Finite Group Databasebeta

← D5×#3249 #12972109

Subgroups — D5×#3249 #12972109

Order 1280.

1898 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#533C212yesmaximal
2C5×#3249C211yesmaximal, characteristic, complemented
3#10615104C212yesmaximal, complemented
4#10612849C214yesmaximal, complemented
5D5×#477C214yesmaximal, complemented
6D5×#467C214yesmaximal
7#10612478C214yesmaximal, complemented
8#10612217C214yesmaximal, complemented
9D5×#462C214yesmaximal, complemented
10#10610768C211yesmaximal, characteristic, complemented
11C2×D5×C4⋊C8C211yesmaximal, characteristic
12C5×#533C2212yes—
13C2×D5×Q16C2214yes—
14C22×C8×D5C2212yes—
15#66062C2212yes—
16#66062C2212yes—
17D5×C4⋊Q8C2212yes—
18D5×C4⋊Q8C2212yes—
19#66022—22no—
20D5×#125—22no—
21D5×C22⋊Q8—22no—
22D5×C22⋊Q8C2214yes—
23#65956—22no—
24#65956C2214yes—
25#65912—22no—
26#65912C2214yes—
27C4×Q8×D5—22no—
28C4×Q8×D5C2214yes—
29D5×#117—22no—
30#65851—22no—
31C2×C42×D5C2211yescharacteristic
32#65804C2212yes—
33C5×#477C2214yescomplemented
34C5×#467C2214yes—
35C5×#462C2214yescomplemented
36C2×C5×C4⋊C8C2211yescharacteristic
37#65477C2214yes—
38#65471C2218yescomplemented
39#65405C2214yes—
40C2×#3946C2212yes—
41#65380C2214yescomplemented
42#65379C2214yescomplemented
43#65340C2218yes—
44#65320C2214yescomplemented
45#65310C2214yescomplemented
46C2×C4⋊#168C2211yescharacteristic
47#65176C2214yes—
48D5×#107C2214yes—
49#65164C2218yescomplemented
50#65139C2214yescomplemented
51#65138C2214yescomplemented
52D5×C4⋊C8C2214yes—
53D5×C4⋊C8C2214yes—
54#65102C2218yes—
55D5×#103C2218yes—
56#65083C2218yescomplemented
57#65082C2218yescomplemented
58C2×#3249—51nomaximal, Sylow
59C2×Q8×D5—22no—
60C2×Q8×D5C2314yes—
61C22×C4×D5D411yescharacteristic
62C22×C4×D5D411yescharacteristic
63C22×C4×D5C2311yescharacteristic
64C2×C5×Q16C2314yes—
65C22×C5×C8C2312yes—
66C5×C4⋊Q8C2312yes—
67C5×C4⋊Q8C2312yes—
68C5×#125—22no—
69C5×C22⋊Q8—22no—
70C5×C22⋊Q8C2314yes—
71C4×C5×Q8—22no—
72C4×C5×Q8C2314yes—
73C5×#117—22no—
74C2×C42×C5C2311yescharacteristic
75#4092—22no—
76#4092C2314yes—
77Q8×Dic5—22no—
78Q8×Dic5C2314yes—
79#4090—22no—
80#4090C2314yes—
81C2×#689C2318yes—
82#4072—22no—
83#4070—22no—
84#4070C2314yes—
85C2×C4×Dic5D411yescharacteristic
86C2×C4×Dic5D411yescharacteristic
87C2×C4×Dic5C2311yescharacteristic
88C22×#168C2312yes—
89D5×Q16—48no—
90C2×Dic20C2314yes—
91C2×C8×D5—44no—
92C2×C8×D5C2314yes—
93C2×C8×D5C2314yes—
94#4043—22no—
95#4043C2314yes—
96#4042—24no—
97#4042—24no—
98#4038—22no—
99#4038—22no—
100D5×C4⋊C4—22no—
101D5×C4⋊C4—22no—
102D5×C4⋊C4—22no—
