Finite Group Databasebeta

← D5×#5500 #14481756

Subgroups — D5×#5500 #14481756

Order 1920.

2160 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756259C211yesmaximal, characteristic, complemented
2C5×#5500C211yesmaximal, characteristic, complemented
3C2×C3×D5×C4⋊C4C211yesmaximal, characteristic
4#11754618C212yesmaximal, complemented
5D5×#926C212yesmaximal
6D5×#924C214yesmaximal
7#11754181C214yesmaximal, complemented
8D5×#890C214yesmaximal, complemented
9D5×#888C214yesmaximal, complemented
10#11753369C214yesmaximal, complemented
11#11753368C214yesmaximal, complemented
12D5×#544—31nomaximal, Hall
13C22×C3×C4×D5C2212yes—
14C22×C3×C4×D5C2211yescharacteristic
15C2×D5×Dic6C2214yes—
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17#91815C2212yes—
18#91813C2214yes—
19#91779C2214yescomplemented
20#91777C2214yescomplemented
21C5×#926C2212yes—
22C5×#924C2214yes—
23C5×#890C2214yescomplemented
24C5×#888C2214yescomplemented
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×#683C2212yes—
27C3×D5×C4⋊C4C2214yes—
28C3×D5×C4⋊C4C2214yes—
29#91574C2218yes—
30D5×#217C2214yes—
31#91528C2214yes—
32#91470C2214yescomplemented
33#91469C2214yescomplemented
34D5×C4⋊Dic3C2214yes—
35D5×C4⋊Dic3C2212yes—
36D5×C4⋊Dic3C2212yes—
37D5×Dic3⋊C4C2218yes—
38C4×D5×Dic3C2214yes—
39#91384C2214yescomplemented
40#91381C2214yescomplemented
41#91360C2214yes—
42#91359C2214yes—
43#91358C2212yes—
44#91358C2212yes—
45#91357C2214yes—
46#91349C2218yes—
47#91335C2218yescomplemented
48#91334C2218yescomplemented
49#91329C2218yescomplemented
50#91328C2218yescomplemented
51C2×#5500—51nomaximal, Hall
52C5×#544—31no—
53#66063—34no—
54D5×C4⋊Q8—34no—
55#66023—34no—
56D5×#125—34no—
57D5×C22⋊Q8—34no—
58#65957—34no—
59D5×#117—32no—
60#65853—32no—
61C2×D5×C4⋊C4D311yescharacteristic
62#65834—31no—
63C23×C3×D5C2311yescharacteristic
64C22×C3×C4×C5C2312yes—
65C22×C3×C4×C5C2311yescharacteristic
66C2×Dic30C2314yes—
67C2×C5×Dic6C2314yes—
68C22×C3×Dic5C2312yes—
69C22×C3×Dic5C2311yescharacteristic
70C2×C3×C4×D5—44no—
71C2×C3×C4×D5Q818yescomplemented
72C2×C3×C4×D5C2314yes—
73C2×C3×C4×D5C2314yes—
74C2×C3×C4×D5C2312yes—
75C2×C3×C4×D5C2312yes—
76C2×#1204C2318yes—
77C2×D5×Dic3C2314yes—
78C2×D5×Dic3C2314yes—
79D5×Dic6—48no—
80C3×C5×C4⋊C4C2314yes—
81#6719C2314yes—
82C4⋊Dic15C2312yes—
83C4⋊Dic15C2314yes—
84C4⋊Dic15C2312yes—
85#6712C2318yes—
86C4×Dic15C2314yes—
87C5×#217C2314yes—
88C5×C4⋊Dic3C2312yes—
89C5×C4⋊Dic3C2314yes—
90C5×C4⋊Dic3C2312yes—
91C5×Dic3⋊C4C2318yes—
92C4×C5×Dic3C2314yes—
93C3×C4⋊Dic5C2314yes—
94C3×#683C2318yes—
95Dic3⋊Dic5C2318yes—
96#6669C2318yes—
97#6668C2314yes—
98#6668C2314yes—
99#6668C2318yes—
100#6665C2314yes—
101#6665C2314yes—
