Finite Group Databasebeta

← C151×#24105 #50329030

Subgroups — C151×#24105 #50329030

Order 38656.

1126 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555×C151C211yesmaximal, characteristic
2C2×#480×C151C211yesmaximal, characteristic, complemented
3C2×#428×C151C211yesmaximal, characteristic, complemented
4#2487×C151C211yesmaximal, characteristic, complemented
5#2057×C151C212yesmaximal, complemented
6#2033×C151C212yesmaximal, complemented
7#1895×C151C211yesmaximal, characteristic, complemented
8#1870×C151C211yesmaximal, characteristic, complemented
9#1864×C151C212yesmaximal, complemented
10#1854×C151C212yesmaximal, complemented
11#1825×C151C211yesmaximal, characteristic, complemented
12C22×M16×C151C2211yescharacteristic
13#555×C151—24no—
14#555×C151—24no—
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×#124×C151—21no—
17C2×#124×C151—21no—
18C2×#121×C151—21no—
19C2×#121×C151C2211yescharacteristic
20C2×#121×C151C2211yescharacteristic
21C2×C4×D4×C151C2211yescharacteristic
22C2×C4×D4×C151C2211yescharacteristic
23C2×#117×C151—21no—
24C2×#117×C151C2211yescharacteristic
25C2×#117×C151C2211yescharacteristic
26C22×C4⋊C4×C151C2211yescharacteristic
27#480×C151—24no—
28#428×C151—24no—
29C2×#107×C151C2212yes—
30C2×#107×C151C2212yes—
31C2×#106×C151C2212yes—
32C2×#102×C151C2212yes—
33C2×#102×C151C2212yes—
34C2×C22⋊C8×C151C2211yescharacteristic
35C2×C22⋊C8×C151C2212yes—
36#400×C151—21no—
37#399×C151—21no—
38#389×C151C2212yes—
39#386×C151—21no—
40#385×C151—21no—
41#380×C151C2211yescharacteristic
42#380×C151C2212yes—
43#340×C151C2212yes—
44#340×C151C2212yes—
45#340×C151C2212yes—
46#336×C151C2212yes—
47C22×Q8×C151—21no—
48C22×D4×C151—21no—
49C22×D4×C151—21no—
50C22×D4×C151D411yescharacteristic
51C22×D4×C151C2311yescharacteristic
52C23×C4×C151—21no—
53C23×C4×C151—21no—
54C23×C4×C151C2311yescharacteristic
55C23×C4×C151C2311yescharacteristic
56C2×M16×C151—22no—
57C2×M16×C151—24no—
58C22×C8×C151C2312yes—
59C22×C8×C151C2312yes—
60C4⋊Q8×C151—22no—
61C4⋊Q8×C151—22no—
62#124×C151—28no—
63#124×C151—28no—
64#121×C151—28no—
65#121×C151—22no—
66#121×C151—22no—
67#121×C151—22no—
68#121×C151D414yes—
69C4×D4×C151—22no—
70C4×D4×C151—22no—
71C4×D4×C151—22no—
72C4×D4×C151—22no—
73#117×C151—28no—
74#117×C151—22no—
75#117×C151—22no—
76#117×C151—22no—
77#117×C151D414yes—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151—22no—
80C2×C4⋊C4×C151—22no—
81C2×C4⋊C4×C151—21no—
82C2×C4⋊C4×C151—22no—
83C2×C4⋊C4×C151D411yescharacteristic
84C2×C4⋊C4×C151C2312yes—
85C2×C4⋊C4×C151C2312yes—
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×C22⋊C4×C151—21no—
95C2×C22⋊C4×C151—22no—
96C2×C22⋊C4×C151D411yescharacteristic
97C2×C22⋊C4×C151D411yescharacteristic
98C2×C42×C151—21no—
99C2×C42×C151—21no—
100C2×C42×C151C2311yescharacteristic
101#107×C151—24no—
102#107×C151—24no—
103#106×C151—24no—
104#102×C151—24no—
105#102×C151—24no—
106C22⋊C8×C151—22no—
107C22⋊C8×C151—24no—
108#95×C151—22no—
109#95×C151—21no—
110#95×C151—22no—
111C24×C151—21no—
112C24×C151C2×D411yescharacteristic
113C24×C151C2×D411yescharacteristic
