximal, characteristic, complemented
12C22×#41×C151C2212yes—
13C22×M16×C151C2212yes—
14#547×C151—24no—
15#547×C151C418yescomplemented
16C2×C4⋊Q8×C151C2211yescharacteristic
17C2×#123×C151—22no—
18C2×C22⋊Q8×C151C2212yes—
19C2×#121×C151—21no—
20C2×C4×D4×C151—21no—
21C2×C4×D4×C151C2211yescharacteristic
22C2×C4×D4×C151C2211yescharacteristic
23C22×C4⋊C4×C151—21no—
24C22×C4⋊C4×C151C2212yes—
25#436×C151—24no—
26C2×C4⋊C8×C151C2211yescharacteristic
27#417×C151—28no—
28C2×#103×C151C2212yes—
29C2×#103×C151C2212yes—
30C2×#103×C151C2212yes—
31C2×#103×C151C2212yes—
32C2×#102×C151C2212yes—
33C2×#102×C151C2211yescharacteristic
34C2×#102×C151C2211yescharacteristic
35C2×C22⋊C8×C151C2212yes—
36C2×C4×C8×C151C2211yescharacteristic
37#389×C151—21no—
38#385×C151—22no—
39#384×C151C2211yescharacteristic
40#380×C151C2212yes—
41C4×C4⋊C4×C151C2211yescharacteristic
42#340×C151C2211yescharacteristic
43#340×C151C2212yes—
44#340×C151C2211yescharacteristic
45C2×#41×C151—24no—
46C2×#41×C151—24no—
47C2×#41×C151—24no—
48C2×#41×C151—42no—
49C2×#41×C151—42no—
50C2×#41×C151C2×C418yescomplemented
51C22×Q8×C151C2312yes—
52C22×D4×C151—22no—
53C22×D4×C151D411yescharacteristic
54C22×D4×C151C2311yescharacteristic
55C23×C4×C151—21no—
56C23×C4×C151—22no—
57C23×C4×C151—22no—
58C23×C4×C151C2312yes—
59C2×M16×C151—24no—
60C2×M16×C151—24no—
61C2×M16×C151—42no—
62C22×C8×C151C2311yescharacteristic
63C22×C8×C151C2312yes—
64C22×C8×C151C2311yescharacteristic
65C4⋊Q8×C151—22no—
66C4⋊Q8×C151C2×C414yescomplemented
67#123×C151—216no—
68C22⋊Q8×C151—24no—
69C22⋊Q8×C151C2×C418yescomplemented
70#121×C151—28no—
71C4×D4×C151—22no—
72C4×D4×C151—44no—
73C4×D4×C151C2×C414yes—
74C4×D4×C151C2×C414yescomplemented
75C4×D4×C151D414yes—
76C2×C4⋊C4×C151—21no—
77C2×C4⋊C4×C151—22no—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151—28no—
80C2×C4⋊C4×C151—24no—
81C2×C4⋊C4×C151—21no—
82C2×C4⋊C4×C151—22no—
83C2×C4⋊C4×C151—21no—
84C2×C4⋊C4×C151—42no—
85C2×C4⋊C4×C151D411yescharacteristic
86C2×C4⋊C4×C151C2311yescharacteristic
87C2×C4⋊C4×C151C2312yes—
88C2×C4⋊C4×C151C2311yescharacteristic
89C2×C4⋊C4×C151C2311yescharacteristic
90C2×C22⋊C4×C151—21no—
91C2×C22⋊C4×C151—22no—
92C2×C22⋊C4×C151—22no—
93C2×C22⋊C4×C151D412yes—
94C2×C42×C151—21no—
95C2×C42×C151C2311yescharacteristic
96C4⋊C8×C151—22no—
97#103×C151—24no—
98#103×C151—24no—
99#103×C151—42no—
100#103×C151—42no—
101#102×C151—24no—
102#102×C151—22no—
103#102×C151D414yescomplemented
104C22⋊C8×C151—24no—
105C4×C8×C151—22no—
106#95×C151—21no—
107#95×C151—22no—
108C24×C151—21no—
109C24×C151C2×D412yes—
110#41×C151—216no—
111#41×C151—48no—
112#41×C151—48no—
113#41×C151—48no—
114C2×Q8×C151—22no—
115C2×Q8×C151—22no—
116C2×Q8×C151—22no—
117C2×Q8×C151—22no—
118C2×Q8×C151—22no—
119C2×Q8×C151C22×C414yes—