102Dic3×Dic5C2314yes—
103Dic3×Dic5C2314yes—
104C22×C3×C4⋊C4—51no—
105C2×#926—52no—
106C2×#924—54no—
107#5500—51no—
108#5500—52no—
109#5500—54no—
110#5500—54no—
111#5500—54no—
112#5500D511yescharacteristic, direct product factor
113C2×#890—54no—
114C2×#888—54no—
115C2×Q8×D5—34no—
116C22×C4×D5C2×D312yes—
117C22×C4×D5C2×D311yescharacteristic
118C5×C4⋊Q8—34no—
119C5×#125—34no—
120C5×C22⋊Q8—34no—
121C5×#117—32no—
122C2×C5×C4⋊C4C2×D311yescharacteristic
123#4092—34no—
124#4090—38no—
125#4072—32no—
126C2×C4⋊Dic5C2×D311yescharacteristic
127#4070—34no—
128C2×#683C2×D312yes—
129#4043—34no—
130#4042—38no—
131#4038—32no—
132#4038—32no—
133D5×C4⋊C4—38no—
134D5×C4⋊C4—32no—
135D5×C4⋊C4—32no—
136D5×C4⋊C4—34no—
137D5×C4⋊C4C2×D314yes—
138D5×C4⋊C4C2×D314yes—
139#4036—34no—
140#4036—34no—
141#4035—38no—
142#4035—38no—
143#4034—34no—
144#4034—34no—
145#4034—38no—
146D5×C22⋊C4—34no—
147#4024—38no—
148#4023—34no—
149#4018—34no—
150C42×D5—34no—
151#4016—34no—
152#4015—34no—
153C2×#544—151noSylow
154C23×C3×C5C2411yescharacteristic
155C22×C3×D5—22no—
156C22×C3×D5—22no—
157C22×C3×D5C2×Q814yes—
158C22×C3×D5C2411yescharacteristic
159C22×C3×D5C2411yescharacteristic
160C2×C3×C4×C5—42no—
161C2×C3×C4×C5C2×Q814yescomplemented
162C2×C3×C4×C5C2414yes—
163C2×C3×C4×C5C2412yes—
164C2×Dic15C2414yes—
165C2×Dic15C2414yes—
166Dic30—44no—
167C2×C5×Dic3C2414yes—
168C2×C5×Dic3C2414yes—
169C5×Dic6—44no—
170C2×C3×Dic5—42no—
171C2×C3×Dic5C2×Q814yescomplemented
172C2×C3×Dic5C2414yes—
173C2×C3×Dic5C2412yes—
174C3×C4×D5—28no—
175C3×C4×D5—44no—
176C3×C4×D5C2×Q818yes—
177C3×C4×D5C2×Q818yescomplemented
178#1204—48no—
179D5×Dic3—28no—
180D5×Dic3—28no—
181C23×C3×C4—51no—
182C23×C3×C4—52no—
183C22×Dic6—54no—
184C2×C3×C4⋊C4—52no—
185C2×C3×C4⋊C4—51no—
186C2×C3×C4⋊C4—54no—
187C2×C3×C4⋊C4—54no—
188C2×C3×C4⋊C4C2×D511yescharacteristic
189C2×#217—54no—
190#926—52no—
191#926—54no—
192#926—52no—
193#926—52no—
194#926—54no—
195#926C2×D512yes—
196C2×C4⋊Dic3—52no—
197C2×C4⋊Dic3—54no—
198C2×C4⋊Dic3—52no—
199#924—54no—
200#924—54no—
201#924—58no—
202#924—54no—
203#924—58no—
204#924C2×D514yes—
205C2×Dic3⋊C4—58no—
206C2×C4×Dic3—54no—
207#890—54no—
208#890—58no—
209#890—54no—
210#890—58no—
211#890—54no—
212#890C2×D514yescomplemented
213#888—58no—
214#888—54no—
215#888—54no—
216#888—58no—
217#888—54no—
218#888C2×D514yescomplemented
219C23×D5C22×D311yescharacteristic
220C2×C5×Q8—34no—
221C22×C4×C5C22×D312yes—
222C22×C4×C5C22×D311yescharacteristic
223C22×Dic5C22×D312yes—
224C22×Dic5C22×D311yescharacteristic
225Q8×D5—128no—
226C2×C4×D5—34no—
227C2×C4×D5—34no—
228C2×C4×D5—44no—
229C2×C4×D5Dic618yescomplemented
230C2×C4×D5C22×D314yes—
231C2×C4×D5C22×D314yes—
232C2×C4×D5C22×D312yes—
