Finite Group Databasebeta

← C151×#23926 #50328851

Subgroups — C151×#23926 #50328851

Order 38656.

1888 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536×C151C211yesmaximal, characteristic, complemented
2C2×#535×C151C211yesmaximal, characteristic
3C2×#448×C151C212yesmaximal, complemented
4C2×#450×C151C212yesmaximal, complemented
5#2000×C151C214yesmaximal, complemented
6#1875×C151C212yesmaximal, complemented
7#1873×C151C212yesmaximal, complemented
8#1780×C151C211yesmaximal, characteristic, complemented
9C22×#41×C151C2211yescharacteristic
10C23×Q8×C151C2211yescharacteristic, complemented
11C23×D4×C151C2211yescharacteristic
12C22×SD16×C151C2214yes—
13#536×C151—28no—
14#535×C151—28no—
15C2×C4⋊Q8×C151—21no—
16C2×#127×C151—21no—
17C2×#124×C151—21no—
18C2×#124×C151—21no—
19C2×#123×C151—21no—
20C2×#123×C151—21no—
21C2×C22⋊Q8×C151C2212yes—
22C2×C22⋊Q8×C151C2212yescomplemented
23C2×#121×C151C2212yes—
24C2×#121×C151C2212yes—
25C2×#120×C151—21no—
26C2×#120×C151—21no—
27C2×#117×C151C2211yescharacteristic
28C2×#117×C151C2211yescharacteristic, complemented
29#450×C151—28no—
30#448×C151—28no—
31C2×#103×C151C2214yescomplemented
32C2×#102×C151C2214yes—
33C2×C22⋊C8×C151C2212yes—
34C2×C22⋊C8×C151C2212yescomplemented
35C2×C22⋊C8×C151C2212yescomplemented
36#340×C151C2214yescomplemented
37C25×C151—21no—
38C2×#41×C151—24no—
39C2×#41×C151—24no—
40C2×#41×C151—42no—
41C22×Q8×C151—21no—
42C22×Q8×C151—24no—
43C22×Q8×C151—24no—
44C22×Q8×C151—42no—
45C22×Q8×C151D411yescharacteristic
46C22×Q8×C151C2312yes—
47C22×D4×C151—24no—
48C22×D4×C151—21no—
49C22×D4×C151—22no—
50C22×D4×C151—22no—
51C22×D4×C151—24no—
52C22×D4×C151—42no—
53C22×D4×C151D411yescharacteristic
54C22×D4×C151D411yescharacteristic
55C22×D4×C151D411yescharacteristic
56C22×D4×C151C2312yes—
57C23×C4×C151—21no—
58C23×C4×C151C2311yescharacteristic
59C23×C4×C151C2311yescharacteristic
60C23×C4×C151C2311yescharacteristic
61C2×SD16×C151—28no—
62C2×SD16×C151—28no—
63C2×SD16×C151—44no—
64C22×C8×C151C2314yes—
65C4⋊Q8×C151—28no—
66#127×C151—28no—
67#124×C151—28no—
68#124×C151—28no—
69#123×C151—28no—
70#123×C151—28no—
71C22⋊Q8×C151—24no—
72C22⋊Q8×C151—24no—
73C22⋊Q8×C151—24no—
74C22⋊Q8×C151D418yescomplemented
75#121×C151—24no—
76#121×C151—24no—
77#121×C151—24no—
78#121×C151D418yes—
79#120×C151—28no—
80#120×C151—28no—
81#117×C151—24no—
82#117×C151—24no—
83C2×C4⋊C4×C151—22no—
84C2×C4⋊C4×C151—22no—
85C2×C4⋊C4×C151C2312yes—
86C2×C4⋊C4×C151C2312yes—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151—22no—
89C2×C22⋊C4×C151—21no—
90C2×C22⋊C4×C151—21no—
91C2×C22⋊C4×C151—22no—
92C2×C22⋊C4×C151—22no—
93C2×C42×C151—21no—
94C2×C42×C151—21no—
95#103×C151—28no—
96#102×C151—28no—
97C22⋊C8×C151—24no—
98C22⋊C8×C151—24no—
99C22⋊C8×C151—42no—
100C24×C151—24no—
101C24×C151—22no—
102C24×C151—42no—
103C24×C151—42no—
104C24×C151—84no—
105C24×C151C2×D411yescharacteristic
106C24×C151C2×D411yescharacteristic
107C24×C151C2×D411yescharacteristic
108#41×C151—416no—
109C2×Q8×C151—24no—
110C2×Q8×C151—22no—
111C2×Q8×C151—22no—
112C2×Q8×C151—22no—
