Finite Group Databasebeta

← C151×#22222 #50327147

Subgroups — C151×#22222 #50327147

Order 38656.

802 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#488×C151C211yesmaximal, characteristic
2C4×#125×C151C211yesmaximal, characteristic, complemented
3C4×#124×C151C211yesmaximal, characteristic
4#1991×C151C211yesmaximal, characteristic, complemented
5#1958×C151C211yesmaximal, characteristic
6#1928×C151C211yesmaximal, characteristic, complemented
7#1927×C151C211yesmaximal, characteristic, complemented
8#1926×C151C211yesmaximal, characteristic
9#1925×C151C211yesmaximal, characteristic
10#1865×C151C211yesmaximal, characteristic, complemented
11#1864×C151C211yesmaximal, characteristic
12#1863×C151C211yesmaximal, characteristic, complemented
13#1862×C151C211yesmaximal, characteristic
14#1846×C151C211yesmaximal, characteristic, complemented
15#1744×C151C211yesmaximal, characteristic, complemented
16C2×#125×C151C2211yescharacteristic
17C2×#124×C151C2211yescharacteristic
18C2×C4×Q8×C151C2211yescharacteristic
19C2×C4×D4×C151C2211yescharacteristic
20#488×C151C418yescomplemented
21C2×#107×C151C2211yescharacteristic
22C2×#107×C151C2211yescharacteristic
23C2×#106×C151C2211yescharacteristic
24C2×#106×C151C2211yescharacteristic
25C2×#103×C151C2211yescharacteristic
26C2×#103×C151C2211yescharacteristic
27C2×#103×C151C2211yescharacteristic
28C2×#103×C151C2211yescharacteristic
29C2×#102×C151C2211yescharacteristic
30C2×#102×C151C2211yescharacteristic
31C2×#102×C151C2211yescharacteristic
32C2×#102×C151C2211yescharacteristic
33C2×C4×C8×C151C2211yescharacteristic
34C2×C4×C8×C151C2211yescharacteristic
35#391×C151C2211yescharacteristic
36#390×C151C2211yescharacteristic
37#389×C151C2211yescharacteristic
38#389×C151C2211yescharacteristic
39#388×C151—21no—
40#388×C151—21no—
41#387×C151—21no—
42#387×C151—21no—
43#382×C151C2211yescharacteristic
44#382×C151C2211yescharacteristic
45C4×C4⋊C4×C151—21no—
46C4×C4⋊C4×C151—21no—
47C4×C4⋊C4×C151C2211yescharacteristic
48C4×C4⋊C4×C151C2211yescharacteristic
49C4×C22⋊C4×C151—21no—
50C4×C22⋊C4×C151—21no—
51C43×C151C2211yescharacteristic
52#340×C151C2211yescharacteristic
53#340×C151C2211yescharacteristic
54#340×C151C2211yescharacteristic
55#340×C151C2211yescharacteristic
56#336×C151C2211yescharacteristic
57#336×C151C2211yescharacteristic
58#336×C151C2211yescharacteristic
59#336×C151C2211yescharacteristic
60C22×Q8×C151C2311yescharacteristic
61C22×D4×C151C2311yescharacteristic
62C23×C4×C151—21no—
63C22×C8×C151C2311yescharacteristic
64C22×C8×C151C2311yescharacteristic
65C22×C8×C151C2311yescharacteristic
66C22×C8×C151C2311yescharacteristic
67#125×C151—22no—
68#125×C151C2×C414yescomplemented
69#124×C151—22no—
70#124×C151C2×C414yes—
71C4×Q8×C151—41no—
72C4×Q8×C151—41no—
73C4×D4×C151—41no—
74C4×D4×C151—41no—
75C2×C4⋊C4×C151—21no—
76C2×C4⋊C4×C151—21no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151—21no—
80C2×C4⋊C4×C151C2311yescharacteristic
81C2×C4⋊C4×C151C2311yescharacteristic
