Finite Group Databasebeta

← C151×#25002 #50329927

Subgroups — C151×#25002 #50329927

Order 38656.

716 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2580×C151C211yesmaximal, characteristic
2Q8×C4⋊C4×C151C211yesmaximal, characteristic, complemented
3#2100×C151C211yesmaximal, characteristic, complemented
4#2099×C151C211yesmaximal, characteristic, complemented
5#2071×C151C211yesmaximal, characteristic
6#2071×C151C211yesmaximal, characteristic
7#1985×C151C211yesmaximal, characteristic, complemented
8#1949×C151C211yesmaximal, characteristic, complemented
9#1926×C151C211yesmaximal, characteristic
10#1925×C151C211yesmaximal, characteristic
11#1921×C151C211yesmaximal, characteristic, complemented
12#1920×C151C211yesmaximal, characteristic, complemented
13#1836×C151C211yesmaximal, characteristic, complemented
14#1802×C151C211yesmaximal, characteristic
15#1762×C151C211yesmaximal, characteristic
16C2×C4×Q8×C151—21no—
17C2×C4×Q8×C151C2211yescharacteristic
18C2×C4×D4×C151C2211yescharacteristic
19C2×#107×C151C2211yescharacteristic
20C2×#106×C151C2211yescharacteristic
21C2×C4⋊C8×C151C2211yescharacteristic
22C2×#102×C151C2211yescharacteristic
23C2×#102×C151C2211yescharacteristic
24C2×#102×C151C2211yescharacteristic
25C2×#102×C151C2211yescharacteristic
26C2×#97×C151C2211yescharacteristic
27#399×C151—21no—
28#393×C151—21no—
29#390×C151C2211yescharacteristic
30#390×C151C2211yescharacteristic
31#389×C151—21no—
32#389×C151—21no—
33#389×C151—21no—
34#389×C151C2211yescharacteristic
35#389×C151C2211yescharacteristic
36#385×C151—21no—
37#385×C151—21no—
38#384×C151C2211yescharacteristic
39#384×C151C2211yescharacteristic
40#384×C151C2211yescharacteristic
41#382×C151C2211yescharacteristic
42C4×C4⋊C4×C151C2211yescharacteristic
43C4×C4⋊C4×C151C2211yescharacteristic
44C4×C4⋊C4×C151C2211yescharacteristic
45#340×C151C2211yescharacteristic
46#340×C151C2211yescharacteristic
47#340×C151C2211yescharacteristic
48#340×C151C2211yescharacteristic
49#340×C151C2211yescharacteristic
50#340×C151C2211yescharacteristic
51#340×C151C2211yescharacteristic
52#340×C151C2211yescharacteristic
53#340×C151C2211yescharacteristic
54#340×C151C2211yescharacteristic
55#336×C151C2211yescharacteristic
56#336×C151C2211yescharacteristic
57#336×C151C2211yescharacteristic
58#336×C151C2211yescharacteristic
59C22×Q8×C151D411yescharacteristic
60C22×D4×C151C2311yescharacteristic
61C23×C4×C151—21no—
62C22×C8×C151C2311yescharacteristic
63C22×C8×C151C2311yescharacteristic
64C22×C8×C151C2311yescharacteristic
65C22×C8×C151C2311yescharacteristic
66C4×Q8×C151—22no—
67C4×Q8×C151—22no—
68C4×Q8×C151—44no—
69C4×D4×C151—22no—
70C4×D4×C151—22no—
71C2×C4⋊C4×C151—21no—
72C2×C4⋊C4×C151—21no—
73C2×C4⋊C4×C151—21no—
74C2×C4⋊C4×C151—21no—
75C2×C4⋊C4×C151—21no—
76C2×C4⋊C4×C151—21no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151D411yescharacteristic
80C2×C4⋊C4×C151D411yescharacteristic
81C2×C4⋊C4×C151D411yescharacteristic
82C2×C4⋊C4×C151C2311yescharacteristic
83C2×C4⋊C4×C151C2311yescharacteristic
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C4⋊C4×C151C2311yescharacteristic
86C2×C4⋊C4×C151C2311yescharacteristic
87C2×C4⋊C4×C151C2311yescharacteristic
88C2×C4⋊C4×C151C2311yescharacteristic
89C2×C22⋊C4×C151—21no—
90C2×C22⋊C4×C151—21no—
91C2×C22⋊C4×C151—21no—
92C2×C22⋊C4×C151—21no—
93C2×C22⋊C4×C151—21no—
94C2×C42×C151—21no—
95C2×C42×C151—21no—
