Finite Group Databasebeta

← C151×#23917 #50328842

Subgroups — C151×#23917 #50328842

Order 38656.

1682 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3486×C151C211yesmaximal, characteristic, complemented
2C2×#536×C151C211yesmaximal, characteristic, complemented
3C2×#456×C151C211yesmaximal, characteristic, complemented
4C2×#454×C151C211yesmaximal, characteristic, complemented
5#3005×C151C212yesmaximal, complemented
6#2005×C151C212yesmaximal, complemented
7#2003×C151C212yesmaximal, complemented
8#1897×C151C212yesmaximal, complemented
9#1888×C151C211yesmaximal, characteristic, complemented
10#1886×C151C211yesmaximal, characteristic, complemented
11#1784×C151C211yesmaximal, characteristic, complemented
12C2×2−1+4×C151C2211yescharacteristic
13C22×#41×C151C2211yescharacteristic
14C23×Q8×C151C2211yescharacteristic
15C2×#137×C151C2212yes—
16C2×AGL(1,ℤ/8)×C151C2212yes—
17D4×Q8×C151—24no—
18#548×C151—24no—
19#536×C151—22no—
20#536×C151—24no—
21#536×C151—24no—
22#536×C151C2212yes—
23C2×C4⋊Q8×C151C2211yescharacteristic
24C2×C4⋊Q8×C151C2211yescharacteristic
25C2×#124×C151C2211yescharacteristic
26C2×#124×C151C2211yescharacteristic
27#525×C151—22no—
28#525×C151C2212yes—
29C2×#123×C151—22no—
30C2×C22⋊Q8×C151—22no—
31#519×C151C2211yescharacteristic
32C2×#117×C151C2211yescharacteristic
33#456×C151—24no—
34#454×C151—24no—
35#452×C151—24no—
36#449×C151—24no—
37C2×#104×C151C2212yes—
38C2×#104×C151C2212yes—
39#419×C151C2212yes—
40#418×C151C2212yes—
41C2×#101×C151C2211yescharacteristic
42C2×#101×C151C2211yescharacteristic
43C2×#101×C151C2212yes—
44#407×C151C2212yes—
45#339×C151C2212yes—
46#339×C151C2212yes—
472−1+4×C151—22no—
482−1+4×C151—22no—
492−1+4×C151—42no—
50C2×#41×C151—22no—
51C2×#41×C151—22no—
52C2×#41×C151—22no—
53C2×#41×C151—24no—
54C2×#41×C151—22no—
55C2×#41×C151—42no—
56C2×#41×C151C2312yes—
57C22×Q8×C151—24no—
58C22×Q8×C151—22no—
59C22×Q8×C151—22no—
60C22×Q8×C151—22no—
61C22×Q8×C151—42no—
62C22×Q8×C151D412yes—
63C22×Q8×C151C2311yescharacteristic
64C22×Q8×C151C2311yescharacteristic
65C22×Q8×C151C2311yescharacteristic
66C22×D4×C151—21no—
67C22×D4×C151C2311yescharacteristic
68C23×C4×C151—21no—
69C23×C4×C151D412yes—
70C23×C4×C151C2311yescharacteristic
71#137×C151—24no—
72#137×C151—42no—
73AGL(1,ℤ/8)×C151—24no—
74AGL(1,ℤ/8)×C151—42no—
75C2×Q16×C151—22no—
76C2×SD16×C151—22no—
77C2×SD16×C151—22no—
78C2×D8×C151—22no—
79C2×M16×C151C2312yes—
80C2×M16×C151C2312yes—
81C4⋊Q8×C151—24no—
82C4⋊Q8×C151—22no—
83C4⋊Q8×C151—22no—
84C4⋊Q8×C151—22no—
85C4⋊Q8×C151—22no—
86C4⋊Q8×C151D414yescomplemented
87#124×C151—24no—
88#124×C151—22no—
89#124×C151—22no—
90#124×C151—22no—
91#124×C151—22no—
92#124×C151D414yescomplemented
93#123×C151—216no—
94#123×C151—24no—
95#123×C151—22no—
