Finite Group Databasebeta

← #14416754

Subgroups — #14416754

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#908C211yesmaximal, characteristic
2#11754455C211yesmaximal, characteristic, complemented
3C2×#91435C211yesmaximal, characteristic, complemented
4M16×Dic15C211yesmaximal, characteristic, complemented
5C3×#65410C211yesmaximal, characteristic, complemented
6#11748386C211yesmaximal, characteristic, complemented
7#11748384C211yesmaximal, characteristic, complemented
8#11747816C212yesmaximal, complemented
9#11747813C212yesmaximal, complemented
10#11747810C212yesmaximal, complemented
11#11747807C212yesmaximal, complemented
12Dic5×AGL(1,ℤ/8)—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C22×C5×D12C2211yescharacteristic
15C22×D3×Dic5—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C4×Dic15C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C5×#908C418yescomplemented
20C2×C5×D24C2212yes—
21C2×C5×#204C2212yes—
22C3×#4072C2211yescharacteristic
23C2×C3×C4⋊Dic5C2211yescharacteristic
24#67⋊Dic5—21no—
25#67⋊Dic5—21no—
26C2×(C2×D3)⋊Dic5—21no—
27#91435—21no—
28#91435—21no—
29#91435C2212yes—
30#91435C2212yes—
31#91435C2211yescharacteristic, complemented
32#91435C2211yescharacteristic, complemented
33C4×D3×Dic5—21no—
34#91371—21no—
35#91344C2211yescharacteristic, complemented
36#91344C2211yescharacteristic, complemented
37#91099C2212yescomplemented
38C8×Dic15C2212yescomplemented
39C3×#3950C2212yescomplemented
40C3×#3949C2212yescomplemented
41#90972C2212yescomplemented
42#90966C2212yes—
43#90966C2212yes—
44#90966C2212yescomplemented
45#79199—51nomaximal, Hall
46C2×C5×AGL(1,ℤ/8)—31no—
47#41×Dic5—31no—
48C2×D4×Dic5—31no—
49#65521—31no—
50#65504—31no—
51#65453—32no—
52SD16×Dic5—32no—
53#65441—32no—
54D8×Dic5—32no—
55#65410D311yescharacteristic, complemented
56M16×Dic5—31no—
57C23×C5×D3—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C22×Dic15D411yescharacteristic
60C22×Dic15D411yescharacteristic
61C2×C5×#67—21no—
62C5×#235—22no—
63C5×#235C2×C414yescomplemented
64C2×C5×D12—21no—
65C2×C5×D12—22no—
66C2×C5×D12C2×C414yescomplemented
67C2×C5×D12C2312yes—
68C2×C5×D12C2311yescharacteristic
69C2×C4×C5×D3—21no—
70C2×C5×Dic6C2311yescharacteristic
71C22×C3×Dic5—21no—
72C2×D3×Dic5—21no—
73C2×D3×Dic5—21no—
74C2×D3×Dic5—22no—
75C2×D3×Dic5—22no—
76C2×D3×Dic5—44no—
77C3×C5×M16C2×C414yescomplemented
78C2×C3×C5×C8C2312yes—
79C4×Dic15D411yescharacteristic
80C4×Dic15D411yescharacteristic
81C4×Dic15C2311yescharacteristic
82C4×Dic15C2311yescharacteristic
83#6710—22no—
84C2×C3⋊#168C2312yes—
85C5×D24C2×C418yescomplemented
86C5×#204C2×C418yescomplemented
87C3×C22⋊Dic5—21no—
88C3×C4⋊Dic5—21no—
89C3×C4⋊Dic5C2312yes—
90C3×C4⋊Dic5C2311yescharacteristic
91C3×C4⋊Dic5C2311yescharacteristic
92C3×C4×Dic5—21no—
93Dic3⋊Dic5—21no—
94Dic3⋊Dic5—21no—
95(C2×D3)⋊Dic5—22no—
96(C2×D3)⋊Dic5—22no—
97(C2×D3)⋊Dic5—21no—
98(C2×D3)⋊Dic5—21no—
99Dic3×Dic5—21no—
100Dic3×Dic5—21no—
