Finite Group Databasebeta

← C3×C5×#2717 #14275751

Subgroups — C3×C5×#2717 #14275751

Order 1920.

1024 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#124C211yesmaximal, characteristic
2C2×C3×C5×#123C211yesmaximal, characteristic
3C3×C5×#397C212yesmaximal, complemented
4C3×C5×#397C212yesmaximal, complemented
5C3×C5×#395C212yesmaximal, complemented
6C3×C5×#387C212yesmaximal, complemented
7C3×C5×#386C211yesmaximal, characteristic
8C3×C5×#385C212yesmaximal, complemented
9C3×C5×#380C212yesmaximal, complemented
10C5×#2717C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C3×C5×#124—24no—
16C3×C5×#123—24no—
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C5×C4⋊C4C2212yes—
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C4⋊C4C2212yes—
21C2×C3×C5×C22⋊C4C2212yes—
22C2×C3×C5×C22⋊C4C2211yescharacteristic
23C2×C3×C5×C22⋊C4C2212yes—
24C2×C3×C5×C22⋊C4C2212yes—
25C2×C3×C5×C22⋊C4C2212yes—
26C2×C3×C5×C22⋊C4C2211yescharacteristic
27C2×C3×C42×C5C2211yescharacteristic
28C3×C5×#95C2212yes—
29C3×C5×#95C2212yes—
30C3×C5×#95C2212yes—
31C3×C5×#95C2212yes—
32C3×C5×#95C2212yes—
33C3×C5×#95C2212yes—
34C3×#2717C511yesmaximal, characteristic, direct product factor, Hall
35C2×C5×#124C2×C311yescharacteristic
36C2×C5×#123C2×C311yescharacteristic
37C5×#397C2×C312yescomplemented
38C5×#397C2×C312yescomplemented
39C5×#395C2×C312yescomplemented
40C5×#387C2×C312yescomplemented
41C5×#386C2×C311yescharacteristic
42C5×#385C2×C312yescomplemented
43C5×#380C2×C312yescomplemented
44C24×C3×C5C2311yescharacteristic
45C24×C3×C5C2311yescharacteristic
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8—41no—
48C2×C3×C5×Q8—41no—
49C2×C3×C5×D4—41no—
50C2×C3×C5×D4—41no—
51C2×C3×C5×D4D414yes—
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—24no—
54C22×C3×C4×C5—41no—
55C22×C3×C4×C5—41no—
56C22×C3×C4×C5—41no—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2312yes—
61C22×C3×C4×C5C2312yes—
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2312yes—
64C3×C5×C4⋊C4—24no—
65C3×C5×C4⋊C4—24no—
66C3×C5×C4⋊C4—42no—
67C3×C5×C4⋊C4—42no—
68C3×C5×C22⋊C4—22no—
69C3×C5×C22⋊C4—24no—
70C3×C5×C22⋊C4—24no—
71C3×C5×C22⋊C4—24no—
72C3×C5×C22⋊C4—24no—
73C3×C5×C22⋊C4—41no—
74C3×C42×C5—22no—
75C2×C3×#124C2×C511yescharacteristic
76C2×C3×#123C2×C511yescharacteristic
77C3×#397C2×C512yescomplemented
78C3×#397C2×C512yescomplemented
79C3×#395C2×C512yescomplemented
80C3×#387C2×C512yescomplemented
81C3×#386C2×C511yescharacteristic
82C3×#385C2×C512yescomplemented
83C3×#380C2×C512yescomplemented
84C22×C5×Q8C22×C311yescharacteristic
85C22×C5×D4C22×C311yescharacteristic
86C23×C4×C5C22×C311yescharacteristic
87C23×C4×C5C22×C311yescharacteristic
88C5×#124—24no—
89C5×#123—24no—
90C2×C5×C4⋊C4C22×C312yes—
91C2×C5×C4⋊C4C22×C312yes—
92C2×C5×C4⋊C4C22×C312yes—
93C2×C5×C4⋊C4C22×C312yes—
94C2×C5×C22⋊C4C22×C312yes—
95C2×C5×C22⋊C4C22×C312yes—
96C2×C5×C22⋊C4C22×C312yes—
97C2×C5×C22⋊C4C22×C312yes—
98C2×C5×C22⋊C4C22×C311yescharacteristic
