Finite Group Databasebeta

← #14439278

Subgroups — #14439278

Order 1920.

1144 conjugacy classes of subgroups.

The subgroup lattice has 604 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#11756591C211yesmaximal, characteristic, complemented
3C2×#91840C211yesmaximal, characteristic, complemented
4C2×#91825C211yesmaximal, characteristic, complemented
5C2×#91821C211yesmaximal, characteristic, complemented
6C2×#91798C211yesmaximal, characteristic, complemented
7#11751269C212yesmaximal, complemented
8#11751268C212yesmaximal, complemented
9#11751256C212yesmaximal, complemented
10#11751255C212yesmaximal, complemented
11M16×Dic15C211yesmaximal, characteristic, complemented
12#10615669—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×C3×C5×D4C2211yescharacteristic
15C2×#6817C2211yescharacteristic
16C22×Dic30C2211yescharacteristic
17C3×C5×AGL(1,ℤ/8)—24no—
18C2×C3×C5×SD16C2212yes—
19C2×C3×C5×D8C2212yes—
20C2×C3×C5×M16C2211yescharacteristic
21C2×#6719—22no—
22#91840—24no—
23#91836C2211yescharacteristic, complemented
24#91833C2211yescharacteristic, complemented
25C2×#6718C2212yescomplemented
26#91829C2211yescharacteristic, complemented
27#91828—22no—
28#91825—22no—
29#91825C2212yes—
30#91825C2212yes—
31#91825C2211yescharacteristic, complemented
32C2×#6716C2212yes—
33C2×#6716C2212yescomplemented
34#91821—24no—
35C2×#6715C2212yes—
36C4×#1220—22no—
37C2×C4×Dic15C2211yescharacteristic
38C2×#6710C2211yescharacteristic
39#91798—24no—
40C2×Dic60C2212yescomplemented
41C2×#6706C2212yescomplemented
42#91099C2212yescomplemented
43C8×Dic15C2212yescomplemented
44#83938—51nomaximal, Hall
45C2×C5×AGL(1,ℤ/8)D311yescharacteristic, complemented
46#66166—31no—
47C2×#4097—31no—
48C2×#4082—31no—
49C2×#4078—31no—
50C2×#4055—31no—
51#65457—32no—
52#65456—32no—
53#65444—32no—
54#65443—32no—
55M16×Dic5—31no—
56C24×C3×C5—21no—
57C3×C5×#41—22no—
58C3×C5×#41—22no—
59C2×C3×C5×Q8C2311yescharacteristic
60C2×C3×C5×D4—21no—
61C2×C3×C5×D4—22no—
62C2×C3×C5×D4—22no—
63C2×C3×C5×D4—21no—
64C2×C3×C5×D4C2312yes—
65C2×C3×C5×D4C2311yescharacteristic
66C22×C3×C4×C5—21no—
67C22×C3×C4×C5C2311yescharacteristic
68C2×#1220—21no—
69C2×#1220—22no—
70C22×Dic15—21no—
71C22×Dic15D412yes—
72#6817—22no—
73#6817—22no—
74C2×D60C2311yescharacteristic
75C2×C4×D15—21no—
76C2×Dic30—22no—
77C2×Dic30—22no—
78C2×Dic30—21no—
79C2×Dic30C2312yes—
80C2×Dic30C2311yescharacteristic
81C3×C5×SD16—24no—
82C3×C5×D8—24no—
83C3×C5×M16D414yescomplemented
84C2×C3×C5×C8C2312yes—
85#6719—22no—
86#6719—24no—
87#6719—24no—
88#6718—24no—
89#6716—24no—
90#6716—24no—
91#6715—24no—
92#6714—22no—
93#6712—22no—
94C4×Dic15D412yes—
95C4×Dic15C2311yescharacteristic
96C4×Dic15C2311yescharacteristic
