Finite Group Databasebeta

← C5×#86581 #14481804

Subgroups — C5×#86581 #14481804

Order 1920.

1992 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#544C211yesmaximal, characteristic, complemented
2C5×#5729C214yesmaximal, complemented
3C5×#5728C214yesmaximal, complemented
4C5×#5689C214yesmaximal, complemented
5C5×#5688C214yesmaximal, complemented
6C5×#5634C214yesmaximal
7C5×#5622C214yesmaximal, complemented
8C5×D3×#117C212yesmaximal
9C5×#5519C212yesmaximal, complemented
10C2×C5×#892C211yesmaximal, characteristic
11C5×#5500C211yesmaximal, characteristic, complemented
12C5×#3458—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C3×C5×C4⋊Q8C2214yescomplemented
15C3×C5×#125C2214yescomplemented
16C3×C5×C22⋊Q8C2214yes—
17C3×C5×#117C2212yes—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C5×#946C2218yes—
20C5×Q8×Dic3C2214yescomplemented
21C5×#944C2214yescomplemented
22C2×C5×(C2×D3)⋊C4C2212yes—
23C5×#926C2212yes—
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C5×#924C2214yes—
26C2×C4×C5×Dic3C2212yes—
27C5×#897C2218yes—
28C5×#896C2218yes—
29C5×#896C2218yes—
30C5×#892C2212yes—
31C5×#892C2214yes—
32C5×#892C2214yes—
33C5×#892C2212yes—
34C5×D3×C4⋊C4C2212yes—
35C5×D3×C4⋊C4C2212yes—
36C5×#890C2214yescomplemented
37C5×#890C2214yescomplemented
38C5×#889C2218yescomplemented
39C5×#888C2214yescomplemented
40C5×#888C2214yescomplemented
41C5×#888C2214yescomplemented
42C5×#887C2214yescomplemented
43C5×#887C2218yescomplemented
44C5×D3×C22⋊C4C2214yes—
45C5×#878C2218yes—
46C5×#877C2214yes—
47C5×#872C2214yes—
48C42×C5×D3C2214yes—
49C5×#870C2214yescomplemented
50C5×#869C2214yescomplemented
51#86581C511yesmaximal, characteristic, direct product factor, Hall
52C5×#544—34no—
53C5×#544—34no—
54C5×#544—32no—
55C5×#544—31no—
56C5×#544D311yescharacteristic, complemented
57C5×#543—34no—
58C5×#533—34no—
59C5×#533—34no—
60C5×#533—34no—
61C2×C5×#117—32no—
62C2×C5×#117—31no—
63C23×C5×D3C2311yescharacteristic
64C2×C3×C5×Q8C2314yes—
65C22×C3×C4×C5C2312yes—
66C22×C3×C4×C5C2311yescharacteristic
67C22×C5×Dic3C2312yes—
68C22×C5×Dic3C2311yescharacteristic
69C2×C4×C5×D3Q818yescomplemented
70C2×C4×C5×D3C2314yes—
71C2×C4×C5×D3C2314yes—
72C2×C4×C5×D3C2312yes—
73C2×C4×C5×D3C2312yes—
74C2×C5×Dic6C2318yes—
75C2×C5×Dic6C2314yes—
76C3×C5×C4⋊C4C2314yes—
77C3×C5×C4⋊C4C2318yes—
78C3×C5×C4⋊C4C2312yes—
79C3×C5×C4⋊C4C2314yes—
80C3×C5×C4⋊C4C2312yes—
81C3×C5×C22⋊C4C2314yes—