99C2×C5×C22⋊C4C22×C311yescharacteristic
100C2×C42×C5C22×C311yescharacteristic
101C5×#95C22×C312yes—
102C5×#95C22×C312yes—
103C5×#95C22×C312yes—
104C5×#95C22×C312yes—
105C5×#95C22×C312yes—
106C5×#95C22×C312yes—
107#2717C3×C511yescharacteristic, direct product factor, Sylow
108C23×C3×C5—22no—
109C23×C3×C5—22no—
110C23×C3×C5—21no—
111C23×C3×C5—21no—
112C23×C3×C5—21no—
113C23×C3×C5—21no—
114C23×C3×C5—22no—
115C23×C3×C5—22no—
116C23×C3×C5C2×D412yes—
117C23×C3×C5C2×D412yes—
118C23×C3×C5C2411yescharacteristic
119C3×C5×Q8—44no—
120C3×C5×D4—44no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—21no—
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—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—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—44no—
141C2×C3×C4×C5—44no—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5#4114yes—
144C22×C3×Q8C22×C511yescharacteristic
145C22×C3×D4C22×C511yescharacteristic
146C23×C3×C4C22×C511yescharacteristic
147C23×C3×C4C22×C511yescharacteristic
148C3×#124—24no—
149C3×#123—24no—
150C2×C3×C4⋊C4C22×C512yes—
151C2×C3×C4⋊C4C22×C512yes—
152C2×C3×C4⋊C4C22×C512yes—
153C2×C3×C4⋊C4C22×C512yes—
154C2×C3×C22⋊C4C22×C512yes—
155C2×C3×C22⋊C4C22×C512yes—
156C2×C3×C22⋊C4C22×C512yes—
157C2×C3×C22⋊C4C22×C512yes—
158C2×C3×C22⋊C4C22×C511yescharacteristic
159C2×C3×C22⋊C4C22×C511yescharacteristic
160C2×C3×C42C22×C511yescharacteristic
161C3×#95C22×C512yes—
162C3×#95C22×C512yes—
163C3×#95C22×C512yes—
164C3×#95C22×C512yes—
165C3×#95C22×C512yes—
166C3×#95C22×C512yes—
167C24×C5C23×C311yescharacteristic
168C24×C5C23×C311yescharacteristic
169C2×C5×Q8—22no—
170C2×C5×Q8—41no—
171C2×C5×Q8—41no—
172C2×C5×D4—41no—
173C2×C5×D4—41no—
174C2×C5×D4C3×D414yes—
175C22×C4×C5—22no—
176C22×C4×C5—24no—
177C22×C4×C5—41no—
178C22×C4×C5—41no—
179C22×C4×C5—41no—
180C22×C4×C5C23×C312yes—
181C22×C4×C5C23×C312yes—
182C22×C4×C5C23×C312yes—
183C22×C4×C5C23×C312yes—
184C22×C4×C5C23×C312yes—
185C22×C4×C5C23×C311yescharacteristic
186C22×C4×C5C23×C312yes—
187C5×C4⋊C4—24no—
188C5×C4⋊C4—24no—
189C5×C4⋊C4—42no—
190C5×C4⋊C4—42no—
191C5×C22⋊C4—22no—
192C5×C22⋊C4—24no—
193C5×C22⋊C4—24no—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—24no—
196C5×C22⋊C4—41no—
197C42×C5—22no—
198C2×#124C2×C3×C511yescharacteristic
199C2×#123C2×C3×C511yescharacteristic
200#397C2×C3×C512yescomplemented
201#397C2×C3×C512yescomplemented
202#395C2×C3×C512yescomplemented
203#387C2×C3×C512yescomplemented
204#386C2×C3×C511yescharacteristic
205#385C2×C3×C512yescomplemented
206#380C2×C3×C512yescomplemented
207C22×C3×C5—22no—
208C22×C3×C5—22no—
209C22×C3×C5—24no—
210C22×C3×C5—24no—
211C22×C3×C5—42no—
212C22×C3×C5—42no—
213C22×C3×C5—41no—
214C22×C3×C5—41no—
215C22×C3×C5—41no—
216C22×C3×C5—41no—
217C22×C3×C5C22×D411yescharacteristic
218C22×C3×C5C2×#4112yes—
219C22×C3×C5H(5,𝔽2)11yescharacteristic
220C22×C3×C5H(5,𝔽2)11yescharacteristic
221C22×C3×C52−1+412yes—
222C3×C4×C5—22no—
223C3×C4×C5—24no—
224C3×C4×C5—42no—
225C3×C4×C5—42no—
226C3×C4×C5—42no—
