Finite Group Databasebeta

← C151×#23908 #50328833

Subgroups — C151×#23908 #50328833

Order 38656.

1814 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3485×C151C211yesmaximal, characteristic, complemented
2C2×#532×C151C211yesmaximal, characteristic, complemented
3C2×#455×C151C211yesmaximal, characteristic, complemented
4C2×#453×C151C211yesmaximal, characteristic, complemented
5#3005×C151C212yesmaximal, complemented
6#2005×C151C212yesmaximal, complemented
7#2003×C151C212yesmaximal, complemented
8#1896×C151C212yesmaximal, complemented
9#1887×C151C211yesmaximal, characteristic, complemented
10#1885×C151C211yesmaximal, characteristic, complemented
11#1785×C151C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)×C151C2211yescharacteristic
13C22×#41×C151C2211yescharacteristic
14C22×#41×C151C2211yescharacteristic, complemented
15C2×#137×C151C2212yes—
16C2×AGL(1,ℤ/8)×C151C2212yes—
17#547×C151—24no—
18#546×C151—24no—
19#536×C151C2212yes—
20#534×C151—22no—
21#532×C151—24no—
22#532×C151—24no—
23C2×C4⋊Q8×C151C2211yescharacteristic, complemented
24C2×#127×C151C2211yescharacteristic
25C2×#124×C151C2211yescharacteristic
26C2×#124×C151C2211yescharacteristic, complemented
27#525×C151C2212yes—
28C2×#123×C151—22no—
29C2×#121×C151—22no—
30#518×C151C2211yescharacteristic
31C2×C4×D4×C151—22no—
32C22×C42×C151C2211yescharacteristic, complemented
33#455×C151—24no—
34#453×C151—24no—
35#452×C151—24no—
36#449×C151—24no—
37C2×#104×C151C2212yes—
38C2×#104×C151C2212yes—
39#419×C151C2212yescomplemented
40#418×C151C2212yescomplemented
41C2×#101×C151C2212yescomplemented
42C2×#100×C151C2211yescharacteristic
43C2×#100×C151C2211yescharacteristic
44#407×C151C2212yescomplemented
45#339×C151C2212yescomplemented
46#339×C151C2212yescomplemented
47H(5,𝔽2)×C151—22no—
48H(5,𝔽2)×C151—22no—
49H(5,𝔽2)×C151—42no—
50C2×#41×C151—22no—
51C2×#41×C151—22no—
52C2×#41×C151—22no—
53C2×#41×C151—24no—
54C2×#41×C151—24no—
55C2×#41×C151—22no—
56C2×#41×C151—22no—
57C2×#41×C151—42no—
58C2×#41×C151—42no—
59C2×#41×C151C2312yes—
60C22×Q8×C151C2311yescharacteristic
61C22×Q8×C151C2311yescharacteristic
62C22×D4×C151—22no—
63C22×D4×C151—22no—
64C22×D4×C151—21no—
65C22×D4×C151—22no—
66C22×D4×C151D412yes—
67C22×D4×C151C2311yescharacteristic
68C22×D4×C151C2311yescharacteristic
69C23×C4×C151—21no—
70C23×C4×C151—21no—
71C23×C4×C151D412yes—
72C23×C4×C151C2311yescharacteristic
73#137×C151—24no—
74#137×C151—42no—
75AGL(1,ℤ/8)×C151—24no—
76AGL(1,ℤ/8)×C151—42no—
77C2×Q16×C151—22no—
78C2×SD16×C151—22no—
79C2×SD16×C151—22no—
80C2×D8×C151—22no—
81C2×M16×C151C2312yes—
82C2×M16×C151C2312yes—
83C4⋊Q8×C151—22no—
84C4⋊Q8×C151—22no—
85C4⋊Q8×C151—22no—
86#127×C151—22no—
87#127×C151D414yescomplemented
88#124×C151—22no—
89#124×C151—22no—
90#124×C151—22no—
91#124×C151—22no—
92#124×C151D414yescomplemented
93#123×C151—28no—
94#123×C151—24no—
95#123×C151—22no—
96#123×C151—24no—
97C22⋊Q8×C151—24no—
98C22⋊Q8×C151D414yescomplemented
99#121×C151—216no—
100#121×C151—28no—
