Finite Group Databasebeta

← C151×#23993 #50328918

Subgroups — C151×#23993 #50328918

Order 38656.

1152 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
1#3510×C151C211yesmaximal, characteristic, complemented
2#3028×C151C211yesmaximal, characteristic, complemented
3#3021×C151C211yesmaximal, characteristic
4#2901×C151C211yesmaximal, characteristic, complemented
5#2031×C151C212yesmaximal, complemented
6#2003×C151C212yesmaximal, complemented
7#1915×C151C211yesmaximal, characteristic, complemented
8#1912×C151C212yesmaximal, complemented
9#1910×C151C211yesmaximal, characteristic, complemented
10#1904×C151C212yesmaximal, complemented
11#1827×C151C211yesmaximal, characteristic, complemented
12#585×C151C2211yescharacteristic
13C2×AGL(1,ℤ/8)×C151C2212yes—
14#568×C151C2211yescharacteristic
15#559×C151—22no—
16#556×C151—22no—
17#551×C151—24no—
18#542×C151—22no—
19#536×C151C2211yescharacteristic
20#536×C151C2212yes—
21#535×C151C2211yescharacteristic
22#533×C151C2211yescharacteristic, complemented
23#529×C151—22no—
24C2×#124×C151C2211yescharacteristic, complemented
25#519×C151C2211yescharacteristic, complemented
26#518×C151C2211yescharacteristic, complemented
27C4×#41×C151C2211yescharacteristic
28#458×C151—24no—
29#454×C151—22no—
30#453×C151—22no—
31#421×C151—22no—
32#421×C151C2212yes—
33#421×C151C2212yes—
34C2×#104×C151—22no—
35C2×#104×C151C2212yes—
36#418×C151C2212yescomplemented
37#413×C151C2211yescharacteristic
38C2×#101×C151C2212yescomplemented
39#410×C151C2211yescharacteristic
40#410×C151C2212yes—
41#344×C151C2212yescomplemented
42#339×C151C2212yescomplemented
43H(3,ℤ/4)×C151C2212yescomplemented
44H(3,ℤ/4)×C151C2212yescomplemented
452−1+4×C151—22no—
462−1+4×C151—21no—
47H(5,𝔽2)×C151—22no—
48H(5,𝔽2)×C151—21no—
49H(5,𝔽2)×C151—22no—
50C2×#41×C151—22no—
51C2×#41×C151—42no—
52C2×#41×C151C2311yescharacteristic
53C2×#41×C151C2312yes—
54C22×Q8×C151C2311yescharacteristic
55C22×D4×C151C2311yescharacteristic
56AGL(1,ℤ/8)×C151—24no—
57AGL(1,ℤ/8)×C151—42no—
58C2×SD16×C151—22no—
59C2×D8×C151—22no—
60#131×C151—42no—
61C2×M16×C151—22no—
62C2×M16×C151C2312yes—
63C2×M16×C151C2312yes—
64C4⋊Q8×C151—21no—
65C4⋊Q8×C151—21no—
66C4⋊Q8×C151—22no—
67#127×C151D412yes—
68#126×C151—24no—
69#126×C151—24no—
70#125×C151—21no—
71#124×C151—24no—
72#124×C151—22no—
73#124×C151—21no—
74#124×C151—22no—
75#124×C151—21no—
76#124×C151—21no—
77#124×C151D412yes—
78#123×C151—24no—
79#123×C151—24no—
80#123×C151—24no—
81#123×C151D414yes—
82C22⋊Q8×C151—24no—
83C22⋊Q8×C151—22no—
84C22⋊Q8×C151—22no—
85C22⋊Q8×C151—42no—
86#121×C151—22no—
87#120×C151—22no—
88C4×Q8×C151—22no—
89C4×Q8×C151—22no—
90C4×Q8×C151—22no—
91C4×Q8×C151—21no—
92C4×D4×C151—24no—
93C4×D4×C151—22no—
94C4×D4×C151—21no—
95C4×D4×C151—22no—
96C4×D4×C151—22no—
97#117×C151—24no—
98#117×C151—22no—
99#117×C151D412yes—
100#117×C151C2312yes—
101#117×C151C2311yescharacteristic
102#117×C151C2311yescharacteristic
103#117×C151C2311yescharacteristic
104C2×C22⋊C4×C151—22no—
105C2×C22⋊C4×C151—22no—
106C2×C42×C151D412yes—
107C2×C42×C151C2311yescharacteristic
108#104×C151—22no—
109#104×C151—24no—
110#104×C151—22no—
111#104×C151—24no—
112#102×C151—42no—
113#101×C151—21no—
114#101×C151—42no—
115#100×C151—21no—
116#99×C151—22no—
117#99×C151—24no—
118C24×C151—21no—
119#41×C151—22no—
120#41×C151—22no—
121#41×C151—24no—
122#41×C151—22no—
123#41×C151—22no—
