Finite Group Databasebeta

← C151×#24270 #50329195

Subgroups — C151×#24270 #50329195

Order 38656.

1452 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#135×C151C211yesmaximal, characteristic, complemented
2C2×#468×C151C212yesmaximal, complemented
3C2×#435×C151C211yesmaximal, characteristic, complemented
4#2407×C151C212yesmaximal, complemented
5#2056×C151C211yesmaximal, characteristic, complemented
6#2054×C151C211yesmaximal, characteristic, complemented
7#2001×C151C211yesmaximal, characteristic, complemented
8#1999×C151C211yesmaximal, characteristic, complemented
9#1973×C151C212yesmaximal, complemented
10#1808×C151C211yesmaximal, characteristic, complemented
11#1788×C151C212yesmaximal, complemented
12C22×#41×C151C2212yes—
13C2×#135×C151—28no—
14C2×#135×C151—24no—
15C22×Q16×C151C2211yescharacteristic
16C22×SD16×C151C2212yes—
17C22×D8×C151C2211yescharacteristic
18C22×M16×C151C2211yescharacteristic
19C23×C8×C151C2211yescharacteristic
20C2×C22⋊Q8×C151C2212yes—
21C2×#121×C151C2212yes—
22C2×C4×D4×C151C2211yescharacteristic
23#468×C151—28no—
24#435×C151—24no—
25C2×#107×C151C2212yes—
26C2×C4⋊C8×C151C2211yescharacteristic
27C2×#103×C151C2212yes—
28C2×#103×C151C2212yes—
29C2×#102×C151C2212yes—
30C2×#102×C151C2212yes—
31C2×C22⋊C8×C151C2211yescharacteristic
32C2×C22⋊C8×C151C2211yescharacteristic
33C2×#97×C151C2211yescharacteristic
34#394×C151—22no—
35#392×C151—22no—
36#391×C151C2212yes—
37#390×C151C2212yes—
38#386×C151—22no—
39#385×C151—22no—
40#380×C151C2212yes—
41#340×C151C2211yescharacteristic
42#340×C151C2211yescharacteristic
43C2×#41×C151—24no—
44C2×#41×C151—24no—
45C2×#41×C151—24no—
46C2×#41×C151—24no—
47C2×#41×C151—42no—
48C2×#41×C151—42no—
49C22×Q8×C151C2312yes—
50C22×D4×C151—22no—
51C22×D4×C151—22no—
52C22×D4×C151D411yescharacteristic
53C22×D4×C151C2312yes—
54C23×C4×C151—22no—
55C23×C4×C151C2311yescharacteristic
56C23×C4×C151C2311yescharacteristic
57#135×C151—232no—
58C2×Q16×C151—22no—
59C2×Q16×C151—24no—
60C2×SD16×C151—24no—
61C2×SD16×C151—24no—
62C2×SD16×C151D418yescomplemented
63C2×D8×C151—24no—
64C2×D8×C151—22no—
65C2×M16×C151—22no—
66C2×M16×C151—24no—
67C22×C8×C151—24no—
68C22×C8×C151D414yescomplemented
69C22×C8×C151C2311yescharacteristic
70C22×C8×C151C2311yescharacteristic
71C22×C8×C151C2311yescharacteristic
72C22×C8×C151C2311yescharacteristic
73C22⋊Q8×C151—24no—
74C22⋊Q8×C151—42no—
75#121×C151—24no—
76#121×C151—42no—
77C4×D4×C151—22no—
78C4×D4×C151—22no—
79C2×C4⋊C4×C151—22no—
80C2×C4⋊C4×C151D411yescharacteristic
81C2×C4⋊C4×C151C2312yes—
82C2×C4⋊C4×C151C2312yes—
83C2×C22⋊C4×C151—22no—
84C2×C22⋊C4×C151—22no—
85C2×C22⋊C4×C151—22no—
86C2×C22⋊C4×C151—22no—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151D411yescharacteristic
89C2×C22⋊C4×C151D411yescharacteristic
90C2×C42×C151C2311yescharacteristic
91#107×C151—24no—
92C4⋊C8×C151—22no—
93#103×C151—24no—
94#103×C151—42no—
95#102×C151—24no—
96#102×C151—42no—
97C22⋊C8×C151—22no—
98C22⋊C8×C151—22no—
99#97×C151—22no—
100#95×C151—22no—
101#95×C151—22no—
102#95×C151—22no—
103C24×C151—22no—
104C24×C151C2×D411yescharacteristic
105C24×C151C2×D411yescharacteristic
106#41×C151—216no—
107#41×C151—48no—
108#41×C151—48no—
109#41×C151—48no—
110C2×Q8×C151—22no—
111C2×Q8×C151—22no—