233C2×C4×D5C22×D312yes—
234C2×Dic10—34no—
235C2×Dic10—38no—
236C5×C4⋊C4—32no—
237C5×C4⋊C4—32no—
238C5×C4⋊C4—34no—
239C5×C4⋊C4—38no—
240C5×C4⋊C4C22×D314yes—
241C5×C22⋊C4—34no—
242C42×C5—34no—
243C22⋊Dic5—34no—
244#685—34no—
245#685—34no—
246C4⋊Dic5—32no—
247C4⋊Dic5—32no—
248C4⋊Dic5—34no—
249C4⋊Dic5—38no—
250C4⋊Dic5C22×D314yes—
251#683—38no—
252#683—38no—
253#683—34no—
254#683—34no—
255#683—38no—
256#683C22×D318yes—
257C4×Dic5—34no—
258C4×Dic5—34no—
259C4×Dic5—34no—
260#544—31no—
261#544—152no—
262#544—154no—
263#544—154no—
264#544—154no—
265#544—151no—
266C2×C4⋊Q8—154no—
267C2×#125—154no—
268C2×C22⋊Q8—154no—
269C2×#117—152no—
270C22×C4⋊C4—51no—
271C22×C3×C5—21no—
272C22×C3×C5—21no—
273C22×C3×C5C22×Q812yes—
274C22×C3×C5C2511yescharacteristic
275C2×C3×D5—21no—
276C2×C3×D5—21no—
277C2×C3×D5—28no—
278C2×C3×D5C22×Q812yes—
279C2×C3×D5C22×Q812yes—
280C2×C3×D5H(5,𝔽2)12yes—
281C2×C3×D52−1+412yes—
282C3×C4×C5—24no—
283C3×C4×C5C22×Q814yes—
284Dic15—24no—
285Dic15—24no—
286C3×Dic5—24no—
287C3×Dic5C22×Q814yes—
288C5×Dic3—24no—
289C5×Dic3—24no—
290C24×C3—51no—
291C22×C3×C4—51no—
292C22×C3×C4—58no—
293C22×C3×C4—52no—
294C22×C3×C4—52no—
295C22×C3×C4—52no—
296C22×C3×C4—54no—
297C22×C3×C4—54no—
298C22×C3×C4—204no—
299C22×C3×C4C22×D512yes—
300C22×C3×C4C22×D511yescharacteristic
301C22×Dic3—54no—
302C22×Dic3—54no—
303C2×Dic6—54no—
304C2×Dic6—58no—
305C2×Dic6—208no—
306C2×Dic6C22×D514yes—
307C3×C4⋊C4—58no—
308C3×C4⋊C4—54no—
309C3×C4⋊C4C22×D514yes—
310#217—54no—
311#217—54no—
312#217—54no—
313#217C22×D514yes—
314C4⋊Dic3—52no—
315C4⋊Dic3—54no—
316C4⋊Dic3—52no—
317C4⋊Dic3—54no—
318C4⋊Dic3—58no—
319C4⋊Dic3—54no—
320C4⋊Dic3C22×D512yes—
321C4⋊Dic3C22×D514yes—
322C4⋊Dic3C22×D512yes—
323Dic3⋊C4—58no—
324Dic3⋊C4—58no—
325Dic3⋊C4—58no—
326Dic3⋊C4C22×D518yes—
327C4×Dic3—54no—
328C4×Dic3—54no—
329C4×Dic3—54no—
330C4×Dic3C22×D514yes—
331C23×C5C23×D311yescharacteristic
332C22×D5—22no—
333C22×D5—22no—
334C22×D5C2×Dic614yes—
335C22×D5C23×D311yescharacteristic
336C22×D5C23×D311yescharacteristic
337C5×Q8—124no—
338C2×C4×C5—34no—
339C2×C4×C5—34no—
340C2×C4×C5—42no—
341C2×C4×C5C2×Dic614yescomplemented
342C2×C4×C5C23×D314yes—
343C2×C4×C5C23×D312yes—
344C2×Dic5—34no—
345C2×Dic5—34no—
346C2×Dic5—42no—
347C2×Dic5C2×Dic614yescomplemented
348C2×Dic5C23×D314yes—
349C2×Dic5C23×D312yes—
350C4×D5—28no—
351C4×D5—44no—
352C4×D5—68no—
353C4×D5—68no—
354C4×D5C2×Dic618yes—
355C4×D5C2×Dic618yescomplemented
356Dic10—128no—
357Dic10—124no—
358C22×Q8—154no—
359C23×C4—51no—
360C23×C4—52no—
361C4⋊Q8—34no—
362C4⋊Q8—154no—
363C4⋊Q8—158no—
364C4⋊Q8—154no—
365C4⋊Q8—158no—
366C4⋊Q8—154no—
367#125—34no—
368#125—154no—
369#125—158no—
370#125—158no—
371#125—154no—