103D5×C4⋊C4—24no—
104D5×C4⋊C4—24no—
105D5×C4⋊C4C2314yes—
106#4036—22no—
107#4036—22no—
108#4035—24no—
109#4035—24no—
110#4034—22no—
111#4034—22no—
112#4034—22no—
113#4034C2314yes—
114#4033—22no—
115#4033—24no—
116#4033—24no—
117#4033C2314yes—
118D5×C22⋊C4—24no—
119#4024—24no—
120#4024—24no—
121#4023—24no—
122#4018—24no—
123C42×D5—24no—
124C42×D5D412yes—
125C42×D5D412yes—
126C42×D5C2312yes—
127C42×D5C2312yes—
128#4016—22no—
129#4015C2312yes—
130#4015C2312yes—
131C4×Dic10—22no—
132C4×Dic10—24no—
133C4×Dic10—24no—
134C4×Dic10C2314yes—
135C5×#107C2314yes—
136C5×C4⋊C8C2314yes—
137C5×#103C2318yes—
138#3967C2318yes—
139#3950C2314yes—
140#3948C2318yes—
141#3946C2318yes—
142#3942C2318yes—
143#3939C2318yes—
144C4⋊#168C2314yes—
145C2×#533—52no—
146#3249—54no—
147#3249—54no—
148#3249—54no—
149#3249—51no—
150#3249—52no—
151#3249D511yescharacteristic, direct product factor
152C2×#477—54no—
153C2×#467—54no—
154C2×#462—54no—
155C22×C4⋊C8—51no—
156C23×D5C2×D411yescharacteristic
157C2×C5×Q8—22no—
158C2×C5×Q8C2414yes—
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2411yescharacteristic
162C22×Dic5C2×D411yescharacteristic
163C22×Dic5C2×D411yescharacteristic
164C22×Dic5C2×D411yescharacteristic
165Q8×D5—28no—
166Q8×D5—28no—
167Q8×D5—48no—
168C2×C4×D5—22no—
169C2×C4×D5—22no—
170C2×C4×D5—24no—
171C2×C4×D5—24no—
172C2×C4×D5—42no—
173C2×C4×D5Q1618yescomplemented
174C2×C4×D5C2×D412yes—
175C2×C4×D5C2×D411yescharacteristic
176C2×C4×D5C2×D411yescharacteristic
177C2×C4×D5C2×D412yes—
178C2×C4×D5C2×D412yes—
179C2×C4×D5C2×D412yes—
180C2×C4×D5C2411yescharacteristic
181C2×C4×D5C2411yescharacteristic
182C2×Dic10—22no—
183C2×Dic10—24no—
184C2×Dic10—24no—
185C2×Dic10C2414yes—
186C5×Q16—44no—
187C2×C5×C8—42no—
188C2×C5×C8C2414yes—
189C5×C4⋊C4—22no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—24no—
193C5×C4⋊C4—24no—
194C5×C4⋊C4C2414yes—
195C5×C22⋊C4—24no—
196C42×C5—24no—
197C42×C5C2×D412yes—
198C42×C5C2412yes—
199C22⋊Dic5—24no—
200#689—48no—
201#685—24no—
202#685—24no—
203C4⋊Dic5—22no—
204C4⋊Dic5—22no—
205C4⋊Dic5—22no—
206C4⋊Dic5—24no—
207C4⋊Dic5—24no—
208C4⋊Dic5C2414yes—
209#683—24no—
210#683—24no—
211#683—24no—
212#683—24no—
213#683—24no—
214#683—24no—
215#683—24no—
216#683—24no—
217C4×Dic5—24no—
218C4×Dic5—24no—
219C4×Dic5—24no—
220C4×Dic5C2×D412yes—
221C4×Dic5C2×D412yes—
222C4×Dic5C2×D412yes—
223C4×Dic5C2×D412yes—
224C4×Dic5C2×D412yes—
225C4×Dic5C2412yes—
226C2×#168—42no—
227C2×#168C2414yes—
228Dic20—44no—
229C8×D5—28no—
230C8×D5—44no—
231C22×Q16—54no—
232C23×C8—52no—
233#533—52no—