96C2×C42×C151—21no—
97C2×C42×C151C2311yescharacteristic
98C2×C42×C151C2311yescharacteristic
99C2×C42×C151C2311yescharacteristic
100#107×C151—22no—
101#106×C151—22no—
102C4⋊C8×C151—22no—
103#102×C151—22no—
104#102×C151—22no—
105#102×C151—22no—
106#102×C151—22no—
107#97×C151—22no—
108#95×C151—21no—
109#95×C151—21no—
110#95×C151—21no—
111#95×C151—21no—
112#95×C151—21no—
113#95×C151—21no—
114#95×C151—21no—
115C24×C151—21no—
116C2×Q8×C151—22no—
117C2×Q8×C151—22no—
118C2×Q8×C151—22no—
119C2×D4×C151—21no—
120C2×D4×C151—21no—
121C2×D4×C151—21no—
122C2×D4×C151—21no—
123C2×D4×C151—21no—
124C2×D4×C151#4112yes—
125C22×C4×C151—21no—
126C22×C4×C151—21no—
127C22×C4×C151—21no—
128C22×C4×C151—21no—
129C22×C4×C151—21no—
130C22×C4×C151—21no—
131C22×C4×C151—21no—
132C22×C4×C151—42no—
133C22×C4×C151—81no—
134C22×C4×C151—81no—
135C22×C4×C151C2×D411yescharacteristic
136C22×C4×C151C2×D411yescharacteristic
137C22×C4×C151C2×D411yescharacteristic
138C22×C4×C151C2×D411yescharacteristic
139C22×C4×C151C2×D411yescharacteristic
140C22×C4×C151C2×D411yescharacteristic
141C22×C4×C151C2411yescharacteristic
142C2×C8×C151—21no—
143C2×C8×C151—21no—
144C2×C8×C151—21no—
145C2×C8×C151—21no—
146C2×C8×C151—21no—
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—
154C4⋊C4×C151—22no—
155C4⋊C4×C151—22no—
156C4⋊C4×C151—22no—
157C4⋊C4×C151—21no—
158C4⋊C4×C151—21no—
159C4⋊C4×C151—21no—
160C4⋊C4×C151—21no—
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—42no—
171C4⋊C4×C151—42no—
172C4⋊C4×C151—42no—
173C4⋊C4×C151—81no—
174C4⋊C4×C151—81no—
175C4⋊C4×C151—81no—
176C4⋊C4×C151—81no—
177C4⋊C4×C151—81no—
178C4⋊C4×C151#4112yes—
179C22⋊C4×C151—42no—
180C22⋊C4×C151—42no—
181C22⋊C4×C151—42no—
182C22⋊C4×C151—81no—
183C22⋊C4×C151—81no—
184C42×C151—21no—
185C42×C151—21no—
186C42×C151—21no—
187C42×C151—21no—
188C42×C151—42no—
189C42×C151—42no—
190C42×C151—42no—
191C42×C151#4112yes—
192C42×C151#4112yes—
193C23×C151—22no—
194C23×C151—41no—
195C23×C151—41no—
196C23×C151—41no—
197C23×C151—41no—
198C23×C151—41no—
199C23×C151—41no—
200C23×C151C22×D411yescharacteristic
201Q8×C151—28no—
202D4×C151—22no—
203D4×C151—22no—
204D4×C151—41no—
205D4×C151—41no—
206C2×C4×C151—21no—
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—21no—
214C2×C4×C151—21no—
215C2×C4×C151—21no—
216C2×C4×C151—22no—
217C2×C4×C151—21no—
218C2×C4×C151—21no—
219C2×C4×C151—21no—
220C2×C4×C151—21no—
221C2×C4×C151—41no—
222C2×C4×C151—41no—
223C2×C4×C151—41no—
224C2×C4×C151—41no—
225C2×C4×C151—41no—
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—84no—
242C2×C4×C151#12312yes—
243C2×C4×C151#12312yes—
244C2×C4×C151AGL(1,ℤ/8)12yes—
245C2×C4×C151#13712yes—
246C2×C4×C151C2×#4111yescharacteristic
247C2×C4×C151C2×#4111yescharacteristic
248C2×C4×C151C2×#4111yescharacteristic
249C2×C4×C151C2×#4111yescharacteristic
250C2×C4×C1512−1+411yescharacteristic
251C2×C4×C1512−1+411yescharacteristic
252C8×C151—41no—
253C8×C151—41no—
254C8×C151—41no—
255C8×C151—41no—
256C22×C151—42no—
257C22×C151—42no—
258C22×C151—42no—
259C22×C151—81no—
260C22×C151—81no—
261C22×C151—81no—
262C22×C151—81no—
263C22×C151C2×#12311yescharacteristic
264C22×C151#54811yescharacteristic
265C22×C151#54811yescharacteristic
266C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C151C2×#13711yescharacteristic
268C22×C151#57611yescharacteristic
269C22×C151#57711yescharacteristic
270C4×C151—22no—
271C4×C151—22no—
