Finite Group Databasebeta

← C151×#24671 #50329596

Subgroups — C151×#24671 #50329596

Order 38656.

786 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3039×C151C211yesmaximal, characteristic, complemented
2C2×#461×C151C211yesmaximal, characteristic, complemented
3C4×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic
4#2925×C151C211yesmaximal, characteristic, complemented
5#2052×C151C212yesmaximal, complemented
6#2019×C151C212yesmaximal
7#1977×C151C212yesmaximal, complemented
8#1966×C151C211yesmaximal, characteristic, complemented
9#1950×C151C211yesmaximal, characteristic, complemented
10#1930×C151C212yesmaximal, complemented
11#1800×C151C211yesmaximal, characteristic, complemented
12C2×#137×C151C2212yes—
13C2×AGL(1,ℤ/8)×C151C2211yescharacteristic
14C22×SD16×C151C2212yes—
15C2×C4⋊Q8×C151C2211yescharacteristic
16C4×#41×C151C2211yescharacteristic
17C2×C4×D4×C151C2211yescharacteristic
18#462×C151—21no—
19#462×C151—21no—
20#461×C151—21no—
21#461×C151—21no—
22#461×C151—21no—
23#461×C151—21no—
24#461×C151—22no—
25#442×C151—21no—
26#440×C151—21no—
27C4×SD16×C151—21no—
28C4×D8×C151—21no—
29#433×C151—22no—
30#432×C151—21no—
31#431×C151—21no—
32C2×#108×C151C2212yes—
33#428×C151C2211yescharacteristic
34#423×C151C2211yescharacteristic
35#423×C151C2211yescharacteristic
36C2×C4⋊C8×C151C2211yescharacteristic
37#419×C151C2212yes—
38#418×C151C2211yescharacteristic
39#417×C151C2211yescharacteristic
40C2×#103×C151C2212yes—
41C2×#101×C151C2212yes—
42C2×#101×C151C2212yes—
43C2×#100×C151C2212yes—
44#408×C151C2212yes—
45#405×C151C2212yes—
46C4×M16×C151C2211yescharacteristic
47#343×C151C2212yes—
48#341×C151C2212yes—
49C2×#41×C151C2311yescharacteristic
50C22×Q8×C151C2312yes—
51C22×D4×C151C2311yescharacteristic
52C23×C4×C151—21no—
53#137×C151—42no—
54#137×C151—42no—
55AGL(1,ℤ/8)×C151—22no—
56AGL(1,ℤ/8)×C151—22no—
57C2×Q16×C151—22no—
58C2×SD16×C151—22no—
59C2×SD16×C151—24no—
60C2×SD16×C151—22no—
61C2×SD16×C151—22no—
62C2×SD16×C151—21no—
63C2×SD16×C151—42no—
64C2×D8×C151—21no—
65C2×M16×C151C2312yes—
66C2×M16×C151C2312yes—
67C2×M16×C151C2311yescharacteristic
68C2×M16×C151C2311yescharacteristic
69C22×C8×C151C2312yes—
70C4⋊Q8×C151—21no—
71C4⋊Q8×C151—21no—
72C4⋊Q8×C151—21no—
73C4⋊Q8×C151—21no—
74C4×Q8×C151D411yescharacteristic
75C4×Q8×C151D411yescharacteristic
76C4×D4×C151—21no—
77C4×D4×C151—21no—
78C4×D4×C151—21no—
79C4×D4×C151—21no—
80C4×D4×C151—21no—
81C4×D4×C151—21no—
82C4×D4×C151D411yescharacteristic
83C4×D4×C151D411yescharacteristic
84#117×C151—21no—
85#117×C151C2311yescharacteristic
86C2×C4⋊C4×C151—21no—
87C2×C4⋊C4×C151—21no—
88C2×C4⋊C4×C151C2311yescharacteristic
89C2×C22⋊C4×C151—21no—
90C2×C42×C151—21no—
91C2×C42×C151C2311yescharacteristic
92#108×C151—42no—
93#107×C151—21no—
94#106×C151—21no—
95C4⋊C8×C151—21no—
96C4⋊C8×C151—21no—
97C4⋊C8×C151—21no—
98C4⋊C8×C151—21no—
99C4⋊C8×C151D411yescharacteristic
100C4⋊C8×C151D411yescharacteristic
101C4⋊C8×C151D412yes—
102#103×C151—22no—
103#103×C151—22no—
104#103×C151—22no—
105#103×C151—22no—
106#103×C151—21no—
107#102×C151—21no—
108#102×C151—21no—
109#102×C151—21no—
110#101×C151—24no—
111#101×C151—42no—
112#100×C151—42no—
113C22⋊C8×C151—22no—
114C22⋊C8×C151—22no—
115#97×C151—22no—
116#97×C151—21no—
117C4×C8×C151—22no—
118C4×C8×C151—21no—
119C24×C151—21no—
120#41×C151—21no—
