Finite Group Databasebeta

← C151×#24157 #50329082

Subgroups — C151×#24157 #50329082

Order 38656.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#533×C151C211yesmaximal, characteristic
2C2×#475×C151C211yesmaximal, characteristic, complemented
3C2×#462×C151C211yesmaximal, characteristic, complemented
4C4×C22⋊Q8×C151C211yesmaximal, characteristic
5#2035×C151C212yesmaximal
6#2001×C151C212yesmaximal
7#1988×C151C211yesmaximal, characteristic, complemented
8#1981×C151C211yesmaximal, characteristic, complemented
9#1878×C151C212yesmaximal, complemented
10#1865×C151C212yesmaximal, complemented
11#1797×C151C211yesmaximal, characteristic
12C22×Q16×C151C2212yes—
13#533×C151—24no—
14#533×C151—24no—
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×C4⋊Q8×C151C2211yescharacteristic
17C2×#125×C151—21no—
18C2×C22⋊Q8×C151—21no—
19C2×C22⋊Q8×C151C2212yes—
20C2×C22⋊Q8×C151C2211yescharacteristic
21C2×C4×Q8×C151—21no—
22C2×C4×Q8×C151C2212yes—
23C2×C4×Q8×C151C2211yescharacteristic
24C2×#117×C151—21no—
25C22×C42×C151C2211yescharacteristic
26#475×C151—24no—
27#462×C151—24no—
28C2×#106×C151C2212yes—
29C2×C4⋊C8×C151C2211yescharacteristic
30C2×C4⋊C8×C151C2211yescharacteristic
31C2×#103×C151C2212yes—
32C2×#103×C151C2212yes—
33C2×#103×C151C2212yes—
34C2×#103×C151C2212yes—
35C2×C22⋊C8×C151C2212yes—
36#391×C151C2211yescharacteristic
37#389×C151C2211yescharacteristic
38#387×C151—21no—
39#385×C151—21no—
40#380×C151C2211yescharacteristic
41C4×C4⋊C4×C151—21no—
42C4×C4⋊C4×C151C2211yescharacteristic
43C4×C22⋊C4×C151—21no—
44#340×C151C2212yes—
45#340×C151C2212yes—
46#336×C151C2212yes—
47C22×Q8×C151—21no—
48C22×Q8×C151C2312yes—
49C22×Q8×C151C2311yescharacteristic
50C23×C4×C151D411yescharacteristic
51C23×C4×C151D411yescharacteristic
52C23×C4×C151C2311yescharacteristic
53C2×Q16×C151—24no—
54C2×Q16×C151—24no—
55C2×Q16×C151—42no—
56C22×C8×C151C2312yes—
57C22×C8×C151C2312yes—
58C4⋊Q8×C151—22no—
59C4⋊Q8×C151—22no—
60C4⋊Q8×C151D414yes—
61C4⋊Q8×C151D414yes—
62#125×C151—28no—
63C22⋊Q8×C151—28no—
64C22⋊Q8×C151—24no—
65C22⋊Q8×C151—22no—
66C22⋊Q8×C151—24no—
67C22⋊Q8×C151—22no—
68C4×Q8×C151—28no—
69C4×Q8×C151—24no—
70C4×Q8×C151—22no—
71C4×Q8×C151—22no—
72C4×Q8×C151—24no—
73#117×C151—28no—
74C2×C4⋊C4×C151—21no—
75C2×C4⋊C4×C151—21no—
76C2×C4⋊C4×C151—21no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151—22no—
79C2×C4⋊C4×C151—22no—
80C2×C4⋊C4×C151—21no—
81C2×C4⋊C4×C151—21no—
82C2×C4⋊C4×C151C2311yescharacteristic
83C2×C4⋊C4×C151C2311yescharacteristic
84C2×C4⋊C4×C151C2312yes—
85C2×C4⋊C4×C151C2311yescharacteristic
86C2×C22⋊C4×C151—21no—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151—21no—
89C2×C42×C151—22no—
90C2×C42×C151—22no—
91C2×C42×C151—22no—
92C2×C42×C151—21no—
93C2×C42×C151—21no—
94C2×C42×C151D411yescharacteristic
95C2×C42×C151D411yescharacteristic
96C2×C42×C151C2311yescharacteristic
97C2×C42×C151C2311yescharacteristic
98#106×C151—24no—
99C4⋊C8×C151—22no—
100C4⋊C8×C151—22no—
101#103×C151—24no—
102#103×C151—24no—
103#103×C151—24no—
104#103×C151—42no—
105C22⋊C8×C151—42no—
106#95×C151—21no—
107#95×C151—21no—
108#95×C151—21no—
109#95×C151—21no—
110#95×C151—21no—
111C24×C151C2×D411yescharacteristic
112C2×Q8×C151—24no—
