Finite Group Databasebeta

← C151×#24253 #50329178

Subgroups — C151×#24253 #50329178

Order 38656.

1424 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547×C151C211yesmaximal, characteristic
2C2×#462×C151C211yesmaximal, characteristic, complemented
3C2×#452×C151C212yesmaximal, complemented
4C2×#440×C151C211yesmaximal, characteristic, complemented
5#2536×C151C211yesmaximal, characteristic, complemented
6#2036×C151C212yesmaximal, complemented
7#2001×C151C212yesmaximal, complemented
8#1960×C151C211yesmaximal, characteristic, complemented
9#1955×C151C211yesmaximal, characteristic, complemented
10#1804×C151C211yesmaximal, characteristic, complemented
11#1788×C151C212yesmaximal, complemented
12C22×#41×C151C2212yes—
13C22×Q16×C151C2212yes—
14C22×SD16×C151C2211yescharacteristic
15#547×C151—24no—
16#547×C151—24no—
17C2×C4⋊Q8×C151C2211yescharacteristic
18C2×#123×C151—22no—
19C2×C22⋊Q8×C151C2212yes—
20C2×C22⋊Q8×C151C2212yes—
21C2×#121×C151—21no—
22C2×C4×Q8×C151C2211yescharacteristic
23C2×C4×D4×C151C2211yescharacteristic
24C2×C4×D4×C151C2211yescharacteristic
25C22×C4⋊C4×C151—21no—
26#462×C151—24no—
27#452×C151—28no—
28#440×C151—24no—
29C2×#107×C151C2211yescharacteristic
30C2×C4⋊C8×C151C2211yescharacteristic
31C2×#103×C151C2211yescharacteristic
32C2×#103×C151C2212yes—
33C2×#103×C151C2212yes—
34C2×#103×C151C2212yes—
35C2×#102×C151C2211yescharacteristic
36C2×#102×C151C2212yes—
37C2×C22⋊C8×C151C2212yes—
38C2×C22⋊C8×C151C2212yes—
39C2×#97×C151C2211yescharacteristic
40#398×C151—21no—
41#394×C151—21no—
42#388×C151—22no—
43#382×C151C2211yescharacteristic
44#380×C151C2212yes—
45#340×C151C2212yes—
46C2×#41×C151—24no—
47C2×#41×C151—24no—
48C2×#41×C151—24no—
49C2×#41×C151—24no—
50C2×#41×C151—42no—
51C2×#41×C151—42no—
52C22×Q8×C151C2312yes—
53C22×Q8×C151C2311yescharacteristic
54C22×D4×C151—22no—
55C22×D4×C151D411yescharacteristic
56C22×D4×C151C2311yescharacteristic
57C23×C4×C151—21no—
58C23×C4×C151—22no—
59C23×C4×C151C2312yes—
60C2×Q16×C151—24no—
61C2×Q16×C151—24no—
62C2×Q16×C151—42no—
63C2×SD16×C151—22no—
64C2×SD16×C151—22no—
65C2×SD16×C151D414yescomplemented
66C22×C8×C151C2312yes—
67C22×C8×C151C2311yescharacteristic
68C22×C8×C151C2311yescharacteristic
69C4⋊Q8×C151—22no—
70C4⋊Q8×C151D414yes—
71#123×C151—216no—
72C22⋊Q8×C151—24no—
73C22⋊Q8×C151—24no—
74C22⋊Q8×C151—24no—
75C22⋊Q8×C151—42no—
76#121×C151—28no—
77C4×Q8×C151—22no—
78C4×Q8×C151—22no—
79C4×D4×C151—22no—
80C4×D4×C151—22no—
81C4×D4×C151—22no—
82C4×D4×C151D414yes—
83C2×C4⋊C4×C151—28no—
84C2×C4⋊C4×C151—21no—
85C2×C4⋊C4×C151—22no—
86C2×C4⋊C4×C151—21no—
87C2×C4⋊C4×C151—22no—
88C2×C4⋊C4×C151—21no—
89C2×C4⋊C4×C151D411yescharacteristic
90C2×C4⋊C4×C151C2312yes—
91C2×C4⋊C4×C151C2311yescharacteristic
92C2×C4⋊C4×C151C2311yescharacteristic
93C2×C22⋊C4×C151—21no—
94C2×C22⋊C4×C151—22no—
95C2×C22⋊C4×C151—22no—
96C2×C22⋊C4×C151—22no—
97C2×C22⋊C4×C151D412yes—
98C2×C42×C151—21no—
99C2×C42×C151C2311yescharacteristic
100#107×C151—22no—
101C4⋊C8×C151—22no—
102#103×C151—24no—
103#103×C151—22no—
104#103×C151—24no—
105#103×C151—42no—
106#102×C151—22no—
107#102×C151—24no—
108C22⋊C8×C151—24no—
109C22⋊C8×C151—42no—
110#97×C151—22no—
111#95×C151—22no—
112#95×C151—21no—
113#95×C151—21no—
114C24×C151—21no—
115C24×C151C2×D412yes—
116#41×C151—48no—
