Finite Group Databasebeta

← C151×#14024 #50318949

Subgroups — C151×#14024 #50318949

Order 38656.

2054 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊Q8×C151C211yesmaximal, characteristic
2C22×C4×Q8×C151C211yesmaximal, characteristic
3C2×#391×C151C211yesmaximal, characteristic
4C2×#391×C151C211yesmaximal, characteristic
5C2×#391×C151C211yesmaximal, characteristic
6#1441×C151C214yesmaximal
7#1436×C151C214yesmaximal
8C4⋊#95×C151C211yesmaximal, characteristic
9C4×#95×C151C211yesmaximal, characteristic
10C23×Q8×C151C2211yescharacteristic
11C23×Q8×C151C2211yescharacteristic
12C2×C4⋊Q8×C151—24no—
13C2×C4⋊Q8×C151—24no—
14C2×C4⋊Q8×C151C418yescomplemented
15C2×C4×Q8×C151—24no—
16C2×C4×Q8×C151—24no—
17C2×C4×Q8×C151—24no—
18C22×C4⋊C4×C151C2212yes—
19C22×C4⋊C4×C151C2211yescharacteristic
20C22×C4⋊C4×C151C2214yes—
21C22×C4⋊C4×C151C2211yescharacteristic
22C22×C42×C151C2212yes—
23C22×C42×C151C2211yescharacteristic
24C22×C42×C151C2211yescharacteristic
25C22×C42×C151C2211yescharacteristic
26#391×C151—24no—
27#391×C151—24no—
28#391×C151—24no—
29C2×#95×C151C2212yes—
30C2×#95×C151C2214yes—
31C2×#95×C151C2212yes—
32C2×#95×C151C2214yes—
33C2×#95×C151C2214yes—
34C2×#95×C151C2212yes—
35C2×#95×C151C2212yes—
36C22×Q8×C151—22no—
37C22×Q8×C151—22no—
38C22×Q8×C151—22no—
39C22×Q8×C151—22no—
40C22×Q8×C151—22no—
41C22×Q8×C151—22no—
42C22×Q8×C151—22no—
43C22×Q8×C151—22no—
44C22×Q8×C151—22no—
45C22×Q8×C151—22no—
46C22×Q8×C151C2×C414yes—
47C22×Q8×C151C2×C414yes—
48C22×Q8×C151D414yes—
49C22×Q8×C151D414yes—
50C23×C4×C151C2312yes—
51C23×C4×C151C2312yes—
52C23×C4×C151C2312yes—
53C23×C4×C151C2311yescharacteristic
54C23×C4×C151C2312yes—
55C23×C4×C151C2312yes—
56C23×C4×C151C2312yes—
57C23×C4×C151C2311yescharacteristic
58C23×C4×C151C2311yescharacteristic
59C4⋊Q8×C151—232no—
60C4×Q8×C151—232no—
61C2×C4⋊C4×C151—24no—
62C2×C4⋊C4×C151—24no—
63C2×C4⋊C4×C151—24no—
64C2×C4⋊C4×C151—22no—
65C2×C4⋊C4×C151—22no—
66C2×C4⋊C4×C151—28no—
67C2×C4⋊C4×C151—28no—
68C2×C4⋊C4×C151—22no—
69C2×C4⋊C4×C151—22no—
70C2×C4⋊C4×C151—22no—
71C2×C4⋊C4×C151C2×C4116yes—
72C2×C4⋊C4×C151D414yes—
73C2×C42×C151—24no—
74C2×C42×C151—24no—
75C2×C42×C151—24no—
76C2×C42×C151—22no—
77C2×C42×C151—22no—
78C2×C42×C151—22no—
79C2×C42×C151—22no—
80C2×C42×C151—22no—
81C2×C42×C151—22no—
82C2×C42×C151—22no—
83C2×C42×C151C2×C414yes—
84C2×C42×C151D414yes—
85#95×C151—28no—
86#95×C151—28no—
87#95×C151—24no—
88#95×C151—24no—
89#95×C151—24no—
90#95×C151—28no—
91#95×C151Q818yescomplemented
92C24×C151C2411yescharacteristic
93C2×Q8×C151—28no—
94C2×Q8×C151—28no—
95C2×Q8×C151—28no—
96C2×Q8×C151—28no—
97C2×Q8×C151—28no—
98C2×Q8×C151—28no—
99C2×Q8×C151—28no—
100C2×Q8×C151—28no—
101C2×Q8×C151—28no—
102C2×Q8×C151—44no—
103C2×Q8×C151—44no—
104C2×Q8×C151—44no—
105C2×Q8×C151—44no—
106C2×Q8×C151C22⋊C4116yes—
107C22×C4×C151—21no—
108C22×C4×C151—21no—
109C22×C4×C151—22no—
110C22×C4×C151—22no—
111C22×C4×C151—22no—
112C22×C4×C151—22no—
113C22×C4×C151—22no—
114C22×C4×C151—22no—
115C22×C4×C151—22no—
116C22×C4×C151—22no—
117C22×C4×C151—22no—
118C22×C4×C151—22no—
119C22×C4×C151—22no—
