Finite Group Databasebeta

← C151×#22489 #50327414

Subgroups — C151×#22489 #50327414

Order 38656.

1206 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555×C151C211yesmaximal, characteristic
2C2×C8×D4×C151C211yesmaximal, characteristic, complemented
3#2354×C151C211yesmaximal, characteristic, complemented
4#1990×C151C211yesmaximal, characteristic, complemented
5#1987×C151C211yesmaximal, characteristic, complemented
6#1893×C151C212yesmaximal, complemented
7#1875×C151C212yesmaximal, complemented
8#1825×C151C212yesmaximal, complemented
9#1802×C151C211yesmaximal, characteristic, complemented
10C4×#102×C151C211yesmaximal, characteristic, complemented
11#1735×C151C212yesmaximal, complemented
12C23×C8×C151C2212yes—
13#555×C151—24no—
14#555×C151C418yescomplemented
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×#124×C151—22no—
17C2×#121×C151—21no—
18C2×#121×C151C2212yes—
19C2×C4×D4×C151C2211yescharacteristic
20C2×C4×D4×C151C2211yescharacteristic
21C2×#117×C151—21no—
22C2×#117×C151C2212yes—
23C2×#117×C151C2212yes—
24C8×D4×C151—24no—
25C2×C4⋊C8×C151C2211yescharacteristic
26C2×#102×C151C2211yescharacteristic
27C2×#102×C151C2211yescharacteristic
28C2×#102×C151C2212yes—
29C2×C22⋊C8×C151C2212yes—
30C2×C4×C8×C151C2211yescharacteristic
31#390×C151—21no—
32#388×C151—22no—
33#384×C151C2211yescharacteristic
34#380×C151C2212yes—
35C4×C4⋊C4×C151—21no—
36C4×C4⋊C4×C151C2211yescharacteristic
37#340×C151C2211yescharacteristic
38#340×C151C2212yes—
39#340×C151C2212yes—
40#340×C151C2212yes—
41#340×C151C2212yes—
42#340×C151C2212yes—
43#340×C151C2211yescharacteristic
44C22×Q8×C151—21no—
45C22×D4×C151—22no—
46C22×D4×C151D411yescharacteristic
47C22×D4×C151C2311yescharacteristic
48C23×C4×C151—21no—
49C23×C4×C151C2312yes—
50C22×C8×C151—24no—
51C22×C8×C151—24no—
52C22×C8×C151—42no—
53C22×C8×C151C2311yescharacteristic
54C22×C8×C151C2312yes—
55C22×C8×C151C2311yescharacteristic
56C4⋊Q8×C151—22no—
57C4⋊Q8×C151C2×C414yes—
58#124×C151—216no—
59#121×C151—28no—
60#121×C151—24no—
61#121×C151C2×C418yes—
62C4×D4×C151—22no—
63C4×D4×C151C2×C414yes—
64C4×D4×C151C2×C414yes—
65C4×D4×C151D414yes—
66#117×C151—24no—
67#117×C151—28no—
68#117×C151—24no—
69#117×C151—42no—
70#117×C151C2×C418yes—
71C2×C4⋊C4×C151—21no—
72C2×C4⋊C4×C151—21no—
73C2×C4⋊C4×C151—21no—
74C2×C4⋊C4×C151D411yescharacteristic
75C2×C4⋊C4×C151C2311yescharacteristic
76C2×C4⋊C4×C151C2312yes—
77C2×C4⋊C4×C151C2311yescharacteristic
78C2×C4⋊C4×C151C2312yes—
79C2×C4⋊C4×C151C2311yescharacteristic
80C2×C22⋊C4×C151—22no—
81C2×C22⋊C4×C151—21no—
82C2×C22⋊C4×C151—22no—
83C2×C22⋊C4×C151—22no—
84C2×C22⋊C4×C151—22no—
85C2×C22⋊C4×C151D412yes—
86C2×C42×C151—22no—
87C2×C42×C151—21no—
88C2×C42×C151—22no—
89C2×C42×C151C2311yescharacteristic
90C4⋊C8×C151—22no—
91#102×C151—22no—
92#102×C151—22no—
93#102×C151—42no—
94C22⋊C8×C151—24no—
95C4×C8×C151—22no—
96#95×C151—22no—
97#95×C151—21no—
98C24×C151—21no—
99C24×C151C2×D412yes—
100C2×Q8×C151—24no—
101C2×Q8×C151—81no—
102C2×Q8×C151—81no—
103C2×D4×C151—28no—
104C2×D4×C151—22no—
105C2×D4×C151—22no—
106C2×D4×C151—21no—
107C2×D4×C151—21no—
108C2×D4×C151—21no—
109C2×D4×C151—21no—
110C2×D4×C151—21no—
111C2×D4×C151—82no—
112C2×D4×C151—82no—
113C2×D4×C151C22⋊C414yes—
