Finite Group Databasebeta

← C151×#24481 #50329406

Subgroups — C151×#24481 #50329406

Order 38656.

1012 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#480×C151C211yesmaximal, characteristic
2C22×#103×C151C211yesmaximal, characteristic, complemented
3#2436×C151C211yesmaximal, characteristic
4#2344×C151C211yesmaximal, characteristic, complemented
5#2068×C151C212yesmaximal, complemented
6#2042×C151C212yesmaximal, complemented
7#1983×C151C212yesmaximal, complemented
8#1928×C151C212yesmaximal, complemented
9#1893×C151C211yesmaximal, characteristic
10#1850×C151C211yesmaximal, characteristic, complemented
11#1787×C151C211yesmaximal, characteristic, complemented
12C23×Q8×C151C2211yescharacteristic
13C23×C8×C151C2211yescharacteristic
14C2×#124×C151C2212yes—
15C2×#121×C151C2211yescharacteristic
16C2×#117×C151—21no—
17C22×C4⋊C4×C151C2211yescharacteristic
18#480×C151—24no—
19C2×#107×C151C2211yescharacteristic
20C2×#107×C151C2211yescharacteristic
21C2×C4⋊C8×C151C2212yes—
22C2×#103×C151—24no—
23C2×#103×C151C2212yes—
24C2×#103×C151C2212yes—
25C2×#103×C151C2212yes—
26C2×#103×C151C2212yes—
27C2×#102×C151C2211yescharacteristic
28C2×#102×C151C2211yescharacteristic
29C2×#102×C151C2212yes—
30C2×C22⋊C8×C151C2211yescharacteristic
31#393×C151—22no—
32#391×C151C2212yes—
33#389×C151C2212yes—
34#387×C151—22no—
35#386×C151—21no—
36#386×C151—21no—
37#381×C151—21no—
38#380×C151C2211yescharacteristic
39C4×C4⋊C4×C151C2212yes—
40C4×C22⋊C4×C151C2211yescharacteristic
41#340×C151C2212yes—
42#336×C151C2212yes—
43C22×Q8×C151—22no—
44C22×Q8×C151—24no—
45C22×Q8×C151—22no—
46C22×Q8×C151—21no—
47C22×Q8×C151—42no—
48C22×Q8×C151C2312yes—
49C22×D4×C151—21no—
50C22×D4×C151C2311yescharacteristic
51C23×C4×C151—21no—
52C23×C4×C151—21no—
53C23×C4×C151C2311yescharacteristic
54C22×C8×C151—22no—
55C22×C8×C151—42no—
56C22×C8×C151C2311yescharacteristic
57C22×C8×C151C2312yes—
58C22×C8×C151C2311yescharacteristic
59#124×C151—24no—
60#124×C151—42no—
61#121×C151—22no—
62#121×C151—22no—
63#117×C151—44no—
64C2×C4⋊C4×C151—21no—
65C2×C4⋊C4×C151—22no—
66C2×C4⋊C4×C151—22no—
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—21no—
75C2×C22⋊C4×C151—21no—
76C2×C22⋊C4×C151D411yescharacteristic
77C2×C22⋊C4×C151D411yescharacteristic
78C2×C42×C151—22no—
79C2×C42×C151C2312yes—
80#107×C151—22no—
81#107×C151—22no—
82C4⋊C8×C151—42no—
83#103×C151—24no—
84#103×C151—24no—
85#103×C151—24no—
86#103×C151—42no—
87#102×C151—22no—
88#102×C151—22no—
89#102×C151—42no—
90C22⋊C8×C151—22no—
91#95×C151—22no—
92#95×C151—21no—
93#95×C151—22no—
94#95×C151—22no—
95#95×C151—22no—
96#95×C151D412yes—
97C24×C151—21no—
98C24×C151C2×D411yescharacteristic
99C2×Q8×C151—22no—
100C2×Q8×C151—28no—
101C2×Q8×C151—22no—
102C2×Q8×C151—22no—
103C2×Q8×C151—22no—
104C2×Q8×C151—22no—
105C2×Q8×C151—44no—
106C2×Q8×C151—48no—
107C2×Q8×C151—44no—
108C2×Q8×C151—42no—
109C2×Q8×C151#4114yes—
110C2×D4×C151—21no—
111C2×D4×C151—21no—
112C2×D4×C151—22no—
113C2×D4×C151—22no—
114C2×D4×C151—42no—
115C2×D4×C151—42no—
116C2×D4×C151—42no—
117C22×C4×C151—22no—
118C22×C4×C151—22no—
119C22×C4×C151—24no—
120C22×C4×C151—21no—
121C22×C4×C151—22no—
122C22×C4×C151—21no—
123C22×C4×C151—21no—
124C22×C4×C151—22no—
125C22×C4×C151—44no—
126C22×C4×C151—42no—
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×C151C2×D412yes—
152C4⋊C4×C151C2×D412yes—
153C22⋊C4×C151—22no—
154C22⋊C4×C151—22no—
155C22⋊C4×C151—44no—
156C22⋊C4×C151—44no—
157C22⋊C4×C151—42no—
158C22⋊C4×C151—42no—
159C42×C151—22no—
160C42×C151—22no—
161C42×C151—44no—
162C23×C151—22no—
