Finite Group Databasebeta

← #14424369

Subgroups — #14424369

Order 1920.

1504 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
1C2×C5×#1004C211yesmaximal, characteristic
2C22×#6716C211yesmaximal, characteristic, complemented
3#11754271C211yesmaximal, characteristic, complemented
4C2×#91322C211yesmaximal, characteristic, complemented
5C3×#65504C211yesmaximal, characteristic, complemented
6#11749056C214yesmaximal, complemented
7#11749054C214yesmaximal, complemented
8#11748382C211yesmaximal, characteristic, complemented
9#11748347C211yesmaximal, characteristic, complemented
10#10615836—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C22×Dic30C2211yescharacteristic, complemented
13C22×C5×#67—22no—
14C5×#1004—24no—
15C5×#1004—24no—
16C2×C5×#237—21no—
17C2×C5×#237C2212yes—
18C2×C5×D3×D4—21no—
19C2×C5×D3×D4C2212yes—
20C2×C5×#235C2211yescharacteristic
21C2×C5×#235C2211yescharacteristic
22C2×#6716—24no—
23C2×#6716C2214yescomplemented
24C22×C3⋊#168C2211yescharacteristic, complemented
25C3×#4072C2211yescharacteristic, complemented
26C2×C3×#681C2211yescharacteristic, complemented
27#91543—22no—
28#91376—21no—
29#91373—21no—
30#91365C2214yescomplemented
31#91322—24no—
32C2×#6663C2212yescomplemented
33C2×#6659C2212yescomplemented
34C3×#3964C2214yescomplemented
35#90972C2212yescomplemented
36#90966C2212yescomplemented
37(C2×D3)⋊#168C2214yescomplemented
38#78929—51nomaximal, Hall
39C2×C5×H(5,𝔽2)—31no—
40#66159—31no—
41C2×#4097—31no—
42C22×#687—31no—
43#65528—34no—
44#65523—31no—
45#65522—31no—
46#65507—34no—
47#65504D311yescharacteristic, complemented
48C24×C3×C5—21no—
49C23×C5×D3—22no—
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—22no—
52C2×C3×C5×D4D414yes—
53C2×C3×C5×D4C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15—21no—
56C2×Dic30—22no—
57C2×Dic30—22no—
58C2×Dic30—21no—
59C2×Dic30C2312yes—
60C2×C5×#67—216no—
61C2×C5×#67—21no—
62C2×C5×#67—22no—
63C2×C5×#67—22no—
64C2×C5×#67D412yes—
65C22×C5×Dic3—22no—
66C5×#237—28no—
67C5×#237—24no—
68C5×#237—24no—
69C5×D3×D4—28no—
70C5×D3×D4—24no—
71C5×D3×D4—24no—
72C5×#235—22no—
73C5×#235—22no—
74C5×#235—22no—
75C5×#235D414yes—
76C2×C5×D12D411yescharacteristic
77C2×C5×D12C2311yescharacteristic
78C2×C4×C5×D3—21no—
79C2×C4×C5×D3C2312yes—
80C2×C5×Dic6D411yescharacteristic
81C2×C5×Dic6C2311yescharacteristic
82#6716—28no—
83C2×C3⋊#168—22no—
84C2×C3⋊#168C2312yes—
85C3×C22⋊Dic5—21no—
86C3×C4⋊Dic5C2312yes—
87C3×C4×Dic5—21no—
88C3×#681—22no—
89C2×C3×#168C2312yes—
90#6668—24no—
91(C2×D3)⋊Dic5—24no—
92#6663—24no—
93#6659—24no—
94C2×#1004D511yescharacteristic
95C22×#214—51no—
96C2×#909—51no—
97#5526—51no—
98C3×#418—51no—
99#5129—51no—
100#5002—51no—
101#4783—54no—
102#4776—54no—
103C5×H(5,𝔽2)—64no—
104C5×H(5,𝔽2)—64no—
105C2×C5×#41—31no—
106C2×C5×#41—32no—
107C2×C5×#41—31no—
108C2×C5×#41—61no—
109C22×C5×D4—32no—
110C22×C5×D4—61no—
111C22×C5×D4—62no—
112C22×C5×D4C2×D311yescharacteristic
113C22×Dic10—31no—
114#4097—64no—
115#4090—61no—
116C2×#689—32no—
117#4082—61no—
118#4081—62no—
119C2×#687—34no—
120C2×#687—32no—
121C2×#687—64no—
122#4072C2×D311yescharacteristic, complemented
123#4070—34no—
124C2×#681C2×D311yescharacteristic, complemented
125C22×#168—31no—
