Finite Group Databasebeta

← C151×#24375 #50329300

Subgroups — C151×#24375 #50329300

Order 38656.

1422 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#137×C151C211yesmaximal, characteristic, complemented
2C2×#464×C151C212yesmaximal, complemented
3C2×#449×C151C211yesmaximal, characteristic
4#2436×C151C211yesmaximal, characteristic, complemented
5#2341×C151C211yesmaximal, characteristic, complemented
6#2055×C151C212yesmaximal, complemented
7#2012×C151C212yesmaximal, complemented
8#1895×C151C211yesmaximal, characteristic, complemented
9#1878×C151C212yesmaximal, complemented
10#1851×C151C211yesmaximal, characteristic, complemented
11#1787×C151C211yesmaximal, characteristic, complemented
12C22×#41×C151C2211yescharacteristic
13C23×Q8×C151C2211yescharacteristic
14C2×#137×C151—24no—
15C2×#137×C151—28no—
16C22×Q16×C151C2212yes—
17C22×SD16×C151C2211yescharacteristic
18C22×SD16×C151C2212yes—
19C22×D8×C151C2211yescharacteristic
20C22×M16×C151C2211yescharacteristic
21C2×#124×C151C2212yes—
22C2×#121×C151C2211yescharacteristic
23C2×C4×Q8×C151C2212yes—
24#464×C151—28no—
25#449×C151—24no—
26C2×C4⋊C8×C151C2212yes—
27C2×#103×C151C2211yescharacteristic
28C2×#103×C151C2212yes—
29C2×#103×C151C2212yes—
30C2×#102×C151C2212yes—
31C2×#102×C151C2211yescharacteristic
32C2×C22⋊C8×C151C2211yescharacteristic
33#393×C151—21no—
34#393×C151—21no—
35#391×C151C2212yes—
36#390×C151—22no—
37#390×C151C2212yes—
38#386×C151—22no—
39#380×C151—21no—
40#380×C151C2211yescharacteristic
41C4×C22⋊C4×C151—21no—
42C4×C22⋊C4×C151C2211yescharacteristic
43#340×C151C2212yes—
44#336×C151C2212yes—
45C2×#41×C151—22no—
46C2×#41×C151—24no—
47C2×#41×C151—42no—
48C2×#41×C151D414yes—
49C22×Q8×C151—22no—
50C22×Q8×C151—24no—
51C22×Q8×C151—22no—
52C22×Q8×C151—21no—
53C22×Q8×C151—42no—
54C22×Q8×C151C2312yes—
55C22×Q8×C151C2311yescharacteristic
56C22×D4×C151—21no—
57C22×D4×C151—21no—
58C22×D4×C151C2311yescharacteristic
59C22×D4×C151C2311yescharacteristic
60C23×C4×C151—21no—
61C23×C4×C151—21no—
62C23×C4×C151C2311yescharacteristic
63#137×C151—232no—
64C2×Q16×C151—24no—
65C2×Q16×C151—24no—
66C2×Q16×C151D418yescomplemented
67C2×SD16×C151—24no—
68C2×SD16×C151—24no—
69C2×SD16×C151—24no—
70C2×SD16×C151—22no—
71C2×SD16×C151—24no—
72C2×D8×C151—22no—
73C2×D8×C151—42no—
74C2×M16×C151—22no—
75C2×M16×C151—24no—
76C22×C8×C151C2311yescharacteristic
77C22×C8×C151C2311yescharacteristic
78C22×C8×C151C2312yes—
79#124×C151—24no—
80#124×C151—42no—
81#121×C151—22no—
82#121×C151—22no—
83C4×Q8×C151—24no—
84C4×Q8×C151—42no—
85C2×C4⋊C4×C151—22no—
86C2×C4⋊C4×C151C2311yescharacteristic
87C2×C4⋊C4×C151C2312yes—
88C2×C22⋊C4×C151—22no—
89C2×C22⋊C4×C151—22no—
90C2×C22⋊C4×C151—22no—
91C2×C22⋊C4×C151—21no—
92C2×C22⋊C4×C151—22no—
93C2×C22⋊C4×C151—21no—
94C2×C22⋊C4×C151D412yes—
95C2×C42×C151—22no—
96C2×C42×C151C2312yes—
97C4⋊C8×C151—24no—
98#103×C151—22no—
99#103×C151—24no—
100#103×C151—42no—
101#102×C151—22no—
102#102×C151—24no—
103C22⋊C8×C151—22no—
104#95×C151—21no—
105#95×C151—22no—
106#95×C151—22no—
107#95×C151D411yescharacteristic
108#95×C151D411yescharacteristic
109C24×C151—21no—
110C24×C151—21no—
111C24×C151C2×D411yescharacteristic
112#41×C151—28no—