113C2×Q8×C151—24no—
114C2×Q8×C151—24no—
115C2×Q8×C151—41no—
116C2×Q8×C151—416no—
117C2×Q8×C151—42no—
118C2×Q8×C151—42no—
119C2×Q8×C151C2×D414yes—
120C2×D4×C151—24no—
121C2×D4×C151—22no—
122C2×D4×C151—22no—
123C2×D4×C151—22no—
124C2×D4×C151—22no—
125C2×D4×C151—28no—
126C2×D4×C151—28no—
127C2×D4×C151—22no—
128C2×D4×C151—22no—
129C2×D4×C151—24no—
130C2×D4×C151—22no—
131C2×D4×C151—24no—
132C2×D4×C151—416no—
133C2×D4×C151—41no—
134C2×D4×C151—41no—
135C2×D4×C151—41no—
136C2×D4×C151—42no—
137C2×D4×C151—42no—
138C2×D4×C151—84no—
139C2×D4×C151—84no—
140C2×D4×C151C2×D414yes—
141C22×C4×C151—22no—
142C22×C4×C151—22no—
143C22×C4×C151—22no—
144C22×C4×C151—22no—
145C22×C4×C151—24no—
146C22×C4×C151—22no—
147C22×C4×C151—22no—
148C22×C4×C151—22no—
149C22×C4×C151—22no—
150C22×C4×C151—22no—
151C22×C4×C151—22no—
152C22×C4×C151—42no—
153C22×C4×C151—42no—
154C22×C4×C151C2×D411yescharacteristic
155C22×C4×C151C2×D411yescharacteristic
156C22×C4×C151C2×D411yescharacteristic
157C22×C4×C151C2×D414yes—
158C22×C4×C151C2×D414yes—
159C22×C4×C151C2411yescharacteristic
160SD16×C151—416no—
161C2×C8×C151—24no—
162C2×C8×C151—24no—
163C2×C8×C151—24no—
164C4⋊C4×C151—28no—
165C4⋊C4×C151—28no—
166C4⋊C4×C151—22no—
167C4⋊C4×C151—22no—
168C4⋊C4×C151C2×D414yes—
169C4⋊C4×C151C2×D414yes—
170C22⋊C4×C151—24no—
171C22⋊C4×C151—24no—
172C22⋊C4×C151—28no—
173C22⋊C4×C151—28no—
174C22⋊C4×C151—28no—
175C22⋊C4×C151—28no—
176C42×C151—24no—
177C42×C151—24no—
178C23×C151—21no—
179C23×C151—21no—
180C23×C151—21no—
181C23×C151—21no—
182C23×C151—24no—
183C23×C151—24no—
184C23×C151—24no—
185C23×C151—21no—
186C23×C151—21no—
187C23×C151—21no—
188C23×C151—24no—
189C23×C151—48no—
190C23×C151—42no—
191C23×C151—42no—
192C23×C151—44no—
193C23×C151—44no—
194C23×C151—88no—
195C23×C151—88no—
196C23×C151—88no—
197C23×C151#12012yes—
198C23×C151#12012yes—
199C23×C151C22×D411yescharacteristic
200Q8×C151—28no—
201Q8×C151—44no—
202Q8×C151—48no—
203Q8×C151—44no—
204D4×C151—28no—
205D4×C151—44no—
206D4×C151—44no—
207D4×C151—48no—
208D4×C151—44no—
209D4×C151—44no—
210D4×C151—88no—
211D4×C151—88no—
212C2×C4×C151—21no—
213C2×C4×C151—21no—
214C2×C4×C151—21no—
215C2×C4×C151—21no—
216C2×C4×C151—28no—
217C2×C4×C151—28no—
218C2×C4×C151—24no—
219C2×C4×C151—21no—
220C2×C4×C151—21no—
221C2×C4×C151—24no—
222C2×C4×C151—24no—
223C2×C4×C151—21no—
224C2×C4×C151—48no—
225C2×C4×C151—42no—
226C2×C4×C151—42no—
227C2×C4×C151—44no—
228C2×C4×C151—44no—
229C2×C4×C151—44no—
230C2×C4×C151#12012yes—
231C2×C4×C151#12012yes—
232C2×C4×C151C22×D412yes—
233C2×C4×C151C22×D412yes—
234C2×C4×C151H(5,𝔽2)12yes—
235C8×C151—44no—
236C22×C151—22no—
237C22×C151—22no—
238C22×C151—24no—
239C22×C151—24no—
240C22×C151—42no—
241C22×C151—42no—
242C22×C151—42no—
243C22×C151—42no—
244C22×C151—41no—
245C22×C151—44no—
246C22×C151—42no—
247C22×C151—42no—
248C22×C151—44no—
249C22×C151—44no—
250C22×C151—42no—
251C22×C151—816no—
252C22×C151—84no—
253C22×C151—84no—
254C22×C151C2×#12011yescharacteristic
255C22×C151C2×#12011yescharacteristic
256C22×C151#53411yescharacteristic
