Finite Group Databasebeta

← C151×#24119 #50329044

Subgroups — C151×#24119 #50329044

Order 38656.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556×C151C211yesmaximal, characteristic, complemented
2C2×#474×C151C211yesmaximal, characteristic, complemented
3C2×C4×SD16×C151C211yesmaximal, characteristic, complemented
4#2537×C151C211yesmaximal, characteristic
5#2036×C151C212yesmaximal, complemented
6#2000×C151C212yesmaximal, complemented
7#1990×C151C211yesmaximal, characteristic, complemented
8#1954×C151C211yesmaximal, characteristic, complemented
9#1875×C151C212yesmaximal, complemented
10#1854×C151C212yesmaximal, complemented
11#1803×C151C211yesmaximal, characteristic, complemented
12C22×SD16×C151C2211yescharacteristic
13#556×C151—24no—
14#556×C151—24no—
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×#126×C151—22no—
17C2×#124×C151—21no—
18C2×C22⋊Q8×C151C2212yes—
19C2×#121×C151—21no—
20C2×#121×C151C2212yes—
21C2×C4×Q8×C151—21no—
22C2×C4×Q8×C151C2211yescharacteristic
23C2×C4×D4×C151C2211yescharacteristic
24C2×C4×D4×C151C2211yescharacteristic
25C2×#117×C151C2212yes—
26#474×C151—24no—
27C4×SD16×C151—24no—
28C2×#107×C151C2212yes—
29C2×#106×C151C2211yescharacteristic
30C2×C4⋊C8×C151C2211yescharacteristic
31C2×#103×C151C2211yescharacteristic
32C2×#103×C151C2212yes—
33C2×#102×C151C2211yescharacteristic
34C2×#102×C151C2212yes—
35C2×C22⋊C8×C151C2212yes—
36C2×C22⋊C8×C151C2212yes—
37C2×C4×C8×C151C2211yescharacteristic
38#394×C151—22no—
39#384×C151C2211yescharacteristic
40#380×C151—21no—
41#380×C151C2212yes—
42#340×C151C2212yes—
43#340×C151C2212yes—
44#340×C151C2212yes—
45C22×Q8×C151—21no—
46C22×Q8×C151C2311yescharacteristic
47C22×D4×C151—22no—
48C22×D4×C151D411yescharacteristic
49C22×D4×C151C2311yescharacteristic
50C23×C4×C151—21no—
51C23×C4×C151C2312yes—
52C2×SD16×C151—22no—
53C2×SD16×C151—22no—
54C2×SD16×C151—22no—
55C22×C8×C151C2311yescharacteristic
56C22×C8×C151C2312yes—
57C22×C8×C151C2311yescharacteristic
58C4⋊Q8×C151—22no—
59C4⋊Q8×C151D414yescomplemented
60#126×C151—216no—
61#124×C151—28no—
62C22⋊Q8×C151—24no—
63C22⋊Q8×C151—24no—
64#121×C151—24no—
65#121×C151—42no—
66#121×C151—44no—
67C4×Q8×C151—28no—
68C4×Q8×C151—22no—
69C4×Q8×C151D414yescomplemented
70C4×D4×C151—22no—
71C4×D4×C151—22no—
72C4×D4×C151—22no—
73C4×D4×C151—22no—
74#117×C151—24no—
75#117×C151—24no—
76C2×C4⋊C4×C151—22no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151—21no—
80C2×C4⋊C4×C151—21no—
81C2×C4⋊C4×C151D411yescharacteristic
82C2×C4⋊C4×C151C2312yes—
83C2×C4⋊C4×C151C2311yescharacteristic
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C4⋊C4×C151C2312yes—
86C2×C22⋊C4×C151—22no—
87C2×C22⋊C4×C151—21no—
88C2×C22⋊C4×C151—22no—
89C2×C22⋊C4×C151—22no—
90C2×C22⋊C4×C151—22no—
91C2×C22⋊C4×C151D412yes—
92C2×C42×C151—22no—
93C2×C42×C151—21no—
94C2×C42×C151C2311yescharacteristic
95#107×C151—24no—
96#106×C151—22no—
97C4⋊C8×C151—22no—
98#103×C151—22no—
99#103×C151—42no—
100#102×C151—22no—
101#102×C151—24no—
102C22⋊C8×C151—24no—
103C22⋊C8×C151—42no—
104C4×C8×C151—22no—
105#95×C151—22no—
106C24×C151—21no—
107C24×C151C2×D412yes—
108C2×Q8×C151—24no—
109C2×Q8×C151—21no—
110C2×Q8×C151—21no—
111C2×Q8×C151—21no—
112C2×Q8×C151—21no—
113C2×Q8×C151—21no—
114C2×Q8×C151—42no—
115C2×Q8×C151—42no—
