Finite Group Databasebeta

← C151×#18770 #50323695

Subgroups — C151×#18770 #50323695

Order 38656.

1288 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#550×C151C211yesmaximal, characteristic, complemented
2#3205×C151C211yesmaximal, characteristic, complemented
3#3103×C151C211yesmaximal, characteristic, complemented
4#3036×C151C211yesmaximal, characteristic, complemented
5#1642×C151C214yesmaximal, complemented
6#1621×C151C214yesmaximal, complemented
7#1545×C151C211yesmaximal, characteristic, complemented
8#1487×C151C211yesmaximal, characteristic, complemented
9#1478×C151C211yesmaximal, characteristic, complemented
10C22×#41×C151—21no—
11C2×AGL(1,ℤ/8)×C151—21no—
12#550×C151—22no—
13#550×C151—22no—
14#550×C151—22no—
15#550×C151C2214yes—
16#532×C151C2211yescharacteristic
17C2×#127×C151—21no—
18C2×#127×C151C2211yescharacteristic
19C2×#121×C151—22no—
20C2×#121×C151—21no—
21C2×C4×Q8×C151C2211yescharacteristic
22C2×C4×D4×C151—21no—
23C2×C4×D4×C151C2211yescharacteristic
24#493×C151C2212yescomplemented
25#486×C151C2212yescomplemented
26#481×C151—21no—
27#475×C151C2212yescomplemented
28#463×C151—22no—
29#461×C151C2212yescomplemented
30#460×C151C2212yescomplemented
31#459×C151—22no—
32#459×C151C2212yescomplemented
33#448×C151—21no—
34#447×C151—22no—
35C4×SD16×C151—22no—
36#431×C151C2211yescharacteristic
37#431×C151C2211yescharacteristic
38#423×C151C2211yescharacteristic
39#417×C151—21no—
40#331×C151C2214yescomplemented
41Q8⋊C8×C151C2214yescomplemented
42D4⋊C8×C151C2214yescomplemented
43C2×#41×C151—24no—
44C2×#41×C151—21no—
45C2×#41×C151—24no—
46C2×#41×C151—48no—
47C22×Q8×C151D411yescharacteristic
48C22×D4×C151—22no—
49C22×D4×C151—22no—
50C22×D4×C151—21no—
51C22×D4×C151—21no—
52C22×D4×C151D411yescharacteristic
53C22×D4×C151D412yes—
54C23×C4×C151—22no—
55C23×C4×C151—21no—
56AGL(1,ℤ/8)×C151—44no—
57C2×SD16×C151—22no—
58C2×SD16×C151—22no—
59C2×D8×C151—22no—
60C2×D8×C151—24no—
61C2×M16×C151—21no—
62C4⋊Q8×C151C2311yescharacteristic
63#127×C151—24no—
64#127×C151—24no—
65#127×C151—21no—
66#127×C151—21no—
67#127×C151D412yes—
68#127×C151C2311yescharacteristic
69#127×C151C2312yes—
70#124×C151—21no—
71#121×C151—24no—
72#121×C151—28no—
73#121×C151—28no—
74#121×C151—24no—
75#121×C151—21no—
76#121×C151—21no—
77C4×Q8×C151—21no—
78C4×Q8×C151—21no—
79C4×Q8×C151D412yes—
80C4×Q8×C151C2312yes—
81C4×D4×C151—24no—
82C4×D4×C151—24no—
83C4×D4×C151—21no—
84C4×D4×C151—21no—
85C4×D4×C151—21no—
86C4×D4×C151—21no—
87C4×D4×C151D412yes—
88C4×D4×C151C2312yes—
89C2×C4⋊C4×C151—21no—
90C2×C4⋊C4×C151D411yescharacteristic
91C2×C4⋊C4×C151D411yescharacteristic
92C2×C22⋊C4×C151—22no—
93C2×C22⋊C4×C151—21no—
94C2×C42×C151—21no—
95C2×C42×C151C2311yescharacteristic
96#106×C151—22no—
97C4⋊C8×C151—21no—
98C4⋊C8×C151—21no—
99C4⋊C8×C151C2312yes—
100C4⋊C8×C151C2312yes—
101#103×C151—22no—
102#103×C151—22no—
103#102×C151—24no—
104#102×C151—22no—
105#102×C151—22no—
106C22⋊C8×C151—21no—
107C22⋊C8×C151—21no—
108C4×C8×C151—21no—
109C4×C8×C151C2312yes—
110C24×C151—21no—
111C24×C151—22no—
112#41×C151—416no—
113#41×C151—48no—
114#41×C151—48no—
115#41×C151—42no—
116#41×C151—42no—
117C2×Q8×C151—24no—
118C2×Q8×C151—21no—
119C2×Q8×C151—21no—
120C2×Q8×C151C2×D412yes—
121C2×D4×C151—28no—
122C2×D4×C151—28no—
123C2×D4×C151—21no—