112C2×Q8×C151—22no—
113C2×Q8×C151—22no—
114C2×Q8×C151—22no—
115C2×Q8×C151C2×D414yes—
116C2×D4×C151—22no—
117C2×D4×C151—22no—
118C2×D4×C151—22no—
119C2×D4×C151—22no—
120C2×D4×C151—22no—
121C2×D4×C151—22no—
122C2×D4×C151—42no—
123C2×D4×C151—44no—
124C2×D4×C151—44no—
125C2×D4×C151—44no—
126C2×D4×C151—44no—
127C2×D4×C151—44no—
128C2×D4×C151—82no—
129C2×D4×C151C2×D414yes—
130C22×C4×C151—22no—
131C22×C4×C151—22no—
132C22×C4×C151—22no—
133C22×C4×C151—21no—
134C22×C4×C151—22no—
135C22×C4×C151—21no—
136C22×C4×C151—22no—
137C22×C4×C151—22no—
138C22×C4×C151—22no—
139C22×C4×C151—44no—
140C22×C4×C151—44no—
141C22×C4×C151—44no—
142C22×C4×C151C2×D411yescharacteristic
143C22×C4×C151C2×D411yescharacteristic
144C22×C4×C151C2×D411yescharacteristic
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2×D412yes—
147C22×C4×C151#4112yes—
148C22×C4×C151C2411yescharacteristic
149Q16×C151—28no—
150SD16×C151—216no—
151D8×C151—28no—
152M16×C151—44no—
153C2×C8×C151—22no—
154C2×C8×C151—28no—
155C2×C8×C151—21no—
156C2×C8×C151—21no—
157C2×C8×C151—22no—
158C2×C8×C151—22no—
159C2×C8×C151C2×D412yes—
160C2×C8×C151C2×D412yes—
161C2×C8×C151C2×D414yes—
162C4⋊C4×C151—22no—
163C4⋊C4×C151—22no—
164C4⋊C4×C151—22no—
165C4⋊C4×C151—22no—
166C4⋊C4×C151—22no—
167C4⋊C4×C151—44no—
168C22⋊C4×C151—22no—
169C22⋊C4×C151—22no—
170C22⋊C4×C151—44no—
171C22⋊C4×C151—44no—
172C22⋊C4×C151—44no—
173C22⋊C4×C151—82no—
174C22⋊C4×C151—82no—
175C42×C151—21no—
176C42×C151—21no—
177C23×C151—22no—
178C23×C151—22no—
179C23×C151—21no—
180C23×C151—21no—
181C23×C151—21no—
182C23×C151—24no—
183C23×C151—22no—
184C23×C151—22no—
185C23×C151—21no—
186C23×C151—21no—
187C23×C151—42no—
188C23×C151—42no—
189C23×C151—42no—
190C23×C151—42no—
191C23×C151—42no—
192C23×C151—42no—
193C23×C151#12112yes—
194C23×C151C22×D411yescharacteristic
195Q8×C151—24no—
196Q8×C151—24no—
197Q8×C151—42no—
198Q8×C151—42no—
199D4×C151—24no—
200D4×C151—24no—
201D4×C151—416no—
202D4×C151—44no—
203D4×C151—42no—
204D4×C151—42no—
205D4×C151—88no—
206C2×C4×C151—22no—
207C2×C4×C151—22no—
208C2×C4×C151—22no—
209C2×C4×C151—24no—
210C2×C4×C151—21no—
211C2×C4×C151—21no—
212C2×C4×C151—21no—
213C2×C4×C151—24no—
214C2×C4×C151—22no—
215C2×C4×C151—22no—
216C2×C4×C151—21no—
217C2×C4×C151—21no—
218C2×C4×C151—416no—
219C2×C4×C151—42no—
220C2×C4×C151—42no—
221C2×C4×C151—41no—
222C2×C4×C151—41no—
223C2×C4×C151—42no—
224C2×C4×C151—42no—
225C2×C4×C151—42no—
226C2×C4×C151—42no—
227C2×C4×C151—42no—
228C2×C4×C151—42no—
229C2×C4×C151—42no—
230C2×C4×C151#12112yes—
231C2×C4×C151C22×D411yescharacteristic
232C2×C4×C151C22×D412yes—
233C2×C4×C151C2×#4111yescharacteristic
234C2×C4×C151H(5,𝔽2)12yes—
235C8×C151—22no—
236C8×C151—41no—
237C8×C151—41no—
238C8×C151#12014yes—
239C22×C151—22no—
240C22×C151—22no—
241C22×C151—22no—
242C22×C151—42no—
243C22×C151—42no—
244C22×C151—44no—
245C22×C151—41no—
246C22×C151—42no—
247C22×C151—44no—
248C22×C151—42no—
249C22×C151—44no—
250C22×C151—41no—
251C22×C151—41no—
252C22×C151—82no—
253C22×C151—82no—
254C22×C151—82no—
255C22×C151—82no—
256C22×C151C2×#12111yescharacteristic
257C22×C151D4212yes—
258C22×C151#57512yes—
259C22×C151#57611yescharacteristic
260C22×C151#57811yescharacteristic
261C4×C151—22no—
262C4×C151—41no—
