Finite Group Databasebeta

← C3×C5×#2544 #14275578

Subgroups — C3×C5×#2544 #14275578

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#127C211yesmaximal, characteristic
2C2×C3×C5×#124C211yesmaximal, characteristic, complemented
3C2×C3×C5×#124C211yesmaximal, characteristic, complemented
4C2×C3×C4×C5×D4C211yesmaximal, characteristic, complemented
5C3×C5×#394C214yesmaximal, complemented
6C3×C5×#391C211yesmaximal, characteristic, complemented
7C3×C5×#390C211yesmaximal, characteristic, complemented
8C3×C5×#386C214yesmaximal, complemented
9C3×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
10C5×#2544C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13C22×C3×C5×D4C2214yes—
14C22×C3×C5×D4C2211yescharacteristic
15C23×C3×C4×C5C2212yes—
16C3×C5×#127—24no—
17C3×C5×#124—24no—
18C3×C5×#124—24no—
19C3×C4×C5×D4—24no—
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2212yes—
22C2×C3×C5×C22⋊C4C2212yes—
23C2×C3×C5×C22⋊C4C2212yes—
24C2×C3×C5×C22⋊C4C2214yes—
25C2×C3×C5×C22⋊C4C2212yes—
26C2×C3×C5×C22⋊C4C2214yes—
27C2×C3×C42×C5C2211yescharacteristic
28C2×C3×C42×C5C2211yescharacteristic
29C2×C3×C42×C5C2211yescharacteristic
30C3×C5×#95C2212yes—
31C3×C5×#95C2214yes—
32C3×#2544C511yesmaximal, characteristic, direct product factor, Hall
33C2×C5×#127C2×C311yescharacteristic
34C2×C5×#124C2×C311yescharacteristic, complemented
35C2×C5×#124C2×C311yescharacteristic, complemented
36C2×C4×C5×D4C2×C311yescharacteristic, complemented
37C5×#394C2×C314yescomplemented
38C5×#391C2×C311yescharacteristic, complemented
39C5×#390C2×C311yescharacteristic, complemented
40C5×#386C2×C314yescomplemented
41C4×C5×C4⋊C4C2×C311yescharacteristic, complemented
42C24×C3×C5C2312yes—
43C24×C3×C5C2312yes—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×D4—28no—
48C2×C3×C5×D4—22no—
49C2×C3×C5×D4—22no—
50C2×C3×C5×D4—22no—
51C2×C3×C5×D4—22no—
52C2×C3×C5×D4—28no—
53C2×C3×C5×D4—22no—
54C2×C3×C5×D4—44no—
55C2×C3×C5×D4D414yes—
56C22×C3×C4×C5—24no—
57C22×C3×C4×C5—44no—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2312yes—
61C22×C3×C4×C5C2311yescharacteristic
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2312yes—
64C22×C3×C4×C5C2311yescharacteristic
65C3×C5×C4⋊C4—22no—
66C3×C5×C4⋊C4—24no—
67C3×C5×C22⋊C4—28no—
68C3×C5×C22⋊C4—24no—
69C3×C5×C22⋊C4—24no—
70C3×C5×C22⋊C4—24no—
71C3×C5×C22⋊C4—44no—
72C3×C42×C5—22no—
73C3×C42×C5—22no—
74C3×C42×C5D414yescomplemented
75C2×C3×#127C2×C511yescharacteristic
76C2×C3×#124C2×C511yescharacteristic, complemented
77C2×C3×#124C2×C511yescharacteristic, complemented
78C2×C3×C4×D4C2×C511yescharacteristic, complemented
79C3×#394C2×C514yescomplemented
