Finite Group Databasebeta

← C151×#18314 #50323239

Subgroups — C151×#18314 #50323239

Order 38656.

958 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3459×C151C211yesmaximal, characteristic
2#3232×C151C212yesmaximal, complemented
3#2633×C151C211yesmaximal, characteristic, complemented
4#1618×C151C212yesmaximal, complemented
5#1614×C151C211yesmaximal, characteristic, complemented
6#1613×C151C211yesmaximal, characteristic, complemented
7#1612×C151C212yesmaximal, complemented
8#1608×C151C211yesmaximal, characteristic, complemented
9#1607×C151C211yesmaximal, characteristic, complemented
10#1521×C151C212yesmaximal, complemented
11#1460×C151C211yesmaximal, characteristic, complemented
12#543×C151C2211yescharacteristic
13#539×C151—22no—
14#539×C151—21no—
15#538×C151—22no—
16#538×C151C2212yes—
17#534×C151—21no—
18#534×C151C2211yescharacteristic
19#529×C151—22no—
20#529×C151C2212yes—
21C2×#125×C151C2211yescharacteristic, complemented
22#525×C151—21no—
23#525×C151—22no—
24C2×C4×D4×C151—21no—
25C2×C4×D4×C151C2211yescharacteristic
26#499×C151—22no—
27#488×C151—22no—
28#484×C151C2212yescomplemented
29#483×C151C2212yescomplemented
30#481×C151C2212yes—
31#480×C151C2212yes—
32#479×C151—22no—
33#478×C151—22no—
34#431×C151C2212yescomplemented
35#399×C151C2211yescharacteristic
36#399×C151C2211yescharacteristic, complemented
37#398×C151—21no—
38#397×C151C2211yescharacteristic
39#397×C151C2211yescharacteristic, complemented
40#394×C151—21no—
41#385×C151—21no—
42#385×C151—21no—
43#383×C151C2211yescharacteristic, complemented
44#328×C151C2212yescomplemented
45#328×C151C2212yescomplemented
46#327×C151C2212yes—
47#327×C151C2212yes—
48#323×C151C2211yescharacteristic
49#323×C151C2211yescharacteristic
50#323×C151C2212yescomplemented
51C2×#41×C151—21no—
52C22×D4×C151—21no—
53C22×D4×C151D411yescharacteristic
54C23×C4×C151—21no—
55C23×C4×C151D411yescharacteristic
56C23×C4×C151D411yescharacteristic
57#126×C151—22no—
58#126×C151—22no—
59#126×C151—22no—
60#125×C151—22no—
61#125×C151—22no—
62#125×C151—22no—
63#125×C151D412yes—
64#124×C151—22no—
65#124×C151D412yes—
66#123×C151—22no—
67#123×C151—22no—
68#123×C151—22no—
69#123×C151—22no—
70#123×C151—22no—
71#123×C151—22no—
72C22⋊Q8×C151—21no—
73C22⋊Q8×C151—22no—
74C22⋊Q8×C151—21no—
75C22⋊Q8×C151—22no—
76C22⋊Q8×C151—22no—
77C22⋊Q8×C151C2312yes—
78#121×C151—22no—
79#121×C151—21no—
80#121×C151—22no—
81#121×C151—22no—
82#121×C151—21no—
83#121×C151C2312yes—
84#120×C151—22no—
85#120×C151—22no—
86#120×C151—22no—
87C4×Q8×C151D412yes—
88C4×D4×C151—22no—
89C4×D4×C151—22no—
90C4×D4×C151—22no—
91C4×D4×C151—22no—
92C4×D4×C151—22no—
93C4×D4×C151—22no—
94C4×D4×C151—22no—
95C4×D4×C151D412yes—
96#117×C151—21no—
97#117×C151—22no—
98C2×C4⋊C4×C151—21no—
99C2×C4⋊C4×C151—21no—
100C2×C4⋊C4×C151—21no—
101C2×C4⋊C4×C151—21no—
102C2×C4⋊C4×C151—21no—
103C2×C4⋊C4×C151D411yescharacteristic
104C2×C4⋊C4×C151C2311yescharacteristic
105C2×C4⋊C4×C151C2311yescharacteristic
106C2×C22⋊C4×C151—21no—
107C2×C22⋊C4×C151—21no—
108C2×C22⋊C4×C151—21no—
109C2×C22⋊C4×C151—21no—
110C2×C22⋊C4×C151—21no—
111C2×C22⋊C4×C151—21no—
112C2×C22⋊C4×C151C2311yescharacteristic
113C2×C22⋊C4×C151C2311yescharacteristic
114C2×C42×C151—21no—
115C2×C42×C151C2311yescharacteristic
116#107×C151—22no—
117#107×C151—22no—
118#106×C151—22no—
119#106×C151—22no—
120C4⋊C8×C151—22no—
121#103×C151—22no—
122#103×C151—22no—
123#102×C151—22no—
124#102×C151—22no—
125C22⋊C8×C151C2312yes—