82C3×C42×C5C2314yes—
83C5×#217C2314yes—
84C5×(C2×D3)⋊C4C2314yes—
85C5×(C2×D3)⋊C4C2314yes—
86C5×(C2×D3)⋊C4C2318yes—
87C5×C4⋊Dic3C2312yes—
88C5×C4⋊Dic3C2314yes—
89C5×C4⋊Dic3C2318yes—
90C5×C4⋊Dic3C2312yes—
91C5×C4⋊Dic3C2314yes—
92C5×Dic3⋊C4C2318yes—
93C5×Dic3⋊C4C2318yes—
94C5×Dic3⋊C4C2314yes—
95C5×Dic3⋊C4C2314yes—
96C5×Dic3⋊C4C2318yes—
97C4×C5×Dic3C2314yes—
98C4×C5×Dic3C2314yes—
99C4×C5×Dic3C2314yes—
100C4×C5×Dic3C2314yes—
101C4×C5×Dic3C2314yes—
102C3×#544C2×C511yescharacteristic, complemented
103#5729C2×C514yescomplemented
104#5728C2×C514yescomplemented
105#5689C2×C514yescomplemented
106#5688C2×C514yescomplemented
107#5634C2×C514yes—
108#5622C2×C514yescomplemented
109D3×#117C2×C512yes—
110#5519C2×C512yescomplemented
111C2×#892C2×C511yescharacteristic
112#5500C2×C511yescharacteristic, complemented
113C23×C4×C5—31no—
114C5×C4⋊Q8—34no—
115C5×C4⋊Q8—34no—
116C5×C4⋊Q8—34no—
117C5×C4⋊Q8—34no—
118C5×C4⋊Q8—34no—
119C5×C4⋊Q8C2×D314yescomplemented
120C5×#125—38no—
121C5×#125—34no—
122C5×#125—34no—
123C5×#125—34no—
124C5×#125C2×D314yescomplemented
125C5×C22⋊Q8—38no—
126C5×C22⋊Q8—34no—
127C5×C22⋊Q8—38no—
128C5×C22⋊Q8—38no—
129C5×C22⋊Q8—38no—
130C5×C22⋊Q8—38no—
131C5×C22⋊Q8C2×D314yes—
132C4×C5×Q8—34no—
133C4×C5×Q8—38no—
134C4×C5×Q8—34no—
135C5×#117—32no—
136C5×#117—32no—
137C5×#117—32no—
138C5×#117—34no—
139C5×#117—34no—
140C5×#117—34no—
141C5×#117—34no—
142C5×#117C2×D312yes—
143C2×C5×C4⋊C4—31no—
144C2×C5×C4⋊C4—32no—
145C2×C5×C4⋊C4—32no—
146C2×C5×C4⋊C4C2×D311yescharacteristic
147C2×C5×C22⋊C4—32no—
148C2×C5×C22⋊C4—34no—
149C2×C42×C5—32no—
150C2×C42×C5—34no—
151#3458—31noSylow
152C23×C3×C5C2411yescharacteristic
153C22×C5×D3—41no—
154C22×C5×D3—41no—
155C22×C5×D3C2×Q814yes—
156C22×C5×D3C2412yes—
157C3×C5×Q8—44no—
158C2×C3×C4×C5—42no—
159C2×C3×C4×C5C2×Q814yes—
160C2×C3×C4×C5C2414yes—
161C2×C3×C4×C5C2414yes—
162C2×C3×C4×C5C2414yes—
163C2×C3×C4×C5C2412yes—
164C2×C5×Dic3—42no—
165C2×C5×Dic3C2×Q814yes—
166C2×C5×Dic3C2414yes—
167C2×C5×Dic3C2414yes—
168C2×C5×Dic3C2414yes—
169C2×C5×Dic3C2411yescharacteristic
170C2×C5×Dic3C2411yescharacteristic
171C4×C5×D3—44no—
172C4×C5×D3—44no—
173C4×C5×D3C2×Q818yes—
174C4×C5×D3C2×Q818yes—
175C5×Dic6—44no—
176C5×Dic6—48no—
177C22×C4×D3C22×C511yescharacteristic
178C3×C4⋊Q8C22×C514yescomplemented
179C3×#125C22×C514yescomplemented
180C3×C22⋊Q8C22×C514yes—
181C3×#117C22×C512yes—
182C2×C3×C4⋊C4C22×C511yescharacteristic
183#946C22×C518yes—
