Finite Group Databasebeta

← C3×C5×#2535 #14275569

Subgroups — C3×C5×#2535 #14275569

Order 1920.

1464 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×#121C212yesmaximal, complemented
2C2×C3×C4×C5×D4C211yesmaximal, characteristic, complemented
3C2×C3×C4×C5×D4C211yesmaximal, characteristic, complemented
4C3×C5×#397C212yesmaximal, complemented
5C3×C5×#392C212yesmaximal, complemented
6C3×C5×#386C214yesmaximal, complemented
7C3×C5×#382C211yesmaximal, characteristic, complemented
8C3×C5×#380C212yesmaximal, complemented
9C5×#2535C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C22×C3×C5×D4C2214yes—
12C22×C3×C5×D4C2211yescharacteristic
13C23×C3×C4×C5C2212yes—
14C23×C3×C4×C5C2212yes—
15C3×C5×#121—28no—
16C3×C4×C5×D4—24no—
17C3×C4×C5×D4—24no—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C4⋊C4C2211yescharacteristic
20C2×C3×C5×C4⋊C4C2212yes—
21C2×C3×C5×C22⋊C4C2212yes—
22C2×C3×C5×C22⋊C4C2212yes—
23C2×C3×C5×C22⋊C4C2214yes—
24C2×C3×C5×C22⋊C4C2214yes—
25C2×C3×C42×C5C2211yescharacteristic
26C3×C5×#95C2214yes—
27C3×C5×#95C2212yes—
28C3×C5×#95C2212yes—
29C3×#2535C511yesmaximal, characteristic, direct product factor, Hall
30C2×C5×#121C2×C312yescomplemented
31C2×C4×C5×D4C2×C311yescharacteristic, complemented
32C2×C4×C5×D4C2×C311yescharacteristic, complemented
33C5×#397C2×C312yescomplemented
34C5×#392C2×C312yescomplemented
35C5×#386C2×C314yescomplemented
36C5×#382C2×C311yescharacteristic, complemented
37C5×#380C2×C312yescomplemented
38C24×C3×C5C2312yes—
39C24×C3×C5C2312yes—
40C2×C3×C5×D4—24no—
41C2×C3×C5×D4—28no—
42C2×C3×C5×D4—28no—
43C2×C3×C5×D4—22no—
44C2×C3×C5×D4—22no—
45C2×C3×C5×D4—22no—
46C2×C3×C5×D4—44no—
47C2×C3×C5×D4D414yescomplemented
48C22×C3×C4×C5—24no—
49C22×C3×C4×C5—24no—
50C22×C3×C4×C5—24no—
51C22×C3×C4×C5—24no—
52C22×C3×C4×C5—44no—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2314yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C3×C5×C4⋊C4—22no—
60C3×C5×C4⋊C4—22no—
61C3×C5×C4⋊C4—24no—
62C3×C5×C22⋊C4—28no—
63C3×C5×C22⋊C4—24no—
64C3×C5×C22⋊C4—24no—
65C3×C5×C22⋊C4—44no—
66C3×C42×C5D414yescomplemented
67C2×C3×#121C2×C512yescomplemented
68C2×C3×C4×D4C2×C511yescharacteristic, complemented
69C2×C3×C4×D4C2×C511yescharacteristic, complemented
70C3×#397C2×C512yescomplemented
71C3×#392C2×C512yescomplemented
72C3×#386C2×C514yescomplemented
73C3×#382C2×C511yescharacteristic, complemented
74C3×#380C2×C512yescomplemented
75C22×C5×D4C22×C311yescharacteristic
76C22×C5×D4C22×C314yes—
77C22×C5×D4C22×C311yescharacteristic
78C23×C4×C5C22×C312yes—
79C23×C4×C5C22×C312yes—
80C5×#121—28no—
81C4×C5×D4—24no—
82C4×C5×D4—24no—
83C2×C5×C4⋊C4C22×C311yescharacteristic
84C2×C5×C4⋊C4C22×C312yes—
85C2×C5×C4⋊C4C22×C311yescharacteristic
86C2×C5×C22⋊C4C22×C312yes—
87C2×C5×C22⋊C4C22×C314yes—
88C2×C5×C22⋊C4C22×C312yes—
89C2×C5×C22⋊C4C22×C314yes—
90C2×C42×C5C22×C311yescharacteristic
91C5×#95C22×C314yes—
92C5×#95C22×C312yes—
93C5×#95C22×C312yes—
94#2535C3×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—22no—
101C23×C3×C5—24no—
102C23×C3×C5—22no—
103C23×C3×C5—24no—
104C23×C3×C5—22no—
105C23×C3×C5C2×D414yes—
106C23×C3×C5#4114yes—