227C3×C4×C5—42no—
228C3×C4×C5—42no—
229C24×C3C23×C511yescharacteristic
230C24×C3C23×C511yescharacteristic
231C2×C3×Q8—22no—
232C2×C3×Q8—41no—
233C2×C3×Q8—41no—
234C2×C3×D4—41no—
235C2×C3×D4—41no—
236C2×C3×D4C5×D414yes—
237C22×C3×C4—22no—
238C22×C3×C4—24no—
239C22×C3×C4—41no—
240C22×C3×C4—41no—
241C22×C3×C4—41no—
242C22×C3×C4C23×C512yes—
243C22×C3×C4C23×C512yes—
244C22×C3×C4C23×C512yes—
245C22×C3×C4C23×C512yes—
246C22×C3×C4C23×C512yes—
247C22×C3×C4C23×C511yescharacteristic
248C22×C3×C4C23×C512yes—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—42no—
252C3×C4⋊C4—42no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—24no—
258C3×C22⋊C4—41no—
259C3×C42—22no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5—21no—
263C23×C5—21no—
264C23×C5—21no—
265C23×C5—21no—
266C23×C5—22no—
267C23×C5—22no—
268C23×C5C2×C3×D412yes—
269C23×C5C2×C3×D412yes—
270C23×C5C24×C311yescharacteristic
271C5×Q8—44no—
272C5×D4—44no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
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—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—44no—
293C2×C4×C5—44no—
294C2×C4×C5C2×C3×D412yes—
295C2×C4×C5C3×#4114yes—
296C22×Q8C22×C3×C511yescharacteristic
297C22×D4C22×C3×C511yescharacteristic
298C23×C4C22×C3×C511yescharacteristic
299C23×C4C22×C3×C511yescharacteristic
300#124—24no—
301#123—24no—
302C2×C4⋊C4C22×C3×C512yes—
303C2×C4⋊C4C22×C3×C512yes—
304C2×C4⋊C4C22×C3×C512yes—
305C2×C4⋊C4C22×C3×C512yes—
306C2×C22⋊C4C22×C3×C512yes—
307C2×C22⋊C4C22×C3×C511yescharacteristic
308C2×C22⋊C4C22×C3×C512yes—
309C2×C22⋊C4C22×C3×C512yes—
310C2×C22⋊C4C22×C3×C512yes—
311C2×C22⋊C4C22×C3×C511yescharacteristic
312C2×C42C22×C3×C511yescharacteristic
313#95C22×C3×C512yes—
314#95C22×C3×C512yes—
315#95C22×C3×C512yes—
316#95C22×C3×C512yes—
317#95C22×C3×C512yes—
318#95C22×C3×C512yes—
319C2×C3×C5—42no—
320C2×C3×C5—42no—
321C2×C3×C5#53411yescharacteristic
322C2×C3×C5#54212yes—
323C2×C3×C5#54712yes—
324C2×C3×C5#55111yescharacteristic
325C2×C3×C5#56311yescharacteristic
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3C2×C5×D412yes—
335C23×C3C2×C5×D412yes—
336C23×C3C24×C511yescharacteristic
337C3×Q8—44no—
338C3×D4—44no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—44no—
359C2×C3×C4—44no—
360C2×C3×C4C2×C5×D412yes—
361C2×C3×C4C5×#4114yes—
362C22×C5—22no—
363C22×C5—22no—
364C22×C5—24no—
365C22×C5—24no—
366C22×C5—41no—
367C22×C5—41no—
368C22×C5—42no—
369C22×C5—42no—
370C22×C5—41no—
371C22×C5—41no—
372C22×C5C22×C3×D411yescharacteristic
373C22×C5C2×C3×#4112yes—
374C22×C5C3×H(5,𝔽2)11yescharacteristic
375C22×C5C3×H(5,𝔽2)11yescharacteristic
376C22×C5C3×2−1+412yes—
377C4×C5—22no—
378C4×C5—24no—
379C4×C5—42no—
380C4×C5—42no—
381C4×C5—42no—
382C4×C5—42no—
383C4×C5—42no—
384C24C23×C3×C511yescharacteristic
385C24C23×C3×C511yescharacteristic
386C2×Q8—22no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4C3×C5×D414yes—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4C23×C3×C512yes—
