Finite Group Databasebeta

← #14424261

Subgroups — #14424261

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×C3×#4149C211yesmaximal, characteristic, complemented
2C5×#5830C211yesmaximal, characteristic, complemented
3C2×#91299C211yesmaximal, characteristic, complemented
4C22×#6652C211yesmaximal, characteristic, complemented
5#11751335C211yesmaximal, characteristic, complemented
6#11748511C214yesmaximal, complemented
7#11748510C214yesmaximal, complemented
8#11748366C211yesmaximal, characteristic, complemented
9#11748364C211yesmaximal, characteristic, complemented
10#10615114—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C5×D12C2211yescharacteristic, complemented
13C3×#4149—24no—
14C3×#4149—24no—
15C2×C3×#713—21no—
16C2×C3×D4×D5—21no—
17C2×C3×D4×D5—22no—
18C2×C3×#709C2211yescharacteristic
19C2×C3×#709C2212yescomplemented
20C22×C3×D20C2211yescharacteristic
21C22×C3×D20C2212yescomplemented
22C2×#6710C2211yescharacteristic, complemented
23C22×C3⋊#168C2211yescharacteristic, complemented
24C5×#947—21no—
25C5×#940—21no—
26C5×#937—22no—
27C5×#930C2214yescomplemented
28C5×#926C2211yescharacteristic, complemented
29C2×#6656C2212yescomplemented
30#91299—24no—
31C2×#6652—24no—
32C2×#6652C2214yescomplemented
33C2×#6652C2212yescomplemented
34#91114C2214yescomplemented
35#90970C2212yescomplemented
36#90964C2212yescomplemented
37#90964C2214yescomplemented
38#84101—51nomaximal, Hall
39C2×#4149D311yescharacteristic, complemented
40C5×#535—31no—
41C2×#4078—31no—
42C22×#686—31no—
43#65523—31no—
44#65326—34no—
45#65325—34no—
46#65291—31no—
47#65290—31no—
48C23×C5×D3—21no—
49C23×C3×D5—21no—
50C23×C3×D5—22no—
51C3×C5×#41—22no—
52C3×C5×#41—22no—
53C2×C3×C5×Q8D411yescharacteristic
54C2×C3×C5×D4D412yes—
55C2×C3×C5×D4D411yescharacteristic
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2311yescharacteristic
58C2×C5×#67—24no—
59C2×C5×D12—22no—
60C2×C5×D12—22no—
61C2×C5×D12—21no—
62C2×C5×D12C2312yes—
63C2×C3×C22⋊D5—21no—
64C2×C3×C22⋊D5—22no—
65C3×#713—28no—
66C3×D4×D5—216no—
67C3×D4×D5—28no—
68C3×#709—22no—
69C3×#709—24no—
70C3×#709—24no—
71C3×#709D414yescomplemented
72C2×C3×D20—21no—
73C2×C3×D20—22no—
74C2×C3×D20—22no—
75C2×C3×D20—24no—
76C2×C3×D20—24no—
77C2×C3×D20D414yescomplemented
78C2×C3×D20C2312yes—
79C2×C3×D20C2311yescharacteristic
80C2×C3×C4×D5—21no—
81C2×C3×C4×D5—22no—
82C2×C3×Dic10C2311yescharacteristic
83#6710—22no—
84C2×C3⋊#168—22no—
85C2×C3⋊#168C2312yes—
86C2×C3⋊#168C2312yes—
87C5×#217—21no—
88C5×(C2×D3)⋊C4—24no—
89C5×C4⋊Dic3C2312yes—
90C4×C5×Dic3—21no—
91#6656—24no—
92#6652—28no—
93#6652—24no—
94C2×C3×H(5,𝔽2)—51no—
95#5830D511yescharacteristic, complemented
96C2×#949—51no—
97C22×#213—51no—
98#5238—54no—
99#5234—51no—
