Finite Group Databasebeta

← C151×#23075 #50328000

Subgroups — C151×#23075 #50328000

Order 38656.

806 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#476×C151C211yesmaximal, characteristic
2#2365×C151C211yesmaximal, characteristic, complemented
3#2357×C151C211yesmaximal, characteristic
4#1993×C151C211yesmaximal, characteristic, complemented
5#1949×C151C211yesmaximal, characteristic, complemented
6#1936×C151C211yesmaximal, characteristic
7#1934×C151C211yesmaximal, characteristic
8#1926×C151C211yesmaximal, characteristic
9#1925×C151C211yesmaximal, characteristic
10#1920×C151C211yesmaximal, characteristic, complemented
11#1919×C151C211yesmaximal, characteristic, complemented
12#1776×C151C211yesmaximal, characteristic, complemented
13C4×#106×C151C211yesmaximal, characteristic, complemented
14#1771×C151C211yesmaximal, characteristic, complemented
15#1762×C151C211yesmaximal, characteristic
16C2×C4⋊Q8×C151C2211yescharacteristic
17C2×C4×Q8×C151—21no—
18C2×C4×D4×C151C2211yescharacteristic
19#476×C151C418yescomplemented
20C2×#107×C151C2211yescharacteristic
21C2×#106×C151C2211yescharacteristic
22C2×#106×C151C2211yescharacteristic
23C2×#106×C151C2211yescharacteristic
24C2×C4⋊C8×C151C2211yescharacteristic
25C2×#102×C151C2211yescharacteristic
26C2×#102×C151C2211yescharacteristic
27C2×#102×C151C2211yescharacteristic
28C2×#102×C151C2211yescharacteristic
29C2×#97×C151C2211yescharacteristic
30C2×#97×C151C2211yescharacteristic
31C2×C4×C8×C151C2211yescharacteristic
32#391×C151—21no—
33#390×C151C2211yescharacteristic
34#390×C151C2211yescharacteristic
35#389×C151C2211yescharacteristic
36#389×C151C2211yescharacteristic
37#388×C151—21no—
38#388×C151—21no—
39#387×C151—21no—
40#387×C151—21no—
41#382×C151C2211yescharacteristic
42#381×C151—21no—
43C4×C4⋊C4×C151C2211yescharacteristic
44C4×C4⋊C4×C151C2211yescharacteristic
45C4×C4⋊C4×C151C2211yescharacteristic
46C4×C4⋊C4×C151C2211yescharacteristic
47C4×C4⋊C4×C151C2211yescharacteristic
48C4×C4⋊C4×C151C2211yescharacteristic
49C4×C22⋊C4×C151—21no—
50#340×C151C2211yescharacteristic
51#340×C151C2211yescharacteristic
52#340×C151C2211yescharacteristic
53#340×C151C2211yescharacteristic
54#340×C151C2211yescharacteristic
55#340×C151C2211yescharacteristic
56#340×C151C2211yescharacteristic
57#340×C151C2211yescharacteristic
58#336×C151C2211yescharacteristic
59#336×C151C2211yescharacteristic
60C22×Q8×C151—21no—
61C22×D4×C151C2311yescharacteristic
62C23×C4×C151—21no—
63C22×C8×C151C2311yescharacteristic
64C22×C8×C151C2311yescharacteristic
65C22×C8×C151C2311yescharacteristic
66C22×C8×C151C2311yescharacteristic
67C4⋊Q8×C151—22no—
68C4⋊Q8×C151C2×C414yes—
69C4×Q8×C151—44no—
70C4×D4×C151—22no—
71C4×D4×C151C2×C414yes—
72C2×C4⋊C4×C151—21no—
73C2×C4⋊C4×C151—21no—
74C2×C4⋊C4×C151—21no—
75C2×C4⋊C4×C151D411yescharacteristic
76C2×C4⋊C4×C151D411yescharacteristic
77C2×C4⋊C4×C151C2311yescharacteristic
78C2×C4⋊C4×C151C2311yescharacteristic