120C2×D4×C151—28no—
121C2×D4×C151—22no—
122C2×D4×C151—22no—
123C2×D4×C151—21no—
124C2×D4×C151—21no—
125C2×D4×C151—21no—
126C2×D4×C151—44no—
127C2×D4×C151—44no—
128C2×D4×C151C22⋊C414yes—
129C2×D4×C151C22×C412yes—
130C2×D4×C151C2×D412yes—
131C2×D4×C151#4112yes—
132C22×C4×C151—22no—
133C22×C4×C151—22no—
134C22×C4×C151—22no—
135C22×C4×C151—22no—
136C22×C4×C151—21no—
137C22×C4×C151—21no—
138C22×C4×C151—22no—
139C22×C4×C151—24no—
140C22×C4×C151—28no—
141C22×C4×C151—21no—
142C22×C4×C151—22no—
143C22×C4×C151—44no—
144C22×C4×C151—44no—
145C22×C4×C151—44no—
146C22×C4×C151—42no—
147C22×C4×C151—42no—
148C22×C4×C151—44no—
149C22×C4×C151—82no—
150C22×C4×C151C22×C414yes—
151C22×C4×C151C2×D411yescharacteristic
152C22×C4×C151C2×D412yes—
153C22×C4×C151C2×D414yes—
154C22×C4×C151C2×D411yescharacteristic
155C22×C4×C151C2411yescharacteristic
156M16×C151—48no—
157C2×C8×C151—21no—
158C2×C8×C151—21no—
159C2×C8×C151—21no—
160C2×C8×C151—22no—
161C2×C8×C151—22no—
162C2×C8×C151—21no—
163C2×C8×C151—21no—
164C2×C8×C151—22no—
165C2×C8×C151C2×D412yes—
166C4⋊C4×C151—24no—
167C4⋊C4×C151—28no—
168C4⋊C4×C151—24no—
169C4⋊C4×C151—21no—
170C4⋊C4×C151—21no—
171C4⋊C4×C151—22no—
172C4⋊C4×C151—21no—
173C4⋊C4×C151—21no—
174C4⋊C4×C151—22no—
175C4⋊C4×C151—44no—
176C4⋊C4×C151—42no—
177C4⋊C4×C151—81no—
178C4⋊C4×C151C22⋊C414yes—
179C4⋊C4×C151C22×C412yes—
180C4⋊C4×C151C2×D412yes—
181C22⋊C4×C151—28no—
182C22⋊C4×C151—24no—
183C22⋊C4×C151—44no—
184C22⋊C4×C151C22⋊C418yes—
185C42×C151—42no—
186C42×C151C22×C412yes—
187C42×C151C2×D412yes—
188C23×C151—22no—
189C23×C151—22no—
190C23×C151—22no—
191C23×C151—22no—
192C23×C151—22no—
193C23×C151—22no—
194C23×C151—22no—
195C23×C151—22no—
196C23×C151—22no—
197C23×C151—41no—
198C23×C151—41no—
199C23×C151—41no—
200C23×C151—41no—
201C23×C151C2×C22⋊C414yes—
202C23×C151C22×D411yescharacteristic
203Q8×C151—24no—
204Q8×C151—24no—
205Q8×C151—42no—
206Q8×C151—42no—
207D4×C151—22no—
208D4×C151—22no—
209D4×C151—22no—
210D4×C151—28no—
211D4×C151—416no—
212D4×C151C4×D414yescomplemented
213C2×C4×C151—22no—
214C2×C4×C151—22no—
215C2×C4×C151—21no—
216C2×C4×C151—21no—
217C2×C4×C151—24no—
218C2×C4×C151—24no—
219C2×C4×C151—24no—
220C2×C4×C151—24no—
221C2×C4×C151—24no—
222C2×C4×C151—22no—
223C2×C4×C151—22no—
224C2×C4×C151—416no—
225C2×C4×C151—42no—
226C2×C4×C151—42no—
227C2×C4×C151—41no—
228C2×C4×C151—41no—
229C2×C4×C151—41no—
230C2×C4×C151—41no—
231C2×C4×C151—48no—
232C2×C4×C151—42no—
233C2×C4×C151—41no—
234C2×C4×C151—41no—
235C2×C4×C151—42no—
236C2×C4×C151—42no—
237C2×C4×C151—41no—
238C2×C4×C151—88no—
239C2×C4×C151C2×C22⋊C412yes—
240C2×C4×C151C2×C22⋊C414yes—
241C2×C4×C151C2×C22⋊C412yes—
242C2×C4×C151AGL(1,ℤ/8)12yes—
243C2×C4×C151#13712yes—
244C2×C4×C151C23×C411yescharacteristic
245C2×C4×C151C22×D411yescharacteristic
