Finite Group Databasebeta

← C151×#15406 #50320331

Subgroups — C151×#15406 #50320331

Order 38656.

1674 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399×C151C211yesmaximal, characteristic
2C2×#389×C151C212yesmaximal, complemented
3C2×#385×C151C211yesmaximal, characteristic
4C2×#385×C151C212yesmaximal
5#1444×C151C212yesmaximal, complemented
6#1444×C151C212yesmaximal, complemented
7#1442×C151C212yesmaximal, complemented
8#1439×C151C212yesmaximal
9#1437×C151C211yesmaximal, characteristic
10C24×C4×C151C2211yescharacteristic
11C22×C4⋊C4×C151C2212yes—
12C22×C4⋊C4×C151C2212yes—
13C22×C4⋊C4×C151C2212yes—
14C22×C4⋊C4×C151C2212yes—
15C22×C4⋊C4×C151C2211yescharacteristic
16C22×C4⋊C4×C151C2212yes—
17C22×C22⋊C4×C151C2211yescharacteristic
18C22×C22⋊C4×C151C2212yes—
19C22×C22⋊C4×C151C2212yes—
20C22×C22⋊C4×C151C2211yescharacteristic
21C22×C42×C151C2211yescharacteristic
22#399×C151—24no—
23#389×C151—28no—
24#385×C151—24no—
25#385×C151—28no—
26C2×#95×C151C2211yescharacteristic
27C2×#95×C151C2212yes—
28C2×#95×C151C2212yes—
29C2×#95×C151C2212yes—
30C2×#95×C151C2212yes—
31C2×#95×C151C2212yes—
32C2×#95×C151C2211yescharacteristic
33C2×#95×C151C2212yes—
34C2×#95×C151C2212yes—
35C25×C151C2311yescharacteristic
36C23×C4×C151—24no—
37C23×C4×C151—22no—
38C23×C4×C151—41no—
39C23×C4×C151—42no—
40C23×C4×C151—41no—
41C23×C4×C151C2312yes—
42C23×C4×C151C2311yescharacteristic
43C23×C4×C151C2312yes—
44C23×C4×C151C2312yes—
45C23×C4×C151C2311yescharacteristic
46C23×C4×C151C2312yes—
47C23×C4×C151C2312yes—
48C23×C4×C151C2312yes—
49C2×C4⋊C4×C151—24no—
50C2×C4⋊C4×C151—24no—
51C2×C4⋊C4×C151—24no—
52C2×C4⋊C4×C151—24no—
53C2×C4⋊C4×C151—24no—
54C2×C4⋊C4×C151—24no—
55C2×C4⋊C4×C151—24no—
56C2×C4⋊C4×C151—24no—
57C2×C4⋊C4×C151—24no—
58C2×C4⋊C4×C151—24no—
59C2×C4⋊C4×C151—22no—
60C2×C4⋊C4×C151—42no—
61C2×C4⋊C4×C151—42no—
62C2×C4⋊C4×C151—42no—
63C2×C4⋊C4×C151—42no—
64C2×C4⋊C4×C151—42no—
65C2×C4⋊C4×C151—41no—
66C2×C4⋊C4×C151—41no—
67C2×C22⋊C4×C151—24no—
68C2×C22⋊C4×C151—24no—
69C2×C22⋊C4×C151—24no—
70C2×C22⋊C4×C151—24no—
71C2×C22⋊C4×C151—22no—
72C2×C22⋊C4×C151—24no—
73C2×C22⋊C4×C151—41no—
74C2×C22⋊C4×C151—41no—
75C2×C22⋊C4×C151—42no—
76C2×C22⋊C4×C151D414yes—
77C2×C22⋊C4×C151Q818yes—
78C2×C42×C151—24no—
79C2×C42×C151—41no—
80#95×C151—24no—
81#95×C151—24no—
82#95×C151—24no—
83#95×C151—22no—
84#95×C151—24no—
85#95×C151—24no—
86#95×C151—41no—
87#95×C151—42no—
88#95×C151D418yes—
89C24×C151—22no—
90C24×C151—22no—
91C24×C151—22no—
92C24×C151—21no—
93C24×C151—21no—
94C24×C151—22no—
95C24×C151—22no—
96C24×C151C2×D412yes—
97C24×C151C2×Q814yes—
98C24×C151C2411yescharacteristic
99C22×C4×C151—22no—
100C22×C4×C151—22no—
101C22×C4×C151—22no—
102C22×C4×C151—22no—
103C22×C4×C151—22no—
104C22×C4×C151—22no—
105C22×C4×C151—22no—
106C22×C4×C151—22no—
107C22×C4×C151—22no—
108C22×C4×C151—22no—
109C22×C4×C151—22no—
110C22×C4×C151—22no—
111C22×C4×C151—22no—