116C2×Q8×C151C2×D412yes—
117C2×D4×C151—22no—
118C2×D4×C151—22no—
119C2×D4×C151—22no—
120C2×D4×C151—21no—
121C2×D4×C151—21no—
122C2×D4×C151—21no—
123C2×D4×C151—21no—
124C2×D4×C151—21no—
125C2×D4×C151—44no—
126C2×D4×C151—44no—
127C2×D4×C151—82no—
128C2×D4×C151#4112yes—
129C22×C4×C151—21no—
130C22×C4×C151—21no—
131C22×C4×C151—22no—
132C22×C4×C151—22no—
133C22×C4×C151—22no—
134C22×C4×C151—22no—
135C22×C4×C151—22no—
136C22×C4×C151—22no—
137C22×C4×C151—22no—
138C22×C4×C151—21no—
139C22×C4×C151—22no—
140C22×C4×C151—42no—
141C22×C4×C151—42no—
142C22×C4×C151—42no—
143C22×C4×C151C2×D412yes—
144C22×C4×C151C2×D411yescharacteristic
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2411yescharacteristic
147SD16×C151—28no—
148C2×C8×C151—21no—
149C2×C8×C151—21no—
150C2×C8×C151—21no—
151C2×C8×C151—22no—
152C2×C8×C151—22no—
153C2×C8×C151—21no—
154C2×C8×C151—22no—
155C2×C8×C151—21no—
156C2×C8×C151#4112yes—
157C4⋊C4×C151—28no—
158C4⋊C4×C151—24no—
159C4⋊C4×C151—24no—
160C4⋊C4×C151—22no—
161C4⋊C4×C151—21no—
162C4⋊C4×C151—22no—
163C4⋊C4×C151—21no—
164C4⋊C4×C151—21no—
165C4⋊C4×C151—22no—
166C4⋊C4×C151—22no—
167C4⋊C4×C151—42no—
168C4⋊C4×C151—81no—
169C4⋊C4×C151C2×D412yes—
170C4⋊C4×C151C2×D414yes—
171C22⋊C4×C151—28no—
172C22⋊C4×C151—28no—
173C22⋊C4×C151—24no—
174C22⋊C4×C151—44no—
175C22⋊C4×C151—44no—
176C22⋊C4×C151—42no—
177C42×C151—28no—
178C42×C151—24no—
179C42×C151C2×D412yes—
180C42×C151C2×D412yes—
181C23×C151—22no—
182C23×C151—22no—
183C23×C151—22no—
184C23×C151—22no—
185C23×C151—22no—
186C23×C151—22no—
187C23×C151—22no—
188C23×C151—22no—
189C23×C151—41no—
190C23×C151—41no—
191C23×C151—41no—
192C23×C151—41no—
193C23×C151—41no—
194C23×C151—41no—
195C23×C151C22×D411yescharacteristic
196Q8×C151—22no—
197Q8×C151—22no—
198Q8×C151—41no—
199Q8×C151—41no—
200Q8×C151—48no—
201D4×C151—22no—
202D4×C151—22no—
203D4×C151—41no—
204D4×C151—41no—
205D4×C151—44no—
206D4×C151—88no—
207C2×C4×C151—21no—
208C2×C4×C151—21no—
209C2×C4×C151—24no—
210C2×C4×C151—22no—
211C2×C4×C151—22no—
212C2×C4×C151—22no—
213C2×C4×C151—22no—
214C2×C4×C151—22no—
215C2×C4×C151—41no—
216C2×C4×C151—41no—
217C2×C4×C151—42no—
218C2×C4×C151—42no—
219C2×C4×C151—41no—
220C2×C4×C151—41no—
221C2×C4×C151—42no—
222C2×C4×C151—42no—
223C2×C4×C151—42no—
224C2×C4×C151—42no—
225C2×C4×C151—41no—
226C2×C4×C151—42no—
227C2×C4×C151—41no—
228C2×C4×C151—41no—
229C2×C4×C151—42no—
230C2×C4×C151—42no—
231C2×C4×C151—84no—
232C2×C4×C151#12012yes—
233C2×C4×C151#12012yes—
234C2×C4×C151#13512yes—
235C2×C4×C151AGL(1,ℤ/8)12yes—
236C2×C4×C151C22×D411yescharacteristic
237C2×C4×C151C22×D411yescharacteristic
238C2×C4×C151C2×#4111yescharacteristic
239C2×C4×C151C2×#4111yescharacteristic
240C2×C4×C151H(5,𝔽2)11yescharacteristic
241C2×C4×C1512−1+411yescharacteristic
242C8×C151—22no—
243C8×C151—41no—
244C8×C151—42no—
245C22×C151—42no—
246C22×C151—42no—
247C22×C151—42no—
248C22×C151—42no—
249C22×C151—42no—
250C22×C151—42no—
251C22×C151—42no—
252C22×C151—42no—
253C22×C151—42no—
254C22×C151—42no—
255C22×C151—81no—
256C22×C151—81no—
257C22×C151—81no—
258C22×C151—81no—
259C22×C151C2×#12011yescharacteristic
260C22×C151#54611yescharacteristic
261C22×C151#54711yescharacteristic
262C22×C151C2×#13511yescharacteristic
263C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C151#57711yescharacteristic
