Finite Group Databasebeta

← C3×C5×#2620 #14275654

Subgroups — C3×C5×#2620 #14275654

Order 1920.

1224 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#124C211yesmaximal, characteristic, complemented
2C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×C3×C5×#121C211yesmaximal, characteristic
4C2×C3×C5×#117C211yesmaximal, characteristic, complemented
5C3×C5×#396C214yesmaximal, complemented
6C3×C5×#394C214yesmaximal, complemented
7C3×C5×#391C211yesmaximal, characteristic, complemented
8C3×C5×#390C211yesmaximal, characteristic, complemented
9C3×C5×#384C211yesmaximal, characteristic, complemented
10C5×#2620C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13C22×C3×C5×D4C2212yes—
14C23×C3×C4×C5C2211yescharacteristic
15C3×C5×#124—24no—
16C3×C5×C22⋊Q8—24no—
17C3×C5×#121—24no—
18C3×C5×#117—24no—
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×C3×C5×C4⋊C4C2212yes—
23C2×C3×C5×C4⋊C4C2214yes—
24C2×C3×C5×C22⋊C4C2212yes—
25C2×C3×C5×C22⋊C4C2212yes—
26C2×C3×C5×C22⋊C4C2212yes—
27C2×C3×C5×C22⋊C4C2214yes—
28C2×C3×C5×C22⋊C4C2214yes—
29C2×C3×C42×C5C2211yescharacteristic
30C2×C3×C42×C5C2211yescharacteristic
31C3×C5×#95C2214yes—
32C3×#2620C511yesmaximal, characteristic, direct product factor, Hall
33C2×C5×#124C2×C311yescharacteristic, complemented
34C2×C5×C22⋊Q8C2×C311yescharacteristic, complemented
35C2×C5×#121C2×C311yescharacteristic
36C2×C5×#117C2×C311yescharacteristic, complemented
37C5×#396C2×C314yescomplemented
38C5×#394C2×C314yescomplemented
39C5×#391C2×C311yescharacteristic, complemented
40C5×#390C2×C311yescharacteristic, complemented
41C5×#384C2×C311yescharacteristic, complemented
42C24×C3×C5C2312yes—
43C24×C3×C5C2311yescharacteristic
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×D4—24no—
48C2×C3×C5×D4—22no—
49C2×C3×C5×D4—22no—
50C2×C3×C5×D4—22no—
51C2×C3×C5×D4—44no—
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5D414yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2311yescharacteristic
61C22×C3×C4×C5C2311yescharacteristic
62C3×C5×C4⋊C4—24no—
63C3×C5×C4⋊C4—22no—
64C3×C5×C4⋊C4—22no—
65C3×C5×C4⋊C4—24no—
66C3×C5×C4⋊C4—44no—
67C3×C5×C22⋊C4—24no—
68C3×C5×C22⋊C4—24no—
69C3×C5×C22⋊C4—24no—
70C3×C5×C22⋊C4—28no—
71C3×C5×C22⋊C4—44no—
72C3×C42×C5—22no—
73C3×C42×C5D414yescomplemented
74C2×C3×#124C2×C511yescharacteristic, complemented
75C2×C3×C22⋊Q8C2×C511yescharacteristic, complemented
76C2×C3×#121C2×C511yescharacteristic
77C2×C3×#117C2×C511yescharacteristic, complemented
78C3×#396C2×C514yescomplemented
79C3×#394C2×C514yescomplemented
80C3×#391C2×C511yescharacteristic, complemented
81C3×#390C2×C511yescharacteristic, complemented
82C3×#384C2×C511yescharacteristic, complemented
83C22×C5×Q8C22×C311yescharacteristic
84C22×C5×D4C22×C311yescharacteristic
85C22×C5×D4C22×C312yes—
86C23×C4×C5C22×C311yescharacteristic
87C5×#124—24no—
88C5×C22⋊Q8—24no—
89C5×#121—24no—
90C5×#117—24no—
91C2×C5×C4⋊C4C22×C311yescharacteristic
92C2×C5×C4⋊C4C22×C311yescharacteristic
93C2×C5×C4⋊C4C22×C312yes—
94C2×C5×C4⋊C4C22×C312yes—
95C2×C5×C4⋊C4C22×C314yes—
96C2×C5×C22⋊C4C22×C312yes—
97C2×C5×C22⋊C4C22×C312yes—
98C2×C5×C22⋊C4C22×C314yes—
99C2×C5×C22⋊C4C22×C312yes—
100C2×C5×C22⋊C4C22×C314yes—
101C2×C42×C5C22×C311yescharacteristic
