Finite Group Databasebeta

← #14424212

Subgroups — #14424212

Order 1920.

1568 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92095C211yesmaximal, characteristic, complemented
2C22×#6715C211yesmaximal, characteristic, complemented
3C2×#91797C211yesmaximal, characteristic, complemented
4#11756284C211yesmaximal, characteristic, complemented
5C3×C5×#418C211yesmaximal, characteristic, complemented
6#11751230C211yesmaximal, characteristic, complemented
7#11751103C211yesmaximal, characteristic, complemented
8#11750882C214yesmaximal, complemented
9#11750874C214yesmaximal, complemented
10#10610659—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C22×#1220—22no—
13#92095—24no—
14#92095—24no—
15C2×#6819—21no—
16C2×#6819C2212yescomplemented
17C2×D4×D15—21no—
18C2×D4×D15C2212yescomplemented
19C2×#6817C2211yescharacteristic
20C2×#6817C2211yescharacteristic
21C22×D60C2211yescharacteristic, complemented
22C2×C3×C5×M16C2211yescharacteristic, complemented
23C3×C5×#117C2211yescharacteristic, complemented
24C2×#6715—24no—
25C2×#6715C2214yescomplemented
26C22×C3⋊#168C2211yescharacteristic, complemented
27#91797—24no—
28C2×D120C2212yescomplemented
29C2×#6706C2212yescomplemented
30#91784C2214yescomplemented
31#91771—22no—
32#91764—21no—
33#91763—21no—
34C3×C5×#102C2214yescomplemented
35#91104C2214yescomplemented
36#91094C2212yescomplemented
37#91093C2212yescomplemented
38#78928—51nomaximal, Hall
39C2×#4144—31no—
40C22×#686—31no—
41C2×#4054—31no—
42#65859—31no—
43C5×#418D311yescharacteristic, complemented
44#65418—31no—
45#65291—31no—
46#65070—34no—
47#65062—34no—
48C24×C3×C5—21no—
49C23×D15—22no—
50C23×D15—21no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4—22no—
53C2×C3×C5×D4D414yescomplemented
54C2×C3×C5×D4C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C2×#1220—216no—
57C2×#1220—21no—
58C2×#1220—22no—
59C2×#1220—22no—
60C2×#1220D412yes—
61C22×Dic15—22no—
62#6819—28no—
63#6819—24no—
64#6819—24no—
65D4×D15—28no—
66D4×D15—24no—
67D4×D15—24no—
68#6817—22no—
69#6817—22no—
70#6817—22no—
71#6817D414yescomplemented
72C2×D60—22no—
73C2×D60—22no—
74C2×D60—21no—
75C2×D60—24no—
76C2×D60D411yescharacteristic
77C2×D60C2312yes—
78C2×D60C2311yescharacteristic
79C2×C4×D15—21no—
80C2×C4×D15C2312yes—
81C2×Dic30D411yescharacteristic
82C2×Dic30C2311yescharacteristic
83C3×C5×M16—22no—
84C2×C3×C5×C8C2312yes—
85C3×C5×C4⋊C4C2312yes—
86C3×C5×C22⋊C4—21no—
87C3×C42×C5—21no—
88#6715—28no—
89#6714—24no—
90C2×C3⋊#168—22no—
91C2×C3⋊#168C2312yes—
92D120—24no—
93#6706—24no—
94C2×#1004—51no—
95C22×#213—51no—
96C2×#908—51no—
97#5525—51no—
98C3×#418D511yescharacteristic, complemented
99#5129—51no—
100#5002—51no—
101#4781—54no—
102#4773—54no—
