Finite Group Databasebeta

← C4×C5×#806 #14295366

Subgroups — C4×C5×#806 #14295366

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×#67C211yesmaximal, characteristic
2C3×C4×C5×C22⋊C4C211yesmaximal, characteristic, complemented
3C4×C5×#217C211yesmaximal, characteristic, complemented
4C2×C5×#806C212yesmaximal
5C5×#4527C212yesmaximal, complemented
6C4×C5×#99—31nomaximal, Hall
7C23×C3×C4×C5C2211yescharacteristic
8C22×C5×#67C2211yescharacteristic
9C22×C4×C5×D3C411yescharacteristic, complemented
10C2×C3×C5×C22⋊C4C2212yes—
11C2×C3×C42×C5—21no—
12C2×C5×#217C2211yescharacteristic
13C2×C5×#217C2212yes—
14C4×C5×#67—21no—
15C4×C5×#67—21no—
16C4×C5×#67—21no—
17C4×C5×#67—21no—
18C2×C5×(C2×D3)⋊C4C411yescharacteristic, complemented
19C2×C5×Dic3⋊C4C411yescharacteristic, complemented
20C2×C4×C5×Dic3—21no—
21C2×C4×C5×Dic3C411yescharacteristic, complemented
22C3×C5×#95—21no—
23C5×C3⋊#95—21no—
24C5×#806C418yesdirect product factor
25C4×#806C511yesmaximal, characteristic, direct product factor, Hall
26C2×C4×C5×D4—31no—
27C2×C5×#99—32no—
28C4×C5×C22⋊C4—31no—
29C4×C5×C22⋊C4D311yescharacteristic, complemented
30C5×#342—32no—
31C24×C3×C5C2311yescharacteristic
32C23×C5×D3C2×C411yescharacteristic
33C22×C3×C4×C5—22no—
34C22×C3×C4×C5—42no—
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×C3×C4×C5C2×C411yescharacteristic
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5D411yescharacteristic
39C22×C3×C4×C5D411yescharacteristic
40C22×C3×C4×C5D411yescharacteristic
41C2×C5×#67—21no—
42C2×C5×#67—21no—
43C2×C5×#67—22no—
44C2×C5×#67C2×C414yes—
45C22×C5×Dic3—22no—
46C22×C5×Dic3C2×C411yescharacteristic
47C22×C5×Dic3C2×C411yescharacteristic
48C22×C5×Dic3C2×C411yescharacteristic
49C2×C4×C5×D3—21no—
50C2×C4×C5×D3—21no—
51C2×C4×C5×D3—42no—
52C3×C5×C22⋊C4—22no—
53C3×C5×C22⋊C4C2×C414yescomplemented
54C3×C42×C5—42no—
55C5×#217—21no—
56C5×#217—21no—
57C5×#217—22no—
58C5×#217C2×C414yescomplemented
59C5×(C2×D3)⋊C4—21no—
60C5×(C2×D3)⋊C4—21no—
61C5×Dic3⋊C4—21no—
62C5×Dic3⋊C4—21no—
63C4×C5×Dic3—21no—
64C4×C5×Dic3—21no—
65C4×C5×Dic3—42no—
66C2×C4×#67C2×C511yescharacteristic
67C3×C4×C22⋊C4C2×C511yescharacteristic, complemented
68C4×#217C2×C511yescharacteristic, complemented
69C2×#806C2×C512yes—
70#4527C2×C512yescomplemented
71C22×C5×D4—31no—
72C23×C4×C5—31no—
73C23×C4×C5C2×D311yescharacteristic
74C4×C5×D4—61no—
75C4×C5×D4—61no—
76C4×C5×D4—61no—
77C4×C5×D4—61no—
78C2×C5×C4⋊C4—31no—
79C2×C5×C22⋊C4—32no—
80C2×C5×C22⋊C4—31no—
81C2×C5×C22⋊C4—31no—
82C2×C5×C22⋊C4C2×D312yes—
83C2×C42×C5—21no—
84C2×C42×C5—31no—
85C2×C42×C5—61no—
86C5×#99—38no—
87C5×#95—21no—
88C5×#95—61no—
89C4×#99—31noSylow
90C23×C3×C5—41no—
91C23×C3×C5—41no—
92C23×C3×C5C22×C411yescharacteristic