126#3967—32no—
127#3964—32no—
128#3964C2×D314yescomplemented
129C22⋊#168—34no—
130#3009—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—24no—
133C23×C3×C5—82no—
134C23×C3×C5C2×D411yescharacteristic
135C22×C5×D3—21no—
136C22×C5×D3—28no—
137C22×C5×D3—84no—
138C22×C5×D3C2×D412yes—
139C3×C5×D4—24no—
140C3×C5×D4—22no—
141C3×C5×D4C2×D414yes—
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2411yescharacteristic
146C2×Dic15—22no—
147C2×Dic15—42no—
148Dic30—22no—
149Dic30—22no—
150Dic30—42no—
151C5×#67—28no—
152C5×#67—28no—
153C5×#67—24no—
154C5×#67—24no—
155C2×C5×Dic3—28no—
156C2×C5×Dic3—21no—
157C2×C5×Dic3C2×D412yes—
158C5×D12—22no—
159C5×D12—21no—
160C5×D12C2×D412yes—
161C4×C5×D3—24no—
162C4×C5×D3—22no—
163C4×C5×D3—22no—
164C5×Dic6—22no—
165C5×Dic6—21no—
166C5×Dic6C2×D412yes—
167C2×C3×Dic5—22no—
168C3⋊#168—22no—
169C3×#168—22no—
170C22×C3×D4C2×D511yescharacteristic
171C22×#67—22no—
172#1004—24no—
173#1004—24no—
174C2×#237—21no—
175C2×#237C2×D512yes—
176C2×D3×D4—21no—
177C2×D3×D4C2×D512yes—
178C2×#235C2×D511yescharacteristic
179C2×#235C2×D511yescharacteristic
180C22×Dic6—51no—
181C2×C3×M16—51no—
182C3×#117—51no—
183C2×#214—54no—
184C2×#214—104no—
185C22×C3⋊C8—51no—
186#909—104no—
187C2×Dic12—52no—
188C2×#204—52no—
189#897—54no—
190#886—102no—
191#875—101no—
192#869—101no—
193C3×#102—54no—
194(C2×D3)⋊C8—54no—
195#810—52no—
196#809—52no—
197C24×C5—21no—
198C24×C5—62no—
199C5×#41—34no—
200C5×#41—64no—
201C5×#41—68no—
202C5×#41—64no—
203C5×#41—62no—
204C5×#41—62no—
205C5×#41—62no—
206C2×C5×Q8—31no—
207C2×C5×Q8—31no—
208C2×C5×D4—21no—
209C2×C5×D4—22no—
210C2×C5×D4—32no—
211C2×C5×D4—31no—
212C2×C5×D4—31no—
213C2×C5×D4—62no—
214C2×C5×D4—64no—
215C2×C5×D4—68no—
216C2×C5×D4—62no—
217C2×C5×D4—616no—
218C2×C5×D4—61no—
219C2×C5×D4—64no—
220C2×C5×D4D1214yes—
221C2×C5×D4C22×D312yes—
222C22×C4×C5—32no—
223C22×C4×C5—62no—
224C22×C4×C5—61no—
225C22×C4×C5C22×D311yescharacteristic
226C22×Dic5—61no—
227C2×Dic10—32no—
228C2×Dic10—62no—
229C2×Dic10—62no—
230C2×Dic10—61no—
231C22⋊Dic5—21no—
232C22⋊Dic5—64no—
233#689—64no—
234#687—64no—
235#687—68no—
236C4⋊Dic5C22×D312yes—
237#683—64no—
238C4×Dic5—21no—
239#681—22no—
240C2×#168—32no—
241C2×#168—62no—
242C2×#168C22×D312yes—
243C2×H(5,𝔽2)—31no—
244C2×#137—151no—
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—51no—
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5—24no—
254C22×C3×C5—44no—
255C22×C3×C5—84no—
256C22×C3×C5#12012yes—
257C22×C3×C5C22×D411yescharacteristic
258C2×C5×D3—22no—
259C2×C5×D3—22no—
260C2×C5×D3—24no—
261C2×C5×D3—42no—
262C2×C5×D3—88no—
263C3×C4×C5#12012yes—
264C3×C4×C5C22×D411yescharacteristic
265C3×C4×C5C22×D411yescharacteristic
266Dic15—42no—
267C3×Dic5—42no—
268C5×Dic3—22no—
269C5×Dic3—22no—
270C24×C3—21no—
271C23×D3—22no—
272C2×C3×D4—21no—
273C2×C3×D4—22no—
274C2×C3×D4C22⋊D514yes—
275C2×C3×D4C22×D512yes—
276C22×C3×C4C22×D511yescharacteristic
277C2×#67—216no—
278C2×#67—21no—
279C2×#67—22no—
280C2×#67—22no—
281C2×#67C22⋊D512yes—
282C22×Dic3—22no—
283C22×Dic3—101no—
284#237—28no—
285#237—24no—
286#237—24no—
287D3×D4—28no—
288D3×D4—24no—
289D3×D4—24no—
290#235—22no—
291#235—22no—
292#235—22no—
293#235C22⋊D514yes—
294C2×D12C22⋊D511yescharacteristic