97#6710D414yescomplemented
98C2×C3⋊#168C2312yes—
99Dic60—24no—
100#6706—24no—
101C2×C3×AGL(1,ℤ/8)D511yescharacteristic, complemented
102#5832—51no—
103C2×#951—51no—
104C2×#936—51no—
105C2×#932—51no—
106C2×#909—51no—
107#5168—52no—
108#5167—52no—
109#5155—52no—
110#5154—52no—
111Dic3×M16—51no—
112C2×C5×#41C2×D311yescharacteristic
113C22×C5×D4C2×D311yescharacteristic
114C2×#709—31no—
115C22×Dic10—31no—
116C5×AGL(1,ℤ/8)—24no—
117C2×C5×SD16C2×D312yes—
118C2×C5×D8C2×D312yes—
119C2×C5×M16C2×D311yescharacteristic
120C2×C22⋊Dic5—62no—
121#4097—64no—
122#4093—31no—
123#4090—31no—
124C2×#689—32no—
125#4086—31no—
126#4085—62no—
127#4082—32no—
128#4082—32no—
129#4082—31no—
130#4082—62no—
131C2×#687—32no—
132C2×#687—32no—
133#4078—64no—
134C2×#686—32no—
135C4×C22⋊D5—62no—
136C2×C4×Dic5—31no—
137C2×#681—31no—
138#4055—64no—
139C2×Dic20—32no—
140C2×#677—32no—
141#3947—32no—
142C8×Dic5—32no—
143#3153—151noSylow
144C23×C3×C5—22no—
145C23×C3×C5—21no—
146C23×C3×C5—44no—
147C23×C3×C5—42no—
148C23×C3×C5C2×D411yescharacteristic
149C22×D15—21no—
150C3×C5×Q8—21no—
151C3×C5×Q8—21no—
152C3×C5×D4—22no—
153C3×C5×D4—21no—
154C3×C5×D4—21no—
155C3×C5×D4—22no—
156C3×C5×D4—42no—
157C3×C5×D4—42no—
158C2×C3×C4×C5—21no—
159C2×C3×C4×C5—42no—
160C2×C3×C4×C5C2×D411yescharacteristic
161C2×C3×C4×C5C2×D412yes—
162C2×C3×C4×C5C2×D412yes—
163C2×C3×C4×C5C2411yescharacteristic
164#1220—44no—
165#1220—42no—
166C2×Dic15—21no—
167C2×Dic15—22no—
168C2×Dic15—42no—
169C2×Dic15—42no—
170C2×Dic15C2×D412yes—
171C2×Dic15C2×D412yes—
172D60—21no—
173D60—21no—
174C4×D15—42no—
175Dic30—22no—
176Dic30—21no—
177Dic30—21no—
178Dic30—22no—
179Dic30—42no—
180C3×C5×C8C2×D414yes—
181C3⋊#168C2×D414yes—
182C2×C3×#41C2×D511yescharacteristic
183C22×C3×D4C2×D511yescharacteristic
184C2×#235—51no—
185C22×Dic6—51no—
186C3×AGL(1,ℤ/8)—24no—
187C2×C3×SD16C2×D512yes—
188C2×C3×D8C2×D512yes—
189C2×C3×M16C2×D511yescharacteristic
190C2×#217—102no—
191#951—104no—
192#947—51no—
193#944—51no—
194C2×#216—52no—
195#940—51no—
196#939—102no—
197#936—52no—
198#936—51no—
199#936—52no—
200#936—102no—
201C2×#214—52no—
202C2×#214—52no—
203#932—104no—
204C2×#213—52no—
205C4×#67—102no—
206C2×C4×Dic3—51no—
207C2×#208—51no—
208#909—104no—
209C2×Dic12—52no—
210C2×#204—52no—
211#815—52no—
212C8×Dic3—52no—
213C24×C5—21no—
214C5×#41—22no—
215C5×#41—22no—
216C2×C5×Q8C22×D311yescharacteristic
217C2×C5×D4—21no—
218C2×C5×D4—22no—
219C2×C5×D4—21no—
220C2×C5×D4—22no—
221C2×C5×D4C22×D311yescharacteristic
222C2×C5×D4C22×D312yes—
223C22×C4×C5—21no—
224C22×C4×C5C22×D311yescharacteristic
225C2×C22⋊D5—62no—
226C2×C22⋊D5—61no—
227C22×Dic5—32no—
228C22×Dic5—61no—
229#709—62no—
230#709—62no—
231C2×D20—31no—
232C2×C4×D5—61no—
233C2×Dic10—32no—