126C22⋊C8×C151C2312yes—
127C4×C8×C151—22no—
128#95×C151—21no—
129#95×C151—21no—
130#95×C151—21no—
131#95×C151—21no—
132#95×C151C2311yescharacteristic
133#95×C151C2311yescharacteristic
134C24×C151—21no—
135C24×C151C2×D411yescharacteristic
136C24×C151C2×D411yescharacteristic
137#41×C151—82no—
138C2×Q8×C151—21no—
139C2×Q8×C151C2×D412yes—
140C2×D4×C151—22no—
141C2×D4×C151—22no—
142C2×D4×C151—22no—
143C2×D4×C151—22no—
144C2×D4×C151—21no—
145C2×D4×C151—22no—
146C2×D4×C151—22no—
147C2×D4×C151—41no—
148C2×D4×C151—42no—
149C2×D4×C151—42no—
150C2×D4×C151C2×D412yes—
151C22×C4×C151—21no—
152C22×C4×C151—21no—
153C22×C4×C151—21no—
154C22×C4×C151—22no—
155C22×C4×C151—21no—
156C22×C4×C151—21no—
157C22×C4×C151—22no—
158C22×C4×C151—21no—
159C22×C4×C151—21no—
160C22×C4×C151—21no—
161C22×C4×C151—21no—
162C22×C4×C151—21no—
163C22×C4×C151—21no—
164C22×C4×C151—81no—
165C22×C4×C151—81no—
166C22×C4×C151—82no—
167C22×C4×C151C2×D411yescharacteristic
168C22×C4×C151C2×D411yescharacteristic
169C22×C4×C151C2×D411yescharacteristic
170C22×C4×C151C2×D411yescharacteristic
171C22×C4×C151C2411yescharacteristic
172C2×C8×C151—22no—
173C2×C8×C151—22no—
174C4⋊C4×C151—22no—
175C4⋊C4×C151—21no—
176C4⋊C4×C151—21no—
177C4⋊C4×C151—22no—
178C4⋊C4×C151—22no—
179C4⋊C4×C151—22no—
180C4⋊C4×C151—22no—
181C4⋊C4×C151—22no—
182C4⋊C4×C151—22no—
183C4⋊C4×C151—22no—
184C4⋊C4×C151—22no—
185C4⋊C4×C151—41no—
186C4⋊C4×C151—42no—
187C4⋊C4×C151—41no—
188C4⋊C4×C151—81no—
189C4⋊C4×C151—81no—
190C4⋊C4×C151C2×D412yes—
191C4⋊C4×C151C2×D412yes—
192C22⋊C4×C151—22no—
193C22⋊C4×C151—22no—
194C22⋊C4×C151—22no—
195C22⋊C4×C151—22no—
196C22⋊C4×C151—22no—
197C22⋊C4×C151—22no—
198C22⋊C4×C151—22no—
199C22⋊C4×C151—22no—
200C22⋊C4×C151—22no—
201C22⋊C4×C151—22no—
202C22⋊C4×C151—22no—
203C22⋊C4×C151—22no—
204C22⋊C4×C151—42no—
205C22⋊C4×C151—42no—
206C22⋊C4×C151—81no—
207C22⋊C4×C151—81no—
208C42×C151—22no—
209C42×C151—22no—
210C42×C151C2×D412yes—
211C42×C151C2×D412yes—
212C23×C151—21no—
213C23×C151—21no—
214C23×C151—22no—
215C23×C151—21no—
216C23×C151—21no—
217C23×C151—21no—
218C23×C151—41no—
219C23×C151—41no—
220C23×C151—41no—
221C23×C151—41no—
222C23×C151—42no—
223C23×C151—81no—
224C23×C151—81no—
225C23×C151#13512yes—
226C23×C151C22×D411yescharacteristic
227Q8×C151—42no—
228Q8×C151—81no—
229D4×C151—44no—
230D4×C151—42no—
231D4×C151—44no—
232D4×C151—42no—
233D4×C151—42no—
234D4×C151—81no—
235D4×C151—82no—
236D4×C151—82no—
237C2×C4×C151—21no—
238C2×C4×C151—21no—
239C2×C4×C151—21no—
240C2×C4×C151—22no—
241C2×C4×C151—22no—
242C2×C4×C151—22no—
243C2×C4×C151—21no—
244C2×C4×C151—41no—
245C2×C4×C151—41no—
246C2×C4×C151—42no—
247C2×C4×C151—44no—
248C2×C4×C151—42no—
249C2×C4×C151—42no—
250C2×C4×C151—41no—
251C2×C4×C151—41no—
252C2×C4×C151—41no—
253C2×C4×C151—41no—
254C2×C4×C151—41no—
255C2×C4×C151—41no—
256C2×C4×C151—41no—
257C2×C4×C151—41no—
258C2×C4×C151—41no—
259C2×C4×C151—41no—
260C2×C4×C151—41no—
261C2×C4×C151—41no—
262C2×C4×C151—41no—
263C2×C4×C151—41no—
264C2×C4×C151—81no—
265C2×C4×C151—81no—
266C2×C4×C151—81no—
267C2×C4×C151—82no—
268C2×C4×C151—81no—
269C2×C4×C151#12011yescharacteristic
270C2×C4×C151#12012yes—
271C2×C4×C151#12011yescharacteristic
272C2×C4×C151C22×D412yes—
273C8×C151—42no—
274C8×C151—42no—
275C22×C151—21no—
276C22×C151—21no—
277C22×C151—22no—
278C22×C151—42no—
279C22×C151—42no—
280C22×C151—41no—
281C22×C151—41no—
282C22×C151—41no—
283C22×C151—42no—
284C22×C151—41no—
285C22×C151—42no—
286C22×C151—42no—