124#41×C151—22no—
125#41×C151—22no—
126#41×C151—44no—
127#41×C151—42no—
128#41×C151—42no—
129#41×C151—42no—
130#41×C151—42no—
131C2×Q8×C151—22no—
132C2×Q8×C151—21no—
133C2×Q8×C151—21no—
134C2×Q8×C151—22no—
135C2×Q8×C151—22no—
136C2×Q8×C151—22no—
137C2×Q8×C151—42no—
138C2×Q8×C151C2×D411yescharacteristic
139C2×Q8×C151#4112yes—
140C2×D4×C151—22no—
141C2×D4×C151—22no—
142C2×D4×C151—24no—
143C2×D4×C151—21no—
144C2×D4×C151—22no—
145C2×D4×C151—22no—
146C2×D4×C151—42no—
147C2×D4×C151—44no—
148C2×D4×C151—42no—
149C2×D4×C151C2×D412yes—
150C2×D4×C151C2×D412yes—
151C2×D4×C151C2×D411yescharacteristic
152C2×D4×C151#4112yes—
153C22×C4×C151—24no—
154C22×C4×C151—22no—
155C22×C4×C151—21no—
156C22×C4×C151—22no—
157C22×C4×C151—42no—
158C22×C4×C151C2×D412yes—
159C22×C4×C151C2×D412yes—
160C22×C4×C151C2411yescharacteristic
161SD16×C151—44no—
162D8×C151—44no—
163M16×C151—22no—
164M16×C151—22no—
165M16×C151—22no—
166M16×C151—22no—
167M16×C151—42no—
168C2×C8×C151—22no—
169C2×C8×C151—22no—
170C2×C8×C151—42no—
171C4⋊C4×C151—22no—
172C4⋊C4×C151—24no—
173C4⋊C4×C151—22no—
174C4⋊C4×C151—22no—
175C4⋊C4×C151—24no—
176C4⋊C4×C151—22no—
177C4⋊C4×C151—21no—
178C4⋊C4×C151—21no—
179C4⋊C4×C151—21no—
180C4⋊C4×C151—22no—
181C4⋊C4×C151—44no—
182C22⋊C4×C151—24no—
183C22⋊C4×C151—24no—
184C22⋊C4×C151—24no—
185C22⋊C4×C151—22no—
186C22⋊C4×C151—24no—
187C22⋊C4×C151—21no—
188C22⋊C4×C151—21no—
189C22⋊C4×C151—24no—
190C22⋊C4×C151—22no—
191C22⋊C4×C151—44no—
192C22⋊C4×C151C2×D412yes—
193C42×C151—24no—
194C42×C151—22no—
195C42×C151—21no—
196C42×C151—21no—
197C42×C151—22no—
198C42×C151C2×D411yescharacteristic
199C42×C151C2×D412yes—
200C42×C151C2×D411yescharacteristic
201C42×C151C2×D412yes—
202C23×C151—22no—
203C23×C151—22no—
204C23×C151—42no—
205C23×C151—42no—
206C23×C151—82no—
207C23×C151#12012yes—
208C23×C151C22×D411yescharacteristic
209Q8×C151—22no—
210Q8×C151—22no—
211Q8×C151—22no—
212Q8×C151—42no—
213Q8×C151—41no—
214Q8×C151—41no—
215Q8×C151—42no—
216Q8×C151—82no—
217D4×C151—24no—
218D4×C151—22no—
219D4×C151—22no—
220D4×C151—22no—
221D4×C151—24no—
222D4×C151—42no—
223D4×C151—44no—
224D4×C151—42no—
225D4×C151—42no—
226D4×C151—41no—
227D4×C151—41no—
228D4×C151—84no—
229C2×C4×C151—22no—
230C2×C4×C151—22no—
231C2×C4×C151—22no—
232C2×C4×C151—21no—
233C2×C4×C151—22no—
234C2×C4×C151—22no—
235C2×C4×C151—22no—
236C2×C4×C151—21no—
237C2×C4×C151—24no—
238C2×C4×C151—22no—
239C2×C4×C151—44no—
240C2×C4×C151—42no—
241C2×C4×C151—42no—
242C2×C4×C151—42no—
243C2×C4×C151—42no—
244C2×C4×C151—82no—
245C2×C4×C151—84no—
246C2×C4×C151—81no—
247C2×C4×C151#12012yes—
248C2×C4×C151C22×D411yescharacteristic
249C2×C4×C151C22×D411yescharacteristic
250C2×C4×C151C2×#4112yes—
251C2×C4×C151H(5,𝔽2)12yes—
252C8×C151—42no—
253C8×C151—42no—
254C22×C151—22no—
255C22×C151—41no—
256C22×C151—42no—
257C22×C151—44no—
258C22×C151—42no—
259C22×C151—42no—
260C22×C151—82no—
261C22×C151—162no—
262C22×C151—162no—
263C22×C151C2×#12011yescharacteristic
264C22×C151#54612yes—
265C4×C151—22no—
266C4×C151—42no—
267C4×C151—41no—
268C4×C151—42no—
269C4×C151—41no—
270C4×C151—42no—
271C4×C151—42no—
272C4×C151—42no—
273C4×C151—82no—
274C4×C151#52511yescharacteristic
275C4×C151#53511yescharacteristic
276C4×C151#54612yes—
277C2×C151—21no—
278C2×C151—22no—
279C2×C151—42no—
280C2×C151—82no—
281C2×C151—82no—
282C2×C151#240411yescharacteristic
283#23993C15111yesmaximal, characteristic, direct product factor, Sylow
