Finite Group Databasebeta

← #14424116

Subgroups — #14424116

Order 1920.

1508 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6817C211yesmaximal, characteristic
2C2×#91813C211yesmaximal, characteristic, complemented
3C2×#91798C212yesmaximal, complemented
4C3×C5×#407C211yesmaximal, characteristic, complemented
5#11751234C212yesmaximal, complemented
6#11750840C212yesmaximal, complemented
7#11750834C214yesmaximal, complemented
8#11750832C212yesmaximal, complemented
9#10610371—31nomaximal, Hall
10C23×C3×C4×C5C2211yescharacteristic
11C22×#1220—21no—
12C2×#6817—21no—
13C2×#6817—21no—
14C2×#6817—28no—
15C2×#6817C2212yes—
16C2×#6817C2212yes—
17C22×D60C2211yescharacteristic
18C22×C4×D15—21no—
19C22×Dic30C2211yescharacteristic
20C22×Dic30C2211yescharacteristic, complemented
21C2×C3×C5×M16C2212yescomplemented
22C2×#6719—21no—
23C2×C4⋊Dic15C2211yescharacteristic, complemented
24#91813—21no—
25#91813—21no—
26#91813C2212yes—
27#91813C2212yescomplemented
28C2×#6712—21no—
29#91798—28no—
30C2×Dic60C2214yescomplemented
31C2×#6706C2214yescomplemented
32C3×C5×C22⋊C8C2212yes—
33C3×C5×C22⋊C8C2212yescomplemented
34#91105C2214yescomplemented
35#91100C2214yescomplemented
36#78640—51nomaximal, Hall
37C22×#709—31no—
38C2×#4070—31no—
39C2×#4055—32no—
40C5×#407D311yescharacteristic, complemented
41#65422—32no—
42#65028—32no—
43#65022—34no—
44#65020—32no—
45C24×C3×C5D411yescharacteristic
46C23×D15—21no—
47C22×C3×C4×C5—24no—
48C22×C3×C4×C5D411yescharacteristic
49C22×C3×C4×C5D411yescharacteristic
50C22×C3×C4×C5D411yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×C4×C5C2311yescharacteristic
54C2×#1220—28no—
55C2×#1220—22no—
56C2×#1220—22no—
57C22×Dic15—21no—
58C22×Dic15—21no—
59C22×Dic15—21no—
60#6817—28no—
61#6817—28no—
62#6817—24no—
63#6817—24no—
64#6817—24no—
65#6817D418yes—
66C2×D60—21no—
67C2×D60—24no—
68C2×D60C2312yes—
69C2×C4×D15—28no—
70C2×C4×D15—22no—
71C2×C4×D15—22no—
72C2×Dic30—21no—
73C2×Dic30—24no—
74C2×Dic30—24no—
75C2×Dic30—21no—
76C2×Dic30C2312yes—
77C2×Dic30C2312yes—
78C3×C5×M16—24no—
79C2×C3×C5×C8C2314yes—
80#6719—22no—
81#6719—22no—
82C4⋊Dic15—21no—
83C4⋊Dic15C2312yes—
84#6712—22no—
85#6712—22no—
86Dic60—28no—
87#6706—28no—
88C22×#235—51no—
89C2×#924—51no—
90C2×#909—52no—
91C3×#407D511yescharacteristic, complemented
92#5133—52no—
93#4739—52no—
94#4733—54no—
95#4731—52no—
96C23×C4×C5C2×D311yescharacteristic
97C22×C22⋊D5—61no—
98C2×#709—32no—
99C2×#709—32no—
100C2×#709—68no—
101C2×#709—61no—
102C2×#709—61no—
103C22×D20—31no—
104C22×C4×D5—61no—
105C22×Dic10—31no—
106C22×Dic10—31no—
107C2×C5×M16C2×D312yescomplemented
108C2×C22⋊Dic5—61no—