93C23×C3×C5C22×C412yes—
94C23×C3×C5C2×D411yescharacteristic
95C23×C3×C5C2×D411yescharacteristic
96C23×C3×C5#4112yes—
97C22×C5×D3—21no—
98C22×C5×D3—21no—
99C22×C5×D3—41no—
100C22×C5×D3—41no—
101C22×C5×D3C4212yescomplemented
102C2×C3×C4×C5—22no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—24no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—42no—
110C2×C3×C4×C5—42no—
111C2×C3×C4×C5C22⋊C411yescharacteristic
112C2×C3×C4×C5C22⋊C411yescharacteristic
113C2×C3×C4×C5C22⋊C411yescharacteristic
114C2×C3×C4×C5C22⋊C411yescharacteristic
115C5×#67—24no—
116C5×#67—24no—
117C2×C5×Dic3—21no—
118C2×C5×Dic3—21no—
119C2×C5×Dic3—21no—
120C2×C5×Dic3—21no—
121C2×C5×Dic3—24no—
122C2×C5×Dic3—42no—
123C2×C5×Dic3—42no—
124C2×C5×Dic3—41no—
125C2×C5×Dic3—41no—
126C2×C5×Dic3C4212yescomplemented
127C4×C5×D3—42no—
128C4×C5×D3—42no—
129C23×C3×C4C22×C511yescharacteristic
130C22×#67C22×C511yescharacteristic
131C22×C4×D3C4×C511yescharacteristic, complemented
132C2×C3×C22⋊C4C22×C512yes—
133C2×C3×C42—21no—
134C2×#217C22×C511yescharacteristic
135C2×#217C22×C512yes—
136C4×#67—21no—
137C4×#67—21no—
138C4×#67—21no—
139C4×#67—21no—
140C2×(C2×D3)⋊C4C4×C511yescharacteristic, complemented
141C2×Dic3⋊C4C4×C511yescharacteristic, complemented
142C2×C4×Dic3—21no—
143C2×C4×Dic3C4×C511yescharacteristic, complemented
144C3×#95—21no—
145C3⋊#95—21no—
146#806C4×C518yesdirect product factor
147C24×C5—31no—
148C24×C5C22×D311yescharacteristic
149C2×C5×D4—34no—
150C2×C5×D4—61no—
151C2×C5×D4—62no—
152C2×C5×D4—61no—
153C22×C4×C5—22no—
154C22×C4×C5—31no—
155C22×C4×C5—31no—
156C22×C4×C5—31no—
157C22×C4×C5—42no—
158C22×C4×C5—62no—
159C22×C4×C5—61no—
160C22×C4×C5—61no—
161C22×C4×C5—122no—
162C22×C4×C5C4×D311yescharacteristic
163C22×C4×C5C4×D311yescharacteristic
164C22×C4×C5D1211yescharacteristic
165C22×C4×C5D1211yescharacteristic
166C22×C4×C5#6711yescharacteristic
167C22×C4×C5#6711yescharacteristic
168C5×C4⋊C4—61no—
169C5×C4⋊C4—61no—
170C5×C22⋊C4—22no—
171C5×C22⋊C4—34no—
172C5×C22⋊C4—61no—
173C5×C22⋊C4—62no—
174C5×C22⋊C4—61no—
175C5×C22⋊C4—61no—
176C5×C22⋊C4—61no—
177C5×C22⋊C4C4×D314yescomplemented
178C42×C5—42no—
179C42×C5—61no—
180C42×C5—61no—
181C42×C5—122no—
182C2×C4×D4—31no—
183C2×#99—32no—
184C4×C22⋊C4—31no—
185C4×C22⋊C4C5×D311yescharacteristic, complemented
186#342—32no—
187C22×C3×C5—21no—
188C22×C3×C5—21no—
189C22×C3×C5—21no—
190C22×C3×C5—21no—
191C22×C3×C5—21no—
192C22×C3×C5—21no—
193C22×C3×C5—42no—
194C22×C3×C5—42no—
195C22×C3×C5C2×C4211yescharacteristic
196C22×C3×C5C2×C22⋊C411yescharacteristic
197C22×C3×C5#11711yescharacteristic
198C22×C3×C5C4×D412yes—
199C22×C3×C5C4×D412yes—
200C2×C5×D3—22no—
201C2×C5×D3—42no—
202C2×C5×D3—41no—
203C2×C5×D3—41no—
