Finite Group Databasebeta

← C3×C5×#2545 #14275579

Subgroups — C3×C5×#2545 #14275579

Order 1920.

1152 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×C5×#124C211yesmaximal, characteristic
3C2×C3×C5×#124C211yesmaximal, characteristic
4C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
5C3×C5×#396C214yesmaximal, complemented
6C3×C5×#391C211yesmaximal, characteristic, complemented
7C3×C5×#390C211yesmaximal, characteristic
8C3×C5×#388C214yesmaximal, complemented
9C3×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
10C5×#2545C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C3×C5×Q8C2211yescharacteristic
13C22×C3×C5×D4C2211yescharacteristic
14C3×C5×C4⋊Q8—24no—
15C3×C5×#124—24no—
16C3×C5×#124—24no—
17C3×C4×C5×Q8—24no—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C4⋊C4C2214yes—
21C2×C3×C5×C4⋊C4C2212yes—
22C2×C3×C5×C22⋊C4C2212yes—
23C2×C3×C5×C22⋊C4C2214yes—
24C2×C3×C5×C22⋊C4C2212yes—
25C2×C3×C5×C22⋊C4C2214yes—
26C2×C3×C42×C5C2212yes—
27C2×C3×C42×C5C2211yescharacteristic
28C2×C3×C42×C5C2211yescharacteristic
29C2×C3×C42×C5C2211yescharacteristic
30C3×C5×#95C2212yes—
31C3×C5×#95C2214yes—
32C3×#2545C511yesmaximal, characteristic, direct product factor, Hall
33C2×C5×C4⋊Q8C2×C311yescharacteristic, complemented
34C2×C5×#124C2×C311yescharacteristic
35C2×C5×#124C2×C311yescharacteristic
36C2×C4×C5×Q8C2×C311yescharacteristic, complemented
37C5×#396C2×C314yescomplemented
38C5×#391C2×C311yescharacteristic, complemented
39C5×#390C2×C311yescharacteristic
40C5×#388C2×C314yescomplemented
41C4×C5×C4⋊C4C2×C311yescharacteristic, complemented
42C24×C3×C5C2312yes—
43C2×C3×C5×Q8—22no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8—22no—
48C2×C3×C5×Q8D414yes—
49C2×C3×C5×D4—22no—
50C2×C3×C5×D4—22no—
51C2×C3×C5×D4—22no—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2311yescharacteristic
60C3×C5×C4⋊C4—24no—
61C3×C5×C4⋊C4—22no—
62C3×C5×C4⋊C4—28no—
63C3×C5×C4⋊C4—24no—
64C3×C5×C22⋊C4—28no—
65C3×C5×C22⋊C4—24no—
66C3×C5×C22⋊C4—24no—
67C3×C5×C22⋊C4—28no—
68C3×C42×C5—24no—
69C3×C42×C5—22no—
70C3×C42×C5—22no—
71C3×C42×C5D414yes—
72C2×C3×C4⋊Q8C2×C511yescharacteristic, complemented
73C2×C3×#124C2×C511yescharacteristic
74C2×C3×#124C2×C511yescharacteristic
75C2×C3×C4×Q8C2×C511yescharacteristic, complemented
76C3×#396C2×C514yescomplemented
77C3×#391C2×C511yescharacteristic, complemented
78C3×#390C2×C511yescharacteristic
79C3×#388C2×C514yescomplemented
80C3×C4×C4⋊C4C2×C511yescharacteristic, complemented
81C22×C5×Q8C22×C311yescharacteristic
82C22×C5×Q8C22×C311yescharacteristic
83C22×C5×D4C22×C311yescharacteristic
84C5×C4⋊Q8—24no—
85C5×#124—24no—
86C5×#124—24no—
87C4×C5×Q8—24no—
88C2×C5×C4⋊C4C22×C312yes—
89C2×C5×C4⋊C4C22×C314yes—
90C2×C5×C4⋊C4C22×C311yescharacteristic
91C2×C5×C4⋊C4C22×C312yes—
92C2×C5×C22⋊C4C22×C312yes—
93C2×C5×C22⋊C4C22×C314yes—
94C2×C5×C22⋊C4C22×C314yes—
95C2×C5×C22⋊C4C22×C312yes—
96C2×C42×C5C22×C311yescharacteristic