96C22⋊Q8×C151—216no—
97C22⋊Q8×C151—24no—
98C22⋊Q8×C151—28no—
99C22⋊Q8×C151—24no—
100C22⋊Q8×C151D414yes—
101#121×C151—24no—
102#121×C151D414yes—
103#120×C151—22no—
104C4×Q8×C151—24no—
105C4×D4×C151—28no—
106C4×D4×C151—24no—
107#117×C151—22no—
108#117×C151—22no—
109#117×C151—22no—
110#117×C151C2312yes—
111C2×C4⋊C4×C151—21no—
112C2×C4⋊C4×C151—22no—
113C2×C4⋊C4×C151—22no—
114C2×C22⋊C4×C151—22no—
115C2×C22⋊C4×C151—22no—
116C2×C22⋊C4×C151—21no—
117C2×C42×C151C2311yescharacteristic
118C2×C42×C151C2311yescharacteristic
119#104×C151—24no—
120#104×C151—24no—
121#103×C151—24no—
122#102×C151—24no—
123#101×C151—22no—
124#101×C151—22no—
125#101×C151—42no—
126C22⋊C8×C151—24no—
127C24×C151—21no—
128C24×C151C2×D411yescharacteristic
129#41×C151—22no—
130#41×C151—22no—
131#41×C151—22no—
132#41×C151—22no—
133#41×C151—48no—
134#41×C151—48no—
135#41×C151—44no—
136#41×C151—44no—
137#41×C151—44no—
138#41×C151—44no—
139C2×Q8×C151—22no—
140C2×Q8×C151—24no—
141C2×Q8×C151—24no—
142C2×Q8×C151—22no—
143C2×Q8×C151—22no—
144C2×Q8×C151—21no—
145C2×Q8×C151—21no—
146C2×Q8×C151—21no—
147C2×Q8×C151—22no—
148C2×Q8×C151—22no—
149C2×Q8×C151—22no—
150C2×Q8×C151—22no—
151C2×Q8×C151—42no—
152C2×Q8×C151—48no—
153C2×Q8×C151—48no—
154C2×Q8×C151—44no—
155C2×Q8×C151—44no—
156C2×Q8×C151C2×D412yes—
157C2×Q8×C151C2×D412yes—
158C2×Q8×C151C2×D412yes—
159C2×Q8×C151C2×D412yes—
160C2×D4×C151—22no—
161C2×D4×C151—24no—
162C2×D4×C151—24no—
163C2×D4×C151—22no—
164C2×D4×C151—22no—
165C2×D4×C151—21no—
166C2×D4×C151—22no—
167C2×D4×C151—42no—
168C2×D4×C151—42no—
169C2×D4×C151C2×D412yes—
170C2×D4×C151C2×D412yes—
171C22×C4×C151—21no—
172C22×C4×C151—21no—
173C22×C4×C151—22no—
174C22×C4×C151—24no—
175C22×C4×C151—24no—
176C22×C4×C151—24no—
177C22×C4×C151—21no—
178C22×C4×C151—21no—
179C22×C4×C151—21no—
180C22×C4×C151—24no—
181C22×C4×C151—22no—
182C22×C4×C151—42no—
183C22×C4×C151—42no—
184C22×C4×C151—42no—
185C22×C4×C151C2×D412yes—
186C22×C4×C151C2×D412yes—
187C22×C4×C151C2×D411yescharacteristic
188C22×C4×C151C2×D414yes—
189C22×C4×C151C2411yescharacteristic
190Q16×C151—44no—
191SD16×C151—44no—
192SD16×C151—44no—
193D8×C151—44no—
194M16×C151—22no—
195M16×C151—22no—
196M16×C151—22no—
197M16×C151—22no—
198C2×C8×C151—22no—
199C2×C8×C151—22no—
200C4⋊C4×C151—24no—
201C4⋊C4×C151—28no—
202C4⋊C4×C151—24no—
203C4⋊C4×C151—28no—
204C4⋊C4×C151—28no—
205C4⋊C4×C151—24no—
206C4⋊C4×C151C2×D414yes—
207C22⋊C4×C151—24no—
208C22⋊C4×C151—28no—
209C22⋊C4×C151—28no—
210C22⋊C4×C151—24no—
211C22⋊C4×C151—22no—
212C22⋊C4×C151—28no—
213C22⋊C4×C151—24no—
214C42×C151—24no—
215C42×C151—21no—
216C42×C151—21no—
217C42×C151—22no—
218C42×C151C2×D412yes—
219C42×C151C2×D412yes—
220C23×C151—21no—
221C23×C151—21no—
222C23×C151—22no—
223C23×C151—22no—
224C23×C151—42no—
