Finite Group Databasebeta

← C151×#24477 #50329402

Subgroups — C151×#24477 #50329402

Order 38656.

1130 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#483×C151C211yesmaximal, characteristic, complemented
2C22×#102×C151C211yesmaximal, characteristic
3#2435×C151C211yesmaximal, characteristic
4#2344×C151C211yesmaximal, characteristic, complemented
5#2065×C151C212yesmaximal, complemented
6#2042×C151C212yesmaximal, complemented
7#1983×C151C212yesmaximal, complemented
8#1926×C151C212yesmaximal, complemented
9#1892×C151C211yesmaximal, characteristic, complemented
10#1850×C151C211yesmaximal, characteristic, complemented
11#1786×C151C211yesmaximal, characteristic
12C23×D4×C151C2211yescharacteristic
13C23×C8×C151C2211yescharacteristic
14C2×#124×C151C2212yes—
15C2×C22⋊Q8×C151C2211yescharacteristic
16C2×#117×C151—21no—
17C22×C4⋊C4×C151C2211yescharacteristic
18#483×C151—24no—
19C2×#106×C151C2211yescharacteristic, complemented
20C2×#106×C151C2211yescharacteristic, complemented
21C2×C4⋊C8×C151C2212yescomplemented
22C2×#103×C151C2211yescharacteristic, complemented
23C2×#103×C151C2211yescharacteristic, complemented
24C2×#103×C151C2212yescomplemented
25C2×#102×C151—24no—
26C2×#102×C151C2212yes—
27C2×#102×C151C2212yes—
28C2×#102×C151C2212yes—
29C2×#102×C151C2212yes—
30C2×C22⋊C8×C151C2211yescharacteristic
31#394×C151—22no—
32#390×C151C2212yes—
33#389×C151C2212yescomplemented
34#387×C151—22no—
35#381×C151—21no—
36#380×C151C2211yescharacteristic
37#379×C151—21no—
38#379×C151—21no—
39C4×C4⋊C4×C151C2212yescomplemented
40C4×C22⋊C4×C151C2211yescharacteristic
41#340×C151C2212yescomplemented
42#336×C151C2212yescomplemented
43C25×C151—21no—
44C22×Q8×C151C2311yescharacteristic
45C22×D4×C151—21no—
46C22×D4×C151—24no—
47C22×D4×C151—22no—
48C22×D4×C151—22no—
49C22×D4×C151—42no—
50C22×D4×C151C2312yes—
51C23×C4×C151—21no—
52C23×C4×C151C2311yescharacteristic
53C22×C8×C151—22no—
54C22×C8×C151—42no—
55C22×C8×C151C2311yescharacteristic
56C22×C8×C151C2312yes—
57C22×C8×C151C2311yescharacteristic
58#124×C151—24no—
59#124×C151—42no—
60C22⋊Q8×C151—22no—
61C22⋊Q8×C151—22no—
62#117×C151—44no—
63C2×C4⋊C4×C151—21no—
64C2×C4⋊C4×C151—22no—
65C2×C4⋊C4×C151—22no—
66C2×C4⋊C4×C151—21no—
67C2×C4⋊C4×C151D414yes—
68C2×C4⋊C4×C151C2311yescharacteristic
69C2×C4⋊C4×C151C2312yes—
70C2×C4⋊C4×C151C2311yescharacteristic
71C2×C4⋊C4×C151C2311yescharacteristic
72C2×C22⋊C4×C151—22no—
73C2×C22⋊C4×C151—22no—
74C2×C22⋊C4×C151—22no—
75C2×C22⋊C4×C151—22no—
76C2×C22⋊C4×C151—21no—
77C2×C22⋊C4×C151—21no—
78C2×C22⋊C4×C151D411yescharacteristic
79C2×C22⋊C4×C151D411yescharacteristic
80C2×C42×C151—22no—
81C2×C42×C151C2312yes—
82#106×C151—22no—
83#106×C151—22no—
84C4⋊C8×C151—42no—
85#103×C151—22no—
86#103×C151—22no—
87#103×C151—42no—
88#102×C151—24no—
89#102×C151—24no—
90#102×C151—24no—
91#102×C151—42no—
92C22⋊C8×C151—22no—
93#95×C151—22no—
94#95×C151—21no—
95#95×C151—22no—
96#95×C151D412yes—
97C24×C151—22no—
98C24×C151—44no—
99C24×C151—42no—
100C24×C151—42no—
101C24×C151—42no—
102C24×C151—82no—
103C24×C151C2×D411yescharacteristic
104C2×Q8×C151—21no—
105C2×Q8×C151—21no—
106C2×Q8×C151—22no—
107C2×Q8×C151—22no—
108C2×D4×C151—28no—