234C2×Dic10—31no—
235C2×Dic10—62no—
236C2×Dic10—62no—
237C2×Dic10—61no—
238C5×SD16—24no—
239C5×D8—24no—
240C5×M16#6714yescomplemented
241C2×C5×C8C22×D312yes—
242C22⋊Dic5—64no—
243C22⋊Dic5—64no—
244C22⋊Dic5—62no—
245#689—64no—
246#687—64no—
247#687—64no—
248#686—64no—
249#685—62no—
250#683—62no—
251C4×Dic5—32no—
252C4×Dic5—31no—
253C4×Dic5—31no—
254#681—34no—
255C2×#168—32no—
256Dic20—64no—
257#677—64no—
258C2×#137—151no—
259C2×#137—151no—
260C2×AGL(1,ℤ/8)—151no—
261C2×AGL(1,ℤ/8)D1511yescharacteristic, complemented
262#532—151no—
263C2×#124—151no—
264#497—152no—
265#496—152no—
266#495—152no—
267#495—152no—
268C4×M16—151no—
269C22×C3×C5—22no—
270C22×C3×C5—41no—
271C22×C3×C5—42no—
272C22×C3×C5—42no—
273C22×C3×C5—44no—
274C22×C3×C5—84no—
275C22×C3×C5#12712yes—
276C22×C3×C5C22×D411yescharacteristic
277C2×D15—41no—
278C2×D15—42no—
279C3×C4×C5—41no—
280C3×C4×C5#12712yes—
281C3×C4×C5C22×D411yescharacteristic
282C3×C4×C5C22×D411yescharacteristic
283Dic15—22no—
284Dic15—22no—
285Dic15—42no—
286Dic15—41no—
287C24×C3—21no—
288C3×#41—22no—
289C3×#41—22no—
290C2×C3×Q8C22×D511yescharacteristic
291C2×C3×D4—21no—
292C2×C3×D4—22no—
293C2×C3×D4—21no—
294C2×C3×D4—22no—
295C2×C3×D4C22×D512yes—
296C2×C3×D4C22×D511yescharacteristic
297C22×C3×C4—21no—
298C22×C3×C4C22×D511yescharacteristic
299C2×#67—102no—
300C2×#67—101no—
301C22×Dic3—52no—
302C22×Dic3—101no—
303#235—102no—
304#235—102no—
305C2×D12—51no—
306C2×C4×D3—101no—
307C2×Dic6—52no—
308C2×Dic6—51no—
309C2×Dic6—102no—
310C2×Dic6—102no—
311C2×Dic6—101no—
312C3×SD16—24no—
313C3×D8—24no—
314C3×M16C22⋊D514yescomplemented
315C2×C3×C8C22×D512yes—
316#217—102no—
317#217—104no—
318#217—104no—
319#216—104no—
320#214—104no—
321#214—104no—
322#213—104no—
323(C2×D3)⋊C4—102no—
324Dic3⋊C4—102no—
325C4×Dic3—52no—
326C4×Dic3—51no—
327C4×Dic3—51no—
328#208—54no—
329C2×C3⋊C8—52no—
330Dic12—104no—
331#204—104no—
332C23×C5—22no—
333C23×C5—21no—
334C23×C5—44no—
335C23×C5—42no—
336C23×C5D3×D411yescharacteristic
337C22×D5—61no—
338C5×Q8—21no—
339C5×Q8—21no—
340C5×D4—22no—
341C5×D4—21no—
342C5×D4—21no—
343C5×D4—22no—
344C5×D4—42no—
345C5×D4—42no—
346C2×C4×C5—21no—
347C2×C4×C5—42no—
348C2×C4×C5D3×D411yescharacteristic
349C2×C4×C5D3×D412yes—
350C2×C4×C5C2×#6712yes—
351C2×C4×C5C23×D311yescharacteristic
352C22⋊D5—124no—
353C22⋊D5—122no—
354C2×Dic5—32no—
355C2×Dic5—32no—
356C2×Dic5—61no—
357C2×Dic5—62no—
358C2×Dic5—122no—
359C2×Dic5—122no—
360D20—61no—
361D20—61no—
362C4×D5—122no—
363Dic10—61no—
364Dic10—62no—
365Dic10—62no—
366Dic10—61no—
367Dic10—122no—
368C5×C8C2×#6714yes—
369#168—34no—
370C2×#41—151no—
371C2×#41C2×D1511yescharacteristic
372C22×Q8—151no—
373C22×D4C2×D1511yescharacteristic
374#137—304no—
