Finite Group Databasebeta

← C3×C5×#2406 #14275440

Subgroups — C3×C5×#2406 #14275440

Order 1920.

1728 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×#41C211yesmaximal, characteristic
2C2×C3×C5×#123C212yesmaximal, complemented
3C22×C3×C5×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×C5×#395C212yesmaximal, complemented
5C3×C5×#394C212yesmaximal, complemented
6C3×C5×#386C212yesmaximal, complemented
7C3×C5×#385C214yesmaximal, complemented
8C3×C5×#381C211yesmaximal, characteristic, complemented
9C5×#2406C311yesmaximal, characteristic, direct product factor, Hall
10C2×C3×C5×#41—28no—
11C2×C3×C5×#41—24no—
12C22×C3×C5×Q8C2211yescharacteristic
13C22×C3×C5×D4C2211yescharacteristic
14C22×C3×C5×D4C2212yes—
15C23×C3×C4×C5C2211yescharacteristic
16C23×C3×C4×C5C2211yescharacteristic
17C23×C3×C4×C5C2212yes—
18C23×C3×C4×C5C2211yescharacteristic
19C3×C5×#123—28no—
20C2×C3×C5×C4⋊C4—24no—
21C2×C3×C5×C4⋊C4C2212yes—
22C2×C3×C5×C4⋊C4C2212yes—
23C2×C3×C5×C4⋊C4C2214yes—
24C2×C3×C5×C22⋊C4C2212yes—
25C2×C3×C5×C22⋊C4C2214yes—
26C2×C3×C5×C22⋊C4C2214yes—
27C3×C5×#95C2212yes—
28C3×C5×#95C2212yes—
29C3×C5×#95C2214yes—
30C3×#2406C511yesmaximal, characteristic, direct product factor, Hall
31C22×C5×#41C2×C311yescharacteristic
32C2×C5×#123C2×C312yescomplemented
33C22×C5×C4⋊C4C2×C311yescharacteristic, complemented
34C5×#395C2×C312yescomplemented
35C5×#394C2×C312yescomplemented
36C5×#386C2×C312yescomplemented
37C5×#385C2×C314yescomplemented
38C5×#381C2×C311yescharacteristic, complemented
39C24×C3×C5C2312yes—
40C24×C3×C5C2311yescharacteristic
41C3×C5×#41—232no—
42C2×C3×C5×Q8—24no—
43C2×C3×C5×Q8—22no—
44C2×C3×C5×D4—24no—
45C2×C3×C5×D4—24no—
46C2×C3×C5×D4—24no—
47C2×C3×C5×D4—22no—
48C2×C3×C5×D4D418yes—
49C22×C3×C4×C5—22no—
50C22×C3×C4×C5—24no—
51C22×C3×C4×C5—24no—
52C22×C3×C4×C5—24no—
53C22×C3×C4×C5—24no—
54C22×C3×C4×C5—24no—
55C22×C3×C4×C5—22no—
56C22×C3×C4×C5—24no—
57C22×C3×C4×C5D414yes—
58C22×C3×C4×C5C2314yes—
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2311yescharacteristic
61C22×C3×C4×C5C2312yes—
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2312yes—
64C3×C5×C4⋊C4—24no—
65C3×C5×C4⋊C4—24no—
66C3×C5×C4⋊C4—28no—
67C3×C5×C22⋊C4—24no—
68C3×C5×C22⋊C4—28no—
69C3×C5×C22⋊C4—28no—
70C22×C3×#41C2×C511yescharacteristic
71C2×C3×#123C2×C512yescomplemented
72C22×C3×C4⋊C4C2×C511yescharacteristic, complemented
73C3×#395C2×C512yescomplemented
74C3×#394C2×C512yescomplemented
75C3×#386C2×C512yescomplemented
76C3×#385C2×C514yescomplemented
77C3×#381C2×C511yescharacteristic, complemented
78C2×C5×#41—28no—
79C2×C5×#41—24no—
80C22×C5×Q8C22×C311yescharacteristic
81C22×C5×D4C22×C311yescharacteristic
82C22×C5×D4C22×C312yes—
83C23×C4×C5C22×C312yes—
84C23×C4×C5C22×C311yescharacteristic
85C23×C4×C5C22×C311yescharacteristic
86C23×C4×C5C22×C311yescharacteristic
87C5×#123—28no—
88C2×C5×C4⋊C4—24no—
89C2×C5×C4⋊C4C22×C312yes—
90C2×C5×C4⋊C4C22×C314yes—
91C2×C5×C4⋊C4C22×C312yes—
92C2×C5×C22⋊C4C22×C312yes—
93C2×C5×C22⋊C4C22×C314yes—
94C2×C5×C22⋊C4C22×C314yes—
95C5×#95C22×C312yes—
96C5×#95C22×C312yes—