117#41×C151—48no—
118#41×C151—48no—
119#41×C151—48no—
120C2×Q8×C151—22no—
121C2×Q8×C151—21no—
122C2×Q8×C151—21no—
123C2×Q8×C151—22no—
124C2×Q8×C151—21no—
125C2×Q8×C151—21no—
126C2×Q8×C151—22no—
127C2×Q8×C151—22no—
128C2×Q8×C151—21no—
129C2×Q8×C151—22no—
130C2×Q8×C151C2×D414yes—
131C2×Q8×C151C2×D412yes—
132C2×D4×C151—28no—
133C2×D4×C151—22no—
134C2×D4×C151—21no—
135C2×D4×C151—21no—
136C2×D4×C151—21no—
137C2×D4×C151—42no—
138C2×D4×C151—44no—
139C2×D4×C151—44no—
140C2×D4×C151C2×D412yes—
141C2×D4×C151C2×D412yes—
142C2×D4×C151#4112yes—
143C22×C4×C151—28no—
144C22×C4×C151—21no—
145C22×C4×C151—21no—
146C22×C4×C151—22no—
147C22×C4×C151—24no—
148C22×C4×C151—22no—
149C22×C4×C151—22no—
150C22×C4×C151—22no—
151C22×C4×C151—22no—
152C22×C4×C151—22no—
153C22×C4×C151—22no—
154C22×C4×C151—22no—
155C22×C4×C151—21no—
156C22×C4×C151—44no—
157C22×C4×C151—44no—
158C22×C4×C151—44no—
159C22×C4×C151C2×D412yes—
160C22×C4×C151C2×D411yescharacteristic
161C22×C4×C151C2×D411yescharacteristic
162C22×C4×C151C2411yescharacteristic
163Q16×C151—48no—
164SD16×C151—28no—
165C2×C8×C151—21no—
166C2×C8×C151—21no—
167C2×C8×C151—21no—
168C2×C8×C151—22no—
169C2×C8×C151—22no—
170C2×C8×C151—21no—
171C2×C8×C151—22no—
172C2×C8×C151—21no—
173C2×C8×C151C2×D412yes—
174C4⋊C4×C151—28no—
175C4⋊C4×C151—22no—
176C4⋊C4×C151—24no—
177C4⋊C4×C151—24no—
178C4⋊C4×C151—21no—
179C4⋊C4×C151—22no—
180C4⋊C4×C151—21no—
181C4⋊C4×C151—21no—
182C4⋊C4×C151—22no—
183C4⋊C4×C151—44no—
184C4⋊C4×C151—42no—
185C4⋊C4×C151C2×D412yes—
186C22⋊C4×C151—28no—
187C22⋊C4×C151—24no—
188C22⋊C4×C151—24no—
189C22⋊C4×C151—44no—
190C22⋊C4×C151—44no—
191C42×C151—42no—
192C42×C151C2×D412yes—
193C42×C151C2×D412yes—
194C23×C151—22no—
195C23×C151—22no—
196C23×C151—22no—
197C23×C151—22no—
198C23×C151—22no—
199C23×C151—22no—
200C23×C151—22no—
201C23×C151—22no—
202C23×C151—22no—
203C23×C151—22no—
204C23×C151—42no—
205C23×C151—42no—
206C23×C151C22×D411yescharacteristic
207Q8×C151—22no—
208Q8×C151—24no—
209Q8×C151—22no—
210Q8×C151—24no—
211Q8×C151—42no—
212Q8×C151—41no—
213Q8×C151—42no—
214Q8×C151—41no—
215D4×C151—22no—
216D4×C151—22no—
217D4×C151—22no—
218D4×C151—44no—
219D4×C151—416no—
220D4×C151#12114yes—
221C2×C4×C151—22no—
222C2×C4×C151—22no—
223C2×C4×C151—24no—
224C2×C4×C151—22no—
225C2×C4×C151—22no—
226C2×C4×C151—22no—
227C2×C4×C151—22no—
228C2×C4×C151—22no—
229C2×C4×C151—48no—
230C2×C4×C151—416no—
231C2×C4×C151—41no—
232C2×C4×C151—42no—
233C2×C4×C151—41no—
234C2×C4×C151—42no—
235C2×C4×C151—42no—
236C2×C4×C151—42no—
237C2×C4×C151—42no—
238C2×C4×C151—42no—
239C2×C4×C151—42no—
240C2×C4×C151—41no—
241C2×C4×C151—42no—
242C2×C4×C151—42no—
243C2×C4×C151—42no—
244C2×C4×C151—42no—
245C2×C4×C151#12012yes—
246C2×C4×C151#12012yes—
247C2×C4×C151C22×D411yescharacteristic
248C2×C4×C151C22×D411yescharacteristic
249C2×C4×C151C22×D411yescharacteristic
250C2×C4×C151C2×#4111yescharacteristic
251C2×C4×C151H(5,𝔽2)11yescharacteristic
252C2×C4×C151H(5,𝔽2)11yescharacteristic
253C8×C151—22no—
254C8×C151—41no—
255C8×C151—42no—
256C22×C151—24no—
257C22×C151—42no—
258C22×C151—42no—
259C22×C151—44no—
260C22×C151—44no—
261C22×C151—42no—
262C22×C151—42no—
263C22×C151—42no—
264C22×C151—42no—
265C22×C151—42no—
266C22×C151—42no—
267C22×C151—42no—
268C22×C151C2×#12011yescharacteristic