184Q8×Dic3C22×C514yescomplemented
185#944C22×C514yescomplemented
186C2×(C2×D3)⋊C4C22×C512yes—
187#926C22×C512yes—
188C2×C4⋊Dic3C22×C511yescharacteristic
189#924C22×C514yes—
190C2×C4×Dic3C22×C512yes—
191#897C22×C518yes—
192#896C22×C518yes—
193#896C22×C518yes—
194#892C22×C512yes—
195#892C22×C514yes—
196#892C22×C514yes—
197#892C22×C512yes—
198D3×C4⋊C4C22×C512yes—
199D3×C4⋊C4C22×C512yes—
200#890C22×C514yescomplemented
201#890C22×C514yescomplemented
202#889C22×C518yescomplemented
203#888C22×C514yescomplemented
204#888C22×C514yescomplemented
205#888C22×C514yescomplemented
206#887C22×C514yescomplemented
207#887C22×C518yescomplemented
208D3×C22⋊C4C22×C514yes—
209#878C22×C518yes—
210#877C22×C514yes—
211#872C22×C514yes—
212C42×D3C22×C514yes—
213#870C22×C514yescomplemented
214#869C22×C514yescomplemented
215C24×C5—31no—
216C2×C5×Q8—34no—
217C2×C5×Q8—38no—
218C2×C5×Q8C22×D314yes—
219C22×C4×C5—31no—
220C22×C4×C5—38no—
221C22×C4×C5—32no—
222C22×C4×C5—32no—
223C22×C4×C5—32no—
224C22×C4×C5—34no—
225C22×C4×C5—34no—
226C22×C4×C5C22×D312yes—
227C22×C4×C5C22×D311yescharacteristic
228C5×C4⋊C4—34no—
229C5×C4⋊C4—34no—
230C5×C4⋊C4—34no—
231C5×C4⋊C4—32no—
232C5×C4⋊C4—32no—
233C5×C4⋊C4—38no—
234C5×C4⋊C4—38no—
235C5×C4⋊C4—38no—
236C5×C4⋊C4—34no—
237C5×C4⋊C4—38no—
238C5×C4⋊C4C22×D312yes—
239C5×C4⋊C4C22×D312yes—
240C5×C4⋊C4C22×D314yes—
241C5×C4⋊C4C22×D318yes—
242C5×C4⋊C4C22×D314yes—
243C5×C22⋊C4—34no—
244C5×C22⋊C4—38no—
245C5×C22⋊C4—34no—
246C5×C22⋊C4—34no—
247C5×C22⋊C4C22×D314yes—
248C42×C5—34no—
249C42×C5—34no—
250C42×C5—34no—
251C42×C5—34no—
252C42×C5—34no—
253C42×C5C22×D314yes—
254#544—34no—
255#544—34no—
256#544—32no—
257#544—31no—
258#544C5×D311yescharacteristic, complemented
259#543—34no—
260#533—34no—
261#533—34no—
262#533—34no—
263C2×#117—32no—
264C2×#117—31no—
265C22×C3×C5—21no—
266C22×C3×C5—21no—
267C22×C3×C5C22×Q812yes—
268C22×C3×C5C2511yescharacteristic
269C2×C5×D3—22no—
270C2×C5×D3—22no—
271C2×C5×D3—44no—
272C2×C5×D3C22×Q814yes—
273C3×C4×C5—24no—
274C3×C4×C5—24no—
275C3×C4×C5—24no—
276C3×C4×C5C22×Q814yes—
277C5×Dic3—24no—
278C5×Dic3—24no—
279C5×Dic3—24no—
280C5×Dic3C22×Q812yes—
281C5×Dic3C22×Q812yes—
282C23×D3C23×C511yescharacteristic
283C2×C3×Q8C23×C514yes—
284C22×C3×C4C23×C512yes—
285C22×C3×C4C23×C511yescharacteristic
286C22×Dic3C23×C512yes—
287C22×Dic3C23×C511yescharacteristic
288C2×C4×D3C5×Q818yescomplemented
289C2×C4×D3C23×C514yes—