272C4×C151—22no—
273C4×C151—22no—
274C4×C151—41no—
275C4×C151—41no—
276C4×C151—42no—
277C4×C151—81no—
278C4×C151—81no—
279C4×C151—81no—
280C4×C151—81no—
281C4×C151—81no—
282C4×C151—81no—
283C4×C151#52911yescharacteristic
284C4×C151#54111yescharacteristic
285C4×C151#55211yescharacteristic
286C4×C151#55611yescharacteristic
287C2×C151—82no—
288C2×C151#242111yescharacteristic
289C2×C151#309211yescharacteristic
290C2×C151#309611yescharacteristic
291C2×C151#336611yescharacteristic
292C2×C151#337211yescharacteristic
293C2×C151#337311yescharacteristic
294C2×C151#337511yescharacteristic
295#25002C15111yesmaximal, characteristic, direct product factor, Sylow
296C151#2500211yescharacteristic, direct product factor, Sylow
297#2580C2×C15111yescharacteristic
298Q8×C4⋊C4C2×C15111yescharacteristic, complemented
299#2100C2×C15111yescharacteristic, complemented
300#2099C2×C15111yescharacteristic, complemented
301#2071C2×C15111yescharacteristic
302#2071C2×C15111yescharacteristic
303#1985C2×C15111yescharacteristic, complemented
304#1949C2×C15111yescharacteristic, complemented
305#1926C2×C15111yescharacteristic
306#1925C2×C15111yescharacteristic
307#1921C2×C15111yescharacteristic, complemented
308#1920C2×C15111yescharacteristic, complemented
309#1836C2×C15111yescharacteristic, complemented
310#1802C2×C15111yescharacteristic
311#1762C2×C15111yescharacteristic
312C2×C4×Q8—21no—
313C2×C4×Q8C22×C15111yescharacteristic
314C2×C4×D4C22×C15111yescharacteristic
315C2×#107C22×C15111yescharacteristic
316C2×#106C22×C15111yescharacteristic
317C2×C4⋊C8C22×C15111yescharacteristic
318C2×#102C22×C15111yescharacteristic
319C2×#102C22×C15111yescharacteristic
320C2×#102C22×C15111yescharacteristic
321C2×#102C22×C15111yescharacteristic
322C2×#97C22×C15111yescharacteristic
323#399—21no—
324#393—21no—
325#390C22×C15111yescharacteristic
326#390C22×C15111yescharacteristic
327#389—21no—
328#389—21no—
329#389—21no—
330#389C22×C15111yescharacteristic
331#389C22×C15111yescharacteristic
332#385—21no—
333#385—21no—
334#384C22×C15111yescharacteristic
335#384C22×C15111yescharacteristic
336#384C22×C15111yescharacteristic
337#382C22×C15111yescharacteristic
338C4×C4⋊C4C22×C15111yescharacteristic
339C4×C4⋊C4C22×C15111yescharacteristic
340C4×C4⋊C4C22×C15111yescharacteristic
341#340C22×C15111yescharacteristic
342#340C22×C15111yescharacteristic
343#340C22×C15111yescharacteristic
344#340C22×C15111yescharacteristic
345#340C22×C15111yescharacteristic
346#340C22×C15111yescharacteristic
347#340C22×C15111yescharacteristic
348#340C22×C15111yescharacteristic
349#340C22×C15111yescharacteristic
350#340C22×C15111yescharacteristic
351#336C22×C15111yescharacteristic
352#336C22×C15111yescharacteristic
353#336C22×C15111yescharacteristic
354#336C22×C15111yescharacteristic
355C22×Q8D4×C15111yescharacteristic
356C22×D4C23×C15111yescharacteristic
357C23×C4—21no—
358C22×C8C23×C15111yescharacteristic
359C22×C8C23×C15111yescharacteristic
360C22×C8C23×C15111yescharacteristic
361C22×C8C23×C15111yescharacteristic
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—44no—
365C4×D4—22no—
366C4×D4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4D4×C15111yescharacteristic
376C2×C4⋊C4D4×C15111yescharacteristic
377C2×C4⋊C4D4×C15111yescharacteristic
378C2×C4⋊C4C23×C15111yescharacteristic
379C2×C4⋊C4C23×C15111yescharacteristic
380C2×C4⋊C4C23×C15111yescharacteristic