109C2×D4×C151—22no—
110C2×D4×C151—22no—
111C2×D4×C151—22no—
112C2×D4×C151—22no—
113C2×D4×C151—22no—
114C2×D4×C151—48no—
115C2×D4×C151—44no—
116C2×D4×C151—44no—
117C2×D4×C151—42no—
118C2×D4×C151#4114yes—
119C22×C4×C151—22no—
120C22×C4×C151—22no—
121C22×C4×C151—24no—
122C22×C4×C151—21no—
123C22×C4×C151—21no—
124C22×C4×C151—22no—
125C22×C4×C151—21no—
126C22×C4×C151—21no—
127C22×C4×C151—42no—
128C22×C4×C151—42no—
129C22×C4×C151C2×D412yes—
130C22×C4×C151C2×D412yes—
131C22×C4×C151C2×D411yescharacteristic
132C22×C4×C151C2×D412yes—
133C22×C4×C151#4112yes—
134C22×C4×C151C2411yescharacteristic
135C2×C8×C151—21no—
136C2×C8×C151—21no—
137C2×C8×C151—22no—
138C2×C8×C151—22no—
139C2×C8×C151—22no—
140C2×C8×C151—22no—
141C2×C8×C151—22no—
142C2×C8×C151—44no—
143C4⋊C4×C151—24no—
144C4⋊C4×C151—21no—
145C4⋊C4×C151—21no—
146C4⋊C4×C151—21no—
147C4⋊C4×C151—22no—
148C4⋊C4×C151—21no—
149C4⋊C4×C151—22no—
150C4⋊C4×C151—44no—
151C4⋊C4×C151—42no—
152C4⋊C4×C151C2×D412yes—
153C4⋊C4×C151C2×D412yes—
154C22⋊C4×C151—22no—
155C22⋊C4×C151—22no—
156C22⋊C4×C151—44no—
157C22⋊C4×C151—44no—
158C22⋊C4×C151—42no—
159C22⋊C4×C151—42no—
160C22⋊C4×C151—82no—
161C22⋊C4×C151—82no—
162C42×C151—22no—
163C42×C151—22no—
164C42×C151—44no—
165C23×C151—22no—
166C23×C151—22no—
167C23×C151—24no—
168C23×C151—48no—
169C23×C151—42no—
170C23×C151—42no—
171C23×C151—42no—
172C23×C151—42no—
173C23×C151—42no—
174C23×C151—42no—
175C23×C151—88no—
176C23×C151—88no—
177C23×C151—84no—
178C23×C151—84no—
179C23×C151#12112yes—
180C23×C151#13512yes—
181C23×C151AGL(1,ℤ/8)12yes—
182C23×C151C22×D411yescharacteristic
183Q8×C151—41no—
184Q8×C151—41no—
185Q8×C151—42no—
186D4×C151—24no—
187D4×C151—24no—
188D4×C151—48no—
189D4×C151—42no—
190D4×C151—42no—
191C2×C4×C151—22no—
192C2×C4×C151—22no—
193C2×C4×C151—22no—
194C2×C4×C151—24no—
195C2×C4×C151—22no—
196C2×C4×C151—22no—
197C2×C4×C151—22no—
198C2×C4×C151—22no—
199C2×C4×C151—22no—
200C2×C4×C151—22no—
201C2×C4×C151—22no—
202C2×C4×C151—22no—
203C2×C4×C151—41no—
204C2×C4×C151—41no—
205C2×C4×C151—41no—
206C2×C4×C151—41no—
207C2×C4×C151—48no—
208C2×C4×C151—41no—
209C2×C4×C151—41no—
210C2×C4×C151—42no—
211C2×C4×C151—42no—
212C2×C4×C151—42no—
213C2×C4×C151—42no—
214C2×C4×C151—42no—
215C2×C4×C151#12112yes—
216C2×C4×C151C22×D411yescharacteristic
217C2×C4×C151C2×#4112yes—
218C2×C4×C151C2×#4111yescharacteristic
219C2×C4×C151H(5,𝔽2)12yes—
220C8×C151—41no—
221C8×C151—41no—
222C8×C151—42no—
223C22×C151—22no—
224C22×C151—24no—
225C22×C151—22no—
226C22×C151—24no—
227C22×C151—44no—
228C22×C151—44no—
229C22×C151—44no—
230C22×C151—816no—
231C22×C151—82no—
232C22×C151—82no—
233C22×C151—82no—
234C22×C151—82no—
235C22×C151#46314yes—
236C22×C151C2×#12111yescharacteristic
237C22×C151#54612yes—
238C22×C151C2×#13511yescharacteristic
239C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
240C22×C151#57712yes—
241C4×C151—22no—
242C4×C151—42no—
243C4×C151—42no—
244C4×C151—42no—
245C4×C151—42no—
246C4×C151—81no—
247C4×C151—81no—
248C4×C151—82no—
249C4×C151#52511yescharacteristic
250C4×C151#53511yescharacteristic
251C4×C151#55112yes—
252C2×C151—24no—
253C2×C151—84no—
254C2×C151#241111yescharacteristic
