Finite Group Databasebeta

← C3×C5×#2664 #14275698

Subgroups — C3×C5×#2664 #14275698

Order 1920.

868 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#126C211yesmaximal, characteristic
2C2×C3×C5×#125C211yesmaximal, characteristic, complemented
3C3×C5×#400C211yesmaximal, characteristic, complemented
4C3×C5×#399C211yesmaximal, characteristic
5C3×C5×#398C212yesmaximal, complemented
6C3×C5×#398C211yesmaximal, characteristic, complemented
7C3×C5×#397C211yesmaximal, characteristic
8C3×C5×#396C212yesmaximal
9C3×C5×#389C212yesmaximal, complemented
10C3×C5×#388C212yesmaximal
11C3×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
12C5×#2664C311yesmaximal, characteristic, direct product factor, Hall
13C3×C5×#126—24no—
14C3×C5×#125—24no—
15C2×C3×C5×C4⋊C4C2212yes—
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C4⋊C4C2211yescharacteristic
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×C3×C5×C4⋊C4C2212yes—
23C2×C3×C5×C4⋊C4C2212yes—
24C2×C3×C5×C4⋊C4C2212yes—
25C2×C3×C5×C22⋊C4C2211yescharacteristic
26C2×C3×C5×C22⋊C4C2211yescharacteristic
27C2×C3×C5×C22⋊C4C2211yescharacteristic
28C2×C3×C5×C22⋊C4C2212yes—
29C2×C3×C5×C22⋊C4C2212yes—
30C2×C3×C42×C5C2212yes—
31C2×C3×C42×C5C2211yescharacteristic
32C3×C5×#95C2211yescharacteristic
33C3×C5×#95C2211yescharacteristic
34C3×C5×#95C2212yes—
35C3×C5×#95C2212yes—
36C3×C5×#95C2212yes—
37C3×C5×#95C2212yes—
38C3×#2664C511yesmaximal, characteristic, direct product factor, Hall
39C2×C5×#126C2×C311yescharacteristic
40C2×C5×#125C2×C311yescharacteristic, complemented
41C5×#400C2×C311yescharacteristic, complemented
42C5×#399C2×C311yescharacteristic
43C5×#398C2×C312yescomplemented
44C5×#398C2×C311yescharacteristic, complemented
45C5×#397C2×C311yescharacteristic
46C5×#396C2×C312yes—
47C5×#389C2×C312yescomplemented
48C5×#388C2×C312yes—
49C4×C5×C4⋊C4C2×C311yescharacteristic, complemented
50C24×C3×C5C2311yescharacteristic
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C22×C3×C4×C5C2312yes—
61C3×C5×C4⋊C4—24no—
62C3×C5×C4⋊C4—24no—
63C3×C5×C4⋊C4—22no—
64C3×C5×C4⋊C4—22no—
65C3×C5×C4⋊C4—22no—
66C3×C5×C4⋊C4—22no—
67C3×C5×C4⋊C4—22no—
68C3×C5×C4⋊C4—24no—
69C3×C5×C4⋊C4—24no—
70C3×C5×C4⋊C4—42no—
71C3×C5×C22⋊C4—22no—
72C3×C5×C22⋊C4—22no—
73C3×C5×C22⋊C4—22no—
74C3×C5×C22⋊C4—24no—
75C3×C5×C22⋊C4—42no—
76C3×C42×C5—42no—
77C3×C42×C5D414yes—
78C2×C3×#126C2×C511yescharacteristic
79C2×C3×#125C2×C511yescharacteristic, complemented
80C3×#400C2×C511yescharacteristic, complemented
81C3×#399C2×C511yescharacteristic
82C3×#398C2×C511yescharacteristic, complemented
83C3×#398C2×C512yescomplemented
84C3×#397C2×C511yescharacteristic
85C3×#396C2×C512yes—
86C3×#389C2×C512yescomplemented
87C3×#388C2×C512yes—
88C3×C4×C4⋊C4C2×C511yescharacteristic, complemented
89C5×#126—24no—
90C5×#125—24no—
91C2×C5×C4⋊C4C22×C312yes—
92C2×C5×C4⋊C4C22×C312yes—
93C2×C5×C4⋊C4C22×C311yescharacteristic
94C2×C5×C4⋊C4C22×C312yes—
95C2×C5×C4⋊C4C22×C312yes—
96C2×C5×C4⋊C4C22×C311yescharacteristic
97C2×C5×C4⋊C4C22×C311yescharacteristic
98C2×C5×C4⋊C4C22×C311yescharacteristic
99C2×C5×C4⋊C4C22×C311yescharacteristic
100C2×C5×C4⋊C4C22×C312yes—