79C2×C4⋊C4×C151C2311yescharacteristic
80C2×C4⋊C4×C151C2311yescharacteristic
81C2×C4⋊C4×C151C2311yescharacteristic
82C2×C4⋊C4×C151C2311yescharacteristic
83C2×C4⋊C4×C151C2311yescharacteristic
84C2×C22⋊C4×C151—21no—
85C2×C22⋊C4×C151—21no—
86C2×C22⋊C4×C151—21no—
87C2×C22⋊C4×C151—21no—
88C2×C22⋊C4×C151—21no—
89C2×C42×C151—21no—
90C2×C42×C151—21no—
91C2×C42×C151—21no—
92C2×C42×C151—21no—
93C2×C42×C151—21no—
94C2×C42×C151C2311yescharacteristic
95C2×C42×C151C2311yescharacteristic
96C2×C42×C151C2311yescharacteristic
97#107×C151—22no—
98#106×C151—22no—
99#106×C151C2×C414yes—
100#106×C151C2×C414yes—
101C4⋊C8×C151C2×C414yes—
102#102×C151—22no—
103#102×C151—22no—
104#102×C151C2×C414yes—
105#102×C151C2×C414yes—
106#97×C151—22no—
107#97×C151—22no—
108C4×C8×C151—22no—
109#95×C151—21no—
110#95×C151—21no—
111#95×C151—21no—
112#95×C151—21no—
113#95×C151—21no—
114#95×C151—21no—
115#95×C151—21no—
116#95×C151—21no—
117#95×C151—21no—
118#95×C151D411yescharacteristic
119#95×C151D411yescharacteristic
120C24×C151—21no—
121C2×Q8×C151—24no—
122C2×Q8×C151—42no—
123C2×Q8×C151—42no—
124C2×D4×C151—21no—
125C2×D4×C151—21no—
126C2×D4×C151—21no—
127C2×D4×C151—21no—
128C2×D4×C151—21no—
129C2×D4×C151C22×C412yes—
130C22×C4×C151—21no—
131C22×C4×C151—21no—
132C22×C4×C151—21no—
133C22×C4×C151—21no—
134C22×C4×C151—24no—
135C22×C4×C151—21no—
136C22×C4×C151—21no—
137C22×C4×C151—21no—
138C22×C4×C151—81no—
139C22×C4×C151—81no—
140C22×C4×C151C2×D411yescharacteristic
141C22×C4×C151C2×D411yescharacteristic
142C22×C4×C151C2×D411yescharacteristic
143C22×C4×C151C2×D411yescharacteristic
144C22×C4×C151C2×D411yescharacteristic
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2411yescharacteristic
147C2×C8×C151—21no—
148C2×C8×C151—21no—
149C2×C8×C151—21no—
150C2×C8×C151—21no—
151C2×C8×C151—21no—
152C2×C8×C151—21no—
153C2×C8×C151—21no—
154C2×C8×C151—21no—
155C2×C8×C151—21no—
156C2×C8×C151C22×C412yes—
157C2×C8×C151C22×C412yes—
158C2×C8×C151#4112yes—
159C4⋊C4×C151—24no—
160C4⋊C4×C151—22no—
161C4⋊C4×C151—21no—
162C4⋊C4×C151—21no—
163C4⋊C4×C151—21no—
164C4⋊C4×C151—21no—
165C4⋊C4×C151—21no—
166C4⋊C4×C151—21no—
167C4⋊C4×C151—21no—
168C4⋊C4×C151—21no—
169C4⋊C4×C151—21no—
170C4⋊C4×C151—21no—
171C4⋊C4×C151—41no—
172C4⋊C4×C151—42no—
173C4⋊C4×C151—42no—
174C4⋊C4×C151C22×C412yes—
175C4⋊C4×C151C22×C412yes—
176C4⋊C4×C151C22×C412yes—
177C4⋊C4×C151#4112yes—
178C22⋊C4×C151—24no—
179C22⋊C4×C151—42no—
180C22⋊C4×C151—42no—
181C22⋊C4×C151—81no—
182C22⋊C4×C151—81no—
183C42×C151—21no—
184C42×C151—21no—
185C42×C151—21no—
186C42×C151—21no—
187C42×C151—42no—
188C42×C151—42no—
189C42×C151—42no—
190C42×C151—81no—
191C42×C151—81no—
192C42×C151C22×C412yes—
193C42×C151#4112yes—
194C23×C151—22no—
195C23×C151—41no—
196C23×C151—41no—
197C23×C151—41no—
198C23×C151—41no—
199C23×C151—41no—
200C23×C151—41no—
201C23×C151C22×D411yescharacteristic
202Q8×C151—48no—
203D4×C151—22no—
204D4×C151—22no—
205D4×C151—41no—
