Finite Group Databasebeta

← C3×C5×#2486 #14275520

Subgroups — C3×C5×#2486 #14275520

Order 1920.

1248 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#123C212yesmaximal
2C2×C3×C4×C5×D4C211yesmaximal, characteristic
3C22×C3×C5×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×C5×#398C212yesmaximal, complemented
5C3×C5×#397C212yesmaximal, complemented
6C3×C5×#389C212yesmaximal, complemented
7C3×C5×#381C212yesmaximal, complemented
8C3×C5×#380C211yesmaximal, characteristic, complemented
9C3×C5×#380C212yesmaximal, complemented
10C5×#2486C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2212yes—
13C23×C3×C4×C5C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C3×C5×#123—28no—
16C3×C4×C5×D4—24no—
17C2×C3×C5×C4⋊C4—24no—
18C2×C3×C5×C4⋊C4C2212yes—
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2214yes—
22C2×C3×C5×C4⋊C4C2212yes—
23C2×C3×C5×C22⋊C4C2211yescharacteristic
24C2×C3×C5×C22⋊C4C2211yescharacteristic
25C2×C3×C5×C22⋊C4C2212yes—
26C2×C3×C5×C22⋊C4C2214yes—
27C2×C3×C42×C5C2211yescharacteristic
28C3×C5×#95C2214yes—
29C3×C5×#95C2212yes—
30C3×C5×#95C2214yes—
31C3×#2486C511yesmaximal, characteristic, direct product factor, Hall
32C2×C5×#123C2×C312yes—
33C2×C4×C5×D4C2×C311yescharacteristic
34C22×C5×C4⋊C4C2×C311yescharacteristic, complemented
35C5×#398C2×C312yescomplemented
36C5×#397C2×C312yescomplemented
37C5×#389C2×C312yescomplemented
38C5×#381C2×C312yescomplemented
39C5×#380C2×C311yescharacteristic, complemented
40C5×#380C2×C312yescomplemented
41C24×C3×C5C2311yescharacteristic
42C24×C3×C5C2311yescharacteristic
43C2×C3×C5×D4—22no—
44C2×C3×C5×D4—22no—
45C2×C3×C5×D4D414yes—
46C22×C3×C4×C5—24no—
47C22×C3×C4×C5—24no—
48C22×C3×C4×C5—24no—
49C22×C3×C4×C5—24no—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—24no—
52C22×C3×C4×C5D414yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2314yes—
55C22×C3×C4×C5C2314yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C3×C5×C4⋊C4—24no—
61C3×C5×C4⋊C4—24no—
62C3×C5×C4⋊C4—22no—
63C3×C5×C4⋊C4—28no—
64C3×C5×C4⋊C4—24no—
65C3×C5×C22⋊C4—22no—
66C3×C5×C22⋊C4—24no—
67C3×C5×C22⋊C4—28no—
68C3×C5×C22⋊C4—22no—
69C3×C42×C5—22no—
70C2×C3×#123C2×C512yes—
71C2×C3×C4×D4C2×C511yescharacteristic
72C22×C3×C4⋊C4C2×C511yescharacteristic, complemented
73C3×#398C2×C512yescomplemented
74C3×#397C2×C512yescomplemented
75C3×#389C2×C512yescomplemented
76C3×#381C2×C512yescomplemented
77C3×#380C2×C512yescomplemented
78C3×#380C2×C511yescharacteristic, complemented
79C22×C5×D4C22×C311yescharacteristic
80C23×C4×C5C22×C311yescharacteristic
81C23×C4×C5C22×C312yes—
82C23×C4×C5C22×C311yescharacteristic
83C5×#123—28no—
84C4×C5×D4—24no—
85C2×C5×C4⋊C4—24no—
86C2×C5×C4⋊C4C22×C312yes—
87C2×C5×C4⋊C4C22×C312yes—
88C2×C5×C4⋊C4C22×C314yes—
89C2×C5×C4⋊C4C22×C312yes—
90C2×C5×C4⋊C4C22×C311yescharacteristic
91C2×C5×C22⋊C4C22×C312yes—
92C2×C5×C22⋊C4C22×C314yes—
93C2×C5×C22⋊C4C22×C311yescharacteristic
94C2×C5×C22⋊C4C22×C311yescharacteristic
95C2×C42×C5C22×C311yescharacteristic
96C5×#95C22×C314yes—
97C5×#95C22×C312yes—
98C5×#95C22×C314yes—
99#2486C3×C511yescharacteristic, direct product factor, Sylow
100C23×C3×C5—22no—
101C23×C3×C5—22no—
102C23×C3×C5—21no—
103C23×C3×C5—22no—
104C23×C3×C5—21no—
105C23×C3×C5—22no—
106C23×C3×C5C2×D412yes—
107C23×C3×C5C2×D412yes—