295C2×D12C22×D511yescharacteristic
296C2×C4×D3—21no—
297C2×C4×D3C22×D512yes—
298C2×Dic6—52no—
299C2×Dic6—102no—
300C2×Dic6—102no—
301C2×Dic6—101no—
302C2×Dic6C22⋊D511yescharacteristic
303C2×Dic6C22×D511yescharacteristic
304C3×M16—102no—
305C2×C3×C8—52no—
306C3×C4⋊C4—52no—
307C3×C22⋊C4—101no—
308C3×C42—101no—
309#214—108no—
310(C2×D3)⋊C4—104no—
311C4⋊Dic3—104no—
312C2×C3⋊C8—52no—
313C2×C3⋊C8—102no—
314Dic12—104no—
315#204—104no—
316C23×C5—22no—
317C23×C5—24no—
318C23×C5—32no—
319C23×C5—68no—
320C23×C5—61no—
321C23×C5—82no—
322C23×C5—244no—
323C23×C5D3×D411yescharacteristic
324C5×Q8—32no—
325C5×Q8—62no—
326C5×Q8—61no—
327C5×D4—24no—
328C5×D4—22no—
329C5×D4—32no—
330C5×D4—68no—
331C5×D4—68no—
332C5×D4—64no—
333C5×D4—64no—
334C5×D4—62no—
335C5×D4—61no—
336C5×D4C2×D1214yes—
337C2×C4×C5—32no—
338C2×C4×C5—68no—
339C2×C4×C5—61no—
340C2×C4×C5—64no—
341C2×C4×C5—62no—
342C2×C4×C5—62no—
343C2×C4×C5C2×D1212yes—
344C2×C4×C5D3×D411yescharacteristic
345C2×C4×C5D3×D412yes—
346C2×C4×C5C23×D311yescharacteristic
347C2×Dic5—22no—
348C2×Dic5—62no—
349C2×Dic5—122no—
350Dic10—62no—
351Dic10—62no—
352Dic10—122no—
353#168—22no—
354#168—62no—
355H(5,𝔽2)—64no—
356H(5,𝔽2)—64no—
357C2×#41—31no—
358C2×#41—32no—
359C2×#41—31no—
360C2×#41—61no—
361C22×Q8—151no—
362C22×D4—32no—
363C22×D4—61no—
364C22×D4—62no—
365C22×D4D3×D511yescharacteristic
366#137—304no—
367C2×Q16—152no—
368C2×SD16—152no—
369C2×SD16—154no—
370C2×SD16—304no—
371C2×M16—51no—
372C22×C8—151no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—51no—
378#103—152no—
379#102—54no—
380#102—152no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5—44no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386C5×D3—42no—
387C5×D3—42no—
388C23×C3—22no—
389C23×C3—24no—
390C23×C3—82no—
391C23×C3C2×C22⋊D511yescharacteristic
392C22×D3—21no—
393C22×D3—28no—
394C22×D3—84no—
395C22×D3C2×C22⋊D512yes—
396C3×D4—24no—
397C3×D4—22no—
398C3×D4C2×C22⋊D514yes—
399C2×C3×C4—102no—
400C2×C3×C4C2×C22⋊D512yes—
401C2×C3×C4C2×C22⋊D511yescharacteristic
402C2×C3×C4C2×C22⋊D512yes—
403C2×C3×C4C23×D511yescharacteristic
404#67—28no—
405#67—28no—
406#67—24no—
407#67—24no—
408C2×Dic3—28no—
409C2×Dic3—21no—
410C2×Dic3—102no—
411C2×Dic3—202no—
412C2×Dic3C2×C22⋊D512yes—
413D12—22no—
414D12—21no—
415D12C2×C22⋊D512yes—
416C4×D3—24no—
417C4×D3—22no—
418C4×D3—22no—
419Dic6—22no—
420Dic6—21no—
421Dic6—102no—
422Dic6—102no—
423Dic6—202no—
424Dic6C2×C22⋊D512yes—
425C3×C8—102no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—24no—
429C22×C5—44no—
430C22×C5—62no—
431C22×C5—62no—
432C22×C5—64no—
433C22×C5—84no—
434C22×C5—122no—
435C22×C5—248no—
436C22×C5#88212yes—
437C22×C5C2×D3×D411yescharacteristic
438C4×C5—62no—
439C4×C5—62no—
440C4×C5#88212yes—
441C4×C5C22×D1211yescharacteristic
442C4×C5C2×D3×D411yescharacteristic
443Dic5—42no—
444Dic5—122no—
445C24—21no—
446C24—62no—
447#41—34no—
448#41—64no—
449#41—68no—
450#41—64no—
451#41—62no—
452#41—62no—
453#41—62no—
454C2×Q8—31no—
455C2×Q8—31no—
456C2×Q8—152no—
457C2×Q8—301no—
458C2×Q8—302no—
459C2×Q8—302no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—32no—
463C2×D4—31no—
464C2×D4—31no—
465C2×D4—62no—
466C2×D4—64no—
467C2×D4—68no—
468C2×D4—62no—
469C2×D4—616no—