114C2×Q8×C151—24no—
115C2×Q8×C151—82no—
116C2×D4×C151—22no—
117C2×D4×C151—24no—
118C2×D4×C151—24no—
119C2×D4×C151—22no—
120C2×D4×C151—21no—
121C2×D4×C151—22no—
122C2×D4×C151—22no—
123C2×D4×C151—41no—
124C2×D4×C151—82no—
125C2×D4×C151—82no—
126C2×D4×C151C2×D412yes—
127C22×C4×C151—22no—
128C22×C4×C151—22no—
129C22×C4×C151—24no—
130C22×C4×C151—21no—
131C22×C4×C151—21no—
132C22×C4×C151—21no—
133C22×C4×C151—21no—
134C22×C4×C151—21no—
135C22×C4×C151—22no—
136C22×C4×C151—22no—
137C22×C4×C151—22no—
138C22×C4×C151—22no—
139C22×C4×C151—42no—
140C22×C4×C151—42no—
141C22×C4×C151—42no—
142C22×C4×C151—82no—
143C22×C4×C151C2×D411yescharacteristic
144C22×C4×C151C2×D411yescharacteristic
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2×D412yes—
147C22×C4×C151C2×D412yes—
148C22×C4×C151C2×D411yescharacteristic
149C22×C4×C151C2411yescharacteristic
150M16×C151—28no—
151C2×C8×C151—22no—
152C2×C8×C151—22no—
153C2×C8×C151—22no—
154C2×C8×C151—22no—
155C2×C8×C151—22no—
156C2×C8×C151#4114yes—
157C4⋊C4×C151—24no—
158C4⋊C4×C151—22no—
159C4⋊C4×C151—21no—
160C4⋊C4×C151—21no—
161C4⋊C4×C151—21no—
162C4⋊C4×C151—22no—
163C4⋊C4×C151—22no—
164C4⋊C4×C151—22no—
165C4⋊C4×C151—22no—
166C4⋊C4×C151—42no—
167C4⋊C4×C151—82no—
168C4⋊C4×C151C2×D412yes—
169C4⋊C4×C151C2×D412yes—
170C4⋊C4×C151C2×D412yes—
171C22⋊C4×C151—24no—
172C22⋊C4×C151—24no—
173C22⋊C4×C151—24no—
174C22⋊C4×C151—24no—
175C22⋊C4×C151—22no—
176C22⋊C4×C151—24no—
177C22⋊C4×C151—22no—
178C22⋊C4×C151—42no—
179C22⋊C4×C151—82no—
180C42×C151—24no—
181C42×C151—24no—
182C42×C151—21no—
183C42×C151—21no—
184C23×C151—22no—
185C23×C151—21no—
186C23×C151—21no—
187C23×C151—22no—
188C23×C151—22no—
189C23×C151—21no—
190C23×C151—22no—
191C23×C151—21no—
192C23×C151—21no—
193C23×C151—22no—
194C23×C151—42no—
195C23×C151—41no—
196C23×C151—41no—
197C23×C151—42no—
198C23×C151#12012yes—
199C23×C151C22×D411yescharacteristic
200Q8×C151—84no—
201D4×C151—24no—
202D4×C151—44no—
203D4×C151—41no—
204D4×C151—41no—
205D4×C151—84no—
206D4×C151—84no—
207C2×C4×C151—21no—
208C2×C4×C151—22no—
209C2×C4×C151—21no—
210C2×C4×C151—22no—
211C2×C4×C151—22no—
212C2×C4×C151—24no—
213C2×C4×C151—21no—
214C2×C4×C151—22no—
215C2×C4×C151—22no—
216C2×C4×C151—21no—
217C2×C4×C151—22no—
218C2×C4×C151—22no—
219C2×C4×C151—22no—
220C2×C4×C151—48no—
221C2×C4×C151—41no—
222C2×C4×C151—41no—
223C2×C4×C151—42no—
224C2×C4×C151—42no—
225C2×C4×C151—41no—
226C2×C4×C151—41no—
227C2×C4×C151—42no—
228C2×C4×C151—42no—
229C2×C4×C151—42no—
230C2×C4×C151—42no—
231C2×C4×C151—42no—
232C2×C4×C151—42no—
233C2×C4×C151—42no—
234C2×C4×C151—84no—
235C2×C4×C151#12012yes—
236C2×C4×C151#13712yes—
237C2×C4×C151C22×D411yescharacteristic
238C2×C4×C151C22×D411yescharacteristic
239C2×C4×C151C2×#4112yes—
240C2×C4×C1512−1+412yes—
241C8×C151—24no—
242C8×C151—42no—
243C22×C151—22no—
244C22×C151—22no—
245C22×C151—22no—
246C22×C151—42no—
247C22×C151—42no—
248C22×C151—42no—
249C22×C151—42no—
250C22×C151—41no—
251C22×C151—42no—
252C22×C151—42no—
253C22×C151—41no—
254C22×C151—41no—
255C22×C151—42no—
256C22×C151—82no—
257C22×C151—82no—