290C2×C4×D3C23×C514yes—
291C2×C4×D3C23×C512yes—
292C2×C4×D3C23×C512yes—
293C2×Dic6C23×C518yes—
294C2×Dic6C23×C514yes—
295C3×C4⋊C4C23×C514yes—
296C3×C4⋊C4C23×C518yes—
297C3×C4⋊C4C23×C512yes—
298C3×C4⋊C4C23×C514yes—
299C3×C4⋊C4C23×C512yes—
300C3×C22⋊C4C23×C514yes—
301C3×C42C23×C514yes—
302#217C23×C514yes—
303(C2×D3)⋊C4C23×C514yes—
304(C2×D3)⋊C4C23×C514yes—
305(C2×D3)⋊C4C23×C518yes—
306C4⋊Dic3C23×C512yes—
307C4⋊Dic3C23×C514yes—
308C4⋊Dic3C23×C518yes—
309C4⋊Dic3C23×C512yes—
310C4⋊Dic3C23×C514yes—
311Dic3⋊C4C23×C518yes—
312Dic3⋊C4C23×C518yes—
313Dic3⋊C4C23×C514yes—
314Dic3⋊C4C23×C514yes—
315Dic3⋊C4C23×C518yes—
316C4×Dic3C23×C514yes—
317C4×Dic3C23×C514yes—
318C4×Dic3C23×C514yes—
319C4×Dic3C23×C514yes—
320C4×Dic3C23×C514yes—
321C23×C5—32no—
322C23×C5—34no—
323C23×C5—121no—
324C23×C5—121no—
325C23×C5C23×D311yescharacteristic
326C5×Q8—44no—
327C5×Q8—124no—
328C5×Q8—128no—
329C2×C4×C5—34no—
330C2×C4×C5—31no—
331C2×C4×C5—31no—
332C2×C4×C5—38no—
333C2×C4×C5—38no—
334C2×C4×C5—34no—
335C2×C4×C5—34no—
336C2×C4×C5—34no—
337C2×C4×C5—42no—
338C2×C4×C5—124no—
339C2×C4×C5—124no—
340C2×C4×C5—122no—
341C2×C4×C5D3×Q814yes—
342C2×C4×C5C23×D314yes—
343C2×C4×C5C23×D314yes—
344C2×C4×C5C23×D314yes—
345C2×C4×C5C23×D312yes—
346C23×C4—31no—
347C4⋊Q8—34no—
348C4⋊Q8—34no—
349C4⋊Q8—34no—
350C4⋊Q8—34no—
351C4⋊Q8—34no—
352C4⋊Q8C2×C5×D314yescomplemented
353#125—38no—
354#125—34no—
355#125—34no—
356#125—34no—
357#125C2×C5×D314yescomplemented
358C22⋊Q8—38no—
359C22⋊Q8—34no—
360C22⋊Q8—38no—
361C22⋊Q8—38no—
362C22⋊Q8—38no—
363C22⋊Q8—38no—
364C22⋊Q8C2×C5×D314yes—
365C4×Q8—34no—
366C4×Q8—38no—
367C4×Q8—34no—
368#117—32no—
369#117—32no—
370#117—32no—
371#117—34no—
372#117—34no—
373#117—34no—
374#117—34no—
375#117C2×C5×D312yes—
376C2×C4⋊C4—31no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—32no—
379C2×C4⋊C4C2×C5×D311yescharacteristic
380C2×C22⋊C4—32no—
381C2×C22⋊C4—34no—
382C2×C42—32no—
383C2×C42—34no—
384C2×C3×C5—22no—
385C2×C3×C5C23×Q811yescharacteristic
386C2×C3×C5#58511yescharacteristic
387C2×C3×C5#58511yescharacteristic
388C5×D3—24no—
389C23×C3C24×C511yescharacteristic
390C22×D3—41no—
391C22×D3—41no—
392C22×D3C2×C5×Q814yes—
393C22×D3C24×C512yes—
394C3×Q8—44no—
395C2×C3×C4—42no—
396C2×C3×C4C2×C5×Q814yes—
397C2×C3×C4C24×C514yes—
398C2×C3×C4C24×C514yes—
399C2×C3×C4C24×C514yes—
400C2×C3×C4C24×C512yes—
401C2×Dic3—42no—
402C2×Dic3C2×C5×Q814yes—
403C2×Dic3C24×C514yes—
