Finite Group Databasebeta

← #14439401

Subgroups — #14439401

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
1C3×C5×#536C211yesmaximal, characteristic, complemented
2C2×#91839C211yesmaximal, characteristic, complemented
3C2×#91838C211yesmaximal, characteristic, complemented
4C2×#91831C211yesmaximal, characteristic, complemented
5C22×#6717C211yesmaximal, characteristic, complemented
6#11756284C211yesmaximal, characteristic, complemented
7#11751187C212yesmaximal, complemented
8#11751186C212yesmaximal, complemented
9#11751160C212yesmaximal, complemented
10#11751159C212yesmaximal, complemented
11#11751098C211yesmaximal, characteristic, complemented
12#10615133—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×C3×C5×Q8C2211yescharacteristic
15C2×#6817C2211yescharacteristic
16C22×D60C2211yescharacteristic
17C3×C5×C4⋊Q8C2212yescomplemented
18C3×C5×#124C2212yescomplemented
19C3×C5×#123—22no—
20C3×C5×C22⋊Q8—22no—
21C3×C5×#117C2211yescharacteristic
22#91839—24no—
23#91838—24no—
24C2×#6718C2211yescharacteristic, complemented
25C2×#6718C2212yescomplemented
26#91831—24no—
27C2×#6717—24no—
28C2×#6717C2212yes—
29C2×#6717C2212yes—
30C2×#6717C2212yes—
31C2×#6717C2211yescharacteristic, complemented
32C2×#6717C2212yescomplemented
33C2×#6716C2211yescharacteristic, complemented
34C2×#6715C2212yes—
35C2×#6715C2211yescharacteristic, complemented
36C2×#6710C2211yescharacteristic
37C22×C3⋊#168C2211yescharacteristic
38#91784—22no—
39#91771—22no—
40#91764C2212yes—
41#91763C2212yes—
42#91087C2212yescomplemented
43C4×C3⋊#168C2212yescomplemented
44#83402—51nomaximal, Hall
45C5×#536D311yescharacteristic, complemented
46C2×#4096—31no—
47C2×#4095—31no—
48C2×#4088—31no—
49C22×#688—31no—
50#65859—31no—
51#65375—32no—
52#65374—32no—
53#65348—32no—
54#65347—32no—
55#65286—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—
70#6817—22no—
71#6817—22no—
72C2×D60—22no—
73C2×D60—22no—
74C2×D60—21no—
75C2×D60—24no—
76C2×D60C2311yescharacteristic
77C2×D60C2312yes—
78C2×C4×D15—21no—
79C2×Dic30C2311yescharacteristic
80C3×C5×C4⋊C4—24no—
81C3×C5×C4⋊C4D412yes—
82C3×C5×C22⋊C4—24no—
83C3×C5×C22⋊C4D412yes—
84C3×C42×C5C2312yes—
85#6718—24no—
86#6718—22no—
87#6717—24no—
88#6717—24no—
89#6717—24no—
90#6717—24no—
91#6717—22no—
92#6716—22no—
93#6715—24no—
94#6715—22no—
95#6714—24no—
96#6710D414yescomplemented
97C2×C3⋊#168D414yescomplemented
98C2×C3⋊#168C2311yescharacteristic
99C2×C3⋊#168C2311yescharacteristic
100C2×C3⋊#168C2312yes—
101C3×#536D511yescharacteristic, complemented
102C2×#950—51no—
103C2×#949—51no—
104C2×#942—51no—
105C22×#215—51no—
106#5525—51no—
107#5086—52no—
108#5085—52no—
109#5059—52no—
110#5058—52no—
111#4997—51no—
112C2×C5×#41C2×D311yescharacteristic
113C22×C5×Q8C2×D311yescharacteristic
114C2×#709—31no—
115C22×D20—31no—
116C5×C4⋊Q8C2×D312yescomplemented
117C5×#124C2×D312yescomplemented
118C5×#123—22no—
119C5×C22⋊Q8—22no—