97C2×C42×C5C22×C311yescharacteristic
98C2×C42×C5C22×C312yes—
99C2×C42×C5C22×C311yescharacteristic
100C5×#95C22×C312yes—
101C5×#95C22×C314yes—
102#2545C3×C511yescharacteristic, direct product factor, Sylow
103C23×C3×C5—22no—
104C23×C3×C5—22no—
105C23×C3×C5—24no—
106C23×C3×C5—22no—
107C23×C3×C5—24no—
108C23×C3×C5C2411yescharacteristic
109C3×C5×Q8—28no—
110C3×C5×Q8—44no—
111C3×C5×D4—44no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—22no—
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—22no—
122C2×C3×C4×C5—24no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5C2×D414yes—
129C2×C3×C4×C5C2×D412yes—
130C2×C3×C4×C5C2×D412yes—
131C2×C3×C4×C5#4114yes—
132C2×C3×C4×C5#4112yes—
133C22×C3×Q8C22×C511yescharacteristic
134C22×C3×Q8C22×C511yescharacteristic
135C22×C3×D4C22×C511yescharacteristic
136C3×C4⋊Q8—24no—
137C3×#124—24no—
138C3×#124—24no—
139C3×C4×Q8—24no—
140C2×C3×C4⋊C4C22×C512yes—
141C2×C3×C4⋊C4C22×C514yes—
142C2×C3×C4⋊C4C22×C511yescharacteristic
143C2×C3×C4⋊C4C22×C512yes—
144C2×C3×C22⋊C4C22×C512yes—
145C2×C3×C22⋊C4C22×C514yes—
146C2×C3×C22⋊C4C22×C514yes—
147C2×C3×C22⋊C4C22×C512yes—
148C2×C3×C42C22×C511yescharacteristic
149C2×C3×C42C22×C511yescharacteristic
150C2×C3×C42C22×C512yes—
151C2×C3×C42C22×C511yescharacteristic
152C3×#95C22×C512yes—
153C3×#95C22×C514yes—
154C24×C5C23×C312yes—
155C2×C5×Q8—22no—
156C2×C5×Q8—22no—
157C2×C5×Q8—22no—
158C2×C5×Q8—22no—
159C2×C5×Q8—22no—
160C2×C5×Q8C3×D414yes—
161C2×C5×D4—22no—
162C2×C5×D4—22no—
163C2×C5×D4—22no—
164C22×C4×C5C23×C312yes—
165C22×C4×C5C23×C312yes—
166C22×C4×C5C23×C312yes—
167C22×C4×C5C23×C311yescharacteristic
168C22×C4×C5C23×C311yescharacteristic
169C22×C4×C5C23×C312yes—
170C22×C4×C5C23×C312yes—
171C22×C4×C5C23×C311yescharacteristic
172C5×C4⋊C4—24no—
173C5×C4⋊C4—22no—
174C5×C4⋊C4—28no—
175C5×C4⋊C4—24no—
176C5×C22⋊C4—28no—
177C5×C22⋊C4—24no—
178C5×C22⋊C4—24no—
179C5×C22⋊C4—28no—
180C42×C5—24no—
181C42×C5—22no—
182C42×C5—22no—
183C42×C5C3×D414yes—
184C2×C4⋊Q8C2×C3×C511yescharacteristic, complemented
185C2×#124C2×C3×C511yescharacteristic
186C2×#124C2×C3×C511yescharacteristic
187C2×C4×Q8C2×C3×C511yescharacteristic, complemented
188#396C2×C3×C514yescomplemented
189#391C2×C3×C511yescharacteristic, complemented
190#390C2×C3×C511yescharacteristic
191#388C2×C3×C514yescomplemented
192C4×C4⋊C4C2×C3×C511yescharacteristic, complemented
193C22×C3×C5—42no—
194C22×C3×C5—42no—
195C22×C3×C5—42no—
196C22×C3×C5—44no—
197C22×C3×C5—44no—
198C22×C3×C5C22×D411yescharacteristic
199C22×C3×C5C22×D411yescharacteristic
200C22×C3×C5C2×#4111yescharacteristic
201C22×C3×C5C2×#4112yes—
202C22×C3×C52−1+412yes—
203C3×C4×C5—24no—
204C3×C4×C5—22no—
205C3×C4×C5—22no—
206C3×C4×C5—24no—
207C3×C4×C5—42no—
208C3×C4×C5—42no—
209C3×C4×C5—42no—
210C3×C4×C5#12114yes—
211C24×C3C23×C512yes—
212C2×C3×Q8—22no—
213C2×C3×Q8—22no—
214C2×C3×Q8—22no—
215C2×C3×Q8—22no—
216C2×C3×Q8—22no—
217C2×C3×Q8C5×D414yes—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C22×C3×C4C23×C512yes—