82C2×C4⋊C4×C151C2311yescharacteristic
83C2×C4⋊C4×C151C2311yescharacteristic
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C4⋊C4×C151C2311yescharacteristic
86C2×C22⋊C4×C151—21no—
87C2×C22⋊C4×C151—21no—
88C2×C22⋊C4×C151—21no—
89C2×C22⋊C4×C151—21no—
90C2×C22⋊C4×C151—21no—
91C2×C42×C151—21no—
92C2×C42×C151—21no—
93C2×C42×C151—21no—
94C2×C42×C151D411yescharacteristic
95C2×C42×C151D411yescharacteristic
96C2×C42×C151D411yescharacteristic
97C2×C42×C151D411yescharacteristic
98C2×C42×C151C2311yescharacteristic
99C2×C42×C151C2311yescharacteristic
100C2×C42×C151C2311yescharacteristic
101#107×C151—41no—
102#107×C151—41no—
103#106×C151—41no—
104#106×C151—41no—
105#103×C151—41no—
106#103×C151—41no—
107#103×C151C2×C414yescomplemented
108#103×C151C2×C414yescomplemented
109#102×C151—41no—
110#102×C151—41no—
111#102×C151C2×C414yes—
112#102×C151C2×C414yes—
113C4×C8×C151—22no—
114C4×C8×C151C2×C414yescomplemented
115#95×C151—21no—
116#95×C151—21no—
117#95×C151—21no—
118#95×C151—21no—
119#95×C151—21no—
120#95×C151—21no—
121#95×C151—21no—
122#95×C151—21no—
123#95×C151—21no—
124#95×C151—21no—
125C24×C151—21no—
126C2×Q8×C151—21no—
127C2×Q8×C151—21no—
128C2×Q8×C151—21no—
129C2×Q8×C151—21no—
130C2×Q8×C151—21no—
131C2×Q8×C151C22×C412yes—
132C2×D4×C151—21no—
133C2×D4×C151—21no—
134C2×D4×C151—21no—
135C2×D4×C151—21no—
136C2×D4×C151—21no—
137C2×D4×C151C22×C412yes—
138C22×C4×C151—21no—
139C22×C4×C151—21no—
140C22×C4×C151—21no—
141C22×C4×C151—21no—
142C22×C4×C151—21no—
143C22×C4×C151—21no—
144C22×C4×C151—21no—
145C22×C4×C151—42no—
146C22×C4×C151—42no—
147C22×C4×C151—42no—
148C22×C4×C151C2×D411yescharacteristic
149C22×C4×C151C2×D411yescharacteristic
150C22×C4×C151C2×D411yescharacteristic
151C22×C4×C151C2×D411yescharacteristic
152C22×C4×C151C2×D411yescharacteristic
153C22×C4×C151C2×D411yescharacteristic
154C22×C4×C151C2411yescharacteristic
155C2×C8×C151—21no—
156C2×C8×C151—21no—
157C2×C8×C151—21no—
158C2×C8×C151—21no—
159C2×C8×C151—21no—
160C2×C8×C151—21no—
161C2×C8×C151—21no—
162C2×C8×C151—21no—
163C2×C8×C151—21no—
164C2×C8×C151—21no—
165C2×C8×C151C22×C412yes—
166C2×C8×C151C22×C412yes—
167C4⋊C4×C151—24no—
168C4⋊C4×C151—24no—
169C4⋊C4×C151—21no—
170C4⋊C4×C151—21no—
171C4⋊C4×C151—21no—
172C4⋊C4×C151—21no—
173C4⋊C4×C151—21no—
174C4⋊C4×C151—21no—
175C4⋊C4×C151—21no—
176C4⋊C4×C151—21no—
177C4⋊C4×C151—21no—
178C4⋊C4×C151—21no—
179C4⋊C4×C151—42no—
180C4⋊C4×C151—42no—
181C4⋊C4×C151—42no—
182C4⋊C4×C151C22×C412yes—
183C4⋊C4×C151C22×C412yes—
184C22⋊C4×C151—24no—
185C22⋊C4×C151—24no—
186C22⋊C4×C151—42no—
187C22⋊C4×C151—42no—
188C22⋊C4×C151—42no—
189C42×C151—22no—
190C42×C151—22no—
191C42×C151—21no—
192C42×C151—21no—
193C42×C151—21no—
194C42×C151—21no—
195C42×C151—41no—
196C42×C151—41no—
197C42×C151—42no—
198C42×C151—42no—
199C42×C151—42no—
200C42×C151C22×C412yes—
201C42×C151#4112yes—
202C23×C151—22no—
203C23×C151—41no—
204C23×C151—41no—
205C23×C151—41no—
206C23×C151—41no—
207C23×C151—41no—
208C23×C151—41no—
209C23×C151C22×D411yescharacteristic