265C22×C151#57811yescharacteristic
266C4×C151—22no—
267C4×C151—42no—
268C4×C151—42no—
269C4×C151—44no—
270C4×C151—42no—
271C4×C151—81no—
272C4×C151—82no—
273C4×C151#44914yes—
274C4×C151#52511yescharacteristic
275C4×C151#53711yescharacteristic
276C4×C151#54811yescharacteristic
277C4×C151#55011yescharacteristic
278C2×C151—82no—
279C2×C151—82no—
280C2×C151#240711yescharacteristic
281C2×C151C2×#44911yescharacteristic
282C2×C151#301311yescharacteristic
283C2×C151#330211yescharacteristic
284C2×C151#332411yescharacteristic
285C2×C151#334011yescharacteristic
286C2×C151#335311yescharacteristic
287#24119C15111yesmaximal, characteristic, direct product factor, Sylow
288C151#2411911yescharacteristic, direct product factor, Sylow
289C2×#556C2×C15111yescharacteristic, complemented
290C2×#474C2×C15111yescharacteristic, complemented
291C2×C4×SD16C2×C15111yescharacteristic, complemented
292#2537C2×C15111yescharacteristic
293#2036C2×C15112yescomplemented
294#2000C2×C15112yescomplemented
295#1990C2×C15111yescharacteristic, complemented
296#1954C2×C15111yescharacteristic, complemented
297#1875C2×C15112yescomplemented
298#1854C2×C15112yescomplemented
299#1803C2×C15111yescharacteristic, complemented
300C22×SD16C22×C15111yescharacteristic
301#556—24no—
302#556—24no—
303C2×C4⋊Q8C22×C15111yescharacteristic
304C2×#126—22no—
305C2×#124—21no—
306C2×C22⋊Q8C22×C15112yes—
307C2×#121—21no—
308C2×#121C22×C15112yes—
309C2×C4×Q8—21no—
310C2×C4×Q8C22×C15111yescharacteristic
311C2×C4×D4C22×C15111yescharacteristic
312C2×C4×D4C22×C15111yescharacteristic
313C2×#117C22×C15112yes—
314#474—24no—
315C4×SD16—24no—
316C2×#107C22×C15112yes—
317C2×#106C22×C15111yescharacteristic
318C2×C4⋊C8C22×C15111yescharacteristic
319C2×#103C22×C15111yescharacteristic
320C2×#103C22×C15112yes—
321C2×#102C22×C15111yescharacteristic
322C2×#102C22×C15112yes—
323C2×C22⋊C8C22×C15112yes—
324C2×C22⋊C8C22×C15112yes—
325C2×C4×C8C22×C15111yescharacteristic
326#394—22no—
327#384C22×C15111yescharacteristic
328#380—21no—
329#380C22×C15112yes—
330#340C22×C15112yes—
331#340C22×C15112yes—
332#340C22×C15112yes—
333C22×Q8—21no—
334C22×Q8C23×C15111yescharacteristic
335C22×D4—22no—
336C22×D4D4×C15111yescharacteristic
337C22×D4C23×C15111yescharacteristic
338C23×C4—21no—
339C23×C4C23×C15112yes—
340C2×SD16—22no—
341C2×SD16—22no—
342C2×SD16—22no—
343C22×C8C23×C15111yescharacteristic
344C22×C8C23×C15112yes—
345C22×C8C23×C15111yescharacteristic
346C4⋊Q8—22no—
347C4⋊Q8D4×C15114yescomplemented
348#126—216no—
349#124—28no—
350C22⋊Q8—24no—
351C22⋊Q8—24no—
352#121—24no—
353#121—42no—
354#121—44no—
355C4×Q8—28no—
356C4×Q8—22no—
357C4×Q8D4×C15114yescomplemented
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362#117—24no—
363#117—24no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4D4×C15111yescharacteristic
370C2×C4⋊C4C23×C15111yescharacteristic
371C2×C4⋊C4C23×C15112yes—
372C2×C4⋊C4C23×C15111yescharacteristic
373C2×C4⋊C4C23×C15112yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4D4×C15112yes—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42C23×C15111yescharacteristic
383#107—24no—
384#106—22no—
385C4⋊C8—22no—
386#103—22no—
387#103—42no—
388#102—22no—
389#102—24no—
390C22⋊C8—24no—
391C22⋊C8—42no—
392C4×C8—22no—
393#95—22no—