121#41×C151—21no—
122#41×C151—21no—
123#41×C151—21no—
124C2×Q8×C151—22no—
125C2×Q8×C151—22no—
126C2×Q8×C151—22no—
127C2×Q8×C151—22no—
128C2×Q8×C151—22no—
129C2×Q8×C151—22no—
130C2×Q8×C151C2×D411yescharacteristic
131C2×D4×C151—21no—
132C2×D4×C151—21no—
133C2×D4×C151—21no—
134C2×D4×C151—21no—
135C2×D4×C151—21no—
136C2×D4×C151—21no—
137C2×D4×C151C2×D411yescharacteristic
138C2×D4×C151#4112yes—
139C22×C4×C151—21no—
140C22×C4×C151—21no—
141C22×C4×C151—21no—
142C22×C4×C151—22no—
143C22×C4×C151—21no—
144C22×C4×C151—41no—
145C22×C4×C151—41no—
146C22×C4×C151—82no—
147C22×C4×C151#4111yescharacteristic
148C22×C4×C151#4111yescharacteristic
149C22×C4×C151C2411yescharacteristic
150Q16×C151—44no—
151SD16×C151—24no—
152SD16×C151—44no—
153SD16×C151—44no—
154SD16×C151—44no—
155D8×C151—24no—
156M16×C151—21no—
157M16×C151—22no—
158M16×C151—22no—
159M16×C151—22no—
160M16×C151—21no—
161M16×C151—21no—
162M16×C151—22no—
163M16×C151#4112yes—
164C2×C8×C151—21no—
165C2×C8×C151—22no—
166C2×C8×C151—22no—
167C2×C8×C151—22no—
168C2×C8×C151—21no—
169C2×C8×C151—22no—
170C2×C8×C151C2×D412yes—
171C2×C8×C151C2×D412yes—
172C4⋊C4×C151—21no—
173C4⋊C4×C151—21no—
174C4⋊C4×C151—21no—
175C4⋊C4×C151—21no—
176C4⋊C4×C151—41no—
177C4⋊C4×C151—41no—
178C4⋊C4×C151—41no—
179C4⋊C4×C151—41no—
180C4⋊C4×C151C2×D411yescharacteristic
181C4⋊C4×C151C2×D411yescharacteristic
182C22⋊C4×C151—21no—
183C22⋊C4×C151—21no—
184C22⋊C4×C151—21no—
185C22⋊C4×C151—41no—
186C22⋊C4×C151—41no—
187C42×C151—21no—
188C42×C151—21no—
189C42×C151—21no—
190C42×C151C2×D411yescharacteristic
191C42×C151C2×D411yescharacteristic
192C42×C151C2×D411yescharacteristic
193C42×C151C2×D411yescharacteristic
194C23×C151—21no—
195C23×C151—22no—
196C23×C151—41no—
197C23×C151—41no—
198C23×C151—81no—
199C23×C151—81no—
200C23×C151C2×#4111yescharacteristic
201Q8×C151—21no—
202Q8×C151—21no—
203Q8×C151—42no—
204Q8×C151—42no—
205Q8×C151—42no—
206Q8×C151—42no—
207D4×C151—21no—
208D4×C151—22no—
209D4×C151—21no—
210D4×C151—22no—
211D4×C151—41no—
212D4×C151—41no—
213D4×C151—41no—
214D4×C151—41no—
215C2×C4×C151—22no—
216C2×C4×C151—21no—
217C2×C4×C151—21no—
218C2×C4×C151—21no—
219C2×C4×C151—41no—
220C2×C4×C151—41no—
221C2×C4×C151—41no—
222C2×C4×C151—42no—
223C2×C4×C151—42no—
224C2×C4×C151—41no—
225C2×C4×C151—42no—
226C2×C4×C151—42no—
227C2×C4×C151—81no—
228C2×C4×C151—82no—
229C2×C4×C151—82no—
230C2×C4×C151—81no—
231C2×C4×C151#12111yescharacteristic
232C2×C4×C151#12111yescharacteristic
233C2×C4×C151#12111yescharacteristic
234C2×C4×C151#12111yescharacteristic
235C2×C4×C151C22×D411yescharacteristic
236C2×C4×C151C22×D411yescharacteristic
237C2×C4×C151C2×#4111yescharacteristic
238C2×C4×C151C2×#4111yescharacteristic
239C2×C4×C151H(5,𝔽2)11yescharacteristic
240C2×C4×C151H(5,𝔽2)11yescharacteristic
241C8×C151—22no—
242C8×C151—22no—
243C8×C151—22no—
244C8×C151—41no—
245C8×C151—42no—
246C8×C151—42no—
247C22×C151—21no—
248C22×C151—21no—
249C22×C151—41no—
250C22×C151—42no—
251C22×C151—41no—
252C22×C151—41no—
253C22×C151—82no—
254C22×C151—82no—
255C22×C151—81no—
256C22×C151—81no—
257C22×C151C2×#12111yescharacteristic
258C22×C151#55311yescharacteristic
259C22×C151#55511yescharacteristic
260C4×C151—21no—
261C4×C151—21no—
262C4×C151—21no—
263C4×C151—41no—
264C4×C151—41no—
265C4×C151—41no—
266C4×C151—82no—
267C4×C151—81no—
