Finite Group Databasebeta

← #14307982

Subgroups — #14307982

Order 1920.

812 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×C3×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#11756592C211yesmaximal, characteristic, complemented
3C2×#91821C211yesmaximal, characteristic, complemented
4C2×#91121C211yesmaximal, characteristic, complemented
5#11751322C211yesmaximal, characteristic, complemented
6C2×#91109C211yesmaximal, characteristic, complemented
7#11746375C211yesmaximal, characteristic, complemented
8#10602891—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×C5×D4C2211yescharacteristic
11C2×#6817C2211yescharacteristic
12C3×C5×AGL(1,ℤ/8)—22no—
13C3×C5×AGL(1,ℤ/8)—22no—
14C2×C3×C5×SD16—21no—
15C2×C3×C5×D8—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×#6719—22no—
18#91835—21no—
19#91828—22no—
20#91827—21no—
21C2×#6716—21no—
22#91821—22no—
23#91821—22no—
24C2×#6715—21no—
25#91819—21no—
26C2×C4⋊Dic15C2211yescharacteristic
27#91813—21no—
28C2×C4×Dic15C2211yescharacteristic
29C2×#6710C2211yescharacteristic
30#91121—22no—
31#91109—22no—
32#72098—51nomaximal, Hall
33C2×C5×AGL(1,ℤ/8)D311yescharacteristic, complemented
34#66167—31no—
35C2×#4078—31no—
36C2×#3969—31no—
37#65510—31no—
38C2×#3957—31no—
39#64774—31no—
40C24×C3×C5—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C3×C5×#41—21no—
45C2×C3×C5×Q8D411yescharacteristic
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4—21no—
48C2×C3×C5×D4—21no—
49C2×C3×C5×D4—21no—
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4D411yescharacteristic
53C2×C3×C5×D4D412yescomplemented
54C22×C3×C4×C5—21no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×#1220—21no—
57C2×#1220—22no—
58C22×Dic15—21no—
59C22×Dic15D412yes—
60#6817—21no—
61#6817—21no—
62#6817—21no—
63#6817—21no—
64C2×D60D411yescharacteristic
65C2×C4×D15—21no—
66C2×Dic30D411yescharacteristic
67C3×C5×SD16—24no—
68C3×C5×D8—24no—
69C3×C5×M16—21no—
70C3×C5×M16D412yescomplemented
71C2×C3×C5×C8—21no—
72#6719—22no—
73#6719—44no—
74#6716—24no—
75#6715—24no—
76#6714—22no—
77C4⋊Dic15—21no—
78C4⋊Dic15—21no—
79#6712—22no—
80C4×Dic15—21no—
81C4×Dic15—21no—
82#6710—21no—
83#6710D412yescomplemented
84C2×C3⋊#168—21no—
85C2×C3×AGL(1,ℤ/8)D511yescharacteristic, complemented
86#5833—51no—
87C2×#932—51no—
88C2×#837—51no—
89#5221—51no—
90C2×#825—51no—
91#4554—51no—
92C2×C5×#41C2×D311yescharacteristic
93C22×C5×D4C2×D311yescharacteristic
94C2×#709—31no—
95C5×AGL(1,ℤ/8)—22no—
96C5×AGL(1,ℤ/8)—22no—
97C2×C5×SD16—21no—
98C2×C5×D8—21no—
99C2×C5×M16C2×D311yescharacteristic
100C2×C22⋊Dic5—62no—
101#4092—61no—
102#4085—62no—
103#4084—61no—
104C2×#687—61no—
105#4078—62no—
106#4078—62no—
107C2×#686—61no—
108#4076—61no—