101#121×C151D414yescomplemented
102#120×C151—24no—
103#120×C151—22no—
104C4×D4×C151—216no—
105C4×D4×C151—24no—
106C4×D4×C151—24no—
107#117×C151C2312yes—
108C2×C4⋊C4×C151—24no—
109C2×C4⋊C4×C151—22no—
110C2×C22⋊C4×C151—24no—
111C2×C22⋊C4×C151—22no—
112C2×C22⋊C4×C151—22no—
113C2×C22⋊C4×C151—22no—
114C2×C42×C151—21no—
115C2×C42×C151—22no—
116C2×C42×C151—22no—
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—42no—
124#100×C151—22no—
125#100×C151—22no—
126C22⋊C8×C151—24no—
127C24×C151—22no—
128C24×C151—21no—
129C24×C151C2×D412yes—
130C24×C151C2×D411yescharacteristic
131#41×C151—22no—
132#41×C151—22no—
133#41×C151—22no—
134#41×C151—22no—
135#41×C151—48no—
136#41×C151—48no—
137#41×C151—48no—
138#41×C151—48no—
139#41×C151—44no—
140#41×C151—44no—
141C2×Q8×C151—22no—
142C2×Q8×C151—21no—
143C2×Q8×C151—21no—
144C2×Q8×C151—21no—
145C2×Q8×C151—22no—
146C2×Q8×C151—22no—
147C2×Q8×C151—22no—
148C2×Q8×C151C2×D412yes—
149C2×Q8×C151C2×D412yes—
150C2×D4×C151—24no—
151C2×D4×C151—24no—
152C2×D4×C151—28no—
153C2×D4×C151—24no—
154C2×D4×C151—24no—
155C2×D4×C151—22no—
156C2×D4×C151—24no—
157C2×D4×C151—22no—
158C2×D4×C151—22no—
159C2×D4×C151—21no—
160C2×D4×C151—22no—
161C2×D4×C151—22no—
162C2×D4×C151—42no—
163C2×D4×C151—48no—
164C2×D4×C151—44no—
165C2×D4×C151—44no—
166C2×D4×C151—42no—
167C2×D4×C151—44no—
168C2×D4×C151—42no—
169C2×D4×C151—44no—
170C2×D4×C151C2×D412yes—
171C2×D4×C151C2×D412yes—
172C2×D4×C151C2×D412yes—
173C2×D4×C151C2×D412yes—
174C22×C4×C151—21no—
175C22×C4×C151—21no—
176C22×C4×C151—21no—
177C22×C4×C151—22no—
178C22×C4×C151—24no—
179C22×C4×C151—22no—
180C22×C4×C151—24no—
181C22×C4×C151—24no—
182C22×C4×C151—21no—
183C22×C4×C151—21no—
184C22×C4×C151—24no—
185C22×C4×C151—24no—
186C22×C4×C151—22no—
187C22×C4×C151—42no—
188C22×C4×C151—44no—
189C22×C4×C151—42no—
190C22×C4×C151—42no—
191C22×C4×C151C2×D412yes—
192C22×C4×C151C2×D411yescharacteristic
193C22×C4×C151C2×D414yes—
194C22×C4×C151C2411yescharacteristic
195Q16×C151—44no—
196SD16×C151—44no—
197SD16×C151—44no—
198D8×C151—44no—
199M16×C151—22no—
200M16×C151—22no—
201M16×C151—22no—
202M16×C151—22no—
203C2×C8×C151—22no—
204C2×C8×C151—22no—
205C4⋊C4×C151—28no—
206C4⋊C4×C151—28no—
207C4⋊C4×C151—24no—
208C4⋊C4×C151C2×D414yes—
209C22⋊C4×C151—24no—
210C22⋊C4×C151—28no—
211C22⋊C4×C151—28no—
212C22⋊C4×C151—28no—
213C22⋊C4×C151—24no—
214C22⋊C4×C151—22no—
215C22⋊C4×C151—24no—
216C42×C151—24no—
217C42×C151—21no—
218C42×C151—21no—
219C42×C151—22no—
220C42×C151C2×D412yes—
221C42×C151C2×D412yes—
222C23×C151—21no—
223C23×C151—22no—
224C23×C151—21no—
225C23×C151—22no—
226C23×C151—22no—
227C23×C151—22no—
228C23×C151—42no—
229C23×C151—44no—
230C23×C151—44no—
231C23×C151—44no—
232C23×C151—41no—
233C23×C151—41no—
234C23×C151—84no—
235C23×C151#12012yes—
236C23×C151#12014yes—
237C23×C151C22×D411yescharacteristic
238Q8×C151—24no—
239Q8×C151—22no—
240Q8×C151—22no—
241Q8×C151—42no—
242Q8×C151—42no—
243Q8×C151—42no—
244D4×C151—24no—
245D4×C151—24no—
246D4×C151—22no—
247D4×C151—22no—
248D4×C151—24no—