109C2×C4⋊Dic5—31no—
110#4070—32no—
111#4070—32no—
112#4070—61no—
113#4070—61no—
114C2×#683—61no—
115#4055—68no—
116C2×Dic20—34no—
117C2×#677—34no—
118C5×C22⋊C8C2×D312yes—
119C5×C22⋊C8C2×D312yescomplemented
120#3953—34no—
121#3948—34no—
122#3007—151noSylow
123C23×C3×C5—22no—
124C23×C3×C5—22no—
125C23×C3×C5—41no—
126C23×C3×C5C2×D411yescharacteristic
127C23×C3×C5C2×D411yescharacteristic
128C23×C3×C5C2×D411yescharacteristic
129C22×D15—24no—
130C22×D15—22no—
131C22×D15—42no—
132C22×D15—42no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D414yes—
142C2×C3×C4×C5C2411yescharacteristic
143#1220—28no—
144#1220—28no—
145C2×Dic15—24no—
146C2×Dic15—22no—
147C2×Dic15—22no—
148C2×Dic15—22no—
149C2×Dic15—42no—
150C2×Dic15—42no—
151D60—24no—
152D60—22no—
153D60C2×D414yes—
154C4×D15—28no—
155C4×D15—28no—
156Dic30—24no—
157Dic30—22no—
158Dic30—24no—
159Dic30—42no—
160Dic30C2×D414yes—
161C3×C5×C8—24no—
162C23×C3×C4C2×D511yescharacteristic
163C22×#67—101no—
164C2×#235—52no—
165C2×#235—52no—
166C2×#235—108no—
167C2×#235—101no—
168C2×#235—101no—
169C22×D12—51no—
170C22×C4×D3—101no—
171C22×Dic6—51no—
172C22×Dic6—51no—
173C2×C3×M16C2×D512yescomplemented
174C2×#217—101no—
175C2×C4⋊Dic3—51no—
176#924—52no—
177#924—52no—
178#924—101no—
179#924—101no—
180C2×Dic3⋊C4—101no—
181#909—108no—
182C2×Dic12—54no—
183C2×#204—54no—
184C3×C22⋊C8C2×D512yes—
185C3×C22⋊C8C2×D512yescomplemented
186#821—54no—
187#816—54no—
188C24×C5D1211yescharacteristic
189C23×D5—61no—
190C22×C4×C5—24no—
191C22×C4×C5D1211yescharacteristic
192C22×C4×C5D1211yescharacteristic
193C22×C4×C5D1211yescharacteristic
194C22×C4×C5C22×D311yescharacteristic
195C22×C4×C5C22×D311yescharacteristic
196C22×C4×C5C22×D311yescharacteristic
197C2×C22⋊D5—68no—
198C2×C22⋊D5—62no—
199C2×C22⋊D5—62no—
200C22×Dic5—61no—
201C22×Dic5—61no—
202C22×Dic5—61no—
203#709—38no—
204#709—64no—
205#709—64no—
206#709—68no—
207#709—68no—
208#709—64no—
209C2×D20—32no—
210C2×D20—64no—
211C2×D20—61no—
212C2×C4×D5—68no—
213C2×C4×D5—62no—
214C2×C4×D5—62no—
215C2×Dic10—32no—
216C2×Dic10—32no—
217C2×Dic10—64no—
218C2×Dic10—61no—
219C2×Dic10—64no—
220C2×Dic10—61no—
221C5×M16—24no—
222C2×C5×C8C22×D314yes—
223C22⋊Dic5—62no—
224C22⋊Dic5—62no—
225C4⋊Dic5—32no—
226C4⋊Dic5—61no—
227#683—62no—
228#683—62no—
229Dic20—68no—
230#677—68no—
231C22×#41—151no—
232C2×#137—152no—
233C2×C22⋊Q8—151no—
234#452—154no—
235#451—152no—
236#450—152no—
237#417—152no—
238#407D1511yescharacteristic, complemented
239C22×C3×C5—21no—
240C22×C3×C5—21no—
241C22×C3×C5—21no—
242C22×C3×C5—21no—
243C22×C3×C5—21no—
244C22×C3×C5—21no—
245C22×C3×C5—21no—
246C22×C3×C5—44no—