120C22×C4×C151—22no—
121C22×C4×C151—22no—
122C22×C4×C151—22no—
123C22×C4×C151—21no—
124C22×C4×C151—22no—
125C22×C4×C151—21no—
126C22×C4×C151—22no—
127C22×C4×C151—22no—
128C22×C4×C151—22no—
129C22×C4×C151—22no—
130C22×C4×C151—24no—
131C22×C4×C151—22no—
132C22×C4×C151—22no—
133C22×C4×C151—22no—
134C22×C4×C151—22no—
135C22×C4×C151—24no—
136C22×C4×C151—22no—
137C22×C4×C151—22no—
138C22×C4×C151—22no—
139C22×C4×C151—22no—
140C22×C4×C151—24no—
141C22×C4×C151—22no—
142C22×C4×C151—24no—
143C22×C4×C151C22×C414yes—
144C22×C4×C151C22×C414yes—
145C22×C4×C151C22×C414yes—
146C22×C4×C151C22×C412yes—
147C22×C4×C151C2×D414yes—
148C22×C4×C151C2×D412yes—
149C22×C4×C151C2×D414yes—
150C22×C4×C151C2×D414yes—
151C22×C4×C151C2×D412yes—
152C22×C4×C151C2×D412yes—
153C22×C4×C151C2×D412yes—
154C22×C4×C151C2×D412yes—
155C22×C4×C151C2×D412yes—
156C22×C4×C151C2×Q814yes—
157C22×C4×C151C2×Q814yes—
158C22×C4×C151C2×Q814yes—
159C22×C4×C151#4114yes—
160C22×C4×C151#4112yes—
161C22×C4×C151#4112yes—
162C4⋊C4×C151—216no—
163C4⋊C4×C151—232no—
164C4⋊C4×C151—28no—
165C4⋊C4×C151—44no—
166C42×C151—216no—
167C42×C151—28no—
168C42×C151—28no—
169C42×C151—44no—
170C23×C151C23×C411yescharacteristic
171C23×C151C22×D411yescharacteristic
172C23×C151C22×D411yescharacteristic
173C23×C151C22×D411yescharacteristic
174C23×C151C22×D411yescharacteristic
175C23×C151C22×Q812yes—
176C23×C151C2×#4111yescharacteristic
177C23×C151C2×#4111yescharacteristic
178C23×C151C2×#4112yes—
179C23×C1512−1+412yes—
180C23×C1512−1+412yes—
181Q8×C151—232no—
182Q8×C151—416no—
183C2×C4×C151—24no—
184C2×C4×C151—24no—
185C2×C4×C151—24no—
186C2×C4×C151—24no—
187C2×C4×C151—24no—
188C2×C4×C151—22no—
189C2×C4×C151—24no—
190C2×C4×C151—24no—
191C2×C4×C151—24no—
192C2×C4×C151—22no—
193C2×C4×C151—24no—
194C2×C4×C151—24no—
195C2×C4×C151—24no—
196C2×C4×C151—24no—
197C2×C4×C151—24no—
198C2×C4×C151—24no—
199C2×C4×C151—24no—
200C2×C4×C151—24no—
201C2×C4×C151—24no—
202C2×C4×C151—24no—
203C2×C4×C151—22no—
204C2×C4×C151—22no—
205C2×C4×C151—24no—
206C2×C4×C151—22no—
207C2×C4×C151—28no—
208C2×C4×C151—24no—
209C2×C4×C151—24no—
210C2×C4×C151—42no—
211C2×C4×C151—42no—
212C2×C4×C151—42no—
213C2×C4×C151—42no—
214C2×C4×C151—42no—
215C2×C4×C151—42no—
216C2×C4×C151—42no—
217C2×C4×C151—42no—
218C2×C4×C151—42no—
219C2×C4×C151—42no—
220C2×C4×C151—42no—
221C2×C4×C151—42no—
222C2×C4×C151—44no—
223C2×C4×C151—42no—
224C2×C4×C151—42no—
225C2×C4×C151—42no—
226C2×C4×C151—44no—
227C2×C4×C151—44no—
228C2×C4×C151C2×C22⋊C418yes—
229C2×C4×C151C2×C22⋊C414yes—
230C2×C4×C151C4×D414yes—
231C2×C4×C151C4×D414yes—
232C2×C4×C151C4×Q818yescomplemented
233C2×C4×C151#12014yes—
234C2×C4×C151#12114yes—
235C2×C4×C151#12114yes—
236C2×C4×C151#12314yes—
237C2×C4×C151C4⋊Q814yescomplemented
238C22×C151C22×C22⋊C411yescharacteristic
239C22×C151C2×C4×D411yescharacteristic
240C22×C151C2×C4×D411yescharacteristic
241C22×C151C2×C4×Q812yes—
242C22×C151#51912yes—
243C22×C151C2×#12011yescharacteristic
244C22×C151C2×#12111yescharacteristic
245C22×C151C2×#12111yescharacteristic
246C22×C151C2×#12311yescharacteristic
247C22×C151C2×C4⋊Q811yescharacteristic