113C2×Q8×C151—22no—
114C2×Q8×C151—21no—
115C2×Q8×C151—21no—
116C2×Q8×C151—21no—
117C2×Q8×C151—22no—
118C2×Q8×C151—22no—
119C2×Q8×C151—22no—
120C2×Q8×C151—21no—
121C2×Q8×C151—22no—
122C2×Q8×C151—21no—
123C2×Q8×C151—42no—
124C2×Q8×C151—42no—
125C2×Q8×C151C2×D414yes—
126C2×Q8×C151#4112yes—
127C22×C4×C151—22no—
128C22×C4×C151—22no—
129C22×C4×C151—22no—
130C22×C4×C151—21no—
131C22×C4×C151—21no—
132C22×C4×C151—22no—
133C22×C4×C151—22no—
134C22×C4×C151—22no—
135C22×C4×C151—22no—
136C22×C4×C151—22no—
137C22×C4×C151—21no—
138C22×C4×C151—21no—
139C22×C4×C151—21no—
140C22×C4×C151—21no—
141C22×C4×C151—21no—
142C22×C4×C151—21no—
143C22×C4×C151—21no—
144C22×C4×C151—21no—
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2×D411yescharacteristic
147C22×C4×C151C2×D411yescharacteristic
148C22×C4×C151C2×D411yescharacteristic
149C22×C4×C151C2×D411yescharacteristic
150C22×C4×C151C2411yescharacteristic
151Q16×C151—48no—
152C2×C8×C151—22no—
153C2×C8×C151—22no—
154C2×C8×C151—22no—
155C2×C8×C151—22no—
156C2×C8×C151—22no—
157C2×C8×C151—22no—
158C4⋊C4×C151—28no—
159C4⋊C4×C151—28no—
160C4⋊C4×C151—24no—
161C4⋊C4×C151—24no—
162C4⋊C4×C151—24no—
163C4⋊C4×C151—24no—
164C4⋊C4×C151—22no—
165C4⋊C4×C151—21no—
166C4⋊C4×C151—21no—
167C4⋊C4×C151—21no—
168C4⋊C4×C151—21no—
169C4⋊C4×C151—21no—
170C4⋊C4×C151—42no—
171C4⋊C4×C151—42no—
172C4⋊C4×C151C2×D414yes—
173C4⋊C4×C151#4112yes—
174C22⋊C4×C151—28no—
175C22⋊C4×C151—24no—
176C22⋊C4×C151—42no—
177C42×C151—28no—
178C42×C151—22no—
179C42×C151—22no—
180C42×C151—21no—
181C42×C151—21no—
182C42×C151—42no—
183C42×C151—42no—
184C42×C151C2×D412yes—
185C42×C151C2×D412yes—
186C23×C151—21no—
187C23×C151—21no—
188C23×C151—21no—
189C23×C151—21no—
190C23×C151—21no—
191C23×C151—21no—
192C23×C151#13712yes—
193C23×C151C22×D411yescharacteristic
194Q8×C151—24no—
195Q8×C151—22no—
196Q8×C151—24no—
197Q8×C151—22no—
198Q8×C151—41no—
199Q8×C151—42no—
200Q8×C151—42no—
201Q8×C151—41no—
202Q8×C151—48no—
203C2×C4×C151—28no—
204C2×C4×C151—21no—
205C2×C4×C151—21no—
206C2×C4×C151—21no—
207C2×C4×C151—21no—
208C2×C4×C151—24no—
209C2×C4×C151—24no—
210C2×C4×C151—21no—
211C2×C4×C151—21no—
212C2×C4×C151—22no—
213C2×C4×C151—22no—
214C2×C4×C151—21no—
215C2×C4×C151—21no—
216C2×C4×C151—21no—
217C2×C4×C151—21no—
218C2×C4×C151—41no—
219C2×C4×C151—42no—
220C2×C4×C151—42no—
221C2×C4×C151—41no—
222C2×C4×C151—42no—
223C2×C4×C151—41no—
224C2×C4×C151—42no—
225C2×C4×C151—41no—
226C2×C4×C151—41no—
227C2×C4×C151—41no—
228C2×C4×C151—41no—
229C2×C4×C151—41no—
230C2×C4×C151—44no—
231C2×C4×C151—41no—
232C2×C4×C151—41no—
233C2×C4×C151—41no—
234C2×C4×C151—41no—
235C2×C4×C151—41no—
236C2×C4×C151—41no—
237C2×C4×C151#12012yes—
238C2×C4×C151#12012yes—
239C2×C4×C151#13512yes—
240C2×C4×C151#13712yes—
241C2×C4×C151#13712yes—
242C2×C4×C151C22×D411yescharacteristic
243C2×C4×C151C22×D411yescharacteristic
244C2×C4×C151C2×#4111yescharacteristic
245C2×C4×C151C2×#4111yescharacteristic
246C2×C4×C151H(5,𝔽2)11yescharacteristic
247C2×C4×C1512−1+411yescharacteristic
248C8×C151—42no—
249C8×C151—42no—
250C22×C151—22no—
251C22×C151—22no—
252C22×C151—22no—
253C22×C151—41no—
254C22×C151—41no—
255C22×C151—41no—
256C22×C151—41no—