225C23×C151—44no—
226C23×C151—41no—
227C23×C151—41no—
228C23×C151#12012yes—
229C23×C151C22×D411yescharacteristic
230Q8×C151—24no—
231Q8×C151—24no—
232Q8×C151—24no—
233Q8×C151—22no—
234Q8×C151—22no—
235Q8×C151—48no—
236Q8×C151—44no—
237Q8×C151—42no—
238Q8×C151—42no—
239Q8×C151—42no—
240D4×C151—22no—
241D4×C151—22no—
242D4×C151—24no—
243D4×C151—44no—
244D4×C151—48no—
245D4×C151—48no—
246D4×C151—42no—
247D4×C151—42no—
248D4×C151—42no—
249C2×C4×C151—21no—
250C2×C4×C151—22no—
251C2×C4×C151—22no—
252C2×C4×C151—21no—
253C2×C4×C151—24no—
254C2×C4×C151—24no—
255C2×C4×C151—22no—
256C2×C4×C151—22no—
257C2×C4×C151—22no—
258C2×C4×C151—22no—
259C2×C4×C151—22no—
260C2×C4×C151—22no—
261C2×C4×C151—24no—
262C2×C4×C151—22no—
263C2×C4×C151—48no—
264C2×C4×C151—48no—
265C2×C4×C151—48no—
266C2×C4×C151—42no—
267C2×C4×C151—42no—
268C2×C4×C151—41no—
269C2×C4×C151—41no—
270C2×C4×C151—42no—
271C2×C4×C151—42no—
272C2×C4×C151#12014yes—
273C2×C4×C151#12012yes—
274C2×C4×C151C22×D411yescharacteristic
275C2×C4×C151C22×D411yescharacteristic
276C2×C4×C151C22×D412yes—
277C2×C4×C151H(5,𝔽2)12yes—
278C8×C151—42no—
279C8×C151—42no—
280C22×C151—22no—
281C22×C151—22no—
282C22×C151—22no—
283C22×C151—22no—
284C22×C151—42no—
285C22×C151—42no—
286C22×C151—44no—
287C22×C151—42no—
288C22×C151—42no—
289C22×C151—44no—
290C22×C151—82no—
291C22×C151—82no—
292C22×C151C2×#12011yescharacteristic
293C22×C151C2×#12011yescharacteristic
294C22×C151#53411yescharacteristic
295C4×C151—22no—
296C4×C151—24no—
297C4×C151—42no—
298C4×C151—42no—
299C4×C151—42no—
300C4×C151—42no—
301C4×C151—42no—
302C4×C151—44no—
303C4×C151D4212yes—
304C4×C151D4212yes—
305C2×C151—21no—
306C2×C151—21no—
307C2×C151—42no—
308C2×C151—82no—
309C2×C151—82no—
310C2×C151#240311yescharacteristic
311C2×C151#302012yes—
312#23917C15111yesmaximal, characteristic, direct product factor, Sylow
313C151#2391711yescharacteristic, direct product factor, Sylow
314#3486C2×C15111yescharacteristic, complemented
315C2×#536C2×C15111yescharacteristic, complemented
316C2×#456C2×C15111yescharacteristic, complemented
317C2×#454C2×C15111yescharacteristic, complemented
318#3005C2×C15112yescomplemented
319#2005C2×C15112yescomplemented
320#2003C2×C15112yescomplemented
321#1897C2×C15112yescomplemented
322#1888C2×C15111yescharacteristic, complemented
323#1886C2×C15111yescharacteristic, complemented
324#1784C2×C15111yescharacteristic, complemented
325C2×2−1+4C22×C15111yescharacteristic
326C22×#41C22×C15111yescharacteristic
327C23×Q8C22×C15111yescharacteristic
328C2×#137C22×C15112yes—
329C2×AGL(1,ℤ/8)C22×C15112yes—
330D4×Q8—24no—
331#548—24no—
332#536—22no—
333#536—24no—
334#536—24no—
335#536C22×C15112yes—
336C2×C4⋊Q8C22×C15111yescharacteristic
337C2×C4⋊Q8C22×C15111yescharacteristic