109C2×C4⋊Dic5—31no—
110#4070—61no—
111C2×C4×Dic5—31no—
112C2×#681—31no—
113#3969—62no—
114#3957—62no—
115#2006—151noSylow
116C23×C3×C5—21no—
117C23×C3×C5—22no—
118C23×C3×C5—42no—
119C23×C3×C5—42no—
120C23×C3×C5—41no—
121C23×C3×C5—41no—
122C23×C3×C5C2×D411yescharacteristic
123C22×D15—21no—
124C3×C5×Q8—21no—
125C3×C5×Q8—21no—
126C3×C5×D4—22no—
127C3×C5×D4—21no—
128C3×C5×D4—22no—
129C3×C5×D4—21no—
130C3×C5×D4—41no—
131C3×C5×D4—41no—
132C3×C5×D4—41no—
133C3×C5×D4—41no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—41no—
136C2×C3×C4×C5—41no—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5#4111yescharacteristic
143#1220—41no—
144#1220—41no—
145#1220—82no—
146C2×Dic15—22no—
147C2×Dic15—21no—
148C2×Dic15—22no—
149C2×Dic15—41no—
150C2×Dic15—41no—
151C2×Dic15—42no—
152D60—21no—
153D60—21no—
154C4×D15—41no—
155C4×D15—41no—
156Dic30—21no—
157Dic30—21no—
158C3×C5×C8—22no—
159C3⋊#168—22no—
160C2×C3×#41C2×D511yescharacteristic
161C22×C3×D4C2×D511yescharacteristic
162C2×#235—51no—
163C3×AGL(1,ℤ/8)—22no—
164C3×AGL(1,ℤ/8)—22no—
165C2×C3×SD16—21no—
166C2×C3×D8—21no—
167C2×C3×M16C2×D511yescharacteristic
168C2×#217—102no—
169#946—101no—
170#939—102no—
171#938—101no—
172C2×#214—101no—
173#932—102no—
174#932—102no—
175C2×#213—101no—
176#930—101no—
177C2×C4⋊Dic3—51no—
178#924—101no—
179C2×C4×Dic3—51no—
180C2×#208—51no—
181#837—102no—
182#825—102no—
183C24×C5—21no—
184C5×#41—21no—
185C5×#41—21no—
186C5×#41—21no—
187C5×#41—21no—
188C2×C5×Q8#6711yescharacteristic
189C2×C5×D4—21no—
190C2×C5×D4—21no—
191C2×C5×D4—21no—
192C2×C5×D4—21no—
193C2×C5×D4—21no—
194C2×C5×D4—21no—
195C2×C5×D4#6711yescharacteristic
196C2×C5×D4#6712yescomplemented
197C22×C4×C5—21no—
198C22×C4×C5C22×D311yescharacteristic
199C2×C22⋊D5—61no—
200C2×C22⋊D5—62no—
201C22×Dic5—32no—
202C22×Dic5—61no—
203#709—61no—
204#709—61no—
205#709—61no—
206#709—61no—
207C2×D20—31no—
208C2×C4×D5—61no—
209C2×Dic10—31no—
210C5×SD16—24no—
211C5×D8—24no—
212C5×M16—21no—
213C5×M16#6712yescomplemented
214C2×C5×C8—21no—
215C22⋊Dic5—62no—
216C22⋊Dic5—124no—
217#687—64no—
218#686—64no—
219#685—62no—
220C4⋊Dic5—61no—
221C4⋊Dic5—61no—
222#683—62no—
223C4×Dic5—61no—
224C4×Dic5—61no—
225#681—32no—
226#681—61no—
227C2×#168—61no—
228C2×AGL(1,ℤ/8)—151no—
229C2×AGL(1,ℤ/8)D1511yescharacteristic, complemented
230#537—151no—
231C2×#104—151no—
232C2×#104—151no—
233#390—151no—
234#343—151no—
235C22×C3×C5—21no—
236C22×C3×C5—21no—
237C22×C3×C5—44no—
238C22×C3×C5—41no—
239C22×C3×C5—42no—
240C22×C3×C5—41no—
241C22×C3×C5—41no—
242C22×C3×C5—82no—
243C22×C3×C5—81no—
244C22×C3×C5—81no—
245C22×C3×C5#12011yescharacteristic
246C22×C3×C5#12111yescharacteristic
247C22×C3×C5#12711yescharacteristic