113#41×C151—44no—
114#41×C151—48no—
115C2×Q8×C151—21no—
116C2×Q8×C151—22no—
117C2×Q8×C151—22no—
118C2×Q8×C151—28no—
119C2×Q8×C151—22no—
120C2×Q8×C151—22no—
121C2×Q8×C151—22no—
122C2×Q8×C151—22no—
123C2×Q8×C151—44no—
124C2×Q8×C151—48no—
125C2×Q8×C151—44no—
126C2×Q8×C151—42no—
127C2×Q8×C151C2×D414yes—
128C2×Q8×C151C2×D412yes—
129C2×Q8×C151C2×D414yes—
130C2×D4×C151—24no—
131C2×D4×C151—22no—
132C2×D4×C151—22no—
133C2×D4×C151—21no—
134C2×D4×C151—21no—
135C2×D4×C151—21no—
136C2×D4×C151—22no—
137C2×D4×C151—22no—
138C2×D4×C151—44no—
139C2×D4×C151—44no—
140C2×D4×C151—42no—
141C2×D4×C151C2×D412yes—
142C22×C4×C151—21no—
143C22×C4×C151—22no—
144C22×C4×C151—22no—
145C22×C4×C151—22no—
146C22×C4×C151—24no—
147C22×C4×C151—21no—
148C22×C4×C151—22no—
149C22×C4×C151—44no—
150C22×C4×C151—44no—
151C22×C4×C151—42no—
152C22×C4×C151C2×D412yes—
153C22×C4×C151C2×D412yes—
154C22×C4×C151C2×D411yescharacteristic
155C22×C4×C151C2×D412yes—
156C22×C4×C151#4112yes—
157C22×C4×C151C2411yescharacteristic
158Q16×C151—216no—
159SD16×C151—216no—
160SD16×C151—44no—
161D8×C151—44no—
162M16×C151—28no—
163C2×C8×C151—22no—
164C2×C8×C151—22no—
165C2×C8×C151—21no—
166C2×C8×C151—21no—
167C2×C8×C151—22no—
168C2×C8×C151—22no—
169C2×C8×C151C2×D414yes—
170C4⋊C4×C151—22no—
171C4⋊C4×C151—21no—
172C4⋊C4×C151—21no—
173C4⋊C4×C151—22no—
174C4⋊C4×C151—44no—
175C22⋊C4×C151—42no—
176C22⋊C4×C151—44no—
177C22⋊C4×C151—44no—
178C22⋊C4×C151—44no—
179C22⋊C4×C151—42no—
180C22⋊C4×C151—82no—
181C22⋊C4×C151—81no—
182C42×C151—22no—
183C42×C151—22no—
184C42×C151—44no—
185C23×C151—22no—
186C23×C151—22no—
187C23×C151—21no—
188C23×C151—21no—
189C23×C151—22no—
190C23×C151—42no—
191C23×C151—42no—
192C23×C151—41no—
193C23×C151—41no—
194C23×C151—41no—
195C23×C151—42no—
196C23×C151—42no—
197C23×C151—41no—
198C23×C151—41no—
199C23×C151#12112yes—
200C23×C151C22×D411yescharacteristic
201Q8×C151—22no—
202Q8×C151—24no—
203Q8×C151—24no—
204Q8×C151—24no—
205Q8×C151—48no—
206Q8×C151—42no—
207Q8×C151—42no—
208Q8×C151#12014yes—
209D4×C151—22no—
210D4×C151—22no—
211D4×C151—48no—
212D4×C151—41no—
213D4×C151—41no—
214D4×C151—42no—
215D4×C151—42no—
216D4×C151—84no—
217C2×C4×C151—22no—
218C2×C4×C151—24no—
219C2×C4×C151—22no—
220C2×C4×C151—22no—
221C2×C4×C151—22no—
222C2×C4×C151—24no—
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—22no—
230C2×C4×C151—48no—
231C2×C4×C151—48no—
232C2×C4×C151—42no—
233C2×C4×C151—42no—
234C2×C4×C151—41no—
235C2×C4×C151—42no—
236C2×C4×C151—42no—
237C2×C4×C151—42no—
238C2×C4×C151—42no—
239C2×C4×C151#12112yes—
240C2×C4×C151C22×D411yescharacteristic
241C2×C4×C151C22×D412yes—
242C2×C4×C151C2×#4111yescharacteristic
243C2×C4×C151H(5,𝔽2)12yes—
244C8×C151—24no—
245C8×C151—41no—
246C8×C151—41no—
247C22×C151—22no—
248C22×C151—22no—
249C22×C151—22no—
250C22×C151—42no—
251C22×C151—42no—
252C22×C151—42no—
253C22×C151—42no—
254C22×C151—42no—
255C22×C151—82no—
256C22×C151—82no—
257C22×C151—81no—
258C22×C151—82no—
259C22×C151—82no—
260C22×C151—81no—
261C22×C151—81no—
262C22×C151C2×#12111yescharacteristic
263C22×C151D4212yes—
264C22×C151#57512yes—
265C22×C151#57711yescharacteristic
266C22×C151#57811yescharacteristic
267C4×C151—22no—