398C22×C4C23×C3×C512yes—
399C22×C4C23×C3×C512yes—
400C22×C4C23×C3×C512yes—
401C22×C4C23×C3×C512yes—
402C22×C4C23×C3×C511yescharacteristic
403C22×C4C23×C3×C512yes—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C22⋊C4—22no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—41no—
414C42—22no—
415C3×C5#271711yescharacteristic, direct product factor, Hall
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3C22×C5×D411yescharacteristic
427C22×C3C2×C5×#4112yes—
428C22×C3C5×H(5,𝔽2)11yescharacteristic
429C22×C3C5×H(5,𝔽2)11yescharacteristic
430C22×C3C5×2−1+412yes—
431C3×C4—22no—
432C3×C4—24no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—42no—
438C2×C5—42no—
439C2×C5—42no—
440C2×C5C3×#53411yescharacteristic
441C2×C5C3×#54212yes—
442C2×C5C3×#54712yes—
443C2×C5C3×#55111yescharacteristic
444C2×C5C3×#56311yescharacteristic
445C23—22no—
446C23—22no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23C2×C3×C5×D412yes—
454C23C2×C3×C5×D412yes—
455C23C24×C3×C511yescharacteristic
456Q8—44no—
457D4—44no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4C2×C3×C5×D412yes—
480C2×C4C3×C5×#4114yes—
481C2×C3—42no—
482C2×C3—42no—
483C2×C3C5×#53411yescharacteristic
484C2×C3C5×#54212yes—
485C2×C3C5×#54712yes—
486C2×C3C5×#55111yescharacteristic
487C2×C3C5×#56311yescharacteristic
488C5C3×#271711yescharacteristic, direct product factor, Sylow
489C22—22no—
490C22—22no—
491C22—24no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22C22×C3×C5×D411yescharacteristic
500C22C2×C3×C5×#4112yes—
501C22C3×C5×H(5,𝔽2)11yescharacteristic
502C22C3×C5×H(5,𝔽2)11yescharacteristic
503C22C3×C5×2−1+412yes—
504C4—22no—
505C4—24no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C3C5×#271711yescharacteristic, direct product factor, Sylow
512C2—42no—
513C2—42no—
514C2C3×C5×#53411yescharacteristic
515C2C3×C5×#54212yes—
516C2C3×C5×#54712yes—
517C2C3×C5×#55111yescharacteristic
518C2C3×C5×#56311yescharacteristic
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
3. C3×C5×#397C2semidirect
4. C3×C5×#397C2semidirect
5. C3×C5×#395C2semidirect
6. C3×C5×#387C2semidirect
8. C3×C5×#385C2semidirect
9. C3×C5×#380C2semidirect
10. C5×#2717C3direct (N × H)
34. C3×#2717C5direct (N × H)
37. C5×#397C2×C3semidirect
38. C5×#397C2×C3semidirect
39. C5×#395C2×C3semidirect
40. C5×#387C2×C3semidirect
42. C5×#385C2×C3semidirect
43. C5×#380C2×C3semidirect
77. C3×#397C2×C5semidirect
78. C3×#397C2×C5semidirect
79. C3×#395C2×C5semidirect
80. C3×#387C2×C5semidirect
82. C3×#385C2×C5semidirect
83. C3×#380C2×C5semidirect
107. #2717C3×C5direct (N × H)
200. #397C2×C3×C5semidirect
201. #397C2×C3×C5semidirect
202. #395C2×C3×C5semidirect
203. #387C2×C3×C5semidirect
205. #385C2×C3×C5semidirect
206. #380C2×C3×C5semidirect
415. C3×C5#2717direct (N × H)
488. C5C3×#2717direct (N × H)
511. C3C5×#2717direct (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×#1232
C3×C5×#3972
C3×C5×#3952
C3×C5×#3872
C3×C5×#3862
C3×C5×#3852
C3×C5×#3802
C5×#27173
C3×#27175