404C2×Dic3C24×C514yes—
405C2×Dic3C24×C514yes—
406C2×Dic3C24×C511yescharacteristic
407C2×Dic3C24×C511yescharacteristic
408C4×D3—44no—
409C4×D3—44no—
410C4×D3C2×C5×Q818yes—
411C4×D3C2×C5×Q818yes—
412Dic6—44no—
413Dic6—48no—
414C22×C5—21no—
415C22×C5—21no—
416C22×C5—34no—
417C22×C5—62no—
418C22×C5—62no—
419C22×C5—124no—
420C22×C5C2×D3×Q812yes—
421C22×C5C24×D311yescharacteristic
422C4×C5—24no—
423C4×C5—24no—
424C4×C5—24no—
425C4×C5—32no—
426C4×C5—32no—
427C4×C5—64no—
428C4×C5—64no—
429C4×C5—64no—
430C4×C5C2×D3×Q814yes—
431C24—31no—
432C2×Q8—34no—
433C2×Q8—38no—
434C2×Q8C22×C5×D314yes—
435C22×C4—31no—
436C22×C4—38no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—32no—
440C22×C4—34no—
441C22×C4—34no—
442C22×C4C22×C5×D312yes—
443C22×C4C22×C5×D311yescharacteristic
444C4⋊C4—34no—
445C4⋊C4—34no—
446C4⋊C4—34no—
447C4⋊C4—32no—
448C4⋊C4—32no—
449C4⋊C4—38no—
450C4⋊C4—38no—
451C4⋊C4—38no—
452C4⋊C4—34no—
453C4⋊C4—38no—
454C4⋊C4C22×C5×D312yes—
455C4⋊C4C22×C5×D312yes—
456C4⋊C4C22×C5×D314yes—
457C4⋊C4C22×C5×D318yes—
458C4⋊C4C22×C5×D314yes—
459C22⋊C4—34no—
460C22⋊C4—38no—
461C22⋊C4—34no—
462C22⋊C4—34no—
463C22⋊C4C22×C5×D314yes—
464C42—34no—
465C42—34no—
466C42—34no—
467C42—34no—
468C42—34no—
469C42C22×C5×D314yes—
470C3×C5#345811yescharacteristic, complemented, Hall
471C22×C3—21no—
472C22×C3—21no—
473C22×C3C22×C5×Q812yes—
474C22×C3C25×C511yescharacteristic
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—44no—
478C2×D3C22×C5×Q814yes—
479C3×C4—24no—
480C3×C4—24no—
481C3×C4—24no—
482C3×C4C22×C5×Q814yes—
483Dic3—24no—
484Dic3—24no—
485Dic3—24no—
486Dic3C22×C5×Q812yes—
487Dic3C22×C5×Q812yes—
488C2×C5—22no—
489C2×C5—64no—
490C2×C5C22×D3×Q811yescharacteristic
491C2×C5#596611yescharacteristic
492C2×C5#596811yescharacteristic
493C23—32no—
494C23—34no—
495C23—121no—
496C23—121no—
497C23C23×C5×D311yescharacteristic
498Q8—44no—
499Q8—124no—
500Q8—128no—
501C2×C4—34no—
502C2×C4—31no—
503C2×C4—31no—
504C2×C4—38no—
505C2×C4—38no—
506C2×C4—34no—
507C2×C4—34no—
508C2×C4—34no—
509C2×C4—42no—
510C2×C4—124no—
511C2×C4—124no—
512C2×C4—122no—
513C2×C4C5×D3×Q814yes—
514C2×C4C23×C5×D314yes—
515C2×C4C23×C5×D314yes—
516C2×C4C23×C5×D314yes—
517C2×C4C23×C5×D312yes—
518C2×C3—22no—
519C2×C3C23×C5×Q811yescharacteristic
520C2×C3C5×#58511yescharacteristic
521C2×C3C5×#58511yescharacteristic
522D3—24no—
523C5#8658111yescharacteristic, direct product factor, Sylow
524C22—21no—
525C22—21no—