268C4×C151#47112yes—
269C4×C151#52511yescharacteristic
270C4×C151#53511yescharacteristic
271C4×C151#54611yescharacteristic
272C4×C151#55011yescharacteristic
273C2×C151—21no—
274C2×C151—21no—
275C2×C151—81no—
276C2×C151—82no—
277C2×C151#241411yescharacteristic
278C2×C151C2×#47111yescharacteristic
279C2×C151#306911yescharacteristic
280#24671C15111yesmaximal, characteristic, direct product factor, Sylow
281C151#2467111yescharacteristic, direct product factor, Sylow
282#3039C2×C15111yescharacteristic, complemented
283C2×#461C2×C15111yescharacteristic, complemented
284C4×AGL(1,ℤ/8)C2×C15111yescharacteristic
285#2925C2×C15111yescharacteristic, complemented
286#2052C2×C15112yescomplemented
287#2019C2×C15112yes—
288#1977C2×C15112yescomplemented
289#1966C2×C15111yescharacteristic, complemented
290#1950C2×C15111yescharacteristic, complemented
291#1930C2×C15112yescomplemented
292#1800C2×C15111yescharacteristic, complemented
293C2×#137C22×C15112yes—
294C2×AGL(1,ℤ/8)C22×C15111yescharacteristic
295C22×SD16C22×C15112yes—
296C2×C4⋊Q8C22×C15111yescharacteristic
297C4×#41C22×C15111yescharacteristic
298C2×C4×D4C22×C15111yescharacteristic
299#462—21no—
300#462—21no—
301#461—21no—
302#461—21no—
303#461—21no—
304#461—21no—
305#461—22no—
306#442—21no—
307#440—21no—
308C4×SD16—21no—
309C4×D8—21no—
310#433—22no—
311#432—21no—
312#431—21no—
313C2×#108C22×C15112yes—
314#428C22×C15111yescharacteristic
315#423C22×C15111yescharacteristic
316#423C22×C15111yescharacteristic
317C2×C4⋊C8C22×C15111yescharacteristic
318#419C22×C15112yes—
319#418C22×C15111yescharacteristic
320#417C22×C15111yescharacteristic
321C2×#103C22×C15112yes—
322C2×#101C22×C15112yes—
323C2×#101C22×C15112yes—
324C2×#100C22×C15112yes—
325#408C22×C15112yes—
326#405C22×C15112yes—
327C4×M16C22×C15111yescharacteristic
328#343C22×C15112yes—
329#341C22×C15112yes—
330C2×#41C23×C15111yescharacteristic
331C22×Q8C23×C15112yes—
332C22×D4C23×C15111yescharacteristic
333C23×C4—21no—
334#137—42no—
335#137—42no—
336AGL(1,ℤ/8)—22no—
337AGL(1,ℤ/8)—22no—
338C2×Q16—22no—
339C2×SD16—21no—
340C2×SD16—24no—
341C2×SD16—22no—
342C2×SD16—22no—
343C2×SD16—22no—
344C2×SD16—42no—
345C2×D8—21no—
346C2×M16C23×C15112yes—
347C2×M16C23×C15112yes—
348C2×M16C23×C15111yescharacteristic
349C2×M16C23×C15111yescharacteristic
350C22×C8C23×C15112yes—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4×Q8D4×C15111yescharacteristic
356C4×Q8D4×C15111yescharacteristic
357C4×D4—21no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4D4×C15111yescharacteristic
364C4×D4D4×C15111yescharacteristic
365#117—21no—
366#117C23×C15111yescharacteristic
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4C23×C15111yescharacteristic
370C2×C22⋊C4—21no—
371C2×C42—21no—
372C2×C42C23×C15111yescharacteristic
373#108—42no—
374#107—21no—
375#106—21no—
376C4⋊C8—21no—
377C4⋊C8—21no—
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8D4×C15111yescharacteristic
381C4⋊C8D4×C15111yescharacteristic
382C4⋊C8D4×C15112yes—
383#103—21no—
384#103—22no—
385#103—22no—
386#103—22no—
387#103—22no—
388#102—21no—
389#102—21no—
390#102—21no—
391#101—24no—
392#101—42no—
393#100—42no—
394C22⋊C8—22no—
395C22⋊C8—22no—
396#97—22no—
397#97—21no—
398C4×C8—22no—
399C4×C8—21no—
400C24—21no—
401#41—21no—
402#41—21no—
403#41—21no—
