Finite Group Databasebeta

← C3×C5×#2377 #14275411

Subgroups — C3×C5×#2377 #14275411

Order 1920.

1152 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×#121C211yesmaximal, characteristic
2C3×C5×#390C212yesmaximal, complemented
3C3×C5×#388C212yesmaximal, complemented
4C3×C5×#387C212yesmaximal, complemented
5C3×C5×#386C212yesmaximal, complemented
6C3×C5×#386C212yesmaximal, complemented
7C3×C5×#385C212yesmaximal, complemented
8C3×C5×#381C211yesmaximal, characteristic, complemented
9C3×C5×#380C211yesmaximal, characteristic, complemented
10C5×#2377C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C22×C3×C5×D4C2212yes—
13C23×C3×C4×C5C2212yes—
14C23×C3×C4×C5C2211yescharacteristic
15C3×C5×#121C418yescomplemented
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C2×C3×C5×C22⋊C4C2211yescharacteristic
21C2×C3×C5×C22⋊C4C2212yes—
22C2×C3×C5×C22⋊C4C2212yes—
23C2×C3×C5×C22⋊C4C2212yes—
24C2×C3×C5×C22⋊C4C2212yes—
25C2×C3×C5×C22⋊C4C2212yes—
26C2×C3×C42×C5C2212yes—
27C3×C5×#95C2212yes—
28C3×C5×#95C2211yescharacteristic
29C3×C5×#95C2212yes—
30C3×C5×#95C2211yescharacteristic
31C3×C5×#95C2212yes—
32C3×C5×#95C2212yes—
33C3×#2377C511yesmaximal, characteristic, direct product factor, Hall
34C2×C5×#121C2×C311yescharacteristic
35C5×#390C2×C312yescomplemented
36C5×#388C2×C312yescomplemented
37C5×#387C2×C312yescomplemented
38C5×#386C2×C312yescomplemented
39C5×#386C2×C312yescomplemented
40C5×#385C2×C312yescomplemented
41C5×#381C2×C311yescharacteristic, complemented
42C5×#380C2×C311yescharacteristic, complemented
43C24×C3×C5C2311yescharacteristic
44C24×C3×C5C2312yes—
45C2×C3×C5×D4—42no—
46C2×C3×C5×D4—42no—
47C2×C3×C5×D4—42no—
48C2×C3×C5×D4C2×C414yes—
49C2×C3×C5×D4C2×C418yescomplemented
50C22×C3×C4×C5—42no—
51C22×C3×C4×C5—42no—
52C22×C3×C4×C5—42no—
53C22×C3×C4×C5—42no—
54C22×C3×C4×C5C2×C414yescomplemented
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—42no—
63C3×C5×C4⋊C4—42no—
64C3×C5×C4⋊C4C2×C414yescomplemented
65C3×C5×C22⋊C4—42no—
66C3×C5×C22⋊C4—41no—
67C3×C5×C22⋊C4—42no—
68C3×C5×C22⋊C4—42no—
69C3×C5×C22⋊C4—42no—
70C3×C5×C22⋊C4—41no—
71C3×C5×C22⋊C4C2×C418yescomplemented
72C3×C42×C5—42no—
73C2×C3×#121C2×C511yescharacteristic
74C3×#390C2×C512yescomplemented
75C3×#388C2×C512yescomplemented
76C3×#387C2×C512yescomplemented
77C3×#386C2×C512yescomplemented
78C3×#386C2×C512yescomplemented
79C3×#385C2×C512yescomplemented
80C3×#381C2×C511yescharacteristic, complemented
81C3×#380C2×C511yescharacteristic, complemented
82C22×C5×D4C22×C311yescharacteristic
83C22×C5×D4C22×C312yes—
84C23×C4×C5C22×C312yes—
85C23×C4×C5C22×C311yescharacteristic
86C5×#121C3×C418yescomplemented
87C2×C5×C4⋊C4C22×C312yes—
88C2×C5×C4⋊C4C22×C312yes—
89C2×C5×C4⋊C4C22×C311yescharacteristic
90C2×C5×C22⋊C4C22×C311yescharacteristic
91C2×C5×C22⋊C4C22×C311yescharacteristic
92C2×C5×C22⋊C4C22×C312yes—
93C2×C5×C22⋊C4C22×C312yes—
94C2×C5×C22⋊C4C22×C312yes—
95C2×C5×C22⋊C4C22×C312yes—
