Finite Group Databasebeta

← C151×#23568 #50328493

Subgroups — C151×#23568 #50328493

Order 38656.

1240 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#533×C151C211yesmaximal, characteristic, complemented
2C2×C4×SD16×C151C212yesmaximal, complemented
3C4×#121×C151C211yesmaximal, characteristic
4#2058×C151C211yesmaximal, characteristic, complemented
5#2043×C151C211yesmaximal, characteristic, complemented
6#2035×C151C212yesmaximal, complemented
7#2023×C151C211yesmaximal, characteristic, complemented
8#2013×C151C211yesmaximal, characteristic, complemented
9#2000×C151C212yesmaximal
10C4×#103×C151C212yesmaximal, complemented
11C4×C22⋊C8×C151C211yesmaximal, characteristic, complemented
12C22×SD16×C151C2212yes—
13#533×C151—28no—
14C2×C4⋊Q8×C151C2211yescharacteristic
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×#125×C151—21no—
17C2×C22⋊Q8×C151—21no—
18C2×C22⋊Q8×C151C2212yes—
19C2×#121×C151C2211yescharacteristic
20C2×C4×Q8×C151—21no—
21C2×C4×Q8×C151C2212yes—
22C2×C4×D4×C151—21no—
23C2×C4×D4×C151C2211yescharacteristic
24C2×#117×C151—21no—
25C22×C42×C151C2211yescharacteristic
26C4×SD16×C151—28no—
27C2×#106×C151C2212yes—
28C2×#103×C151C2212yes—
29C2×#103×C151C2212yes—
30C2×#103×C151C2212yes—
31C2×#102×C151C2212yes—
32C2×C22⋊C8×C151C2212yes—
33C2×C4×C8×C151C2212yes—
34#390×C151C2211yescharacteristic
35#389×C151C2211yescharacteristic
36#388×C151—21no—
37#386×C151—21no—
38#380×C151C2211yescharacteristic
39C4×C4⋊C4×C151C2211yescharacteristic
40C4×C22⋊C4×C151—21no—
41#340×C151C2212yes—
42#340×C151C2212yes—
43#336×C151C2211yescharacteristic
44#336×C151C2211yescharacteristic
45C22×Q8×C151—21no—
46C22×Q8×C151C2312yes—
47C22×D4×C151—21no—
48C22×D4×C151C2311yescharacteristic
49C23×C4×C151—21no—
50C23×C4×C151D411yescharacteristic
51C23×C4×C151D411yescharacteristic
52C23×C4×C151C2311yescharacteristic
53C2×SD16×C151—24no—
54C2×SD16×C151—24no—
55C2×SD16×C151—24no—
56C22×C8×C151C2312yes—
57C22×C8×C151C2312yes—
58C4⋊Q8×C151—24no—
59C4⋊Q8×C151—24no—
60#125×C151—28no—
61C22⋊Q8×C151—28no—
62C22⋊Q8×C151—24no—
63C22⋊Q8×C151—24no—
64#121×C151—22no—
65#121×C151—22no—
66C4×Q8×C151—28no—
67C4×Q8×C151—24no—
68C4×Q8×C151D418yescomplemented
69C4×D4×C151—24no—
70C4×D4×C151—44no—
71#117×C151—28no—
72C2×C4⋊C4×C151—21no—
73C2×C4⋊C4×C151—21no—
74C2×C4⋊C4×C151—21no—
75C2×C4⋊C4×C151—22no—
76C2×C4⋊C4×C151—22no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151C2311yescharacteristic
79C2×C4⋊C4×C151C2311yescharacteristic
80C2×C4⋊C4×C151C2312yes—
81C2×C4⋊C4×C151C2311yescharacteristic
82C2×C22⋊C4×C151—21no—
83C2×C22⋊C4×C151—21no—
84C2×C22⋊C4×C151—21no—
85C2×C22⋊C4×C151—21no—
86C2×C22⋊C4×C151—22no—
87C2×C22⋊C4×C151—21no—
88C2×C42×C151—22no—
89C2×C42×C151—24no—
90C2×C42×C151—21no—
91C2×C42×C151D411yescharacteristic
92C2×C42×C151D411yescharacteristic
93C2×C42×C151C2311yescharacteristic
94C2×C42×C151C2311yescharacteristic
95#106×C151—24no—
96#103×C151—24no—
97#103×C151—24no—
98#103×C151—24no—
99#102×C151—24no—
100C22⋊C8×C151—24no—
101C4×C8×C151—24no—
102#95×C151—21no—
103#95×C151—21no—
104#95×C151—21no—
105C24×C151—21no—
106C24×C151C2×D411yescharacteristic
107C2×Q8×C151—24no—
108C2×Q8×C151—22no—
109C2×Q8×C151—22no—
110C2×Q8×C151—22no—
111C2×Q8×C151—22no—
112C2×Q8×C151—22no—