287C22×C151—41no—
288C22×C151—81no—
289C22×C151—81no—
290C22×C151—84no—
291C22×C151C2×#12011yescharacteristic
292C22×C151C2×#13511yescharacteristic
293C22×C151#57511yescharacteristic
294C4×C151—22no—
295C4×C151—21no—
296C4×C151—21no—
297C4×C151—22no—
298C4×C151—42no—
299C4×C151—41no—
300C4×C151—41no—
301C4×C151—42no—
302C4×C151—42no—
303C4×C151—81no—
304C4×C151—81no—
305C4×C151—81no—
306C4×C151—81no—
307C2×C151—21no—
308C2×C151—21no—
309C2×C151—42no—
310C2×C151—81no—
311C2×C151—82no—
312C2×C151#300211yescharacteristic
313C2×C151#301911yescharacteristic
314C2×C151#302811yescharacteristic
315#18314C15111yesmaximal, characteristic, direct product factor, Sylow
316C151#1831411yescharacteristic, direct product factor, Sylow
317#3459C2×C15111yescharacteristic
318#3232C2×C15112yescomplemented
319#2633C2×C15111yescharacteristic, complemented
320#1618C2×C15112yescomplemented
321#1614C2×C15111yescharacteristic, complemented
322#1613C2×C15111yescharacteristic, complemented
323#1612C2×C15112yescomplemented
324#1608C2×C15111yescharacteristic, complemented
325#1607C2×C15111yescharacteristic, complemented
326#1521C2×C15112yescomplemented
327#1460C2×C15111yescharacteristic, complemented
328#543C22×C15111yescharacteristic
329#539—22no—
330#539—21no—
331#538—22no—
332#538C22×C15112yes—
333#534—21no—
334#534C22×C15111yescharacteristic
335#529—22no—
336#529C22×C15112yes—
337C2×#125C22×C15111yescharacteristic, complemented
338#525—21no—
339#525—22no—
340C2×C4×D4—21no—
341C2×C4×D4C22×C15111yescharacteristic
342#499—22no—
343#488—22no—
344#484C22×C15112yescomplemented
345#483C22×C15112yescomplemented
346#481C22×C15112yes—
347#480C22×C15112yes—
348#479—22no—
349#478—22no—
350#431C22×C15112yescomplemented
351#399C22×C15111yescharacteristic
352#399C22×C15111yescharacteristic, complemented
353#398—21no—
354#397C22×C15111yescharacteristic
355#397C22×C15111yescharacteristic, complemented
356#394—21no—
357#385—21no—
358#385—21no—
359#383C22×C15111yescharacteristic, complemented
360#328C22×C15112yescomplemented
361#328C22×C15112yescomplemented
362#327C22×C15112yes—
363#327C22×C15112yes—
364#323C22×C15111yescharacteristic
365#323C22×C15111yescharacteristic
366#323C22×C15112yescomplemented
367C2×#41—21no—
368C22×D4—21no—
369C22×D4D4×C15111yescharacteristic
370C23×C4—21no—
371C23×C4D4×C15111yescharacteristic
372C23×C4D4×C15111yescharacteristic
373#126—22no—
374#126—22no—
375#126—22no—
376#125—22no—
377#125—22no—
378#125—22no—
379#125D4×C15112yes—
380#124—22no—
381#124D4×C15112yes—
382#123—22no—
383#123—22no—
384#123—22no—
385#123—22no—
386#123—22no—
387#123—22no—
388C22⋊Q8—21no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—21no—
392C22⋊Q8—22no—
393C22⋊Q8C23×C15112yes—
394#121—22no—
395#121—21no—
396#121—22no—
397#121—22no—
398#121—21no—
399#121C23×C15112yes—
400#120—22no—
401#120—22no—
402#120—22no—
403C4×Q8D4×C15112yes—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4D4×C15112yes—
412#117—21no—
413#117—22no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4D4×C15111yescharacteristic
420C2×C4⋊C4C23×C15111yescharacteristic
421C2×C4⋊C4C23×C15111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4C23×C15111yescharacteristic
429C2×C22⋊C4C23×C15111yescharacteristic
430C2×C42—21no—
431C2×C42C23×C15111yescharacteristic
432#107—22no—
433#107—22no—
434#106—22no—
435#106—22no—
436C4⋊C8—22no—
437#103—22no—
438#103—22no—
439#102—22no—
440#102—22no—
441C22⋊C8C23×C15112yes—