246C2×C4×C151C22×D411yescharacteristic
247C2×C4×C151C2×#4111yescharacteristic
248C2×C4×C151H(5,𝔽2)11yescharacteristic
249C2×C4×C151H(5,𝔽2)11yescharacteristic
250C8×C151—22no—
251C8×C151—42no—
252C8×C151—41no—
253C22×C151—24no—
254C22×C151—24no—
255C22×C151—24no—
256C22×C151—24no—
257C22×C151—42no—
258C22×C151—42no—
259C22×C151—42no—
260C22×C151—42no—
261C22×C151—42no—
262C22×C151—42no—
263C22×C151—42no—
264C22×C151—42no—
265C22×C151—42no—
266C22×C151—42no—
267C22×C151C22×C22⋊C411yescharacteristic
268C22×C151D4211yescharacteristic
269C22×C151#54611yescharacteristic
270C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C151C2×#13711yescharacteristic
272C22×C151#57711yescharacteristic
273C22×C151#57811yescharacteristic
274C4×C151—24no—
275C4×C151—22no—
276C4×C151—44no—
277C4×C151—42no—
278C4×C151—42no—
279C4×C151—81no—
280C4×C151—82no—
281C4×C151#41714yescomplemented
282C4×C151C2×C4×D411yescharacteristic
283C4×C151#51811yescharacteristic
284C4×C151D4211yescharacteristic
285C4×C151#54611yescharacteristic
286C2×C151—44no—
287C2×C151—44no—
288C2×C151D4×C22⋊C411yescharacteristic
289C2×C151C2×#41711yescharacteristic
290C2×C151#289811yescharacteristic
291C2×C151#327211yescharacteristic
292C2×C151#327611yescharacteristic
293C2×C151#333011yescharacteristic
294C2×C151#333111yescharacteristic
295#22431C15111yesmaximal, characteristic, direct product factor, Sylow
296C151#2243111yescharacteristic, direct product factor, Sylow
297C2×#547C2×C15111yescharacteristic
298C2×#436C2×C15111yescharacteristic, complemented
299C2×#417C2×C15112yescomplemented
300D4×C4⋊C4C2×C15111yescharacteristic, complemented
301#1981C2×C15111yescharacteristic, complemented
302#1956C2×C15111yescharacteristic, complemented
303#1894C2×C15112yescomplemented
304#1872C2×C15112yescomplemented
305#1802C2×C15111yescharacteristic, complemented
306#1788C2×C15112yescomplemented
307C4×#102C2×C15111yescharacteristic, complemented
308C22×#41C22×C15112yes—
309C22×M16C22×C15112yes—
310#547—24no—
311#547C4×C15118yescomplemented
312C2×C4⋊Q8C22×C15111yescharacteristic
313C2×#123—22no—
314C2×C22⋊Q8C22×C15112yes—
315C2×#121—21no—
316C2×C4×D4—21no—
317C2×C4×D4C22×C15111yescharacteristic
318C2×C4×D4C22×C15111yescharacteristic
319C22×C4⋊C4—21no—
320C22×C4⋊C4C22×C15112yes—
321#436—24no—
322C2×C4⋊C8C22×C15111yescharacteristic
323#417—28no—
324C2×#103C22×C15112yes—
325C2×#103C22×C15112yes—
326C2×#103C22×C15112yes—
327C2×#103C22×C15112yes—
328C2×#102C22×C15112yes—
329C2×#102C22×C15111yescharacteristic
330C2×#102C22×C15111yescharacteristic
331C2×C22⋊C8C22×C15112yes—
332C2×C4×C8C22×C15111yescharacteristic
333#389—21no—
334#385—22no—
335#384C22×C15111yescharacteristic
336#380C22×C15112yes—
337C4×C4⋊C4C22×C15111yescharacteristic