101C2×#908D511yescharacteristic
102#5522—51no—
103C2×#893—51no—
104C3×#428—51no—
105Dic3×M16—51no—
106#5001—51no—
107#4966—51no—
108#4889—52no—
109#4872—52no—
110#4869—52no—
111#4852—52no—
112C2×C5×#41—31no—
113C22×C5×D4—31no—
114C23×Dic5—61no—
115C5×AGL(1,ℤ/8)—38no—
116C2×C5×SD16—32no—
117C2×C5×D8—32no—
118C2×C5×M16C2×D311yescharacteristic
119C2×C22⋊Dic5—61no—
120Q8×Dic5—31no—
121Q8×Dic5—31no—
122D4×Dic5—32no—
123D4×Dic5—32no—
124D4×Dic5—31no—
125D4×Dic5—31no—
126D4×Dic5—61no—
127D4×Dic5—61no—
128D4×Dic5—61no—
129D4×Dic5—61no—
130#4072—61no—
131#4072C2×D311yescharacteristic
132C2×C4⋊Dic5C2×D311yescharacteristic
133C2×C4×Dic5—31no—
134C2×C4×Dic5—61no—
135C2×#681—31no—
136#3967—32no—
137#3964—32no—
138#3964—32no—
139#3964—32no—
140#3950C2×D312yescomplemented
141#3949C2×D312yescomplemented
142#3947—32no—
143C8×Dic5—32no—
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5C2×D411yescharacteristic
146C22×C5×D3—24no—
147C22×C5×D3—21no—
148C22×C5×D3—22no—
149C22×C5×D3—42no—
150C22×C5×D3—42no—
151C2×C3×C4×C5C22×C412yes—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5#4112yes—
154C2×C3×C4×C5C2411yescharacteristic
155C2×Dic15—22no—
156C2×Dic15—41no—
157C2×Dic15C2×D411yescharacteristic
158C2×Dic15C2×D411yescharacteristic
159C2×Dic15C2×D411yescharacteristic
160C2×Dic15C2×D411yescharacteristic
161C5×#67—24no—
162C2×C5×Dic3—21no—
163C5×D12—24no—
164C5×D12—22no—
165C5×D12—21no—
166C5×D12C22×C414yes—
167C5×D12C22×C412yescomplemented
168C4×C5×D3—24no—
169C5×Dic6—21no—
170C5×Dic6C22×C412yescomplemented
171C2×C3×Dic5—21no—
172C2×C3×Dic5—21no—
173C2×C3×Dic5—22no—
174C2×C3×Dic5—42no—
175D3×Dic5—42no—
176D3×Dic5—44no—
177D3×Dic5—42no—
178D3×Dic5—44no—
179C3×C5×C8C22×C414yescomplemented
180C3⋊#168—22no—
181C2×#235C2×D511yescharacteristic
182C22×D12C2×D511yescharacteristic
183C22×C4×D3—101no—
184C2×C3×M16C2×D511yescharacteristic
185C3×#117—51no—
186C2×C3×C4⋊C4—51no—
187C2×(C2×D3)⋊C4—101no—
188C2×C4×Dic3—51no—
189C2×#208—51no—
190#908Dic518yescomplemented
191C2×D24C2×D512yes—
192C2×#204C2×D512yes—
193#893—51no—
194#893—52no—
195#893—51no—
196#893—52no—
197#893—101no—
198#893—101no—
199#887—51no—
200#887—51no—
201#67⋊C4—101no—
202#67⋊C4—101no—
203#872—101no—
204C42×D3—101no—
205C3×#107—52no—
206C3×#106—52no—
207#815—52no—
208C8×Dic3—52no—
209#810—52no—
210#809—52no—
211#809—52no—
212#809—52no—
213C24×C5—61no—
214C5×#41—34no—
215C5×#41—62no—
216C2×C5×Q8—31no—
217C2×C5×D4—34no—
218C2×C5×D4—31no—
219C2×C5×D4—32no—
220C2×C5×D4—61no—
221C2×C5×D4—61no—
222C2×C5×D4—62no—
223C22×C4×C5—61no—
224C22×C4×C5C22×D311yescharacteristic
225C22×Dic5—21no—
226C22×Dic5—31no—
227C22×Dic5—31no—
228C22×Dic5—62no—
229C22×Dic5—62no—
230C22×Dic5—61no—
231C22×Dic5—61no—
232C22×Dic5—124no—
233C5×SD16—38no—
234C5×D8—38no—
235C5×M16C4×D314yescomplemented
236C2×C5×C8C22×D312yes—
237C22⋊Dic5—21no—
238C22⋊Dic5—62no—
239C22⋊Dic5—62no—
240C22⋊Dic5—61no—
241C22⋊Dic5—61no—
242C4⋊Dic5—21no—
243C4⋊Dic5—61no—
244C4⋊Dic5—61no—