210Q8×C151—22no—
211Q8×C151—22no—
212Q8×C151—41no—
213Q8×C151—41no—
214D4×C151—22no—
215D4×C151—22no—
216D4×C151—41no—
217D4×C151—41no—
218C2×C4×C151—21no—
219C2×C4×C151—21no—
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—22no—
227C2×C4×C151—22no—
228C2×C4×C151—22no—
229C2×C4×C151—21no—
230C2×C4×C151—21no—
231C2×C4×C151—21no—
232C2×C4×C151—21no—
233C2×C4×C151—21no—
234C2×C4×C151—21no—
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—41no—
244C2×C4×C151—41no—
245C2×C4×C151—41no—
246C2×C4×C151—41no—
247C2×C4×C151—41no—
248C2×C4×C151—41no—
249C2×C4×C151—41no—
250C2×C4×C151—41no—
251C2×C4×C151—41no—
252C2×C4×C151—41no—
253C2×C4×C151—84no—
254C2×C4×C151C4×D412yes—
255C2×C4×C151C4×D412yes—
256C2×C4×C151#13512yes—
257C2×C4×C151#13512yes—
258C2×C4×C151C23×C411yescharacteristic
259C2×C4×C151C2×#4111yescharacteristic
260C2×C4×C151H(5,𝔽2)11yescharacteristic
261C2×C4×C151H(5,𝔽2)11yescharacteristic
262C2×C4×C1512−1+411yescharacteristic
263C2×C4×C1512−1+411yescharacteristic
264C8×C151—22no—
265C8×C151—22no—
266C8×C151—41no—
267C8×C151—41no—
268C22×C151—42no—
269C22×C151—42no—
270C22×C151—42no—
271C22×C151—81no—
272C22×C151—81no—
273C22×C151—81no—
274C22×C151—81no—
275C22×C151C2×C4×D411yescharacteristic
276C22×C151#53511yescharacteristic
277C22×C151#53611yescharacteristic
278C22×C151C2×#13511yescharacteristic
279C22×C151C2×#13511yescharacteristic
280C22×C151#57511yescharacteristic
281C22×C151#57511yescharacteristic
282C4×C151—22no—
283C4×C151—22no—
284C4×C151—22no—
285C4×C151—22no—
286C4×C151—42no—
287C4×C151—42no—
288C4×C151—42no—
289C4×C151—41no—
290C4×C151—41no—
291C4×C151—81no—
292C4×C151—81no—
293C4×C151—81no—
294C4×C151—81no—
295C4×C151#52011yescharacteristic
296C4×C151#52011yescharacteristic
297C4×C151#54211yescharacteristic
298C4×C151#54211yescharacteristic
299C2×C151—82no—
300C2×C151#232611yescharacteristic
301C2×C151C4×#13511yescharacteristic
302C2×C151#294311yescharacteristic
303C2×C151#321111yescharacteristic
304C2×C151#321211yescharacteristic
305C2×C151#323211yescharacteristic
306C2×C151#323311yescharacteristic
307#22222C15111yesmaximal, characteristic, direct product factor, Sylow
308C151#2222211yescharacteristic, direct product factor, Sylow
309C2×#488C2×C15111yescharacteristic
310C4×#125C2×C15111yescharacteristic, complemented
311C4×#124C2×C15111yescharacteristic
312#1991C2×C15111yescharacteristic, complemented
313#1958C2×C15111yescharacteristic
314#1928C2×C15111yescharacteristic, complemented
315#1927C2×C15111yescharacteristic, complemented
316#1926C2×C15111yescharacteristic
317#1925C2×C15111yescharacteristic
318#1865C2×C15111yescharacteristic, complemented
319#1864C2×C15111yescharacteristic
320#1863C2×C15111yescharacteristic, complemented
321#1862C2×C15111yescharacteristic
322#1846C2×C15111yescharacteristic, complemented
323#1744C2×C15111yescharacteristic, complemented
324C2×#125C22×C15111yescharacteristic
325C2×#124C22×C15111yescharacteristic
326C2×C4×Q8C22×C15111yescharacteristic