124C2×D4×C151—22no—
125C2×D4×C151—22no—
126C2×D4×C151—22no—
127C2×D4×C151—22no—
128C2×D4×C151—22no—
129C2×D4×C151—22no—
130C2×D4×C151—22no—
131C2×D4×C151—21no—
132C2×D4×C151—21no—
133C2×D4×C151—21no—
134C2×D4×C151—48no—
135C2×D4×C151—44no—
136C2×D4×C151—41no—
137C2×D4×C151—42no—
138C2×D4×C151—42no—
139C2×D4×C151—82no—
140C2×D4×C151—84no—
141C2×D4×C151C2×D412yes—
142C2×D4×C151C2×D414yes—
143C22×C4×C151—21no—
144C22×C4×C151—28no—
145C22×C4×C151—21no—
146C22×C4×C151—21no—
147C22×C4×C151—22no—
148C22×C4×C151—22no—
149C22×C4×C151—48no—
150C22×C4×C151—44no—
151C22×C4×C151—41no—
152C22×C4×C151C2×D411yescharacteristic
153C22×C4×C151C2×D411yescharacteristic
154C22×C4×C151C2×D411yescharacteristic
155SD16×C151—44no—
156SD16×C151—44no—
157D8×C151—44no—
158D8×C151—84no—
159M16×C151—42no—
160C2×C8×C151—22no—
161C2×C8×C151—22no—
162C2×C8×C151—22no—
163C4⋊C4×C151—22no—
164C4⋊C4×C151—21no—
165C4⋊C4×C151—21no—
166C4⋊C4×C151—22no—
167C4⋊C4×C151—21no—
168C4⋊C4×C151—21no—
169C4⋊C4×C151C2×D412yes—
170C4⋊C4×C151C2×D412yes—
171C22⋊C4×C151—28no—
172C22⋊C4×C151—22no—
173C22⋊C4×C151—22no—
174C22⋊C4×C151—41no—
175C22⋊C4×C151—41no—
176C42×C151—22no—
177C42×C151—22no—
178C42×C151C2×D411yescharacteristic
179C42×C151C2×D411yescharacteristic
180C42×C151C2×D411yescharacteristic
181C42×C151C2411yescharacteristic
182C23×C151—22no—
183C23×C151—24no—
184C23×C151—44no—
185C23×C151—42no—
186C23×C151—44no—
187C23×C151—42no—
188C23×C151—42no—
189C23×C151—44no—
190C23×C151—41no—
191C23×C151#12011yescharacteristic
192Q8×C151—24no—
193Q8×C151—22no—
194Q8×C151—41no—
195Q8×C151—41no—
196Q8×C151AGL(1,ℤ/8)14yescomplemented
197D4×C151—22no—
198D4×C151—22no—
199D4×C151—44no—
200D4×C151—416no—
201D4×C151—44no—
202D4×C151—42no—
203D4×C151—42no—
204D4×C151—41no—
205D4×C151—44no—
206D4×C151—42no—
207D4×C151—41no—
208D4×C151—42no—
209D4×C151—42no—
210D4×C151—44no—
211D4×C151—82no—
212D4×C151—88no—
213D4×C151—82no—
214C2×C4×C151—24no—
215C2×C4×C151—22no—
216C2×C4×C151—22no—
217C2×C4×C151—22no—
218C2×C4×C151—22no—
219C2×C4×C151—41no—
220C2×C4×C151—416no—
221C2×C4×C151—44no—
222C2×C4×C151—42no—
223C2×C4×C151—42no—
224C2×C4×C151—44no—
225C2×C4×C151—41no—
226C2×C4×C151—82no—
227C2×C4×C151#12011yescharacteristic
228C2×C4×C151#12011yescharacteristic
229C2×C4×C151#12011yescharacteristic
230C2×C4×C151#13512yes—
231C2×C4×C151AGL(1,ℤ/8)12yes—
232C2×C4×C151C22×D411yescharacteristic
233C2×C4×C151C22×D411yescharacteristic
234C2×C4×C151C22×D411yescharacteristic
235C8×C151—42no—
236C8×C151—42no—
237C8×C151—42no—
238C22×C151—21no—
239C22×C151—21no—
240C22×C151—42no—
241C22×C151—44no—
242C22×C151—44no—
243C22×C151—42no—
244C22×C151—44no—
245C22×C151—42no—
246C22×C151—84no—
247C22×C151—88no—
248C22×C151—81no—
249C22×C151—81no—
250C22×C151—81no—
251C22×C151#44912yes—
252C22×C151C2×#12011yescharacteristic
253C4×C151—24no—
254C4×C151—21no—
255C4×C151—21no—
256C4×C151—21no—
257C4×C151—42no—
258C4×C151—42no—
259C4×C151—81no—
260C4×C151#44912yes—
261C4×C151C2×#13511yescharacteristic
262C4×C151C2×AGL(1,ℤ/8)11yescharacteristic
263C4×C151C2×AGL(1,ℤ/8)12yes—
264C2×C151—22no—
265C2×C151—82no—
266C2×C151—84no—
267C2×C151—161no—
268C2×C151C2×#44911yescharacteristic
269C2×C151#300311yescharacteristic
270C2×C151#301911yescharacteristic