108C23×C3×C5C2×D412yes—
109C23×C3×C5#4112yes—
110C23×C3×C5C2411yescharacteristic
111C3×C5×D4—28no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—28no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—24no—
118C2×C3×C4×C5—24no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—24no—
121C2×C3×C4×C5—24no—
122C2×C3×C4×C5—24no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—24no—
126C2×C3×C4×C5—48no—
127C2×C3×C4×C5—44no—
128C2×C3×C4×C5C2×D412yes—
129C2×C3×C4×C5C2×D412yes—
130C2×C3×C4×C5C2×D412yes—
131C2×C3×C4×C5#4114yes—
132C22×C3×D4C22×C511yescharacteristic
133C23×C3×C4C22×C511yescharacteristic
134C23×C3×C4C22×C512yes—
135C23×C3×C4C22×C511yescharacteristic
136C3×#123—28no—
137C3×C4×D4—24no—
138C2×C3×C4⋊C4—24no—
139C2×C3×C4⋊C4C22×C512yes—
140C2×C3×C4⋊C4C22×C512yes—
141C2×C3×C4⋊C4C22×C514yes—
142C2×C3×C4⋊C4C22×C512yes—
143C2×C3×C4⋊C4C22×C511yescharacteristic
144C2×C3×C22⋊C4C22×C512yes—
145C2×C3×C22⋊C4C22×C514yes—
146C2×C3×C22⋊C4C22×C511yescharacteristic
147C2×C3×C22⋊C4C22×C511yescharacteristic
148C2×C3×C42C22×C511yescharacteristic
149C3×#95C22×C514yes—
150C3×#95C22×C512yes—
151C3×#95C22×C514yes—
152C24×C5C23×C311yescharacteristic
153C24×C5C23×C311yescharacteristic
154C2×C5×D4—22no—
155C2×C5×D4—22no—
156C2×C5×D4C3×D414yes—
157C22×C4×C5—24no—
158C22×C4×C5—24no—
159C22×C4×C5—24no—
160C22×C4×C5—24no—
161C22×C4×C5—22no—
162C22×C4×C5—24no—
163C22×C4×C5C3×D414yes—
164C22×C4×C5C23×C311yescharacteristic
165C22×C4×C5C23×C314yes—
166C22×C4×C5C23×C314yes—
167C22×C4×C5C23×C311yescharacteristic
168C22×C4×C5C23×C311yescharacteristic
169C22×C4×C5C23×C311yescharacteristic
170C22×C4×C5C23×C311yescharacteristic
171C5×C4⋊C4—24no—
172C5×C4⋊C4—24no—
173C5×C4⋊C4—22no—
174C5×C4⋊C4—28no—
175C5×C4⋊C4—24no—
176C5×C22⋊C4—22no—
177C5×C22⋊C4—24no—
178C5×C22⋊C4—28no—
179C5×C22⋊C4—22no—
180C42×C5—22no—
181C2×#123C2×C3×C512yes—
182C2×C4×D4C2×C3×C511yescharacteristic
183C22×C4⋊C4C2×C3×C511yescharacteristic, complemented
184#398C2×C3×C512yescomplemented
185#397C2×C3×C512yescomplemented
186#389C2×C3×C512yescomplemented
187#381C2×C3×C512yescomplemented
188#380C2×C3×C512yescomplemented
189#380C2×C3×C511yescharacteristic, complemented
190C22×C3×C5—22no—
191C22×C3×C5—22no—
192C22×C3×C5—24no—
193C22×C3×C5—24no—
194C22×C3×C5—22no—
195C22×C3×C5—24no—
196C22×C3×C5—41no—
197C22×C3×C5—41no—
198C22×C3×C5—42no—
199C22×C3×C5#12114yes—
200C22×C3×C5C22×D411yescharacteristic
201C22×C3×C5C22×D411yescharacteristic
202C22×C3×C5C2×#4111yescharacteristic
203C22×C3×C5C2×#4112yes—
204C22×C3×C52−1+412yes—
205C3×C4×C5—22no—
206C3×C4×C5—22no—
207C3×C4×C5—44no—
208C3×C4×C5—41no—
209C3×C4×C5—41no—
210C3×C4×C5—44no—
211C3×C4×C5#12314yescomplemented
212C24×C3C23×C511yescharacteristic
213C24×C3C23×C511yescharacteristic
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4C5×D414yes—
217C22×C3×C4—24no—
218C22×C3×C4—24no—
219C22×C3×C4—24no—
220C22×C3×C4—24no—
221C22×C3×C4—22no—
222C22×C3×C4—24no—
223C22×C3×C4C5×D414yes—
224C22×C3×C4C23×C511yescharacteristic
225C22×C3×C4C23×C511yescharacteristic
226C22×C3×C4C23×C514yes—
227C22×C3×C4C23×C514yes—
228C22×C3×C4C23×C511yescharacteristic
229C22×C3×C4C23×C511yescharacteristic
230C22×C3×C4C23×C511yescharacteristic
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—24no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—28no—