234#533—52no—
235#533—54no—
236#533—54no—
237#533—52no—
238#533—102no—
239#533—102no—
240#533—102no—
241#533—102no—
242#533C2×D512yes—
243C2×C4⋊Q8—52no—
244C2×C4⋊Q8—52no—
245C2×#125—102no—
246C2×C22⋊Q8—54no—
247C2×C22⋊Q8—102no—
248C2×C4×Q8—54no—
249C2×C4×Q8—102no—
250C2×#117—102no—
251C22×C42—51no—
252#477—54no—
253#477—54no—
254#477—58no—
255#477—54no—
256#477—58no—
257#477C2×D514yescomplemented
258#467—54no—
259#467—54no—
260#467—58no—
261#467—54no—
262#467—58no—
263#467C2×D514yes—
264#462—58no—
265#462—54no—
266#462—54no—
267#462—54no—
268#462—58no—
269#462C2×D514yescomplemented
270C2×#107—54no—
271C2×C4⋊C8—52no—
272C2×C4⋊C8—51no—
273C2×C4⋊C8—54no—
274C2×C4⋊C8—54no—
275C2×C4⋊C8C2×D511yescharacteristic
276C2×#103—58no—
277C23×C5C22×D411yescharacteristic
278C22×D5—22no—
279C22×D5—22no—
280C22×D5C2×Q1614yes—
281C22×D5C22×D411yescharacteristic
282C22×D5C22×D411yescharacteristic
283C5×Q8—24no—
284C5×Q8—24no—
285C5×Q8—44no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—24no—
289C2×C4×C5—24no—
290C2×C4×C5—41no—
291C2×C4×C5C2×Q1614yescomplemented
292C2×C4×C5C22×D411yescharacteristic
293C2×C4×C5C22×D412yes—
294C2×C4×C5C22×D412yes—
295C2×C4×C5C2511yescharacteristic
296C2×Dic5—22no—
297C2×Dic5—24no—
298C2×Dic5—24no—
299C2×Dic5—41no—
300C2×Dic5C2×Q1614yescomplemented
301C2×Dic5C22×D411yescharacteristic
302C2×Dic5C22×D412yes—
303C2×Dic5C22×D411yescharacteristic
304C2×Dic5C22×D412yes—
305C4×D5—22no—
306C4×D5—24no—
307C4×D5—21no—
308C4×D5—21no—
309C4×D5—22no—
310C4×D5—42no—
311C4×D5—48no—
312C4×D5—48no—
313C4×D5C2×Q1618yes—
314C4×D5C2×Q1618yescomplemented
315C4×D5H(5,𝔽2)12yes—
316C4×D52−1+412yes—
317Dic10—24no—
318Dic10—24no—
319Dic10—48no—
320Dic10—44no—
321Dic10—48no—
322C5×C8—24no—
323#168—24no—
324C22×Q8—54no—
325C22×Q8—102no—
326C23×C4—51no—
327C23×C4—51no—
328C23×C4—51no—
329C2×Q16—58no—
330C2×Q16—54no—
331C2×Q16—208no—
332C2×Q16C22×D514yes—
333C22×C8—52no—
334C22×C8—54no—
335C22×C8—54no—
336C22×C8—204no—
337C22×C8C22×D512yes—
338C4⋊Q8—54no—
339C4⋊Q8—54no—
340C4⋊Q8—52no—
341C4⋊Q8—52no—
342C4⋊Q8—102no—
343C4⋊Q8—102no—
344C4⋊Q8—102no—
345C4⋊Q8—102no—
346C4⋊Q8C22×D512yes—
347C4⋊Q8C22×D512yes—
348#125—22no—
349#125—104no—
350#125—104no—
351#125—102no—
352#125—102no—
353#125—102no—
354C22⋊Q8—22no—
355C22⋊Q8—54no—
356C22⋊Q8—54no—
357C22⋊Q8—54no—
358C22⋊Q8—104no—
359C22⋊Q8—104no—
360C22⋊Q8—104no—
361C22⋊Q8—104no—
362C22⋊Q8—102no—
363C22⋊Q8—102no—
364C22⋊Q8—102no—
365C22⋊Q8C22×D514yes—
366C4×Q8—22no—
367C4×Q8—54no—
368C4×Q8—54no—
369C4×Q8—54no—
370C4×Q8—104no—