257C22×C151#57714yes—
258C4×C151—22no—
259C4×C151—22no—
260C4×C151—41no—
261C4×C151—44no—
262C4×C151—44no—
263C4×C151—44no—
264C4×C151D4214yes—
265C2×C151—42no—
266C2×C151—42no—
267C2×C151—81no—
268C2×C151—84no—
269C2×C151#240311yescharacteristic
270C2×C151#300912yes—
271C2×C151#301012yes—
272C2×C151#320012yes—
273#23926C15111yesmaximal, characteristic, direct product factor, Sylow
274C151#2392611yescharacteristic, direct product factor, Sylow
275C2×#536C2×C15111yescharacteristic, complemented
276C2×#535C2×C15111yescharacteristic
277C2×#448C2×C15112yescomplemented
278C2×#450C2×C15112yescomplemented
279#2000C2×C15114yescomplemented
280#1875C2×C15112yescomplemented
281#1873C2×C15112yescomplemented
282#1780C2×C15111yescharacteristic, complemented
283C22×#41C22×C15111yescharacteristic
284C23×Q8C22×C15111yescharacteristic, complemented
285C23×D4C22×C15111yescharacteristic
286C22×SD16C22×C15114yes—
287#536—28no—
288#535—28no—
289C2×C4⋊Q8—21no—
290C2×#127—21no—
291C2×#124—21no—
292C2×#124—21no—
293C2×#123—21no—
294C2×#123—21no—
295C2×C22⋊Q8C22×C15112yes—
296C2×C22⋊Q8C22×C15112yescomplemented
297C2×#121C22×C15112yes—
298C2×#121C22×C15112yes—
299C2×#120—21no—
300C2×#120—21no—
301C2×#117C22×C15111yescharacteristic
302C2×#117C22×C15111yescharacteristic, complemented
303#450—28no—
304#448—28no—
305C2×#103C22×C15114yescomplemented
306C2×#102C22×C15114yes—
307C2×C22⋊C8C22×C15112yes—
308C2×C22⋊C8C22×C15112yescomplemented
309C2×C22⋊C8C22×C15112yescomplemented
310#340C22×C15114yescomplemented
311C25—21no—
312C2×#41—24no—
313C2×#41—24no—
314C2×#41—42no—
315C22×Q8—24no—
316C22×Q8—21no—
317C22×Q8—24no—
318C22×Q8—42no—
319C22×Q8D4×C15111yescharacteristic
320C22×Q8C23×C15112yes—
321C22×D4—22no—
322C22×D4—22no—
323C22×D4—24no—
324C22×D4—21no—
325C22×D4—24no—
326C22×D4—42no—
327C22×D4D4×C15111yescharacteristic
328C22×D4D4×C15111yescharacteristic
329C22×D4D4×C15111yescharacteristic
330C22×D4C23×C15112yes—
331C23×C4—21no—
332C23×C4C23×C15111yescharacteristic
333C23×C4C23×C15111yescharacteristic
334C23×C4C23×C15111yescharacteristic
335C2×SD16—28no—
336C2×SD16—28no—
337C2×SD16—44no—
338C22×C8C23×C15114yes—
339C4⋊Q8—28no—
340#127—28no—
341#124—28no—
342#124—28no—
343#123—28no—
344#123—28no—
345C22⋊Q8—24no—
346C22⋊Q8—24no—
347C22⋊Q8—24no—
348C22⋊Q8D4×C15118yescomplemented
349#121—24no—
350#121—24no—
351#121—24no—
352#121D4×C15118yes—
353#120—28no—
354#120—28no—
355#117—24no—
356#117—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4C23×C15112yes—
360C2×C4⋊C4C23×C15112yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C42—21no—
368C2×C42—21no—
369#103—28no—
370#102—28no—
371C22⋊C8—24no—
372C22⋊C8—24no—
373C22⋊C8—42no—
374C24—24no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—84no—
379C24C2×D4×C15111yescharacteristic
380C24C2×D4×C15111yescharacteristic
381C24C2×D4×C15111yescharacteristic
382#41—416no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—41no—
