Finite Group Databasebeta

← C151×#9996 #50314921

Subgroups — C151×#9996 #50314921

Order 38656.

686 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3100×C151C211yesmaximal, characteristic, complemented
2#3036×C151C211yesmaximal, characteristic, complemented
3#2535×C151C211yesmaximal, characteristic, complemented
4#1517×C151C211yesmaximal, characteristic, complemented
5#1486×C151C211yesmaximal, characteristic, complemented
6#1478×C151C211yesmaximal, characteristic, complemented
7#1286×C151C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C151—21no—
9#532×C151C2211yescharacteristic
10C2×#121×C151—21no—
11C2×C4×D4×C151C2211yescharacteristic
12C2×C4×D4×C151C2211yescharacteristic
13#480×C151—21no—
14#474×C151—21no—
15#463×C151—22no—
16#461×C151—21no—
17#459×C151—21no—
18#459×C151—21no—
19#447×C151—21no—
20C4×D8×C151—22no—
21#431×C151C2211yescharacteristic
22#431×C151C2211yescharacteristic
23#423×C151C2211yescharacteristic
24#417×C151—21no—
25#397×C151—21no—
26#392×C151—21no—
27#386×C151—22no—
28#382×C151C2211yescharacteristic
29#380×C151—21no—
30#327×C151—22no—
31D4⋊C8×C151—22no—
32D4⋊C8×C151—22no—
33C2×#41×C151—21no—
34C22×D4×C151—22no—
35C22×D4×C151D411yescharacteristic
36C22×D4×C151D411yescharacteristic
37C23×C4×C151—21no—
38C23×C4×C151—21no—
39AGL(1,ℤ/8)×C151—42no—
40AGL(1,ℤ/8)×C151—42no—
41C2×SD16×C151—22no—
42C2×D8×C151—22no—
43C2×D8×C151—22no—
44C2×M16×C151—21no—
45C4⋊Q8×C151D411yescharacteristic, complemented
46#127×C151D411yescharacteristic, complemented
47#124×C151—21no—
48#121×C151—21no—
49#121×C151—21no—
50#121×C151—42no—
51#121×C151—42no—
52C4×D4×C151—21no—
53C4×D4×C151—21no—
54C4×D4×C151—21no—
55C4×D4×C151—21no—
56C4×D4×C151—21no—
57C4×D4×C151—21no—
58C4×D4×C151—21no—
59C4×D4×C151—21no—
60C4×D4×C151—21no—
61C4×D4×C151—21no—
62C2×C4⋊C4×C151—21no—
63C2×C4⋊C4×C151D411yescharacteristic
64C2×C4⋊C4×C151D411yescharacteristic
65C2×C22⋊C4×C151—22no—
66C2×C22⋊C4×C151—21no—
67C2×C22⋊C4×C151—21no—
68C2×C22⋊C4×C151—22no—
69C2×C42×C151C2311yescharacteristic
70#107×C151—22no—
71C4⋊C8×C151—21no—
72C4⋊C8×C151—21no—
73C4⋊C8×C151—21no—
74C4⋊C8×C151D412yescomplemented
75#103×C151—22no—
76#102×C151—22no—
77#102×C151—22no—
78#102×C151—22no—
79C22⋊C8×C151—21no—
80C22⋊C8×C151—21no—
81C4×C8×C151—21no—
82C4×C8×C151D412yescomplemented
83#95×C151—22no—
84#95×C151—21no—
85#95×C151D412yes—
86C24×C151—21no—
87C24×C151—21no—
88#41×C151—41no—
89#41×C151—41no—
90#41×C151—41no—
91#41×C151—41no—
92C2×Q8×C151—21no—
93C2×D4×C151—21no—
94C2×D4×C151—21no—
95C2×D4×C151—21no—
96C2×D4×C151—21no—
97C2×D4×C151—21no—
98C2×D4×C151—21no—
99C2×D4×C151—21no—
100C2×D4×C151—21no—
101C2×D4×C151—21no—
102C2×D4×C151—21no—
103C2×D4×C151—42no—
104C2×D4×C151—44no—
105C2×D4×C151—44no—
106C2×D4×C151—41no—
107C2×D4×C151—82no—
108C2×D4×C151D812yescomplemented
109C22×C4×C151—22no—
110C22×C4×C151—21no—
111C22×C4×C151—21no—
112C22×C4×C151—41no—
113C22×C4×C151—41no—
114C22×C4×C151—41no—
115C22×C4×C151—41no—
116C22×C4×C151—41no—
117C22×C4×C151—41no—
118C22×C4×C151—41no—
119C22×C4×C151—41no—
120C22×C4×C151—41no—
121C22×C4×C151—82no—
122C22×C4×C151C2×D411yescharacteristic
123C22×C4×C151C2×D411yescharacteristic
124C22×C4×C151C2×D411yescharacteristic
125SD16×C151—44no—
126D8×C151—44no—
127D8×C151—44no—
128M16×C151—41no—
129M16×C151—41no—
130C2×C8×C151—22no—
131C2×C8×C151—22no—
132C2×C8×C151—41no—
133C4⋊C4×C151—21no—
134C4⋊C4×C151—21no—
135C4⋊C4×C151—21no—
136C4⋊C4×C151—21no—
137C4⋊C4×C151—21no—
138C4⋊C4×C151—21no—
139C4⋊C4×C151—41no—
140C4⋊C4×C151—41no—
141C22⋊C4×C151—44no—
142C22⋊C4×C151—41no—
143C22⋊C4×C151—41no—
144C22⋊C4×C151—41no—
145C22⋊C4×C151—41no—
146C22⋊C4×C151—41no—
147C22⋊C4×C151—41no—
148C22⋊C4×C151—82no—
149C42×C151C2×D411yescharacteristic