245C4⋊Dic5C22×D311yescharacteristic
246C4⋊Dic5C22×D312yes—
247C4⋊Dic5C22×D311yescharacteristic
248C4×Dic5—21no—
249C4×Dic5—31no—
250C4×Dic5—31no—
251C4×Dic5—31no—
252C4×Dic5—31no—
253C4×Dic5—61no—
254C4×Dic5—61no—
255#681—62no—
256C2×#168—32no—
257C2×AGL(1,ℤ/8)—31no—
258C4×#41—151no—
259C2×C4×D4—151no—
260#442—152no—
261#440—152no—
262C4×SD16—152no—
263C4×D8—152no—
264#428—51no—
265#418—151no—
266#417—151no—
267C4×M16—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5C4×D412yes—
271C22×C3×C5C22×D411yescharacteristic
272C2×C5×D3—22no—
273C2×C5×D3—24no—
274C2×C5×D3—48no—
275C2×C5×D3—42no—
276C2×C5×D3—41no—
277C2×C5×D3—41no—
278C2×C5×D3—42no—
279C3×C4×C5C4×D412yes—
280C3×C4×C5C23×C411yescharacteristic
281C3×C4×C5C2×#4111yescharacteristic
282Dic15—21no—
283Dic15—21no—
284Dic15—21no—
285Dic15AGL(1,ℤ/8)12yescomplemented
286C3×Dic5—42no—
287C3×Dic5—41no—
288C3×Dic5—41no—
289C5×Dic3—22no—
290C23×D3—21no—
291C22×C3×C4—101no—
292C22×C3×C4C22×D511yescharacteristic
293C2×#67—21no—
294C22×Dic3—51no—
295C22×Dic3—51no—
296#235—22no—
297#235C2×Dic514yescomplemented
298C2×D12—21no—
299C2×D12—22no—
300C2×D12C2×Dic514yescomplemented
301C2×D12C22×D512yes—
302C2×D12C22×D511yescharacteristic
303C2×C4×D3—21no—
304C2×C4×D3—101no—
305C2×C4×D3—101no—
306C2×C4×D3—102no—
307C2×C4×D3—102no—
308C2×C4×D3—204no—
309C2×Dic6C22×D511yescharacteristic
310C3×M16C2×Dic514yescomplemented
311C2×C3×C8C22×D512yes—
312C3×C4⋊C4—51no—
313C3×C4⋊C4—51no—
314C3×C4⋊C4—52no—
315C3×C4⋊C4—101no—
316C3×C22⋊C4—101no—
317C3×C42—101no—
318(C2×D3)⋊C4—101no—
319(C2×D3)⋊C4—101no—
320(C2×D3)⋊C4—102no—
321(C2×D3)⋊C4—102no—
322Dic3⋊C4—101no—
323Dic3⋊C4—101no—
324C4×Dic3—51no—
325C4×Dic3—51no—
326C4×Dic3—51no—
327C4×Dic3—51no—
328C4×Dic3—101no—
329C4×Dic3—101no—
330#208—102no—
331C2×C3⋊C8—52no—
332D24C2×Dic518yescomplemented
333#204C2×Dic518yescomplemented
334C23×C5—64no—
335C23×C5—61no—
336C23×C5—62no—
337C23×C5—122no—
338C23×C5—122no—
339C23×C5D3×D411yescharacteristic
340C5×Q8—32no—
341C5×Q8—61no—
342C5×D4—32no—
343C5×D4—34no—
344C5×D4—64no—
345C5×D4—64no—
346C5×D4—62no—
347C5×D4—61no—
348C2×C4×C5—61no—
349C2×C4×C5—64no—
350C2×C4×C5C2×C4×D312yes—
351C2×C4×C5D3×D411yescharacteristic
352C2×C4×C5#23912yes—
353C2×C4×C5C23×D311yescharacteristic
354C2×Dic5—21no—
355C2×Dic5—21no—
356C2×Dic5—22no—
357C2×Dic5—31no—
358C2×Dic5—31no—
359C2×Dic5—31no—
360C2×Dic5—31no—
361C2×Dic5—42no—
362C2×Dic5—62no—
363C2×Dic5—122no—
364C2×Dic5—124no—
365C2×Dic5—122no—
366C2×Dic5—124no—
367C2×Dic5—121no—
368C5×C8C2×C4×D314yescomplemented
369#168—62no—
370C2×#41—31no—
371C22×D4—31no—
372C23×C4—301no—
373AGL(1,ℤ/8)—38no—
374C2×SD16—32no—
375C2×D8—32no—
376C2×M16—151no—
377C2×M16D3×D511yescharacteristic
378C4×Q8—151no—
379C4×Q8—151no—
380C4×D4—152no—
381C4×D4—152no—
382C4×D4—151no—
383C4×D4—151no—
384C4×D4—301no—
385C4×D4—301no—
386C4×D4—301no—
387C4×D4—301no—
388#117—51no—
389#117—301no—
390C2×C4⋊C4—51no—
391C2×C22⋊C4—301no—
392C2×C42—151no—