247C22×C3×C5#12012yes—
248C22×C3×C5#13712yescomplemented
249C22×C3×C5C22×D411yescharacteristic
250C2×D15—24no—
251C2×D15—48no—
252C2×D15—42no—
253C2×D15—42no—
254C3×C4×C5—21no—
255C3×C4×C5—21no—
256C3×C4×C5#12012yes—
257C3×C4×C5C22×D412yes—
258Dic15—24no—
259Dic15—42no—
260Dic15—42no—
261C24×C3D2011yescharacteristic
262C23×D3—101no—
263C22×C3×C4—24no—
264C22×C3×C4D2011yescharacteristic
265C22×C3×C4D2011yescharacteristic
266C22×C3×C4D2011yescharacteristic
267C22×C3×C4C22×D511yescharacteristic
268C22×C3×C4C22×D511yescharacteristic
269C22×C3×C4C22×D511yescharacteristic
270C2×#67—108no—
271C2×#67—102no—
272C2×#67—102no—
273C22×Dic3—101no—
274C22×Dic3—101no—
275C22×Dic3—101no—
276#235—58no—
277#235—104no—
278#235—104no—
279#235—108no—
280#235—108no—
281#235—104no—
282C2×D12—52no—
283C2×D12—104no—
284C2×D12—101no—
285C2×C4×D3—108no—
286C2×C4×D3—102no—
287C2×C4×D3—102no—
288C2×Dic6—52no—
289C2×Dic6—52no—
290C2×Dic6—104no—
291C2×Dic6—101no—
292C2×Dic6—104no—
293C2×Dic6—101no—
294C3×M16—24no—
295C2×C3×C8C22×D514yes—
296#217—102no—
297#217—102no—
298C4⋊Dic3—52no—
299C4⋊Dic3—101no—
300Dic3⋊C4—102no—
301Dic3⋊C4—102no—
302Dic12—108no—
303#204—108no—
304C23×C5—22no—
305C23×C5—22no—
306C23×C5—41no—
307C23×C5C2×D1211yescharacteristic
308C23×C5C2×D1211yescharacteristic
309C23×C5C2×D1211yescharacteristic
310C22×D5—64no—
311C22×D5—62no—
312C22×D5—122no—
313C22×D5—122no—
314C2×C4×C5—22no—
315C2×C4×C5—22no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5C2×D1211yescharacteristic
320C2×C4×C5C2×D1211yescharacteristic
321C2×C4×C5C2×D1211yescharacteristic
322C2×C4×C5D3×D414yes—
323C2×C4×C5C23×D311yescharacteristic
324C22⋊D5—68no—
325C22⋊D5—68no—
326C2×Dic5—64no—
327C2×Dic5—62no—
328C2×Dic5—62no—
329C2×Dic5—62no—
330C2×Dic5—122no—
331C2×Dic5—122no—
332D20—34no—
333D20—64no—
334D20—62no—
335C4×D5—68no—
336C4×D5—68no—
337Dic10—34no—
338Dic10—64no—
339Dic10—62no—
340Dic10—64no—
341Dic10—122no—
342C5×C8—24no—
343C2×#41—152no—
344C2×#41—152no—
345C2×#41—308no—
346C2×#41—301no—
347C2×#41—301no—
348C22×Q8—151no—
349C22×Q8—151no—
350C22×D4—151no—
351C22×D4—301no—
352C23×C4—301no—
353C23×C4C2×D1511yescharacteristic
354#137—308no—
355C2×Q16—154no—
356C2×SD16—154no—
357C2×M16C2×D1512yescomplemented
358C22⋊Q8—152no—
359C22⋊Q8—152no—
360C22⋊Q8—301no—
361C22⋊Q8—301no—
362C2×C4⋊C4—151no—
363C2×C4⋊C4—301no—
364C2×C22⋊C4—301no—
365#103—154no—
366#102—154no—
367C22⋊C8C2×D1512yes—
368C22⋊C8C2×D1512yescomplemented
369C2×C3×C5—22no—
370C2×C3×C5—22no—
371C2×C3×C5—41no—
372C2×C3×C5C2×#12011yescharacteristic
373C2×C3×C5C2×#13711yescharacteristic
374C2×C3×C5#57511yescharacteristic