371C4×Q8—104no—
372C4×Q8—104no—
373C4×Q8—102no—
374C4×Q8—104no—
375C4×Q8—102no—
376C4×Q8—102no—
377C4×Q8C22×D514yes—
378#117—22no—
379#117—104no—
380#117—104no—
381#117—102no—
382#117—102no—
383#117—102no—
384C2×C4⋊C4—54no—
385C2×C4⋊C4—102no—
386C2×C4⋊C4—102no—
387C2×C4⋊C4—104no—
388C2×C4⋊C4—104no—
389C2×C4⋊C4—102no—
390C2×C22⋊C4—104no—
391C2×C42—51no—
392C2×C42—51no—
393C2×C42—51no—
394C2×C42—52no—
395C2×C42—52no—
396C2×C42—52no—
397C2×C42—52no—
398C2×C42—104no—
399C2×C42C22×D511yescharacteristic
400#107—54no—
401#107—58no—
402#107C22×D514yes—
403C4⋊C8—58no—
404C4⋊C8—54no—
405C4⋊C8C22×D514yes—
406#103—58no—
407#103—58no—
408#103—58no—
409#103C22×D518yes—
410C22×C5—21no—
411C22×C5—21no—
412C22×C5C22×Q1612yes—
413C22×C5C23×D411yescharacteristic
414C2×D5—21no—
415C2×D5—21no—
416C2×D5—28no—
417C2×D5#53712yes—
418C2×D5C22×Q1612yes—
419C2×D5C22×Q1612yes—
420C2×D5#57512yes—
421C4×C5—21no—
422C4×C5—22no—
423C4×C5—44no—
424C4×C5—44no—
425C4×C5C22×Q1614yes—
426C4×C5C2×H(5,𝔽2)11yescharacteristic
427C4×C5C2×2−1+411yescharacteristic
428Dic5—21no—
429Dic5—22no—
430Dic5—44no—
431Dic5—44no—
432Dic5#53712yes—
433Dic5C22×Q1614yes—
434C24—51no—
435C2×Q8—22no—
436C2×Q8—54no—
437C2×Q8—104no—
438C2×Q8—104no—
439C2×Q8—108no—
440C2×Q8—108no—
441C2×Q8—102no—
442C2×Q8—208no—
443C2×Q8C23×D514yes—
444C22×C4—51no—
445C22×C4—52no—
446C22×C4—52no—
447C22×C4—51no—
448C22×C4—58no—
449C22×C4—51no—
450C22×C4—51no—
451C22×C4—51no—
452C22×C4—52no—
453C22×C4—51no—
454C22×C4—51no—
455C22×C4—52no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—104no—
459C22×C4—104no—
460C22×C4—202no—
461C22×C4D4×D511yescharacteristic
462C22×C4D4×D511yescharacteristic
463C22×C4C23×D511yescharacteristic
464Q16—44no—
465Q16—204no—
466Q16—208no—
467C2×C8—42no—
468C2×C8—54no—
469C2×C8—108no—
470C2×C8—204no—
471C2×C8—202no—
472C2×C8C23×D514yes—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—54no—
479C4⋊C4—104no—
480C4⋊C4—104no—
481C4⋊C4—104no—
482C4⋊C4—104no—
483C4⋊C4—104no—
484C4⋊C4—104no—
485C4⋊C4—104no—
486C4⋊C4—104no—
487C4⋊C4—102no—
488C4⋊C4—102no—
489C4⋊C4—104no—
490C4⋊C4—104no—
491C4⋊C4—102no—
492C4⋊C4C23×D514yes—
493C22⋊C4—24no—
494C22⋊C4—104no—
495C22⋊C4—104no—
496C22⋊C4—104no—
497C42—24no—
498C42—52no—
499C42—52no—
500C42—52no—
501C42—52no—
502C42—52no—
503C42—52no—
504C42—104no—
505C42—104no—
506C42—104no—
507C42D4×D512yes—
508C42C23×D512yes—
509C2×C5—22no—
510C2×C5C2×#53711yescharacteristic
511C2×C5C23×Q1611yescharacteristic
512C2×C5C2×#57511yescharacteristic
513D5—22no—
514D5#324914yesdirect product factor