338#340C22×C15112yes—
339#340C22×C15111yescharacteristic
340#340C22×C15111yescharacteristic
341C2×#41—24no—
342C2×#41—24no—
343C2×#41—24no—
344C2×#41—42no—
345C2×#41—42no—
346C2×#41C2×C4×C15118yescomplemented
347C22×Q8C23×C15112yes—
348C22×D4—22no—
349C22×D4D4×C15111yescharacteristic
350C22×D4C23×C15111yescharacteristic
351C23×C4—21no—
352C23×C4—22no—
353C23×C4—22no—
354C23×C4C23×C15112yes—
355C2×M16—24no—
356C2×M16—24no—
357C2×M16—42no—
358C22×C8C23×C15112yes—
359C22×C8C23×C15111yescharacteristic
360C22×C8C23×C15111yescharacteristic
361C4⋊Q8—22no—
362C4⋊Q8C2×C4×C15114yescomplemented
363#123—216no—
364C22⋊Q8—24no—
365C22⋊Q8C2×C4×C15118yescomplemented
366#121—28no—
367C4×D4—22no—
368C4×D4—44no—
369C4×D4C2×C4×C15114yes—
370C4×D4C2×C4×C15114yescomplemented
371C4×D4D4×C15114yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—28no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4D4×C15111yescharacteristic
382C2×C4⋊C4C23×C15112yes—
383C2×C4⋊C4C23×C15111yescharacteristic
384C2×C4⋊C4C23×C15111yescharacteristic
385C2×C4⋊C4C23×C15111yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4D4×C15112yes—
390C2×C42—21no—
391C2×C42C23×C15111yescharacteristic
392C4⋊C8—22no—
393#103—24no—
394#103—24no—
395#103—42no—
396#103—42no—
397#102—22no—
398#102—24no—
399#102D4×C15114yescomplemented
400C22⋊C8—24no—
401C4×C8—22no—
402#95—21no—
403#95—22no—
404C24—21no—
405C24C2×D4×C15112yes—
406#41—216no—
407#41—48no—
408#41—48no—
409#41—48no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8C22×C4×C15114yes—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4C22⋊C4×C15114yes—
425C2×D4C22×C4×C15112yes—
426C2×D4C2×D4×C15112yes—
427C2×D4#41×C15112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—82no—
446C22×C4C22×C4×C15114yes—
447C22×C4C2×D4×C15111yescharacteristic
448C22×C4C2×D4×C15112yes—
449C22×C4C2×D4×C15114yes—
450C22×C4C2×D4×C15111yescharacteristic
451C22×C4C24×C15111yescharacteristic
452M16—48no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—22no—
461C2×C8C2×D4×C15112yes—
462C4⋊C4—24no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4C22⋊C4×C15114yes—
475C4⋊C4C22×C4×C15112yes—
476C4⋊C4C2×D4×C15112yes—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—44no—
480C22⋊C4C22⋊C4×C15118yes—
481C42—42no—
482C42C22×C4×C15112yes—
483C42C2×D4×C15112yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×C22⋊C4×C15114yes—
498C23C22×D4×C15111yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—42no—
502Q8—42no—
503D4—28no—
504D4—22no—
505D4—22no—
506D4—22no—
507D4—416no—
508D4C4×D4×C15114yescomplemented
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—416no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—88no—
535C2×C4C2×C22⋊C4×C15112yes—
536C2×C4C2×C22⋊C4×C15114yes—