206D4×C151—41no—
207C2×C4×C151—21no—
208C2×C4×C151—21no—
209C2×C4×C151—21no—
210C2×C4×C151—21no—
211C2×C4×C151—21no—
212C2×C4×C151—21no—
213C2×C4×C151—22no—
214C2×C4×C151—22no—
215C2×C4×C151—22no—
216C2×C4×C151—21no—
217C2×C4×C151—21no—
218C2×C4×C151—21no—
219C2×C4×C151—22no—
220C2×C4×C151—21no—
221C2×C4×C151—21no—
222C2×C4×C151—21no—
223C2×C4×C151—21no—
224C2×C4×C151—21no—
225C2×C4×C151—21no—
226C2×C4×C151—41no—
227C2×C4×C151—41no—
228C2×C4×C151—41no—
229C2×C4×C151—41no—
230C2×C4×C151—41no—
231C2×C4×C151—41no—
232C2×C4×C151—41no—
233C2×C4×C151—41no—
234C2×C4×C151—41no—
235C2×C4×C151—41no—
236C2×C4×C151—41no—
237C2×C4×C151—41no—
238C2×C4×C151—41no—
239C2×C4×C151—41no—
240C2×C4×C151—41no—
241C2×C4×C151—41no—
242C2×C4×C151—41no—
243C2×C4×C151—84no—
244C2×C4×C151C4×D412yes—
245C2×C4×C151C4×D412yes—
246C2×C4×C151AGL(1,ℤ/8)12yes—
247C2×C4×C151C23×C411yescharacteristic
248C2×C4×C151C2×#4111yescharacteristic
249C2×C4×C151C2×#4111yescharacteristic
250C2×C4×C151C2×#4111yescharacteristic
251C2×C4×C1512−1+411yescharacteristic
252C2×C4×C1512−1+411yescharacteristic
253C8×C151—22no—
254C8×C151—22no—
255C8×C151—22no—
256C8×C151—41no—
257C22×C151—42no—
258C22×C151—42no—
259C22×C151—42no—
260C22×C151—81no—
261C22×C151—81no—
262C22×C151—81no—
263C22×C151—81no—
264C22×C151C2×C4×D411yescharacteristic
265C22×C151#54711yescharacteristic
266C22×C151#54811yescharacteristic
267C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C151#57511yescharacteristic
269C22×C151#57711yescharacteristic
270C22×C151#57711yescharacteristic
271C4×C151—22no—
272C4×C151—22no—
273C4×C151—22no—
274C4×C151—41no—
275C4×C151—41no—
276C4×C151—41no—
277C4×C151—42no—
278C4×C151—42no—
279C4×C151—42no—
280C4×C151—42no—
281C4×C151—81no—
282C4×C151—81no—
283C4×C151—81no—
284C4×C151C4×#4111yescharacteristic
285C4×C151#51911yescharacteristic
286C4×C151#55211yescharacteristic
287C4×C151#55611yescharacteristic
288C2×C151—82no—
289C2×C151#236211yescharacteristic
290C2×C151C4×AGL(1,ℤ/8)11yescharacteristic
291C2×C151#294911yescharacteristic
292C2×C151#334511yescharacteristic
293C2×C151#335011yescharacteristic
294C2×C151#337111yescharacteristic
295C2×C151#337311yescharacteristic
296#23075C15111yesmaximal, characteristic, direct product factor, Sylow
297C151#2307511yescharacteristic, direct product factor, Sylow
298C2×#476C2×C15111yescharacteristic
299#2365C2×C15111yescharacteristic, complemented
300#2357C2×C15111yescharacteristic
301#1993C2×C15111yescharacteristic, complemented
302#1949C2×C15111yescharacteristic, complemented
303#1936C2×C15111yescharacteristic
304#1934C2×C15111yescharacteristic
305#1926C2×C15111yescharacteristic
306#1925C2×C15111yescharacteristic
307#1920C2×C15111yescharacteristic, complemented
308#1919C2×C15111yescharacteristic, complemented
309#1776C2×C15111yescharacteristic, complemented
310C4×#106C2×C15111yescharacteristic, complemented
311#1771C2×C15111yescharacteristic, complemented