107C23×C3×C5C2411yescharacteristic
108C3×C5×D4—28no—
109C3×C5×D4—28no—
110C3×C5×D4—416no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—24no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—24no—
119C2×C3×C4×C5—24no—
120C2×C3×C4×C5—24no—
121C2×C3×C4×C5—48no—
122C2×C3×C4×C5—48no—
123C2×C3×C4×C5C2×D412yes—
124C2×C3×C4×C5C2×D412yes—
125C2×C3×C4×C5C2×D412yes—
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5#4112yes—
128C22×C3×D4C22×C511yescharacteristic
129C22×C3×D4C22×C514yes—
130C22×C3×D4C22×C511yescharacteristic
131C23×C3×C4C22×C512yes—
132C23×C3×C4C22×C512yes—
133C3×#121—28no—
134C3×C4×D4—24no—
135C3×C4×D4—24no—
136C2×C3×C4⋊C4C22×C511yescharacteristic
137C2×C3×C4⋊C4C22×C512yes—
138C2×C3×C4⋊C4C22×C511yescharacteristic
139C2×C3×C22⋊C4C22×C512yes—
140C2×C3×C22⋊C4C22×C514yes—
141C2×C3×C22⋊C4C22×C512yes—
142C2×C3×C22⋊C4C22×C514yes—
143C2×C3×C42C22×C511yescharacteristic
144C3×#95C22×C514yes—
145C3×#95C22×C512yes—
146C3×#95C22×C512yes—
147C24×C5C23×C312yes—
148C24×C5C23×C312yes—
149C2×C5×D4—24no—
150C2×C5×D4—28no—
151C2×C5×D4—28no—
152C2×C5×D4—22no—
153C2×C5×D4—22no—
154C2×C5×D4—22no—
155C2×C5×D4—44no—
156C2×C5×D4C3×D414yescomplemented
157C22×C4×C5—24no—
158C22×C4×C5—24no—
159C22×C4×C5—24no—
160C22×C4×C5—24no—
161C22×C4×C5—44no—
162C22×C4×C5C23×C312yes—
163C22×C4×C5C23×C312yes—
164C22×C4×C5C23×C314yes—
165C22×C4×C5C23×C311yescharacteristic
166C22×C4×C5C23×C311yescharacteristic
167C22×C4×C5C23×C311yescharacteristic
168C5×C4⋊C4—22no—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—24no—
171C5×C22⋊C4—28no—
172C5×C22⋊C4—24no—
173C5×C22⋊C4—24no—
174C5×C22⋊C4—44no—
175C42×C5C3×D414yescomplemented
176C2×#121C2×C3×C512yescomplemented
177C2×C4×D4C2×C3×C511yescharacteristic, complemented
178C2×C4×D4C2×C3×C511yescharacteristic, complemented
179#397C2×C3×C512yescomplemented
180#392C2×C3×C512yescomplemented
181#386C2×C3×C514yescomplemented
182#382C2×C3×C511yescharacteristic, complemented
183#380C2×C3×C512yescomplemented
184C22×C3×C5—24no—
185C22×C3×C5—24no—
186C22×C3×C5—24no—
187C22×C3×C5—28no—
188C22×C3×C5—24no—
189C22×C3×C5—42no—
190C22×C3×C5—42no—
191C22×C3×C5—42no—
192C22×C3×C5—42no—
193C22×C3×C5—42no—
194C22×C3×C5—42no—
195C22×C3×C5—44no—
196C22×C3×C5C22×D411yescharacteristic
197C22×C3×C5C22×D411yescharacteristic
198C22×C3×C5C2×#4111yescharacteristic
199C22×C3×C5C2×#4112yes—
200C22×C3×C5H(5,𝔽2)12yes—
201C3×C4×C5—22no—
202C3×C4×C5—22no—
203C3×C4×C5—42no—
204C3×C4×C5—42no—
205C3×C4×C5—44no—
206C3×C4×C5#12114yes—
207C24×C3C23×C512yes—
208C24×C3C23×C512yes—
209C2×C3×D4—24no—
210C2×C3×D4—28no—
211C2×C3×D4—28no—
212C2×C3×D4—22no—
213C2×C3×D4—22no—
214C2×C3×D4—22no—
215C2×C3×D4—44no—
216C2×C3×D4C5×D414yescomplemented
217C22×C3×C4—24no—
218C22×C3×C4—24no—
219C22×C3×C4—24no—
220C22×C3×C4—24no—
221C22×C3×C4—44no—
222C22×C3×C4C23×C514yes—
223C22×C3×C4C23×C512yes—
224C22×C3×C4C23×C512yes—
225C22×C3×C4C23×C511yescharacteristic
226C22×C3×C4C23×C511yescharacteristic
227C22×C3×C4C23×C511yescharacteristic
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—24no—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—44no—