284C151#2399311yescharacteristic, direct product factor, Sylow
285#3510C2×C15111yescharacteristic, complemented
286#3028C2×C15111yescharacteristic, complemented
287#3021C2×C15111yescharacteristic
288#2901C2×C15111yescharacteristic, complemented
289#2031C2×C15112yescomplemented
290#2003C2×C15112yescomplemented
291#1915C2×C15111yescharacteristic, complemented
292#1912C2×C15112yescomplemented
293#1910C2×C15111yescharacteristic, complemented
294#1904C2×C15112yescomplemented
295#1827C2×C15111yescharacteristic, complemented
296#585C22×C15111yescharacteristic
297C2×AGL(1,ℤ/8)C22×C15112yes—
298#568C22×C15111yescharacteristic
299#559—22no—
300#556—22no—
301#551—24no—
302#542—22no—
303#536C22×C15111yescharacteristic
304#536C22×C15112yes—
305#535C22×C15111yescharacteristic
306#533C22×C15111yescharacteristic, complemented
307#529—22no—
308C2×#124C22×C15111yescharacteristic, complemented
309#519C22×C15111yescharacteristic, complemented
310#518C22×C15111yescharacteristic, complemented
311C4×#41C22×C15111yescharacteristic
312#458—24no—
313#454—22no—
314#453—22no—
315#421—22no—
316#421C22×C15112yes—
317#421C22×C15112yes—
318C2×#104—22no—
319C2×#104C22×C15112yes—
320#418C22×C15112yescomplemented
321#413C22×C15111yescharacteristic
322C2×#101C22×C15112yescomplemented
323#410C22×C15111yescharacteristic
324#410C22×C15112yes—
325#344C22×C15112yescomplemented
326#339C22×C15112yescomplemented
327H(3,ℤ/4)C22×C15112yescomplemented
328H(3,ℤ/4)C22×C15112yescomplemented
3292−1+4—22no—
3302−1+4—21no—
331H(5,𝔽2)—22no—
332H(5,𝔽2)—22no—
333H(5,𝔽2)—21no—
334C2×#41—22no—
335C2×#41—42no—
336C2×#41C23×C15112yes—
337C2×#41C23×C15111yescharacteristic
338C22×Q8C23×C15111yescharacteristic
339C22×D4C23×C15111yescharacteristic
340AGL(1,ℤ/8)—24no—
341AGL(1,ℤ/8)—42no—
342C2×SD16—22no—
343C2×D8—22no—
344#131—42no—
345C2×M16—22no—
346C2×M16C23×C15112yes—
347C2×M16C23×C15112yes—
348C4⋊Q8—21no—
349C4⋊Q8—22no—
350C4⋊Q8—21no—
351#127D4×C15112yes—
352#126—24no—
353#126—24no—
354#125—21no—
355#124—24no—
356#124—22no—
357#124—21no—
358#124—22no—
359#124—21no—
360#124—21no—
361#124D4×C15112yes—
362#123—24no—
363#123—24no—
364#123—24no—
365#123D4×C15114yes—
366C22⋊Q8—24no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—42no—
370#121—22no—
371#120—22no—
372C4×Q8—21no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×D4—24no—
377C4×D4—22no—
378C4×D4—21no—
379C4×D4—22no—
380C4×D4—22no—
381#117—24no—
382#117—22no—
383#117D4×C15112yes—
384#117C23×C15112yes—
385#117C23×C15111yescharacteristic
386#117C23×C15111yescharacteristic
387#117C23×C15111yescharacteristic
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C42D4×C15112yes—
391C2×C42C23×C15111yescharacteristic
392#104—22no—
393#104—24no—
394#104—22no—
395#104—24no—
396#102—42no—
397#101—21no—
398#101—42no—
399#100—21no—
400#99—22no—
401#99—24no—
402C24—21no—
403#41—22no—
404#41—22no—
405#41—24no—
406#41—22no—
407#41—22no—
408#41—22no—
409#41—22no—
410#41—44no—
411#41—42no—
412#41—42no—
413#41—42no—
414#41—42no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—42no—
422C2×Q8C2×D4×C15111yescharacteristic
423C2×Q8#41×C15112yes—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4C2×D4×C15112yes—
434C2×D4C2×D4×C15112yes—
435C2×D4C2×D4×C15111yescharacteristic
436C2×D4#41×C15112yes—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4C2×D4×C15112yes—