338C2×#124C22×C15111yescharacteristic
339C2×#124C22×C15111yescharacteristic
340#525—22no—
341#525C22×C15112yes—
342C2×#123—22no—
343C2×C22⋊Q8—22no—
344#519C22×C15111yescharacteristic
345C2×#117C22×C15111yescharacteristic
346#456—24no—
347#454—24no—
348#452—24no—
349#449—24no—
350C2×#104C22×C15112yes—
351C2×#104C22×C15112yes—
352#419C22×C15112yes—
353#418C22×C15112yes—
354C2×#101C22×C15111yescharacteristic
355C2×#101C22×C15111yescharacteristic
356C2×#101C22×C15112yes—
357#407C22×C15112yes—
358#339C22×C15112yes—
359#339C22×C15112yes—
3602−1+4—22no—
3612−1+4—22no—
3622−1+4—42no—
363C2×#41—22no—
364C2×#41—22no—
365C2×#41—22no—
366C2×#41—22no—
367C2×#41—24no—
368C2×#41—42no—
369C2×#41C23×C15112yes—
370C22×Q8—22no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—24no—
374C22×Q8—42no—
375C22×Q8D4×C15112yes—
376C22×Q8C23×C15111yescharacteristic
377C22×Q8C23×C15111yescharacteristic
378C22×Q8C23×C15111yescharacteristic
379C22×D4—21no—
380C22×D4C23×C15111yescharacteristic
381C23×C4—21no—
382C23×C4D4×C15112yes—
383C23×C4C23×C15111yescharacteristic
384#137—24no—
385#137—42no—
386AGL(1,ℤ/8)—24no—
387AGL(1,ℤ/8)—42no—
388C2×Q16—22no—
389C2×SD16—22no—
390C2×SD16—22no—
391C2×D8—22no—
392C2×M16C23×C15112yes—
393C2×M16C23×C15112yes—
394C4⋊Q8—24no—
395C4⋊Q8—22no—
396C4⋊Q8—22no—
397C4⋊Q8—22no—
398C4⋊Q8—22no—
399C4⋊Q8D4×C15114yescomplemented
400#124—24no—
401#124—22no—
402#124—22no—
403#124—22no—
404#124—22no—
405#124D4×C15114yescomplemented
406#123—216no—
407#123—24no—
408#123—22no—
409C22⋊Q8—216no—
410C22⋊Q8—24no—
411C22⋊Q8—28no—
412C22⋊Q8—24no—
413C22⋊Q8D4×C15114yes—
414#121—24no—
415#121D4×C15114yes—
416#120—22no—
417C4×Q8—24no—
418C4×D4—28no—
419C4×D4—24no—
420#117—22no—
421#117—22no—
422#117—22no—
423#117C23×C15112yes—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—22no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C42C23×C15111yescharacteristic
431C2×C42C23×C15111yescharacteristic
432#104—24no—
433#104—24no—
434#103—24no—
435#102—24no—
436#101—22no—
437#101—22no—
438#101—42no—
439C22⋊C8—24no—
440C24—21no—
441C24C2×D4×C15111yescharacteristic
442#41—22no—
443#41—22no—
444#41—22no—
445#41—22no—
446#41—48no—
447#41—48no—
448#41—44no—
449#41—44no—
450#41—44no—
451#41—44no—
452C2×Q8—22no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—42no—
465C2×Q8—48no—
466C2×Q8—48no—
467C2×Q8—44no—
468C2×Q8—44no—
469C2×Q8C2×D4×C15112yes—
470C2×Q8C2×D4×C15112yes—
471C2×Q8C2×D4×C15112yes—
472C2×Q8C2×D4×C15112yes—
473C2×D4—22no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—22no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4C2×D4×C15112yes—
483C2×D4C2×D4×C15112yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4C2×D4×C15112yes—