515C23—51no—
516C23—54no—
517C23—51no—
518C23—102no—
519C23—102no—
520C23C2×D4×D511yescharacteristic
521Q8—24no—
522Q8—24no—
523Q8—44no—
524Q8—104no—
525Q8—104no—
526Q8—204no—
527Q8—208no—
528Q8—208no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—54no—
535C2×C4—52no—
536C2×C4—51no—
537C2×C4—58no—
538C2×C4—58no—
539C2×C4—52no—
540C2×C4—52no—
541C2×C4—51no—
542C2×C4—52no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—104no—
546C2×C4—102no—
547C2×C4—101no—
548C2×C4—101no—
549C2×C4—104no—
550C2×C4—104no—
551C2×C4—202no—
552C2×C4—208no—
553C2×C4—208no—
554C2×C4—201no—
555C2×C4D5×Q1614yescomplemented
556C2×C4C2×D4×D511yescharacteristic
557C2×C4C2×D4×D512yes—
558C2×C4C2×D4×D512yes—
559C2×C4C24×D511yescharacteristic
560C8—24no—
561C8—104no—
562C5C2×#324911yescharacteristic, complemented, Sylow
563C22—21no—
564C22—21no—
565C22—52no—
566C22—52no—
567C22—52no—
568C22—52no—
569C22—108no—
570C22—101no—
571C22—101no—
572C22C2×D5×Q1612yes—
573C22C22×D4×D511yescharacteristic
574C4—21no—
575C4—22no—
576C4—44no—
577C4—44no—
578C4—52no—
579C4—54no—
580C4—101no—
581C4—102no—
582C4—204no—
583C4—204no—
584C4C2×D5×Q1614yes—
585C4D5×H(5,𝔽2)11yescharacteristic
586C4D5×2−1+411yescharacteristic
587C2—22no—
588C2—54no—
589C2—102no—
590C2D5×#53711yescharacteristic
591C2C22×D5×Q1611yescharacteristic
592C2D5×#57511yescharacteristic
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#3249C2
socleC22×C5C23×D4
prime radicalC23×D5C2×D4
hypercenter#3249D5
hypocenterC5C2×#3249
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
2. C5×#3249C2semidirect
3. #10615104C2semidirect
4. #10612849C2semidirect
5. D5×#477C2semidirect
7. #10612478C2semidirect
8. #10612217C2semidirect
9. D5×#462C2semidirect
10. #10610768C2semidirect
33. C5×#477C22semidirect
35. C5×#462C22semidirect
38. #65471C22semidirect
41. #65380C22semidirect
42. #65379C22semidirect
44. #65320C22semidirect
45. #65310C22semidirect
49. #65164C22semidirect
50. #65139C22semidirect
51. #65138C22semidirect
56. #65083C22semidirect
57. #65082C22semidirect
151. #3249D5direct (N × H)
173. C2×C4×D5Q16semidirect
257. #477C2×D5semidirect
269. #462C2×D5semidirect
291. C2×C4×C5C2×Q16semidirect
300. C2×Dic5C2×Q16semidirect
314. C4×D5C2×Q16semidirect
514. D5#3249direct (N × H)
555. C2×C4D5×Q16semidirect
562. C5C2×#3249semidirect

Maximal subgroups

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

subgroupindex
D5×#5332
C5×#32492
#106151042
#106128492
D5×#4772
D5×#4672
#106124782
#106122172
D5×#4622
#106107682
C2×D5×C4⋊C82
C2×#32495