443C22×C4C2×D4×C15112yes—
444C22×C4C24×C15111yescharacteristic
445SD16—44no—
446D8—44no—
447M16—22no—
448M16—22no—
449M16—22no—
450M16—22no—
451M16—42no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—42no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—44no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—24no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—44no—
476C22⋊C4C2×D4×C15112yes—
477C42—24no—
478C42—22no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42C2×D4×C15111yescharacteristic
483C42C2×D4×C15112yes—
484C42C2×D4×C15111yescharacteristic
485C42C2×D4×C15112yes—
486C23—22no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—82no—
491C23#120×C15112yes—
492C23C22×D4×C15111yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—22no—
496Q8—42no—
497Q8—41no—
498Q8—41no—
499Q8—42no—
500Q8—82no—
501D4—24no—
502D4—22no—
503D4—22no—
504D4—22no—
505D4—24no—
506D4—42no—
507D4—44no—
508D4—42no—
509D4—42no—
510D4—41no—
511D4—41no—
512D4—84no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—82no—
529C2×C4—84no—
530C2×C4—81no—
531C2×C4#120×C15112yes—
532C2×C4C22×D4×C15111yescharacteristic
533C2×C4C22×D4×C15111yescharacteristic
534C2×C4C2×#41×C15112yes—
535C2×C4H(5,𝔽2)×C15112yes—
536C8—42no—
537C8—42no—
538C22—22no—
539C22—41no—
540C22—42no—
541C22—44no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—162no—
546C22—162no—
547C22C2×#120×C15111yescharacteristic
548C22#546×C15112yes—
549C4—22no—
550C4—42no—
551C4—41no—
552C4—42no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—82no—
558C4#525×C15111yescharacteristic
559C4#535×C15111yescharacteristic
560C4#546×C15112yes—
561C2—21no—
562C2—22no—
563C2—42no—
564C2—82no—
565C2—82no—
566C2#2404×C15111yescharacteristic
subgroupG/H
center, socleC2×C151#2404
commutator, Frattini, cosocleC22×C4C24×C151
prime radical#3021×C151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C151 ◁ C22×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3510×C151C2semidirect
2. #3028×C151C2semidirect
4. #2901×C151C2semidirect
5. #2031×C151C2semidirect
6. #2003×C151C2semidirect
7. #1915×C151C2semidirect
8. #1912×C151C2semidirect
9. #1910×C151C2semidirect
10. #1904×C151C2semidirect
11. #1827×C151C2semidirect
22. #533×C151C22semidirect
24. C2×#124×C151C22semidirect
25. #519×C151C22semidirect
26. #518×C151C22semidirect
36. #418×C151C22semidirect
38. C2×#101×C151C22semidirect
41. #344×C151C22semidirect
42. #339×C151C22semidirect
43. H(3,ℤ/4)×C151C22semidirect
44. H(3,ℤ/4)×C151C22semidirect
283. #23993C151direct (N × H)
284. C151#23993direct (N × H)
285. #3510C2×C151semidirect
286. #3028C2×C151semidirect
288. #2901C2×C151semidirect
289. #2031C2×C151semidirect
290. #2003C2×C151semidirect
291. #1915C2×C151semidirect
292. #1912C2×C151semidirect
293. #1910C2×C151semidirect
294. #1904C2×C151semidirect
295. #1827C2×C151semidirect
306. #533C22×C151semidirect
308. C2×#124C22×C151semidirect
309. #519C22×C151semidirect
310. #518C22×C151semidirect
320. #418C22×C151semidirect
322. C2×#101C22×C151semidirect
325. #344C22×C151semidirect
326. #339C22×C151semidirect
327. H(3,ℤ/4)C22×C151semidirect
328. H(3,ℤ/4)C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3510×C1512
#3028×C1512
#3021×C1512
#2901×C1512
#2031×C1512
#2003×C1512
#1915×C1512
#1912×C1512
#1910×C1512
#1904×C1512
#1827×C1512
#23993151