112C22×C4×C151—22no—
113C22×C4×C151—22no—
114C22×C4×C151—22no—
115C22×C4×C151—22no—
116C22×C4×C151—22no—
117C22×C4×C151—22no—
118C22×C4×C151—22no—
119C22×C4×C151—21no—
120C22×C4×C151—22no—
121C22×C4×C151—22no—
122C22×C4×C151—22no—
123C22×C4×C151—21no—
124C22×C4×C151—21no—
125C22×C4×C151—21no—
126C22×C4×C151—22no—
127C22×C4×C151—22no—
128C22×C4×C151—22no—
129C22×C4×C151—22no—
130C22×C4×C151—22no—
131C22×C4×C151—22no—
132C22×C4×C151—22no—
133C22×C4×C151—22no—
134C22×C4×C151—22no—
135C22×C4×C151—22no—
136C22×C4×C151—22no—
137C22×C4×C151—22no—
138C22×C4×C151—22no—
139C22×C4×C151—22no—
140C22×C4×C151—22no—
141C22×C4×C151—22no—
142C22×C4×C151—44no—
143C22×C4×C151—48no—
144C22×C4×C151—48no—
145C22×C4×C151—44no—
146C22×C4×C151—44no—
147C22×C4×C151C2×D412yes—
148C22×C4×C151C2×D414yes—
149C22×C4×C151C2×D414yes—
150C22×C4×C151C2×D412yes—
151C22×C4×C151C2×D414yes—
152C22×C4×C151C2×Q814yes—
153C22×C4×C151C2×Q814yes—
154C22×C4×C151#4114yes—
155C22×C4×C151#4114yes—
156C4⋊C4×C151—216no—
157C4⋊C4×C151—48no—
158C4⋊C4×C151—48no—
159C4⋊C4×C151—48no—
160C4⋊C4×C151—44no—
161C4⋊C4×C151—48no—
162C22⋊C4×C151—28no—
163C22⋊C4×C151—216no—
164C22⋊C4×C151—44no—
165C22⋊C4×C151—48no—
166C42×C151—44no—
167C23×C151—24no—
168C23×C151—22no—
169C23×C151—22no—
170C23×C151—24no—
171C23×C151—24no—
172C23×C151—24no—
173C23×C151—24no—
174C23×C151—24no—
175C23×C151—24no—
176C23×C151—24no—
177C23×C151—42no—
178C23×C151—42no—
179C23×C151—42no—
180C23×C151—42no—
181C23×C151—41no—
182C23×C151—42no—
183C23×C151—41no—
184C23×C151—42no—
185C23×C151—41no—
186C23×C151—41no—
187C23×C151C4⋊Q814yes—
188C23×C151C22×D412yes—
189C23×C151C22×D411yescharacteristic
190C23×C151C22×Q812yes—
191C23×C151C2×#4112yes—
192C23×C151C2×#4112yes—
193C23×C151H(5,𝔽2)11yescharacteristic
194C23×C151H(5,𝔽2)11yescharacteristic
195C23×C151H(5,𝔽2)12yes—
196C23×C1512−1+412yes—
197C2×C4×C151—24no—
198C2×C4×C151—24no—
199C2×C4×C151—24no—
200C2×C4×C151—24no—
201C2×C4×C151—24no—
202C2×C4×C151—24no—
203C2×C4×C151—24no—
204C2×C4×C151—24no—
205C2×C4×C151—24no—
206C2×C4×C151—24no—
207C2×C4×C151—22no—
208C2×C4×C151—22no—
209C2×C4×C151—24no—
210C2×C4×C151—24no—
211C2×C4×C151—24no—
212C2×C4×C151—24no—
213C2×C4×C151—24no—
214C2×C4×C151—24no—
215C2×C4×C151—24no—
216C2×C4×C151—24no—
217C2×C4×C151—24no—
218C2×C4×C151—24no—
219C2×C4×C151—416no—
220C2×C4×C151—42no—
221C2×C4×C151—42no—
222C2×C4×C151—42no—
223C2×C4×C151—42no—
224C2×C4×C151—42no—
225C2×C4×C151—42no—
226C2×C4×C151—42no—
227C2×C4×C151—42no—
228C2×C4×C151—41no—
229C2×C4×C151—42no—
230C2×C4×C151—42no—
231C2×C4×C151—42no—
232C2×C4×C151—42no—
233C2×C4×C151—42no—
234C2×C4×C151—42no—
235C2×C4×C151—42no—
236C2×C4×C151—42no—
237C2×C4×C151—42no—
238C2×C4×C151—41no—
239C2×C4×C151—41no—
240C2×C4×C151—42no—
241C2×C4×C151—41no—
242C2×C4×C151—42no—
243C2×C4×C151—42no—
244C2×C4×C151—42no—
245C2×C4×C151—42no—
246C2×C4×C151—42no—