249D4×C151—416no—
250D4×C151—48no—
251D4×C151—48no—
252D4×C151—48no—
253D4×C151—42no—
254D4×C151—42no—
255D4×C151—42no—
256D4×C151—84no—
257C2×C4×C151—24no—
258C2×C4×C151—21no—
259C2×C4×C151—22no—
260C2×C4×C151—21no—
261C2×C4×C151—24no—
262C2×C4×C151—24no—
263C2×C4×C151—22no—
264C2×C4×C151—22no—
265C2×C4×C151—22no—
266C2×C4×C151—22no—
267C2×C4×C151—22no—
268C2×C4×C151—24no—
269C2×C4×C151—22no—
270C2×C4×C151—48no—
271C2×C4×C151—48no—
272C2×C4×C151—48no—
273C2×C4×C151—44no—
274C2×C4×C151—42no—
275C2×C4×C151—41no—
276C2×C4×C151—41no—
277C2×C4×C151—42no—
278C2×C4×C151—42no—
279C2×C4×C151—84no—
280C2×C4×C151#12012yes—
281C2×C4×C151C22×D411yescharacteristic
282C2×C4×C151C22×D411yescharacteristic
283C2×C4×C151C22×D412yes—
284C2×C4×C151H(5,𝔽2)12yes—
285C8×C151—42no—
286C8×C151—42no—
287C22×C151—22no—
288C22×C151—22no—
289C22×C151—24no—
290C22×C151—24no—
291C22×C151—44no—
292C22×C151—42no—
293C22×C151—42no—
294C22×C151—44no—
295C22×C151—44no—
296C22×C151—42no—
297C22×C151—42no—
298C22×C151—44no—
299C22×C151—82no—
300C22×C151—82no—
301C22×C151—82no—
302C22×C151—164no—
303C22×C151C2×#12011yescharacteristic
304C22×C151C2×#12011yescharacteristic
305C22×C151#53411yescharacteristic
306C4×C151—24no—
307C4×C151—22no—
308C4×C151—42no—
309C4×C151—42no—
310C4×C151—42no—
311C4×C151—42no—
312C4×C151—44no—
313C4×C151D4212yes—
314C4×C151D4212yes—
315C2×C151—21no—
316C2×C151—21no—
317C2×C151—42no—
318C2×C151—44no—
319C2×C151—82no—
320C2×C151—82no—
321C2×C151#240311yescharacteristic
322C2×C151#301812yes—
323#23908C15111yesmaximal, characteristic, direct product factor, Sylow
324C151#2390811yescharacteristic, direct product factor, Sylow
325#3485C2×C15111yescharacteristic, complemented
326C2×#532C2×C15111yescharacteristic, complemented
327C2×#455C2×C15111yescharacteristic, complemented
328C2×#453C2×C15111yescharacteristic, complemented
329#3005C2×C15112yescomplemented
330#2005C2×C15112yescomplemented
331#2003C2×C15112yescomplemented
332#1896C2×C15112yescomplemented
333#1887C2×C15111yescharacteristic, complemented
334#1885C2×C15111yescharacteristic, complemented
335#1785C2×C15111yescharacteristic, complemented
336C2×H(5,𝔽2)C22×C15111yescharacteristic
337C22×#41C22×C15111yescharacteristic
338C22×#41C22×C15111yescharacteristic, complemented
339C2×#137C22×C15112yes—
340C2×AGL(1,ℤ/8)C22×C15112yes—
341#547—24no—
342#546—24no—
343#536C22×C15112yes—
344#534—22no—
345#532—24no—
346#532—24no—
347C2×C4⋊Q8C22×C15111yescharacteristic, complemented
348C2×#127C22×C15111yescharacteristic
349C2×#124C22×C15111yescharacteristic
350C2×#124C22×C15111yescharacteristic, complemented
351#525C22×C15112yes—
352C2×#123—22no—
353C2×#121—22no—
354#518C22×C15111yescharacteristic
355C2×C4×D4—22no—
356C22×C42C22×C15111yescharacteristic, complemented
357#455—24no—
358#453—24no—
359#452—24no—
360#449—24no—
361C2×#104C22×C15112yes—
362C2×#104C22×C15112yes—
363#419C22×C15112yescomplemented
364#418C22×C15112yescomplemented
365C2×#101C22×C15112yescomplemented