381C2×C4⋊C4C23×C15111yescharacteristic
382C2×C4⋊C4C23×C15111yescharacteristic
383C2×C4⋊C4C23×C15111yescharacteristic
384C2×C4⋊C4C23×C15111yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42C23×C15111yescharacteristic
394C2×C42C23×C15111yescharacteristic
395C2×C42C23×C15111yescharacteristic
396#107—22no—
397#106—22no—
398C4⋊C8—22no—
399#102—22no—
400#102—22no—
401#102—22no—
402#102—22no—
403#97—22no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411C24—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4#41×C15112yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4C2×D4×C15111yescharacteristic
432C22×C4C2×D4×C15111yescharacteristic
433C22×C4C2×D4×C15111yescharacteristic
434C22×C4C2×D4×C15111yescharacteristic
435C22×C4C2×D4×C15111yescharacteristic
436C22×C4C2×D4×C15111yescharacteristic
437C22×C4C24×C15111yescharacteristic
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
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—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4#41×C15112yes—
475C22⋊C4—42no—
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#41×C15112yes—
488C42#41×C15112yes—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C22×D4×C15111yescharacteristic
497Q8—28no—
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
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—84no—
538C2×C4#123×C15112yes—
539C2×C4#123×C15112yes—
540C2×C4AGL(1,ℤ/8)×C15112yes—
541C2×C4#137×C15112yes—
542C2×C4C2×#41×C15111yescharacteristic
543C2×C4C2×#41×C15111yescharacteristic
544C2×C4C2×#41×C15111yescharacteristic
545C2×C4C2×#41×C15111yescharacteristic
546C2×C42−1+4×C15111yescharacteristic
547C2×C42−1+4×C15111yescharacteristic
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×#123×C15111yescharacteristic
560C22#548×C15111yescharacteristic
561C22#548×C15111yescharacteristic
562C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
563C22C2×#137×C15111yescharacteristic
564C22#576×C15111yescharacteristic
565C22#577×C15111yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4#529×C15111yescharacteristic
580C4#541×C15111yescharacteristic
581C4#552×C15111yescharacteristic
582C4#556×C15111yescharacteristic
583C2—82no—
584C2#2421×C15111yescharacteristic
585C2#3092×C15111yescharacteristic
586C2#3096×C15111yescharacteristic
587C2#3366×C15111yescharacteristic
588C2#3372×C15111yescharacteristic
589C2#3373×C15111yescharacteristic
590C2#3375×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC22×D4×C151C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C4×Q8×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊C4×C151C2semidirect
3. #2100×C151C2semidirect
4. #2099×C151C2semidirect
7. #1985×C151C2semidirect
8. #1949×C151C2semidirect
11. #1921×C151C2semidirect
12. #1920×C151C2semidirect
13. #1836×C151C2semidirect
295. #25002C151direct (N × H)
296. C151#25002direct (N × H)
298. Q8×C4⋊C4C2×C151semidirect
299. #2100C2×C151semidirect
300. #2099C2×C151semidirect
303. #1985C2×C151semidirect
304. #1949C2×C151semidirect
307. #1921C2×C151semidirect
308. #1920C2×C151semidirect
309. #1836C2×C151semidirect

Maximal subgroups

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

subgroupindex
#2580×C1512
Q8×C4⋊C4×C1512
#2100×C1512
#2099×C1512
#2071×C1512
#1985×C1512
#1949×C1512
#1926×C1512
#1925×C1512
#1921×C1512
#1920×C1512
#1836×C1512
#1802×C1512
#1762×C1512
#25002151