393C2×C42—301no—
394#107—52no—
395#106—52no—
396#103—152no—
397#102—152no—
398#102—152no—
399#102—152no—
400#97—152no—
401C4×C8—152no—
402C2×C3×C5—22no—
403C2×C3×C5C2×C4×D411yescharacteristic
404C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C3×C5#57511yescharacteristic
406C5×D3—42no—
407C5×D3—44no—
408C23×C3D4×D511yescharacteristic
409C22×D3—21no—
410C22×D3—22no—
411C22×D3—24no—
412C22×D3—42no—
413C22×D3—42no—
414C2×C3×C4—101no—
415C2×C3×C4—101no—
416C2×C3×C4—102no—
417C2×C3×C4—202no—
418C2×C3×C4D4×D511yescharacteristic
419C2×C3×C4#71112yes—
420C2×C3×C4C22×Dic512yes—
421C2×C3×C4C23×D511yescharacteristic
422#67—24no—
423C2×Dic3—21no—
424C2×Dic3—51no—
425C2×Dic3—51no—
426C2×Dic3—51no—
427C2×Dic3—51no—
428C2×Dic3—102no—
429C2×Dic3—201no—
430D12—22no—
431D12—21no—
432D12—24no—
433D12C22×Dic514yes—
434D12C22×Dic512yescomplemented
435C4×D3—24no—
436C4×D3—202no—
437C4×D3—204no—
438C4×D3—202no—
439C4×D3—204no—
440Dic6—21no—
441Dic6C22×Dic512yescomplemented
442C3×C8C22×Dic514yescomplemented
443C3⋊C8—102no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—62no—
447C22×C5—64no—
448C22×C5—128no—
449C22×C5—122no—
450C22×C5—121no—
451C22×C5—121no—
452C22×C5—122no—
453C22×C5#89312yes—
454C22×C5C2×D3×D411yescharacteristic
455C4×C5—62no—
456C4×C5#89312yes—
457C4×C5C22×C4×D311yescharacteristic
458C4×C5C2×#23911yescharacteristic
459Dic5—32no—
460Dic5—42no—
461Dic5—41no—
462Dic5—41no—
463Dic5—61no—
464Dic5—61no—
465Dic5—61no—
466C24—61no—
467#41—34no—
468#41—62no—
469C2×Q8—31no—
470C2×D4—34no—
471C2×D4—31no—
472C2×D4—32no—
473C2×D4—61no—
474C2×D4—62no—
475C2×D4—61no—
476C22×C4—61no—
477C22×C4—101no—
478C22×C4—151no—
479C22×C4—151no—
480C22×C4—301no—
481C22×C4—301no—
482C22×C4—302no—
483C22×C4—302no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—38no—
487D8—38no—
488M16—302no—
489M16D3×Dic514yescomplemented
490C2×C8—152no—
491C2×C8C2×D3×D512yes—
492C4⋊C4—51no—
493C4⋊C4—51no—
494C4⋊C4—52no—
495C4⋊C4—101no—
496C4⋊C4—301no—
497C4⋊C4—301no—
498C22⋊C4—101no—
499C22⋊C4—302no—
500C22⋊C4—302no—
501C22⋊C4—301no—
502C22⋊C4—301no—
503C42—101no—
504C42—151no—
505C42—151no—
506C42—151no—
507C42—151no—
508C42—301no—
509C42—301no—
510C3×C5C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
511C22×C3—21no—
512C22×C3—21no—
513C22×C3D4×Dic512yes—
514C22×C3C2×D4×D511yescharacteristic
515C2×D3—22no—
516C2×D3—24no—
517C2×D3—48no—
518C2×D3—42no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—42no—
522C3×C4—202no—
523C3×C4—201no—
524C3×C4—201no—
525C3×C4D4×Dic512yes—
526C3×C4C2×#71111yescharacteristic
527C3×C4C23×Dic511yescharacteristic
528Dic3—22no—
529Dic3—52no—
530Dic3—101no—
531Dic3—101no—
532Dic3—101no—
533C2×C5—22no—
534C2×C5—124no—
535C2×C5—122no—
536C2×C5C2×#89311yescharacteristic
537C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
538C2×C5#577711yescharacteristic
539C23—64no—
540C23—61no—
541C23—62no—
542C23—122no—
543C23—122no—
544C23#677711yescharacteristic