150C42×C151C2×D411yescharacteristic
151C42×C151C2×D411yescharacteristic
152C42×C151#4111yescharacteristic
153C23×C151—22no—
154C23×C151—22no—
155C23×C151—41no—
156C23×C151—42no—
157C23×C151—42no—
158C23×C151—41no—
159C23×C151—41no—
160C23×C151—41no—
161C23×C151—41no—
162C23×C151—41no—
163C23×C151—41no—
164C23×C151—41no—
165C23×C151—41no—
166C23×C151#12011yescharacteristic
167Q8×C151—41no—
168Q8×C151—41no—
169D4×C151—22no—
170D4×C151—22no—
171D4×C151—22no—
172D4×C151—22no—
173D4×C151—41no—
174D4×C151—41no—
175D4×C151—42no—
176D4×C151—41no—
177D4×C151—41no—
178D4×C151—41no—
179D4×C151—41no—
180D4×C151—81no—
181D4×C151—88no—
182D4×C151—81no—
183D4×C151—81no—
184C2×C4×C151—41no—
185C2×C4×C151—41no—
186C2×C4×C151—42no—
187C2×C4×C151—42no—
188C2×C4×C151—42no—
189C2×C4×C151—41no—
190C2×C4×C151—41no—
191C2×C4×C151—41no—
192C2×C4×C151—41no—
193C2×C4×C151—42no—
194C2×C4×C151—41no—
195C2×C4×C151—82no—
196C2×C4×C151—82no—
197C2×C4×C151—81no—
198C2×C4×C151—81no—
199C2×C4×C151—81no—
200C2×C4×C151—81no—
201C2×C4×C151—81no—
202C2×C4×C151—81no—
203C2×C4×C151#12011yescharacteristic
204C2×C4×C151#12011yescharacteristic
205C2×C4×C151#12111yescharacteristic
206C2×C4×C151#12111yescharacteristic
207C2×C4×C151#12111yescharacteristic
208C2×C4×C151#12711yescharacteristic
209C2×C4×C151C2×D811yescharacteristic
210C2×C4×C151#13511yescharacteristic
211C2×C4×C151AGL(1,ℤ/8)11yescharacteristic
212C2×C4×C151AGL(1,ℤ/8)11yescharacteristic
213C8×C151—42no—
214C8×C151—42no—
215C8×C151—81no—
216C22×C151—21no—
217C22×C151—21no—
218C22×C151—42no—
219C22×C151—44no—
220C22×C151—42no—
221C22×C151—42no—
222C22×C151—42no—
223C22×C151—81no—
224C22×C151—82no—
225C22×C151—81no—
226C22×C151—81no—
227C22×C151—81no—
228C22×C151—81no—
229C22×C151—81no—
230C22×C151—81no—
231C22×C151—81no—
232C22×C151—81no—
233C22×C151#39211yescharacteristic
234C22×C151#44711yescharacteristic
235C22×C151#44911yescharacteristic
236C4×C151—21no—
237C4×C151—21no—
238C4×C151—21no—
239C4×C151—21no—
240C4×C151—82no—
241C4×C151—81no—
242C4×C151—81no—
243C4×C151—81no—
244C4×C151#44711yescharacteristic
245C4×C151#44911yescharacteristic
246C4×C151#46611yescharacteristic
247C4×C151#46911yescharacteristic
248C2×C151—21no—
249C2×C151—21no—
250C2×C151—82no—
251C2×C151—82no—
252C2×C151—161no—
253C2×C151#199911yescharacteristic
254C2×C151#200811yescharacteristic
255C2×C151#201011yescharacteristic
256#9996C15111yesmaximal, characteristic, direct product factor, Sylow
257C151#999611yescharacteristic, direct product factor, Sylow
258#3100C2×C15111yescharacteristic, complemented
259#3036C2×C15111yescharacteristic, complemented
260#2535C2×C15111yescharacteristic, complemented
261#1517C2×C15111yescharacteristic, complemented
262#1486C2×C15111yescharacteristic, complemented
263#1478C2×C15111yescharacteristic, complemented
264#1286C2×C15111yescharacteristic, complemented
265C2×AGL(1,ℤ/8)—21no—
266#532C22×C15111yescharacteristic
267C2×#121—21no—
268C2×C4×D4C22×C15111yescharacteristic
269C2×C4×D4C22×C15111yescharacteristic
270#480—21no—
271#474—21no—
272#463—22no—
273#461—21no—
274#459—21no—
275#459—21no—
276#447—21no—
277C4×D8—22no—
278#431C22×C15111yescharacteristic
279#431C22×C15111yescharacteristic
280#423C22×C15111yescharacteristic
281#417—21no—
282#397—21no—
283#392—21no—
284#386—22no—
285#382C22×C15111yescharacteristic
286#380—21no—
287#327—22no—
288D4⋊C8—22no—
289D4⋊C8—22no—
290C2×#41—21no—
291C22×D4—22no—
292C22×D4D4×C15111yescharacteristic
293C22×D4D4×C15111yescharacteristic
294C23×C4—21no—
295C23×C4—21no—
296AGL(1,ℤ/8)—42no—
297AGL(1,ℤ/8)—42no—
298C2×SD16—22no—
299C2×D8—22no—
300C2×D8—22no—
301C2×M16—21no—
302C4⋊Q8D4×C15111yescharacteristic, complemented
