Finite Group Databasebeta

← #14439389

Subgroups — #14439389

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 596 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#11756584C211yesmaximal, characteristic, complemented
3C2×#91839C211yesmaximal, characteristic, complemented
4#11756564C211yesmaximal, characteristic, complemented
5C22×#6716C211yesmaximal, characteristic, complemented
6C2×#91798C211yesmaximal, characteristic, complemented
7#11751276C212yesmaximal, complemented
8#11751268C212yesmaximal, complemented
9#11751256C212yesmaximal, complemented
10#11751255C212yesmaximal, complemented
11#11751223C211yesmaximal, characteristic, complemented
12#10615670—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
21#91842—22no—
22#91839—24no—
23#91833C2212yescomplemented
24C2×#6718C2211yescharacteristic, complemented
25C2×#6717C2211yescharacteristic, complemented
26#91829C2212yes—
27#91827—22no—
28#91825C2212yes—
29#91825C2212yescomplemented
30#91824—22no—
31C2×#6716—24no—
32C2×#6716C2212yes—
33C2×#6716C2212yes—
34C2×#6716C2211yescharacteristic, complemented
35C2×#6715C2211yescharacteristic, complemented
36#91815C2211yescharacteristic
37#91813—22no—
38C22×C3⋊#168C2211yescharacteristic
39#91798—24no—
40C2×Dic60C2212yescomplemented
41C2×#6706C2212yescomplemented
42#91099C2212yescomplemented
43C8×Dic15C2212yescomplemented
44#83939—51nomaximal, Hall
45C2×C5×AGL(1,ℤ/8)D311yescharacteristic, complemented
46#66159—31no—
47C2×#4096—31no—
48#66139—31no—
49C22×#687—31no—
50C2×#4055—31no—
51#65464—32no—
52#65456—32no—
53#65444—32no—
54#65443—32no—
55#65411—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×D4C2311yescharacteristic
65C2×C3×C5×D4C2312yes—
66C22×C3×C4×C5—21no—
67C22×C3×C4×C5C2311yescharacteristic
68C2×#1220—21no—
69C2×#1220—24no—
70C22×Dic15—21no—
71#6817—22no—
72#6817—22no—
73C2×D60C2311yescharacteristic
74C2×C4×D15—21no—
75C2×Dic30—22no—
76C2×Dic30—22no—
77C2×Dic30—21no—
78C2×Dic30C2312yes—
79C2×Dic30C2311yescharacteristic
80C3×C5×SD16—24no—
81C3×C5×D8—24no—
82C3×C5×M16D414yescomplemented
83C2×C3×C5×C8C2312yes—
84#6719—24no—
85#6719—24no—
86#6719D412yes—
87#6718—22no—
88#6717—22no—
89#6716—24no—
90#6716—24no—
91#6716—22no—
92#6715—22no—
93C4⋊Dic15D412yes—
94#6712—24no—
95C4×Dic15C2312yes—
96C2×C3⋊#168D414yescomplemented
97C2×C3⋊#168C2311yescharacteristic
98C2×C3⋊#168C2311yescharacteristic
99Dic60—24no—
100#6706—24no—
101C2×C3×AGL(1,ℤ/8)D511yescharacteristic, complemented
102#5825—51no—
103C2×#950—51no—
104#5805—51no—
105C22×#214—51no—
106C2×#909—51no—
107#5175—52no—
108#5167—52no—
109#5155—52no—
110#5154—52no—
111#5122—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
120#4099—62no—
121#4096—64no—
122#4090—32no—
123C2×#689—31no—
124C2×#688—31no—
125#4086—32no—
126#4084—62no—
127#4082—32no—
128#4082—32no—
129#4081—62no—
130C2×#687—32no—
131C2×#687—32no—
132C2×#687—31no—
133C2×#687—64no—
134C2×#686—31no—
135#4072—31no—
136#4070—62no—
137C22×#168—31no—
138#4055—64no—
139C2×Dic20—32no—
140C2×#677—32no—
141#3947—32no—
142C8×Dic5—32no—
143#3154—151noSylow
144C23×C3×C5—21no—
145C23×C3×C5—22no—
146C23×C3×C5—42no—