537C2×C4C2×C22⋊C4×C15112yes—
538C2×C4AGL(1,ℤ/8)×C15112yes—
539C2×C4#137×C15112yes—
540C2×C4C23×C4×C15111yescharacteristic
541C2×C4C22×D4×C15111yescharacteristic
542C2×C4C22×D4×C15111yescharacteristic
543C2×C4C2×#41×C15111yescharacteristic
544C2×C4H(5,𝔽2)×C15111yescharacteristic
545C2×C4H(5,𝔽2)×C15111yescharacteristic
546C8—22no—
547C8—42no—
548C8—41no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22C22×C22⋊C4×C15111yescharacteristic
564C22D42×C15111yescharacteristic
565C22#546×C15111yescharacteristic
566C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
567C22C2×#137×C15111yescharacteristic
568C22#577×C15111yescharacteristic
569C22#578×C15111yescharacteristic
570C4—24no—
571C4—22no—
572C4—44no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—81no—
577C4#417×C15114yescomplemented
578C4C2×C4×D4×C15111yescharacteristic
579C4#518×C15111yescharacteristic
580C4D42×C15111yescharacteristic
581C4#546×C15111yescharacteristic
582C2—44no—
583C2—44no—
584C2D4×C22⋊C4×C15111yescharacteristic
585C2C2×#417×C15111yescharacteristic
586C2#2898×C15111yescharacteristic
587C2#3272×C15111yescharacteristic
588C2#3276×C15111yescharacteristic
589C2#3330×C15111yescharacteristic
590C2#3331×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutatorC2×C4C23×C4×C151
Frattini, cosocleC22×C4C24×C151
prime radicalC2×#547×C151C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C4×D4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#436×C151C2semidirect
3. C2×#417×C151C2semidirect
4. D4×C4⋊C4×C151C2semidirect
5. #1981×C151C2semidirect
6. #1956×C151C2semidirect
7. #1894×C151C2semidirect
8. #1872×C151C2semidirect
9. #1802×C151C2semidirect
10. #1788×C151C2semidirect
11. C4×#102×C151C2semidirect
15. #547×C151C4semidirect
50. C2×#41×C151C2×C4semidirect
66. C4⋊Q8×C151C2×C4semidirect
69. C22⋊Q8×C151C2×C4semidirect
74. C4×D4×C151C2×C4semidirect
103. #102×C151D4semidirect
212. D4×C151C4×D4semidirect
281. C4×C151#417semidirect
295. #22431C151direct (N × H)
296. C151#22431direct (N × H)
298. C2×#436C2×C151semidirect
299. C2×#417C2×C151semidirect
300. D4×C4⋊C4C2×C151semidirect
301. #1981C2×C151semidirect
302. #1956C2×C151semidirect
303. #1894C2×C151semidirect
304. #1872C2×C151semidirect
305. #1802C2×C151semidirect
306. #1788C2×C151semidirect
307. C4×#102C2×C151semidirect
311. #547C4×C151semidirect
346. C2×#41C2×C4×C151semidirect
362. C4⋊Q8C2×C4×C151semidirect
365. C22⋊Q8C2×C4×C151semidirect
370. C4×D4C2×C4×C151semidirect
399. #102D4×C151semidirect
508. D4C4×D4×C151semidirect
577. C4#417×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#547×C1512
C2×#436×C1512
C2×#417×C1512
D4×C4⋊C4×C1512
#1981×C1512
#1956×C1512
#1894×C1512
#1872×C1512
#1802×C1512
#1788×C1512
C4×#102×C1512
#22431151