269C22×C151D4211yescharacteristic
270C22×C151#54611yescharacteristic
271C22×C151#57512yes—
272C22×C151#57812yes—
273C4×C151—22no—
274C4×C151—22no—
275C4×C151—42no—
276C4×C151—44no—
277C4×C151—42no—
278C4×C151—42no—
279C4×C151—81no—
280C4×C151—82no—
281C4×C151C2×#12111yescharacteristic
282C4×C151#53511yescharacteristic
283C4×C151D4211yescharacteristic
284C4×C151#54611yescharacteristic
285C2×C151—44no—
286C2×C151—82no—
287C2×C151#240811yescharacteristic
288C2×C151#300511yescharacteristic
289C2×C151#301111yescharacteristic
290C2×C151#326912yes—
291C2×C151#331512yes—
292#24253C15111yesmaximal, characteristic, direct product factor, Sylow
293C151#2425311yescharacteristic, direct product factor, Sylow
294C2×#547C2×C15111yescharacteristic
295C2×#462C2×C15111yescharacteristic, complemented
296C2×#452C2×C15112yescomplemented
297C2×#440C2×C15111yescharacteristic, complemented
298#2536C2×C15111yescharacteristic, complemented
299#2036C2×C15112yescomplemented
300#2001C2×C15112yescomplemented
301#1960C2×C15111yescharacteristic, complemented
302#1955C2×C15111yescharacteristic, complemented
303#1804C2×C15111yescharacteristic, complemented
304#1788C2×C15112yescomplemented
305C22×#41C22×C15112yes—
306C22×Q16C22×C15112yes—
307C22×SD16C22×C15111yescharacteristic
308#547—24no—
309#547—24no—
310C2×C4⋊Q8C22×C15111yescharacteristic
311C2×#123—22no—
312C2×C22⋊Q8C22×C15112yes—
313C2×C22⋊Q8C22×C15112yes—
314C2×#121—21no—
315C2×C4×Q8C22×C15111yescharacteristic
316C2×C4×D4C22×C15111yescharacteristic
317C2×C4×D4C22×C15111yescharacteristic
318C22×C4⋊C4—21no—
319#462—24no—
320#452—28no—
321#440—24no—
322C2×#107C22×C15111yescharacteristic
323C2×C4⋊C8C22×C15111yescharacteristic
324C2×#103C22×C15111yescharacteristic
325C2×#103C22×C15112yes—
326C2×#103C22×C15112yes—
327C2×#103C22×C15112yes—
328C2×#102C22×C15112yes—
329C2×#102C22×C15111yescharacteristic
330C2×C22⋊C8C22×C15112yes—
331C2×C22⋊C8C22×C15112yes—
332C2×#97C22×C15111yescharacteristic
333#398—21no—
334#394—21no—
335#388—22no—
336#382C22×C15111yescharacteristic
337#380C22×C15112yes—
338#340C22×C15112yes—
339C2×#41—24no—
340C2×#41—24no—
341C2×#41—24no—
342C2×#41—24no—
343C2×#41—42no—
344C2×#41—42no—
345C22×Q8C23×C15111yescharacteristic
346C22×Q8C23×C15112yes—
347C22×D4—22no—
348C22×D4D4×C15111yescharacteristic
349C22×D4C23×C15111yescharacteristic
350C23×C4—21no—
351C23×C4—22no—
352C23×C4C23×C15112yes—
353C2×Q16—24no—
354C2×Q16—24no—
355C2×Q16—42no—
356C2×SD16—22no—
357C2×SD16—22no—
358C2×SD16D4×C15114yescomplemented
359C22×C8C23×C15112yes—
360C22×C8C23×C15111yescharacteristic
361C22×C8C23×C15111yescharacteristic
362C4⋊Q8—22no—
363C4⋊Q8D4×C15114yes—
364#123—216no—
365C22⋊Q8—24no—
366C22⋊Q8—24no—
367C22⋊Q8—24no—
368C22⋊Q8—42no—
369#121—28no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4D4×C15114yes—
376C2×C4⋊C4—28no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4D4×C15111yescharacteristic
383C2×C4⋊C4C23×C15111yescharacteristic
384C2×C4⋊C4C23×C15112yes—
385C2×C4⋊C4C23×C15111yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4D4×C15112yes—
391C2×C42—21no—
392C2×C42C23×C15111yescharacteristic
393#107—22no—
394C4⋊C8—22no—
395#103—24no—
396#103—24no—
397#103—22no—
398#103—42no—
399#102—22no—
400#102—24no—
401C22⋊C8—24no—
402C22⋊C8—42no—
