Finite Group Databasebeta

← C3×C5×#2618 #14275652

Subgroups — C3×C5×#2618 #14275652

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#123C212yesmaximal, complemented
2C2×C3×C5×#121C211yesmaximal, characteristic
3C22×C3×C5×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×C5×#398C212yesmaximal, complemented
5C3×C5×#396C214yesmaximal, complemented
6C3×C5×#394C212yesmaximal, complemented
7C3×C5×#386C212yesmaximal, complemented
8C3×C5×#380C211yesmaximal, characteristic, complemented
9C5×#2618C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2212yes—
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C3×C5×#123—28no—
16C3×C5×#121—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⋊C4C2212yes—
24C2×C3×C5×C22⋊C4C2212yes—
25C2×C3×C5×C22⋊C4C2214yes—
26C2×C3×C5×C22⋊C4C2214yes—
27C3×C5×#95C2212yes—
28C3×C5×#95C2214yes—
29C3×#2618C511yesmaximal, characteristic, direct product factor, Hall
30C2×C5×#123C2×C312yescomplemented
31C2×C5×#121C2×C311yescharacteristic
32C22×C5×C4⋊C4C2×C311yescharacteristic, complemented
33C5×#398C2×C312yescomplemented
34C5×#396C2×C314yescomplemented
35C5×#394C2×C312yescomplemented
36C5×#386C2×C312yescomplemented
37C5×#380C2×C311yescharacteristic, complemented
38C24×C3×C5C2312yes—
39C24×C3×C5C2311yescharacteristic
40C2×C3×C5×D4—24no—
41C2×C3×C5×D4—24no—
42C2×C3×C5×D4—24no—
43C2×C3×C5×D4—22no—
44C2×C3×C5×D4—42no—
45C22×C3×C4×C5—22no—
46C22×C3×C4×C5—22no—
47C22×C3×C4×C5—22no—
48C22×C3×C4×C5—42no—
49C22×C3×C4×C5—42no—
50C22×C3×C4×C5D414yes—
51C22×C3×C4×C5D414yescomplemented
52C22×C3×C4×C5C2314yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C3×C5×C4⋊C4—24no—
59C3×C5×C4⋊C4—24no—
60C3×C5×C4⋊C4—22no—
61C3×C5×C4⋊C4—28no—
62C3×C5×C4⋊C4—42no—
63C3×C5×C22⋊C4—24no—
64C3×C5×C22⋊C4—24no—
65C3×C5×C22⋊C4—28no—
66C3×C5×C22⋊C4—44no—
67C2×C3×#123C2×C512yescomplemented
68C2×C3×#121C2×C511yescharacteristic
69C22×C3×C4⋊C4C2×C511yescharacteristic, complemented
70C3×#398C2×C512yescomplemented
71C3×#396C2×C514yescomplemented
72C3×#394C2×C512yescomplemented
73C3×#386C2×C512yescomplemented
74C3×#380C2×C511yescharacteristic, complemented
75C22×C5×D4C22×C312yes—
76C22×C5×D4C22×C311yescharacteristic
77C23×C4×C5C22×C311yescharacteristic
78C23×C4×C5C22×C311yescharacteristic
79C23×C4×C5C22×C311yescharacteristic
80C5×#123—28no—
81C5×#121—24no—
82C2×C5×C4⋊C4—24no—
83C2×C5×C4⋊C4C22×C312yes—
84C2×C5×C4⋊C4C22×C312yes—
85C2×C5×C4⋊C4C22×C314yes—
86C2×C5×C4⋊C4C22×C311yescharacteristic
87C2×C5×C4⋊C4C22×C312yes—
88C2×C5×C22⋊C4C22×C312yes—
89C2×C5×C22⋊C4C22×C314yes—
90C2×C5×C22⋊C4C22×C312yes—
91C2×C5×C22⋊C4C22×C314yes—
92C5×#95C22×C312yes—
93C5×#95C22×C314yes—
94#2618C3×C511yescharacteristic, direct product factor, Sylow
95C23×C3×C5—22no—
96C23×C3×C5—22no—
97C23×C3×C5—22no—
98C23×C3×C5—22no—
99C23×C3×C5—22no—
100C23×C3×C5—24no—
101C23×C3×C5—24no—
102C23×C3×C5C2×D412yes—
103C23×C3×C5C2×D412yes—
104C23×C3×C5#4112yes—
105C23×C3×C5C2411yescharacteristic