80C3×#391C2×C511yescharacteristic, complemented
81C3×#390C2×C511yescharacteristic, complemented
82C3×#386C2×C514yescomplemented
83C3×C4×C4⋊C4C2×C511yescharacteristic, complemented
84C22×C5×Q8C22×C311yescharacteristic
85C22×C5×D4C22×C311yescharacteristic
86C22×C5×D4C22×C314yes—
87C22×C5×D4C22×C311yescharacteristic
88C23×C4×C5C22×C312yes—
89C5×#127—24no—
90C5×#124—24no—
91C5×#124—24no—
92C4×C5×D4—24no—
93C2×C5×C4⋊C4C22×C312yes—
94C2×C5×C4⋊C4C22×C311yescharacteristic
95C2×C5×C22⋊C4C22×C312yes—
96C2×C5×C22⋊C4C22×C314yes—
97C2×C5×C22⋊C4C22×C312yes—
98C2×C5×C22⋊C4C22×C312yes—
99C2×C5×C22⋊C4C22×C314yes—
100C2×C42×C5C22×C311yescharacteristic
101C2×C42×C5C22×C311yescharacteristic
102C2×C42×C5C22×C311yescharacteristic
103C5×#95C22×C312yes—
104C5×#95C22×C314yes—
105#2544C3×C511yescharacteristic, direct product factor, Sylow
106C23×C3×C5—22no—
107C23×C3×C5—22no—
108C23×C3×C5—24no—
109C23×C3×C5—24no—
110C23×C3×C5—22no—
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5—24no—
114C23×C3×C5—24no—
115C23×C3×C5C2×D414yes—
116C23×C3×C5C2411yescharacteristic
117C3×C5×Q8—44no—
118C3×C5×D4—28no—
119C3×C5×D4—416no—
120C3×C5×D4—44no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—24no—
133C2×C3×C4×C5—48no—
134C2×C3×C4×C5C2×D412yes—
135C2×C3×C4×C5C2×D412yes—
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5#4114yes—
139C2×C3×C4×C5#4112yes—
140C22×C3×Q8C22×C511yescharacteristic
141C22×C3×D4C22×C511yescharacteristic
142C22×C3×D4C22×C514yes—
143C22×C3×D4C22×C511yescharacteristic
144C23×C3×C4C22×C512yes—
145C3×#127—24no—
146C3×#124—24no—
147C3×#124—24no—
148C3×C4×D4—24no—
149C2×C3×C4⋊C4C22×C512yes—
150C2×C3×C4⋊C4C22×C511yescharacteristic
151C2×C3×C22⋊C4C22×C512yes—
152C2×C3×C22⋊C4C22×C514yes—
153C2×C3×C22⋊C4C22×C512yes—
154C2×C3×C22⋊C4C22×C512yes—
155C2×C3×C22⋊C4C22×C514yes—
156C2×C3×C42C22×C511yescharacteristic
157C2×C3×C42C22×C511yescharacteristic
158C2×C3×C42C22×C511yescharacteristic
159C3×#95C22×C512yes—
160C3×#95C22×C514yes—
161C24×C5C23×C312yes—
162C24×C5C23×C312yes—
163C2×C5×Q8—22no—
164C2×C5×Q8—22no—
165C2×C5×Q8—22no—
166C2×C5×D4—28no—
167C2×C5×D4—22no—
168C2×C5×D4—22no—
169C2×C5×D4—22no—
170C2×C5×D4—22no—
171C2×C5×D4—28no—
172C2×C5×D4—22no—
173C2×C5×D4—44no—
174C2×C5×D4C3×D414yes—
175C22×C4×C5—24no—
176C22×C4×C5—44no—
177C22×C4×C5C23×C312yes—
178C22×C4×C5C23×C312yes—
179C22×C4×C5C23×C312yes—
180C22×C4×C5C23×C311yescharacteristic
181C22×C4×C5C23×C311yescharacteristic
182C22×C4×C5C23×C312yes—
183C22×C4×C5C23×C311yescharacteristic
184C5×C4⋊C4—22no—
185C5×C4⋊C4—24no—
186C5×C22⋊C4—28no—
187C5×C22⋊C4—24no—
188C5×C22⋊C4—24no—
189C5×C22⋊C4—24no—
190C5×C22⋊C4—44no—
191C42×C5—22no—