258C22×C151C2×#12011yescharacteristic
259C22×C151#54712yes—
260C22×C151C2×#13711yescharacteristic
261C22×C151#57511yescharacteristic
262C22×C151#57612yes—
263C4×C151—22no—
264C4×C151—22no—
265C4×C151—41no—
266C4×C151—42no—
267C4×C151—41no—
268C4×C151—42no—
269C4×C151—42no—
270C4×C151—82no—
271C4×C151—82no—
272C4×C151#52511yescharacteristic
273C4×C151#53611yescharacteristic
274C4×C151#54712yes—
275C2×C151—42no—
276C2×C151—81no—
277C2×C151—82no—
278C2×C151#240611yescharacteristic
279C2×C151#300711yescharacteristic
280C2×C151#300811yescharacteristic
281C2×C151#334712yes—
282C2×C151#335712yes—
283#24105C15111yesmaximal, characteristic, direct product factor, Sylow
284C151#2410511yescharacteristic, direct product factor, Sylow
285C2×#555C2×C15111yescharacteristic
286C2×#480C2×C15111yescharacteristic, complemented
287C2×#428C2×C15111yescharacteristic, complemented
288#2487C2×C15111yescharacteristic, complemented
289#2057C2×C15112yescomplemented
290#2033C2×C15112yescomplemented
291#1895C2×C15111yescharacteristic, complemented
292#1870C2×C15111yescharacteristic, complemented
293#1864C2×C15112yescomplemented
294#1854C2×C15112yescomplemented
295#1825C2×C15111yescharacteristic, complemented
296C22×M16C22×C15111yescharacteristic
297#555—24no—
298#555—24no—
299C2×C4⋊Q8C22×C15111yescharacteristic
300C2×#124—21no—
301C2×#124—21no—
302C2×#121—21no—
303C2×#121C22×C15111yescharacteristic
304C2×#121C22×C15111yescharacteristic
305C2×C4×D4C22×C15111yescharacteristic
306C2×C4×D4C22×C15111yescharacteristic
307C2×#117—21no—
308C2×#117C22×C15111yescharacteristic
309C2×#117C22×C15111yescharacteristic
310C22×C4⋊C4C22×C15111yescharacteristic
311#480—24no—
312#428—24no—
313C2×#107C22×C15112yes—
314C2×#107C22×C15112yes—
315C2×#106C22×C15112yes—
316C2×#102C22×C15112yes—
317C2×#102C22×C15112yes—
318C2×C22⋊C8C22×C15111yescharacteristic
319C2×C22⋊C8C22×C15112yes—
320#400—21no—
321#399—21no—
322#389C22×C15112yes—
323#386—21no—
324#385—21no—
325#380C22×C15111yescharacteristic
326#380C22×C15112yes—
327#340C22×C15112yes—
328#340C22×C15112yes—
329#340C22×C15112yes—
330#336C22×C15112yes—
331C22×Q8—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4D4×C15111yescharacteristic
335C22×D4C23×C15111yescharacteristic
336C23×C4—21no—
337C23×C4—21no—
338C23×C4C23×C15111yescharacteristic
339C23×C4C23×C15111yescharacteristic
340C2×M16—22no—
341C2×M16—24no—
342C22×C8C23×C15112yes—
343C22×C8C23×C15112yes—
344C4⋊Q8—22no—
345C4⋊Q8—22no—
346#124—28no—
347#124—28no—
348#121—28no—
349#121—22no—
350#121—22no—
351#121—22no—
352#121D4×C15114yes—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357#117—28no—
358#117—22no—
359#117—22no—
360#117—22no—
361#117D4×C15114yes—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4D4×C15111yescharacteristic
368C2×C4⋊C4C23×C15112yes—
369C2×C4⋊C4C23×C15112yes—
370C2×C4⋊C4C23×C15111yescharacteristic
371C2×C4⋊C4C23×C15111yescharacteristic
372C2×C4⋊C4C23×C15111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4D4×C15111yescharacteristic
381C2×C22⋊C4D4×C15111yescharacteristic
382C2×C42—21no—