114C2×D4×C151C22×C412yes—
115C22×C4×C151—22no—
116C22×C4×C151—22no—
117C22×C4×C151—22no—
118C22×C4×C151—22no—
119C22×C4×C151—21no—
120C22×C4×C151—21no—
121C22×C4×C151—22no—
122C22×C4×C151—24no—
123C22×C4×C151—21no—
124C22×C4×C151—22no—
125C22×C4×C151—42no—
126C22×C4×C151—42no—
127C22×C4×C151C22×C414yes—
128C22×C4×C151C2×D412yes—
129C22×C4×C151C2×D411yescharacteristic
130C22×C4×C151C2×D414yes—
131C22×C4×C151C2×D411yescharacteristic
132C22×C4×C151C2411yescharacteristic
133C2×C8×C151—21no—
134C2×C8×C151—21no—
135C2×C8×C151—21no—
136C2×C8×C151—22no—
137C2×C8×C151—21no—
138C2×C8×C151—22no—
139C2×C8×C151—22no—
140C2×C8×C151—21no—
141C2×C8×C151—48no—
142C2×C8×C151#4112yes—
143C4⋊C4×C151—24no—
144C4⋊C4×C151—21no—
145C4⋊C4×C151—22no—
146C4⋊C4×C151—21no—
147C4⋊C4×C151—21no—
148C4⋊C4×C151—22no—
149C4⋊C4×C151—22no—
150C4⋊C4×C151—21no—
151C4⋊C4×C151—42no—
152C4⋊C4×C151—81no—
153C4⋊C4×C151C22⋊C414yes—
154C4⋊C4×C151C22×C412yes—
155C4⋊C4×C151C22×C414yes—
156C4⋊C4×C151#4112yes—
157C22⋊C4×C151—28no—
158C22⋊C4×C151—28no—
159C22⋊C4×C151—24no—
160C22⋊C4×C151—44no—
161C22⋊C4×C151—82no—
162C22⋊C4×C151C22⋊C418yes—
163C42×C151—28no—
164C42×C151—44no—
165C42×C151—42no—
166C42×C151C22×C412yes—
167C42×C151C2×D412yes—
168C23×C151—22no—
169C23×C151—22no—
170C23×C151—22no—
171C23×C151—22no—
172C23×C151—22no—
173C23×C151—22no—
174C23×C151—22no—
175C23×C151—41no—
176C23×C151—41no—
177C23×C151—41no—
178C23×C151—41no—
179C23×C151—41no—
180C23×C151—41no—
181C23×C151C2×C22⋊C414yes—
182C23×C151C22×D411yescharacteristic
183Q8×C151—84no—
184D4×C151—22no—
185D4×C151—22no—
186D4×C151—28no—
187D4×C151—41no—
188D4×C151—41no—
189D4×C151—88no—
190C2×C4×C151—22no—
191C2×C4×C151—22no—
192C2×C4×C151—21no—
193C2×C4×C151—21no—
194C2×C4×C151—24no—
195C2×C4×C151—24no—
196C2×C4×C151—24no—
197C2×C4×C151—24no—
198C2×C4×C151—24no—
199C2×C4×C151—22no—
200C2×C4×C151—22no—
201C2×C4×C151—41no—
202C2×C4×C151—41no—
203C2×C4×C151—41no—
204C2×C4×C151—42no—
205C2×C4×C151—42no—
206C2×C4×C151—41no—
207C2×C4×C151—48no—
208C2×C4×C151—42no—
209C2×C4×C151—41no—
210C2×C4×C151—41no—
211C2×C4×C151—42no—
212C2×C4×C151—41no—
213C2×C4×C151—42no—
214C2×C4×C151C2×C22⋊C412yes—
215C2×C4×C151C2×C22⋊C414yes—
216C2×C4×C151C2×C22⋊C412yes—
217C2×C4×C151#13512yes—
218C2×C4×C151#13512yes—
219C2×C4×C151C23×C411yescharacteristic
220C2×C4×C151C22×D411yescharacteristic
221C2×C4×C151C2×#4111yescharacteristic
222C2×C4×C151C2×#4111yescharacteristic
223C2×C4×C151H(5,𝔽2)11yescharacteristic
224C2×C4×C1512−1+411yescharacteristic
225C8×C151—22no—
226C8×C151—42no—
227C8×C151—41no—
228C22×C151—24no—
229C22×C151—24no—
230C22×C151—24no—
231C22×C151—42no—
232C22×C151—42no—
233C22×C151—42no—
234C22×C151—42no—
235C22×C151—42no—
236C22×C151—42no—
237C22×C151—42no—
238C22×C151—81no—
239C22×C151—81no—
240C22×C151—81no—
241C22×C151—81no—
242C22×C151C22×C22⋊C411yescharacteristic
243C22×C151#54611yescharacteristic
244C22×C151#54711yescharacteristic
245C22×C151C2×#13511yescharacteristic
246C22×C151C2×#13511yescharacteristic
247C22×C151#57611yescharacteristic
248C22×C151#57711yescharacteristic
249C4×C151—24no—
250C4×C151—22no—
251C4×C151—42no—
252C4×C151—44no—
253C4×C151—42no—
254C4×C151—81no—
255C4×C151—82no—
256C4×C151#41614yes—