103C22×C5×D4C2×D311yescharacteristic
104C22×C22⋊D5—62no—
105#4144—64no—
106#4144—64no—
107C2×#711—32no—
108C2×#711—61no—
109C2×D4×D5—32no—
110C2×D4×D5—61no—
111C2×#709—31no—
112C2×#709—31no—
113C22×D20—31no—
114C2×C5×M16C2×D311yescharacteristic, complemented
115C5×#117C2×D311yescharacteristic, complemented
116C2×#686—34no—
117C2×#686—64no—
118C22×#168—31no—
119#4054—64no—
120C2×D40—32no—
121C2×#677—32no—
122#4041—34no—
123#4028—62no—
124#4021—61no—
125#4020—61no—
126C5×#102C2×D314yescomplemented
127#3952—34no—
128#3942—32no—
129#3941—32no—
130#3008—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—24no—
133C23×C3×C5—82no—
134C23×C3×C5C2×D411yescharacteristic
135C22×D15—21no—
136C22×D15—28no—
137C22×D15—22no—
138C22×D15—42no—
139C22×D15—84no—
140C22×D15—82no—
141C22×D15C2×D412yes—
142C3×C5×D4—24no—
143C3×C5×D4—22no—
144C3×C5×D4C2×D414yes—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5C2411yescharacteristic
150#1220—28no—
151#1220—28no—
152#1220—24no—
153#1220—24no—
154C2×Dic15—28no—
155C2×Dic15—21no—
156C2×Dic15C2×D412yes—
157D60—22no—
158D60—21no—
159D60—22no—
160D60—22no—
161D60—42no—
162D60—84no—
163D60C2×D412yes—
164C4×D15—24no—
165C4×D15—22no—
166C4×D15—22no—
167Dic30—22no—
168Dic30—21no—
169Dic30C2×D412yes—
170C3×C5×C8—22no—
171C3⋊#168—22no—
172C22×C3×D4C2×D511yescharacteristic
173C22×#67—102no—
174#1004—104no—
175#1004—104no—
176C2×#237—52no—
177C2×#237—101no—
178C2×D3×D4—52no—
179C2×D3×D4—101no—
180C2×#235—51no—
181C2×#235—51no—
182C22×D12—51no—
183C2×C3×M16C2×D511yescharacteristic, complemented
184C3×#117C2×D511yescharacteristic, complemented
185C2×#213—54no—
186C2×#213—104no—
187C22×C3⋊C8—51no—
188#908—104no—
189C2×D24—52no—
190C2×#204—52no—
191#895—54no—
192#882—102no—
193#875—101no—
194#874—101no—
195C3×#102C2×D514yescomplemented
196(C2×D3)⋊C8—54no—
197#810—52no—
198#809—52no—
199C24×C5—21no—
200C23×D5—62no—
201C23×D5—61no—
202C2×C5×D4—21no—
203C2×C5×D4—22no—
204C2×C5×D4D1214yescomplemented
205C2×C5×D4C22×D312yes—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—32no—
208C2×C22⋊D5—62no—
209C2×C22⋊D5—62no—
210C2×C22⋊D5—616no—
211C2×C22⋊D5—61no—
212C22×Dic5—62no—
213#711—64no—
214#711—68no—
215#711—64no—
216D4×D5—64no—
217D4×D5—68no—
218D4×D5—64no—
219#709—34no—
220#709—62no—
221#709—62no—
222#709—62no—
223C2×D20—31no—
224C2×D20—31no—
225C2×D20—32no—
226C2×D20—62no—
227C2×D20—62no—
228C2×D20—61no—
229C2×D20—64no—
230C2×C4×D5—32no—
231C2×C4×D5—61no—
232C2×Dic10—31no—
233C2×Dic10—31no—
234C5×M16—22no—
235C2×C5×C8C22×D312yes—
236C5×C4⋊C4C22×D312yes—
237C5×C22⋊C4—21no—
238C42×C5—21no—
239#686—68no—
240#685—64no—
241C2×#168—32no—
242C2×#168—62no—
243D40—64no—
244#677—64no—
245C2×H(5,𝔽2)—151no—
246C2×AGL(1,ℤ/8)—151no—