235C3×C42C5×D414yescomplemented
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—22no—
239C23×C5—22no—
240C23×C5—22no—
241C23×C5—22no—
242C23×C5—24no—
243C23×C5—22no—
244C23×C5—24no—
245C23×C5—22no—
246C23×C5C2×C3×D414yes—
247C23×C5C3×#4114yes—
248C23×C5C24×C311yescharacteristic
249C5×D4—28no—
250C5×D4—28no—
251C5×D4—416no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—24no—
258C2×C4×C5—22no—
259C2×C4×C5—24no—
260C2×C4×C5—24no—
261C2×C4×C5—24no—
262C2×C4×C5—48no—
263C2×C4×C5—48no—
264C2×C4×C5C2×C3×D412yes—
265C2×C4×C5C2×C3×D412yes—
266C2×C4×C5C2×C3×D412yes—
267C2×C4×C5C2×C3×D412yes—
268C2×C4×C5C3×#4112yes—
269C22×D4C22×C3×C511yescharacteristic
270C22×D4C22×C3×C514yes—
271C22×D4C22×C3×C511yescharacteristic
272C23×C4C22×C3×C512yes—
273C23×C4C22×C3×C512yes—
274#121—28no—
275C4×D4—24no—
276C4×D4—24no—
277C2×C4⋊C4C22×C3×C511yescharacteristic
278C2×C4⋊C4C22×C3×C511yescharacteristic
279C2×C4⋊C4C22×C3×C512yes—
280C2×C22⋊C4C22×C3×C512yes—
281C2×C22⋊C4C22×C3×C512yes—
282C2×C22⋊C4C22×C3×C514yes—
283C2×C22⋊C4C22×C3×C514yes—
284C2×C42C22×C3×C511yescharacteristic
285#95C22×C3×C514yes—
286#95C22×C3×C512yes—
287#95C22×C3×C512yes—
288C2×C3×C5—44no—
289C2×C3×C5—44no—
290C2×C3×C5C2×#12111yescharacteristic
291C2×C3×C5#52511yescharacteristic
292C2×C3×C5#53511yescharacteristic
293C2×C3×C5#54612yes—
294C2×C3×C5#55312yes—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—22no—
305C23×C3C2×C5×D414yes—
306C23×C3C5×#4114yes—
307C23×C3C24×C511yescharacteristic
308C3×D4—28no—
309C3×D4—28no—
310C3×D4—416no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—48no—
322C2×C3×C4—48no—
323C2×C3×C4C2×C5×D412yes—
324C2×C3×C4C2×C5×D412yes—
325C2×C3×C4C2×C5×D412yes—
326C2×C3×C4C2×C5×D412yes—
327C2×C3×C4C5×#4112yes—
328C22×C5—24no—
329C22×C5—24no—
330C22×C5—24no—
331C22×C5—28no—
332C22×C5—24no—
333C22×C5—42no—
334C22×C5—42no—
335C22×C5—42no—
336C22×C5—42no—
337C22×C5—42no—
338C22×C5—42no—
339C22×C5—44no—
340C22×C5C22×C3×D411yescharacteristic
341C22×C5C22×C3×D411yescharacteristic
342C22×C5C2×C3×#4111yescharacteristic
343C22×C5C2×C3×#4112yes—
344C22×C5C3×H(5,𝔽2)12yes—
345C4×C5—22no—
346C4×C5—22no—
347C4×C5—42no—
348C4×C5—42no—
349C4×C5—44no—
350C4×C5C3×#12114yes—
351C24C23×C3×C512yes—
352C24C23×C3×C512yes—
353C2×D4—24no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—44no—
360C2×D4C3×C5×D414yescomplemented
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—44no—
366C22×C4C23×C3×C514yes—
367C22×C4C23×C3×C512yes—
368C22×C4C23×C3×C512yes—
369C22×C4C23×C3×C511yescharacteristic
370C22×C4C23×C3×C511yescharacteristic
371C22×C4C23×C3×C511yescharacteristic
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C22⋊C4—28no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—44no—
379C42C3×C5×D414yescomplemented
380C3×C5#253511yescharacteristic, direct product factor, Hall
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—28no—
385C22×C3—24no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—44no—
393C22×C3C22×C5×D411yescharacteristic
394C22×C3C22×C5×D411yescharacteristic
395C22×C3C2×C5×#4111yescharacteristic
396C22×C3C2×C5×#4112yes—