499C22×C4C2×D4×C15112yes—
500C22×C4C2×D4×C15111yescharacteristic
501C22×C4C2×D4×C15114yes—
502C22×C4C24×C15111yescharacteristic
503Q16—44no—
504SD16—44no—
505SD16—44no—
506D8—44no—
507M16—22no—
508M16—22no—
509M16—22no—
510M16—22no—
511C2×C8—22no—
512C2×C8—22no—
513C4⋊C4—24no—
514C4⋊C4—28no—
515C4⋊C4—24no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—24no—
519C4⋊C4C2×D4×C15114yes—
520C22⋊C4—24no—
521C22⋊C4—28no—
522C22⋊C4—28no—
523C22⋊C4—24no—
524C22⋊C4—22no—
525C22⋊C4—28no—
526C22⋊C4—24no—
527C42—24no—
528C42—21no—
529C42—21no—
530C42—22no—
531C42C2×D4×C15112yes—
532C42C2×D4×C15112yes—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—42no—
538C23—44no—
539C23—41no—
540C23—41no—
541C23#120×C15112yes—
542C23C22×D4×C15111yescharacteristic
543Q8—24no—
544Q8—24no—
545Q8—22no—
546Q8—24no—
547Q8—22no—
548Q8—48no—
549Q8—44no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553D4—22no—
554D4—24no—
555D4—22no—
556D4—44no—
557D4—48no—
558D4—48no—
559D4—42no—
560D4—42no—
561D4—42no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4#120×C15114yes—
586C2×C4#120×C15112yes—
587C2×C4C22×D4×C15111yescharacteristic
588C2×C4C22×D4×C15111yescharacteristic
589C2×C4C22×D4×C15112yes—
590C2×C4H(5,𝔽2)×C15112yes—
591C8—42no—
592C8—42no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—82no—
604C22—82no—
605C22C2×#120×C15111yescharacteristic
606C22C2×#120×C15111yescharacteristic
607C22#534×C15111yescharacteristic
608C4—22no—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4D42×C15112yes—
617C4D42×C15112yes—
618C2—21no—
619C2—21no—
620C2—42no—
621C2—82no—
622C2—82no—
623C2#2403×C15111yescharacteristic
624C2#3020×C15112yes—
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3486×C151C2semidirect
2. C2×#536×C151C2semidirect
3. C2×#456×C151C2semidirect
4. C2×#454×C151C2semidirect
5. #3005×C151C2semidirect
6. #2005×C151C2semidirect
7. #2003×C151C2semidirect
8. #1897×C151C2semidirect
9. #1888×C151C2semidirect
10. #1886×C151C2semidirect
11. #1784×C151C2semidirect
86. C4⋊Q8×C151D4semidirect
92. #124×C151D4semidirect
312. #23917C151direct (N × H)
313. C151#23917direct (N × H)
314. #3486C2×C151semidirect
315. C2×#536C2×C151semidirect
316. C2×#456C2×C151semidirect
317. C2×#454C2×C151semidirect
318. #3005C2×C151semidirect
319. #2005C2×C151semidirect
320. #2003C2×C151semidirect
321. #1897C2×C151semidirect
322. #1888C2×C151semidirect
323. #1886C2×C151semidirect
324. #1784C2×C151semidirect
399. C4⋊Q8D4×C151semidirect
405. #124D4×C151semidirect

Maximal subgroups

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

subgroupindex
#3486×C1512
C2×#536×C1512
C2×#456×C1512
C2×#454×C1512
#3005×C1512
#2005×C1512
#2003×C1512
#1897×C1512
#1888×C1512
#1886×C1512
#1784×C1512
#23917151