268C4×C151—24no—
269C4×C151—44no—
270C4×C151—42no—
271C4×C151—42no—
272C4×C151—82no—
273C4×C151—81no—
274C4×C151C2×#12011yescharacteristic
275C4×C151#53411yescharacteristic
276C4×C151#54612yes—
277C2×C151—42no—
278C2×C151—82no—
279C2×C151—161no—
280C2×C151#241011yescharacteristic
281C2×C151#304011yescharacteristic
282C2×C151#304611yescharacteristic
283C2×C151#327112yes—
284C2×C151#327712yes—
285#24375C15111yesmaximal, characteristic, direct product factor, Sylow
286C151#2437511yescharacteristic, direct product factor, Sylow
287C22×#137C2×C15111yescharacteristic, complemented
288C2×#464C2×C15112yescomplemented
289C2×#449C2×C15111yescharacteristic
290#2436C2×C15111yescharacteristic, complemented
291#2341C2×C15111yescharacteristic, complemented
292#2055C2×C15112yescomplemented
293#2012C2×C15112yescomplemented
294#1895C2×C15111yescharacteristic, complemented
295#1878C2×C15112yescomplemented
296#1851C2×C15111yescharacteristic, complemented
297#1787C2×C15111yescharacteristic, complemented
298C22×#41C22×C15111yescharacteristic
299C23×Q8C22×C15111yescharacteristic
300C2×#137—24no—
301C2×#137—28no—
302C22×Q16C22×C15112yes—
303C22×SD16C22×C15112yes—
304C22×SD16C22×C15111yescharacteristic
305C22×D8C22×C15111yescharacteristic
306C22×M16C22×C15111yescharacteristic
307C2×#124C22×C15112yes—
308C2×#121C22×C15111yescharacteristic
309C2×C4×Q8C22×C15112yes—
310#464—28no—
311#449—24no—
312C2×C4⋊C8C22×C15112yes—
313C2×#103C22×C15112yes—
314C2×#103C22×C15111yescharacteristic
315C2×#103C22×C15112yes—
316C2×#102C22×C15111yescharacteristic
317C2×#102C22×C15112yes—
318C2×C22⋊C8C22×C15111yescharacteristic
319#393—21no—
320#393—21no—
321#391C22×C15112yes—
322#390—22no—
323#390C22×C15112yes—
324#386—22no—
325#380—21no—
326#380C22×C15111yescharacteristic
327C4×C22⋊C4—21no—
328C4×C22⋊C4C22×C15111yescharacteristic
329#340C22×C15112yes—
330#336C22×C15112yes—
331C2×#41—22no—
332C2×#41—24no—
333C2×#41—42no—
334C2×#41D4×C15114yes—
335C22×Q8—21no—
336C22×Q8—24no—
337C22×Q8—22no—
338C22×Q8—22no—
339C22×Q8—42no—
340C22×Q8C23×C15112yes—
341C22×Q8C23×C15111yescharacteristic
342C22×D4—21no—
343C22×D4—21no—
344C22×D4C23×C15111yescharacteristic
345C22×D4C23×C15111yescharacteristic
346C23×C4—21no—
347C23×C4—21no—
348C23×C4C23×C15111yescharacteristic
349#137—232no—
350C2×Q16—24no—
351C2×Q16—24no—
352C2×Q16D4×C15118yescomplemented
353C2×SD16—24no—
354C2×SD16—24no—
355C2×SD16—22no—
356C2×SD16—24no—
357C2×SD16—24no—
358C2×D8—22no—
359C2×D8—42no—
360C2×M16—22no—
361C2×M16—24no—
362C22×C8C23×C15111yescharacteristic
363C22×C8C23×C15111yescharacteristic
364C22×C8C23×C15112yes—
365#124—24no—
366#124—42no—
367#121—22no—
368#121—22no—
369C4×Q8—24no—
370C4×Q8—42no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4C23×C15112yes—
373C2×C4⋊C4C23×C15111yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4D4×C15112yes—
381C2×C42—22no—
382C2×C42C23×C15112yes—
383C4⋊C8—24no—
384#103—22no—
385#103—24no—
386#103—42no—
387#102—22no—
388#102—24no—
389C22⋊C8—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95D4×C15111yescharacteristic
394#95D4×C15111yescharacteristic
395C24—21no—
396C24—21no—
397C24C2×D4×C15111yescharacteristic
