Finite Group Databasebeta

← C3×C5×#2376 #14275410

Subgroups — C3×C5×#2376 #14275410

Order 1920.

1024 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×#123C211yesmaximal, characteristic
2C3×C5×#390C211yesmaximal, characteristic
3C3×C5×#389C212yesmaximal, complemented
4C3×C5×#388C212yesmaximal, complemented
5C3×C5×#388C212yesmaximal, complemented
6C3×C5×#387C212yesmaximal, complemented
7C3×C5×#386C211yesmaximal, characteristic
8C3×C5×#385C212yesmaximal, complemented
9C3×C5×#380C212yesmaximal, complemented
10C5×#2376C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C3×C5×#123C418yescomplemented
15C2×C3×C5×C4⋊C4C2212yes—
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C22⋊C4C2212yes—
21C2×C3×C5×C22⋊C4C2212yes—
22C2×C3×C5×C22⋊C4C2212yes—
23C2×C3×C5×C22⋊C4C2211yescharacteristic
24C2×C3×C5×C22⋊C4C2212yes—
25C2×C3×C5×C22⋊C4C2211yescharacteristic
26C2×C3×C42×C5C2211yescharacteristic
27C2×C3×C42×C5C2212yes—
28C3×C5×#95C2212yes—
29C3×C5×#95C2212yes—
30C3×C5×#95C2212yes—
31C3×C5×#95C2212yes—
32C3×C5×#95C2212yes—
33C3×#2376C511yesmaximal, characteristic, direct product factor, Hall
34C2×C5×#123C2×C311yescharacteristic
35C5×#390C2×C311yescharacteristic
36C5×#389C2×C312yescomplemented
37C5×#388C2×C312yescomplemented
38C5×#388C2×C312yescomplemented
39C5×#387C2×C312yescomplemented
40C5×#386C2×C311yescharacteristic
41C5×#385C2×C312yescomplemented
42C5×#380C2×C312yescomplemented
43C24×C3×C5C2311yescharacteristic
44C24×C3×C5C2311yescharacteristic
45C2×C3×C5×D4—41no—
46C2×C3×C5×D4—41no—
47C2×C3×C5×D4C2×C414yes—
48C22×C3×C4×C5—41no—
49C22×C3×C4×C5—41no—
50C22×C3×C4×C5—42no—
51C22×C3×C4×C5—41no—
52C22×C3×C4×C5C2×C414yescomplemented
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2312yes—
60C3×C5×C4⋊C4—42no—
61C3×C5×C4⋊C4—42no—
62C3×C5×C4⋊C4—42no—
63C3×C5×C4⋊C4—41no—
64C3×C5×C4⋊C4C2×C418yescomplemented
65C3×C5×C22⋊C4—42no—
66C3×C5×C22⋊C4—42no—
67C3×C5×C22⋊C4—42no—
68C3×C5×C22⋊C4—41no—
69C3×C5×C22⋊C4C2×C418yes—
70C3×C5×C22⋊C4C2×C414yescomplemented
71C3×C42×C5—42no—
72C3×C42×C5—41no—
73C2×C3×#123C2×C511yescharacteristic
74C3×#390C2×C511yescharacteristic
75C3×#389C2×C512yescomplemented
76C3×#388C2×C512yescomplemented
77C3×#388C2×C512yescomplemented
78C3×#387C2×C512yescomplemented
79C3×#386C2×C511yescharacteristic
80C3×#385C2×C512yescomplemented
81C3×#380C2×C512yescomplemented
82C22×C5×D4C22×C311yescharacteristic
83C23×C4×C5C22×C311yescharacteristic
84C23×C4×C5C22×C311yescharacteristic
85C5×#123C3×C418yescomplemented
86C2×C5×C4⋊C4C22×C312yes—
87C2×C5×C4⋊C4C22×C312yes—
88C2×C5×C4⋊C4C22×C312yes—
89C2×C5×C4⋊C4C22×C311yescharacteristic
90C2×C5×C4⋊C4C22×C312yes—
91C2×C5×C22⋊C4C22×C312yes—
92C2×C5×C22⋊C4C22×C312yes—
93C2×C5×C22⋊C4C22×C312yes—
94C2×C5×C22⋊C4C22×C311yescharacteristic
95C2×C5×C22⋊C4C22×C312yes—
96C2×C5×C22⋊C4C22×C311yescharacteristic
97C2×C42×C5C22×C311yescharacteristic
98C2×C42×C5C22×C312yes—
99C5×#95C22×C312yes—
100C5×#95C22×C312yes—
101C5×#95C22×C312yes—
102C5×#95C22×C312yes—
103C5×#95C22×C312yes—
104#2376C3×C511yescharacteristic, direct product factor, Sylow
105C23×C3×C5—22no—
106C23×C3×C5—22no—
107C23×C3×C5—21no—
108C23×C3×C5—21no—
109C23×C3×C5—21no—
110C23×C3×C5—21no—