120C5×#117C2×D311yescharacteristic
121#4096—64no—
122#4095—64no—
123C2×#689—32no—
124C2×#689—31no—
125#4088—64no—
126C2×#688—32no—
127C2×#688—32no—
128C2×#688—32no—
129C2×#688—31no—
130C2×#688—32no—
131C2×#688—64no—
132C2×#687—31no—
133C2×#686—31no—
134C2×#686—32no—
135C2×#681—31no—
136C22×#168—31no—
137#4041—62no—
138#4028—62no—
139#4021—32no—
140#4020—32no—
141#3935—32no—
142C4×#168—32no—
143#3154—151noSylow
144C23×C3×C5—21no—
145C23×C3×C5C2×D411yescharacteristic
146C22×D15—22no—
147C22×D15—21no—
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×D414yes—
163C2×C3×C4×C5C2×D411yescharacteristic
164C2×C3×C4×C5C2×D412yes—
165C2×C3×C4×C5C2×D412yes—
166C2×C3×C4×C5C2411yescharacteristic
167#1220—42no—
168C2×Dic15—21no—
169D60—22no—
170D60—21no—
171D60—22no—
172D60—21no—
173D60—42no—
174D60—84no—
175C4×D15—42no—
176Dic30—21no—
177Dic30—21no—
178C3⋊#168C2×D412yes—
179C3⋊#168C2×D412yes—
180C3⋊#168C2×D414yes—
181C2×C3×#41C2×D511yescharacteristic
182C22×C3×Q8C2×D511yescharacteristic
183C2×#235—51no—
184C22×D12—51no—
185C3×C4⋊Q8C2×D512yescomplemented
186C3×#124C2×D512yescomplemented
187C3×#123—22no—
188C3×C22⋊Q8—22no—
189C3×#117C2×D511yescharacteristic
190#950—104no—
191#949—104no—
192C2×#216—52no—
193C2×#216—51no—
194#942—104no—
195C2×#215—52no—
196C2×#215—52no—
197C2×#215—52no—
198C2×#215—52no—
199C2×#215—51no—
200C2×#215—104no—
201C2×#214—51no—
202C2×#213—52no—
203C2×#213—51no—
204C2×#208—51no—
205C22×C3⋊C8—51no—
206#895—102no—
207#882—102no—
208#875—52no—
209#874—52no—
210#803—52no—
211C4×C3⋊C8—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—61no—
226#709—62no—
227#709—62no—
228C2×D20—32no—
229C2×D20—31no—
230C2×D20—62no—
231C2×D20—62no—
232C2×D20—61no—
233C2×D20—64no—
234C2×C4×D5—61no—
235C2×Dic10—31no—
236C5×C4⋊C4—24no—
237C5×C4⋊C4#6712yes—
238C5×C22⋊C4—24no—
239C5×C22⋊C4#6712yes—
240C42×C5C22×D312yes—
241#689—62no—
242#689—64no—
243#688—62no—
244#688—64no—
245#688—64no—
246#688—64no—
247#688—64no—
248#687—62no—
249#686—62no—
250#686—64no—
251#685—64no—
252#681—34no—
253C2×#168—34no—
254C2×#168—31no—
255C2×#168—32no—
256C2×#168—31no—
257C2×#137—151no—
258C2×AGL(1,ℤ/8)—151no—
259C2×#135—151no—
260C22×SD16—151no—
261#536D1511yescharacteristic, complemented
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—42no—
276C2×D15—44no—
277C2×D15—84no—
278C3×C4×C5—24no—
279C3×C4×C5—41no—
280C3×C4×C5—42no—
281C3×C4×C5#12712yes—
282C3×C4×C5C22×D411yescharacteristic
283C3×C4×C5C22×D411yescharacteristic
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—101no—
299#235—102no—
300#235—102no—
301C2×D12—51no—
302C2×D12—52no—
303C2×D12—102no—
304C2×D12—102no—
305C2×D12—101no—
306C2×D12—104no—
307C2×C4×D3—101no—
308C2×Dic6—51no—
309C3×C4⋊C4—24no—
310C3×C4⋊C4C22⋊D512yes—
311C3×C22⋊C4—24no—
312C3×C22⋊C4C22⋊D512yes—
313C3×C42C22×D512yes—
314#216—104no—
315#216—102no—
316#215—104no—
317#215—104no—