375D15—44no—
376C23×C3—22no—
377C23×C3—22no—
378C23×C3—41no—
379C23×C3C2×D2011yescharacteristic
380C23×C3C2×D2011yescharacteristic
381C23×C3C2×D2011yescharacteristic
382C22×D3—104no—
383C22×D3—102no—
384C22×D3—202no—
385C22×D3—202no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4C2×D2011yescharacteristic
392C2×C3×C4C2×D2011yescharacteristic
393C2×C3×C4C2×D2011yescharacteristic
394C2×C3×C4D4×D514yes—
395C2×C3×C4C23×D511yescharacteristic
396#67—108no—
397#67—108no—
398C2×Dic3—104no—
399C2×Dic3—102no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—202no—
403C2×Dic3—202no—
404D12—54no—
405D12—104no—
406D12—102no—
407C4×D3—108no—
408C4×D3—108no—
409Dic6—54no—
410Dic6—104no—
411Dic6—102no—
412Dic6—104no—
413Dic6—202no—
414C3×C8—24no—
415C22×C5—21no—
416C22×C5—21no—
417C22×C5—21no—
418C22×C5—21no—
419C22×C5—21no—
420C22×C5—21no—
421C22×C5—21no—
422C22×C5—44no—
423C22×C5#88212yes—
424C22×C5#90912yescomplemented
425C22×C5C22×D1211yescharacteristic
426C2×D5—64no—
427C2×D5—128no—
428C2×D5—122no—
429C2×D5—122no—
430C4×C5—21no—
431C4×C5—21no—
432C4×C5#88212yes—
433C4×C5C2×D3×D412yes—
434Dic5—64no—
435Dic5—122no—
436Dic5—122no—
437C24—301no—
438C24D6011yescharacteristic
439#41—158no—
440#41—304no—
441#41—304no—
442#41—308no—
443#41—308no—
444#41—304no—
445C2×Q8—152no—
446C2×Q8—152no—
447C2×Q8—304no—
448C2×Q8—301no—
449C2×Q8—301no—
450C2×Q8—304no—
451C2×D4—152no—
452C2×D4—304no—
453C2×D4—308no—
454C2×D4—302no—
455C2×D4—302no—
456C2×D4—301no—
457C22×C4—24no—
458C22×C4—308no—
459C22×C4—301no—
460C22×C4—302no—
461C22×C4—302no—
462C22×C4—301no—
463C22×C4—301no—
464C22×C4D6011yescharacteristic
465C22×C4D6011yescharacteristic
466C22×C4D6011yescharacteristic
467C22×C4C22×D1511yescharacteristic
468C22×C4C22×D1511yescharacteristic
469C22×C4C22×D1511yescharacteristic
470Q16—308no—
471SD16—308no—
472M16—24no—
473C2×C8C22×D1514yes—
474C4⋊C4—152no—
475C4⋊C4—301no—
476C4⋊C4—302no—
477C4⋊C4—302no—
478C22⋊C4—302no—
479C22⋊C4—302no—
480C3×C5#300711yescharacteristic, complemented, Hall
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—44no—
489C22×C3#402812yes—
490C22×C3#405512yescomplemented
491C22×C3C22×D2011yescharacteristic
492C2×D3—104no—
493C2×D3—208no—
494C2×D3—202no—
495C2×D3—202no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4#402812yes—
499C3×C4C2×D4×D512yes—
500Dic3—104no—
501Dic3—202no—
502Dic3—202no—
503C2×C5—22no—
504C2×C5—22no—
505C2×C5—41no—
506C2×C5C2×#88211yescharacteristic
507C2×C5C2×#90911yescharacteristic
508C2×C5#574811yescharacteristic
509D5—124no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—304no—
514C23—302no—