257C22×C151C2×#12011yescharacteristic
258C22×C151#52511yescharacteristic
259C22×C151#53711yescharacteristic
260C22×C151C2×#13511yescharacteristic
261C22×C151C2×#13711yescharacteristic
262C22×C151C2×#13711yescharacteristic
263C22×C151C2×#13711yescharacteristic
264C4×C151—22no—
265C4×C151—22no—
266C4×C151—42no—
267C4×C151—41no—
268C4×C151—44no—
269C4×C151—42no—
270C4×C151—41no—
271C4×C151—44no—
272C4×C151—81no—
273C4×C151—81no—
274C4×C151#45214yes—
275C4×C151#54611yescharacteristic
276C4×C151#54711yescharacteristic
277C4×C151#54811yescharacteristic
278C4×C151#55011yescharacteristic
279C2×C151—42no—
280C2×C151#240711yescharacteristic
281C2×C151C2×#45211yescharacteristic
282C2×C151#300411yescharacteristic
283C2×C151#310711yescharacteristic
284C2×C151#311311yescharacteristic
285C2×C151#324811yescharacteristic
286C2×C151#325511yescharacteristic
287#24157C15111yesmaximal, characteristic, direct product factor, Sylow
288C151#2415711yescharacteristic, direct product factor, Sylow
289C2×#533C2×C15111yescharacteristic
290C2×#475C2×C15111yescharacteristic, complemented
291C2×#462C2×C15111yescharacteristic, complemented
292C4×C22⋊Q8C2×C15111yescharacteristic
293#2035C2×C15112yes—
294#2001C2×C15112yes—
295#1988C2×C15111yescharacteristic, complemented
296#1981C2×C15111yescharacteristic, complemented
297#1878C2×C15112yescomplemented
298#1865C2×C15112yescomplemented
299#1797C2×C15111yescharacteristic
300C22×Q16C22×C15112yes—
301#533—24no—
302#533—24no—
303C2×C4⋊Q8C22×C15111yescharacteristic
304C2×C4⋊Q8C22×C15111yescharacteristic
305C2×#125—21no—
306C2×C22⋊Q8—21no—
307C2×C22⋊Q8C22×C15112yes—
308C2×C22⋊Q8C22×C15111yescharacteristic
309C2×C4×Q8—21no—
310C2×C4×Q8C22×C15111yescharacteristic
311C2×C4×Q8C22×C15112yes—
312C2×#117—21no—
313C22×C42C22×C15111yescharacteristic
314#475—24no—
315#462—24no—
316C2×#106C22×C15112yes—
317C2×C4⋊C8C22×C15111yescharacteristic
318C2×C4⋊C8C22×C15111yescharacteristic
319C2×#103C22×C15112yes—
320C2×#103C22×C15112yes—
321C2×#103C22×C15112yes—
322C2×#103C22×C15112yes—
323C2×C22⋊C8C22×C15112yes—
324#391C22×C15111yescharacteristic
325#389C22×C15111yescharacteristic
326#387—21no—
327#385—21no—
328#380C22×C15111yescharacteristic
329C4×C4⋊C4—21no—
330C4×C4⋊C4C22×C15111yescharacteristic
331C4×C22⋊C4—21no—
332#340C22×C15112yes—
333#340C22×C15112yes—
334#336C22×C15112yes—
335C22×Q8—21no—
336C22×Q8C23×C15111yescharacteristic
337C22×Q8C23×C15112yes—
338C23×C4D4×C15111yescharacteristic
339C23×C4D4×C15111yescharacteristic
340C23×C4C23×C15111yescharacteristic
341C2×Q16—24no—
342C2×Q16—24no—
343C2×Q16—42no—
344C22×C8C23×C15112yes—
345C22×C8C23×C15112yes—
346C4⋊Q8—22no—
347C4⋊Q8—22no—
348C4⋊Q8D4×C15114yes—
349C4⋊Q8D4×C15114yes—
350#125—28no—
351C22⋊Q8—28no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8—22no—
355C22⋊Q8—24no—
356C4×Q8—28no—
357C4×Q8—24no—
358C4×Q8—24no—
359C4×Q8—22no—
360C4×Q8—22no—
361#117—28no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4C23×C15111yescharacteristic
371C2×C4⋊C4C23×C15111yescharacteristic
372C2×C4⋊C4C23×C15111yescharacteristic
373C2×C4⋊C4C23×C15112yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42D4×C15111yescharacteristic
383C2×C42D4×C15111yescharacteristic
