Finite Group Databasebeta

← #14439404

Subgroups — #14439404

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×#137C211yesmaximal, characteristic, complemented
2#11756589C211yesmaximal, characteristic, complemented
3C2×#91839C211yesmaximal, characteristic, complemented
4#11756575C211yesmaximal, characteristic, complemented
5C22×#6717C211yesmaximal, characteristic, complemented
6C2×#91797C211yesmaximal, characteristic, complemented
7#11751292C212yesmaximal, complemented
8#11751291C212yesmaximal, complemented
9#11751277C212yesmaximal, complemented
10#11751276C212yesmaximal, complemented
11#11751223C211yesmaximal, characteristic, complemented
12#10615705—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×C3×C5×Q8C2211yescharacteristic
15C2×#6817C2211yescharacteristic
16C22×D60C2211yescharacteristic
17C3×C5×#137—24no—
18C2×C3×C5×Q16C2212yescomplemented
19C2×C3×C5×SD16C2212yescomplemented
20C2×C3×C5×M16C2211yescharacteristic
21#91839—24no—
22#91836C2212yes—
23#91836C2212yescomplemented
24#91835—22no—
25#91833C2212yescomplemented
26C2×#6718C2211yescharacteristic, complemented
27C2×#6717—24no—
28C2×#6717C2212yes—
29C2×#6717C2212yes—
30C2×#6717C2211yescharacteristic, complemented
31#91829C2212yes—
32#91826—22no—
33C2×#6716C2211yescharacteristic, complemented
34C2×#6715C2211yescharacteristic, complemented
35#91819—22no—
36#91818—22no—
37#91815C2211yescharacteristic
38C22×C3⋊#168C2211yescharacteristic
39#91797—24no—
40C2×D120C2212yes—
41C2×#6706C2212yes—
42#91099C2212yescomplemented
43C8×Dic15C2212yescomplemented
44#83974—51nomaximal, Hall
45C2×C5×#137D311yescharacteristic, complemented
46#66164—31no—
47C2×#4096—31no—
48#66150—31no—
49C22×#688—31no—
50C2×#4054—31no—
51#65480—32no—
52#65479—32no—
53#65465—32no—
54#65464—32no—
55#65411—31no—
56C23×D15—21no—
57C3×C5×#41—22no—
58C3×C5×#41—22no—
59C2×C3×C5×Q8—21no—
60C2×C3×C5×Q8—22no—
61C2×C3×C5×Q8—22no—
62C2×C3×C5×Q8C2312yes—
63C2×C3×C5×Q8C2311yescharacteristic
64C2×C3×C5×D4—21no—
65C2×C3×C5×D4C2311yescharacteristic
66C22×C3×C4×C5—21no—
67C22×C3×C4×C5—21no—
68C22×C3×C4×C5C2311yescharacteristic
69C2×#1220—21no—
70C2×#1220—24no—
71#6817—22no—
72#6817—22no—
73C2×D60—22no—
74C2×D60—22no—
75C2×D60—21no—
76C2×D60C2312yes—
77C2×D60C2311yescharacteristic
78C2×C4×D15—21no—
79C2×Dic30C2311yescharacteristic
80C3×C5×Q16—24no—
81C3×C5×SD16—24no—
82C3×C5×M16D414yescomplemented
83C2×C3×C5×C8C2312yes—
84#6719D412yes—
85#6718—22no—
86#6717—24no—
87#6717—24no—
88#6717—22no—
89#6716—22no—
90#6715—22no—
91#6714—24no—
92#6714—24no—
93C4⋊Dic15D412yes—
94#6712—24no—
95C4×Dic15C2312yes—
96C2×C3⋊#168D414yescomplemented
97C2×C3⋊#168C2311yescharacteristic
98C2×C3⋊#168C2311yescharacteristic
99D120—24no—
100#6706—24no—
101C2×C3×#137D511yescharacteristic, complemented
102#5830—51no—
103C2×#950—51no—
104#5816—51no—
105C22×#215—51no—
106C2×#908—51no—
107#5191—52no—
108#5190—52no—
109#5176—52no—
110#5175—52no—
111#5122—51no—
112C2×C5×#41C2×D311yescharacteristic
113C22×C5×Q8C2×D311yescharacteristic
114C2×#709—31no—
115C22×D20—31no—
116C5×#137—24no—
117C2×C5×Q16C2×D312yescomplemented
118C2×C5×SD16C2×D312yescomplemented
119C2×C5×M16C2×D311yescharacteristic
120#4096—64no—