545Q8—32no—
546Q8—61no—
547D4—32no—
548D4—34no—
549D4—64no—
550D4—64no—
551D4—62no—
552D4—61no—
553C2×C4—61no—
554C2×C4—64no—
555C2×C4—101no—
556C2×C4—101no—
557C2×C4—102no—
558C2×C4—151no—
559C2×C4—151no—
560C2×C4—151no—
561C2×C4—151no—
562C2×C4—202no—
563C2×C4—302no—
564C2×C4—604no—
565C2×C4—602no—
566C2×C4—604no—
567C2×C4—602no—
568C2×C4—601no—
569C2×C4#676812yes—
570C2×C4#677711yescharacteristic
571C2×C4C2×D3×Dic512yes—
572C2×C4C22×D3×D511yescharacteristic
573C8—302no—
574C8C2×D3×Dic514yescomplemented
575C2×C3—22no—
576C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
577C2×C3#6610711yescharacteristic
578C2×C3C2×D4×Dic511yescharacteristic
579D3—44no—
580D3—42no—
581C5#7919911yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—62no—
585C22—64no—
586C22—128no—
587C22—122no—
588C22—121no—
589C22—121no—
590C22—122no—
591C22#9143512yes—
592C22C2×#677711yescharacteristic
593C4—62no—
594C4—152no—
595C4—202no—
596C4—201no—
597C4—201no—
598C4—301no—
599C4—301no—
600C4—301no—
601C4#9143512yes—
602C4C2×#676811yescharacteristic
603C4C22×D3×Dic511yescharacteristic
604C3Dic5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
605C2—22no—
606C2—124no—
607C2—122no—
608C2#1175403311yescharacteristic
609C2#1175403511yescharacteristic
610C2C2×#9143511yescharacteristic
subgroupG/H
centerC22C2×#6777
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×#908C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #11754455C2semidirect
3. C2×#91435C2semidirect
4. M16×Dic15C2semidirect
5. C3×#65410C2semidirect
6. #11748386C2semidirect
7. #11748384C2semidirect
8. #11747816C2semidirect
9. #11747813C2semidirect
10. #11747810C2semidirect
11. #11747807C2semidirect
19. C5×#908C4semidirect
31. #91435C22semidirect
32. #91435C22semidirect
35. #91344C22semidirect
36. #91344C22semidirect
37. #91099C22semidirect
38. C8×Dic15C22semidirect
39. C3×#3950C22semidirect
40. C3×#3949C22semidirect
41. #90972C22semidirect
44. #90966C22semidirect
55. #65410D3semidirect
63. C5×#235C2×C4semidirect
66. C2×C5×D12C2×C4semidirect
77. C3×C5×M16C2×C4semidirect
85. C5×D24C2×C4semidirect
86. C5×#204C2×C4semidirect
140. #3950C2×D3semidirect
141. #3949C2×D3semidirect
167. C5×D12C22×C4semidirect
170. C5×Dic6C22×C4semidirect
179. C3×C5×C8C22×C4semidirect
190. #908Dic5semidirect
235. C5×M16C4×D3semidirect
285. Dic15AGL(1,ℤ/8)semidirect
297. #235C2×Dic5semidirect
300. C2×D12C2×Dic5semidirect
310. C3×M16C2×Dic5semidirect
332. D24C2×Dic5semidirect
333. #204C2×Dic5semidirect
368. C5×C8C2×C4×D3semidirect
434. D12C22×Dic5semidirect
441. Dic6C22×Dic5semidirect
442. C3×C8C22×Dic5semidirect
489. M16D3×Dic5semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
574. C8C2×D3×Dic5semidirect
581. C5#79199semidirect
604. C3Dic5×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9082
#117544552
C2×#914352
M16×Dic152
C3×#654102
#117483862
#117483842
#117478162
#117478132
#117478102
#117478072
Dic5×AGL(1,ℤ/8)3
#791995