318#215—104no—
319#215—104no—
320#215—102no—
321#214—102no—
322#213—104no—
323#213—102no—
324(C2×D3)⋊C4—104no—
325#208—54no—
326C2×C3⋊C8—54no—
327C2×C3⋊C8—51no—
328C2×C3⋊C8—52no—
329C2×C3⋊C8—51no—
330C23×C5—21no—
331C23×C5C2×#6711yescharacteristic
332C22×D5—62no—
333C22×D5—61no—
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×D412yes—
349C2×C4×C5D3×D412yes—
350C2×C4×C5C2×#6714yes—
351C2×C4×C5C2×#6711yescharacteristic
352C2×C4×C5C23×D311yescharacteristic
353C22⋊D5—122no—
354C2×Dic5—61no—
355D20—62no—
356D20—61no—
357D20—62no—
358D20—61no—
359D20—122no—
360D20—244no—
361C4×D5—122no—
362Dic10—61no—
363Dic10—61no—
364#168—32no—
365#168—32no—
366#168—34no—
367C2×#41—151no—
368C2×#41C2×D1511yescharacteristic
369C22×Q8C2×D1511yescharacteristic
370C22×D4—151no—
371#137—304no—
372AGL(1,ℤ/8)—304no—
373#135—304no—
374C2×Q16—151no—
375C2×Q16—152no—
376C2×SD16—152no—
377C2×SD16—152no—
378C2×SD16—151no—
379C2×SD16—152no—
380C2×SD16—152no—
381C2×SD16—151no—
382C2×SD16—304no—
383C2×D8—151no—
384C2×D8—152no—
385C2×M16—151no—
386C22×C8—151no—
387C4⋊Q8C2×D1512yescomplemented
388#127—152no—
389#124—152no—
390#124C2×D1512yescomplemented
391#123—22no—
392C22⋊Q8—22no—
393#121—302no—
394#120—302no—
395#117C2×D1511yescharacteristic
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×C3C2×C22⋊D511yescharacteristic
406C22×D3—102no—
407C22×D3—101no—
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×D512yes—
424C2×C3×C4C2×C22⋊D514yes—
425C2×C3×C4C2×C22⋊D511yescharacteristic
426C2×C3×C4C23×D511yescharacteristic
427#67—202no—
428C2×Dic3—101no—
429D12—102no—
430D12—101no—
431D12—102no—
432D12—101no—
433D12—202no—
434D12—404no—
435C4×D3—202no—
436Dic6—101no—
437Dic6—101no—
438C3⋊C8—52no—
439C3⋊C8—52no—
440C3⋊C8—54no—
441C22×C5—22no—
442C22×C5—41no—
443C22×C5—42no—
444C22×C5#94012yes—
445C22×C5C22×#6711yescharacteristic
446C2×D5—121no—
447C2×D5—122no—
448C2×D5—122no—
449C2×D5—124no—
450C2×D5—244no—
451C4×C5—24no—
452C4×C5—41no—
453C4×C5—42no—
454C4×C5#94012yes—
455C4×C5C2×D3×D411yescharacteristic
456C4×C5C2×D3×D411yescharacteristic
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×D1512yes—
468C2×Q8C22×D1511yescharacteristic
469C2×D4—21no—
470C2×D4—152no—
471C2×D4—151no—
472C2×D4—301no—
473C2×D4—302no—
474C2×D4—302no—
475C2×D4—301no—
476C2×D4—304no—
477C2×D4C22×D1511yescharacteristic
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—301no—
481C22×C4C22×D1511yescharacteristic
482Q16—302no—
483Q16—304no—
484SD16—302no—
485SD16—304no—
486SD16—304no—
487SD16—302no—
488SD16—304no—
489SD16—304no—
490D8—302no—
491D8—304no—
492M16—154no—
493C2×C8—154no—
494C2×C8—151no—
495C2×C8—152no—
496C2×C8—151no—
497C4⋊C4—24no—
498C4⋊C4#122012yes—
499C22⋊C4—24no—
500C22⋊C4—304no—
501C22⋊C4#122012yes—
502C42C22×D1512yes—
503C3×C5#315411yescharacteristic, complemented, Hall
504C22×C3—22no—
505C22×C3—41no—
506C22×C3—42no—