366C2×#100C22×C15111yescharacteristic
367C2×#100C22×C15111yescharacteristic
368#407C22×C15112yescomplemented
369#339C22×C15112yescomplemented
370#339C22×C15112yescomplemented
371H(5,𝔽2)—22no—
372H(5,𝔽2)—22no—
373H(5,𝔽2)—42no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41—24no—
380C2×#41—24no—
381C2×#41—42no—
382C2×#41—42no—
383C2×#41C23×C15112yes—
384C22×Q8C23×C15111yescharacteristic
385C22×Q8C23×C15111yescharacteristic
386C22×D4—22no—
387C22×D4—21no—
388C22×D4—22no—
389C22×D4—22no—
390C22×D4D4×C15112yes—
391C22×D4C23×C15111yescharacteristic
392C22×D4C23×C15111yescharacteristic
393C23×C4—21no—
394C23×C4—21no—
395C23×C4D4×C15112yes—
396C23×C4C23×C15111yescharacteristic
397#137—24no—
398#137—42no—
399AGL(1,ℤ/8)—24no—
400AGL(1,ℤ/8)—42no—
401C2×Q16—22no—
402C2×SD16—22no—
403C2×SD16—22no—
404C2×D8—22no—
405C2×M16C23×C15112yes—
406C2×M16C23×C15112yes—
407C4⋊Q8—22no—
408C4⋊Q8—22no—
409C4⋊Q8—22no—
410#127—22no—
411#127D4×C15114yescomplemented
412#124—22no—
413#124—22no—
414#124—22no—
415#124—22no—
416#124D4×C15114yescomplemented
417#123—28no—
418#123—24no—
419#123—22no—
420#123—24no—
421C22⋊Q8—24no—
422C22⋊Q8D4×C15114yescomplemented
423#121—216no—
424#121—28no—
425#121D4×C15114yescomplemented
426#120—24no—
427#120—22no—
428C4×D4—216no—
429C4×D4—24no—
430C4×D4—24no—
431#117C23×C15112yes—
432C2×C4⋊C4—24no—
433C2×C4⋊C4—22no—
434C2×C22⋊C4—24no—
435C2×C22⋊C4—22no—
436C2×C22⋊C4—22no—
437C2×C22⋊C4—22no—
438C2×C42—21no—
439C2×C42—22no—
440C2×C42—22no—
441C2×C42C23×C15111yescharacteristic
442C2×C42C23×C15111yescharacteristic
443#104—24no—
444#104—24no—
445#103—24no—
446#102—24no—
447#101—42no—
448#100—22no—
449#100—22no—
450C22⋊C8—24no—
451C24—22no—
452C24—21no—
453C24C2×D4×C15112yes—
454C24C2×D4×C15111yescharacteristic
455#41—22no—
456#41—22no—
457#41—22no—
458#41—22no—
459#41—48no—
460#41—48no—
461#41—48no—
462#41—48no—
463#41—44no—
464#41—44no—
465C2×Q8—22no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8C2×D4×C15112yes—
473C2×Q8C2×D4×C15112yes—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—28no—
477C2×D4—24no—
478C2×D4—24no—
479C2×D4—22no—
480C2×D4—24no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—42no—
487C2×D4—48no—
488C2×D4—44no—
489C2×D4—44no—
490C2×D4—42no—
491C2×D4—44no—
492C2×D4—42no—
493C2×D4—44no—
494C2×D4C2×D4×C15112yes—
495C2×D4C2×D4×C15112yes—
496C2×D4C2×D4×C15112yes—
497C2×D4C2×D4×C15112yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—22no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—44no—
514C22×C4—42no—
515C22×C4C2×D4×C15112yes—
516C22×C4C2×D4×C15111yescharacteristic
517C22×C4C2×D4×C15114yes—
518C22×C4C24×C15111yescharacteristic
519Q16—44no—
520SD16—44no—
521SD16—44no—
522D8—44no—
523M16—22no—
524M16—22no—
525M16—22no—
526M16—22no—
527C2×C8—22no—
528C2×C8—22no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4C2×D4×C15114yes—
533C22⋊C4—24no—
534C22⋊C4—28no—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—24no—
538C22⋊C4—22no—
539C22⋊C4—24no—
540C42—24no—