248C2×D15—41no—
249C2×D15—41no—
250C2×D15—41no—
251C3×C4×C5—41no—
252C3×C4×C5#12011yescharacteristic
253C3×C4×C5#12011yescharacteristic
254C3×C4×C5#12111yescharacteristic
255C3×C4×C5#12111yescharacteristic
256Dic15—42no—
257Dic15—41no—
258Dic15—42no—
259C24×C3—21no—
260C3×#41—21no—
261C3×#41—21no—
262C3×#41—21no—
263C3×#41—21no—
264C2×C3×Q8C22⋊D511yescharacteristic
265C2×C3×D4—21no—
266C2×C3×D4—21no—
267C2×C3×D4—21no—
268C2×C3×D4—21no—
269C2×C3×D4—21no—
270C2×C3×D4—21no—
271C2×C3×D4C22⋊D511yescharacteristic
272C2×C3×D4C22⋊D512yescomplemented
273C22×C3×C4—21no—
274C22×C3×C4C22×D511yescharacteristic
275C2×#67—101no—
276C2×#67—102no—
277C22×Dic3—52no—
278C22×Dic3—101no—
279#235—101no—
280#235—101no—
281#235—101no—
282#235—101no—
283C2×D12—51no—
284C2×C4×D3—101no—
285C2×Dic6—51no—
286C3×SD16—24no—
287C3×D8—24no—
288C3×M16—21no—
289C3×M16C22⋊D512yescomplemented
290C2×C3×C8—21no—
291#217—102no—
292#217—204no—
293#214—104no—
294#213—104no—
295(C2×D3)⋊C4—102no—
296C4⋊Dic3—101no—
297C4⋊Dic3—101no—
298Dic3⋊C4—102no—
299C4×Dic3—101no—
300C4×Dic3—101no—
301#208—52no—
302#208—101no—
303C2×C3⋊C8—101no—
304C23×C5—21no—
305C23×C5—22no—
306C23×C5—42no—
307C23×C5—42no—
308C23×C5—41no—
309C23×C5—41no—
310C23×C5D3×D411yescharacteristic
311C22×D5—61no—
312C5×Q8—21no—
313C5×Q8—21no—
314C5×D4—22no—
315C5×D4—21no—
316C5×D4—21no—
317C5×D4—22no—
318C5×D4—41no—
319C5×D4—41no—
320C5×D4—41no—
321C5×D4—41no—
322C2×C4×C5—21no—
323C2×C4×C5—41no—
324C2×C4×C5—41no—
325C2×C4×C5D3×D411yescharacteristic
326C2×C4×C5D3×D411yescharacteristic
327C2×C4×C5#23711yescharacteristic
328C2×C4×C5C2×#6711yescharacteristic
329C2×C4×C5C2×#6711yescharacteristic
330C2×C4×C5C2×#6711yescharacteristic
331C22⋊D5—121no—
332C22⋊D5—121no—
333C22⋊D5—242no—
334C2×Dic5—62no—
335C2×Dic5—61no—
336C2×Dic5—62no—
337C2×Dic5—121no—
338C2×Dic5—121no—
339C2×Dic5—122no—
340D20—61no—
341D20—61no—
342C4×D5—121no—
343C4×D5—121no—
344Dic10—61no—
345Dic10—61no—
346C5×C8—22no—
347#168—62no—
348C2×#41—151no—
349C2×#41C2×D1511yescharacteristic
350C22×D4C2×D1511yescharacteristic
351AGL(1,ℤ/8)—22no—
352AGL(1,ℤ/8)—22no—
353AGL(1,ℤ/8)—302no—
354AGL(1,ℤ/8)—302no—
355C2×SD16—21no—
356C2×SD16—301no—
357C2×D8—21no—
358C2×D8—301no—
359C2×M16—151no—
360C2×M16C2×D1511yescharacteristic
361C22⋊Q8—301no—
362C22⋊Q8—301no—
363#121—301no—
364#121—302no—
365#121—301no—
366C2×C4⋊C4—151no—
367C2×C22⋊C4—302no—
368C2×C42—151no—
369#104—302no—
370#104—302no—
371C2×C3×C5—21no—
372C2×C3×C5—21no—
373C2×C3×C5—81no—
374C2×C3×C5—82no—
375C2×C3×C5#39211yescharacteristic
376C2×C3×C5#45311yescharacteristic
377C2×C3×C5#45411yescharacteristic
378D15—81no—
379C23×C3—21no—
380C23×C3—22no—
381C23×C3—42no—