384C2×C42C23×C15111yescharacteristic
385C2×C42C23×C15111yescharacteristic
386#106—24no—
387C4⋊C8—22no—
388C4⋊C8—22no—
389#103—24no—
390#103—24no—
391#103—24no—
392#103—42no—
393C22⋊C8—42no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399C24C2×D4×C15111yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8C2×D4×C15114yes—
414C2×Q8#41×C15112yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4C2×D4×C15111yescharacteristic
434C22×C4C2×D4×C15111yescharacteristic
435C22×C4C2×D4×C15111yescharacteristic
436C22×C4C2×D4×C15111yescharacteristic
437C22×C4C2×D4×C15111yescharacteristic
438C22×C4C24×C15111yescharacteristic
439Q16—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4C2×D4×C15114yes—
461C4⋊C4#41×C15112yes—
462C22⋊C4—28no—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C42—28no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—42no—
471C42—42no—
472C42C2×D4×C15112yes—
473C42C2×D4×C15112yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23#137×C15112yes—
481C23C22×D4×C15111yescharacteristic
482Q8—22no—
483Q8—24no—
484Q8—24no—
485Q8—22no—
486Q8—48no—
487Q8—41no—
488Q8—42no—
489Q8—42no—
490Q8—41no—
491C2×C4—28no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4#120×C15112yes—
526C2×C4#120×C15112yes—
527C2×C4#135×C15112yes—
528C2×C4#137×C15112yes—
529C2×C4#137×C15112yes—
530C2×C4C22×D4×C15111yescharacteristic
531C2×C4C22×D4×C15111yescharacteristic
532C2×C4C2×#41×C15111yescharacteristic
533C2×C4C2×#41×C15111yescharacteristic
534C2×C4H(5,𝔽2)×C15111yescharacteristic
535C2×C42−1+4×C15111yescharacteristic
536C8—42no—
537C8—42no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×#120×C15111yescharacteristic
546C22#525×C15111yescharacteristic
547C22#537×C15111yescharacteristic
548C22C2×#135×C15111yescharacteristic
549C22C2×#137×C15111yescharacteristic
550C22C2×#137×C15111yescharacteristic
551C22C2×#137×C15111yescharacteristic
552C4—22no—
553C4—22no—
554C4—42no—
555C4—41no—
556C4—44no—
557C4—42no—
558C4—41no—
559C4—44no—
560C4—81no—
561C4—81no—
562C4#452×C15114yes—
563C4#546×C15111yescharacteristic
564C4#547×C15111yescharacteristic
565C4#548×C15111yescharacteristic
566C4#550×C15111yescharacteristic
567C2—42no—
568C2#2407×C15111yescharacteristic
569C2C2×#452×C15111yescharacteristic
570C2#3004×C15111yescharacteristic
571C2#3107×C15111yescharacteristic
572C2#3113×C15111yescharacteristic
573C2#3248×C15111yescharacteristic
574C2#3255×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC24×C151C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C22×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#475×C151C2semidirect
3. C2×#462×C151C2semidirect
7. #1988×C151C2semidirect
8. #1981×C151C2semidirect
9. #1878×C151C2semidirect
10. #1865×C151C2semidirect
287. #24157C151direct (N × H)
288. C151#24157direct (N × H)
290. C2×#475C2×C151semidirect
291. C2×#462C2×C151semidirect
295. #1988C2×C151semidirect
296. #1981C2×C151semidirect
297. #1878C2×C151semidirect
298. #1865C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#533×C1512
C2×#475×C1512
C2×#462×C1512
C4×C22⋊Q8×C1512
#2035×C1512
#2001×C1512
#1988×C1512
#1981×C1512
#1878×C1512
#1865×C1512
#1797×C1512
#24157151