515C23—602no—
516C23—602no—
517C23C2×D6011yescharacteristic
518C23C2×D6011yescharacteristic
519C23C2×D6011yescharacteristic
520Q8—154no—
521Q8—304no—
522Q8—302no—
523Q8—304no—
524Q8—602no—
525D4—154no—
526D4—304no—
527D4—308no—
528D4—308no—
529D4—302no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—308no—
536C2×C4—304no—
537C2×C4—302no—
538C2×C4—308no—
539C2×C4—302no—
540C2×C4—302no—
541C2×C4—602no—
542C2×C4—602no—
543C2×C4C2×D6011yescharacteristic
544C2×C4C2×D6011yescharacteristic
545C2×C4C2×D6011yescharacteristic
546C2×C4D4×D1514yes—
547C2×C4C23×D1511yescharacteristic
548C8—24no—
549C2×C3—22no—
550C2×C3—22no—
551C2×C3—41no—
552C2×C3C2×#402811yescharacteristic
553C2×C3C2×#405511yescharacteristic
554C2×C3#6608211yescharacteristic
555D3—204no—
556C5#7864011yescharacteristic, complemented, Sylow
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—44no—
565C22—304no—
566C22—608no—
567C22—602no—
568C22—602no—
569C22#9177112yes—
570C22#9179812yescomplemented
571C22C22×D6011yescharacteristic
572C4—21no—
573C4—21no—
574C4—304no—
575C4—602no—
576C4—602no—
577C4#9177112yes—
578C4C2×D4×D1512yes—
579C3#1061037111yescharacteristic, complemented, Sylow
580C2—22no—
581C2—22no—
582C2—41no—
583C2—604no—
584C2C2×#9177111yescharacteristic
585C2C2×#9179811yescharacteristic
586C2#1175650711yescharacteristic
subgroupG/H
centerC22C22×D60
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#407C2
socleC22×C3×C5C22×D4
prime radicalC22×#6817C2
hypercenter#407D15
hypocenterC3×C5#3007
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#91813C2semidirect
3. C2×#91798C2semidirect
4. C3×C5×#407C2semidirect
5. #11751234C2semidirect
6. #11750840C2semidirect
7. #11750834C2semidirect
8. #11750832C2semidirect
20. C22×Dic30C22semidirect
21. C2×C3×C5×M16C22semidirect
23. C2×C4⋊Dic15C22semidirect
27. #91813C22semidirect
30. C2×Dic60C22semidirect
31. C2×#6706C22semidirect
33. C3×C5×C22⋊C8C22semidirect
34. #91105C22semidirect
35. #91100C22semidirect
40. C5×#407D3semidirect
91. C3×#407D5semidirect
107. C2×C5×M16C2×D3semidirect
119. C5×C22⋊C8C2×D3semidirect
173. C2×C3×M16C2×D5semidirect
185. C3×C22⋊C8C2×D5semidirect
238. #407D15semidirect
248. C22×C3×C5#137semidirect
357. C2×M16C2×D15semidirect
368. C22⋊C8C2×D15semidirect
424. C22×C5#909semidirect
480. C3×C5#3007semidirect
490. C22×C3#4055semidirect
556. C5#78640semidirect
570. C22#91798semidirect
579. C3#10610371semidirect

Maximal subgroups

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

subgroupindex
C22×#68172
C2×#918132
C2×#917982
C3×C5×#4072
#117512342
#117508402
#117508342
#117508322
#106103713
#786405