100#5233—51no—
101#5217—54no—
102#5215—51no—
103C2×C5×#41C2×D311yescharacteristic
104C22×C5×D4—31no—
105#4149—24no—
106#4149—24no—
107C2×#713—21no—
108C2×D4×D5—21no—
109C2×D4×D5—22no—
110C2×#709C2×D311yescharacteristic
111C2×#709C2×D312yescomplemented
112C22×D20C2×D311yescharacteristic
113C22×D20C2×D312yescomplemented
114C5×#127—61no—
115C5×#124—61no—
116C5×#121—34no—
117C5×#120—62no—
118C5×#117—31no—
119C2×#687—32no—
120#4078—64no—
121C2×#686—32no—
122C2×#686—34no—
123C2×#686—64no—
124C2×#681—31no—
125C22×#168—31no—
126C22⋊#168—34no—
127#3942—32no—
128#3941—34no—
129#3941—32no—
130#3008—151noSylow
131C23×C3×C5C2×D412yes—
132C23×C3×C5C2×D411yescharacteristic
133C22×C5×D3—22no—
134C22×C5×D3—42no—
135C22×C5×D3—82no—
136C22×C3×D5—21no—
137C22×C3×D5—22no—
138C22×C3×D5—28no—
139C22×C3×D5—24no—
140C22×C3×D5—82no—
141C22×C3×D5—84no—
142C3×C5×Q8—22no—
143C3×C5×D4—24no—
144C3×C5×D4—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2×D412yes—
150C2×C3×C4×C5C2×D412yes—
151C2×C3×C4×C5C2411yescharacteristic
152C5×#67—84no—
153C2×C5×Dic3—22no—
154C5×D12—22no—
155C5×D12—22no—
156C5×D12—42no—
157C3×C22⋊D5—28no—
158C3×C22⋊D5—24no—
159C2×C3×Dic5—21no—
160C3×D20—28no—
161C3×D20—24no—
162C3×D20—21no—
163C3×D20—22no—
164C3×D20C2×D412yes—
165C3×D20C2×D414yes—
166C3×C4×D5—28no—
167C3×C4×D5—24no—
168C3×Dic10—21no—
169C3×Dic10C2×D412yes—
170C3⋊#168—22no—
171C3⋊#168—22no—
172C3×H(5,𝔽2)—104no—
173C3×H(5,𝔽2)—104no—
174C2×C3×#41—51no—
175C2×C3×#41—52no—
176C2×C3×#41—101no—
177C2×C3×#41C2×D511yescharacteristic
178C22×C3×D4—51no—
179C22×C3×D4—52no—
180C22×C3×D4—102no—
181C22×C3×D4—101no—
182C22×D12C2×D511yescharacteristic, complemented
183#949—104no—
184#947—21no—
185C2×#215—52no—
186#940—21no—
187#937—22no—
188C2×#213—52no—
189C2×#213—54no—
190C2×#213—104no—
191#930C2×D514yescomplemented
192#926C2×D511yescharacteristic, complemented
193C2×#208—51no—
194C22×C3⋊C8—51no—
195#835—52no—
196#832—54no—
197#832—52no—
198#830—54no—
199C24×C5—61no—
200C23×D5—21no—
201C23×D5—22no—
202C5×#41—22no—
203C5×#41—22no—
204C2×C5×Q8#6711yescharacteristic
205C2×C5×D4—32no—
206C2×C5×D4—64no—
207C2×C5×D4—62no—
208C2×C5×D4—62no—
209C2×C5×D4—61no—
210C2×C5×D4#6712yes—
211C2×C5×D4#6711yescharacteristic
212C22×C4×C5C22×D312yes—
213C22×C4×C5C22×D311yescharacteristic
214C2×C22⋊D5—21no—
215C2×C22⋊D5—22no—
216#713—28no—
217D4×D5—216no—
218D4×D5—28no—
219#709—22no—
220#709—24no—
221#709—24no—
222#709#6714yescomplemented
223C2×D20—21no—
224C2×D20—22no—
225C2×D20—22no—
226C2×D20—24no—
227C2×D20—24no—
228C2×D20#6714yescomplemented
229C2×D20C22×D312yes—
230C2×D20C22×D311yescharacteristic
231C2×C4×D5—21no—
232C2×C4×D5—22no—
233C2×Dic10C22×D311yescharacteristic
234C5×C4⋊C4—32no—