372#125—154no—
373C22⋊Q8—34no—
374C22⋊Q8—158no—
375C22⋊Q8—158no—
376C22⋊Q8—154no—
377C22⋊Q8—154no—
378C22⋊Q8—154no—
379#117—32no—
380#117—154no—
381#117—154no—
382#117—152no—
383#117—152no—
384#117—152no—
385C2×C4⋊C4—52no—
386C2×C4⋊C4—51no—
387C2×C4⋊C4—54no—
388C2×C4⋊C4—54no—
389C2×C4⋊C4—152no—
390C2×C4⋊C4—152no—
391C2×C4⋊C4—154no—
392C2×C4⋊C4—158no—
393C2×C4⋊C4D3×D511yescharacteristic
394C2×C22⋊C4—154no—
395C2×C42—154no—
396C2×C3×C5—22no—
397C2×C3×C5C23×Q811yescharacteristic
398C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
399C2×C3×C5C2×2−1+411yescharacteristic
400C3×D5—22no—
401C3×D5#54414yescomplemented
402C23×C3—51no—
403C23×C3—54no—
404C23×C3—51no—
405C23×C3—102no—
406C23×C3—102no—
407C23×C3C23×D511yescharacteristic
408C2×C3×C4—42no—
409C2×C3×C4—54no—
410C2×C3×C4—52no—
411C2×C3×C4—58no—
412C2×C3×C4—58no—
413C2×C3×C4—54no—
414C2×C3×C4—108no—
415C2×C3×C4—204no—
416C2×C3×C4—202no—
417C2×C3×C4Q8×D514yescomplemented
418C2×C3×C4C23×D514yes—
419C2×C3×C4C23×D512yes—
420C2×Dic3—54no—
421C2×Dic3—54no—
422C2×Dic3—108no—
423C2×Dic3—108no—
424C2×Dic3C23×D514yes—
425C2×Dic3C23×D514yes—
426Dic6—44no—
427Dic6—208no—
428Dic6—204no—
429C22×C5—21no—
430C22×C5—21no—
431C22×C5C22×Dic612yes—
432C22×C5C24×D311yescharacteristic
433C2×D5—21no—
434C2×D5—21no—
435C2×D5—28no—
436C2×D5C22×Dic612yes—
437C2×D5C22×Dic612yes—
438C2×D5#100412yes—
439C2×D5#100712yes—
440C4×C5—24no—
441C4×C5—64no—
442C4×C5—64no—
443C4×C5C22×Dic614yes—
444Dic5—24no—
445Dic5—64no—
446Dic5—64no—
447Dic5C22×Dic614yes—
448C24—51no—
449C2×Q8—34no—
450C2×Q8—158no—
451C2×Q8—154no—
452C2×Q8—608no—
453C22×C4—51no—
454C22×C4—58no—
455C22×C4—52no—
456C22×C4—52no—
457C22×C4—52no—
458C22×C4—54no—
459C22×C4—54no—
460C22×C4—154no—
461C22×C4—154no—
462C22×C4—204no—
463C22×C4C2×D3×D512yes—
464C22×C4C2×D3×D511yescharacteristic
465C4⋊C4—32no—
466C4⋊C4—32no—
467C4⋊C4—34no—
468C4⋊C4—38no—
469C4⋊C4—58no—
470C4⋊C4—54no—
471C4⋊C4—154no—
472C4⋊C4—154no—
473C4⋊C4—158no—
474C4⋊C4—158no—
475C4⋊C4—158no—
476C4⋊C4—152no—
477C4⋊C4—152no—
478C4⋊C4—154no—
479C4⋊C4—158no—
480C4⋊C4C2×D3×D514yes—
481C22⋊C4—34no—
482C22⋊C4—154no—
483C22⋊C4—154no—
484C22⋊C4—154no—
485C42—34no—
486C42—154no—
487C42—154no—
488C42—154no—
489C3×C5C2×#54411yescharacteristic, complemented, Hall
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—52no—
493C22×C3—52no—
494C22×C3—52no—
495C22×C3—52no—
496C22×C3—108no—
497C22×C3—101no—
498C22×C3—101no—
499C22×C3C2×Q8×D512yes—
500C22×C3C24×D511yescharacteristic
501C3×C4—24no—
502C3×C4—54no—
503C3×C4—104no—
504C3×C4C2×Q8×D514yes—
505Dic3—24no—
506Dic3—24no—
507Dic3—104no—
508Dic3—104no—
509C2×C5—22no—
510C2×C5C23×Dic611yescharacteristic