247C2×C4×C151—42no—
248C2×C4×C151—42no—
249C2×C4×C151C22⋊Q818yes—
250C22×C151—24no—
251C22×C151—28no—
252C22×C151—44no—
253C22×C151—42no—
254C22×C151—44no—
255C22×C151—44no—
256C22×C151—44no—
257C22×C151—42no—
258C22×C151—44no—
259C22×C151C2×C22⋊Q812yes—
260C22×C151#52512yes—
261C22×C151#52511yescharacteristic
262C22×C151C2×C4⋊Q811yescharacteristic
263C22×C151#53411yescharacteristic
264C22×C151#53512yes—
265C22×C151#53912yes—
266C22×C151#53912yes—
267C22×C151#54412yes—
268C22×C151D4212yes—
269C22×C151D4211yescharacteristic
270C22×C151#54712yes—
271C22×C151#54712yes—
272C22×C151D4×Q812yes—
273C22×C151#55412yes—
274C22×C151#55412yes—
275C22×C151#55412yes—
276C22×C151#55911yescharacteristic
277C22×C151#55912yes—
278C22×C151#56011yescharacteristic
279C22×C151#56311yescharacteristic
280C4×C151—28no—
281C4×C151—44no—
282C4×C151—42no—
283C4×C151—44no—
284C4×C151—44no—
285C4×C151—44no—
286C4×C151—42no—
287C4×C151—82no—
288C2×C151—44no—
289C2×C151#244012yes—
290C2×C151#244912yes—
291C2×C151#245512yes—
292C2×C151#258012yes—
293C2×C151#258311yescharacteristic
294C2×C151#266911yescharacteristic
295C2×C151#272012yes—
296C2×C151#272411yescharacteristic
297C2×C151#280712yes—
298#15406C15111yesmaximal, characteristic, direct product factor, Sylow
299C151#1540611yescharacteristic, direct product factor, Sylow
300C2×#399C2×C15111yescharacteristic
301C2×#389C2×C15112yescomplemented
302C2×#385C2×C15111yescharacteristic
303C2×#385C2×C15112yes—
304#1444C2×C15112yescomplemented
305#1444C2×C15112yescomplemented
306#1442C2×C15112yescomplemented
307#1439C2×C15112yes—
308#1437C2×C15111yescharacteristic
309C24×C4C22×C15111yescharacteristic
310C22×C4⋊C4C22×C15112yes—
311C22×C4⋊C4C22×C15112yes—
312C22×C4⋊C4C22×C15112yes—
313C22×C4⋊C4C22×C15112yes—
314C22×C4⋊C4C22×C15111yescharacteristic
315C22×C4⋊C4C22×C15112yes—
316C22×C22⋊C4C22×C15111yescharacteristic
317C22×C22⋊C4C22×C15112yes—
318C22×C22⋊C4C22×C15112yes—
319C22×C22⋊C4C22×C15111yescharacteristic
320C22×C42C22×C15111yescharacteristic
321#399—24no—
322#389—28no—
323#385—24no—
324#385—28no—
325C2×#95C22×C15111yescharacteristic
326C2×#95C22×C15112yes—
327C2×#95C22×C15112yes—
328C2×#95C22×C15112yes—
329C2×#95C22×C15112yes—
330C2×#95C22×C15112yes—
331C2×#95C22×C15111yescharacteristic
332C2×#95C22×C15112yes—
333C2×#95C22×C15112yes—
334C25C23×C15111yescharacteristic
335C23×C4—22no—
336C23×C4—24no—
337C23×C4—41no—
338C23×C4—42no—
339C23×C4—41no—
340C23×C4C23×C15112yes—
341C23×C4C23×C15112yes—
342C23×C4C23×C15112yes—
343C23×C4C23×C15111yescharacteristic
344C23×C4C23×C15112yes—
345C23×C4C23×C15111yescharacteristic
346C23×C4C23×C15112yes—
347C23×C4C23×C15112yes—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—42no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—41no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—41no—
373C2×C22⋊C4—41no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4D4×C15114yes—