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5C22×C412yes—
114C23×C3×C5C22×C412yes—
115C23×C3×C5C2411yescharacteristic
116C3×C5×D4—44no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—21no—
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—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—44no—
136C2×C3×C4×C5—44no—
137C2×C3×C4×C5C22×C414yes—
138C2×C3×C4×C5C22×C414yes—
139C2×C3×C4×C5C22×C412yes—
140C22×C3×D4C22×C511yescharacteristic
141C23×C3×C4C22×C511yescharacteristic
142C23×C3×C4C22×C511yescharacteristic
143C3×#123C4×C518yescomplemented
144C2×C3×C4⋊C4C22×C512yes—
145C2×C3×C4⋊C4C22×C512yes—
146C2×C3×C4⋊C4C22×C512yes—
147C2×C3×C4⋊C4C22×C511yescharacteristic
148C2×C3×C4⋊C4C22×C512yes—
149C2×C3×C22⋊C4C22×C512yes—
150C2×C3×C22⋊C4C22×C512yes—
151C2×C3×C22⋊C4C22×C512yes—
152C2×C3×C22⋊C4C22×C511yescharacteristic
153C2×C3×C22⋊C4C22×C512yes—
154C2×C3×C22⋊C4C22×C511yescharacteristic
155C2×C3×C42C22×C511yescharacteristic
156C2×C3×C42C22×C512yes—
157C3×#95C22×C512yes—
158C3×#95C22×C512yes—
159C3×#95C22×C512yes—
160C3×#95C22×C512yes—
161C3×#95C22×C512yes—
162C24×C5C23×C311yescharacteristic
163C24×C5C23×C311yescharacteristic
164C2×C5×D4—41no—
165C2×C5×D4—41no—
166C2×C5×D4C2×C3×C414yes—
167C22×C4×C5—41no—
168C22×C4×C5—41no—
169C22×C4×C5—42no—
170C22×C4×C5—41no—
171C22×C4×C5C2×C3×C414yescomplemented
172C22×C4×C5C23×C312yes—
173C22×C4×C5C23×C312yes—
174C22×C4×C5C23×C312yes—
175C22×C4×C5C23×C312yes—
176C22×C4×C5C23×C312yes—
177C22×C4×C5C23×C311yescharacteristic
178C22×C4×C5C23×C312yes—
179C5×C4⋊C4—42no—
180C5×C4⋊C4—42no—
181C5×C4⋊C4—42no—
182C5×C4⋊C4—41no—
183C5×C4⋊C4C2×C3×C418yescomplemented
184C5×C22⋊C4—42no—
185C5×C22⋊C4—42no—
186C5×C22⋊C4—42no—
187C5×C22⋊C4—41no—
188C5×C22⋊C4C2×C3×C418yes—
189C5×C22⋊C4C2×C3×C414yescomplemented
190C42×C5—42no—
191C42×C5—41no—
192C2×#123C2×C3×C511yescharacteristic
193#390C2×C3×C511yescharacteristic
194#389C2×C3×C512yescomplemented
195#388C2×C3×C512yescomplemented
196#388C2×C3×C512yescomplemented
197#387C2×C3×C512yescomplemented
198#386C2×C3×C511yescharacteristic
199#385C2×C3×C512yescomplemented
200#380C2×C3×C512yescomplemented
201C22×C3×C5—22no—
202C22×C3×C5—22no—
203C22×C3×C5—24no—
204C22×C3×C5—24no—
205C22×C3×C5—41no—
206C22×C3×C5—41no—
207C22×C3×C5—42no—
208C22×C3×C5—42no—
209C22×C3×C5—41no—
210C22×C3×C5—41no—
211C22×C3×C5C23×C411yescharacteristic
212C22×C3×C5H(5,𝔽2)11yescharacteristic
213C22×C3×C5H(5,𝔽2)11yescharacteristic
214C22×C3×C5H(5,𝔽2)12yes—
215C22×C3×C52−1+412yes—
216C3×C4×C5—24no—
217C3×C4×C5—22no—
218C3×C4×C5—24no—
219C3×C4×C5—42no—
220C3×C4×C5—42no—
221C3×C4×C5—42no—
222C3×C4×C5—42no—
223C24×C3C23×C511yescharacteristic
224C24×C3C23×C511yescharacteristic
225C2×C3×D4—41no—
226C2×C3×D4—41no—
227C2×C3×D4C2×C4×C514yes—
228C22×C3×C4—41no—
229C22×C3×C4—41no—
230C22×C3×C4—42no—
231C22×C3×C4—41no—
232C22×C3×C4C2×C4×C514yescomplemented
233C22×C3×C4C23×C512yes—
234C22×C3×C4C23×C512yes—
235C22×C3×C4C23×C512yes—
236C22×C3×C4C23×C512yes—
237C22×C3×C4C23×C512yes—
238C22×C3×C4C23×C511yescharacteristic
239C22×C3×C4C23×C512yes—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4C2×C4×C518yescomplemented
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—42no—