102C2×C42×C5C22×C311yescharacteristic
103C5×#95C22×C314yes—
104#2620C3×C511yescharacteristic, direct product factor, Sylow
105C23×C3×C5—21no—
106C23×C3×C5—21no—
107C23×C3×C5—22no—
108C23×C3×C5—22no—
109C23×C3×C5—22no—
110C23×C3×C5—22no—
111C23×C3×C5—22no—
112C23×C3×C5—24no—
113C23×C3×C5—24no—
114C23×C3×C5C2×D412yes—
115C23×C3×C5C2411yescharacteristic
116C3×C5×Q8—44no—
117C3×C5×D4—48no—
118C3×C5×D4—44no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—28no—
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—24no—
132C2×C3×C4×C5—24no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C2×D414yes—
135C2×C3×C4×C5C2×D412yes—
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5#4112yes—
139C22×C3×Q8C22×C511yescharacteristic
140C22×C3×D4C22×C511yescharacteristic
141C22×C3×D4C22×C512yes—
142C23×C3×C4C22×C511yescharacteristic
143C3×#124—24no—
144C3×C22⋊Q8—24no—
145C3×#121—24no—
146C3×#117—24no—
147C2×C3×C4⋊C4C22×C511yescharacteristic
148C2×C3×C4⋊C4C22×C511yescharacteristic
149C2×C3×C4⋊C4C22×C512yes—
150C2×C3×C4⋊C4C22×C512yes—
151C2×C3×C4⋊C4C22×C514yes—
152C2×C3×C22⋊C4C22×C512yes—
153C2×C3×C22⋊C4C22×C512yes—
154C2×C3×C22⋊C4C22×C514yes—
155C2×C3×C22⋊C4C22×C512yes—
156C2×C3×C22⋊C4C22×C514yes—
157C2×C3×C42C22×C511yescharacteristic
158C2×C3×C42C22×C511yescharacteristic
159C3×#95C22×C514yes—
160C24×C5C23×C312yes—
161C24×C5C23×C311yescharacteristic
162C2×C5×Q8—22no—
163C2×C5×Q8—22no—
164C2×C5×Q8—22no—
165C2×C5×D4—24no—
166C2×C5×D4—22no—
167C2×C5×D4—22no—
168C2×C5×D4—22no—
169C2×C5×D4—44no—
170C22×C4×C5—22no—
171C22×C4×C5—22no—
172C22×C4×C5C3×D414yes—
173C22×C4×C5C23×C312yes—
174C22×C4×C5C23×C312yes—
175C22×C4×C5C23×C312yes—
176C22×C4×C5C23×C312yes—
177C22×C4×C5C23×C312yes—
178C22×C4×C5C23×C311yescharacteristic
179C22×C4×C5C23×C311yescharacteristic
180C5×C4⋊C4—24no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—24no—
184C5×C4⋊C4—44no—
185C5×C22⋊C4—24no—
186C5×C22⋊C4—24no—
187C5×C22⋊C4—24no—
188C5×C22⋊C4—28no—
189C5×C22⋊C4—44no—
190C42×C5—22no—
191C42×C5C3×D414yescomplemented
192C2×#124C2×C3×C511yescharacteristic, complemented
193C2×C22⋊Q8C2×C3×C511yescharacteristic, complemented
194C2×#121C2×C3×C511yescharacteristic
195C2×#117C2×C3×C511yescharacteristic, complemented
196#396C2×C3×C514yescomplemented
197#394C2×C3×C514yescomplemented
198#391C2×C3×C511yescharacteristic, complemented
199#390C2×C3×C511yescharacteristic, complemented
200#384C2×C3×C511yescharacteristic, complemented
201C22×C3×C5—22no—
202C22×C3×C5—22no—
203C22×C3×C5—22no—
204C22×C3×C5—42no—
205C22×C3×C5—42no—
206C22×C3×C5—42no—
207C22×C3×C5—42no—
208C22×C3×C5—42no—
209C22×C3×C5—44no—
210C22×C3×C5—44no—
211C22×C3×C5C22×D411yescharacteristic
212C22×C3×C5C22×D411yescharacteristic
213C22×C3×C5C2×#4111yescharacteristic
214C22×C3×C5H(5,𝔽2)12yes—
215C22×C3×C52−1+412yes—
216C3×C4×C5—22no—
217C3×C4×C5—24no—
218C3×C4×C5—42no—
219C3×C4×C5—42no—
220C3×C4×C5—42no—
221C3×C4×C5—42no—
222C3×C4×C5#12114yes—
223C24×C3C23×C512yes—
224C24×C3C23×C511yescharacteristic
225C2×C3×Q8—22no—
226C2×C3×Q8—22no—
227C2×C3×Q8—22no—
228C2×C3×D4—24no—
229C2×C3×D4—22no—
230C2×C3×D4—22no—
231C2×C3×D4—22no—
232C2×C3×D4—44no—
233C22×C3×C4—22no—
234C22×C3×C4—22no—
235C22×C3×C4C5×D414yes—
236C22×C3×C4C23×C512yes—