257C4×C151C4×#4111yescharacteristic
258C4×C151#52011yescharacteristic
259C4×C151#54611yescharacteristic
260C4×C151#54711yescharacteristic
261C2×C151—44no—
262C2×C151—82no—
263C2×C151#233911yescharacteristic
264C2×C151C2×#41611yescharacteristic
265C2×C151#289711yescharacteristic
266C2×C151#330311yescharacteristic
267C2×C151#330511yescharacteristic
268C2×C151#334011yescharacteristic
269C2×C151#334111yescharacteristic
270#22489C15111yesmaximal, characteristic, direct product factor, Sylow
271C151#2248911yescharacteristic, direct product factor, Sylow
272C2×#555C2×C15111yescharacteristic
273C2×C8×D4C2×C15111yescharacteristic, complemented
274#2354C2×C15111yescharacteristic, complemented
275#1990C2×C15111yescharacteristic, complemented
276#1987C2×C15111yescharacteristic, complemented
277#1893C2×C15112yescomplemented
278#1875C2×C15112yescomplemented
279#1825C2×C15112yescomplemented
280#1802C2×C15111yescharacteristic, complemented
281C4×#102C2×C15111yescharacteristic, complemented
282#1735C2×C15112yescomplemented
283C23×C8C22×C15112yes—
284#555—24no—
285#555C4×C15118yescomplemented
286C2×C4⋊Q8C22×C15111yescharacteristic
287C2×#124—22no—
288C2×#121—21no—
289C2×#121C22×C15112yes—
290C2×C4×D4C22×C15111yescharacteristic
291C2×C4×D4C22×C15111yescharacteristic
292C2×#117—21no—
293C2×#117C22×C15112yes—
294C2×#117C22×C15112yes—
295C8×D4—24no—
296C2×C4⋊C8C22×C15111yescharacteristic
297C2×#102C22×C15111yescharacteristic
298C2×#102C22×C15111yescharacteristic
299C2×#102C22×C15112yes—
300C2×C22⋊C8C22×C15112yes—
301C2×C4×C8C22×C15111yescharacteristic
302#390—21no—
303#388—22no—
304#384C22×C15111yescharacteristic
305#380C22×C15112yes—
306C4×C4⋊C4—21no—
307C4×C4⋊C4C22×C15111yescharacteristic
308#340C22×C15112yes—
309#340C22×C15111yescharacteristic
310#340C22×C15112yes—
311#340C22×C15112yes—
312#340C22×C15112yes—
313#340C22×C15112yes—
314#340C22×C15111yescharacteristic
315C22×Q8—21no—
316C22×D4—22no—
317C22×D4D4×C15111yescharacteristic
318C22×D4C23×C15111yescharacteristic
319C23×C4—21no—
320C23×C4C23×C15112yes—
321C22×C8—24no—
322C22×C8—24no—
323C22×C8—42no—
324C22×C8C23×C15112yes—
325C22×C8C23×C15111yescharacteristic
326C22×C8C23×C15111yescharacteristic
327C4⋊Q8—22no—
328C4⋊Q8C2×C4×C15114yes—
329#124—216no—
330#121—28no—
331#121—24no—
332#121C2×C4×C15118yes—
333C4×D4—22no—
334C4×D4C2×C4×C15114yes—
335C4×D4C2×C4×C15114yes—
336C4×D4D4×C15114yes—
337#117—24no—
338#117—28no—
339#117—24no—
340#117—42no—
341#117C2×C4×C15118yes—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4D4×C15111yescharacteristic
346C2×C4⋊C4C23×C15112yes—
347C2×C4⋊C4C23×C15111yescharacteristic
348C2×C4⋊C4C23×C15111yescharacteristic
349C2×C4⋊C4C23×C15111yescharacteristic
350C2×C4⋊C4C23×C15112yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4D4×C15112yes—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42C23×C15111yescharacteristic
361C4⋊C8—22no—
362#102—22no—
363#102—22no—
364#102—42no—
365C22⋊C8—24no—
366C4×C8—22no—
367#95—22no—
368#95—21no—
369C24—21no—
370C24C2×D4×C15112yes—
371C2×Q8—24no—
372C2×Q8—81no—
373C2×Q8—81no—
374C2×D4—28no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4C22⋊C4×C15114yes—
385C2×D4C22×C4×C15112yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4C22×C4×C15114yes—