192C42×C5—22no—
193C42×C5C3×D414yescomplemented
194C2×#127C2×C3×C511yescharacteristic
195C2×#124C2×C3×C511yescharacteristic, complemented
196C2×#124C2×C3×C511yescharacteristic, complemented
197C2×C4×D4C2×C3×C511yescharacteristic, complemented
198#394C2×C3×C514yescomplemented
199#391C2×C3×C511yescharacteristic, complemented
200#390C2×C3×C511yescharacteristic, complemented
201#386C2×C3×C514yescomplemented
202C4×C4⋊C4C2×C3×C511yescharacteristic, complemented
203C22×C3×C5—24no—
204C22×C3×C5—28no—
205C22×C3×C5—42no—
206C22×C3×C5—42no—
207C22×C3×C5—44no—
208C22×C3×C5—42no—
209C22×C3×C5—42no—
210C22×C3×C5—42no—
211C22×C3×C5—44no—
212C22×C3×C5—44no—
213C22×C3×C5C22×D411yescharacteristic
214C22×C3×C5C22×D411yescharacteristic
215C22×C3×C5C2×#4111yescharacteristic
216C22×C3×C5C2×#4112yes—
217C22×C3×C5H(5,𝔽2)12yes—
218C3×C4×C5—22no—
219C3×C4×C5—22no—
220C3×C4×C5—24no—
221C3×C4×C5—42no—
222C3×C4×C5—42no—
223C3×C4×C5—42no—
224C3×C4×C5#12114yes—
225C24×C3C23×C512yes—
226C24×C3C23×C512yes—
227C2×C3×Q8—22no—
228C2×C3×Q8—22no—
229C2×C3×Q8—22no—
230C2×C3×D4—28no—
231C2×C3×D4—22no—
232C2×C3×D4—22no—
233C2×C3×D4—22no—
234C2×C3×D4—22no—
235C2×C3×D4—28no—
236C2×C3×D4—22no—
237C2×C3×D4—44no—
238C2×C3×D4C5×D414yes—
239C22×C3×C4—24no—
240C22×C3×C4—44no—
241C22×C3×C4C23×C512yes—
242C22×C3×C4C23×C512yes—
243C22×C3×C4C23×C512yes—
244C22×C3×C4C23×C511yescharacteristic
245C22×C3×C4C23×C511yescharacteristic
246C22×C3×C4C23×C512yes—
247C22×C3×C4C23×C511yescharacteristic
248C3×C4⋊C4—22no—
249C3×C4⋊C4—24no—
250C3×C22⋊C4—28no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—44no—
255C3×C42—22no—
256C3×C42—22no—
257C3×C42C5×D414yescomplemented
258C23×C5—22no—
259C23×C5—22no—
260C23×C5—24no—
261C23×C5—24no—
262C23×C5—22no—
263C23×C5—22no—
264C23×C5—22no—
265C23×C5—24no—
266C23×C5—24no—
267C23×C5C2×C3×D414yes—
268C23×C5C24×C311yescharacteristic
269C5×Q8—44no—
270C5×D4—28no—
271C5×D4—416no—
272C5×D4—44no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—24no—
285C2×C4×C5—48no—
286C2×C4×C5C2×C3×D412yes—
287C2×C4×C5C2×C3×D412yes—
288C2×C4×C5C2×C3×D412yes—
289C2×C4×C5C2×C3×D412yes—
290C2×C4×C5C3×#4114yes—
291C2×C4×C5C3×#4112yes—
292C22×Q8C22×C3×C511yescharacteristic
293C22×D4C22×C3×C511yescharacteristic
294C22×D4C22×C3×C514yes—
295C22×D4C22×C3×C511yescharacteristic
296C23×C4C22×C3×C512yes—
297#127—24no—
298#124—24no—
299#124—24no—
300C4×D4—24no—
301C2×C4⋊C4C22×C3×C511yescharacteristic
302C2×C4⋊C4C22×C3×C512yes—
303C2×C22⋊C4C22×C3×C512yes—
304C2×C22⋊C4C22×C3×C512yes—
305C2×C22⋊C4C22×C3×C514yes—
306C2×C22⋊C4C22×C3×C512yes—
307C2×C22⋊C4C22×C3×C514yes—
308C2×C42C22×C3×C511yescharacteristic