263C4×C151—44no—
264C4×C151—41no—
265C4×C151—41no—
266C4×C151—82no—
267C4×C151—82no—
268C4×C151C2×#12011yescharacteristic
269C4×C151#53511yescharacteristic
270C4×C151#54612yes—
271C2×C151—42no—
272C2×C151—81no—
273C2×C151—84no—
274C2×C151#240811yescharacteristic
275C2×C151#305411yescharacteristic
276C2×C151#306111yescharacteristic
277C2×C151#326812yes—
278C2×C151#326912yes—
279#24270C15111yesmaximal, characteristic, direct product factor, Sylow
280C151#2427011yescharacteristic, direct product factor, Sylow
281C22×#135C2×C15111yescharacteristic, complemented
282C2×#468C2×C15112yescomplemented
283C2×#435C2×C15111yescharacteristic, complemented
284#2407C2×C15112yescomplemented
285#2056C2×C15111yescharacteristic, complemented
286#2054C2×C15111yescharacteristic, complemented
287#2001C2×C15111yescharacteristic, complemented
288#1999C2×C15111yescharacteristic, complemented
289#1973C2×C15112yescomplemented
290#1808C2×C15111yescharacteristic, complemented
291#1788C2×C15112yescomplemented
292C22×#41C22×C15112yes—
293C2×#135—28no—
294C2×#135—24no—
295C22×Q16C22×C15111yescharacteristic
296C22×SD16C22×C15112yes—
297C22×D8C22×C15111yescharacteristic
298C22×M16C22×C15111yescharacteristic
299C23×C8C22×C15111yescharacteristic
300C2×C22⋊Q8C22×C15112yes—
301C2×#121C22×C15112yes—
302C2×C4×D4C22×C15111yescharacteristic
303#468—28no—
304#435—24no—
305C2×#107C22×C15112yes—
306C2×C4⋊C8C22×C15111yescharacteristic
307C2×#103C22×C15112yes—
308C2×#103C22×C15112yes—
309C2×#102C22×C15112yes—
310C2×#102C22×C15112yes—
311C2×C22⋊C8C22×C15111yescharacteristic
312C2×C22⋊C8C22×C15111yescharacteristic
313C2×#97C22×C15111yescharacteristic
314#394—22no—
315#392—22no—
316#391C22×C15112yes—
317#390C22×C15112yes—
318#386—22no—
319#385—22no—
320#380C22×C15112yes—
321#340C22×C15111yescharacteristic
322#340C22×C15111yescharacteristic
323C2×#41—24no—
324C2×#41—24no—
325C2×#41—24no—
326C2×#41—24no—
327C2×#41—42no—
328C2×#41—42no—
329C22×Q8C23×C15112yes—
330C22×D4—22no—
331C22×D4—22no—
332C22×D4D4×C15111yescharacteristic
333C22×D4C23×C15112yes—
334C23×C4—22no—
335C23×C4C23×C15111yescharacteristic
336C23×C4C23×C15111yescharacteristic
337#135—232no—
338C2×Q16—24no—
339C2×Q16—22no—
340C2×SD16—24no—
341C2×SD16—24no—
342C2×SD16D4×C15118yescomplemented
343C2×D8—24no—
344C2×D8—22no—
345C2×M16—22no—
346C2×M16—24no—
347C22×C8—24no—
348C22×C8D4×C15114yescomplemented
349C22×C8C23×C15111yescharacteristic
350C22×C8C23×C15111yescharacteristic
351C22×C8C23×C15111yescharacteristic
352C22×C8C23×C15111yescharacteristic
353C22⋊Q8—24no—
354C22⋊Q8—42no—
355#121—24no—
356#121—42no—
357C4×D4—22no—
358C4×D4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4D4×C15111yescharacteristic
361C2×C4⋊C4C23×C15112yes—
362C2×C4⋊C4C23×C15112yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4D4×C15111yescharacteristic
369C2×C22⋊C4D4×C15111yescharacteristic
370C2×C42C23×C15111yescharacteristic
371#107—24no—
372C4⋊C8—22no—
373#103—24no—
374#103—42no—
375#102—24no—
376#102—42no—
377C22⋊C8—22no—
378C22⋊C8—22no—
379#97—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383C24—22no—
384C24C2×D4×C15111yescharacteristic
385C24C2×D4×C15111yescharacteristic
386#41—216no—
387#41—48no—
388#41—48no—