235C5×C22⋊C4—61no—
236C5×C22⋊C4—64no—
237C42×C5—61no—
238#687—64no—
239#686—68no—
240#686—64no—
241#681—62no—
242C2×#168—32no—
243C2×#168—32no—
244C2×#168—62no—
245C2×H(5,𝔽2)—51no—
246C2×AGL(1,ℤ/8)—151no—
247C22×D8—151no—
248#535—31no—
249#449—154no—
250#447—154no—
251#418—151no—
252#416—151no—
253#408—151no—
254C22×C3×C5—22no—
255C22×C3×C5—22no—
256C22×C3×C5—24no—
257C22×C3×C5#12012yes—
258C22×C3×C5C22×D411yescharacteristic
259C2×C5×D3—42no—
260C2×C5×D3—44no—
261C2×C5×D3—84no—
262C2×C3×D5—24no—
263C2×C3×D5—22no—
264C2×C3×D5—44no—
265C2×C3×D5—42no—
266C2×C3×D5—84no—
267C2×C3×D5—88no—
268C3×C4×C5—22no—
269C3×C4×C5#12012yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C22×D411yescharacteristic
272C3×Dic5—22no—
273C5×Dic3—42no—
274C24×C3—102no—
275C24×C3—101no—
276C23×D3—21no—
277C3×#41—22no—
278C3×#41—22no—
279C3×#41—54no—
280C3×#41—104no—
281C3×#41—104no—
282C3×#41—108no—
283C3×#41—102no—
284C2×C3×Q8—51no—
285C2×C3×Q8C22⋊D511yescharacteristic
286C2×C3×D4—54no—
287C2×C3×D4—52no—
288C2×C3×D4—51no—
289C2×C3×D4—104no—
290C2×C3×D4—104no—
291C2×C3×D4—1016no—
292C2×C3×D4—108no—
293C2×C3×D4—102no—
294C2×C3×D4—101no—
295C2×C3×D4—102no—
296C2×C3×D4—102no—
297C2×C3×D4—101no—
298C2×C3×D4C22⋊D512yes—
299C2×C3×D4C22⋊D511yescharacteristic
300C22×C3×C4—102no—
301C22×C3×C4—101no—
302C22×C3×C4C22×D512yes—
303C22×C3×C4C22×D511yescharacteristic
304C2×#67—24no—
305C2×D12—22no—
306C2×D12—22no—
307C2×D12—21no—
308C2×D12C22×D512yes—
309#217—21no—
310#215—104no—
311#213—108no—
312#213—104no—
313(C2×D3)⋊C4—24no—
314C4⋊Dic3C22×D512yes—
315C4×Dic3—21no—
316#208—102no—
317C2×C3⋊C8—52no—
318C2×C3⋊C8—52no—
319C2×C3⋊C8—102no—
320C23×C5—62no—
321C23×C5—122no—
322C23×C5—242no—
323C23×C5C2×#6712yes—
324C23×C5C2×#6711yescharacteristic
325C22×D5—21no—
326C22×D5—22no—
327C22×D5—28no—
328C22×D5—24no—
329C22×D5—82no—
330C22×D5—84no—
331C5×Q8—22no—
332C5×D4—24no—
333C5×D4—22no—
334C5×D4—62no—
335C5×D4—62no—
336C5×D4—122no—
337C5×D4—244no—
338C2×C4×C5—22no—
339C2×C4×C5—22no—
340C2×C4×C5—62no—
341C2×C4×C5C2×#6712yes—
342C2×C4×C5C2×#6711yescharacteristic
343C2×C4×C5C2×#6712yes—
344C2×C4×C5C2×#6712yes—
345C2×C4×C5C23×D311yescharacteristic
346C22⋊D5—28no—
347C22⋊D5—24no—
348C2×Dic5—21no—
349D20—28no—
350D20—24no—
351D20—21no—
352D20—22no—
353D20C2×#6712yes—
354D20C2×#6714yes—
355C4×D5—28no—
356C4×D5—24no—
357Dic10—21no—
358Dic10C2×#6712yes—
359#168—62no—
360#168—62no—
361H(5,𝔽2)—104no—
362H(5,𝔽2)—104no—
363C2×#41—51no—
364C2×#41—52no—
365C2×#41—101no—
366C2×#41D3×D511yescharacteristic
367C22×D4—31no—
368C22×D4—51no—
369C22×D4—52no—
370C22×D4—102no—
371C22×D4—101no—
372AGL(1,ℤ/8)—304no—
373C2×SD16—152no—
374C2×D8—154no—
375C2×D8—152no—
376C2×D8—304no—
377C2×M16—151no—