121#4093—32no—
122#4093—32no—
123#4092—62no—
124#4090—32no—
125C2×#689—31no—
126C2×#688—32no—
127C2×#688—32no—
128C2×#688—31no—
129C2×#688—64no—
130#4086—32no—
131#4083—62no—
132C2×#687—31no—
133C2×#686—31no—
134#4076—62no—
135#4075—62no—
136#4072—31no—
137C22×#168—31no—
138#4054—64no—
139C2×D40—32no—
140C2×#677—32no—
141#3947—32no—
142C8×Dic5—32no—
143#3154—151noSylow
144C23×C3×C5—21no—
145C23×C3×C5C2×D411yescharacteristic
146C22×D15—21no—
147C22×D15—22no—
148C22×D15—42no—
149C22×D15—82no—
150C3×C5×Q8—22no—
151C3×C5×Q8—21no—
152C3×C5×Q8—22no—
153C3×C5×Q8—21no—
154C3×C5×Q8—42no—
155C3×C5×D4—21no—
156C3×C5×D4—21no—
157C3×C5×D4—42no—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5—21no—
160C2×C3×C4×C5—42no—
161C2×C3×C4×C5—42no—
162C2×C3×C4×C5C2×D411yescharacteristic
163C2×C3×C4×C5C2×D412yes—
164C2×C3×C4×C5C2×D412yes—
165C2×C3×C4×C5C2411yescharacteristic
166#1220—42no—
167#1220—84no—
168C2×Dic15—21no—
169C2×Dic15C2×D414yes—
170D60—22no—
171D60—21no—
172D60—21no—
173D60—22no—
174D60—42no—
175C4×D15—42no—
176Dic30—21no—
177Dic30—21no—
178C3×C5×C8C2×D414yes—
179C3⋊#168C2×D412yes—
180C3⋊#168C2×D412yes—
181C2×C3×#41C2×D511yescharacteristic
182C22×C3×Q8C2×D511yescharacteristic
183C2×#235—51no—
184C22×D12—51no—
185C3×#137—24no—
186C2×C3×Q16C2×D512yescomplemented
187C2×C3×SD16C2×D512yescomplemented
188C2×C3×M16C2×D511yescharacteristic
189#950—104no—
190#947—52no—
191#947—52no—
192#946—102no—
193#944—52no—
194C2×#216—51no—
195C2×#215—52no—
196C2×#215—52no—
197C2×#215—51no—
198C2×#215—104no—
199#940—52no—
200#937—102no—
201C2×#214—51no—
202C2×#213—51no—
203#930—102no—
204#929—102no—
205#926—51no—
206C22×C3⋊C8—51no—
207#908—104no—
208C2×D24—52no—
209C2×#204—52no—
210#815—52no—
211C8×Dic3—52no—
212C23×D5—61no—
213C5×#41—22no—
214C5×#41—22no—
215C2×C5×Q8—21no—
216C2×C5×Q8—22no—
217C2×C5×Q8—22no—
218C2×C5×Q8C22×D312yes—
219C2×C5×Q8C22×D311yescharacteristic
220C2×C5×D4—21no—
221C2×C5×D4C22×D311yescharacteristic
222C22×C4×C5—21no—
223C22×C4×C5—21no—
224C22×C4×C5C22×D311yescharacteristic
225C2×C22⋊D5—64no—
226C2×C22⋊D5—61no—
227#709—62no—
228#709—62no—
229C2×D20—32no—
230C2×D20—31no—
231C2×D20—62no—
232C2×D20—62no—
233C2×D20—61no—
234C2×C4×D5—61no—
235C2×Dic10—31no—
236C5×Q16—24no—
237C5×SD16—24no—
238C5×M16#6714yescomplemented
239C2×C5×C8C22×D312yes—
240C22⋊Dic5—32no—
241#689—62no—
242#688—62no—
243#688—64no—
244#688—64no—
245#687—62no—
246#686—62no—
247#685—64no—
248#685—64no—
249C4⋊Dic5—32no—
250#683—64no—
251C4×Dic5—32no—
252C2×#168—34no—
253C2×#168—31no—
254C2×#168—31no—
255D40—64no—
256#677—64no—
257C2×#137D1511yescharacteristic, complemented
258C2×AGL(1,ℤ/8)—151no—
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#12712yes—
272C22×C3×C5C22×D411yescharacteristic
273C2×D15—41no—
274C2×D15—42no—
275C2×D15—44no—
276C2×D15—42no—
277C2×D15—84no—
278C3×C4×C5—41no—
279C3×C4×C5—42no—
280C3×C4×C5#12712yes—
281C3×C4×C5C22×D411yescharacteristic
282C3×C4×C5C22×D411yescharacteristic
283Dic15—24no—
284Dic15—41no—
285C23×D3—101no—
286C3×#41—22no—
287C3×#41—22no—
288C2×C3×Q8—21no—
289C2×C3×Q8—22no—
290C2×C3×Q8—22no—
291C2×C3×Q8C22×D512yes—
292C2×C3×Q8C22×D511yescharacteristic
293C2×C3×D4—21no—