442C22⋊C8C23×C15112yes—
443C4×C8—22no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95C23×C15111yescharacteristic
449#95C23×C15111yescharacteristic
450C24—21no—
451C24C2×D4×C15111yescharacteristic
452C24C2×D4×C15111yescharacteristic
453#41—82no—
454C2×Q8—21no—
455C2×Q8C2×D4×C15112yes—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—41no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×D4×C15112yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4—82no—
483C22×C4C2×D4×C15111yescharacteristic
484C22×C4C2×D4×C15111yescharacteristic
485C22×C4C2×D4×C15111yescharacteristic
486C22×C4C2×D4×C15111yescharacteristic
487C22×C4C24×C15111yescharacteristic
488C2×C8—22no—
489C2×C8—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—41no—
502C4⋊C4—42no—
503C4⋊C4—41no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4C2×D4×C15112yes—
507C4⋊C4C2×D4×C15112yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C42—22no—
525C42—22no—
526C42C2×D4×C15112yes—
527C42C2×D4×C15112yes—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—81no—
540C23—81no—
541C23#135×C15112yes—
542C23C22×D4×C15111yescharacteristic
543Q8—42no—
544Q8—81no—
545D4—44no—
546D4—42no—
547D4—44no—
548D4—42no—
549D4—42no—
550D4—81no—
551D4—82no—
552D4—82no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4#120×C15111yescharacteristic
586C2×C4#120×C15112yes—
587C2×C4#120×C15111yescharacteristic
588C2×C4C22×D4×C15112yes—
589C8—42no—
590C8—42no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—41no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—81no—
605C22—81no—
606C22—84no—
607C22C2×#120×C15111yescharacteristic
608C22C2×#135×C15111yescharacteristic
609C22#575×C15111yescharacteristic
610C4—22no—
611C4—21no—
612C4—21no—
613C4—22no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C2—21no—
624C2—21no—
625C2—42no—
626C2—81no—
627C2—82no—
628C2#3002×C15111yescharacteristic
629C2#3019×C15111yescharacteristic
630C2#3028×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
prime radical#3459×C151C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3232×C151C2semidirect
3. #2633×C151C2semidirect
4. #1618×C151C2semidirect
5. #1614×C151C2semidirect
6. #1613×C151C2semidirect
7. #1612×C151C2semidirect
8. #1608×C151C2semidirect
9. #1607×C151C2semidirect
10. #1521×C151C2semidirect
11. #1460×C151C2semidirect
21. C2×#125×C151C22semidirect
28. #484×C151C22semidirect
29. #483×C151C22semidirect
34. #431×C151C22semidirect
36. #399×C151C22semidirect
39. #397×C151C22semidirect
43. #383×C151C22semidirect
44. #328×C151C22semidirect
45. #328×C151C22semidirect
50. #323×C151C22semidirect
315. #18314C151direct (N × H)
316. C151#18314direct (N × H)
318. #3232C2×C151semidirect
319. #2633C2×C151semidirect
320. #1618C2×C151semidirect
321. #1614C2×C151semidirect
322. #1613C2×C151semidirect
323. #1612C2×C151semidirect
324. #1608C2×C151semidirect
325. #1607C2×C151semidirect
326. #1521C2×C151semidirect
327. #1460C2×C151semidirect
337. C2×#125C22×C151semidirect
344. #484C22×C151semidirect
345. #483C22×C151semidirect
350. #431C22×C151semidirect
352. #399C22×C151semidirect
355. #397C22×C151semidirect
359. #383C22×C151semidirect
360. #328C22×C151semidirect
361. #328C22×C151semidirect
366. #323C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3459×C1512
#3232×C1512
#2633×C1512
#1618×C1512
#1614×C1512
#1613×C1512
#1612×C1512
#1608×C1512
#1607×C1512
#1521×C1512
#1460×C1512
#18314151