255C2×C151C2×#46311yescharacteristic
256C2×C151#304311yescharacteristic
257C2×C151#329612yes—
258C2×C151#330612yes—
259#24477C15111yesmaximal, characteristic, direct product factor, Sylow
260C151#2447711yescharacteristic, direct product factor, Sylow
261C2×#483C2×C15111yescharacteristic, complemented
262C22×#102C2×C15111yescharacteristic
263#2435C2×C15111yescharacteristic
264#2344C2×C15111yescharacteristic, complemented
265#2065C2×C15112yescomplemented
266#2042C2×C15112yescomplemented
267#1983C2×C15112yescomplemented
268#1926C2×C15112yescomplemented
269#1892C2×C15111yescharacteristic, complemented
270#1850C2×C15111yescharacteristic, complemented
271#1786C2×C15111yescharacteristic
272C23×D4C22×C15111yescharacteristic
273C23×C8C22×C15111yescharacteristic
274C2×#124C22×C15112yes—
275C2×C22⋊Q8C22×C15111yescharacteristic
276C2×#117—21no—
277C22×C4⋊C4C22×C15111yescharacteristic
278#483—24no—
279C2×#106C22×C15111yescharacteristic, complemented
280C2×#106C22×C15111yescharacteristic, complemented
281C2×C4⋊C8C22×C15112yescomplemented
282C2×#103C22×C15111yescharacteristic, complemented
283C2×#103C22×C15111yescharacteristic, complemented
284C2×#103C22×C15112yescomplemented
285C2×#102—24no—
286C2×#102C22×C15112yes—
287C2×#102C22×C15112yes—
288C2×#102C22×C15112yes—
289C2×#102C22×C15112yes—
290C2×C22⋊C8C22×C15111yescharacteristic
291#394—22no—
292#390C22×C15112yes—
293#389C22×C15112yescomplemented
294#387—22no—
295#381—21no—
296#380C22×C15111yescharacteristic
297#379—21no—
298#379—21no—
299C4×C4⋊C4C22×C15112yescomplemented
300C4×C22⋊C4C22×C15111yescharacteristic
301#340C22×C15112yescomplemented
302#336C22×C15112yescomplemented
303C25—21no—
304C22×Q8C23×C15111yescharacteristic
305C22×D4—24no—
306C22×D4—22no—
307C22×D4—22no—
308C22×D4—21no—
309C22×D4—42no—
310C22×D4C23×C15112yes—
311C23×C4—21no—
312C23×C4C23×C15111yescharacteristic
313C22×C8—22no—
314C22×C8—42no—
315C22×C8C23×C15112yes—
316C22×C8C23×C15111yescharacteristic
317C22×C8C23×C15111yescharacteristic
318#124—24no—
319#124—42no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322#117—44no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4D4×C15114yes—
328C2×C4⋊C4C23×C15111yescharacteristic
329C2×C4⋊C4C23×C15112yes—
330C2×C4⋊C4C23×C15111yescharacteristic
331C2×C4⋊C4C23×C15111yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4D4×C15111yescharacteristic
339C2×C22⋊C4D4×C15111yescharacteristic
340C2×C42—22no—
341C2×C42C23×C15112yes—
342#106—22no—
343#106—22no—
344C4⋊C8—42no—
345#103—22no—
346#103—22no—
347#103—42no—
348#102—24no—
349#102—24no—
350#102—24no—
351#102—42no—
352C22⋊C8—22no—
353#95—22no—
354#95—21no—
355#95—22no—
356#95D4×C15112yes—
357C24—22no—
358C24—44no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—82no—
363C24C2×D4×C15111yescharacteristic
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—48no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4#41×C15114yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4C2×D4×C15112yes—
390C22×C4C2×D4×C15112yes—
391C22×C4C2×D4×C15111yescharacteristic
392C22×C4C2×D4×C15112yes—
393C22×C4#41×C15112yes—
394C22×C4C24×C15111yescharacteristic