248C3×C22⋊C4—41no—
249C3×C22⋊C4C2×C4×C518yes—
250C3×C22⋊C4C2×C4×C514yescomplemented
251C3×C42—42no—
252C3×C42—41no—
253C23×C5—22no—
254C23×C5—22no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—22no—
260C23×C5—22no—
261C23×C5C22×C3×C412yes—
262C23×C5C22×C3×C412yes—
263C23×C5C24×C311yescharacteristic
264C5×D4—44no—
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—22no—
272C2×C4×C5—22no—
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—44no—
284C2×C4×C5—44no—
285C2×C4×C5C22×C3×C414yes—
286C2×C4×C5C22×C3×C414yes—
287C2×C4×C5C22×C3×C412yes—
288C22×D4C22×C3×C511yescharacteristic
289C23×C4C22×C3×C511yescharacteristic
290C23×C4C22×C3×C511yescharacteristic
291#123C3×C4×C518yescomplemented
292C2×C4⋊C4C22×C3×C512yes—
293C2×C4⋊C4C22×C3×C512yes—
294C2×C4⋊C4C22×C3×C512yes—
295C2×C4⋊C4C22×C3×C511yescharacteristic
296C2×C4⋊C4C22×C3×C512yes—
297C2×C22⋊C4C22×C3×C512yes—
298C2×C22⋊C4C22×C3×C512yes—
299C2×C22⋊C4C22×C3×C512yes—
300C2×C22⋊C4C22×C3×C511yescharacteristic
301C2×C22⋊C4C22×C3×C512yes—
302C2×C22⋊C4C22×C3×C511yescharacteristic
303C2×C42C22×C3×C511yescharacteristic
304C2×C42C22×C3×C512yes—
305#95C22×C3×C512yes—
306#95C22×C3×C512yes—
307#95C22×C3×C512yes—
308#95C22×C3×C512yes—
309#95C22×C3×C512yes—
310C2×C3×C5—42no—
311C2×C3×C5—42no—
312C2×C3×C5#51811yescharacteristic
313C2×C3×C5#52012yes—
314C2×C3×C5#56011yescharacteristic
315C2×C3×C5#56212yes—
316C2×C3×C5#56311yescharacteristic
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3C22×C4×C512yes—
326C23×C3C22×C4×C512yes—
327C23×C3C24×C511yescharacteristic
328C3×D4—44no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
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—22no—
347C2×C3×C4—44no—
348C2×C3×C4—44no—
349C2×C3×C4C22×C4×C514yes—
350C2×C3×C4C22×C4×C514yes—
351C2×C3×C4C22×C4×C512yes—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—24no—
355C22×C5—24no—
356C22×C5—41no—
357C22×C5—41no—
358C22×C5—42no—
359C22×C5—42no—
360C22×C5—41no—
361C22×C5—41no—
362C22×C5C23×C3×C411yescharacteristic
363C22×C5C3×H(5,𝔽2)12yes—
364C22×C5C3×H(5,𝔽2)11yescharacteristic
365C22×C5C3×H(5,𝔽2)11yescharacteristic
366C22×C5C3×2−1+412yes—
367C4×C5—24no—
368C4×C5—22no—
369C4×C5—24no—
370C4×C5—42no—
371C4×C5—42no—
372C4×C5—42no—
373C4×C5—42no—
374C24C23×C3×C511yescharacteristic
375C24C23×C3×C511yescharacteristic
376C2×D4—41no—
377C2×D4—41no—
378C2×D4C2×C3×C4×C514yes—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4C2×C3×C4×C514yescomplemented
384C22×C4C23×C3×C512yes—
385C22×C4C23×C3×C512yes—
386C22×C4C23×C3×C512yes—
387C22×C4C23×C3×C512yes—
388C22×C4C23×C3×C512yes—
389C22×C4C23×C3×C511yescharacteristic
390C22×C4C23×C3×C512yes—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4C2×C3×C4×C518yescomplemented
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—41no—
400C22⋊C4C2×C3×C4×C518yes—
401C22⋊C4C2×C3×C4×C514yescomplemented
402C42—42no—
403C42—41no—
404C3×C5#237611yescharacteristic, direct product factor, Hall
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3C23×C4×C511yescharacteristic
416C22×C3C5×H(5,𝔽2)11yescharacteristic
417C22×C3C5×H(5,𝔽2)11yescharacteristic
418C22×C3C5×H(5,𝔽2)12yes—
419C22×C3C5×2−1+412yes—
420C3×C4—24no—
421C3×C4—22no—