541C42—21no—
542C42—21no—
543C42—22no—
544C42C2×D4×C15112yes—
545C42C2×D4×C15112yes—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—41no—
557C23—41no—
558C23—84no—
559C23#120×C15112yes—
560C23#120×C15114yes—
561C23C22×D4×C15111yescharacteristic
562Q8—22no—
563Q8—24no—
564Q8—22no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568D4—24no—
569D4—24no—
570D4—22no—
571D4—24no—
572D4—22no—
573D4—416no—
574D4—48no—
575D4—48no—
576D4—48no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—84no—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—84no—
604C2×C4#120×C15112yes—
605C2×C4C22×D4×C15111yescharacteristic
606C2×C4C22×D4×C15111yescharacteristic
607C2×C4C22×D4×C15112yes—
608C2×C4H(5,𝔽2)×C15112yes—
609C8—42no—
610C8—42no—
611C22—22no—
612C22—22no—
613C22—24no—
614C22—24no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—164no—
627C22C2×#120×C15111yescharacteristic
628C22C2×#120×C15111yescharacteristic
629C22#534×C15111yescharacteristic
630C4—24no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—44no—
637C4D42×C15112yes—
638C4D42×C15112yes—
639C2—21no—
640C2—21no—
641C2—42no—
642C2—44no—
643C2—82no—
644C2—82no—
645C2#2403×C15111yescharacteristic
646C2#3018×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. #3485×C151C2semidirect
2. C2×#532×C151C2semidirect
3. C2×#455×C151C2semidirect
4. C2×#453×C151C2semidirect
5. #3005×C151C2semidirect
6. #2005×C151C2semidirect
7. #2003×C151C2semidirect
8. #1896×C151C2semidirect
9. #1887×C151C2semidirect
10. #1885×C151C2semidirect
11. #1785×C151C2semidirect
14. C22×#41×C151C22semidirect
23. C2×C4⋊Q8×C151C22semidirect
26. C2×#124×C151C22semidirect
32. C22×C42×C151C22semidirect
39. #419×C151C22semidirect
40. #418×C151C22semidirect
41. C2×#101×C151C22semidirect
44. #407×C151C22semidirect
45. #339×C151C22semidirect
46. #339×C151C22semidirect
87. #127×C151D4semidirect
92. #124×C151D4semidirect
98. C22⋊Q8×C151D4semidirect
101. #121×C151D4semidirect
323. #23908C151direct (N × H)
324. C151#23908direct (N × H)
325. #3485C2×C151semidirect
326. C2×#532C2×C151semidirect
327. C2×#455C2×C151semidirect
328. C2×#453C2×C151semidirect
329. #3005C2×C151semidirect
330. #2005C2×C151semidirect
331. #2003C2×C151semidirect
332. #1896C2×C151semidirect
333. #1887C2×C151semidirect
334. #1885C2×C151semidirect
335. #1785C2×C151semidirect
338. C22×#41C22×C151semidirect
347. C2×C4⋊Q8C22×C151semidirect
350. C2×#124C22×C151semidirect
356. C22×C42C22×C151semidirect
363. #419C22×C151semidirect
364. #418C22×C151semidirect
365. C2×#101C22×C151semidirect
368. #407C22×C151semidirect
369. #339C22×C151semidirect
370. #339C22×C151semidirect
411. #127D4×C151semidirect
416. #124D4×C151semidirect
422. C22⋊Q8D4×C151semidirect
425. #121D4×C151semidirect

Maximal subgroups

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

subgroupindex
#3485×C1512
C2×#532×C1512
C2×#455×C1512
C2×#453×C1512
#3005×C1512
#2005×C1512
#2003×C1512
#1896×C1512
#1887×C1512
#1885×C1512
#1785×C1512
#23908151