294C2×C3×D4C22×D511yescharacteristic
295C22×C3×C4—21no—
296C22×C3×C4—21no—
297C22×C3×C4C22×D511yescharacteristic
298C2×#67—104no—
299C2×#67—101no—
300#235—102no—
301#235—102no—
302C2×D12—52no—
303C2×D12—51no—
304C2×D12—102no—
305C2×D12—102no—
306C2×D12—101no—
307C2×C4×D3—101no—
308C2×Dic6—51no—
309C3×Q16—24no—
310C3×SD16—24no—
311C3×M16C22⋊D514yescomplemented
312C2×C3×C8C22×D512yes—
313#217—52no—
314#216—102no—
315#215—104no—
316#215—104no—
317#215—102no—
318#214—102no—
319#213—102no—
320(C2×D3)⋊C4—104no—
321(C2×D3)⋊C4—104no—
322C4⋊Dic3—52no—
323Dic3⋊C4—104no—
324C4×Dic3—52no—
325C2×C3⋊C8—54no—
326C2×C3⋊C8—51no—
327C2×C3⋊C8—51no—
328D24—104no—
329#204—104no—
330C23×C5—21no—
331C23×C5D3×D411yescharacteristic
332C22×D5—61no—
333C22×D5—62no—
334C22×D5—122no—
335C22×D5—242no—
336C5×Q8—22no—
337C5×Q8—21no—
338C5×Q8—22no—
339C5×Q8—21no—
340C5×Q8—42no—
341C5×D4—21no—
342C5×D4—21no—
343C5×D4—42no—
344C2×C4×C5—22no—
345C2×C4×C5—21no—
346C2×C4×C5—42no—
347C2×C4×C5—42no—
348C2×C4×C5D3×D411yescharacteristic
349C2×C4×C5D3×D412yes—
350C2×C4×C5C2×#6712yes—
351C2×C4×C5C23×D311yescharacteristic
352C22⋊D5—122no—
353C22⋊D5—244no—
354C2×Dic5—34no—
355C2×Dic5—61no—
356D20—61no—
357D20—62no—
358D20—62no—
359D20—61no—
360D20—122no—
361C4×D5—122no—
362Dic10—61no—
363Dic10—61no—
364C5×C8C2×#6714yes—
365#168—32no—
366#168—32no—
367C2×#41—151no—
368C2×#41C2×D1511yescharacteristic
369C22×Q8C2×D1511yescharacteristic
370C22×D4—151no—
371#137—24no—
372AGL(1,ℤ/8)—304no—
373#135—304no—
374C2×Q16—151no—
375C2×Q16C2×D1512yescomplemented
376C2×SD16—152no—
377C2×SD16—152no—
378C2×SD16—151no—
379C2×SD16—151no—
380C2×SD16—152no—
381C2×SD16—304no—
382C2×SD16C2×D1512yescomplemented
383C2×D8—151no—
384C2×D8—152no—
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×C5C2×#12711yescharacteristic
401C2×C3×C5#57712yes—
402D15—81no—
403D15—82no—
404C23×C3—21no—
405C23×C3D4×D511yescharacteristic
406C22×D3—101no—
407C22×D3—102no—
408C22×D3—202no—
409C22×D3—402no—
410C3×Q8—22no—
411C3×Q8—21no—
412C3×Q8—22no—
413C3×Q8—21no—
414C3×Q8—42no—
415C3×D4—21no—
416C3×D4—21no—
417C3×D4—42no—
418C2×C3×C4—22no—
419C2×C3×C4—21no—
420C2×C3×C4—42no—
421C2×C3×C4—42no—
422C2×C3×C4D4×D512yes—
423C2×C3×C4D4×D511yescharacteristic
424C2×C3×C4C2×C22⋊D512yes—
425C2×C3×C4C23×D511yescharacteristic
426#67—202no—
427#67—404no—
428C2×Dic3—54no—
429C2×Dic3—101no—
430D12—101no—
431D12—102no—
432D12—102no—
433D12—101no—
434D12—202no—
435C4×D3—202no—
436Dic6—101no—
437Dic6—101no—
438C3×C8C2×C22⋊D514yes—
439C3⋊C8—52no—
440C3⋊C8—52no—
441C22×C5—22no—
442C22×C5—42no—
443C22×C5—41no—
444C22×C5#94012yes—
445C22×C5C2×D3×D411yescharacteristic
446C2×D5—124no—
447C2×D5—122no—
448C2×D5—121no—
449C2×D5—122no—
450C2×D5—244no—
451C4×C5—41no—
452C4×C5—42no—
453C4×C5#94012yes—
454C4×C5C2×D3×D411yescharacteristic
455C4×C5C22×#6711yescharacteristic
456Dic5—64no—
457Dic5—121no—
458C24—301no—
459#41—22no—
460#41—22no—
461#41—302no—
462#41—302no—
463C2×Q8—21no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—151no—
467C2×Q8C22×D1511yescharacteristic
468C2×Q8C22×D1512yes—
469C2×D4—21no—
470C2×D4—151no—
471C2×D4—152no—
472C2×D4—304no—
473C2×D4—301no—
474C2×D4—302no—
475C2×D4—301no—
476C2×D4—302no—