147C23×C3×C5—82no—
148C23×C3×C5C2×D411yescharacteristic
149C22×D15—21no—
150C3×C5×Q8—21no—
151C3×C5×Q8—21no—
152C3×C5×D4—21no—
153C3×C5×D4—22no—
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—42no—
165#1220—84no—
166C2×Dic15—22no—
167C2×Dic15—21no—
168C2×Dic15—42no—
169C2×Dic15C2×D414yes—
170D60—21no—
171D60—21no—
172C4×D15—42no—
173Dic30—22no—
174Dic30—21no—
175Dic30—22no—
176Dic30—21no—
177Dic30—42no—
178C3×C5×C8C2×D414yes—
179C3⋊#168C2×D412yes—
180C3⋊#168C2×D412yes—
181C2×C3×#41C2×D511yescharacteristic
182C22×C3×D4C2×D511yescharacteristic
183C2×#235—51no—
184C22×Dic6—51no—
185C3×AGL(1,ℤ/8)—24no—
186C2×C3×SD16C2×D512yes—
187C2×C3×D8C2×D512yes—
188C2×C3×M16C2×D511yescharacteristic
189#953—102no—
190#950—104no—
191#944—52no—
192C2×#216—51no—
193C2×#215—51no—
194#940—52no—
195#938—102no—
196#936—52no—
197#936—52no—
198#935—102no—
199C2×#214—52no—
200C2×#214—51no—
201C2×#214—52no—
202C2×#214—104no—
203C2×#213—51no—
204#926—51no—
205#924—102no—
206C22×C3⋊C8—51no—
207#909—104no—
208C2×Dic12—52no—
209C2×#204—52no—
210#815—52no—
211C8×Dic3—52no—
212C24×C5—21no—
213C5×#41—22no—
214C5×#41—22no—
215C2×C5×Q8C22×D311yescharacteristic
216C2×C5×D4—21no—
217C2×C5×D4—22no—
218C2×C5×D4—21no—
219C2×C5×D4—22no—
220C2×C5×D4C22×D311yescharacteristic
221C2×C5×D4C22×D312yes—
222C22×C4×C5—21no—
223C22×C4×C5C22×D311yescharacteristic
224C2×C22⋊D5—64no—
225C2×C22⋊D5—61no—
226C22×Dic5—61no—
227#709—62no—
228#709—62no—
229C2×D20—31no—
230C2×C4×D5—61no—
231C2×Dic10—32no—
232C2×Dic10—31no—
233C2×Dic10—62no—
234C2×Dic10—62no—
235C2×Dic10—61no—
236C5×SD16—24no—
237C5×D8—24no—
238C5×M16#6714yescomplemented
239C2×C5×C8C22×D312yes—
240C22⋊Dic5—32no—
241C22⋊Dic5—64no—
242C22⋊Dic5—64no—
243#689—62no—
244#688—62no—
245#687—62no—
246#687—64no—
247#687—64no—
248#686—62no—
249C4⋊Dic5—32no—
250#683—64no—
251C4×Dic5—32no—
252C2×#168—34no—
253C2×#168—31no—
254C2×#168—31no—
255Dic20—64no—
256#677—64no—
257C2×#137—151no—
258C2×AGL(1,ℤ/8)D1511yescharacteristic, complemented
259C2×#135—151no—
260C22×SD16—151no—
261#536—151no—
262#535—151no—
263#497—152no—
264#496—152no—
265#495—152no—
266#492—152no—
267#405—151no—
268C22×C3×C5—22no—
269C22×C3×C5—41no—
270C22×C3×C5—42no—
271C22×C3×C5—44no—
272C22×C3×C5—42no—
273C22×C3×C5—84no—
274C22×C3×C5#12712yes—
275C22×C3×C5C22×D411yescharacteristic
276C2×D15—41no—
277C2×D15—42no—
278C3×C4×C5—41no—
279C3×C4×C5#12712yes—
280C3×C4×C5C22×D411yescharacteristic
281C3×C4×C5C22×D411yescharacteristic
282Dic15—24no—
283Dic15—42no—
284Dic15—41no—
285C24×C3—21no—
286C3×#41—22no—
287C3×#41—22no—
288C2×C3×Q8C22×D511yescharacteristic
289C2×C3×D4—21no—
290C2×C3×D4—22no—
291C2×C3×D4—21no—
292C2×C3×D4—22no—
293C2×C3×D4C22×D511yescharacteristic
294C2×C3×D4C22×D512yes—
295C22×C3×C4—21no—
296C22×C3×C4C22×D511yescharacteristic
297C2×#67—104no—
298C2×#67—101no—
299C22×Dic3—101no—
300#235—102no—
301#235—102no—
302C2×D12—51no—
303C2×C4×D3—101no—
304C2×Dic6—52no—
305C2×Dic6—51no—
306C2×Dic6—102no—
307C2×Dic6—102no—
308C2×Dic6—101no—
309C3×SD16—24no—
310C3×D8—24no—
311C3×M16C22⋊D514yescomplemented
312C2×C3×C8C22×D512yes—
313#217—52no—
314#217—104no—