113C2×Q8×C151—42no—
114C2×Q8×C151—42no—
115C2×Q8×C151C2×D414yes—
116C2×D4×C151—24no—
117C2×D4×C151—22no—
118C2×D4×C151—22no—
119C2×D4×C151—21no—
120C2×D4×C151—42no—
121C2×D4×C151—42no—
122C2×D4×C151#4112yes—
123C22×C4×C151—22no—
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—22no—
131C22×C4×C151—22no—
132C22×C4×C151—21no—
133C22×C4×C151—21no—
134C22×C4×C151—21no—
135C22×C4×C151—21no—
136C22×C4×C151—41no—
137C22×C4×C151—42no—
138C22×C4×C151—42no—
139C22×C4×C151—42no—
140C22×C4×C151SD1614yescomplemented
141C22×C4×C151C2×D411yescharacteristic
142C22×C4×C151C2×D411yescharacteristic
143C22×C4×C151C2×D411yescharacteristic
144C22×C4×C151C2×D411yescharacteristic
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2411yescharacteristic
147SD16×C151—216no—
148C2×C8×C151—22no—
149C2×C8×C151—22no—
150C2×C8×C151—22no—
151C2×C8×C151—22no—
152C2×C8×C151—22no—
153C2×C8×C151#4114yes—
154C4⋊C4×C151—24no—
155C4⋊C4×C151—24no—
156C4⋊C4×C151—28no—
157C4⋊C4×C151—24no—
158C4⋊C4×C151—28no—
159C4⋊C4×C151—22no—
160C4⋊C4×C151—21no—
161C4⋊C4×C151—22no—
162C4⋊C4×C151—22no—
163C4⋊C4×C151—42no—
164C4⋊C4×C151C2×D414yes—
165C4⋊C4×C151#4112yes—
166C22⋊C4×C151—28no—
167C22⋊C4×C151—24no—
168C22⋊C4×C151—42no—
169C22⋊C4×C151—42no—
170C22⋊C4×C151—42no—
171C22⋊C4×C151—42no—
172C42×C151—22no—
173C42×C151—22no—
174C42×C151—28no—
175C42×C151—22no—
176C42×C151—42no—
177C42×C151C2×D414yes—
178C23×C151—21no—
179C23×C151—21no—
180C23×C151—21no—
181C23×C151—21no—
182C23×C151—21no—
183C23×C151—21no—
184C23×C151—22no—
185C23×C151—41no—
186C23×C151—41no—
187C23×C151—41no—
188C23×C151—41no—
189C23×C151—41no—
190C23×C151—41no—
191C23×C151C2×SD1612yes—
192C23×C151C22×D411yescharacteristic
193Q8×C151—24no—
194Q8×C151—24no—
195Q8×C151—48no—
196Q8×C151—42no—
197Q8×C151—42no—
198D4×C151—24no—
199D4×C151—48no—
200D4×C151—42no—
201C2×C4×C151—21no—
202C2×C4×C151—21no—
203C2×C4×C151—21no—
204C2×C4×C151—21no—
205C2×C4×C151—21no—
206C2×C4×C151—21no—
207C2×C4×C151—28no—
208C2×C4×C151—24no—
209C2×C4×C151—24no—
210C2×C4×C151—21no—
211C2×C4×C151—21no—
212C2×C4×C151—22no—
213C2×C4×C151—21no—
214C2×C4×C151—44no—
215C2×C4×C151—42no—
216C2×C4×C151—42no—
217C2×C4×C151—41no—
218C2×C4×C151—42no—
219C2×C4×C151—42no—
220C2×C4×C151—41no—
221C2×C4×C151—42no—
222C2×C4×C151—42no—
223C2×C4×C151—41no—
224C2×C4×C151—41no—
225C2×C4×C151—41no—
226C2×C4×C151—41no—
227C2×C4×C151—41no—
228C2×C4×C151—41no—
229C2×C4×C151—84no—
230C2×C4×C151#12012yes—
231C2×C4×C151#12012yes—
232C2×C4×C151C2×SD1612yes—
233C2×C4×C151C2×SD1612yes—
234C2×C4×C151#13512yes—
235C2×C4×C151#13712yes—
236C2×C4×C151C22×D412yes—
237C2×C4×C151C2×#4111yescharacteristic
238C2×C4×C151C2×#4112yes—
239C2×C4×C151C2×#4111yescharacteristic
240C8×C151—24no—
241C8×C151—42no—
242C22×C151—22no—
243C22×C151—22no—
244C22×C151—22no—
245C22×C151—41no—
246C22×C151—41no—
247C22×C151—41no—
248C22×C151—41no—
249C22×C151—42no—
250C22×C151—42no—
251C22×C151—42no—
252C22×C151—81no—
253C22×C151—81no—
254C22×C151—81no—
255C22×C151—81no—
256C22×C151C2×#12011yescharacteristic
257C22×C151#52511yescharacteristic
258C22×C151#53211yescharacteristic
259C22×C151C22×SD1611yescharacteristic
260C22×C151C2×#13511yescharacteristic
261C22×C151C2×#13711yescharacteristic
262C22×C151#57511yescharacteristic