97C5×#95C22×C314yes—
98#2406C3×C511yescharacteristic, direct product factor, Sylow
99C23×C3×C5—22no—
100C23×C3×C5—22no—
101C23×C3×C5—22no—
102C23×C3×C5—22no—
103C23×C3×C5—24no—
104C23×C3×C5—24no—
105C23×C3×C5C2×D414yes—
106C23×C3×C5C2×D414yes—
107C23×C3×C5C2×D412yes—
108C23×C3×C5C2411yescharacteristic
109C3×C5×Q8—28no—
110C3×C5×D4—28no—
111C3×C5×D4—216no—
112C2×C3×C4×C5—28no—
113C2×C3×C4×C5—216no—
114C2×C3×C4×C5—28no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—24no—
122C2×C3×C4×C5—24no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—24no—
125C2×C3×C4×C5—24no—
126C2×C3×C4×C5—44no—
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5C2×D414yes—
129C2×C3×C4×C5C2×D412yes—
130C2×C3×C4×C5#4114yescomplemented
131C2×C3×#41—28no—
132C2×C3×#41—24no—
133C22×C3×Q8C22×C511yescharacteristic
134C22×C3×D4C22×C511yescharacteristic
135C22×C3×D4C22×C512yes—
136C23×C3×C4C22×C512yes—
137C23×C3×C4C22×C511yescharacteristic
138C23×C3×C4C22×C511yescharacteristic
139C23×C3×C4C22×C511yescharacteristic
140C3×#123—28no—
141C2×C3×C4⋊C4—24no—
142C2×C3×C4⋊C4C22×C512yes—
143C2×C3×C4⋊C4C22×C514yes—
144C2×C3×C4⋊C4C22×C512yes—
145C2×C3×C22⋊C4C22×C512yes—
146C2×C3×C22⋊C4C22×C514yes—
147C2×C3×C22⋊C4C22×C514yes—
148C3×#95C22×C512yes—
149C3×#95C22×C512yes—
150C3×#95C22×C514yes—
151C24×C5C23×C312yes—
152C24×C5C23×C311yescharacteristic
153C5×#41—232no—
154C2×C5×Q8—24no—
155C2×C5×Q8—22no—
156C2×C5×D4—24no—
157C2×C5×D4—24no—
158C2×C5×D4—24no—
159C2×C5×D4—22no—
160C2×C5×D4C3×D418yes—
161C22×C4×C5—22no—
162C22×C4×C5—24no—
163C22×C4×C5—24no—
164C22×C4×C5—24no—
165C22×C4×C5—24no—
166C22×C4×C5—24no—
167C22×C4×C5—22no—
168C22×C4×C5—24no—
169C22×C4×C5C3×D414yes—
170C22×C4×C5C23×C314yes—
171C22×C4×C5C23×C312yes—
172C22×C4×C5C23×C311yescharacteristic
173C22×C4×C5C23×C312yes—
174C22×C4×C5C23×C311yescharacteristic
175C22×C4×C5C23×C312yes—
176C5×C4⋊C4—24no—
177C5×C4⋊C4—24no—
178C5×C4⋊C4—28no—
179C5×C22⋊C4—24no—
180C5×C22⋊C4—28no—
181C5×C22⋊C4—28no—
182C22×#41C2×C3×C511yescharacteristic
183C2×#123C2×C3×C512yescomplemented
184C22×C4⋊C4C2×C3×C511yescharacteristic, complemented
185#395C2×C3×C512yescomplemented
186#394C2×C3×C512yescomplemented
187#386C2×C3×C512yescomplemented
188#385C2×C3×C514yescomplemented
189#381C2×C3×C511yescharacteristic, complemented
190C22×C3×C5—22no—
191C22×C3×C5—22no—
192C22×C3×C5—28no—
193C22×C3×C5—24no—
194C22×C3×C5—28no—
195C22×C3×C5—42no—
196C22×C3×C5—42no—
197C22×C3×C5—44no—
198C22×C3×C5#12014yes—
199C22×C3×C5C22×D411yescharacteristic
200C22×C3×C5C22×D412yes—
201C22×C3×C5C2×#4112yes—
202C22×C3×C52−1+412yes—
203C3×C4×C5—22no—
204C3×C4×C5—24no—
205C3×C4×C5—22no—
206C3×C4×C5—24no—
207C3×C4×C5—42no—
208C3×C4×C5—44no—
209C24×C3C23×C512yes—
210C24×C3C23×C511yescharacteristic
211C3×#41—232no—
212C2×C3×Q8—24no—
213C2×C3×Q8—22no—
214C2×C3×D4—24no—
215C2×C3×D4—24no—
216C2×C3×D4—24no—
217C2×C3×D4—22no—
218C2×C3×D4C5×D418yes—
219C22×C3×C4—22no—
220C22×C3×C4—24no—
221C22×C3×C4—24no—
222C22×C3×C4—24no—
223C22×C3×C4—24no—
224C22×C3×C4—24no—
225C22×C3×C4—22no—
226C22×C3×C4—24no—
227C22×C3×C4C5×D414yes—