382C23×C3—42no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3D4×D511yescharacteristic
386C22×D3—101no—
387C3×Q8—21no—
388C3×Q8—21no—
389C3×D4—21no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—21no—
393C3×D4—41no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4D4×D511yescharacteristic
401C2×C3×C4D4×D511yescharacteristic
402C2×C3×C4#71111yescharacteristic
403C2×C3×C4C2×C22⋊D511yescharacteristic
404C2×C3×C4C2×C22⋊D511yescharacteristic
405C2×C3×C4C2×C22⋊D511yescharacteristic
406#67—201no—
407#67—201no—
408#67—402no—
409C2×Dic3—102no—
410C2×Dic3—101no—
411C2×Dic3—102no—
412C2×Dic3—201no—
413C2×Dic3—201no—
414C2×Dic3—202no—
415D12—101no—
416D12—101no—
417C4×D3—201no—
418C4×D3—201no—
419Dic6—101no—
420Dic6—101no—
421C3×C8—22no—
422C3⋊C8—102no—
423C22×C5—21no—
424C22×C5—21no—
425C22×C5—44no—
426C22×C5—41no—
427C22×C5—41no—
428C22×C5—41no—
429C22×C5—42no—
430C22×C5—82no—
431C22×C5—81no—
432C22×C5—81no—
433C22×C5#93711yescharacteristic
434C22×C5#93911yescharacteristic
435C22×C5#94011yescharacteristic
436C2×D5—121no—
437C2×D5—121no—
438C2×D5—121no—
439C4×C5—41no—
440C4×C5#93711yescharacteristic
441C4×C5#93811yescharacteristic
442C4×C5#93911yescharacteristic
443C4×C5#95311yescharacteristic
444Dic5—122no—
445Dic5—122no—
446Dic5—121no—
447C24—21no—
448#41—21no—
449#41—21no—
450#41—21no—
451#41—21no—
452#41—301no—
453#41—301no—
454#41—301no—
455#41—301no—
456C2×Q8—151no—
457C2×Q8#122011yescharacteristic
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—151no—
465C2×D4—301no—
466C2×D4—302no—
467C2×D4#122011yescharacteristic
468C2×D4#122012yescomplemented
469C22×C4—21no—
470C22×C4—152no—
471C22×C4—301no—
472C22×C4—301no—
473C22×C4C22×D1511yescharacteristic
474SD16—24no—
475SD16—304no—
476D8—24no—
477D8—304no—
478M16—21no—
479M16—152no—
480M16—301no—
481M16#122012yescomplemented
482C2×C8—21no—
483C2×C8—301no—
484C4⋊C4—302no—
485C4⋊C4—301no—
486C4⋊C4—301no—
487C22⋊C4—302no—
488C22⋊C4—302no—
489C22⋊C4—604no—
490C42—301no—
491C42—301no—
492C3×C5#200611yescharacteristic, complemented, Hall
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—44no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—42no—
500C22×C3—82no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3#408311yescharacteristic
504C22×C3#408511yescharacteristic
505C22×C3#408611yescharacteristic
506C2×D3—201no—
507C2×D3—201no—
508C2×D3—201no—
509C3×C4—41no—
510C3×C4#408311yescharacteristic
511C3×C4#408411yescharacteristic
512C3×C4#408511yescharacteristic
513C3×C4#409911yescharacteristic
514Dic3—202no—
515Dic3—202no—
516Dic3—201no—
517C2×C5—21no—
518C2×C5—21no—
519C2×C5—82no—
520C2×C5—81no—
521C2×C5#519811yescharacteristic
522C2×C5#520111yescharacteristic