247C22×D8—151no—
248#535—151no—
249#449—154no—
250#447—154no—
251#418D1511yescharacteristic, complemented
252#416—151no—
253#408—151no—
254C22×C3×C5—22no—
255C22×C3×C5—24no—
256C22×C3×C5—44no—
257C22×C3×C5—84no—
258C22×C3×C5#12012yes—
259C22×C3×C5C22×D411yescharacteristic
260C2×D15—22no—
261C2×D15—22no—
262C2×D15—24no—
263C2×D15—42no—
264C2×D15—42no—
265C2×D15—44no—
266C2×D15—88no—
267C2×D15—84no—
268C3×C4×C5—42no—
269C3×C4×C5#12012yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C22×D411yescharacteristic
272Dic15—22no—
273Dic15—22no—
274C24×C3—21no—
275C23×D3—102no—
276C23×D3—101no—
277C2×C3×D4—21no—
278C2×C3×D4—22no—
279C2×C3×D4D2014yescomplemented
280C2×C3×D4C22×D512yes—
281C22×C3×C4C22×D511yescharacteristic
282C2×#67—52no—
283C2×#67—102no—
284C2×#67—102no—
285C2×#67—1016no—
286C2×#67—101no—
287C22×Dic3—102no—
288#237—104no—
289#237—108no—
290#237—104no—
291D3×D4—104no—
292D3×D4—108no—
293D3×D4—104no—
294#235—54no—
295#235—102no—
296#235—102no—
297#235—102no—
298C2×D12—51no—
299C2×D12—51no—
300C2×D12—52no—
301C2×D12—102no—
302C2×D12—102no—
303C2×D12—101no—
304C2×D12—104no—
305C2×C4×D3—52no—
306C2×C4×D3—101no—
307C2×Dic6—51no—
308C2×Dic6—51no—
309C3×M16—22no—
310C2×C3×C8C22×D512yes—
311C3×C4⋊C4C22×D512yes—
312C3×C22⋊C4—21no—
313C3×C42—21no—
314#213—108no—
315(C2×D3)⋊C4—104no—
316C2×C3⋊C8—52no—
317C2×C3⋊C8—102no—
318D24—104no—
319#204—104no—
320C23×C5—22no—
321C23×C5—24no—
322C23×C5—82no—
323C23×C5D3×D411yescharacteristic
324C22×D5—32no—
325C22×D5—61no—
326C22×D5—68no—
327C22×D5—62no—
328C22×D5—122no—
329C22×D5—244no—
330C22×D5—242no—
331C5×D4—24no—
332C5×D4—22no—
333C5×D4C2×D1214yes—
334C2×C4×C5—22no—
335C2×C4×C5C2×D1212yes—
336C2×C4×C5D3×D411yescharacteristic
337C2×C4×C5D3×D412yes—
338C2×C4×C5C23×D311yescharacteristic
339C22⋊D5—68no—
340C22⋊D5—68no—
341C22⋊D5—64no—
342C22⋊D5—64no—
343C2×Dic5—32no—
344C2×Dic5—68no—
345C2×Dic5—61no—
346D20—32no—
347D20—62no—
348D20—61no—
349D20—62no—
350D20—62no—
351D20—122no—
352D20—244no—
353C4×D5—64no—
354C4×D5—62no—
355C4×D5—62no—
356Dic10—32no—
357Dic10—62no—
358Dic10—61no—
359C5×C8—22no—
360#168—62no—
361H(5,𝔽2)—304no—
362H(5,𝔽2)—304no—
363C2×#41—151no—
364C2×#41—152no—
365C2×#41—151no—
366C2×#41—301no—
367C22×D4—152no—
368C22×D4—151no—
369C22×D4—301no—
370C22×D4—302no—
371C22×D4C2×D1511yescharacteristic
372AGL(1,ℤ/8)—304no—
373C2×SD16—152no—
374C2×D8—154no—
375C2×D8—152no—
376C2×D8—304no—
377C2×M16C2×D1511yescharacteristic, complemented
378C22×C8—151no—
379#127—301no—
380#124—301no—
381#121—154no—
382#120—302no—
383#117C2×D1511yescharacteristic, complemented
384#103—152no—
385#102—152no—
386#102C2×D1514yescomplemented
387C22⋊C8—154no—
388C2×C3×C5—22no—
389C2×C3×C5—44no—