376C2×C22⋊C4Q8×C15118yes—
377C2×C42—24no—
378C2×C42—41no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—41no—
386#95—42no—
387#95D4×C15118yes—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—21no—
392C24—21no—
393C24—22no—
394C24—22no—
395C24C2×D4×C15112yes—
396C24C2×Q8×C15114yes—
397C24C24×C15111yescharacteristic
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4C2×D4×C15112yes—
447C22×C4C2×D4×C15114yes—
448C22×C4C2×D4×C15114yes—
449C22×C4C2×D4×C15112yes—
450C22×C4C2×D4×C15114yes—
451C22×C4C2×Q8×C15114yes—
452C22×C4C2×Q8×C15114yes—
453C22×C4#41×C15114yes—
454C22×C4#41×C15114yes—
455C4⋊C4—216no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C22⋊C4—28no—
462C22⋊C4—216no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C42—44no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23C4⋊Q8×C15114yes—
487C23C22×D4×C15112yes—
488C23C22×D4×C15111yescharacteristic
489C23C22×Q8×C15112yes—
490C23C2×#41×C15112yes—
491C23C2×#41×C15112yes—
492C23H(5,𝔽2)×C15111yescharacteristic
493C23H(5,𝔽2)×C15111yescharacteristic
494C23H(5,𝔽2)×C15112yes—
495C232−1+4×C15112yes—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—416no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4C22⋊Q8×C15118yes—
549C22—28no—
550C22—24no—
551C22—42no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—44no—
557C22—44no—
558C22C2×C22⋊Q8×C15112yes—
559C22#525×C15112yes—
560C22#525×C15111yescharacteristic
561C22C2×C4⋊Q8×C15111yescharacteristic
562C22#534×C15111yescharacteristic
563C22#535×C15112yes—
564C22#539×C15112yes—
565C22#539×C15112yes—
566C22#544×C15112yes—
567C22D42×C15112yes—
568C22D42×C15111yescharacteristic
569C22#547×C15112yes—
570C22#547×C15112yes—
571C22D4×Q8×C15112yes—
572C22#554×C15112yes—
573C22#554×C15112yes—
574C22#554×C15112yes—
575C22#559×C15112yes—
576C22#559×C15111yescharacteristic
577C22#560×C15111yescharacteristic
578C22#563×C15111yescharacteristic
579C4—28no—
580C4—44no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—42no—
585C4—44no—
586C4—82no—
587C2—44no—
588C2#2440×C15112yes—
589C2#2449×C15112yes—
590C2#2455×C15112yes—
591C2#2580×C15112yes—
592C2#2583×C15111yescharacteristic
593C2#2669×C15111yescharacteristic
594C2#2720×C15112yes—
595C2#2724×C15111yescharacteristic
596C2#2807×C15112yes—
subgroupG/H
center, socleC24×C151C24
commutator, Frattini, cosocleC24C24×C151
prime radicalC25×C151C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#389×C151C2semidirect
5. #1444×C151C2semidirect
6. #1444×C151C2semidirect
7. #1442×C151C2semidirect
298. #15406C151direct (N × H)
299. C151#15406direct (N × H)
301. C2×#389C2×C151semidirect
304. #1444C2×C151semidirect
305. #1444C2×C151semidirect
306. #1442C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#399×C1512
C2×#389×C1512
C2×#385×C1512
#1444×C1512
#1442×C1512
#1439×C1512
#1437×C1512
#15406151