507C22×C3#408612yes—
508C22×C3C22×C22⋊D511yescharacteristic
509C2×D3—201no—
510C2×D3—202no—
511C2×D3—202no—
512C2×D3—204no—
513C2×D3—404no—
514C3×C4—24no—
515C3×C4—41no—
516C3×C4—42no—
517C3×C4#408612yes—
518C3×C4C2×D4×D511yescharacteristic
519C3×C4C2×D4×D511yescharacteristic
520Dic3—201no—
521C2×C5—22no—
522C2×C5—81no—
523C2×C5C2×#94011yescharacteristic
524C2×C5#583712yes—
525D5—242no—
526D5—241no—
527C23—21no—
528C23—302no—
529C23—301no—
530C23—602no—
531C23—1202no—
532C23C2×#122011yescharacteristic
533Q8—22no—
534Q8—21no—
535Q8—22no—
536Q8—21no—
537Q8—42no—
538Q8—301no—
539Q8—301no—
540D4—21no—
541D4—21no—
542D4—42no—
543D4—302no—
544D4—301no—
545D4—302no—
546D4—301no—
547D4—602no—
548D4—602no—
549D4—1204no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—301no—
555C2×C4—602no—
556C2×C4D4×D1512yes—
557C2×C4D4×D1512yes—
558C2×C4C2×#122014yes—
559C2×C4C2×#122011yescharacteristic
560C2×C4C23×D1511yescharacteristic
561C8—152no—
562C8—152no—
563C8—154no—
564C2×C3—22no—
565C2×C3—81no—
566C2×C3C2×#408611yescharacteristic
567C2×C3#6617112yes—
568D3—402no—
569D3—401no—
570C5#8340211yescharacteristic, complemented, Sylow
571C22—22no—
572C22—41no—
573C22—42no—
574C22—601no—
575C22—602no—
576C22—602no—
577C22—604no—
578C22—1204no—
579C22#9182912yes—
580C22C22×#122011yescharacteristic
581C4—24no—
582C4—41no—
583C4—42no—
584C4—601no—
585C4#9182912yes—
586C4C2×D4×D1511yescharacteristic
587C4C2×D4×D1511yescharacteristic
588C3#1061513311yescharacteristic, complemented, Sylow
589C2—22no—
590C2—81no—
591C2—1202no—
592C2—1201no—
593C2C2×#9182911yescharacteristic
594C2#1175659612yes—
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#536C2
socleC22×C3×C5C22×D4
hypercenter#536D15
hypocenterC3×C5#3154
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C3×C5×#536C2semidirect
2. C2×#91839C2semidirect
3. C2×#91838C2semidirect
4. C2×#91831C2semidirect
5. C22×#6717C2semidirect
6. #11756284C2semidirect
7. #11751187C2semidirect
8. #11751186C2semidirect
9. #11751160C2semidirect
10. #11751159C2semidirect
11. #11751098C2semidirect
17. C3×C5×C4⋊Q8C22semidirect
18. C3×C5×#124C22semidirect
24. C2×#6718C22semidirect
25. C2×#6718C22semidirect
31. C2×#6717C22semidirect
32. C2×#6717C22semidirect
33. C2×#6716C22semidirect
35. C2×#6715C22semidirect
42. #91087C22semidirect
43. C4×C3⋊#168C22semidirect
45. C5×#536D3semidirect
96. #6710D4semidirect
97. C2×C3⋊#168D4semidirect
101. C3×#536D5semidirect
116. C5×C4⋊Q8C2×D3semidirect
117. C5×#124C2×D3semidirect
185. C3×C4⋊Q8C2×D5semidirect
186. C3×#124C2×D5semidirect
261. #536D15semidirect
387. C4⋊Q8C2×D15semidirect
390. #124C2×D15semidirect
503. C3×C5#3154semidirect
570. C5#83402semidirect
588. C3#10615133semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5362
C2×#918392
C2×#918382
C2×#918312
C22×#67172
#117562842
#117511872
#117511862
#117511602
#117511592
#117510982
#106151333
#834025