394C24—21no—
395C24C2×D4×C15112yes—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8C2×D4×C15112yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—82no—
416C2×D4#41×C15112yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4C2×D4×C15112yes—
432C22×C4C2×D4×C15111yescharacteristic
433C22×C4C2×D4×C15111yescharacteristic
434C22×C4C24×C15111yescharacteristic
435SD16—28no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—21no—
444C2×C8#41×C15112yes—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4C2×D4×C15112yes—
458C4⋊C4C2×D4×C15114yes—
459C22⋊C4—28no—
460C22⋊C4—28no—
461C22⋊C4—24no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—42no—
465C42—28no—
466C42—24no—
467C42C2×D4×C15112yes—
468C42C2×D4×C15112yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23C22×D4×C15111yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—48no—
487Q8—41no—
488Q8—41no—
489D4—22no—
490D4—22no—
491D4—44no—
492D4—41no—
493D4—41no—
494D4—88no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4#120×C15112yes—
521C2×C4#120×C15112yes—
522C2×C4#135×C15112yes—
523C2×C4AGL(1,ℤ/8)×C15112yes—
524C2×C4C22×D4×C15111yescharacteristic
525C2×C4C22×D4×C15111yescharacteristic
526C2×C4C2×#41×C15111yescharacteristic
527C2×C4C2×#41×C15111yescharacteristic
528C2×C4H(5,𝔽2)×C15111yescharacteristic
529C2×C42−1+4×C15111yescharacteristic
530C8—22no—
531C8—41no—
532C8—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#120×C15111yescharacteristic
548C22#546×C15111yescharacteristic
549C22#547×C15111yescharacteristic
550C22C2×#135×C15111yescharacteristic
551C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
552C22#577×C15111yescharacteristic
553C22#578×C15111yescharacteristic
554C4—22no—
555C4—42no—
556C4—42no—
557C4—44no—
558C4—42no—
559C4—81no—
560C4—82no—
561C4#449×C15114yes—
562C4#525×C15111yescharacteristic
563C4#537×C15111yescharacteristic
564C4#548×C15111yescharacteristic
565C4#550×C15111yescharacteristic
566C2—82no—
567C2—82no—
568C2#2407×C15111yescharacteristic
569C2C2×#449×C15111yescharacteristic
570C2#3013×C15111yescharacteristic
571C2#3302×C15111yescharacteristic
572C2#3324×C15111yescharacteristic
573C2#3340×C15111yescharacteristic
574C2#3353×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radical#2537×C151C2
LCSG ▷ C22×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
1. C2×#556×C151C2semidirect
2. C2×#474×C151C2semidirect
3. C2×C4×SD16×C151C2semidirect
5. #2036×C151C2semidirect
6. #2000×C151C2semidirect
7. #1990×C151C2semidirect
8. #1954×C151C2semidirect
9. #1875×C151C2semidirect
10. #1854×C151C2semidirect
11. #1803×C151C2semidirect
59. C4⋊Q8×C151D4semidirect
69. C4×Q8×C151D4semidirect
287. #24119C151direct (N × H)
288. C151#24119direct (N × H)
289. C2×#556C2×C151semidirect
290. C2×#474C2×C151semidirect
291. C2×C4×SD16C2×C151semidirect
293. #2036C2×C151semidirect
294. #2000C2×C151semidirect
295. #1990C2×C151semidirect
296. #1954C2×C151semidirect
297. #1875C2×C151semidirect
298. #1854C2×C151semidirect
299. #1803C2×C151semidirect
347. C4⋊Q8D4×C151semidirect
357. C4×Q8D4×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#556×C1512
C2×#474×C1512
C2×C4×SD16×C1512
#2537×C1512
#2036×C1512
#2000×C1512
#1990×C1512
#1954×C1512
#1875×C1512
#1854×C1512
#1803×C1512
#24119151