248C22×C151#53211yescharacteristic
249C22×C151#53312yes—
250C22×C151#53611yescharacteristic
251C22×C151#53611yescharacteristic
252C22×C151#54112yes—
253C22×C151#54712yes—
254C22×C151#54812yes—
255C22×C151#54812yes—
256C22×C151D4×Q812yes—
257C22×C151D4×Q812yes—
258C22×C151D4×Q812yes—
259C22×C151#55612yes—
260C22×C151#55712yes—
261C4×C151—28no—
262C4×C151—24no—
263C4×C151—28no—
264C4×C151—44no—
265C4×C151—42no—
266C4×C151—44no—
267C4×C151—44no—
268C4×C151—82no—
269C4×C151#38618yes—
270C2×C151C2×#38611yescharacteristic
271C2×C151C4×C4⋊Q811yescharacteristic
272C2×C151#232711yescharacteristic
273C2×C151Q8×C22⋊C412yes—
274C2×C151#236512yes—
275C2×C151#242912yes—
276C2×C151#243211yescharacteristic
277C2×C151#243411yescharacteristic
278C2×C151#244612yes—
279C2×C151#254511yescharacteristic
280C2×C151#254611yescharacteristic
281#14024C15111yesmaximal, characteristic, direct product factor, Sylow
282C151#1402411yescharacteristic, direct product factor, Sylow
283C22×C4⋊Q8C2×C15111yescharacteristic
284C22×C4×Q8C2×C15111yescharacteristic
285C2×#391C2×C15111yescharacteristic
286C2×#391C2×C15111yescharacteristic
287C2×#391C2×C15111yescharacteristic
288#1441C2×C15114yes—
289#1436C2×C15114yes—
290C4⋊#95C2×C15111yescharacteristic
291C4×#95C2×C15111yescharacteristic
292C23×Q8C22×C15111yescharacteristic
293C23×Q8C22×C15111yescharacteristic
294C2×C4⋊Q8—24no—
295C2×C4⋊Q8—24no—
296C2×C4⋊Q8C4×C15118yescomplemented
297C2×C4×Q8—24no—
298C2×C4×Q8—24no—
299C2×C4×Q8—24no—
300C22×C4⋊C4C22×C15112yes—
301C22×C4⋊C4C22×C15111yescharacteristic
302C22×C4⋊C4C22×C15114yes—
303C22×C4⋊C4C22×C15111yescharacteristic
304C22×C42C22×C15112yes—
305C22×C42C22×C15111yescharacteristic
306C22×C42C22×C15111yescharacteristic
307C22×C42C22×C15111yescharacteristic
308#391—24no—
309#391—24no—
310#391—24no—
311C2×#95C22×C15112yes—
312C2×#95C22×C15114yes—
313C2×#95C22×C15112yes—
314C2×#95C22×C15114yes—
315C2×#95C22×C15114yes—
316C2×#95C22×C15112yes—
317C2×#95C22×C15112yes—
318C22×Q8—22no—
319C22×Q8—22no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8C2×C4×C15114yes—
329C22×Q8C2×C4×C15114yes—
330C22×Q8D4×C15114yes—
331C22×Q8D4×C15114yes—
332C23×C4C23×C15112yes—
333C23×C4C23×C15112yes—
334C23×C4C23×C15112yes—
335C23×C4C23×C15112yes—
336C23×C4C23×C15111yescharacteristic
337C23×C4C23×C15111yescharacteristic
338C23×C4C23×C15112yes—
339C23×C4C23×C15112yes—
340C23×C4C23×C15111yescharacteristic
341C4⋊Q8—232no—
342C4×Q8—232no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—28no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4C2×C4×C151116yes—
354C2×C4⋊C4D4×C15114yes—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42C2×C4×C15114yes—
366C2×C42D4×C15114yes—
367#95—24no—
368#95—28no—
369#95—28no—
370#95—28no—
371#95—24no—
372#95—24no—
373#95Q8×C15118yescomplemented
374C24C24×C15111yescharacteristic
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8C22⋊C4×C151116yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4C22×C4×C15114yes—
426C22×C4C22×C4×C15114yes—
427C22×C4C22×C4×C15114yes—
428C22×C4C22×C4×C15112yes—