523C2×C5#522311yescharacteristic
524D5—241no—
525C23—21no—
526C23—22no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—301no—
532C23D4×D1511yescharacteristic
533Q8—21no—
534Q8—21no—
535Q8—301no—
536Q8—301no—
537D4—22no—
538D4—21no—
539D4—21no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—301no—
546D4—301no—
547D4—601no—
548D4—601no—
549D4—1202no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—302no—
554C2×C4—302no—
555C2×C4—301no—
556C2×C4—601no—
557C2×C4—601no—
558C2×C4—601no—
559C2×C4—601no—
560C2×C4—602no—
561C2×C4D4×D1511yescharacteristic
562C2×C4D4×D1511yescharacteristic
563C2×C4#681911yescharacteristic
564C2×C4C2×#122011yescharacteristic
565C2×C4C2×#122011yescharacteristic
566C2×C4C2×#122011yescharacteristic
567C8—22no—
568C8—302no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—82no—
572C2×C3—81no—
573C2×C3#6548711yescharacteristic
574C2×C3#6549011yescharacteristic
575C2×C3#6551211yescharacteristic
576D3—401no—
577C5#7209811yescharacteristic, complemented, Sylow
578C22—21no—
579C22—21no—
580C22—44no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—42no—
585C22—82no—
586C22—81no—
587C22—81no—
588C22—601no—
589C22—601no—
590C22—601no—
591C22#9182611yescharacteristic
592C22#9182811yescharacteristic
593C22#9182911yescharacteristic
594C4—41no—
595C4—602no—
596C4—602no—
597C4—601no—
598C4#9182611yescharacteristic
599C4#9182711yescharacteristic
600C4#9182811yescharacteristic
601C4#9184211yescharacteristic
602C3#1060289111yescharacteristic, complemented, Sylow
603C2—21no—
604C2—21no—
605C2—82no—
606C2—81no—
607C2—1201no—
608C2#1175129911yescharacteristic
609C2#1175130211yescharacteristic
610C2#1175132411yescharacteristic
subgroupG/H
centerC22#91826
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C22×D15
FittingC2×C3×C5×AGL(1,ℤ/8)C2
socleC22×C3×C5#120
hypercenterC2×AGL(1,ℤ/8)D15
hypocenterC3×C5#2006
LCSG ▷ C22×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×AGL(1,ℤ/8)C2semidirect
2. #11756592C2semidirect
3. C2×#91821C2semidirect
4. C2×#91121C2semidirect
5. #11751322C2semidirect
6. C2×#91109C2semidirect
7. #11746375C2semidirect
33. C2×C5×AGL(1,ℤ/8)D3semidirect
53. C2×C3×C5×D4D4semidirect
70. C3×C5×M16D4semidirect
83. #6710D4semidirect
85. C2×C3×AGL(1,ℤ/8)D5semidirect
196. C2×C5×D4#67semidirect
213. C5×M16#67semidirect
229. C2×AGL(1,ℤ/8)D15semidirect
272. C2×C3×D4C22⋊D5semidirect
289. C3×M16C22⋊D5semidirect
468. C2×D4#1220semidirect
481. M16#1220semidirect
492. C3×C5#2006semidirect
577. C5#72098semidirect
602. C3#10602891semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×AGL(1,ℤ/8)2
#117565922
C2×#918212
C2×#911212
#117513222
C2×#911092
#117463752
#106028913
#720985