106C3×C5×D4—44no—
107C3×C5×D4—48no—
108C2×C3×C4×C5—28no—
109C2×C3×C4×C5—22no—
110C2×C3×C4×C5—22no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—24no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—24no—
119C2×C3×C4×C5—24no—
120C2×C3×C4×C5—24no—
121C2×C3×C4×C5—44no—
122C2×C3×C4×C5—44no—
123C2×C3×C4×C5C2×D412yes—
124C2×C3×C4×C5C2×D414yes—
125C2×C3×C4×C5C2×D412yes—
126C22×C3×D4C22×C512yes—
127C22×C3×D4C22×C511yescharacteristic
128C23×C3×C4C22×C511yescharacteristic
129C23×C3×C4C22×C511yescharacteristic
130C23×C3×C4C22×C511yescharacteristic
131C3×#123—28no—
132C3×#121—24no—
133C2×C3×C4⋊C4—24no—
134C2×C3×C4⋊C4C22×C512yes—
135C2×C3×C4⋊C4C22×C512yes—
136C2×C3×C4⋊C4C22×C514yes—
137C2×C3×C4⋊C4C22×C511yescharacteristic
138C2×C3×C4⋊C4C22×C512yes—
139C2×C3×C22⋊C4C22×C512yes—
140C2×C3×C22⋊C4C22×C514yes—
141C2×C3×C22⋊C4C22×C512yes—
142C2×C3×C22⋊C4C22×C514yes—
143C3×#95C22×C512yes—
144C3×#95C22×C514yes—
145C24×C5C23×C312yes—
146C24×C5C23×C311yescharacteristic
147C2×C5×D4—24no—
148C2×C5×D4—24no—
149C2×C5×D4—24no—
150C2×C5×D4—22no—
151C2×C5×D4—42no—
152C22×C4×C5—22no—
153C22×C4×C5—22no—
154C22×C4×C5—22no—
155C22×C4×C5—42no—
156C22×C4×C5—42no—
157C22×C4×C5C3×D414yes—
158C22×C4×C5C3×D414yescomplemented
159C22×C4×C5C23×C314yes—
160C22×C4×C5C23×C312yes—
161C22×C4×C5C23×C312yes—
162C22×C4×C5C23×C312yes—
163C22×C4×C5C23×C311yescharacteristic
164C22×C4×C5C23×C311yescharacteristic
165C5×C4⋊C4—24no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—28no—
169C5×C4⋊C4—42no—
170C5×C22⋊C4—24no—
171C5×C22⋊C4—24no—
172C5×C22⋊C4—28no—
173C5×C22⋊C4—44no—
174C2×#123C2×C3×C512yescomplemented
175C2×#121C2×C3×C511yescharacteristic
176C22×C4⋊C4C2×C3×C511yescharacteristic, complemented
177#398C2×C3×C512yescomplemented
178#396C2×C3×C514yescomplemented
179#394C2×C3×C512yescomplemented
180#386C2×C3×C512yescomplemented
181#380C2×C3×C511yescharacteristic, complemented
182C22×C3×C5—22no—
183C22×C3×C5—22no—
184C22×C3×C5—24no—
185C22×C3×C5—24no—
186C22×C3×C5—42no—
187C22×C3×C5—42no—
188C22×C3×C5—42no—
189C22×C3×C5—44no—
190C22×C3×C5—44no—
191C22×C3×C5#12114yes—
192C22×C3×C5C22×D411yescharacteristic
193C22×C3×C5C22×D411yescharacteristic
194C22×C3×C5C2×#4111yescharacteristic
195C22×C3×C5H(5,𝔽2)12yes—
196C22×C3×C52−1+412yes—
197C3×C4×C5—24no—
198C3×C4×C5—22no—
199C3×C4×C5—22no—
200C3×C4×C5—42no—
201C3×C4×C5—42no—
202C3×C4×C5—44no—
203C24×C3C23×C512yes—
204C24×C3C23×C511yescharacteristic
205C2×C3×D4—24no—
206C2×C3×D4—24no—
207C2×C3×D4—24no—
208C2×C3×D4—22no—
209C2×C3×D4—42no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4C5×D414yes—
216C22×C3×C4C5×D414yescomplemented
217C22×C3×C4C23×C514yes—
218C22×C3×C4C23×C512yes—
219C22×C3×C4C23×C512yes—
220C22×C3×C4C23×C511yescharacteristic
221C22×C3×C4C23×C512yes—
222C22×C3×C4C23×C511yescharacteristic
223C3×C4⋊C4—24no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—28no—
227C3×C4⋊C4—42no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—44no—