378C22×C8—151no—
379#127—61no—
380#124—61no—
381#121—34no—
382#120—62no—
383#117—31no—
384#103—152no—
385#102—152no—
386#102—154no—
387C22⋊C8—154no—
388C2×C3×C5—22no—
389C2×C3×C5—42no—
390C2×C3×C5C2×#12011yescharacteristic
391C2×C3×C5#57612yes—
392C3×D5—44no—
393C3×D5—42no—
394C5×D3—82no—
395C23×C3—108no—
396C23×C3—102no—
397C23×C3—101no—
398C23×C3—104no—
399C23×C3—402no—
400C23×C3—404no—
401C23×C3C2×C22⋊D512yes—
402C23×C3C2×C22⋊D511yescharacteristic
403C22×D3—22no—
404C22×D3—42no—
405C22×D3—82no—
406C3×Q8—22no—
407C3×Q8—52no—
408C3×Q8—101no—
409C3×D4—24no—
410C3×D4—22no—
411C3×D4—54no—
412C3×D4—52no—
413C3×D4—108no—
414C3×D4—108no—
415C3×D4—104no—
416C3×D4—101no—
417C3×D4—102no—
418C3×D4—104no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—108no—
422C2×C3×C4—101no—
423C2×C3×C4—104no—
424C2×C3×C4D4×D512yes—
425C2×C3×C4D4×D512yes—
426C2×C3×C4C2×C22⋊D512yes—
427C2×C3×C4C2×C22⋊D511yescharacteristic
428C2×C3×C4C23×D511yescharacteristic
429#67—84no—
430C2×Dic3—22no—
431D12—22no—
432D12—22no—
433D12—42no—
434C3⋊C8—102no—
435C3⋊C8—102no—
436C22×C5—22no—
437C22×C5—24no—
438C22×C5—22no—
439C22×C5—124no—
440C22×C5—122no—
441C22×C5—244no—
442C22×C5#95312yes—
443C22×C5C22×#6711yescharacteristic
444C2×D5—24no—
445C2×D5—22no—
446C2×D5—44no—
447C2×D5—42no—
448C2×D5—84no—
449C2×D5—88no—
450C4×C5—22no—
451C4×C5—122no—
452C4×C5#95312yes—
453C4×C5C22×#6711yescharacteristic
454C4×C5C22×#6711yescharacteristic
455Dic5—22no—
456C24—61no—
457C24—102no—
458C24—101no—
459#41—22no—
460#41—22no—
461#41—54no—
462#41—104no—
463#41—104no—
464#41—108no—
465#41—102no—
466C2×Q8—51no—
467C2×Q8#120111yescharacteristic
468C2×D4—32no—
469C2×D4—54no—
470C2×D4—52no—
471C2×D4—51no—
472C2×D4—64no—
473C2×D4—61no—
474C2×D4—62no—
475C2×D4—62no—
476C2×D4—104no—
477C2×D4—104no—
478C2×D4—1016no—
479C2×D4—108no—
480C2×D4—102no—
481C2×D4—101no—
482C2×D4—102no—
483C2×D4—101no—
484C2×D4—102no—
485C2×D4#120112yes—
486C2×D4#120111yescharacteristic
487C22×C4—102no—
488C22×C4—101no—
489C22×C4C2×D3×D512yes—
490C22×C4C2×D3×D511yescharacteristic
491SD16—304no—
492D8—308no—
493D8—304no—
494M16—302no—
495C2×C8—152no—
496C2×C8—152no—
497C2×C8—302no—
498C4⋊C4—32no—
499C22⋊C4—61no—
500C22⋊C4—64no—
501C42—61no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—24no—
506C22×C3—104no—
507C22×C3—102no—
508C22×C3—204no—
509C22×C3—202no—
510C22×C3—404no—
511C22×C3—408no—
512C22×C3#408312yes—
513C22×C3C22×C22⋊D511yescharacteristic
514C2×D3—42no—
515C2×D3—44no—
516C2×D3—84no—
517C3×C4—22no—
518C3×C4—102no—
519C3×C4#408312yes—
520C3×C4C2×D4×D511yescharacteristic
521C3×C4C2×D4×D511yescharacteristic
522Dic3—42no—
523C2×C5—22no—
524C2×C5—42no—
525C2×C5—242no—
526C2×C5#583512yes—