477C2×D4C22×D1511yescharacteristic
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—301no—
481C22×C4C22×D1511yescharacteristic
482Q16—24no—
483Q16—302no—
484SD16—24no—
485SD16—302no—
486SD16—304no—
487SD16—304no—
488SD16—302no—
489SD16—304no—
490D8—302no—
491D8—304no—
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—42no—
506C22×C3—41no—
507C22×C3#408612yes—
508C22×C3C2×D4×D511yescharacteristic
509C2×D3—204no—
510C2×D3—202no—
511C2×D3—201no—
512C2×D3—202no—
513C2×D3—404no—
514C3×C4—41no—
515C3×C4—42no—
516C3×C4#408612yes—
517C3×C4C2×D4×D511yescharacteristic
518C3×C4C22×C22⋊D511yescharacteristic
519Dic3—104no—
520Dic3—201no—
521C2×C5—22no—
522C2×C5—81no—
523C2×C5#577812yes—
524C2×C5C2×#94011yescharacteristic
525D5—242no—
526D5—241no—
527C23—21no—
528C23—301no—
529C23—302no—
530C23—602no—
531C23—1202no—
532C23D4×D1511yescharacteristic
533Q8—22no—
534Q8—21no—
535Q8—22no—
536Q8—21no—
537Q8—42no—
538Q8—301no—
539Q8—301no—
540D4—21no—
541D4—21no—
542D4—42no—
543D4—301no—
544D4—302no—
545D4—301no—
546D4—302no—
547D4—602no—
548D4—602no—
549D4—1204no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—154no—
555C2×C4—301no—
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—81no—
566C2×C3#6611212yes—
567C2×C3C2×#408611yescharacteristic
568D3—402no—
569D3—401no—
570C5#8397411yescharacteristic, complemented, Sylow
571C22—22no—
572C22—41no—
573C22—42no—
574C22—601no—
575C22—602no—
576C22—604no—
577C22—602no—
578C22—1204no—
579C22#9182912yes—
580C22C2×D4×D1511yescharacteristic
581C4—41no—
582C4—42no—
583C4—304no—
584C4—601no—
585C4#9182912yes—
586C4C2×D4×D1511yescharacteristic
587C4C22×#122011yescharacteristic
588C3#1061570511yescharacteristic, complemented, Sylow
589C2—22no—
590C2—81no—
591C2—1202no—
592C2—1201no—
593C2#1175653712yes—
594C2C2×#9182911yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×#137C2
socleC22×C3×C5C22×D4
hypercenterC2×#137D15
hypocenterC3×C5#3154
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#137C2semidirect
2. #11756589C2semidirect
3. C2×#91839C2semidirect
4. #11756575C2semidirect
5. C22×#6717C2semidirect
6. C2×#91797C2semidirect
7. #11751292C2semidirect
8. #11751291C2semidirect
9. #11751277C2semidirect
10. #11751276C2semidirect
11. #11751223C2semidirect
18. C2×C3×C5×Q16C22semidirect
19. C2×C3×C5×SD16C22semidirect
23. #91836C22semidirect
25. #91833C22semidirect
26. C2×#6718C22semidirect
30. C2×#6717C22semidirect
33. C2×#6716C22semidirect
34. C2×#6715C22semidirect
42. #91099C22semidirect
43. C8×Dic15C22semidirect
45. C2×C5×#137D3semidirect
82. C3×C5×M16D4semidirect
96. C2×C3⋊#168D4semidirect
101. C2×C3×#137D5semidirect
117. C2×C5×Q16C2×D3semidirect
118. C2×C5×SD16C2×D3semidirect
186. C2×C3×Q16C2×D5semidirect
187. C2×C3×SD16C2×D5semidirect
238. C5×M16#67semidirect
257. C2×#137D15semidirect
311. C3×M16C22⋊D5semidirect
375. C2×Q16C2×D15semidirect
382. C2×SD16C2×D15semidirect
492. M16#1220semidirect
503. C3×C5#3154semidirect
570. C5#83974semidirect
588. C3#10615705semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1372
#117565892
C2×#918392
#117565752
C22×#67172
C2×#917972
#117512922
#117512912
#117512772
#117512762
#117512232
#106157053
#839745