228C22×C3×C4C23×C514yes—
229C22×C3×C4C23×C512yes—
230C22×C3×C4C23×C511yescharacteristic
231C22×C3×C4C23×C512yes—
232C22×C3×C4C23×C511yescharacteristic
233C22×C3×C4C23×C512yes—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—28no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—28no—
239C3×C22⋊C4—28no—
240C23×C5—22no—
241C23×C5—22no—
242C23×C5—22no—
243C23×C5—22no—
244C23×C5—24no—
245C23×C5—24no—
246C23×C5C2×C3×D414yes—
247C23×C5C2×C3×D414yes—
248C23×C5C2×C3×D412yes—
249C23×C5C24×C311yescharacteristic
250C5×Q8—28no—
251C5×D4—28no—
252C5×D4—216no—
253C2×C4×C5—28no—
254C2×C4×C5—216no—
255C2×C4×C5—28no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—24no—
263C2×C4×C5—24no—
264C2×C4×C5—22no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—44no—
268C2×C4×C5C2×C3×D412yes—
269C2×C4×C5C2×C3×D414yes—
270C2×C4×C5C2×C3×D412yes—
271C2×C4×C5C3×#4114yescomplemented
272C2×#41—28no—
273C2×#41—24no—
274C22×Q8C22×C3×C511yescharacteristic
275C22×D4C22×C3×C511yescharacteristic
276C22×D4C22×C3×C512yes—
277C23×C4C22×C3×C511yescharacteristic
278C23×C4C22×C3×C511yescharacteristic
279C23×C4C22×C3×C512yes—
280C23×C4C22×C3×C511yescharacteristic
281#123—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4C22×C3×C512yes—
284C2×C4⋊C4C22×C3×C512yes—
285C2×C4⋊C4C22×C3×C514yes—
286C2×C22⋊C4C22×C3×C512yes—
287C2×C22⋊C4C22×C3×C514yes—
288C2×C22⋊C4C22×C3×C514yes—
289#95C22×C3×C512yes—
290#95C22×C3×C512yes—
291#95C22×C3×C514yes—
292C2×C3×C5—24no—
293C2×C3×C5—44no—
294C2×C3×C5C2×#12011yescharacteristic
295C2×C3×C5#52511yescharacteristic
296C2×C3×C5#53611yescharacteristic
297C2×C3×C5#54714yes—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3C2×C5×D414yes—
305C23×C3C2×C5×D414yes—
306C23×C3C2×C5×D412yes—
307C23×C3C24×C511yescharacteristic
308C3×Q8—28no—
309C3×D4—28no—
310C3×D4—216no—
311C2×C3×C4—28no—
312C2×C3×C4—216no—
313C2×C3×C4—28no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—44no—
326C2×C3×C4C2×C5×D412yes—
327C2×C3×C4C2×C5×D414yes—
328C2×C3×C4C2×C5×D412yes—
329C2×C3×C4C5×#4114yescomplemented
330C22×C5—22no—
331C22×C5—22no—
332C22×C5—28no—
333C22×C5—24no—
334C22×C5—28no—
335C22×C5—42no—
336C22×C5—42no—
337C22×C5—44no—
338C22×C5C3×#12014yes—
339C22×C5C22×C3×D411yescharacteristic
340C22×C5C22×C3×D412yes—
341C22×C5C2×C3×#4112yes—
342C22×C5C3×2−1+412yes—
343C4×C5—22no—
344C4×C5—24no—
345C4×C5—22no—
346C4×C5—24no—
347C4×C5—42no—
348C4×C5—44no—
349C24C23×C3×C512yes—
350C24C23×C3×C511yescharacteristic
351#41—232no—
352C2×Q8—24no—
353C2×Q8—22no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4C3×C5×D418yes—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4C3×C5×D414yes—
368C22×C4C23×C3×C514yes—
369C22×C4C23×C3×C512yes—
370C22×C4C23×C3×C511yescharacteristic
371C22×C4C23×C3×C512yes—
372C22×C4C23×C3×C511yescharacteristic
373C22×C4C23×C3×C512yes—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C22⋊C4—24no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C3×C5#240611yescharacteristic, direct product factor, Hall
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—28no—