389#41—48no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C2×D4×C15114yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—82no—
409C2×D4C2×D4×C15114yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4C2×D4×C15111yescharacteristic
423C22×C4C2×D4×C15111yescharacteristic
424C22×C4C2×D4×C15111yescharacteristic
425C22×C4C2×D4×C15111yescharacteristic
426C22×C4C2×D4×C15112yes—
427C22×C4#41×C15112yes—
428C22×C4C24×C15111yescharacteristic
429Q16—28no—
430SD16—216no—
431D8—28no—
432M16—44no—
433C2×C8—22no—
434C2×C8—28no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8C2×D4×C15112yes—
440C2×C8C2×D4×C15112yes—
441C2×C8C2×D4×C15114yes—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—44no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C42—21no—
456C42—21no—
457C23—22no—
458C23—22no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23#121×C15112yes—
474C23C22×D4×C15111yescharacteristic
475Q8—24no—
476Q8—24no—
477Q8—42no—
478Q8—42no—
479D4—24no—
480D4—24no—
481D4—44no—
482D4—416no—
483D4—42no—
484D4—42no—
485D4—88no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—416no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4#121×C15112yes—
511C2×C4C22×D4×C15111yescharacteristic
512C2×C4C22×D4×C15112yes—
513C2×C4C2×#41×C15111yescharacteristic
514C2×C4H(5,𝔽2)×C15112yes—
515C8—22no—
516C8—41no—
517C8—41no—
518C8#120×C15114yes—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—44no—
525C22—41no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22—44no—
530C22—41no—
531C22—41no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22C2×#121×C15111yescharacteristic
537C22D42×C15112yes—
538C22#575×C15112yes—
539C22#576×C15111yescharacteristic
540C22#578×C15111yescharacteristic
541C4—22no—
542C4—41no—
543C4—44no—
544C4—41no—
545C4—41no—
546C4—82no—
547C4—82no—
548C4C2×#120×C15111yescharacteristic
549C4#535×C15111yescharacteristic
550C4#546×C15112yes—
551C2—42no—
552C2—81no—
553C2—84no—
554C2#2408×C15111yescharacteristic
555C2#3054×C15111yescharacteristic
556C2#3061×C15111yescharacteristic
557C2#3268×C15112yes—
558C2#3269×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×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
1. C22×#135×C151C2semidirect
2. C2×#468×C151C2semidirect
3. C2×#435×C151C2semidirect
4. #2407×C151C2semidirect
5. #2056×C151C2semidirect
6. #2054×C151C2semidirect
7. #2001×C151C2semidirect
8. #1999×C151C2semidirect
9. #1973×C151C2semidirect
10. #1808×C151C2semidirect
11. #1788×C151C2semidirect
62. C2×SD16×C151D4semidirect
68. C22×C8×C151D4semidirect
279. #24270C151direct (N × H)
280. C151#24270direct (N × H)
281. C22×#135C2×C151semidirect
282. C2×#468C2×C151semidirect
283. C2×#435C2×C151semidirect
284. #2407C2×C151semidirect
285. #2056C2×C151semidirect
286. #2054C2×C151semidirect
287. #2001C2×C151semidirect
288. #1999C2×C151semidirect
289. #1973C2×C151semidirect
290. #1808C2×C151semidirect
291. #1788C2×C151semidirect
342. C2×SD16D4×C151semidirect
348. C22×C8D4×C151semidirect

Maximal subgroups

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

subgroupindex
C22×#135×C1512
C2×#468×C1512
C2×#435×C1512
#2407×C1512
#2056×C1512
#2054×C1512
#2001×C1512
#1999×C1512
#1973×C1512
#1808×C1512
#1788×C1512
#24270151