390C2×C3×C5C2×#12011yescharacteristic
391C2×C3×C5#57612yes—
392D15—42no—
393D15—42no—
394D15—82no—
395C23×C3—22no—
396C23×C3—24no—
397C23×C3—82no—
398C23×C3D4×D511yescharacteristic
399C22×D3—52no—
400C22×D3—101no—
401C22×D3—108no—
402C22×D3—102no—
403C22×D3—202no—
404C22×D3—404no—
405C22×D3—402no—
406C3×D4—22no—
407C3×D4—24no—
408C3×D4C2×D2014yes—
409C2×C3×C4—22no—
410C2×C3×C4C2×D2012yes—
411C2×C3×C4D4×D512yes—
412C2×C3×C4D4×D511yescharacteristic
413C2×C3×C4C23×D511yescharacteristic
414#67—108no—
415#67—108no—
416#67—104no—
417#67—104no—
418C2×Dic3—52no—
419C2×Dic3—108no—
420C2×Dic3—101no—
421D12—52no—
422D12—101no—
423D12—102no—
424D12—102no—
425D12—102no—
426D12—202no—
427D12—404no—
428C4×D3—102no—
429C4×D3—102no—
430C4×D3—104no—
431Dic6—52no—
432Dic6—101no—
433Dic6—102no—
434C3×C8—22no—
435C3⋊C8—102no—
436C22×C5—22no—
437C22×C5—24no—
438C22×C5—44no—
439C22×C5—84no—
440C22×C5#88212yes—
441C22×C5C2×D3×D411yescharacteristic
442C2×D5—62no—
443C2×D5—62no—
444C2×D5—64no—
445C2×D5—122no—
446C2×D5—122no—
447C2×D5—124no—
448C2×D5—248no—
449C2×D5—244no—
450C4×C5—42no—
451C4×C5#88212yes—
452C4×C5C22×D1211yescharacteristic
453C4×C5C2×D3×D411yescharacteristic
454Dic5—62no—
455Dic5—62no—
456C24—21no—
457C24—302no—
458C24—301no—
459#41—154no—
460#41—304no—
461#41—308no—
462#41—304no—
463#41—302no—
464#41—302no—
465#41—302no—
466C2×Q8—151no—
467C2×Q8—151no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—152no—
471C2×D4—151no—
472C2×D4—151no—
473C2×D4—152no—
474C2×D4—304no—
475C2×D4—302no—
476C2×D4—304no—
477C2×D4—308no—
478C2×D4—302no—
479C2×D4—3016no—
480C2×D4—301no—
481C2×D4—304no—
482C2×D4—302no—
483C2×D4—302no—
484C2×D4—301no—
485C2×D4D6014yescomplemented
486C2×D4C22×D1512yes—
487C22×C4—152no—
488C22×C4—302no—
489C22×C4—301no—
490C22×C4C22×D1511yescharacteristic
491SD16—304no—
492D8—304no—
493D8—308no—
494M16—22no—
495C2×C8—152no—
496C2×C8—302no—
497C2×C8C22×D1512yes—
498C4⋊C4C22×D1512yes—
499C22⋊C4—21no—
500C22⋊C4—304no—
501C42—21no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—24no—
505C22×C3—44no—
506C22×C3—84no—
507C22×C3#402812yes—
508C22×C3C2×D4×D511yescharacteristic
509C2×D3—102no—
510C2×D3—102no—
511C2×D3—104no—
512C2×D3—202no—
513C2×D3—202no—
514C2×D3—204no—
515C2×D3—408no—
516C2×D3—404no—
517C3×C4—42no—
518C3×C4#402812yes—
519C3×C4C22×D2011yescharacteristic
520C3×C4C2×D4×D511yescharacteristic
521Dic3—102no—
522Dic3—102no—
523C2×C5—22no—
524C2×C5—44no—
525C2×C5C2×#88211yescharacteristic
526C2×C5#577412yes—
527D5—122no—
528D5—122no—
529D5—242no—
530C23—22no—
531C23—24no—
532C23—82no—
533C23—152no—
534C23—308no—
535C23—301no—
536C23—302no—
537C23—602no—
538C23—1202no—
539C23—1204no—
540C23D4×D1511yescharacteristic