375#137—304no—
376AGL(1,ℤ/8)—24no—
377AGL(1,ℤ/8)—304no—
378C2×Q16—152no—
379C2×Q16—152no—
380C2×SD16—152no—
381C2×SD16—152no—
382C2×SD16—152no—
383C2×SD16C2×D1512yes—
384C2×D8—152no—
385C2×D8C2×D1512yes—
386C2×M16—151no—
387C2×M16C2×D1511yescharacteristic
388C4⋊Q8—151no—
389#127—151no—
390#124—151no—
391#124—152no—
392#124—152no—
393#124—151no—
394#124—302no—
395#121—302no—
396C4×D4—302no—
397C2×C22⋊C4—302no—
398C2×C42—151no—
399#97—152no—
400C4×C8—152no—
401C2×C3×C5—22no—
402C2×C3×C5—81no—
403C2×C3×C5—82no—
404C2×C3×C5C2×#12711yescharacteristic
405C2×C3×C5#57512yes—
406D15—81no—
407C23×C3—22no—
408C23×C3—21no—
409C23×C3—44no—
410C23×C3—42no—
411C23×C3D4×D511yescharacteristic
412C22×D3—101no—
413C3×Q8—21no—
414C3×Q8—21no—
415C3×D4—22no—
416C3×D4—21no—
417C3×D4—21no—
418C3×D4—22no—
419C3×D4—42no—
420C3×D4—42no—
421C2×C3×C4—21no—
422C2×C3×C4—42no—
423C2×C3×C4D4×D512yes—
424C2×C3×C4D4×D511yescharacteristic
425C2×C3×C4C2×C22⋊D512yes—
426C2×C3×C4C23×D511yescharacteristic
427#67—204no—
428#67—202no—
429C2×Dic3—52no—
430C2×Dic3—52no—
431C2×Dic3—101no—
432C2×Dic3—102no—
433C2×Dic3—202no—
434C2×Dic3—202no—
435D12—101no—
436D12—101no—
437C4×D3—202no—
438Dic6—101no—
439Dic6—102no—
440Dic6—102no—
441Dic6—101no—
442Dic6—202no—
443C3×C8C2×C22⋊D514yes—
444C3⋊C8—54no—
445C22×C5—22no—
446C22×C5—42no—
447C22×C5—44no—
448C22×C5—41no—
449C22×C5—42no—
450C22×C5—84no—
451C22×C5#94012yes—
452C22×C5C2×D3×D411yescharacteristic
453C2×D5—122no—
454C2×D5—121no—
455C4×C5—41no—
456C4×C5#94012yes—
457C4×C5C2×D3×D411yescharacteristic
458C4×C5C22×#6711yescharacteristic
459Dic5—62no—
460Dic5—62no—
461Dic5—122no—
462Dic5—121no—
463C24—21no—
464#41—22no—
465#41—22no—
466#41—302no—
467#41—302no—
468C2×Q8—152no—
469C2×Q8—151no—
470C2×Q8—301no—
471C2×Q8—302no—
472C2×Q8—302no—
473C2×Q8C22×D1511yescharacteristic
474C2×D4—21no—
475C2×D4—22no—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—151no—
479C2×D4—302no—
480C2×D4—301no—
481C2×D4C22×D1512yes—
482C2×D4C22×D1511yescharacteristic
483C22×C4—21no—
484C22×C4—152no—
485C22×C4—301no—
486C22×C4—301no—
487C22×C4C22×D1511yescharacteristic
488Q16—304no—
489Q16—304no—
490SD16—24no—
491SD16—304no—
492SD16—304no—
493SD16—304no—
494D8—24no—
495D8—304no—
496M16—154no—
497M16#122014yescomplemented
498C2×C8—152no—
499C2×C8C22×D1512yes—
500C4⋊C4—302no—
501C22⋊C4—302no—
502C22⋊C4—304no—
503C22⋊C4—304no—
504C22⋊C4—302no—
505C42—152no—
506C42—151no—
507C42—151no—
508C3×C5#315311yescharacteristic, complemented, Hall
509C22×C3—22no—
510C22×C3—42no—
511C22×C3—44no—
512C22×C3—41no—
513C22×C3—42no—
514C22×C3—84no—
515C22×C3#408612yes—
516C22×C3C2×D4×D511yescharacteristic
517C2×D3—202no—
518C2×D3—201no—
519C3×C4—41no—