398#41—28no—
399#41—44no—
400#41—48no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—28no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8C2×D4×C15114yes—
414C2×Q8C2×D4×C15112yes—
415C2×Q8C2×D4×C15114yes—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4C2×D4×C15112yes—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4C2×D4×C15112yes—
439C22×C4C2×D4×C15112yes—
440C22×C4C2×D4×C15111yescharacteristic
441C22×C4C2×D4×C15112yes—
442C22×C4#41×C15112yes—
443C22×C4C24×C15111yescharacteristic
444Q16—216no—
445SD16—216no—
446SD16—44no—
447D8—44no—
448M16—28no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8C2×D4×C15114yes—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—44no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—42no—
466C22⋊C4—82no—
467C22⋊C4—81no—
468C42—22no—
469C42—22no—
470C42—44no—
471C23—22no—
472C23—22no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23#121×C15112yes—
486C23C22×D4×C15111yescharacteristic
487Q8—22no—
488Q8—24no—
489Q8—24no—
490Q8—24no—
491Q8—48no—
492Q8—42no—
493Q8—42no—
494Q8#120×C15114yes—
495D4—22no—
496D4—22no—
497D4—48no—
498D4—41no—
499D4—41no—
500D4—42no—
501D4—42no—
502D4—84no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4#121×C15112yes—
526C2×C4C22×D4×C15111yescharacteristic
527C2×C4C22×D4×C15112yes—
528C2×C4C2×#41×C15111yescharacteristic
529C2×C4H(5,𝔽2)×C15112yes—
530C8—24no—
531C8—41no—
532C8—41no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—82no—
542C22—82no—
543C22—81no—
544C22—82no—
545C22—82no—
546C22—81no—
547C22—81no—
548C22C2×#121×C15111yescharacteristic
549C22D42×C15112yes—
550C22#575×C15112yes—
551C22#577×C15111yescharacteristic
552C22#578×C15111yescharacteristic
553C4—22no—
554C4—24no—
555C4—44no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—81no—
560C4C2×#120×C15111yescharacteristic
561C4#534×C15111yescharacteristic
562C4#546×C15112yes—
563C2—42no—
564C2—82no—
565C2—161no—
566C2#2410×C15111yescharacteristic
567C2#3040×C15111yescharacteristic
568C2#3046×C15111yescharacteristic
569C2#3271×C15112yes—
570C2#3277×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×#449×C151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C4×C22⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#137×C151C2semidirect
2. C2×#464×C151C2semidirect
4. #2436×C151C2semidirect
5. #2341×C151C2semidirect
6. #2055×C151C2semidirect
7. #2012×C151C2semidirect
8. #1895×C151C2semidirect
9. #1878×C151C2semidirect
10. #1851×C151C2semidirect
11. #1787×C151C2semidirect
66. C2×Q16×C151D4semidirect
285. #24375C151direct (N × H)
286. C151#24375direct (N × H)
287. C22×#137C2×C151semidirect
288. C2×#464C2×C151semidirect
290. #2436C2×C151semidirect
291. #2341C2×C151semidirect
292. #2055C2×C151semidirect
293. #2012C2×C151semidirect
294. #1895C2×C151semidirect
295. #1878C2×C151semidirect
296. #1851C2×C151semidirect
297. #1787C2×C151semidirect
352. C2×Q16D4×C151semidirect

Maximal subgroups

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

subgroupindex
C22×#137×C1512
C2×#464×C1512
C2×#449×C1512
#2436×C1512
#2341×C1512
#2055×C1512
#2012×C1512
#1895×C1512
#1878×C1512
#1851×C1512
#1787×C1512
#24375151