429C22×C4C2×D4×C15114yes—
430C22×C4C2×D4×C15112yes—
431C22×C4C2×D4×C15114yes—
432C22×C4C2×D4×C15114yes—
433C22×C4C2×D4×C15112yes—
434C22×C4C2×D4×C15112yes—
435C22×C4C2×D4×C15112yes—
436C22×C4C2×D4×C15112yes—
437C22×C4C2×D4×C15112yes—
438C22×C4C2×Q8×C15114yes—
439C22×C4C2×Q8×C15114yes—
440C22×C4C2×Q8×C15114yes—
441C22×C4#41×C15114yes—
442C22×C4#41×C15112yes—
443C22×C4#41×C15112yes—
444C4⋊C4—216no—
445C4⋊C4—232no—
446C4⋊C4—28no—
447C4⋊C4—44no—
448C42—216no—
449C42—28no—
450C42—28no—
451C42—44no—
452C23C23×C4×C15111yescharacteristic
453C23C22×D4×C15111yescharacteristic
454C23C22×D4×C15111yescharacteristic
455C23C22×D4×C15111yescharacteristic
456C23C22×D4×C15111yescharacteristic
457C23C22×Q8×C15112yes—
458C23C2×#41×C15111yescharacteristic
459C23C2×#41×C15111yescharacteristic
460C23C2×#41×C15112yes—
461C232−1+4×C15112yes—
462C232−1+4×C15112yes—
463Q8—232no—
464Q8—416no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4C2×C22⋊C4×C15118yes—
511C2×C4C2×C22⋊C4×C15114yes—
512C2×C4C4×D4×C15114yes—
513C2×C4C4×D4×C15114yes—
514C2×C4C4×Q8×C15118yescomplemented
515C2×C4#120×C15114yes—
516C2×C4#121×C15114yes—
517C2×C4#121×C15114yes—
518C2×C4#123×C15114yes—
519C2×C4C4⋊Q8×C15114yescomplemented
520C22C22×C22⋊C4×C15111yescharacteristic
521C22C2×C4×D4×C15111yescharacteristic
522C22C2×C4×D4×C15111yescharacteristic
523C22C2×C4×Q8×C15112yes—
524C22#519×C15112yes—
525C22C2×#120×C15111yescharacteristic
526C22C2×#121×C15111yescharacteristic
527C22C2×#121×C15111yescharacteristic
528C22C2×#123×C15111yescharacteristic
529C22C2×C4⋊Q8×C15111yescharacteristic
530C22#532×C15111yescharacteristic
531C22#533×C15112yes—
532C22#536×C15111yescharacteristic
533C22#536×C15111yescharacteristic
534C22#541×C15112yes—
535C22#547×C15112yes—
536C22#548×C15112yes—
537C22#548×C15112yes—
538C22D4×Q8×C15112yes—
539C22D4×Q8×C15112yes—
540C22D4×Q8×C15112yes—
541C22#556×C15112yes—
542C22#557×C15112yes—
543C4—28no—
544C4—24no—
545C4—28no—
546C4—44no—
547C4—42no—
548C4—44no—
549C4—44no—
550C4—82no—
551C4#386×C15118yes—
552C2C2×#386×C15111yescharacteristic
553C2C4×C4⋊Q8×C15111yescharacteristic
554C2#2327×C15111yescharacteristic
555C2Q8×C22⋊C4×C15112yes—
556C2#2365×C15112yes—
557C2#2429×C15112yes—
558C2#2432×C15111yescharacteristic
559C2#2434×C15111yescharacteristic
560C2#2446×C15112yes—
561C2#2545×C15111yescharacteristic
562C2#2546×C15111yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C151C24
commutatorC23C23×C4×C151
Frattini, cosocleC24C24×C151
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C151 ◁ G

Complements

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

NH ≅ G/Ntype
14. C2×C4⋊Q8×C151C4semidirect
91. #95×C151Q8semidirect
232. C2×C4×C151C4×Q8semidirect
237. C2×C4×C151C4⋊Q8semidirect
281. #14024C151direct (N × H)
282. C151#14024direct (N × H)
296. C2×C4⋊Q8C4×C151semidirect
373. #95Q8×C151semidirect
514. C2×C4C4×Q8×C151semidirect
519. C2×C4C4⋊Q8×C151semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q8×C1512
C22×C4×Q8×C1512
C2×#391×C1512
#1441×C1512
#1436×C1512
C4⋊#95×C1512
C4×#95×C1512
#14024151