237C22×C3×C4C23×C512yes—
238C22×C3×C4C23×C512yes—
239C22×C3×C4C23×C512yes—
240C22×C3×C4C23×C512yes—
241C22×C3×C4C23×C511yescharacteristic
242C22×C3×C4C23×C511yescharacteristic
243C3×C4⋊C4—24no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—44no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—28no—
252C3×C22⋊C4—44no—
253C3×C42—22no—
254C3×C42C5×D414yescomplemented
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—22no—
258C23×C5—22no—
259C23×C5—22no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5—24no—
263C23×C5—24no—
264C23×C5C2×C3×D412yes—
265C23×C5C24×C311yescharacteristic
266C5×Q8—44no—
267C5×D4—48no—
268C5×D4—44no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—28no—
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—24no—
282C2×C4×C5—24no—
283C2×C4×C5—22no—
284C2×C4×C5C2×C3×D414yes—
285C2×C4×C5C2×C3×D412yes—
286C2×C4×C5C2×C3×D412yes—
287C2×C4×C5C2×C3×D412yes—
288C2×C4×C5C3×#4112yes—
289C22×Q8C22×C3×C511yescharacteristic
290C22×D4C22×C3×C511yescharacteristic
291C22×D4C22×C3×C512yes—
292C23×C4C22×C3×C511yescharacteristic
293#124—24no—
294C22⋊Q8—24no—
295#121—24no—
296#117—24no—
297C2×C4⋊C4C22×C3×C512yes—
298C2×C4⋊C4C22×C3×C511yescharacteristic
299C2×C4⋊C4C22×C3×C511yescharacteristic
300C2×C4⋊C4C22×C3×C512yes—
301C2×C4⋊C4C22×C3×C514yes—
302C2×C22⋊C4C22×C3×C512yes—
303C2×C22⋊C4C22×C3×C512yes—
304C2×C22⋊C4C22×C3×C512yes—
305C2×C22⋊C4C22×C3×C514yes—
306C2×C22⋊C4C22×C3×C514yes—
307C2×C42C22×C3×C511yescharacteristic
308C2×C42C22×C3×C511yescharacteristic
309#95C22×C3×C514yes—
310C2×C3×C5—42no—
311C2×C3×C5—82no—
312C2×C3×C5C2×#12111yescharacteristic
313C2×C3×C5#53511yescharacteristic
314C2×C3×C5#53611yescharacteristic
315C2×C3×C5#53712yes—
316C2×C3×C5#54212yes—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—24no—
325C23×C3—24no—
326C23×C3C2×C5×D412yes—
327C23×C3C24×C511yescharacteristic
328C3×Q8—44no—
329C3×D4—48no—
330C3×D4—44no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—28no—
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—24no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4C2×C5×D414yes—
347C2×C3×C4C2×C5×D412yes—
348C2×C3×C4C2×C5×D412yes—
349C2×C3×C4C2×C5×D412yes—
350C2×C3×C4C5×#4112yes—
351C22×C5—22no—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—42no—
355C22×C5—42no—
356C22×C5—42no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5—44no—
360C22×C5—44no—
361C22×C5C22×C3×D411yescharacteristic
362C22×C5C22×C3×D411yescharacteristic
363C22×C5C2×C3×#4111yescharacteristic
364C22×C5C3×H(5,𝔽2)12yes—
365C22×C5C3×2−1+412yes—
366C4×C5—22no—
367C4×C5—24no—
368C4×C5—42no—
369C4×C5—42no—
370C4×C5—42no—
371C4×C5—42no—
372C4×C5C3×#12114yes—
373C24C23×C3×C512yes—
374C24C23×C3×C511yescharacteristic
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—44no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4C3×C5×D414yes—
386C22×C4C23×C3×C512yes—
387C22×C4C23×C3×C512yes—
388C22×C4C23×C3×C512yes—
389C22×C4C23×C3×C512yes—
390C22×C4C23×C3×C512yes—
391C22×C4C23×C3×C511yescharacteristic
392C22×C4C23×C3×C511yescharacteristic
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—44no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—28no—
402C22⋊C4—44no—
403C42—22no—
404C42C3×C5×D414yescomplemented