404#41—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8C2×D4×C15111yescharacteristic
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×D4×C15111yescharacteristic
419C2×D4#41×C15112yes—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—82no—
428C22×C4#41×C15111yescharacteristic
429C22×C4#41×C15111yescharacteristic
430C22×C4C24×C15111yescharacteristic
431Q16—44no—
432SD16—24no—
433SD16—44no—
434SD16—44no—
435SD16—44no—
436D8—24no—
437M16—21no—
438M16—22no—
439M16—22no—
440M16—22no—
441M16—21no—
442M16—21no—
443M16—22no—
444M16#41×C15112yes—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—22no—
451C2×C8C2×D4×C15112yes—
452C2×C8C2×D4×C15112yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4C2×D4×C15111yescharacteristic
462C4⋊C4C2×D4×C15111yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42C2×D4×C15111yescharacteristic
472C42C2×D4×C15111yescharacteristic
473C42C2×D4×C15111yescharacteristic
474C42C2×D4×C15111yescharacteristic
475C23—21no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—81no—
480C23—81no—
481C23C2×#41×C15111yescharacteristic
482Q8—21no—
483Q8—21no—
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488D4—21no—
489D4—22no—
490D4—21no—
491D4—22no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4#121×C15111yescharacteristic
513C2×C4#121×C15111yescharacteristic
514C2×C4#121×C15111yescharacteristic
515C2×C4#121×C15111yescharacteristic
516C2×C4C22×D4×C15111yescharacteristic
517C2×C4C22×D4×C15111yescharacteristic
518C2×C4C2×#41×C15111yescharacteristic
519C2×C4C2×#41×C15111yescharacteristic
520C2×C4H(5,𝔽2)×C15111yescharacteristic
521C2×C4H(5,𝔽2)×C15111yescharacteristic
522C8—22no—
523C8—22no—
524C8—22no—
525C8—42no—
526C8—41no—
527C8—42no—
528C22—21no—
529C22—21no—
530C22—41no—
531C22—42no—
532C22—41no—
533C22—41no—
534C22—82no—
535C22—82no—
536C22—81no—
537C22—81no—
538C22C2×#121×C15111yescharacteristic
539C22#553×C15111yescharacteristic
540C22#555×C15111yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—82no—
548C4—81no—
549C4#471×C15112yes—
550C4#525×C15111yescharacteristic
551C4#535×C15111yescharacteristic
552C4#546×C15111yescharacteristic
553C4#550×C15111yescharacteristic
554C2—21no—
555C2—21no—
556C2—81no—
557C2—82no—
558C2#2414×C15111yescharacteristic
559C2C2×#471×C15111yescharacteristic
560C2#3069×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#121
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×AGL(1,ℤ/8)×C151C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3039×C151C2semidirect
2. C2×#461×C151C2semidirect
4. #2925×C151C2semidirect
5. #2052×C151C2semidirect
7. #1977×C151C2semidirect
8. #1966×C151C2semidirect
9. #1950×C151C2semidirect
10. #1930×C151C2semidirect
11. #1800×C151C2semidirect
280. #24671C151direct (N × H)
281. C151#24671direct (N × H)
282. #3039C2×C151semidirect
283. C2×#461C2×C151semidirect
285. #2925C2×C151semidirect
286. #2052C2×C151semidirect
288. #1977C2×C151semidirect
289. #1966C2×C151semidirect
290. #1950C2×C151semidirect
291. #1930C2×C151semidirect
292. #1800C2×C151semidirect

Maximal subgroups

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

subgroupindex
#3039×C1512
C2×#461×C1512
C4×AGL(1,ℤ/8)×C1512
#2925×C1512
#2052×C1512
#2019×C1512
#1977×C1512
#1966×C1512
#1950×C1512
#1930×C1512
#1800×C1512
#24671151