101C2×C5×C22⋊C4C22×C311yescharacteristic
102C2×C5×C22⋊C4C22×C312yes—
103C2×C5×C22⋊C4C22×C311yescharacteristic
104C2×C5×C22⋊C4C22×C311yescharacteristic
105C2×C5×C22⋊C4C22×C312yes—
106C2×C42×C5C22×C312yes—
107C2×C42×C5C22×C311yescharacteristic
108C5×#95C22×C311yescharacteristic
109C5×#95C22×C311yescharacteristic
110C5×#95C22×C312yes—
111C5×#95C22×C312yes—
112C5×#95C22×C312yes—
113C5×#95C22×C312yes—
114#2664C3×C511yescharacteristic, direct product factor, Sylow
115C23×C3×C5—21no—
116C23×C3×C5—22no—
117C23×C3×C5—21no—
118C23×C3×C5—22no—
119C23×C3×C5—21no—
120C23×C3×C5C2411yescharacteristic
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—21no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—21no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5#4114yes—
147C3×#126—24no—
148C3×#125—24no—
149C2×C3×C4⋊C4C22×C512yes—
150C2×C3×C4⋊C4C22×C512yes—
151C2×C3×C4⋊C4C22×C511yescharacteristic
152C2×C3×C4⋊C4C22×C512yes—
153C2×C3×C4⋊C4C22×C512yes—
154C2×C3×C4⋊C4C22×C511yescharacteristic
155C2×C3×C4⋊C4C22×C511yescharacteristic
156C2×C3×C4⋊C4C22×C511yescharacteristic
157C2×C3×C4⋊C4C22×C511yescharacteristic
158C2×C3×C4⋊C4C22×C512yes—
159C2×C3×C22⋊C4C22×C511yescharacteristic
160C2×C3×C22⋊C4C22×C512yes—
161C2×C3×C22⋊C4C22×C511yescharacteristic
162C2×C3×C22⋊C4C22×C511yescharacteristic
163C2×C3×C22⋊C4C22×C512yes—
164C2×C3×C42C22×C512yes—
165C2×C3×C42C22×C511yescharacteristic
166C3×#95C22×C511yescharacteristic
167C3×#95C22×C511yescharacteristic
168C3×#95C22×C512yes—
169C3×#95C22×C512yes—
170C3×#95C22×C512yes—
171C3×#95C22×C512yes—
172C24×C5C23×C311yescharacteristic
173C22×C4×C5C23×C312yes—
174C22×C4×C5C23×C311yescharacteristic
175C22×C4×C5C23×C311yescharacteristic
176C22×C4×C5C23×C312yes—
177C22×C4×C5C23×C312yes—
178C22×C4×C5C23×C311yescharacteristic
179C22×C4×C5C23×C311yescharacteristic
180C22×C4×C5C23×C311yescharacteristic
181C22×C4×C5C23×C311yescharacteristic
182C22×C4×C5C23×C312yes—
183C5×C4⋊C4—24no—
184C5×C4⋊C4—24no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—22no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4—22no—
189C5×C4⋊C4—22no—
190C5×C4⋊C4—24no—
191C5×C4⋊C4—24no—
192C5×C4⋊C4—42no—
193C5×C22⋊C4—22no—
194C5×C22⋊C4—22no—
195C5×C22⋊C4—22no—
196C5×C22⋊C4—24no—
197C5×C22⋊C4—42no—
198C42×C5—42no—
199C42×C5C3×D414yes—
200C2×#126C2×C3×C511yescharacteristic
201C2×#125C2×C3×C511yescharacteristic, complemented
202#400C2×C3×C511yescharacteristic, complemented
203#399C2×C3×C511yescharacteristic
204#398C2×C3×C511yescharacteristic, complemented
205#398C2×C3×C512yescomplemented
206#397C2×C3×C511yescharacteristic
207#396C2×C3×C512yes—
208#389C2×C3×C512yescomplemented
209#388C2×C3×C512yes—
210C4×C4⋊C4C2×C3×C511yescharacteristic, complemented
211C22×C3×C5—41no—
212C22×C3×C5—41no—
213C22×C3×C5—41no—
214C22×C3×C5—42no—
215C22×C3×C5—42no—
216C22×C3×C5C22×D411yescharacteristic
217C22×C3×C5C2×#4112yes—
218C22×C3×C5H(5,𝔽2)11yescharacteristic
219C22×C3×C52−1+412yes—
220C22×C3×C52−1+411yescharacteristic
221C3×C4×C5—22no—
222C3×C4×C5—22no—
223C3×C4×C5—22no—
224C3×C4×C5—24no—
225C3×C4×C5—41no—
226C3×C4×C5—42no—