222C22×C3×C4C23×C512yes—
223C22×C3×C4C23×C512yes—
224C22×C3×C4C23×C511yescharacteristic
225C22×C3×C4C23×C511yescharacteristic
226C22×C3×C4C23×C512yes—
227C22×C3×C4C23×C512yes—
228C22×C3×C4C23×C511yescharacteristic
229C3×C4⋊C4—24no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—28no—
232C3×C4⋊C4—24no—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—28no—
237C3×C42—24no—
238C3×C42—22no—
239C3×C42—22no—
240C3×C42C5×D414yes—
241C23×C5—22no—
242C23×C5—22no—
243C23×C5—24no—
244C23×C5—22no—
245C23×C5—24no—
246C23×C5C24×C311yescharacteristic
247C5×Q8—28no—
248C5×Q8—44no—
249C5×D4—44no—
250C2×C4×C5—22no—
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—24no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5C2×C3×D412yes—
265C2×C4×C5C2×C3×D412yes—
266C2×C4×C5C2×C3×D414yes—
267C2×C4×C5C2×C3×D412yes—
268C2×C4×C5C2×C3×D412yes—
269C2×C4×C5C3×#4114yes—
270C2×C4×C5C3×#4112yes—
271C22×Q8C22×C3×C511yescharacteristic
272C22×Q8C22×C3×C511yescharacteristic
273C22×D4C22×C3×C511yescharacteristic
274C4⋊Q8—24no—
275#124—24no—
276#124—24no—
277C4×Q8—24no—
278C2×C4⋊C4C22×C3×C511yescharacteristic
279C2×C4⋊C4C22×C3×C512yes—
280C2×C4⋊C4C22×C3×C514yes—
281C2×C4⋊C4C22×C3×C512yes—
282C2×C22⋊C4C22×C3×C512yes—
283C2×C22⋊C4C22×C3×C514yes—
284C2×C22⋊C4C22×C3×C512yes—
285C2×C22⋊C4C22×C3×C514yes—
286C2×C42C22×C3×C512yes—
287C2×C42C22×C3×C511yescharacteristic
288C2×C42C22×C3×C511yescharacteristic
289C2×C42C22×C3×C511yescharacteristic
290#95C22×C3×C512yes—
291#95C22×C3×C514yes—
292C2×C3×C5—82no—
293C2×C3×C5C2×#12111yescharacteristic
294C2×C3×C5#53211yescharacteristic
295C2×C3×C5#53611yescharacteristic
296C2×C3×C5#54812yes—
297C2×C3×C5#55612yes—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—22no—
302C23×C3—24no—
303C23×C3C24×C511yescharacteristic
304C3×Q8—28no—
305C3×Q8—44no—
306C3×D4—44no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4C2×C5×D412yes—
322C2×C3×C4C2×C5×D412yes—
323C2×C3×C4C2×C5×D414yes—
324C2×C3×C4C2×C5×D412yes—
325C2×C3×C4C2×C5×D412yes—
326C2×C3×C4C5×#4114yes—
327C2×C3×C4C5×#4112yes—
328C22×C5—42no—
329C22×C5—42no—
330C22×C5—42no—
331C22×C5—44no—
332C22×C5—44no—
333C22×C5C22×C3×D411yescharacteristic
334C22×C5C22×C3×D411yescharacteristic
335C22×C5C2×C3×#4111yescharacteristic
336C22×C5C2×C3×#4112yes—
337C22×C5C3×2−1+412yes—
338C4×C5—24no—
339C4×C5—22no—
340C4×C5—22no—
341C4×C5—24no—
342C4×C5—42no—
343C4×C5—42no—
344C4×C5—42no—
345C4×C5C3×#12114yes—
346C24C23×C3×C512yes—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8C3×C5×D414yes—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C22×C4C23×C3×C512yes—
357C22×C4C23×C3×C512yes—
358C22×C4C23×C3×C512yes—
359C22×C4C23×C3×C511yescharacteristic
360C22×C4C23×C3×C511yescharacteristic
361C22×C4C23×C3×C512yes—
362C22×C4C23×C3×C512yes—
363C22×C4C23×C3×C511yescharacteristic
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—28no—
372C42—24no—
373C42—22no—