163C23×C151—22no—
164C23×C151—41no—
165C23×C151—42no—
166C23×C151—41no—
167C23×C151—41no—
168C23×C151—42no—
169C23×C151#12112yes—
170C23×C151#13512yes—
171C23×C151AGL(1,ℤ/8)12yes—
172C23×C151C22×D411yescharacteristic
173Q8×C151—24no—
174Q8×C151—24no—
175Q8×C151—48no—
176Q8×C151—42no—
177Q8×C151—42no—
178D4×C151—41no—
179D4×C151—41no—
180D4×C151—42no—
181D4×C151—84no—
182C2×C4×C151—22no—
183C2×C4×C151—22no—
184C2×C4×C151—22no—
185C2×C4×C151—24no—
186C2×C4×C151—22no—
187C2×C4×C151—22no—
188C2×C4×C151—24no—
189C2×C4×C151—22no—
190C2×C4×C151—22no—
191C2×C4×C151—22no—
192C2×C4×C151—22no—
193C2×C4×C151—22no—
194C2×C4×C151—22no—
195C2×C4×C151—48no—
196C2×C4×C151—42no—
197C2×C4×C151—42no—
198C2×C4×C151—41no—
199C2×C4×C151—41no—
200C2×C4×C151—41no—
201C2×C4×C151—41no—
202C2×C4×C151—48no—
203C2×C4×C151—41no—
204C2×C4×C151—42no—
205C2×C4×C151—41no—
206C2×C4×C151—42no—
207C2×C4×C151—41no—
208C2×C4×C151—42no—
209C2×C4×C151#12112yes—
210C2×C4×C151C22×D411yescharacteristic
211C2×C4×C151C2×#4112yes—
212C2×C4×C151C2×#4111yescharacteristic
213C2×C4×C151H(5,𝔽2)12yes—
214C8×C151—41no—
215C8×C151—41no—
216C8×C151—42no—
217C22×C151—22no—
218C22×C151—22no—
219C22×C151—24no—
220C22×C151—24no—
221C22×C151—81no—
222C22×C151—81no—
223C22×C151—81no—
224C22×C151—82no—
225C22×C151—82no—
226C22×C151#46314yes—
227C22×C151C2×#12111yescharacteristic
228C22×C151#54612yes—
229C22×C151C2×#13511yescharacteristic
230C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
231C22×C151#57812yes—
232C4×C151—22no—
233C4×C151—44no—
234C4×C151—42no—
235C4×C151—42no—
236C4×C151—42no—
237C4×C151—42no—
238C4×C151—81no—
239C4×C151—82no—
240C4×C151#52511yescharacteristic
241C4×C151#53511yescharacteristic
242C4×C151#55112yes—
243C2×C151—24no—
244C2×C151—161no—
245C2×C151#241111yescharacteristic
246C2×C151C2×#46311yescharacteristic
247C2×C151#304411yescharacteristic
248C2×C151#330012yes—
249C2×C151#330812yes—
250#24481C15111yesmaximal, characteristic, direct product factor, Sylow
251C151#2448111yescharacteristic, direct product factor, Sylow
252C2×#480C2×C15111yescharacteristic
253C22×#103C2×C15111yescharacteristic, complemented
254#2436C2×C15111yescharacteristic
255#2344C2×C15111yescharacteristic, complemented
256#2068C2×C15112yescomplemented
257#2042C2×C15112yescomplemented
258#1983C2×C15112yescomplemented
259#1928C2×C15112yescomplemented
260#1893C2×C15111yescharacteristic
261#1850C2×C15111yescharacteristic, complemented
262#1787C2×C15111yescharacteristic, complemented
263C23×Q8C22×C15111yescharacteristic
264C23×C8C22×C15111yescharacteristic
265C2×#124C22×C15112yes—
266C2×#121C22×C15111yescharacteristic
267C2×#117—21no—
268C22×C4⋊C4C22×C15111yescharacteristic
269#480—24no—
270C2×#107C22×C15111yescharacteristic
271C2×#107C22×C15111yescharacteristic
272C2×C4⋊C8C22×C15112yes—
273C2×#103—24no—
274C2×#103C22×C15112yes—
275C2×#103C22×C15112yes—
276C2×#103C22×C15112yes—
277C2×#103C22×C15112yes—
278C2×#102C22×C15111yescharacteristic
279C2×#102C22×C15111yescharacteristic
280C2×#102C22×C15112yes—
281C2×C22⋊C8C22×C15111yescharacteristic
282#393—22no—
283#391C22×C15112yes—
284#389C22×C15112yes—
285#387—22no—
286#386—21no—
287#386—21no—
288#381—21no—
289#380C22×C15111yescharacteristic
290C4×C4⋊C4C22×C15112yes—
291C4×C22⋊C4C22×C15111yescharacteristic
292#340C22×C15112yes—
293#336C22×C15112yes—
294C22×Q8—21no—
295C22×Q8—24no—
296C22×Q8—22no—
297C22×Q8—22no—
298C22×Q8—42no—
299C22×Q8C23×C15112yes—
300C22×D4—21no—
301C22×D4C23×C15111yescharacteristic
302C23×C4—21no—