204C2×C5×D3—42no—
205C3×C4×C5—21no—
206C3×C4×C5—21no—
207C3×C4×C5—21no—
208C3×C4×C5—44no—
209C3×C4×C5#9912yescomplemented
210C5×Dic3—22no—
211C5×Dic3—44no—
212C5×Dic3—41no—
213C5×Dic3—41no—
214C24×C3C23×C511yescharacteristic
215C23×D3C2×C4×C511yescharacteristic
216C22×C3×C4—22no—
217C22×C3×C4—42no—
218C22×C3×C4C2×C4×C511yescharacteristic
219C22×C3×C4C2×C4×C511yescharacteristic
220C22×C3×C4C5×D411yescharacteristic
221C22×C3×C4C5×D411yescharacteristic
222C22×C3×C4C5×D411yescharacteristic
223C22×C3×C4C5×D411yescharacteristic
224C2×#67—21no—
225C2×#67—21no—
226C2×#67—22no—
227C2×#67C2×C4×C514yes—
228C22×Dic3—22no—
229C22×Dic3C2×C4×C511yescharacteristic
230C22×Dic3C2×C4×C511yescharacteristic
231C22×Dic3C2×C4×C511yescharacteristic
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—42no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4C2×C4×C514yescomplemented
237C3×C42—42no—
238#217—21no—
239#217—21no—
240#217—22no—
241#217C2×C4×C514yescomplemented
242(C2×D3)⋊C4—21no—
243(C2×D3)⋊C4—21no—
244Dic3⋊C4—21no—
245Dic3⋊C4—21no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—42no—
249C23×C5—32no—
250C23×C5—41no—
251C23×C5—41no—
252C23×C5—61no—
253C23×C5—61no—
254C23×C5—121no—
255C23×C5—121no—
256C23×C5C2×C4×D311yescharacteristic
257C23×C5C2×C4×D312yes—
258C23×C5C2×D1211yescharacteristic
259C23×C5#23512yes—
260C23×C5C2×#6711yescharacteristic
261C5×D4—64no—
262C5×D4—64no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—24no—
267C2×C4×C5—32no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—61no—
274C2×C4×C5—61no—
275C2×C4×C5—64no—
276C2×C4×C5—61no—
277C2×C4×C5—61no—
278C2×C4×C5—122no—
279C2×C4×C5—122no—
280C2×C4×C5—122no—
281C2×C4×C5—122no—
282C2×C4×C5—121no—
283C2×C4×C5—121no—
284C2×C4×C5(C2×D3)⋊C411yescharacteristic
285C2×C4×C5(C2×D3)⋊C411yescharacteristic
286C2×C4×C5(C2×D3)⋊C411yescharacteristic
287C2×C4×C5(C2×D3)⋊C411yescharacteristic
288C22×D4—31no—
289C23×C4—31no—
290C23×C4C2×C5×D311yescharacteristic
291C4×D4—61no—
292C4×D4—61no—
293C4×D4—61no—
294C4×D4—61no—
295C2×C4⋊C4—31no—
296C2×C22⋊C4—32no—
297C2×C22⋊C4—31no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4C2×C5×D312yes—
300C2×C42—21no—
301C2×C42—31no—
302C2×C42—61no—
303#99—38no—
304#95—21no—
305#95—61no—
306C2×C3×C5—22no—
307C2×C3×C5—22no—
308C2×C3×C5—21no—
309C2×C3×C5—21no—
310C2×C3×C5C4×C22⋊C411yescharacteristic
311C2×C3×C5C2×#9911yescharacteristic
312C2×C3×C5#41011yescharacteristic
313C5×D3—42no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3C22×C4×C511yescharacteristic
317C23×C3C22×C4×C512yes—
318C23×C3C2×C5×D411yescharacteristic
319C23×C3C2×C5×D411yescharacteristic
320C23×C3C5×#4112yes—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3C42×C512yescomplemented