232C23×C5—22no—
233C23×C5—22no—
234C23×C5—22no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—24no—
238C23×C5—24no—
239C23×C5C2×C3×D412yes—
240C23×C5C2×C3×D412yes—
241C23×C5C3×#4112yes—
242C23×C5C24×C311yescharacteristic
243C5×D4—44no—
244C5×D4—48no—
245C2×C4×C5—28no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—24no—
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—24no—
256C2×C4×C5—24no—
257C2×C4×C5—24no—
258C2×C4×C5—44no—
259C2×C4×C5—44no—
260C2×C4×C5C2×C3×D412yes—
261C2×C4×C5C2×C3×D414yes—
262C2×C4×C5C2×C3×D412yes—
263C22×D4C22×C3×C512yes—
264C22×D4C22×C3×C511yescharacteristic
265C23×C4C22×C3×C511yescharacteristic
266C23×C4C22×C3×C511yescharacteristic
267C23×C4C22×C3×C511yescharacteristic
268#123—28no—
269#121—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4C22×C3×C512yes—
272C2×C4⋊C4C22×C3×C512yes—
273C2×C4⋊C4C22×C3×C511yescharacteristic
274C2×C4⋊C4C22×C3×C514yes—
275C2×C4⋊C4C22×C3×C512yes—
276C2×C22⋊C4C22×C3×C512yes—
277C2×C22⋊C4C22×C3×C512yes—
278C2×C22⋊C4C22×C3×C514yes—
279C2×C22⋊C4C22×C3×C514yes—
280#95C22×C3×C512yes—
281#95C22×C3×C514yes—
282C2×C3×C5—24no—
283C2×C3×C5—82no—
284C2×C3×C5C2×#12111yescharacteristic
285C2×C3×C5#53411yescharacteristic
286C2×C3×C5#53611yescharacteristic
287C2×C3×C5#53712yes—
288C2×C3×C5#53912yes—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3C2×C5×D412yes—
297C23×C3C2×C5×D412yes—
298C23×C3C5×#4112yes—
299C23×C3C24×C511yescharacteristic
300C3×D4—44no—
301C3×D4—48no—
302C2×C3×C4—28no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—44no—
316C2×C3×C4—44no—
317C2×C3×C4C2×C5×D412yes—
318C2×C3×C4C2×C5×D414yes—
319C2×C3×C4C2×C5×D412yes—
320C22×C5—22no—
321C22×C5—22no—
322C22×C5—24no—
323C22×C5—24no—
324C22×C5—42no—
325C22×C5—42no—
326C22×C5—42no—
327C22×C5—44no—
328C22×C5—44no—
329C22×C5C3×#12114yes—
330C22×C5C22×C3×D411yescharacteristic
331C22×C5C22×C3×D411yescharacteristic
332C22×C5C2×C3×#4111yescharacteristic
333C22×C5C3×H(5,𝔽2)12yes—
334C22×C5C3×2−1+412yes—
335C4×C5—24no—
336C4×C5—22no—
337C4×C5—22no—
338C4×C5—42no—
339C4×C5—42no—
340C4×C5—44no—
341C24C23×C3×C512yes—
342C24C23×C3×C511yescharacteristic
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4—42no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4C3×C5×D414yes—
354C22×C4C3×C5×D414yescomplemented
355C22×C4C23×C3×C514yes—
356C22×C4C23×C3×C512yes—
357C22×C4C23×C3×C512yes—
358C22×C4C23×C3×C511yescharacteristic
359C22×C4C23×C3×C512yes—
360C22×C4C23×C3×C511yescharacteristic
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—28no—
365C4⋊C4—42no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—28no—
369C22⋊C4—44no—
370C3×C5#261811yescharacteristic, direct product factor, Hall
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—24no—
374C22×C3—24no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—44no—
379C22×C3—44no—
380C22×C3C5×#12114yes—
381C22×C3C22×C5×D411yescharacteristic
382C22×C3C22×C5×D411yescharacteristic