303C23×C4—21no—
304C23×C4C23×C15111yescharacteristic
305C22×C8—22no—
306C22×C8—42no—
307C22×C8C23×C15112yes—
308C22×C8C23×C15111yescharacteristic
309C22×C8C23×C15111yescharacteristic
310#124—24no—
311#124—42no—
312#121—22no—
313#121—22no—
314#117—44no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4D4×C15114yes—
319C2×C4⋊C4C23×C15111yescharacteristic
320C2×C4⋊C4C23×C15112yes—
321C2×C4⋊C4C23×C15111yescharacteristic
322C2×C4⋊C4C23×C15111yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4D4×C15111yescharacteristic
328C2×C22⋊C4D4×C15111yescharacteristic
329C2×C42—22no—
330C2×C42C23×C15112yes—
331#107—22no—
332#107—22no—
333C4⋊C8—42no—
334#103—24no—
335#103—24no—
336#103—24no—
337#103—42no—
338#102—22no—
339#102—22no—
340#102—42no—
341C22⋊C8—22no—
342#95—22no—
343#95—21no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95D4×C15112yes—
348C24—21no—
349C24C2×D4×C15111yescharacteristic
350C2×Q8—22no—
351C2×Q8—28no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—42no—
360C2×Q8#41×C15114yes—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4C2×D4×C15112yes—
381C22×C4C2×D4×C15112yes—
382C22×C4C2×D4×C15111yescharacteristic
383C22×C4C2×D4×C15112yes—
384C22×C4#41×C15112yes—
385C22×C4C24×C15111yescharacteristic
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—44no—
394C4⋊C4—24no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—44no—
402C4⋊C4C2×D4×C15112yes—
403C4⋊C4C2×D4×C15112yes—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C42—22no—
411C42—22no—
412C42—44no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23#121×C15112yes—
421C23#135×C15112yes—
422C23AGL(1,ℤ/8)×C15112yes—
423C23C22×D4×C15111yescharacteristic
424Q8—24no—
425Q8—24no—
426Q8—48no—
427Q8—42no—
428Q8—42no—
429D4—41no—
430D4—41no—
431D4—42no—
432D4—84no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4#121×C15112yes—
461C2×C4C22×D4×C15111yescharacteristic
462C2×C4C2×#41×C15112yes—
463C2×C4C2×#41×C15111yescharacteristic
464C2×C4H(5,𝔽2)×C15112yes—
465C8—41no—
466C8—42no—
467C8—41no—
468C22—22no—
469C22—22no—
470C22—24no—
471C22—24no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—82no—
476C22—82no—
477C22#463×C15114yes—
478C22C2×#121×C15111yescharacteristic
479C22#546×C15112yes—
480C22C2×#135×C15111yescharacteristic
481C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
482C22#578×C15112yes—
483C4—22no—
484C4—44no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—81no—
490C4—82no—
491C4#525×C15111yescharacteristic
492C4#535×C15111yescharacteristic
493C4#551×C15112yes—
494C2—24no—
495C2—161no—
496C2#2411×C15111yescharacteristic
497C2C2×#463×C15111yescharacteristic
498C2#3044×C15111yescharacteristic
499C2#3300×C15112yes—
500C2#3308×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×#121×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
2. C22×#103×C151C2semidirect
4. #2344×C151C2semidirect
5. #2068×C151C2semidirect
6. #2042×C151C2semidirect
7. #1983×C151C2semidirect
8. #1928×C151C2semidirect
10. #1850×C151C2semidirect
11. #1787×C151C2semidirect
250. #24481C151direct (N × H)
251. C151#24481direct (N × H)
253. C22×#103C2×C151semidirect
255. #2344C2×C151semidirect
256. #2068C2×C151semidirect
257. #2042C2×C151semidirect
258. #1983C2×C151semidirect
259. #1928C2×C151semidirect
261. #1850C2×C151semidirect
262. #1787C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#480×C1512
C22×#103×C1512
#2436×C1512
#2344×C1512
#2068×C1512
#2042×C1512
#1983×C1512
#1928×C1512
#1893×C1512
#1850×C1512
#1787×C1512
#24481151