384C22×C3—24no—
385C22×C3—28no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—44no—
389C22×C3C5×#12014yes—
390C22×C3C22×C5×D411yescharacteristic
391C22×C3C22×C5×D412yes—
392C22×C3C2×C5×#4112yes—
393C22×C3C5×2−1+412yes—
394C3×C4—22no—
395C3×C4—24no—
396C3×C4—22no—
397C3×C4—24no—
398C3×C4—42no—
399C3×C4—44no—
400C2×C5—24no—
401C2×C5—44no—
402C2×C5C2×C3×#12011yescharacteristic
403C2×C5C3×#52511yescharacteristic
404C2×C5C3×#53611yescharacteristic
405C2×C5C3×#54714yes—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23C2×C3×C5×D414yes—
413C23C2×C3×C5×D414yes—
414C23C2×C3×C5×D412yes—
415C23C24×C3×C511yescharacteristic
416Q8—28no—
417D4—28no—
418D4—216no—
419C2×C4—216no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—44no—
434C2×C4C2×C3×C5×D412yes—
435C2×C4C2×C3×C5×D414yes—
436C2×C4C2×C3×C5×D412yes—
437C2×C4C3×C5×#4114yescomplemented
438C2×C3—24no—
439C2×C3—44no—
440C2×C3C2×C5×#12011yescharacteristic
441C2×C3C5×#52511yescharacteristic
442C2×C3C5×#53611yescharacteristic
443C2×C3C5×#54714yes—
444C5C3×#240611yescharacteristic, direct product factor, Sylow
445C22—22no—
446C22—22no—
447C22—28no—
448C22—24no—
449C22—28no—
450C22—42no—
451C22—42no—
452C22—44no—
453C22C3×C5×#12014yes—
454C22C22×C3×C5×D411yescharacteristic
455C22C22×C3×C5×D412yes—
456C22C2×C3×C5×#4112yes—
457C22C3×C5×2−1+412yes—
458C4—22no—
459C4—24no—
460C4—22no—
461C4—24no—
462C4—42no—
463C4—44no—
464C3C5×#240611yescharacteristic, direct product factor, Sylow
465C2—24no—
466C2—44no—
467C2C2×C3×C5×#12011yescharacteristic
468C2C3×C5×#52511yescharacteristic
469C2C3×C5×#53611yescharacteristic
470C2C3×C5×#54714yes—
subgroupG/H
center, socleC23×C3×C5C24
commutator, Frattini, cosocleC23C24×C3×C5
prime radicalC22×C3×C5×#41C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#123C2semidirect
3. C22×C3×C5×C4⋊C4C2semidirect
4. C3×C5×#395C2semidirect
5. C3×C5×#394C2semidirect
6. C3×C5×#386C2semidirect
7. C3×C5×#385C2semidirect
8. C3×C5×#381C2semidirect
9. C5×#2406C3direct (N × H)
30. C3×#2406C5direct (N × H)
32. C2×C5×#123C2×C3semidirect
33. C22×C5×C4⋊C4C2×C3semidirect
34. C5×#395C2×C3semidirect
35. C5×#394C2×C3semidirect
36. C5×#386C2×C3semidirect
37. C5×#385C2×C3semidirect
38. C5×#381C2×C3semidirect
71. C2×C3×#123C2×C5semidirect
72. C22×C3×C4⋊C4C2×C5semidirect
73. C3×#395C2×C5semidirect
74. C3×#394C2×C5semidirect
75. C3×#386C2×C5semidirect
76. C3×#385C2×C5semidirect
77. C3×#381C2×C5semidirect
98. #2406C3×C5direct (N × H)
130. C2×C3×C4×C5#41semidirect
183. C2×#123C2×C3×C5semidirect
184. C22×C4⋊C4C2×C3×C5semidirect
185. #395C2×C3×C5semidirect
186. #394C2×C3×C5semidirect
187. #386C2×C3×C5semidirect
188. #385C2×C3×C5semidirect
189. #381C2×C3×C5semidirect
271. C2×C4×C5C3×#41semidirect
329. C2×C3×C4C5×#41semidirect
380. C3×C5#2406direct (N × H)
437. C2×C4C3×C5×#41semidirect
444. C5C3×#2406direct (N × H)
464. C3C5×#2406direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×#412
C2×C3×C5×#1232
C22×C3×C5×C4⋊C42
C3×C5×#3952
C3×C5×#3942
C3×C5×#3862
C3×C5×#3852
C3×C5×#3812
C5×#24063
C3×#24065