309C2×C42C22×C3×C511yescharacteristic
310C2×C42C22×C3×C511yescharacteristic
311#95C22×C3×C512yes—
312#95C22×C3×C514yes—
313C2×C3×C5—44no—
314C2×C3×C5—82no—
315C2×C3×C5C2×#12111yescharacteristic
316C2×C3×C5#53211yescharacteristic
317C2×C3×C5#53511yescharacteristic
318C2×C3×C5#54612yes—
319C2×C3×C5#55912yes—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—24no—
323C23×C3—24no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3C2×C5×D414yes—
330C23×C3C24×C511yescharacteristic
331C3×Q8—44no—
332C3×D4—28no—
333C3×D4—416no—
334C3×D4—44no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—48no—
348C2×C3×C4C2×C5×D412yes—
349C2×C3×C4C2×C5×D412yes—
350C2×C3×C4C2×C5×D412yes—
351C2×C3×C4C2×C5×D412yes—
352C2×C3×C4C5×#4114yes—
353C2×C3×C4C5×#4112yes—
354C22×C5—24no—
355C22×C5—28no—
356C22×C5—44no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5—44no—
360C22×C5—42no—
361C22×C5—42no—
362C22×C5—42no—
363C22×C5—44no—
364C22×C5C22×C3×D411yescharacteristic
365C22×C5C22×C3×D411yescharacteristic
366C22×C5C2×C3×#4111yescharacteristic
367C22×C5C2×C3×#4112yes—
368C22×C5C3×H(5,𝔽2)12yes—
369C4×C5—22no—
370C4×C5—22no—
371C4×C5—24no—
372C4×C5—42no—
373C4×C5—42no—
374C4×C5—42no—
375C4×C5C3×#12114yes—
376C24C23×C3×C512yes—
377C24C23×C3×C512yes—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×D4—28no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—28no—
387C2×D4—22no—
388C2×D4—44no—
389C2×D4C3×C5×D414yes—
390C22×C4—24no—
391C22×C4—44no—
392C22×C4C23×C3×C512yes—
393C22×C4C23×C3×C512yes—
394C22×C4C23×C3×C512yes—
395C22×C4C23×C3×C511yescharacteristic
396C22×C4C23×C3×C511yescharacteristic
397C22×C4C23×C3×C512yes—
398C22×C4C23×C3×C511yescharacteristic
399C4⋊C4—22no—
400C4⋊C4—24no—
401C22⋊C4—28no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—44no—
406C42—22no—
407C42—22no—
408C42C3×C5×D414yescomplemented
409C3×C5#254411yescharacteristic, direct product factor, Hall
410C22×C3—24no—
411C22×C3—28no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—44no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3C22×C5×D411yescharacteristic
421C22×C3C22×C5×D411yescharacteristic
422C22×C3C2×C5×#4111yescharacteristic
423C22×C3C2×C5×#4112yes—
424C22×C3C5×H(5,𝔽2)12yes—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—24no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4C5×#12114yes—
432C2×C5—44no—
433C2×C5—82no—
434C2×C5C2×C3×#12111yescharacteristic
435C2×C5C3×#53211yescharacteristic
436C2×C5C3×#53511yescharacteristic
437C2×C5C3×#54612yes—
438C2×C5C3×#55912yes—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23C2×C3×C5×D414yes—
449C23C24×C3×C511yescharacteristic
450Q8—44no—