383C2×C42—21no—
384C2×C42C23×C15111yescharacteristic
385#107—24no—
386#107—24no—
387#106—24no—
388#102—24no—
389#102—24no—
390C22⋊C8—22no—
391C22⋊C8—24no—
392#95—22no—
393#95—21no—
394#95—22no—
395C24—21no—
396C24C2×D4×C15111yescharacteristic
397C24C2×D4×C15111yescharacteristic
398C2×Q8—24no—
399C2×Q8—82no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—41no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4C2×D4×C15112yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—82no—
427C22×C4C2×D4×C15111yescharacteristic
428C22×C4C2×D4×C15111yescharacteristic
429C22×C4C2×D4×C15111yescharacteristic
430C22×C4C2×D4×C15112yes—
431C22×C4C2×D4×C15112yes—
432C22×C4C2×D4×C15111yescharacteristic
433C22×C4C24×C15111yescharacteristic
434M16—28no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8#41×C15114yes—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—82no—
452C4⋊C4C2×D4×C15112yes—
453C4⋊C4C2×D4×C15112yes—
454C4⋊C4C2×D4×C15112yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—42no—
463C22⋊C4—82no—
464C42—24no—
465C42—24no—
466C42—21no—
467C42—21no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23#120×C15112yes—
483C23C22×D4×C15111yescharacteristic
484Q8—84no—
485D4—24no—
486D4—44no—
487D4—41no—
488D4—41no—
489D4—84no—
490D4—84no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4#120×C15112yes—
520C2×C4#137×C15112yes—
521C2×C4C22×D4×C15111yescharacteristic
522C2×C4C22×D4×C15111yescharacteristic
523C2×C4C2×#41×C15112yes—
524C2×C42−1+4×C15112yes—
525C8—24no—
526C8—42no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—41no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22C2×#120×C15111yescharacteristic
543C22#547×C15112yes—
544C22C2×#137×C15111yescharacteristic
545C22#575×C15111yescharacteristic
546C22#576×C15112yes—
547C4—22no—
548C4—22no—
549C4—41no—
550C4—42no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4#525×C15111yescharacteristic
557C4#536×C15111yescharacteristic
558C4#547×C15112yes—
559C2—42no—
560C2—81no—
561C2—82no—
562C2#2406×C15111yescharacteristic
563C2#3007×C15111yescharacteristic
564C2#3008×C15111yescharacteristic
565C2#3347×C15112yes—
566C2#3357×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×#555×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
2. C2×#480×C151C2semidirect
3. C2×#428×C151C2semidirect
4. #2487×C151C2semidirect
5. #2057×C151C2semidirect
6. #2033×C151C2semidirect
7. #1895×C151C2semidirect
8. #1870×C151C2semidirect
9. #1864×C151C2semidirect
10. #1854×C151C2semidirect
11. #1825×C151C2semidirect
283. #24105C151direct (N × H)
284. C151#24105direct (N × H)
286. C2×#480C2×C151semidirect
287. C2×#428C2×C151semidirect
288. #2487C2×C151semidirect
289. #2057C2×C151semidirect
290. #2033C2×C151semidirect
291. #1895C2×C151semidirect
292. #1870C2×C151semidirect
293. #1864C2×C151semidirect
294. #1854C2×C151semidirect
295. #1825C2×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×#480×C1512
C2×#428×C1512
#2487×C1512
#2057×C1512
#2033×C1512
#1895×C1512
#1870×C1512
#1864×C1512
#1854×C1512
#1825×C1512
#24105151