390C2×Q8—416no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8C2×D4×C15114yes—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—416no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4C2×D4×C15114yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4C2×D4×C15111yescharacteristic
429C22×C4C2×D4×C15111yescharacteristic
430C22×C4C2×D4×C15111yescharacteristic
431C22×C4C2×D4×C15114yes—
432C22×C4C2×D4×C15114yes—
433C22×C4C24×C15111yescharacteristic
434SD16—416no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4C2×D4×C15114yes—
443C4⋊C4C2×D4×C15114yes—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C42—24no—
451C42—24no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—24no—
457C23—24no—
458C23—21no—
459C23—24no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—48no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—88no—
469C23—88no—
470C23—88no—
471C23#120×C15112yes—
472C23#120×C15112yes—
473C23C22×D4×C15111yescharacteristic
474Q8—28no—
475Q8—44no—
476Q8—48no—
477Q8—44no—
478D4—28no—
479D4—44no—
480D4—44no—
481D4—48no—
482D4—44no—
483D4—44no—
484D4—88no—
485D4—88no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4#120×C15112yes—
505C2×C4#120×C15112yes—
506C2×C4C22×D4×C15112yes—
507C2×C4C22×D4×C15112yes—
508C2×C4H(5,𝔽2)×C15112yes—
509C8—44no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—41no—
519C22—44no—
520C22—42no—
521C22—42no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—816no—
526C22—84no—
527C22—84no—
528C22C2×#120×C15111yescharacteristic
529C22C2×#120×C15111yescharacteristic
530C22#534×C15111yescharacteristic
531C22#577×C15114yes—
532C4—22no—
533C4—22no—
534C4—41no—
535C4—44no—
536C4—44no—
537C4—44no—
538C4D42×C15114yes—
539C2—42no—
540C2—42no—
541C2—81no—
542C2—84no—
543C2#2403×C15111yescharacteristic
544C2#3009×C15112yes—
545C2#3010×C15112yes—
546C2#3200×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×#535×C151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C22×#41×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#536×C151C2semidirect
3. C2×#448×C151C2semidirect
4. C2×#450×C151C2semidirect
5. #2000×C151C2semidirect
6. #1875×C151C2semidirect
7. #1873×C151C2semidirect
8. #1780×C151C2semidirect
10. C23×Q8×C151C22semidirect
22. C2×C22⋊Q8×C151C22semidirect
28. C2×#117×C151C22semidirect
31. C2×#103×C151C22semidirect
34. C2×C22⋊C8×C151C22semidirect
35. C2×C22⋊C8×C151C22semidirect
36. #340×C151C22semidirect
74. C22⋊Q8×C151D4semidirect
273. #23926C151direct (N × H)
274. C151#23926direct (N × H)
275. C2×#536C2×C151semidirect
277. C2×#448C2×C151semidirect
278. C2×#450C2×C151semidirect
279. #2000C2×C151semidirect
280. #1875C2×C151semidirect
281. #1873C2×C151semidirect
282. #1780C2×C151semidirect
284. C23×Q8C22×C151semidirect
296. C2×C22⋊Q8C22×C151semidirect
302. C2×#117C22×C151semidirect
305. C2×#103C22×C151semidirect
308. C2×C22⋊C8C22×C151semidirect
309. C2×C22⋊C8C22×C151semidirect
310. #340C22×C151semidirect
348. C22⋊Q8D4×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#536×C1512
C2×#535×C1512
C2×#448×C1512
C2×#450×C1512
#2000×C1512
#1875×C1512
#1873×C1512
#1780×C1512
#23926151