263C4×C151—22no—
264C4×C151—22no—
265C4×C151—22no—
266C4×C151—42no—
267C4×C151—41no—
268C4×C151—44no—
269C4×C151—44no—
270C4×C151—81no—
271C4×C151—81no—
272C4×C151#44814yes—
273C4×C151#52512yes—
274C4×C151#53212yes—
275C2×C151—42no—
276C2×C151—82no—
277C2×C151#238811yescharacteristic
278C2×C151C2×#44811yescharacteristic
279C2×C151#300211yescharacteristic
280C2×C151#310111yescharacteristic
281C2×C151#310711yescharacteristic
282C2×C151#316711yescharacteristic
283C2×C151#317211yescharacteristic
284#23568C15111yesmaximal, characteristic, direct product factor, Sylow
285C151#2356811yescharacteristic, direct product factor, Sylow
286C2×#533C2×C15111yescharacteristic, complemented
287C2×C4×SD16C2×C15112yescomplemented
288C4×#121C2×C15111yescharacteristic
289#2058C2×C15111yescharacteristic, complemented
290#2043C2×C15111yescharacteristic, complemented
291#2035C2×C15112yescomplemented
292#2023C2×C15111yescharacteristic, complemented
293#2013C2×C15111yescharacteristic, complemented
294#2000C2×C15112yes—
295C4×#103C2×C15112yescomplemented
296C4×C22⋊C8C2×C15111yescharacteristic, complemented
297C22×SD16C22×C15112yes—
298#533—28no—
299C2×C4⋊Q8C22×C15111yescharacteristic
300C2×C4⋊Q8C22×C15111yescharacteristic
301C2×#125—21no—
302C2×C22⋊Q8—21no—
303C2×C22⋊Q8C22×C15112yes—
304C2×#121C22×C15111yescharacteristic
305C2×C4×Q8—21no—
306C2×C4×Q8C22×C15112yes—
307C2×C4×D4—21no—
308C2×C4×D4C22×C15111yescharacteristic
309C2×#117—21no—
310C22×C42C22×C15111yescharacteristic
311C4×SD16—28no—
312C2×#106C22×C15112yes—
313C2×#103C22×C15112yes—
314C2×#103C22×C15112yes—
315C2×#103C22×C15112yes—
316C2×#102C22×C15112yes—
317C2×C22⋊C8C22×C15112yes—
318C2×C4×C8C22×C15112yes—
319#390C22×C15111yescharacteristic
320#389C22×C15111yescharacteristic
321#388—21no—
322#386—21no—
323#380C22×C15111yescharacteristic
324C4×C4⋊C4C22×C15111yescharacteristic
325C4×C22⋊C4—21no—
326#340C22×C15112yes—
327#340C22×C15112yes—
328#336C22×C15111yescharacteristic
329#336C22×C15111yescharacteristic
330C22×Q8—21no—
331C22×Q8C23×C15112yes—
332C22×D4—21no—
333C22×D4C23×C15111yescharacteristic
334C23×C4—21no—
335C23×C4D4×C15111yescharacteristic
336C23×C4D4×C15111yescharacteristic
337C23×C4C23×C15111yescharacteristic
338C2×SD16—24no—
339C2×SD16—24no—
340C2×SD16—24no—
341C22×C8C23×C15112yes—
342C22×C8C23×C15112yes—
343C4⋊Q8—24no—
344C4⋊Q8—24no—
345#125—28no—
346C22⋊Q8—28no—
347C22⋊Q8—24no—
348C22⋊Q8—24no—
349#121—22no—
350#121—22no—
351C4×Q8—28no—
352C4×Q8—24no—
353C4×Q8D4×C15118yescomplemented
354C4×D4—24no—
355C4×D4—44no—
356#117—28no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C23×C15111yescharacteristic
364C2×C4⋊C4C23×C15111yescharacteristic
365C2×C4⋊C4C23×C15111yescharacteristic
366C2×C4⋊C4C23×C15112yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C42—22no—
374C2×C42—24no—
375C2×C42—21no—
376C2×C42D4×C15111yescharacteristic
377C2×C42D4×C15111yescharacteristic
378C2×C42C23×C15111yescharacteristic
379C2×C42C23×C15111yescharacteristic
380#106—24no—
381#103—24no—
382#103—24no—
383#103—24no—
384#102—24no—
385C22⋊C8—24no—
386C4×C8—24no—
387#95—21no—
388#95—21no—
389#95—21no—
390C24—21no—
391C24C2×D4×C15111yescharacteristic