271#18770C15111yesmaximal, characteristic, direct product factor, Sylow
272C151#1877011yescharacteristic, direct product factor, Sylow
273C2×#550C2×C15111yescharacteristic, complemented
274#3205C2×C15111yescharacteristic, complemented
275#3103C2×C15111yescharacteristic, complemented
276#3036C2×C15111yescharacteristic, complemented
277#1642C2×C15114yescomplemented
278#1621C2×C15114yescomplemented
279#1545C2×C15111yescharacteristic, complemented
280#1487C2×C15111yescharacteristic, complemented
281#1478C2×C15111yescharacteristic, complemented
282C22×#41—21no—
283C2×AGL(1,ℤ/8)—21no—
284#550—22no—
285#550—22no—
286#550—22no—
287#550C22×C15114yes—
288#532C22×C15111yescharacteristic
289C2×#127—21no—
290C2×#127C22×C15111yescharacteristic
291C2×#121—22no—
292C2×#121—21no—
293C2×C4×Q8C22×C15111yescharacteristic
294C2×C4×D4—21no—
295C2×C4×D4C22×C15111yescharacteristic
296#493C22×C15112yescomplemented
297#486C22×C15112yescomplemented
298#481—21no—
299#475C22×C15112yescomplemented
300#463—22no—
301#461C22×C15112yescomplemented
302#460C22×C15112yescomplemented
303#459—22no—
304#459C22×C15112yescomplemented
305#448—21no—
306#447—22no—
307C4×SD16—22no—
308#431C22×C15111yescharacteristic
309#431C22×C15111yescharacteristic
310#423C22×C15111yescharacteristic
311#417—21no—
312#331C22×C15114yescomplemented
313Q8⋊C8C22×C15114yescomplemented
314D4⋊C8C22×C15114yescomplemented
315C2×#41—24no—
316C2×#41—24no—
317C2×#41—21no—
318C2×#41—48no—
319C22×Q8D4×C15111yescharacteristic
320C22×D4—22no—
321C22×D4—22no—
322C22×D4—21no—
323C22×D4—21no—
324C22×D4D4×C15111yescharacteristic
325C22×D4D4×C15112yes—
326C23×C4—22no—
327C23×C4—21no—
328AGL(1,ℤ/8)—44no—
329C2×SD16—22no—
330C2×SD16—22no—
331C2×D8—22no—
332C2×D8—24no—
333C2×M16—21no—
334C4⋊Q8C23×C15111yescharacteristic
335#127—24no—
336#127—24no—
337#127—21no—
338#127—21no—
339#127D4×C15112yes—
340#127C23×C15111yescharacteristic
341#127C23×C15112yes—
342#124—21no—
343#121—24no—
344#121—28no—
345#121—28no—
346#121—24no—
347#121—21no—
348#121—21no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8D4×C15112yes—
352C4×Q8C23×C15112yes—
353C4×D4—24no—
354C4×D4—24no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4D4×C15112yes—
360C4×D4C23×C15112yes—
361C2×C4⋊C4—21no—
362C2×C4⋊C4D4×C15111yescharacteristic
363C2×C4⋊C4D4×C15111yescharacteristic
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C42—21no—
367C2×C42C23×C15111yescharacteristic
368#106—22no—
369C4⋊C8—21no—
370C4⋊C8—21no—
371C4⋊C8C23×C15112yes—
372C4⋊C8C23×C15112yes—
373#103—22no—
374#103—22no—
375#102—24no—
376#102—22no—
377#102—22no—
378C22⋊C8—21no—
379C22⋊C8—21no—
380C4×C8—21no—
381C4×C8C23×C15112yes—
382C24—21no—
383C24—22no—
384#41—416no—
385#41—48no—
386#41—48no—
387#41—42no—
388#41—42no—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8C2×D4×C15112yes—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—41no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—82no—
412C2×D4—84no—
413C2×D4C2×D4×C15112yes—
414C2×D4C2×D4×C15114yes—
415C22×C4—21no—
416C22×C4—28no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—41no—
424C22×C4C2×D4×C15111yescharacteristic
425C22×C4C2×D4×C15111yescharacteristic
426C22×C4C2×D4×C15111yescharacteristic
427SD16—44no—
428SD16—44no—