399C22×C4C2×D4×C15112yes—
400C22×C4C2×D4×C15111yescharacteristic
401C22×C4C2×D4×C15114yes—
402C22×C4C2×D4×C15111yescharacteristic
403C22×C4C24×C15111yescharacteristic
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—48no—
413C2×C8#41×C15112yes—
414C4⋊C4—24no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—42no—
423C4⋊C4—81no—
424C4⋊C4C22⋊C4×C15114yes—
425C4⋊C4C22×C4×C15112yes—
426C4⋊C4C22×C4×C15114yes—
427C4⋊C4#41×C15112yes—
428C22⋊C4—28no—
429C22⋊C4—28no—
430C22⋊C4—24no—
431C22⋊C4—44no—
432C22⋊C4—82no—
433C22⋊C4C22⋊C4×C15118yes—
434C42—28no—
435C42—44no—
436C42—42no—
437C42C22×C4×C15112yes—
438C42C2×D4×C15112yes—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23C2×C22⋊C4×C15114yes—
453C23C22×D4×C15111yescharacteristic
454Q8—84no—
455D4—28no—
456D4—22no—
457D4—22no—
458D4—41no—
459D4—41no—
460D4—88no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—48no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4C2×C22⋊C4×C15114yes—
486C2×C4C2×C22⋊C4×C15112yes—
487C2×C4C2×C22⋊C4×C15112yes—
488C2×C4#135×C15112yes—
489C2×C4#135×C15112yes—
490C2×C4C23×C4×C15111yescharacteristic
491C2×C4C22×D4×C15111yescharacteristic
492C2×C4C2×#41×C15111yescharacteristic
493C2×C4C2×#41×C15111yescharacteristic
494C2×C4H(5,𝔽2)×C15111yescharacteristic
495C2×C42−1+4×C15111yescharacteristic
496C8—22no—
497C8—42no—
498C8—41no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22C22×C22⋊C4×C15111yescharacteristic
514C22#546×C15111yescharacteristic
515C22#547×C15111yescharacteristic
516C22C2×#135×C15111yescharacteristic
517C22C2×#135×C15111yescharacteristic
518C22#576×C15111yescharacteristic
519C22#577×C15111yescharacteristic
520C4—24no—
521C4—22no—
522C4—42no—
523C4—44no—
524C4—42no—
525C4—82no—
526C4—81no—
527C4#416×C15114yes—
528C4C4×#41×C15111yescharacteristic
529C4#520×C15111yescharacteristic
530C4#546×C15111yescharacteristic
531C4#547×C15111yescharacteristic
532C2—44no—
533C2—82no—
534C2#2339×C15111yescharacteristic
535C2C2×#416×C15111yescharacteristic
536C2#2897×C15111yescharacteristic
537C2#3303×C15111yescharacteristic
538C2#3305×C15111yescharacteristic
539C2#3340×C15111yescharacteristic
540C2#3341×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutatorC2×C4C23×C4×C151
Frattini, cosocleC22×C4C24×C151
prime radicalC2×#555×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×C8×D4×C151C2semidirect
3. #2354×C151C2semidirect
4. #1990×C151C2semidirect
5. #1987×C151C2semidirect
6. #1893×C151C2semidirect
7. #1875×C151C2semidirect
8. #1825×C151C2semidirect
9. #1802×C151C2semidirect
10. C4×#102×C151C2semidirect
11. #1735×C151C2semidirect
14. #555×C151C4semidirect
270. #22489C151direct (N × H)
271. C151#22489direct (N × H)
273. C2×C8×D4C2×C151semidirect
274. #2354C2×C151semidirect
275. #1990C2×C151semidirect
276. #1987C2×C151semidirect
277. #1893C2×C151semidirect
278. #1875C2×C151semidirect
279. #1825C2×C151semidirect
280. #1802C2×C151semidirect
281. C4×#102C2×C151semidirect
282. #1735C2×C151semidirect
285. #555C4×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#555×C1512
C2×C8×D4×C1512
#2354×C1512
#1990×C1512
#1987×C1512
#1893×C1512
#1875×C1512
#1825×C1512
#1802×C1512
C4×#102×C1512
#1735×C1512
#22489151