239C3×C22⋊C4—22no—
240C3×C42—22no—
241C23×C5—22no—
242C23×C5—22no—
243C23×C5—21no—
244C23×C5—22no—
245C23×C5—21no—
246C23×C5—22no—
247C23×C5C2×C3×D412yes—
248C23×C5C2×C3×D412yes—
249C23×C5C2×C3×D412yes—
250C23×C5C3×#4112yes—
251C23×C5C24×C311yescharacteristic
252C5×D4—28no—
253C2×C4×C5—21no—
254C2×C4×C5—22no—
255C2×C4×C5—21no—
256C2×C4×C5—28no—
257C2×C4×C5—21no—
258C2×C4×C5—24no—
259C2×C4×C5—24no—
260C2×C4×C5—21no—
261C2×C4×C5—24no—
262C2×C4×C5—24no—
263C2×C4×C5—24no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—24no—
267C2×C4×C5—48no—
268C2×C4×C5—44no—
269C2×C4×C5C2×C3×D412yes—
270C2×C4×C5C2×C3×D412yes—
271C2×C4×C5C2×C3×D412yes—
272C2×C4×C5C3×#4114yes—
273C22×D4C22×C3×C511yescharacteristic
274C23×C4C22×C3×C512yes—
275C23×C4C22×C3×C511yescharacteristic
276C23×C4C22×C3×C511yescharacteristic
277#123—28no—
278C4×D4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4C22×C3×C512yes—
281C2×C4⋊C4C22×C3×C512yes—
282C2×C4⋊C4C22×C3×C511yescharacteristic
283C2×C4⋊C4C22×C3×C514yes—
284C2×C4⋊C4C22×C3×C512yes—
285C2×C22⋊C4C22×C3×C511yescharacteristic
286C2×C22⋊C4C22×C3×C511yescharacteristic
287C2×C22⋊C4C22×C3×C512yes—
288C2×C22⋊C4C22×C3×C514yes—
289C2×C42C22×C3×C511yescharacteristic
290#95C22×C3×C514yes—
291#95C22×C3×C512yes—
292#95C22×C3×C514yes—
293C2×C3×C5—24no—
294C2×C3×C5—42no—
295C2×C3×C5C2×#12111yescharacteristic
296C2×C3×C5C2×#12311yescharacteristic
297C2×C3×C5#53611yescharacteristic
298C2×C3×C5#54712yes—
299C2×C3×C5#55212yes—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3C2×C5×D412yes—
307C23×C3C2×C5×D412yes—
308C23×C3C2×C5×D412yes—
309C23×C3C5×#4112yes—
310C23×C3C24×C511yescharacteristic
311C3×D4—28no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—28no—
316C2×C3×C4—21no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—21no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—48no—
327C2×C3×C4—44no—
328C2×C3×C4C2×C5×D412yes—
329C2×C3×C4C2×C5×D412yes—
330C2×C3×C4C2×C5×D412yes—
331C2×C3×C4C5×#4114yes—
332C22×C5—22no—
333C22×C5—22no—
334C22×C5—24no—
335C22×C5—24no—
336C22×C5—22no—
337C22×C5—24no—
338C22×C5—41no—
339C22×C5—41no—
340C22×C5—42no—
341C22×C5C3×#12114yes—
342C22×C5C22×C3×D411yescharacteristic
343C22×C5C22×C3×D411yescharacteristic
344C22×C5C2×C3×#4111yescharacteristic
345C22×C5C2×C3×#4112yes—
346C22×C5C3×2−1+412yes—
347C4×C5—22no—
348C4×C5—22no—
349C4×C5—44no—
350C4×C5—41no—
351C4×C5—41no—
352C4×C5—44no—
353C4×C5C3×#12314yescomplemented
354C24C23×C3×C511yescharacteristic
355C24C23×C3×C511yescharacteristic
356C2×D4—22no—
357C2×D4—22no—
358C2×D4C3×C5×D414yes—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4C3×C5×D414yes—
366C22×C4C23×C3×C511yescharacteristic
367C22×C4C23×C3×C511yescharacteristic
368C22×C4C23×C3×C514yes—
369C22×C4C23×C3×C514yes—
370C22×C4C23×C3×C511yescharacteristic
371C22×C4C23×C3×C511yescharacteristic
372C22×C4C23×C3×C511yescharacteristic
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C22⋊C4—22no—
379C22⋊C4—24no—
380C22⋊C4—28no—
381C22⋊C4—22no—
382C42—22no—
383C3×C5#248611yescharacteristic, direct product factor, Hall
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—22no—
389C22×C3—24no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—42no—
393C22×C3C5×#12114yes—