315#217—104no—
316#216—102no—
317#215—102no—
318#214—104no—
319#214—104no—
320#214—102no—
321#213—102no—
322C4⋊Dic3—52no—
323Dic3⋊C4—104no—
324C4×Dic3—52no—
325C2×C3⋊C8—54no—
326C2×C3⋊C8—51no—
327C2×C3⋊C8—51no—
328Dic12—104no—
329#204—104no—
330C23×C5—21no—
331C23×C5—22no—
332C23×C5—42no—
333C23×C5—82no—
334C23×C5D3×D411yescharacteristic
335C22×D5—61no—
336C5×Q8—21no—
337C5×Q8—21no—
338C5×D4—21no—
339C5×D4—22no—
340C5×D4—21no—
341C5×D4—22no—
342C5×D4—42no—
343C5×D4—42no—
344C2×C4×C5—21no—
345C2×C4×C5—42no—
346C2×C4×C5D3×D411yescharacteristic
347C2×C4×C5D3×D412yes—
348C2×C4×C5C2×#6712yes—
349C2×C4×C5C23×D311yescharacteristic
350C22⋊D5—122no—
351C22⋊D5—244no—
352C2×Dic5—34no—
353C2×Dic5—62no—
354C2×Dic5—61no—
355C2×Dic5—122no—
356D20—61no—
357D20—61no—
358C4×D5—122no—
359Dic10—62no—
360Dic10—61no—
361Dic10—62no—
362Dic10—61no—
363Dic10—122no—
364C5×C8C2×#6714yes—
365#168—32no—
366#168—32no—
367C2×#41—151no—
368C2×#41C2×D1511yescharacteristic
369C22×Q8—151no—
370C22×D4C2×D1511yescharacteristic
371#137—304no—
372AGL(1,ℤ/8)—24no—
373#135—304no—
374C2×Q16—151no—
375C2×Q16—152no—
376C2×SD16—151no—
377C2×SD16—152no—
378C2×SD16—152no—
379C2×SD16—151no—
380C2×SD16—152no—
381C2×SD16—304no—
382C2×SD16C2×D1512yes—
383C2×D8—151no—
384C2×D8C2×D1512yes—
385C2×M16C2×D1511yescharacteristic
386C22×C8—151no—
387C4⋊Q8—152no—
388#127—152no—
389#124—152no—
390#124—152no—
391#123—302no—
392C22⋊Q8—302no—
393#121—302no—
394#120—302no—
395#117—151no—
396#97—152no—
397C4×C8—152no—
398C2×C3×C5—22no—
399C2×C3×C5—81no—
400C2×C3×C5—82no—
401C2×C3×C5C2×#12711yescharacteristic
402C2×C3×C5#57712yes—
403D15—81no—
404C23×C3—21no—
405C23×C3—22no—
406C23×C3—42no—
407C23×C3—82no—
408C23×C3D4×D511yescharacteristic
409C22×D3—101no—
410C3×Q8—21no—
411C3×Q8—21no—
412C3×D4—21no—
413C3×D4—22no—
414C3×D4—21no—
415C3×D4—22no—
416C3×D4—42no—
417C3×D4—42no—
418C2×C3×C4—21no—
419C2×C3×C4—42no—
420C2×C3×C4D4×D512yes—
421C2×C3×C4D4×D511yescharacteristic
422C2×C3×C4C2×C22⋊D512yes—
423C2×C3×C4C23×D511yescharacteristic
424#67—202no—
425#67—404no—
426C2×Dic3—54no—
427C2×Dic3—102no—
428C2×Dic3—101no—
429C2×Dic3—202no—
430D12—101no—
431D12—101no—
432C4×D3—202no—
433Dic6—102no—
434Dic6—101no—
435Dic6—102no—
436Dic6—101no—
437Dic6—202no—
438C3×C8C2×C22⋊D514yes—
439C3⋊C8—52no—
440C3⋊C8—52no—
441C22×C5—22no—
442C22×C5—44no—
443C22×C5—42no—
444C22×C5—41no—
445C22×C5—42no—
446C22×C5—84no—
447C22×C5#94012yes—
448C22×C5C2×D3×D411yescharacteristic
449C2×D5—122no—
450C2×D5—121no—
451C4×C5—41no—
452C4×C5#94012yes—
453C4×C5C2×D3×D411yescharacteristic
454C4×C5C22×#6711yescharacteristic
455Dic5—64no—
456Dic5—122no—
457Dic5—121no—
458C24—21no—
459#41—22no—
460#41—22no—
461#41—302no—
462#41—302no—
463C2×Q8—151no—
464C2×Q8—152no—
465C2×Q8—301no—
466C2×Q8—302no—
467C2×Q8—302no—
468C2×Q8C22×D1511yescharacteristic
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—151no—
474C2×D4—304no—
475C2×D4—301no—
476C2×D4C22×D1511yescharacteristic
477C2×D4C22×D1512yes—
478C22×C4—21no—
479C22×C4—301no—
480C22×C4—301no—
481C22×C4C22×D1511yescharacteristic
482Q16—302no—
483Q16—304no—
484SD16—24no—
485SD16—304no—