312#1762C2×C15111yescharacteristic
313C2×C4⋊Q8C22×C15111yescharacteristic
314C2×C4×Q8—21no—
315C2×C4×D4C22×C15111yescharacteristic
316#476C4×C15118yescomplemented
317C2×#107C22×C15111yescharacteristic
318C2×#106C22×C15111yescharacteristic
319C2×#106C22×C15111yescharacteristic
320C2×#106C22×C15111yescharacteristic
321C2×C4⋊C8C22×C15111yescharacteristic
322C2×#102C22×C15111yescharacteristic
323C2×#102C22×C15111yescharacteristic
324C2×#102C22×C15111yescharacteristic
325C2×#102C22×C15111yescharacteristic
326C2×#97C22×C15111yescharacteristic
327C2×#97C22×C15111yescharacteristic
328C2×C4×C8C22×C15111yescharacteristic
329#391—21no—
330#390C22×C15111yescharacteristic
331#390C22×C15111yescharacteristic
332#389C22×C15111yescharacteristic
333#389C22×C15111yescharacteristic
334#388—21no—
335#388—21no—
336#387—21no—
337#387—21no—
338#382C22×C15111yescharacteristic
339#381—21no—
340C4×C4⋊C4C22×C15111yescharacteristic
341C4×C4⋊C4C22×C15111yescharacteristic
342C4×C4⋊C4C22×C15111yescharacteristic
343C4×C4⋊C4C22×C15111yescharacteristic
344C4×C4⋊C4C22×C15111yescharacteristic
345C4×C4⋊C4C22×C15111yescharacteristic
346C4×C22⋊C4—21no—
347#340C22×C15111yescharacteristic
348#340C22×C15111yescharacteristic
349#340C22×C15111yescharacteristic
350#340C22×C15111yescharacteristic
351#340C22×C15111yescharacteristic
352#340C22×C15111yescharacteristic
353#340C22×C15111yescharacteristic
354#340C22×C15111yescharacteristic
355#336C22×C15111yescharacteristic
356#336C22×C15111yescharacteristic
357C22×Q8—21no—
358C22×D4C23×C15111yescharacteristic
359C23×C4—21no—
360C22×C8C23×C15111yescharacteristic
361C22×C8C23×C15111yescharacteristic
362C22×C8C23×C15111yescharacteristic
363C22×C8C23×C15111yescharacteristic
364C4⋊Q8—22no—
365C4⋊Q8C2×C4×C15114yes—
366C4×Q8—44no—
367C4×D4—22no—
368C4×D4C2×C4×C15114yes—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4D4×C15111yescharacteristic
373C2×C4⋊C4D4×C15111yescharacteristic
374C2×C4⋊C4C23×C15111yescharacteristic
375C2×C4⋊C4C23×C15111yescharacteristic
376C2×C4⋊C4C23×C15111yescharacteristic
377C2×C4⋊C4C23×C15111yescharacteristic
378C2×C4⋊C4C23×C15111yescharacteristic
379C2×C4⋊C4C23×C15111yescharacteristic
380C2×C4⋊C4C23×C15111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42C23×C15111yescharacteristic
392C2×C42C23×C15111yescharacteristic
393C2×C42C23×C15111yescharacteristic
394#107—22no—
395#106—22no—
396#106C2×C4×C15114yes—
397#106C2×C4×C15114yes—
398C4⋊C8C2×C4×C15114yes—
399#102—22no—
400#102—22no—
401#102C2×C4×C15114yes—
402#102C2×C4×C15114yes—
403#97—22no—
404#97—22no—
405C4×C8—22no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95D4×C15111yescharacteristic
416#95D4×C15111yescharacteristic
417C24—21no—
418C2×Q8—24no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C22×C4×C15112yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4C2×D4×C15111yescharacteristic
438C22×C4C2×D4×C15111yescharacteristic