227C3×C4×C5—41no—
228C3×C4×C5—41no—
229C3×C4×C5—42no—
230C3×C4×C5—42no—
231C24×C3C23×C511yescharacteristic
232C22×C3×C4C23×C512yes—
233C22×C3×C4C23×C511yescharacteristic
234C22×C3×C4C23×C511yescharacteristic
235C22×C3×C4C23×C511yescharacteristic
236C22×C3×C4C23×C512yes—
237C22×C3×C4C23×C512yes—
238C22×C3×C4C23×C511yescharacteristic
239C22×C3×C4C23×C511yescharacteristic
240C22×C3×C4C23×C511yescharacteristic
241C22×C3×C4C23×C512yes—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—42no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—42no—
257C3×C42—42no—
258C3×C42C5×D414yes—
259C23×C5—21no—
260C23×C5—22no—
261C23×C5—21no—
262C23×C5—22no—
263C23×C5—21no—
264C23×C5C24×C311yescharacteristic
265C2×C4×C5—22no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—21no—
270C2×C4×C5—22no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
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—21no—
283C2×C4×C5—22no—
284C2×C4×C5—21no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5C2×C3×D412yes—
288C2×C4×C5C2×C3×D412yes—
289C2×C4×C5C2×C3×D412yes—
290C2×C4×C5C3×#4114yes—
291#126—24no—
292#125—24no—
293C2×C4⋊C4C22×C3×C512yes—
294C2×C4⋊C4C22×C3×C512yes—
295C2×C4⋊C4C22×C3×C511yescharacteristic
296C2×C4⋊C4C22×C3×C511yescharacteristic
297C2×C4⋊C4C22×C3×C511yescharacteristic
298C2×C4⋊C4C22×C3×C511yescharacteristic
299C2×C4⋊C4C22×C3×C511yescharacteristic
300C2×C4⋊C4C22×C3×C512yes—
301C2×C4⋊C4C22×C3×C512yes—
302C2×C4⋊C4C22×C3×C512yes—
303C2×C22⋊C4C22×C3×C511yescharacteristic
304C2×C22⋊C4C22×C3×C511yescharacteristic
305C2×C22⋊C4C22×C3×C511yescharacteristic
306C2×C22⋊C4C22×C3×C512yes—
307C2×C22⋊C4C22×C3×C512yes—
308C2×C42C22×C3×C512yes—
309C2×C42C22×C3×C511yescharacteristic
310#95C22×C3×C511yescharacteristic
311#95C22×C3×C511yescharacteristic
312#95C22×C3×C512yes—
313#95C22×C3×C512yes—
314#95C22×C3×C512yes—
315#95C22×C3×C512yes—
316C2×C3×C5—81no—
317C2×C3×C5#53211yescharacteristic
318C2×C3×C5#53611yescharacteristic
319C2×C3×C5#53711yescharacteristic
320C2×C3×C5#53912yes—
321C2×C3×C5#54112yes—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—21no—
327C23×C3C24×C511yescharacteristic
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
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—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4C2×C5×D412yes—
351C2×C3×C4C2×C5×D412yes—
352C2×C3×C4C2×C5×D412yes—
353C2×C3×C4C5×#4114yes—
354C22×C5—41no—
355C22×C5—41no—
356C22×C5—41no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5C22×C3×D411yescharacteristic
360C22×C5C2×C3×#4112yes—
361C22×C5C3×H(5,𝔽2)11yescharacteristic
362C22×C5C3×2−1+412yes—
363C22×C5C3×2−1+411yescharacteristic
364C4×C5—22no—
365C4×C5—22no—
366C4×C5—22no—
367C4×C5—24no—
368C4×C5—41no—
369C4×C5—42no—
370C4×C5—41no—
371C4×C5—41no—
372C4×C5—42no—
373C4×C5—42no—
374C24C23×C3×C511yescharacteristic
375C22×C4C23×C3×C512yes—
376C22×C4C23×C3×C511yescharacteristic
377C22×C4C23×C3×C511yescharacteristic
378C22×C4C23×C3×C511yescharacteristic
379C22×C4C23×C3×C512yes—
380C22×C4C23×C3×C512yes—
381C22×C4C23×C3×C511yescharacteristic
382C22×C4C23×C3×C511yescharacteristic