526C22—34no—
527C22—62no—
528C22—62no—
529C22—124no—
530C22C2×C5×D3×Q812yes—
531C22C24×C5×D311yescharacteristic
532C4—24no—
533C4—24no—
534C4—24no—
535C4—32no—
536C4—32no—
537C4—64no—
538C4—64no—
539C4—64no—
540C4C2×C5×D3×Q814yes—
541C3C5×#345811yescharacteristic, complemented, Sylow
542C2—22no—
543C2—64no—
544C2C22×C5×D3×Q811yescharacteristic
545C2C5×#596611yescharacteristic
546C2C5×#596811yescharacteristic
subgroupG/H
centerC22×C5C24×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC3×C5×#544C2
socleC22×C3×C5C25
prime radicalC23×C5×D3C23
hypercenterC5×#544D3
hypocenterC3C5×#3458
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C5×#544

Complements

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

NH ≅ G/Ntype
1. C3×C5×#544C2semidirect
2. C5×#5729C2semidirect
3. C5×#5728C2semidirect
4. C5×#5689C2semidirect
5. C5×#5688C2semidirect
7. C5×#5622C2semidirect
9. C5×#5519C2semidirect
11. C5×#5500C2semidirect
14. C3×C5×C4⋊Q8C22semidirect
15. C3×C5×#125C22semidirect
20. C5×Q8×Dic3C22semidirect
21. C5×#944C22semidirect
36. C5×#890C22semidirect
37. C5×#890C22semidirect
38. C5×#889C22semidirect
39. C5×#888C22semidirect
40. C5×#888C22semidirect
41. C5×#888C22semidirect
42. C5×#887C22semidirect
43. C5×#887C22semidirect
49. C5×#870C22semidirect
50. C5×#869C22semidirect
51. #86581C5direct (N × H)
56. C5×#544D3semidirect
69. C2×C4×C5×D3Q8semidirect
102. C3×#544C2×C5semidirect
103. #5729C2×C5semidirect
104. #5728C2×C5semidirect
105. #5689C2×C5semidirect
106. #5688C2×C5semidirect
108. #5622C2×C5semidirect
110. #5519C2×C5semidirect
112. #5500C2×C5semidirect
119. C5×C4⋊Q8C2×D3semidirect
124. C5×#125C2×D3semidirect
178. C3×C4⋊Q8C22×C5semidirect
179. C3×#125C22×C5semidirect
184. Q8×Dic3C22×C5semidirect
185. #944C22×C5semidirect
200. #890C22×C5semidirect
201. #890C22×C5semidirect
202. #889C22×C5semidirect
203. #888C22×C5semidirect
204. #888C22×C5semidirect
205. #888C22×C5semidirect
206. #887C22×C5semidirect
207. #887C22×C5semidirect
213. #870C22×C5semidirect
214. #869C22×C5semidirect
258. #544C5×D3semidirect
288. C2×C4×D3C5×Q8semidirect
352. C4⋊Q8C2×C5×D3semidirect
357. #125C2×C5×D3semidirect
470. C3×C5#3458semidirect
523. C5#86581direct (N × H)
541. C3C5×#3458semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5442
C5×#57292
C5×#57282
C5×#56892
C5×#56882
C5×#56342
C5×#56222
C5×D3×#1172
C5×#55192
C2×C5×#8922
C5×#55002
C5×#34583
#865815