397C22×C3C5×H(5,𝔽2)12yes—
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—44no—
403C3×C4C5×#12114yes—
404C2×C5—44no—
405C2×C5—44no—
406C2×C5C2×C3×#12111yescharacteristic
407C2×C5C3×#52511yescharacteristic
408C2×C5C3×#53511yescharacteristic
409C2×C5C3×#54612yes—
410C2×C5C3×#55312yes—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—24no—
420C23—22no—
421C23C2×C3×C5×D414yes—
422C23C3×C5×#4114yes—
423C23C24×C3×C511yescharacteristic
424D4—28no—
425D4—28no—
426D4—416no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4C2×C3×C5×D412yes—
440C2×C4C2×C3×C5×D412yes—
441C2×C4C2×C3×C5×D412yes—
442C2×C4C2×C3×C5×D412yes—
443C2×C4C3×C5×#4112yes—
444C2×C3—44no—
445C2×C3—44no—
446C2×C3C2×C5×#12111yescharacteristic
447C2×C3C5×#52511yescharacteristic
448C2×C3C5×#53511yescharacteristic
449C2×C3C5×#54612yes—
450C2×C3C5×#55312yes—
451C5C3×#253511yescharacteristic, direct product factor, Sylow
452C22—24no—
453C22—24no—
454C22—24no—
455C22—28no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22C22×C3×C5×D411yescharacteristic
465C22C22×C3×C5×D411yescharacteristic
466C22C2×C3×C5×#4111yescharacteristic
467C22C2×C3×C5×#4112yes—
468C22C3×C5×H(5,𝔽2)12yes—
469C4—22no—
470C4—22no—
471C4—42no—
472C4—42no—
473C4—44no—
474C4C3×C5×#12114yes—
475C3C5×#253511yescharacteristic, direct product factor, Sylow
476C2—44no—
477C2—44no—
478C2C2×C3×C5×#12111yescharacteristic
479C2C3×C5×#52511yescharacteristic
480C2C3×C5×#53511yescharacteristic
481C2C3×C5×#54612yes—
482C2C3×C5×#55312yes—
subgroupG/H
center, socleC23×C3×C5C24
commutator, Frattini, cosocleC23C24×C3×C5
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×#121C2semidirect
2. C2×C3×C4×C5×D4C2semidirect
3. C2×C3×C4×C5×D4C2semidirect
4. C3×C5×#397C2semidirect
5. C3×C5×#392C2semidirect
6. C3×C5×#386C2semidirect
7. C3×C5×#382C2semidirect
8. C3×C5×#380C2semidirect
9. C5×#2535C3direct (N × H)
29. C3×#2535C5direct (N × H)
30. C2×C5×#121C2×C3semidirect
31. C2×C4×C5×D4C2×C3semidirect
32. C2×C4×C5×D4C2×C3semidirect
33. C5×#397C2×C3semidirect
34. C5×#392C2×C3semidirect
35. C5×#386C2×C3semidirect
36. C5×#382C2×C3semidirect
37. C5×#380C2×C3semidirect
47. C2×C3×C5×D4D4semidirect
66. C3×C42×C5D4semidirect
67. C2×C3×#121C2×C5semidirect
68. C2×C3×C4×D4C2×C5semidirect
69. C2×C3×C4×D4C2×C5semidirect
70. C3×#397C2×C5semidirect
71. C3×#392C2×C5semidirect
72. C3×#386C2×C5semidirect
73. C3×#382C2×C5semidirect
74. C3×#380C2×C5semidirect
94. #2535C3×C5direct (N × H)
156. C2×C5×D4C3×D4semidirect
175. C42×C5C3×D4semidirect
176. C2×#121C2×C3×C5semidirect
177. C2×C4×D4C2×C3×C5semidirect
178. C2×C4×D4C2×C3×C5semidirect
179. #397C2×C3×C5semidirect
180. #392C2×C3×C5semidirect
181. #386C2×C3×C5semidirect
182. #382C2×C3×C5semidirect
183. #380C2×C3×C5semidirect
216. C2×C3×D4C5×D4semidirect
235. C3×C42C5×D4semidirect
360. C2×D4C3×C5×D4semidirect
379. C42C3×C5×D4semidirect
380. C3×C5#2535direct (N × H)
451. C5C3×#2535direct (N × H)
475. C3C5×#2535direct (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×#1212
C2×C3×C4×C5×D42
C3×C5×#3972
C3×C5×#3922
C3×C5×#3862
C3×C5×#3822
C3×C5×#3802
C5×#25353
C3×#25355