327C2×C4×D4C22×C15111yescharacteristic
328#488C4×C15118yescomplemented
329C2×#107C22×C15111yescharacteristic
330C2×#107C22×C15111yescharacteristic
331C2×#106C22×C15111yescharacteristic
332C2×#106C22×C15111yescharacteristic
333C2×#103C22×C15111yescharacteristic
334C2×#103C22×C15111yescharacteristic
335C2×#103C22×C15111yescharacteristic
336C2×#103C22×C15111yescharacteristic
337C2×#102C22×C15111yescharacteristic
338C2×#102C22×C15111yescharacteristic
339C2×#102C22×C15111yescharacteristic
340C2×#102C22×C15111yescharacteristic
341C2×C4×C8C22×C15111yescharacteristic
342C2×C4×C8C22×C15111yescharacteristic
343#391C22×C15111yescharacteristic
344#390C22×C15111yescharacteristic
345#389C22×C15111yescharacteristic
346#389C22×C15111yescharacteristic
347#388—21no—
348#388—21no—
349#387—21no—
350#387—21no—
351#382C22×C15111yescharacteristic
352#382C22×C15111yescharacteristic
353C4×C4⋊C4—21no—
354C4×C4⋊C4—21no—
355C4×C4⋊C4C22×C15111yescharacteristic
356C4×C4⋊C4C22×C15111yescharacteristic
357C4×C22⋊C4—21no—
358C4×C22⋊C4—21no—
359C43C22×C15111yescharacteristic
360#340C22×C15111yescharacteristic
361#340C22×C15111yescharacteristic
362#340C22×C15111yescharacteristic
363#340C22×C15111yescharacteristic
364#336C22×C15111yescharacteristic
365#336C22×C15111yescharacteristic
366#336C22×C15111yescharacteristic
367#336C22×C15111yescharacteristic
368C22×Q8C23×C15111yescharacteristic
369C22×D4C23×C15111yescharacteristic
370C23×C4—21no—
371C22×C8C23×C15111yescharacteristic
372C22×C8C23×C15111yescharacteristic
373C22×C8C23×C15111yescharacteristic
374C22×C8C23×C15111yescharacteristic
375#125—22no—
376#125C2×C4×C15114yescomplemented
377#124—22no—
378#124C2×C4×C15114yes—
379C4×Q8—41no—
380C4×Q8—41no—
381C4×D4—41no—
382C4×D4—41no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4C23×C15111yescharacteristic
389C2×C4⋊C4C23×C15111yescharacteristic
390C2×C4⋊C4C23×C15111yescharacteristic
391C2×C4⋊C4C23×C15111yescharacteristic
392C2×C4⋊C4C23×C15111yescharacteristic
393C2×C4⋊C4C23×C15111yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42D4×C15111yescharacteristic
403C2×C42D4×C15111yescharacteristic
404C2×C42D4×C15111yescharacteristic
405C2×C42D4×C15111yescharacteristic
406C2×C42C23×C15111yescharacteristic
407C2×C42C23×C15111yescharacteristic
408C2×C42C23×C15111yescharacteristic
409#107—41no—
410#107—41no—
411#106—41no—
412#106—41no—
413#103—41no—
414#103—41no—
415#103C2×C4×C15114yescomplemented
416#103C2×C4×C15114yescomplemented
417#102—41no—
418#102—41no—
419#102C2×C4×C15114yes—
420#102C2×C4×C15114yes—
421C4×C8—22no—
422C4×C8C2×C4×C15114yescomplemented
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433C24—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C22×C4×C15112yes—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C22×C4×C15112yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4C2×D4×C15111yescharacteristic
457C22×C4C2×D4×C15111yescharacteristic
458C22×C4C2×D4×C15111yescharacteristic
459C22×C4C2×D4×C15111yescharacteristic