541Q8—152no—
542Q8—302no—
543Q8—301no—
544D4—24no—
545D4—22no—
546D4—152no—
547D4—308no—
548D4—308no—
549D4—304no—
550D4—304no—
551D4—302no—
552D4—301no—
553D4—302no—
554D4—302no—
555D4—602no—
556D4—1204no—
557D4C2×D6014yes—
558C2×C4—22no—
559C2×C4—152no—
560C2×C4—308no—
561C2×C4—301no—
562C2×C4—304no—
563C2×C4—302no—
564C2×C4—302no—
565C2×C4C2×D6012yes—
566C2×C4D4×D1511yescharacteristic
567C2×C4D4×D1512yes—
568C2×C4C23×D1511yescharacteristic
569C8—22no—
570C8—302no—
571C2×C3—22no—
572C2×C3—44no—
573C2×C3C2×#402811yescharacteristic
574C2×C3#6610812yes—
575D3—202no—
576D3—202no—
577D3—402no—
578C5#7892811yescharacteristic, complemented, Sylow
579C22—22no—
580C22—24no—
581C22—44no—
582C22—84no—
583C22—302no—
584C22—302no—
585C22—304no—
586C22—602no—
587C22—602no—
588C22—604no—
589C22—1204no—
590C22—1208no—
591C22#9177112yes—
592C22C2×D4×D1511yescharacteristic
593C4—42no—
594C4—302no—
595C4—302no—
596C4#9177112yes—
597C4C22×D6011yescharacteristic
598C4C2×D4×D1511yescharacteristic
599C3#1061065911yescharacteristic, complemented, Sylow
600C2—22no—
601C2—44no—
602C2—602no—
603C2—602no—
604C2—1202no—
605C2C2×#9177111yescharacteristic
606C2#1175653312yes—
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#418C2
socleC22×C3×C5C22×D4
hypercenter#418D15
hypocenterC3×C5#3008
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#92095C2semidirect
2. C22×#6715C2semidirect
3. C2×#91797C2semidirect
4. #11756284C2semidirect
5. C3×C5×#418C2semidirect
6. #11751230C2semidirect
7. #11751103C2semidirect
8. #11750882C2semidirect
9. #11750874C2semidirect
16. C2×#6819C22semidirect
18. C2×D4×D15C22semidirect
21. C22×D60C22semidirect
22. C2×C3×C5×M16C22semidirect
23. C3×C5×#117C22semidirect
25. C2×#6715C22semidirect
26. C22×C3⋊#168C22semidirect
28. C2×D120C22semidirect
29. C2×#6706C22semidirect
30. #91784C22semidirect
34. C3×C5×#102C22semidirect
35. #91104C22semidirect
36. #91094C22semidirect
37. #91093C22semidirect
43. C5×#418D3semidirect
53. C2×C3×C5×D4D4semidirect
71. #6817D4semidirect
98. C3×#418D5semidirect
114. C2×C5×M16C2×D3semidirect
115. C5×#117C2×D3semidirect
126. C5×#102C2×D3semidirect
183. C2×C3×M16C2×D5semidirect
184. C3×#117C2×D5semidirect
195. C3×#102C2×D5semidirect
204. C2×C5×D4D12semidirect
251. #418D15semidirect
279. C2×C3×D4D20semidirect
377. C2×M16C2×D15semidirect
383. #117C2×D15semidirect
386. #102C2×D15semidirect
485. C2×D4D60semidirect
502. C3×C5#3008semidirect
578. C5#78928semidirect
599. C3#10610659semidirect

Maximal subgroups

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

subgroupindex
C2×#920952
C22×#67152
C2×#917972
#117562842
C3×C5×#4182
#117512302
#117511032
#117508822
#117508742
#106106593
#789285