527C2×C5C2×#95311yescharacteristic
528D5—44no—
529D5—42no—
530C23—62no—
531C23—108no—
532C23—102no—
533C23—101no—
534C23—104no—
535C23—122no—
536C23—242no—
537C23—402no—
538C23—404no—
539C23C2×#120112yes—
540C23C2×#120111yescharacteristic
541Q8—22no—
542Q8—52no—
543Q8—101no—
544D4—24no—
545D4—22no—
546D4—54no—
547D4—52no—
548D4—62no—
549D4—62no—
550D4—108no—
551D4—108no—
552D4—104no—
553D4—101no—
554D4—102no—
555D4—104no—
556D4—122no—
557D4—244no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—62no—
561C2×C4—108no—
562C2×C4—101no—
563C2×C4—104no—
564C2×C4C2×#120112yes—
565C2×C4C2×#120111yescharacteristic
566C2×C4D5×#6712yes—
567C2×C4D5×#6712yes—
568C2×C4C22×D3×D511yescharacteristic
569C8—302no—
570C8—302no—
571C2×C3—22no—
572C2×C3—42no—
573C2×C3—204no—
574C2×C3—202no—
575C2×C3C2×#408311yescharacteristic
576C2×C3#6616912yes—
577D3—82no—
578C5#8410111yescharacteristic, complemented, Sylow
579C22—22no—
580C22—24no—
581C22—22no—
582C22—104no—
583C22—102no—
584C22—124no—
585C22—122no—
586C22—204no—
587C22—202no—
588C22—244no—
589C22—404no—
590C22—408no—
591C22#9156312yes—
592C22C22×#120111yescharacteristic
593C4—22no—
594C4—102no—
595C4—122no—
596C4#9156312yes—
597C4C2×D5×#6711yescharacteristic
598C4C2×D5×#6711yescharacteristic
599C3#1061511411yescharacteristic, complemented, Sylow
600C2—22no—
601C2—42no—
602C2—204no—
603C2—202no—
604C2—242no—
605C2#1175530112yes—
606C2C2×#9156311yescharacteristic
subgroupG/H
centerC22C22×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
hypercenterC2×#41D3×D5
hypocenterC3×C5#3008
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4149C2semidirect
2. C5×#5830C2semidirect
3. C2×#91299C2semidirect
4. C22×#6652C2semidirect
5. #11751335C2semidirect
6. #11748511C2semidirect
7. #11748510C2semidirect
8. #11748366C2semidirect
9. #11748364C2semidirect
12. C22×C5×D12C22semidirect
19. C2×C3×#709C22semidirect
21. C22×C3×D20C22semidirect
22. C2×#6710C22semidirect
23. C22×C3⋊#168C22semidirect
27. C5×#930C22semidirect
28. C5×#926C22semidirect
29. C2×#6656C22semidirect
32. C2×#6652C22semidirect
33. C2×#6652C22semidirect
34. #91114C22semidirect
35. #90970C22semidirect
36. #90964C22semidirect
37. #90964C22semidirect
39. C2×#4149D3semidirect
71. C3×#709D4semidirect
77. C2×C3×D20D4semidirect
95. #5830D5semidirect
111. C2×#709C2×D3semidirect
113. C22×D20C2×D3semidirect
182. C22×D12C2×D5semidirect
191. #930C2×D5semidirect
192. #926C2×D5semidirect
222. #709#67semidirect
228. C2×D20#67semidirect
502. C3×C5#3008semidirect
578. C5#84101semidirect
599. C3#10615114semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41492
C5×#58302
C2×#912992
C22×#66522
#117513352
#117485112
#117485102
#117483662
#117483642
#106151143
#841015