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4C5×C22⋊C411yescharacteristic
336C2×C3×C4C5×C22⋊C411yescharacteristic
337C2×C3×C4C5×C22⋊C411yescharacteristic
338C2×C3×C4C5×C22⋊C411yescharacteristic
339#67—24no—
340#67—24no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—24no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3C42×C512yescomplemented
351C4×D3—42no—
352C4×D3—42no—
353C22×C5—21no—
354C22×C5—21no—
355C22×C5—21no—
356C22×C5—21no—
357C22×C5—21no—
358C22×C5—21no—
359C22×C5—42no—
360C22×C5—42no—
361C22×C5—62no—
362C22×C5—122no—
363C22×C5—122no—
364C22×C5—121no—
365C22×C5—121no—
366C22×C5C42×D311yescharacteristic
367C22×C5#87211yescharacteristic
368C22×C5C4×D1212yes—
369C22×C5C2×(C2×D3)⋊C411yescharacteristic
370C22×C5C4×#6712yes—
371C4×C5—21no—
372C4×C5—21no—
373C4×C5—21no—
374C4×C5—44no—
375C4×C5—62no—
376C4×C5—124no—
377C4×C5—121no—
378C4×C5—121no—
379C4×C5#80612yesdirect product factor
380C24—31no—
381C24C22×C5×D311yescharacteristic
382C2×D4—34no—
383C2×D4—61no—
384C2×D4—62no—
385C2×D4—61no—
386C22×C4—22no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—42no—
391C22×C4—62no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—122no—
395C22×C4C4×C5×D311yescharacteristic
396C22×C4C4×C5×D311yescharacteristic
397C22×C4C5×D1211yescharacteristic
398C22×C4C5×D1211yescharacteristic
399C22×C4C5×#6711yescharacteristic
400C22×C4C5×#6711yescharacteristic
401C4⋊C4—61no—
402C4⋊C4—61no—
403C22⋊C4—22no—
404C22⋊C4—34no—
405C22⋊C4—61no—
406C22⋊C4—62no—
407C22⋊C4—61no—
408C22⋊C4—61no—
409C22⋊C4—61no—
410C22⋊C4C4×C5×D314yescomplemented
411C42—42no—
412C42—61no—
413C42—61no—
414C42—122no—
415C3×C5C4×#9911yescharacteristic, complemented, Hall
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3C2×C42×C511yescharacteristic
425C22×C3C2×C5×C22⋊C411yescharacteristic
426C22×C3C5×#11711yescharacteristic
427C22×C3C4×C5×D412yes—
428C22×C3C4×C5×D412yes—
429C2×D3—22no—
430C2×D3—42no—
431C2×D3—41no—
432C2×D3—41no—
433C2×D3—42no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—44no—
438C3×C4C5×#9912yescomplemented
439Dic3—22no—
440Dic3—44no—
441Dic3—41no—
442Dic3—41no—
443C2×C5—22no—
444C2×C5—22no—
445C2×C5—21no—
446C2×C5—21no—
447C2×C5—122no—
448C2×C5C4×(C2×D3)⋊C411yescharacteristic
449C2×C5C2×#80611yescharacteristic
450C2×C5#500611yescharacteristic
451C23—32no—
452C23—41no—
453C23—41no—
454C23—61no—
455C23—61no—
456C23—121no—
457C23—121no—
458C23C2×C4×C5×D311yescharacteristic
459C23C2×C4×C5×D312yes—
460C23C2×C5×D1211yescharacteristic
461C23C5×#23512yes—
462C23C2×C5×#6711yescharacteristic
463D4—64no—