383C22×C3C2×C5×#4111yescharacteristic
384C22×C3C5×H(5,𝔽2)12yes—
385C22×C3C5×2−1+412yes—
386C3×C4—24no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4—44no—
392C2×C5—24no—
393C2×C5—82no—
394C2×C5C2×C3×#12111yescharacteristic
395C2×C5C3×#53411yescharacteristic
396C2×C5C3×#53611yescharacteristic
397C2×C5C3×#53712yes—
398C2×C5C3×#53912yes—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—24no—
406C23C2×C3×C5×D412yes—
407C23C2×C3×C5×D412yes—
408C23C3×C5×#4112yes—
409C23C24×C3×C511yescharacteristic
410D4—44no—
411D4—48no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4C2×C3×C5×D412yes—
428C2×C4C2×C3×C5×D414yes—
429C2×C4C2×C3×C5×D412yes—
430C2×C3—24no—
431C2×C3—82no—
432C2×C3C2×C5×#12111yescharacteristic
433C2×C3C5×#53411yescharacteristic
434C2×C3C5×#53611yescharacteristic
435C2×C3C5×#53712yes—
436C2×C3C5×#53912yes—
437C5C3×#261811yescharacteristic, direct product factor, Sylow
438C22—22no—
439C22—22no—
440C22—24no—
441C22—24no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—44no—
447C22C3×C5×#12114yes—
448C22C22×C3×C5×D411yescharacteristic
449C22C22×C3×C5×D411yescharacteristic
450C22C2×C3×C5×#4111yescharacteristic
451C22C3×C5×H(5,𝔽2)12yes—
452C22C3×C5×2−1+412yes—
453C4—24no—
454C4—22no—
455C4—22no—
456C4—42no—
457C4—42no—
458C4—44no—
459C3C5×#261811yescharacteristic, direct product factor, Sylow
460C2—24no—
461C2—82no—
462C2C2×C3×C5×#12111yescharacteristic
463C2C3×C5×#53411yescharacteristic
464C2C3×C5×#53611yescharacteristic
465C2C3×C5×#53712yes—
466C2C3×C5×#53912yes—
subgroupG/H
center, socleC23×C3×C5C24
commutator, Frattini, cosocleC23C24×C3×C5
prime radicalC2×C3×C5×#121C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#123C2semidirect
3. C22×C3×C5×C4⋊C4C2semidirect
4. C3×C5×#398C2semidirect
5. C3×C5×#396C2semidirect
6. C3×C5×#394C2semidirect
7. C3×C5×#386C2semidirect
8. C3×C5×#380C2semidirect
9. C5×#2618C3direct (N × H)
29. C3×#2618C5direct (N × H)
30. C2×C5×#123C2×C3semidirect
32. C22×C5×C4⋊C4C2×C3semidirect
33. C5×#398C2×C3semidirect
34. C5×#396C2×C3semidirect
35. C5×#394C2×C3semidirect
36. C5×#386C2×C3semidirect
37. C5×#380C2×C3semidirect
51. C22×C3×C4×C5D4semidirect
67. C2×C3×#123C2×C5semidirect
69. C22×C3×C4⋊C4C2×C5semidirect
70. C3×#398C2×C5semidirect
71. C3×#396C2×C5semidirect
72. C3×#394C2×C5semidirect
73. C3×#386C2×C5semidirect
74. C3×#380C2×C5semidirect
94. #2618C3×C5direct (N × H)
158. C22×C4×C5C3×D4semidirect
174. C2×#123C2×C3×C5semidirect
176. C22×C4⋊C4C2×C3×C5semidirect
177. #398C2×C3×C5semidirect
178. #396C2×C3×C5semidirect
179. #394C2×C3×C5semidirect
180. #386C2×C3×C5semidirect
181. #380C2×C3×C5semidirect
216. C22×C3×C4C5×D4semidirect
354. C22×C4C3×C5×D4semidirect
370. C3×C5#2618direct (N × H)
437. C5C3×#2618direct (N × H)
459. C3C5×#2618direct (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×C5×#1212
C22×C3×C5×C4⋊C42
C3×C5×#3982
C3×C5×#3962
C3×C5×#3942
C3×C5×#3862
C3×C5×#3802
C5×#26183
C3×#26185