374C42—22no—
375C42C3×C5×D414yes—
376C3×C5#254511yescharacteristic, direct product factor, Hall
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—44no—
381C22×C3—44no—
382C22×C3C22×C5×D411yescharacteristic
383C22×C3C22×C5×D411yescharacteristic
384C22×C3C2×C5×#4111yescharacteristic
385C22×C3C2×C5×#4112yes—
386C22×C3C5×2−1+412yes—
387C3×C4—24no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—24no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4C5×#12114yes—
395C2×C5—82no—
396C2×C5C2×C3×#12111yescharacteristic
397C2×C5C3×#53211yescharacteristic
398C2×C5C3×#53611yescharacteristic
399C2×C5C3×#54812yes—
400C2×C5C3×#55612yes—
401C23—22no—
402C23—22no—
403C23—24no—
404C23—22no—
405C23—24no—
406C23C24×C3×C511yescharacteristic
407Q8—28no—
408Q8—44no—
409D4—44no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4C2×C3×C5×D412yes—
425C2×C4C2×C3×C5×D412yes—
426C2×C4C2×C3×C5×D414yes—
427C2×C4C2×C3×C5×D412yes—
428C2×C4C2×C3×C5×D412yes—
429C2×C4C3×C5×#4114yes—
430C2×C4C3×C5×#4112yes—
431C2×C3—82no—
432C2×C3C2×C5×#12111yescharacteristic
433C2×C3C5×#53211yescharacteristic
434C2×C3C5×#53611yescharacteristic
435C2×C3C5×#54812yes—
436C2×C3C5×#55612yes—
437C5C3×#254511yescharacteristic, direct product factor, Sylow
438C22—42no—
439C22—42no—
440C22—42no—
441C22—44no—
442C22—44no—
443C22C22×C3×C5×D411yescharacteristic
444C22C22×C3×C5×D411yescharacteristic
445C22C2×C3×C5×#4111yescharacteristic
446C22C2×C3×C5×#4112yes—
447C22C3×C5×2−1+412yes—
448C4—24no—
449C4—22no—
450C4—22no—
451C4—24no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4C3×C5×#12114yes—
456C3C5×#254511yescharacteristic, direct product factor, Sylow
457C2—82no—
458C2C2×C3×C5×#12111yescharacteristic
459C2C3×C5×#53211yescharacteristic
460C2C3×C5×#53611yescharacteristic
461C2C3×C5×#54812yes—
462C2C3×C5×#55612yes—
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
1. C2×C3×C5×C4⋊Q8C2semidirect
4. C2×C3×C4×C5×Q8C2semidirect
5. C3×C5×#396C2semidirect
6. C3×C5×#391C2semidirect
8. C3×C5×#388C2semidirect
9. C3×C4×C5×C4⋊C4C2semidirect
10. C5×#2545C3direct (N × H)
32. C3×#2545C5direct (N × H)
33. C2×C5×C4⋊Q8C2×C3semidirect
36. C2×C4×C5×Q8C2×C3semidirect
37. C5×#396C2×C3semidirect
38. C5×#391C2×C3semidirect
40. C5×#388C2×C3semidirect
41. C4×C5×C4⋊C4C2×C3semidirect
72. C2×C3×C4⋊Q8C2×C5semidirect
75. C2×C3×C4×Q8C2×C5semidirect
76. C3×#396C2×C5semidirect
77. C3×#391C2×C5semidirect
79. C3×#388C2×C5semidirect
80. C3×C4×C4⋊C4C2×C5semidirect
102. #2545C3×C5direct (N × H)
184. C2×C4⋊Q8C2×C3×C5semidirect
187. C2×C4×Q8C2×C3×C5semidirect
188. #396C2×C3×C5semidirect
189. #391C2×C3×C5semidirect
191. #388C2×C3×C5semidirect
192. C4×C4⋊C4C2×C3×C5semidirect
376. C3×C5#2545direct (N × H)
437. C5C3×#2545direct (N × H)
456. C3C5×#2545direct (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×C4⋊Q82
C2×C3×C5×#1242
C2×C3×C4×C5×Q82
C3×C5×#3962
C3×C5×#3912
C3×C5×#3902
C3×C5×#3882
C3×C4×C5×C4⋊C42
C5×#25453
C3×#25455