429D8—44no—
430D8—84no—
431M16—42no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4C2×D4×C15112yes—
442C4⋊C4C2×D4×C15112yes—
443C22⋊C4—28no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C42—22no—
449C42—22no—
450C42C2×D4×C15111yescharacteristic
451C42C2×D4×C15111yescharacteristic
452C42C2×D4×C15111yescharacteristic
453C42C24×C15111yescharacteristic
454C23—22no—
455C23—24no—
456C23—44no—
457C23—42no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—41no—
463C23#120×C15111yescharacteristic
464Q8—24no—
465Q8—22no—
466Q8—41no—
467Q8—41no—
468Q8AGL(1,ℤ/8)×C15114yescomplemented
469D4—22no—
470D4—22no—
471D4—44no—
472D4—416no—
473D4—44no—
474D4—42no—
475D4—42no—
476D4—44no—
477D4—42no—
478D4—41no—
479D4—44no—
480D4—42no—
481D4—41no—
482D4—42no—
483D4—82no—
484D4—88no—
485D4—82no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—416no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—82no—
499C2×C4#120×C15111yescharacteristic
500C2×C4#120×C15111yescharacteristic
501C2×C4#120×C15111yescharacteristic
502C2×C4#135×C15112yes—
503C2×C4AGL(1,ℤ/8)×C15112yes—
504C2×C4C22×D4×C15111yescharacteristic
505C2×C4C22×D4×C15111yescharacteristic
506C2×C4C22×D4×C15111yescharacteristic
507C8—42no—
508C8—42no—
509C8—42no—
510C22—21no—
511C22—21no—
512C22—42no—
513C22—44no—
514C22—44no—
515C22—42no—
516C22—44no—
517C22—42no—
518C22—84no—
519C22—88no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22#449×C15112yes—
524C22C2×#120×C15111yescharacteristic
525C4—24no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—42no—
530C4—42no—
531C4—81no—
532C4#449×C15112yes—
533C4C2×#135×C15111yescharacteristic
534C4C2×AGL(1,ℤ/8)×C15111yescharacteristic
535C4C2×AGL(1,ℤ/8)×C15112yes—
536C2—22no—
537C2—82no—
538C2—84no—
539C2—161no—
540C2C2×#449×C15111yescharacteristic
541C2#3003×C15111yescharacteristic
542C2#3019×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC42C24×C151
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#550×C151C2semidirect
2. #3205×C151C2semidirect
3. #3103×C151C2semidirect
4. #3036×C151C2semidirect
5. #1642×C151C2semidirect
6. #1621×C151C2semidirect
7. #1545×C151C2semidirect
8. #1487×C151C2semidirect
9. #1478×C151C2semidirect
24. #493×C151C22semidirect
25. #486×C151C22semidirect
27. #475×C151C22semidirect
29. #461×C151C22semidirect
30. #460×C151C22semidirect
32. #459×C151C22semidirect
40. #331×C151C22semidirect
41. Q8⋊C8×C151C22semidirect
42. D4⋊C8×C151C22semidirect
196. Q8×C151AGL(1,ℤ/8)semidirect
271. #18770C151direct (N × H)
272. C151#18770direct (N × H)
273. C2×#550C2×C151semidirect
274. #3205C2×C151semidirect
275. #3103C2×C151semidirect
276. #3036C2×C151semidirect
277. #1642C2×C151semidirect
278. #1621C2×C151semidirect
279. #1545C2×C151semidirect
280. #1487C2×C151semidirect
281. #1478C2×C151semidirect
296. #493C22×C151semidirect
297. #486C22×C151semidirect
299. #475C22×C151semidirect
301. #461C22×C151semidirect
302. #460C22×C151semidirect
304. #459C22×C151semidirect
312. #331C22×C151semidirect
313. Q8⋊C8C22×C151semidirect
314. D4⋊C8C22×C151semidirect
468. Q8AGL(1,ℤ/8)×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#550×C1512
#3205×C1512
#3103×C1512
#3036×C1512
#1642×C1512
#1621×C1512
#1545×C1512
#1487×C1512
#1478×C1512
#18770151