460C22×C4C2×D4×C15111yescharacteristic
461C22×C4C2×D4×C15111yescharacteristic
462C22×C4C24×C15111yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C22×C4×C15112yes—
474C2×C8C22×C4×C15112yes—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4C22×C4×C15112yes—
491C4⋊C4C22×C4×C15112yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C42—22no—
498C42—22no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—41no—
504C42—41no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42C22×C4×C15112yes—
509C42#41×C15112yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C22×D4×C15111yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—84no—
562C2×C4C4×D4×C15112yes—
563C2×C4C4×D4×C15112yes—
564C2×C4#135×C15112yes—
565C2×C4#135×C15112yes—
566C2×C4C23×C4×C15111yescharacteristic
567C2×C4C2×#41×C15111yescharacteristic
568C2×C4H(5,𝔽2)×C15111yescharacteristic
569C2×C4H(5,𝔽2)×C15111yescharacteristic
570C2×C42−1+4×C15111yescharacteristic
571C2×C42−1+4×C15111yescharacteristic
572C8—22no—
573C8—22no—
574C8—41no—
575C8—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C2×C4×D4×C15111yescharacteristic
584C22#535×C15111yescharacteristic
585C22#536×C15111yescharacteristic
586C22C2×#135×C15111yescharacteristic
587C22C2×#135×C15111yescharacteristic
588C22#575×C15111yescharacteristic
589C22#575×C15111yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—42no—
595C4—42no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4#520×C15111yescharacteristic
604C4#520×C15111yescharacteristic
605C4#542×C15111yescharacteristic
606C4#542×C15111yescharacteristic
607C2—82no—
608C2#2326×C15111yescharacteristic
609C2C4×#135×C15111yescharacteristic
610C2#2943×C15111yescharacteristic
611C2#3211×C15111yescharacteristic
612C2#3212×C15111yescharacteristic
613C2#3232×C15111yescharacteristic
614C2#3233×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 ◁ C43×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#125×C151C2semidirect
4. #1991×C151C2semidirect
6. #1928×C151C2semidirect
7. #1927×C151C2semidirect
10. #1865×C151C2semidirect
12. #1863×C151C2semidirect
14. #1846×C151C2semidirect
15. #1744×C151C2semidirect
20. #488×C151C4semidirect
68. #125×C151C2×C4semidirect
107. #103×C151C2×C4semidirect
108. #103×C151C2×C4semidirect
114. C4×C8×C151C2×C4semidirect
307. #22222C151direct (N × H)
308. C151#22222direct (N × H)
310. C4×#125C2×C151semidirect
312. #1991C2×C151semidirect
314. #1928C2×C151semidirect
315. #1927C2×C151semidirect
318. #1865C2×C151semidirect
320. #1863C2×C151semidirect
322. #1846C2×C151semidirect
323. #1744C2×C151semidirect
328. #488C4×C151semidirect
376. #125C2×C4×C151semidirect
415. #103C2×C4×C151semidirect
416. #103C2×C4×C151semidirect
422. C4×C8C2×C4×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#488×C1512
C4×#125×C1512
C4×#124×C1512
#1991×C1512
#1958×C1512
#1928×C1512
#1927×C1512
#1926×C1512
#1925×C1512
#1865×C1512
#1864×C1512
#1863×C1512
#1862×C1512
#1846×C1512
#1744×C1512
#22222151