422C3×C4—24no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—42no—
427C2×C5—42no—
428C2×C5—42no—
429C2×C5C3×#51811yescharacteristic
430C2×C5C3×#52012yes—
431C2×C5C3×#56011yescharacteristic
432C2×C5C3×#56212yes—
433C2×C5C3×#56311yescharacteristic
434C23—22no—
435C23—22no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23C22×C3×C4×C512yes—
443C23C22×C3×C4×C512yes—
444C23C24×C3×C511yescharacteristic
445D4—44no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
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—44no—
465C2×C4—44no—
466C2×C4C22×C3×C4×C514yes—
467C2×C4C22×C3×C4×C514yes—
468C2×C4C22×C3×C4×C512yes—
469C2×C3—42no—
470C2×C3—42no—
471C2×C3C5×#51811yescharacteristic
472C2×C3C5×#52012yes—
473C2×C3C5×#56011yescharacteristic
474C2×C3C5×#56212yes—
475C2×C3C5×#56311yescharacteristic
476C5C3×#237611yescharacteristic, direct product factor, Sylow
477C22—22no—
478C22—22no—
479C22—24no—
480C22—24no—
481C22—41no—
482C22—41no—
483C22—42no—
484C22—42no—
485C22—41no—
486C22—41no—
487C22C23×C3×C4×C511yescharacteristic
488C22C3×C5×H(5,𝔽2)11yescharacteristic
489C22C3×C5×H(5,𝔽2)11yescharacteristic
490C22C3×C5×H(5,𝔽2)12yes—
491C22C3×C5×2−1+412yes—
492C4—24no—
493C4—22no—
494C4—24no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C3C5×#237611yescharacteristic, direct product factor, Sylow
500C2—42no—
501C2—42no—
502C2C3×C5×#51811yescharacteristic
503C2C3×C5×#52012yes—
504C2C3×C5×#56011yescharacteristic
505C2C3×C5×#56212yes—
506C2C3×C5×#56311yescharacteristic
subgroupG/H
center, socleC23×C3×C5C24
commutatorC22C23×C3×C4×C5
Frattini, cosocleC23C24×C3×C5
prime radicalC22×C3×C5×D4C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C3×C5×#389C2semidirect
4. C3×C5×#388C2semidirect
5. C3×C5×#388C2semidirect
6. C3×C5×#387C2semidirect
8. C3×C5×#385C2semidirect
9. C3×C5×#380C2semidirect
10. C5×#2376C3direct (N × H)
14. C3×C5×#123C4semidirect
33. C3×#2376C5direct (N × H)
36. C5×#389C2×C3semidirect
37. C5×#388C2×C3semidirect
38. C5×#388C2×C3semidirect
39. C5×#387C2×C3semidirect
41. C5×#385C2×C3semidirect
42. C5×#380C2×C3semidirect
52. C22×C3×C4×C5C2×C4semidirect
64. C3×C5×C4⋊C4C2×C4semidirect
70. C3×C5×C22⋊C4C2×C4semidirect
75. C3×#389C2×C5semidirect
76. C3×#388C2×C5semidirect
77. C3×#388C2×C5semidirect
78. C3×#387C2×C5semidirect
80. C3×#385C2×C5semidirect
81. C3×#380C2×C5semidirect
85. C5×#123C3×C4semidirect
104. #2376C3×C5direct (N × H)
143. C3×#123C4×C5semidirect
171. C22×C4×C5C2×C3×C4semidirect
183. C5×C4⋊C4C2×C3×C4semidirect
189. C5×C22⋊C4C2×C3×C4semidirect
194. #389C2×C3×C5semidirect
195. #388C2×C3×C5semidirect
196. #388C2×C3×C5semidirect
197. #387C2×C3×C5semidirect
199. #385C2×C3×C5semidirect
200. #380C2×C3×C5semidirect
232. C22×C3×C4C2×C4×C5semidirect
244. C3×C4⋊C4C2×C4×C5semidirect
250. C3×C22⋊C4C2×C4×C5semidirect
291. #123C3×C4×C5semidirect
383. C22×C4C2×C3×C4×C5semidirect
395. C4⋊C4C2×C3×C4×C5semidirect
401. C22⋊C4C2×C3×C4×C5semidirect
404. C3×C5#2376direct (N × H)
476. C5C3×#2376direct (N × H)
499. C3C5×#2376direct (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×#1232
C3×C5×#3902
C3×C5×#3892
C3×C5×#3882
C3×C5×#3872
C3×C5×#3862
C3×C5×#3852
C3×C5×#3802
C5×#23763
C3×#23765