303#127D4×C15111yescharacteristic, complemented
304#124—21no—
305#121—21no—
306#121—21no—
307#121—42no—
308#121—42no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4D4×C15111yescharacteristic
321C2×C4⋊C4D4×C15111yescharacteristic
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C42C23×C15111yescharacteristic
327#107—22no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8D4×C15112yescomplemented
332#103—22no—
333#102—22no—
334#102—22no—
335#102—22no—
336C22⋊C8—21no—
337C22⋊C8—21no—
338C4×C8—21no—
339C4×C8D4×C15112yescomplemented
340#95—22no—
341#95—21no—
342#95D4×C15112yes—
343C24—21no—
344C24—21no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349C2×Q8—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—41no—
364C2×D4—82no—
365C2×D4D8×C15112yescomplemented
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—82no—
379C22×C4C2×D4×C15111yescharacteristic
380C22×C4C2×D4×C15111yescharacteristic
381C22×C4C2×D4×C15111yescharacteristic
382SD16—44no—
383D8—44no—
384D8—44no—
385M16—41no—
386M16—41no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—41no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C22⋊C4—44no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—82no—
406C42C2×D4×C15111yescharacteristic
407C42C2×D4×C15111yescharacteristic
408C42C2×D4×C15111yescharacteristic
409C42#41×C15111yescharacteristic
410C23—22no—
411C23—22no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23#120×C15111yescharacteristic
424Q8—41no—
425Q8—41no—
426D4—22no—
427D4—22no—
428D4—22no—
429D4—22no—
430D4—41no—
431D4—41no—
432D4—42no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—88no—
439D4—81no—
440D4—81no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4#120×C15111yescharacteristic
461C2×C4#120×C15111yescharacteristic
462C2×C4#121×C15111yescharacteristic
463C2×C4#121×C15111yescharacteristic
464C2×C4#121×C15111yescharacteristic
465C2×C4#127×C15111yescharacteristic
466C2×C4C2×D8×C15111yescharacteristic
467C2×C4#135×C15111yescharacteristic
468C2×C4AGL(1,ℤ/8)×C15111yescharacteristic
469C2×C4AGL(1,ℤ/8)×C15111yescharacteristic
470C8—42no—
471C8—42no—
472C8—81no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#392×C15111yescharacteristic
491C22#447×C15111yescharacteristic
492C22#449×C15111yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—82no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4#447×C15111yescharacteristic
502C4#449×C15111yescharacteristic
503C4#466×C15111yescharacteristic
504C4#469×C15111yescharacteristic
505C2—21no—
506C2—21no—
507C2—82no—
508C2—82no—
509C2—161no—
510C2#1999×C15111yescharacteristic
511C2#2008×C15111yescharacteristic
512C2#2010×C15111yescharacteristic
subgroupG/H
center, socleC22×C151#392
commutator, Frattini, cosocleC2×C42C23×C151
LCSG ▷ C2×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. #3100×C151C2semidirect
2. #3036×C151C2semidirect
3. #2535×C151C2semidirect
4. #1517×C151C2semidirect
5. #1486×C151C2semidirect
6. #1478×C151C2semidirect
7. #1286×C151C2semidirect
45. C4⋊Q8×C151D4semidirect
46. #127×C151D4semidirect
74. C4⋊C8×C151D4semidirect
82. C4×C8×C151D4semidirect
108. C2×D4×C151D8semidirect
256. #9996C151direct (N × H)
257. C151#9996direct (N × H)
258. #3100C2×C151semidirect
259. #3036C2×C151semidirect
260. #2535C2×C151semidirect
261. #1517C2×C151semidirect
262. #1486C2×C151semidirect
263. #1478C2×C151semidirect
264. #1286C2×C151semidirect
302. C4⋊Q8D4×C151semidirect
303. #127D4×C151semidirect
331. C4⋊C8D4×C151semidirect
339. C4×C8D4×C151semidirect
365. C2×D4D8×C151semidirect

Maximal subgroups

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

subgroupindex
#3100×C1512
#3036×C1512
#2535×C1512
#1517×C1512
#1486×C1512
#1478×C1512
#1286×C1512
#9996151