405C3×C5#262011yescharacteristic, direct product factor, Hall
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3C22×C5×D411yescharacteristic
417C22×C3C22×C5×D411yescharacteristic
418C22×C3C2×C5×#4111yescharacteristic
419C22×C3C5×H(5,𝔽2)12yes—
420C22×C3C5×2−1+412yes—
421C3×C4—22no—
422C3×C4—24no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4C5×#12114yes—
428C2×C5—42no—
429C2×C5—82no—
430C2×C5C2×C3×#12111yescharacteristic
431C2×C5C3×#53511yescharacteristic
432C2×C5C3×#53611yescharacteristic
433C2×C5C3×#53712yes—
434C2×C5C3×#54212yes—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23C2×C3×C5×D412yes—
445C23C24×C3×C511yescharacteristic
446Q8—44no—
447D4—48no—
448D4—44no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—28no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4C2×C3×C5×D414yes—
465C2×C4C2×C3×C5×D412yes—
466C2×C4C2×C3×C5×D412yes—
467C2×C4C2×C3×C5×D412yes—
468C2×C4C3×C5×#4112yes—
469C2×C3—42no—
470C2×C3—82no—
471C2×C3C2×C5×#12111yescharacteristic
472C2×C3C5×#53511yescharacteristic
473C2×C3C5×#53611yescharacteristic
474C2×C3C5×#53712yes—
475C2×C3C5×#54212yes—
476C5C3×#262011yescharacteristic, direct product factor, Sylow
477C22—22no—
478C22—22no—
479C22—22no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—44no—
487C22C22×C3×C5×D411yescharacteristic
488C22C22×C3×C5×D411yescharacteristic
489C22C2×C3×C5×#4111yescharacteristic
490C22C3×C5×H(5,𝔽2)12yes—
491C22C3×C5×2−1+412yes—
492C4—22no—
493C4—24no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4C3×C5×#12114yes—
499C3C5×#262011yescharacteristic, direct product factor, Sylow
500C2—42no—
501C2—82no—
502C2C2×C3×C5×#12111yescharacteristic
503C2C3×C5×#53511yescharacteristic
504C2C3×C5×#53611yescharacteristic
505C2C3×C5×#53712yes—
506C2C3×C5×#54212yes—
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×#124C2semidirect
2. C2×C3×C5×C22⋊Q8C2semidirect
4. C2×C3×C5×#117C2semidirect
5. C3×C5×#396C2semidirect
6. C3×C5×#394C2semidirect
7. C3×C5×#391C2semidirect
8. C3×C5×#390C2semidirect
9. C3×C5×#384C2semidirect
10. C5×#2620C3direct (N × H)
32. C3×#2620C5direct (N × H)
33. C2×C5×#124C2×C3semidirect
34. C2×C5×C22⋊Q8C2×C3semidirect
36. C2×C5×#117C2×C3semidirect
37. C5×#396C2×C3semidirect
38. C5×#394C2×C3semidirect
39. C5×#391C2×C3semidirect
40. C5×#390C2×C3semidirect
41. C5×#384C2×C3semidirect
73. C3×C42×C5D4semidirect
74. C2×C3×#124C2×C5semidirect
75. C2×C3×C22⋊Q8C2×C5semidirect
77. C2×C3×#117C2×C5semidirect
78. C3×#396C2×C5semidirect
79. C3×#394C2×C5semidirect
80. C3×#391C2×C5semidirect
81. C3×#390C2×C5semidirect
82. C3×#384C2×C5semidirect
104. #2620C3×C5direct (N × H)
191. C42×C5C3×D4semidirect
192. C2×#124C2×C3×C5semidirect
193. C2×C22⋊Q8C2×C3×C5semidirect
195. C2×#117C2×C3×C5semidirect
196. #396C2×C3×C5semidirect
197. #394C2×C3×C5semidirect
198. #391C2×C3×C5semidirect
199. #390C2×C3×C5semidirect
200. #384C2×C3×C5semidirect
254. C3×C42C5×D4semidirect
404. C42C3×C5×D4semidirect
405. C3×C5#2620direct (N × H)
476. C5C3×#2620direct (N × H)
499. C3C5×#2620direct (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×#1242
C2×C3×C5×C22⋊Q82
C2×C3×C5×#1212
C2×C3×C5×#1172
C3×C5×#3962
C3×C5×#3942
C3×C5×#3912
C3×C5×#3902
C3×C5×#3842
C5×#26203
C3×#26205