470C2×D4—61no—
471C2×D4—64no—
472C2×D4#120314yes—
473C2×D4C2×D3×D512yes—
474C22×C4—32no—
475C22×C4—62no—
476C22×C4—61no—
477C22×C4—301no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—304no—
480SD16—304no—
481SD16—308no—
482M16—102no—
483C2×C8—52no—
484C2×C8—152no—
485C2×C8—302no—
486C4⋊C4—52no—
487C4⋊C4—304no—
488C22⋊C4—101no—
489C22⋊C4—304no—
490C42—101no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—24no—
494C22×C3—44no—
495C22×C3—84no—
496C22×C3#409912yes—
497C22×C3C22×C22⋊D511yescharacteristic
498C2×D3—24no—
499C2×D3—22no—
500C2×D3—22no—
501C2×D3—42no—
502C2×D3—88no—
503C3×C4—202no—
504C3×C4#409912yes—
505C3×C4C22×C22⋊D511yescharacteristic
506C3×C4C22×C22⋊D511yescharacteristic
507Dic3—22no—
508Dic3—22no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—44no—
512C2×C5—122no—
513C2×C5—122no—
514C2×C5C2×#88211yescharacteristic
515C2×C5#577512yes—
516C23—22no—
517C23—24no—
518C23—32no—
519C23—68no—
520C23—61no—
521C23—82no—
522C23—244no—
523C23D3×C22⋊D511yescharacteristic
524Q8—32no—
525Q8—61no—
526Q8—62no—
527Q8—302no—
528Q8—302no—
529Q8—602no—
530D4—22no—
531D4—24no—
532D4—32no—
533D4—68no—
534D4—68no—
535D4—64no—
536D4—64no—
537D4—61no—
538D4—62no—
539D4C2×#120314yes—
540C2×C4—32no—
541C2×C4—68no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—64no—
546C2×C4—102no—
547C2×C4—302no—
548C2×C4—602no—
549C2×C4C2×#120312yes—
550C2×C4D3×C22⋊D512yes—
551C2×C4D3×C22⋊D511yescharacteristic
552C2×C4C22×D3×D511yescharacteristic
553C8—102no—
554C8—302no—
555C2×C3—22no—
556C2×C3—44no—
557C2×C3#6617012yes—
558C2×C3C2×#409911yescharacteristic
559D3—42no—
560D3—42no—
561C5#7892911yescharacteristic, complemented, Sylow
562C22—22no—
563C22—24no—
564C22—44no—
565C22—62no—
566C22—62no—
567C22—64no—
568C22—84no—
569C22—122no—
570C22—248no—
571C22#9157012yes—
572C22C2×D3×C22⋊D511yescharacteristic
573C4—62no—
574C4—62no—
575C4—202no—
576C4—602no—
577C4#9157012yes—
578C4C22×#120311yescharacteristic
579C4C2×D3×C22⋊D511yescharacteristic
580C3#1061583611yescharacteristic, complemented, Sylow
581C2—22no—
582C2—44no—
583C2—122no—
584C2—122no—
585C2#1175472012yes—
586C2C2×#9157011yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×D4C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×#1004C2
hypercenterC22×D4D3×D5
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C22×#6716C2semidirect
3. #11754271C2semidirect
4. C2×#91322C2semidirect
5. C3×#65504C2semidirect
6. #11749056C2semidirect
7. #11749054C2semidirect
8. #11748382C2semidirect
9. #11748347C2semidirect
12. C22×Dic30C22semidirect
23. C2×#6716C22semidirect
24. C22×C3⋊#168C22semidirect
25. C3×#4072C22semidirect
26. C2×C3×#681C22semidirect
30. #91365C22semidirect
32. C2×#6663C22semidirect
33. C2×#6659C22semidirect
34. C3×#3964C22semidirect
35. #90972C22semidirect
36. #90966C22semidirect
37. (C2×D3)⋊#168C22semidirect
47. #65504D3semidirect
122. #4072C2×D3semidirect
124. C2×#681C2×D3semidirect
128. #3964C2×D3semidirect
491. C3×C5#3009semidirect
561. C5#78929semidirect
580. C3#10615836semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#10042
C22×#67162
#117542712
C2×#913222
C3×#655042
#117490562
#117490542
#117483822
#117483472
#106158363
#789295