486SD16—304no—
487SD16—302no—
488SD16—302no—
489SD16—304no—
490D8—24no—
491D8—302no—
492M16#122014yescomplemented
493C2×C8—154no—
494C2×C8—151no—
495C2×C8—151no—
496C2×C8C22×D1512yes—
497C4⋊C4—152no—
498C4⋊C4—304no—
499C22⋊C4—152no—
500C22⋊C4—304no—
501C22⋊C4—304no—
502C42—152no—
503C3×C5#315411yescharacteristic, complemented, Hall
504C22×C3—22no—
505C22×C3—44no—
506C22×C3—42no—
507C22×C3—41no—
508C22×C3—42no—
509C22×C3—84no—
510C22×C3#408612yes—
511C22×C3C2×D4×D511yescharacteristic
512C2×D3—202no—
513C2×D3—201no—
514C3×C4—41no—
515C3×C4#408612yes—
516C3×C4C2×D4×D511yescharacteristic
517C3×C4C22×C22⋊D511yescharacteristic
518Dic3—104no—
519Dic3—202no—
520Dic3—201no—
521C2×C5—22no—
522C2×C5—82no—
523C2×C5—81no—
524C2×C5#577512yes—
525C2×C5C2×#94011yescharacteristic
526D5—241no—
527C23—21no—
528C23—22no—
529C23—42no—
530C23—82no—
531C23—301no—
532C23D4×D1511yescharacteristic
533Q8—21no—
534Q8—21no—
535Q8—302no—
536Q8—301no—
537Q8—302no—
538Q8—301no—
539Q8—602no—
540D4—21no—
541D4—22no—
542D4—21no—
543D4—22no—
544D4—42no—
545D4—42no—
546D4—301no—
547D4—301no—
548D4—602no—
549D4—1204no—
550C2×C4—21no—
551C2×C4—42no—
552C2×C4—154no—
553C2×C4—301no—
554C2×C4—302no—
555C2×C4—602no—
556C2×C4—602no—
557C2×C4D4×D1511yescharacteristic
558C2×C4D4×D1512yes—
559C2×C4C2×#122012yes—
560C2×C4C23×D1511yescharacteristic
561C8—152no—
562C8—152no—
563C8C2×#122014yes—
564C2×C3—22no—
565C2×C3—82no—
566C2×C3—81no—
567C2×C3#6610912yes—
568C2×C3C2×#408611yescharacteristic
569D3—401no—
570C5#8393911yescharacteristic, complemented, Sylow
571C22—22no—
572C22—41no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—84no—
577C22—601no—
578C22—602no—
579C22#9182912yes—
580C22C2×D4×D1511yescharacteristic
581C4—41no—
582C4—304no—
583C4—601no—
584C4—602no—
585C4#9182912yes—
586C4C2×D4×D1511yescharacteristic
587C4C22×#122011yescharacteristic
588C3#1061567011yescharacteristic, complemented, Sylow
589C2—22no—
590C2—82no—
591C2—81no—
592C2—1201no—
593C2#1175653412yes—
594C2C2×#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#3154
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. #11756584C2semidirect
3. C2×#91839C2semidirect
4. #11756564C2semidirect
5. C22×#6716C2semidirect
6. C2×#91798C2semidirect
7. #11751276C2semidirect
8. #11751268C2semidirect
9. #11751256C2semidirect
10. #11751255C2semidirect
11. #11751223C2semidirect
23. #91833C22semidirect
24. C2×#6718C22semidirect
25. C2×#6717C22semidirect
29. #91825C22semidirect
34. C2×#6716C22semidirect
35. C2×#6715C22semidirect
40. C2×Dic60C22semidirect
41. C2×#6706C22semidirect
42. #91099C22semidirect
43. C8×Dic15C22semidirect
45. C2×C5×AGL(1,ℤ/8)D3semidirect
82. C3×C5×M16D4semidirect
96. C2×C3⋊#168D4semidirect
101. C2×C3×AGL(1,ℤ/8)D5semidirect
238. C5×M16#67semidirect
258. C2×AGL(1,ℤ/8)D15semidirect
311. C3×M16C22⋊D5semidirect
492. M16#1220semidirect
503. C3×C5#3154semidirect
570. C5#83939semidirect
588. C3#10615670semidirect

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
#117565842
C2×#918392
#117565642
C22×#67162
C2×#917982
#117512762
#117512682
#117512562
#117512552
#117512232
#106156703
#839395