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—44no—
403C4⋊C4—24no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4C2×D4×C15112yes—
413C4⋊C4C2×D4×C15112yes—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C42—22no—
423C42—22no—
424C42—44no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—48no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—88no—
436C23—88no—
437C23—84no—
438C23—84no—
439C23#121×C15112yes—
440C23#135×C15112yes—
441C23AGL(1,ℤ/8)×C15112yes—
442C23C22×D4×C15111yescharacteristic
443Q8—41no—
444Q8—41no—
445Q8—42no—
446D4—24no—
447D4—24no—
448D4—48no—
449D4—42no—
450D4—42no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—48no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4#121×C15112yes—
476C2×C4C22×D4×C15111yescharacteristic
477C2×C4C2×#41×C15112yes—
478C2×C4C2×#41×C15111yescharacteristic
479C2×C4H(5,𝔽2)×C15112yes—
480C8—41no—
481C8—42no—
482C8—41no—
483C22—22no—
484C22—24no—
485C22—22no—
486C22—24no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—816no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22#463×C15114yes—
496C22C2×#121×C15111yescharacteristic
497C22#546×C15112yes—
498C22C2×#135×C15111yescharacteristic
499C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
500C22#577×C15112yes—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—81no—
507C4—81no—
508C4—82no—
509C4#525×C15111yescharacteristic
510C4#535×C15111yescharacteristic
511C4#551×C15112yes—
512C2—24no—
513C2—84no—
514C2#2411×C15111yescharacteristic
515C2C2×#463×C15111yescharacteristic
516C2#3043×C15111yescharacteristic
517C2#3296×C15112yes—
518C2#3306×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC23×D4×C151C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C4×C22⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#483×C151C2semidirect
4. #2344×C151C2semidirect
5. #2065×C151C2semidirect
6. #2042×C151C2semidirect
7. #1983×C151C2semidirect
8. #1926×C151C2semidirect
9. #1892×C151C2semidirect
10. #1850×C151C2semidirect
19. C2×#106×C151C22semidirect
20. C2×#106×C151C22semidirect
21. C2×C4⋊C8×C151C22semidirect
22. C2×#103×C151C22semidirect
23. C2×#103×C151C22semidirect
24. C2×#103×C151C22semidirect
33. #389×C151C22semidirect
39. C4×C4⋊C4×C151C22semidirect
41. #340×C151C22semidirect
42. #336×C151C22semidirect
259. #24477C151direct (N × H)
260. C151#24477direct (N × H)
261. C2×#483C2×C151semidirect
264. #2344C2×C151semidirect
265. #2065C2×C151semidirect
266. #2042C2×C151semidirect
267. #1983C2×C151semidirect
268. #1926C2×C151semidirect
269. #1892C2×C151semidirect
270. #1850C2×C151semidirect
279. C2×#106C22×C151semidirect
280. C2×#106C22×C151semidirect
281. C2×C4⋊C8C22×C151semidirect
282. C2×#103C22×C151semidirect
283. C2×#103C22×C151semidirect
284. C2×#103C22×C151semidirect
293. #389C22×C151semidirect
299. C4×C4⋊C4C22×C151semidirect
301. #340C22×C151semidirect
302. #336C22×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#483×C1512
C22×#102×C1512
#2435×C1512
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#1892×C1512
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#1786×C1512
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