451D4—28no—
452D4—416no—
453D4—44no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—48no—
467C2×C4C2×C3×C5×D412yes—
468C2×C4C2×C3×C5×D412yes—
469C2×C4C2×C3×C5×D412yes—
470C2×C4C2×C3×C5×D412yes—
471C2×C4C3×C5×#4114yes—
472C2×C4C3×C5×#4112yes—
473C2×C3—44no—
474C2×C3—82no—
475C2×C3C2×C5×#12111yescharacteristic
476C2×C3C5×#53211yescharacteristic
477C2×C3C5×#53511yescharacteristic
478C2×C3C5×#54612yes—
479C2×C3C5×#55912yes—
480C5C3×#254411yescharacteristic, direct product factor, Sylow
481C22—24no—
482C22—28no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22C22×C3×C5×D411yescharacteristic
492C22C22×C3×C5×D411yescharacteristic
493C22C2×C3×C5×#4111yescharacteristic
494C22C2×C3×C5×#4112yes—
495C22C3×C5×H(5,𝔽2)12yes—
496C4—22no—
497C4—22no—
498C4—24no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4C3×C5×#12114yes—
503C3C5×#254411yescharacteristic, direct product factor, Sylow
504C2—44no—
505C2—82no—
506C2C2×C3×C5×#12111yescharacteristic
507C2C3×C5×#53211yescharacteristic
508C2C3×C5×#53511yescharacteristic
509C2C3×C5×#54612yes—
510C2C3×C5×#55912yes—
subgroupG/H
center, socleC23×C3×C5C24
commutator, Frattini, cosocleC23C24×C3×C5
prime radicalC2×C3×C5×#127C2
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×#124C2semidirect
3. C2×C3×C5×#124C2semidirect
4. C2×C3×C4×C5×D4C2semidirect
5. C3×C5×#394C2semidirect
6. C3×C5×#391C2semidirect
7. C3×C5×#390C2semidirect
8. C3×C5×#386C2semidirect
9. C3×C4×C5×C4⋊C4C2semidirect
10. C5×#2544C3direct (N × H)
32. C3×#2544C5direct (N × H)
34. C2×C5×#124C2×C3semidirect
35. C2×C5×#124C2×C3semidirect
36. C2×C4×C5×D4C2×C3semidirect
37. C5×#394C2×C3semidirect
38. C5×#391C2×C3semidirect
39. C5×#390C2×C3semidirect
40. C5×#386C2×C3semidirect
41. C4×C5×C4⋊C4C2×C3semidirect
74. C3×C42×C5D4semidirect
76. C2×C3×#124C2×C5semidirect
77. C2×C3×#124C2×C5semidirect
78. C2×C3×C4×D4C2×C5semidirect
79. C3×#394C2×C5semidirect
80. C3×#391C2×C5semidirect
81. C3×#390C2×C5semidirect
82. C3×#386C2×C5semidirect
83. C3×C4×C4⋊C4C2×C5semidirect
105. #2544C3×C5direct (N × H)
193. C42×C5C3×D4semidirect
195. C2×#124C2×C3×C5semidirect
196. C2×#124C2×C3×C5semidirect
197. C2×C4×D4C2×C3×C5semidirect
198. #394C2×C3×C5semidirect
199. #391C2×C3×C5semidirect
200. #390C2×C3×C5semidirect
201. #386C2×C3×C5semidirect
202. C4×C4⋊C4C2×C3×C5semidirect
257. C3×C42C5×D4semidirect
408. C42C3×C5×D4semidirect
409. C3×C5#2544direct (N × H)
480. C5C3×#2544direct (N × H)
503. C3C5×#2544direct (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×#1272
C2×C3×C5×#1242
C2×C3×C4×C5×D42
C3×C5×#3942
C3×C5×#3912
C3×C5×#3902
C3×C5×#3862
C3×C4×C5×C4⋊C42
C5×#25443
C3×#25445