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8C2×D4×C15114yes—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4#41×C15112yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4SD16×C15114yescomplemented
426C22×C4C2×D4×C15111yescharacteristic
427C22×C4C2×D4×C15111yescharacteristic
428C22×C4C2×D4×C15111yescharacteristic
429C22×C4C2×D4×C15111yescharacteristic
430C22×C4C2×D4×C15111yescharacteristic
431C22×C4C24×C15111yescharacteristic
432SD16—216no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8#41×C15114yes—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—28no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4C2×D4×C15114yes—
450C4⋊C4#41×C15112yes—
451C22⋊C4—28no—
452C22⋊C4—24no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C42—22no—
458C42—22no—
459C42—28no—
460C42—22no—
461C42—42no—
462C42C2×D4×C15114yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23C2×SD16×C15112yes—
477C23C22×D4×C15111yescharacteristic
478Q8—24no—
479Q8—24no—
480Q8—48no—
481Q8—42no—
482Q8—42no—
483D4—24no—
484D4—48no—
485D4—42no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—28no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4#120×C15112yes—
516C2×C4#120×C15112yes—
517C2×C4C2×SD16×C15112yes—
518C2×C4C2×SD16×C15112yes—
519C2×C4#135×C15112yes—
520C2×C4#137×C15112yes—
521C2×C4C22×D4×C15112yes—
522C2×C4C2×#41×C15111yescharacteristic
523C2×C4C2×#41×C15112yes—
524C2×C4C2×#41×C15111yescharacteristic
525C8—24no—
526C8—42no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C2×#120×C15111yescharacteristic
542C22#525×C15111yescharacteristic
543C22#532×C15111yescharacteristic
544C22C22×SD16×C15111yescharacteristic
545C22C2×#135×C15111yescharacteristic
546C22C2×#137×C15111yescharacteristic
547C22#575×C15111yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—41no—
553C4—44no—
554C4—44no—
555C4—81no—
556C4—81no—
557C4#448×C15114yes—
558C4#525×C15112yes—
559C4#532×C15112yes—
560C2—42no—
561C2—82no—
562C2#2388×C15111yescharacteristic
563C2C2×#448×C15111yescharacteristic
564C2#3002×C15111yescharacteristic
565C2#3101×C15111yescharacteristic
566C2#3107×C15111yescharacteristic
567C2#3167×C15111yescharacteristic
568C2#3172×C15111yescharacteristic
subgroupG/H
centerC22×C4×C151C2×D4
commutator, Frattini, cosocleC22×C4C24×C151
socleC23×C151C22×D4
prime radicalC2×#121×C151C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C151 ◁ C22×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#533×C151C2semidirect
2. C2×C4×SD16×C151C2semidirect
4. #2058×C151C2semidirect
5. #2043×C151C2semidirect
6. #2035×C151C2semidirect
7. #2023×C151C2semidirect
8. #2013×C151C2semidirect
10. C4×#103×C151C2semidirect
11. C4×C22⋊C8×C151C2semidirect
68. C4×Q8×C151D4semidirect
140. C22×C4×C151SD16semidirect
284. #23568C151direct (N × H)
285. C151#23568direct (N × H)
286. C2×#533C2×C151semidirect
287. C2×C4×SD16C2×C151semidirect
289. #2058C2×C151semidirect
290. #2043C2×C151semidirect
291. #2035C2×C151semidirect
292. #2023C2×C151semidirect
293. #2013C2×C151semidirect
295. C4×#103C2×C151semidirect
296. C4×C22⋊C8C2×C151semidirect
353. C4×Q8D4×C151semidirect
425. C22×C4SD16×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×C4×SD16×C1512
C4×#121×C1512
#2058×C1512
#2043×C1512
#2035×C1512
#2023×C1512
#2013×C1512
#2000×C1512
C4×#103×C1512
C4×C22⋊C8×C1512
#23568151