383C22×C4C23×C3×C511yescharacteristic
384C22×C4C23×C3×C512yes—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—42no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—24no—
399C22⋊C4—42no—
400C42—42no—
401C42C3×C5×D414yes—
402C3×C5#266411yescharacteristic, direct product factor, Hall
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3C22×C5×D411yescharacteristic
409C22×C3C2×C5×#4112yes—
410C22×C3C5×H(5,𝔽2)11yescharacteristic
411C22×C3C5×2−1+412yes—
412C22×C3C5×2−1+411yescharacteristic
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—24no—
417C3×C4—41no—
418C3×C4—42no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—42no—
422C3×C4—42no—
423C2×C5—81no—
424C2×C5C3×#53211yescharacteristic
425C2×C5C3×#53611yescharacteristic
426C2×C5C3×#53711yescharacteristic
427C2×C5C3×#53912yes—
428C2×C5C3×#54112yes—
429C23—21no—
430C23—22no—
431C23—21no—
432C23—22no—
433C23—21no—
434C23C24×C3×C511yescharacteristic
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4C2×C3×C5×D412yes—
458C2×C4C2×C3×C5×D412yes—
459C2×C4C2×C3×C5×D412yes—
460C2×C4C3×C5×#4114yes—
461C2×C3—81no—
462C2×C3C5×#53211yescharacteristic
463C2×C3C5×#53611yescharacteristic
464C2×C3C5×#53711yescharacteristic
465C2×C3C5×#53912yes—
466C2×C3C5×#54112yes—
467C5C3×#266411yescharacteristic, direct product factor, Sylow
468C22—41no—
469C22—41no—
470C22—41no—
471C22—42no—
472C22—42no—
473C22C22×C3×C5×D411yescharacteristic
474C22C2×C3×C5×#4112yes—
475C22C3×C5×H(5,𝔽2)11yescharacteristic
476C22C3×C5×2−1+412yes—
477C22C3×C5×2−1+411yescharacteristic
478C4—22no—
479C4—22no—
480C4—22no—
481C4—24no—
482C4—41no—
483C4—42no—
484C4—41no—
485C4—41no—
486C4—42no—
487C4—42no—
488C3C5×#266411yescharacteristic, direct product factor, Sylow
489C2—81no—
490C2C3×C5×#53211yescharacteristic
491C2C3×C5×#53611yescharacteristic
492C2C3×C5×#53711yescharacteristic
493C2C3×C5×#53912yes—
494C2C3×C5×#54112yes—
subgroupG/H
center, socleC23×C3×C5C24
commutator, Frattini, cosocleC23C24×C3×C5
prime radicalC24×C3×C5C23
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×#125C2semidirect
3. C3×C5×#400C2semidirect
5. C3×C5×#398C2semidirect
6. C3×C5×#398C2semidirect
9. C3×C5×#389C2semidirect
11. C3×C4×C5×C4⋊C4C2semidirect
12. C5×#2664C3direct (N × H)
38. C3×#2664C5direct (N × H)
40. C2×C5×#125C2×C3semidirect
41. C5×#400C2×C3semidirect
43. C5×#398C2×C3semidirect
44. C5×#398C2×C3semidirect
47. C5×#389C2×C3semidirect
49. C4×C5×C4⋊C4C2×C3semidirect
79. C2×C3×#125C2×C5semidirect
80. C3×#400C2×C5semidirect
82. C3×#398C2×C5semidirect
83. C3×#398C2×C5semidirect
86. C3×#389C2×C5semidirect
88. C3×C4×C4⋊C4C2×C5semidirect
114. #2664C3×C5direct (N × H)
201. C2×#125C2×C3×C5semidirect
202. #400C2×C3×C5semidirect
204. #398C2×C3×C5semidirect
205. #398C2×C3×C5semidirect
208. #389C2×C3×C5semidirect
210. C4×C4⋊C4C2×C3×C5semidirect
402. C3×C5#2664direct (N × H)
467. C5C3×#2664direct (N × H)
488. C3C5×#2664direct (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×#1262
C2×C3×C5×#1252
C3×C5×#4002
C3×C5×#3992
C3×C5×#3982
C3×C5×#3972
C3×C5×#3962
C3×C5×#3892
C3×C5×#3882
C3×C4×C5×C4⋊C42
C5×#26643
C3×#26645