394C22×C3C22×C5×D411yescharacteristic
395C22×C3C22×C5×D411yescharacteristic
396C22×C3C2×C5×#4111yescharacteristic
397C22×C3C2×C5×#4112yes—
398C22×C3C5×2−1+412yes—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—44no—
402C3×C4—41no—
403C3×C4—41no—
404C3×C4—44no—
405C3×C4C5×#12314yescomplemented
406C2×C5—24no—
407C2×C5—42no—
408C2×C5C2×C3×#12111yescharacteristic
409C2×C5C2×C3×#12311yescharacteristic
410C2×C5C3×#53611yescharacteristic
411C2×C5C3×#54712yes—
412C2×C5C3×#55212yes—
413C23—22no—
414C23—22no—
415C23—21no—
416C23—22no—
417C23—21no—
418C23—22no—
419C23C2×C3×C5×D412yes—
420C23C2×C3×C5×D412yes—
421C23C2×C3×C5×D412yes—
422C23C3×C5×#4112yes—
423C23C24×C3×C511yescharacteristic
424D4—28no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—28no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—48no—
440C2×C4—44no—
441C2×C4C2×C3×C5×D412yes—
442C2×C4C2×C3×C5×D412yes—
443C2×C4C2×C3×C5×D412yes—
444C2×C4C3×C5×#4114yes—
445C2×C3—24no—
446C2×C3—42no—
447C2×C3C2×C5×#12111yescharacteristic
448C2×C3C2×C5×#12311yescharacteristic
449C2×C3C5×#53611yescharacteristic
450C2×C3C5×#54712yes—
451C2×C3C5×#55212yes—
452C5C3×#248611yescharacteristic, direct product factor, Sylow
453C22—22no—
454C22—22no—
455C22—24no—
456C22—24no—
457C22—22no—
458C22—24no—
459C22—41no—
460C22—41no—
461C22—42no—
462C22C3×C5×#12114yes—
463C22C22×C3×C5×D411yescharacteristic
464C22C22×C3×C5×D411yescharacteristic
465C22C2×C3×C5×#4111yescharacteristic
466C22C2×C3×C5×#4112yes—
467C22C3×C5×2−1+412yes—
468C4—22no—
469C4—22no—
470C4—44no—
471C4—41no—
472C4—41no—
473C4—44no—
474C4C3×C5×#12314yescomplemented
475C3C5×#248611yescharacteristic, direct product factor, Sylow
476C2—24no—
477C2—42no—
478C2C2×C3×C5×#12111yescharacteristic
479C2C2×C3×C5×#12311yescharacteristic
480C2C3×C5×#53611yescharacteristic
481C2C3×C5×#54712yes—
482C2C3×C5×#55212yes—
subgroupG/H
center, socleC23×C3×C5C24
commutator, Frattini, cosocleC23C24×C3×C5
prime radicalC22×C3×C5×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
3. C22×C3×C5×C4⋊C4C2semidirect
4. C3×C5×#398C2semidirect
5. C3×C5×#397C2semidirect
6. C3×C5×#389C2semidirect
7. C3×C5×#381C2semidirect
8. C3×C5×#380C2semidirect
9. C3×C5×#380C2semidirect
10. C5×#2486C3direct (N × H)
31. C3×#2486C5direct (N × H)
34. C22×C5×C4⋊C4C2×C3semidirect
35. C5×#398C2×C3semidirect
36. C5×#397C2×C3semidirect
37. C5×#389C2×C3semidirect
38. C5×#381C2×C3semidirect
39. C5×#380C2×C3semidirect
40. C5×#380C2×C3semidirect
72. C22×C3×C4⋊C4C2×C5semidirect
73. C3×#398C2×C5semidirect
74. C3×#397C2×C5semidirect
75. C3×#389C2×C5semidirect
76. C3×#381C2×C5semidirect
77. C3×#380C2×C5semidirect
78. C3×#380C2×C5semidirect
99. #2486C3×C5direct (N × H)
183. C22×C4⋊C4C2×C3×C5semidirect
184. #398C2×C3×C5semidirect
185. #397C2×C3×C5semidirect
186. #389C2×C3×C5semidirect
187. #381C2×C3×C5semidirect
188. #380C2×C3×C5semidirect
189. #380C2×C3×C5semidirect
211. C3×C4×C5#123semidirect
353. C4×C5C3×#123semidirect
383. C3×C5#2486direct (N × H)
405. C3×C4C5×#123semidirect
452. C5C3×#2486direct (N × H)
474. C4C3×C5×#123semidirect
475. C3C5×#2486direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1232
C2×C3×C4×C5×D42
C22×C3×C5×C4⋊C42
C3×C5×#3982
C3×C5×#3972
C3×C5×#3892
C3×C5×#3812
C3×C5×#3802
C5×#24863
C3×#24865