464D4—64no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—32no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—64no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4C5×(C2×D3)⋊C411yescharacteristic
487C2×C4C5×(C2×D3)⋊C411yescharacteristic
488C2×C4C5×(C2×D3)⋊C411yescharacteristic
489C2×C4C5×(C2×D3)⋊C411yescharacteristic
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3C4×C5×C22⋊C411yescharacteristic
495C2×C3C2×C5×#9911yescharacteristic
496C2×C3C5×#41011yescharacteristic
497D3—42no—
498C5C4×#80611yescharacteristic, direct product factor, Sylow
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—42no—
506C22—42no—
507C22—62no—
508C22—122no—
509C22—122no—
510C22—121no—
511C22—121no—
512C22C42×C5×D311yescharacteristic
513C22C5×#87211yescharacteristic
514C22C4×C5×D1212yes—
515C22C2×C5×(C2×D3)⋊C411yescharacteristic
516C22C4×C5×#6712yes—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—44no—
521C4—62no—
522C4—124no—
523C4—121no—
524C4—121no—
525C4C5×#80612yesdirect product factor
526C3C4×C5×#9911yescharacteristic, complemented, Sylow
527C2—22no—
528C2—22no—
529C2—21no—
530C2—21no—
531C2—122no—
532C2C4×C5×(C2×D3)⋊C411yescharacteristic
533C2C2×C5×#80611yescharacteristic
534C2C5×#500611yescharacteristic
subgroupG/H
centerC2×C4×C5(C2×D3)⋊C4
commutatorC22×C3C2×C42×C5
FrattiniC24C22×C5×D3
FittingC3×C4×C5×C22⋊C4C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C5×#67C22
cosocleC24×C3C23×C5
hypercenterC4×C5×C22⋊C4D3
hypocenterC3C4×C5×#99
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C4×C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C5×C22⋊C4C2semidirect
3. C4×C5×#217C2semidirect
5. C5×#4527C2semidirect
9. C22×C4×C5×D3C4semidirect
18. C2×C5×(C2×D3)⋊C4C4semidirect
19. C2×C5×Dic3⋊C4C4semidirect
21. C2×C4×C5×Dic3C4semidirect
24. C5×#806C4direct (N × H)
25. C4×#806C5direct (N × H)
29. C4×C5×C22⋊C4D3semidirect
53. C3×C5×C22⋊C4C2×C4semidirect
58. C5×#217C2×C4semidirect
67. C3×C4×C22⋊C4C2×C5semidirect
68. C4×#217C2×C5semidirect
70. #4527C2×C5semidirect
101. C22×C5×D3C42semidirect
126. C2×C5×Dic3C42semidirect
131. C22×C4×D3C4×C5semidirect
140. C2×(C2×D3)⋊C4C4×C5semidirect
141. C2×Dic3⋊C4C4×C5semidirect
143. C2×C4×Dic3C4×C5semidirect
146. #806C4×C5direct (N × H)
177. C5×C22⋊C4C4×D3semidirect
185. C4×C22⋊C4C5×D3semidirect
209. C3×C4×C5#99semidirect
236. C3×C22⋊C4C2×C4×C5semidirect
241. #217C2×C4×C5semidirect
325. C22×D3C42×C5semidirect
350. C2×Dic3C42×C5semidirect
379. C4×C5#806direct (N × H)
410. C22⋊C4C4×C5×D3semidirect
415. C3×C5C4×#99semidirect
438. C3×C4C5×#99semidirect
498. C5C4×#806direct (N × H)
525. C4C5×#806direct (N × H)
526. C3C4×C5×#99semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×#672
C3×C4×C5×C22⋊C42
C4×C5×#2172
C2×C5×#8062
C5×#45272
C4×C5×#993
C4×#8065