439C22×C4C2×D4×C15111yescharacteristic
440C22×C4C2×D4×C15111yescharacteristic
441C22×C4C2×D4×C15111yescharacteristic
442C22×C4C2×D4×C15111yescharacteristic
443C22×C4C24×C15111yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8C22×C4×C15112yes—
454C2×C8C22×C4×C15112yes—
455C2×C8#41×C15112yes—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4C22×C4×C15112yes—
472C4⋊C4C22×C4×C15112yes—
473C4⋊C4C22×C4×C15112yes—
474C4⋊C4#41×C15112yes—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—81no—
488C42—81no—
489C42C22×C4×C15112yes—
490C42#41×C15112yes—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23C22×D4×C15111yescharacteristic
499Q8—48no—
500D4—22no—
501D4—22no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4C4×D4×C15112yes—
542C2×C4C4×D4×C15112yes—
543C2×C4AGL(1,ℤ/8)×C15112yes—
544C2×C4C23×C4×C15111yescharacteristic
545C2×C4C2×#41×C15111yescharacteristic
546C2×C4C2×#41×C15111yescharacteristic
547C2×C4C2×#41×C15111yescharacteristic
548C2×C42−1+4×C15111yescharacteristic
549C2×C42−1+4×C15111yescharacteristic
550C8—22no—
551C8—22no—
552C8—22no—
553C8—41no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22C2×C4×D4×C15111yescharacteristic
562C22#547×C15111yescharacteristic
563C22#548×C15111yescharacteristic
564C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
565C22#575×C15111yescharacteristic
566C22#577×C15111yescharacteristic
567C22#577×C15111yescharacteristic
568C4—22no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4C4×#41×C15111yescharacteristic
582C4#519×C15111yescharacteristic
583C4#552×C15111yescharacteristic
584C4#556×C15111yescharacteristic
585C2—82no—
586C2#2362×C15111yescharacteristic
587C2C4×AGL(1,ℤ/8)×C15111yescharacteristic
588C2#2949×C15111yescharacteristic
589C2#3345×C15111yescharacteristic
590C2#3350×C15111yescharacteristic
591C2#3371×C15111yescharacteristic
592C2#3373×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutatorC2×C4C23×C4×C151
Frattini, cosocleC22×C4C24×C151
prime radicalC22×D4×C151C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C4×C4⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2365×C151C2semidirect
4. #1993×C151C2semidirect
5. #1949×C151C2semidirect
10. #1920×C151C2semidirect
11. #1919×C151C2semidirect
12. #1776×C151C2semidirect
13. C4×#106×C151C2semidirect
14. #1771×C151C2semidirect
19. #476×C151C4semidirect
296. #23075C151direct (N × H)
297. C151#23075direct (N × H)
299. #2365C2×C151semidirect
301. #1993C2×C151semidirect
302. #1949C2×C151semidirect
307. #1920C2×C151semidirect
308. #1919C2×C151semidirect
309. #1776C2×C151semidirect
310. C4×#106C2×C151semidirect
311. #1771C2×C151semidirect
316. #476C4×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#476×C1512
#2365×C1512
#2357×C1512
#1993×C1512
#1949×C1512
#1936×C1512
#1934×C1512
#1926×C1512
#1925×C1512
#1920×C1512
#1919×C1512
#1776×C1512
C4×#106×C1512
#1771×C1512
#1762×C1512
#23075151