520C3×C4#408612yes—
521C3×C4C2×D4×D511yescharacteristic
522C3×C4C22×C22⋊D511yescharacteristic
523Dic3—102no—
524Dic3—102no—
525Dic3—202no—
526Dic3—201no—
527C2×C5—22no—
528C2×C5—82no—
529C2×C5—81no—
530C2×C5#577312yes—
531C2×C5C2×#94011yescharacteristic
532D5—241no—
533C23—22no—
534C23—21no—
535C23—44no—
536C23—42no—
537C23—301no—
538C23D4×D1511yescharacteristic
539Q8—21no—
540Q8—21no—
541Q8—301no—
542Q8—302no—
543Q8—302no—
544Q8—301no—
545Q8—602no—
546D4—22no—
547D4—21no—
548D4—21no—
549D4—22no—
550D4—42no—
551D4—42no—
552D4—301no—
553D4—301no—
554D4—604no—
555D4—602no—
556C2×C4—21no—
557C2×C4—42no—
558C2×C4—152no—
559C2×C4—152no—
560C2×C4—302no—
561C2×C4—301no—
562C2×C4—602no—
563C2×C4—602no—
564C2×C4—602no—
565C2×C4D4×D1511yescharacteristic
566C2×C4D4×D1512yes—
567C2×C4C2×#122012yes—
568C2×C4C23×D1511yescharacteristic
569C8—154no—
570C8C2×#122014yes—
571C2×C3—22no—
572C2×C3—82no—
573C2×C3—81no—
574C2×C3#6610712yes—
575C2×C3C2×#408611yescharacteristic
576D3—401no—
577C5#8393811yescharacteristic, complemented, Sylow
578C22—22no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—84no—
584C22—601no—
585C22—602no—
586C22#9182912yes—
587C22C2×D4×D1511yescharacteristic
588C4—41no—
589C4—302no—
590C4—302no—
591C4—601no—
592C4—602no—
593C4#9182912yes—
594C4C2×D4×D1511yescharacteristic
595C4C22×#122011yescharacteristic
596C3#1061566911yescharacteristic, complemented, Sylow
597C2—22no—
598C2—82no—
599C2—81no—
600C2—1201no—
601C2#1175653212yes—
602C2C2×#9182911yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×AGL(1,ℤ/8)C2
socleC22×C3×C5C22×D4
hypercenterC2×AGL(1,ℤ/8)D15
hypocenterC3×C5#3153
LCSG ▷ C2×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. #11756591C2semidirect
3. C2×#91840C2semidirect
4. C2×#91825C2semidirect
5. C2×#91821C2semidirect
6. C2×#91798C2semidirect
7. #11751269C2semidirect
8. #11751268C2semidirect
9. #11751256C2semidirect
10. #11751255C2semidirect
11. M16×Dic15C2semidirect
23. #91836C22semidirect
24. #91833C22semidirect
25. C2×#6718C22semidirect
26. #91829C22semidirect
31. #91825C22semidirect
33. C2×#6716C22semidirect
40. C2×Dic60C22semidirect
41. C2×#6706C22semidirect
42. #91099C22semidirect
43. C8×Dic15C22semidirect
45. C2×C5×AGL(1,ℤ/8)D3semidirect
83. C3×C5×M16D4semidirect
97. #6710D4semidirect
101. C2×C3×AGL(1,ℤ/8)D5semidirect
240. C5×M16#67semidirect
261. C2×AGL(1,ℤ/8)D15semidirect
314. C3×M16C22⋊D5semidirect
497. M16#1220semidirect
508. C3×C5#3153semidirect
577. C5#83938semidirect
596. C3#10615669semidirect

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
#117565912
C2×#918402
C2×#918252
C2×#918212
C2×#917982
#117512692
#117512682
#117512562
#117512552
M16×Dic152
#106156693
#839385