511C2×C5C2×#100411yescharacteristic
512C2×C5C2×#100711yescharacteristic
513D5—22no—
514D5#550014yesdirect product factor
515C23—51no—
516C23—54no—
517C23—51no—
518C23—102no—
519C23—102no—
520C23C22×D3×D511yescharacteristic
521Q8—124no—
522Q8—604no—
523Q8—608no—
524C2×C4—34no—
525C2×C4—34no—
526C2×C4—42no—
527C2×C4—54no—
528C2×C4—52no—
529C2×C4—58no—
530C2×C4—58no—
531C2×C4—54no—
532C2×C4—108no—
533C2×C4—154no—
534C2×C4—154no—
535C2×C4—204no—
536C2×C4—202no—
537C2×C4—308no—
538C2×C4—308no—
539C2×C4D5×Dic614yescomplemented
540C2×C4C22×D3×D514yes—
541C2×C4C22×D3×D512yes—
542C2×C3—22no—
543C2×C3—54no—
544C2×C3—102no—
545C2×C3C22×Q8×D511yescharacteristic
546C2×C3D5×H(5,𝔽2)11yescharacteristic
547C2×C3D5×2−1+411yescharacteristic
548C5C2×#550011yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—52no—
552C22—52no—
553C22—52no—
554C22—52no—
555C22—108no—
556C22—101no—
557C22—101no—
558C22C2×D5×Dic612yes—
559C22C23×D3×D511yescharacteristic
560C4—24no—
561C4—54no—
562C4—64no—
563C4—64no—
564C4—104no—
565C4—304no—
566C4—304no—
567C4C2×D5×Dic614yes—
568C3D5×#54411yescharacteristic, complemented, Sylow
569C2—22no—
570C2—54no—
571C2—102no—
572C2C22×D5×Dic611yescharacteristic
573C2D5×#100411yescharacteristic
574C2D5×#100711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC2×C3×C5×C4⋊C4C22
prime radicalC23×C3×D5C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5C2×#544
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11756259C2semidirect
2. C5×#5500C2semidirect
4. #11754618C2semidirect
7. #11754181C2semidirect
8. D5×#890C2semidirect
9. D5×#888C2semidirect
10. #11753369C2semidirect
11. #11753368C2semidirect
19. #91779C22semidirect
20. #91777C22semidirect
23. C5×#890C22semidirect
24. C5×#888C22semidirect
32. #91470C22semidirect
33. #91469C22semidirect
39. #91384C22semidirect
40. #91381C22semidirect
47. #91335C22semidirect
48. #91334C22semidirect
49. #91329C22semidirect
50. #91328C22semidirect
71. C2×C3×C4×D5Q8semidirect
112. #5500D5direct (N × H)
161. C2×C3×C4×C5C2×Q8semidirect
171. C2×C3×Dic5C2×Q8semidirect
177. C3×C4×D5C2×Q8semidirect
212. #890C2×D5semidirect
218. #888C2×D5semidirect
229. C2×C4×D5Dic6semidirect
341. C2×C4×C5C2×Dic6semidirect
347. C2×Dic5C2×Dic6semidirect
355. C4×D5C2×Dic6semidirect
401. C3×D5#544semidirect
417. C2×C3×C4Q8×D5semidirect
489. C3×C5C2×#544semidirect
514. D5#5500direct (N × H)
539. C2×C4D5×Dic6semidirect
548. C5C2×#5500semidirect
568. C3D5×#544semidirect

Maximal subgroups

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

subgroupindex
#117562592
C5×#55002
C2×C3×D5×C4⋊C42
#117546182
D5×#9262
D5×#9242
#117541812
D5×#8902
D5×#8882
#117533692
#117533682
D5×#5443
C2×#55005