403#97—22no—
404#95—22no—
405#95—21no—
406#95—21no—
407C24—21no—
408C24C2×D4×C15112yes—
409#41—48no—
410#41—48no—
411#41—48no—
412#41—48no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8C2×D4×C15114yes—
424C2×Q8C2×D4×C15112yes—
425C2×D4—28no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4C2×D4×C15112yes—
434C2×D4C2×D4×C15112yes—
435C2×D4#41×C15112yes—
436C22×C4—28no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4C2×D4×C15112yes—
453C22×C4C2×D4×C15111yescharacteristic
454C22×C4C2×D4×C15111yescharacteristic
455C22×C4C24×C15111yescharacteristic
456Q16—48no—
457SD16—28no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—21no—
464C2×C8—22no—
465C2×C8—21no—
466C2×C8C2×D4×C15112yes—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4C2×D4×C15112yes—
479C22⋊C4—28no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C42—42no—
485C42C2×D4×C15112yes—
486C42C2×D4×C15112yes—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23C22×D4×C15111yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—24no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506Q8—41no—
507Q8—41no—
508D4—22no—
509D4—22no—
510D4—22no—
511D4—44no—
512D4—416no—
513D4#121×C15114yes—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—416no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4#120×C15112yes—
539C2×C4#120×C15112yes—
540C2×C4C22×D4×C15111yescharacteristic
541C2×C4C22×D4×C15111yescharacteristic
542C2×C4C22×D4×C15111yescharacteristic
543C2×C4C2×#41×C15111yescharacteristic
544C2×C4H(5,𝔽2)×C15111yescharacteristic
545C2×C4H(5,𝔽2)×C15111yescharacteristic
546C8—22no—
547C8—41no—
548C8—42no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22C2×#120×C15111yescharacteristic
562C22D42×C15111yescharacteristic
563C22#546×C15111yescharacteristic
564C22#575×C15112yes—
565C22#578×C15112yes—
566C4—22no—
567C4—22no—
568C4—42no—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—82no—
574C4C2×#121×C15111yescharacteristic
575C4#535×C15111yescharacteristic
576C4D42×C15111yescharacteristic
577C4#546×C15111yescharacteristic
578C2—44no—
579C2—82no—
580C2#2408×C15111yescharacteristic
581C2#3005×C15111yescharacteristic
582C2#3011×C15111yescharacteristic
583C2#3269×C15112yes—
584C2#3315×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×#547×C151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C4×D4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#462×C151C2semidirect
3. C2×#452×C151C2semidirect
4. C2×#440×C151C2semidirect
5. #2536×C151C2semidirect
6. #2036×C151C2semidirect
7. #2001×C151C2semidirect
8. #1960×C151C2semidirect
9. #1955×C151C2semidirect
10. #1804×C151C2semidirect
11. #1788×C151C2semidirect
65. C2×SD16×C151D4semidirect
292. #24253C151direct (N × H)
293. C151#24253direct (N × H)
295. C2×#462C2×C151semidirect
296. C2×#452C2×C151semidirect
297. C2×#440C2×C151semidirect
298. #2536C2×C151semidirect
299. #2036C2×C151semidirect
300. #2001C2×C151semidirect
301. #1960C2×C151semidirect
302. #1955C2×C151semidirect
303. #1804C2×C151semidirect
304. #1788C2×C151semidirect
358. C2×SD16D4×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#547×C1512
C2×#462×C1512
C2×#452×C1512
C2×#440×C1512
#2536×C1512
#2036×C1512
#2001×C1512
#1960×C1512
#1955×C1512
#1804×C1512
#1788×C1512
#24253151