96C2×C5×C22⋊C4C22×C312yes—
97C2×C42×C5C22×C312yes—
98C5×#95C22×C312yes—
99C5×#95C22×C311yescharacteristic
100C5×#95C22×C312yes—
101C5×#95C22×C311yescharacteristic
102C5×#95C22×C312yes—
103C5×#95C22×C312yes—
104#2377C3×C511yescharacteristic, direct product factor, Sylow
105C23×C3×C5—21no—
106C23×C3×C5—22no—
107C23×C3×C5—22no—
108C23×C3×C5—22no—
109C23×C3×C5—22no—
110C23×C3×C5—22no—
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5—21no—
114C23×C3×C5—22no—
115C23×C3×C5C22×C414yes—
116C23×C3×C5C22×C412yes—
117C23×C3×C5C2411yescharacteristic
118C3×C5×D4—44no—
119C3×C5×D4—48no—
120C2×C3×C4×C5—22no—
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—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—44no—
137C2×C3×C4×C5—48no—
138C2×C3×C4×C5C22×C412yes—
139C2×C3×C4×C5C22×C414yes—
140C2×C3×C4×C5C22×C412yes—
141C22×C3×D4C22×C511yescharacteristic
142C22×C3×D4C22×C512yes—
143C23×C3×C4C22×C512yes—
144C23×C3×C4C22×C511yescharacteristic
145C3×#121C4×C518yescomplemented
146C2×C3×C4⋊C4C22×C512yes—
147C2×C3×C4⋊C4C22×C512yes—
148C2×C3×C4⋊C4C22×C511yescharacteristic
149C2×C3×C22⋊C4C22×C511yescharacteristic
150C2×C3×C22⋊C4C22×C511yescharacteristic
151C2×C3×C22⋊C4C22×C512yes—
152C2×C3×C22⋊C4C22×C512yes—
153C2×C3×C22⋊C4C22×C512yes—
154C2×C3×C22⋊C4C22×C512yes—
155C2×C3×C22⋊C4C22×C512yes—
156C2×C3×C42C22×C512yes—
157C3×#95C22×C512yes—
158C3×#95C22×C511yescharacteristic
159C3×#95C22×C512yes—
160C3×#95C22×C511yescharacteristic
161C3×#95C22×C512yes—
162C3×#95C22×C512yes—
163C24×C5C23×C311yescharacteristic
164C24×C5C23×C312yes—
165C2×C5×D4—42no—
166C2×C5×D4—42no—
167C2×C5×D4—42no—
168C2×C5×D4C2×C3×C414yes—
169C2×C5×D4C2×C3×C418yescomplemented
170C22×C4×C5—42no—
171C22×C4×C5—42no—
172C22×C4×C5—42no—
173C22×C4×C5—42no—
174C22×C4×C5C2×C3×C414yescomplemented
175C22×C4×C5C23×C312yes—
176C22×C4×C5C23×C312yes—
177C22×C4×C5C23×C312yes—
178C22×C4×C5C23×C312yes—
179C22×C4×C5C23×C312yes—
180C22×C4×C5C23×C311yescharacteristic
181C22×C4×C5C23×C311yescharacteristic
182C5×C4⋊C4—42no—
183C5×C4⋊C4—42no—
184C5×C4⋊C4C2×C3×C414yescomplemented
185C5×C22⋊C4—42no—
186C5×C22⋊C4—41no—
187C5×C22⋊C4—42no—
188C5×C22⋊C4—42no—
189C5×C22⋊C4—42no—
190C5×C22⋊C4—41no—
191C5×C22⋊C4C2×C3×C418yescomplemented
192C42×C5—42no—
193C2×#121C2×C3×C511yescharacteristic
194#390C2×C3×C512yescomplemented
195#388C2×C3×C512yescomplemented
196#387C2×C3×C512yescomplemented
197#386C2×C3×C512yescomplemented
198#386C2×C3×C512yescomplemented
199#385C2×C3×C512yescomplemented
200#381C2×C3×C511yescharacteristic, complemented
201#380C2×C3×C511yescharacteristic, complemented
202C22×C3×C5—22no—
203C22×C3×C5—22no—
204C22×C3×C5—24no—
205C22×C3×C5—24no—
206C22×C3×C5—22no—
207C22×C3×C5—24no—
208C22×C3×C5—42no—
209C22×C3×C5—42no—
210C22×C3×C5—42no—
211C22×C3×C5—42no—
212C22×C3×C5—42no—
213C22×C3×C5—42no—
214C22×C3×C5C23×C411yescharacteristic
215C22×C3×C5H(5,𝔽2)12yes—
216C22×C3×C5H(5,𝔽2)12yes—
217C22×C3×C5H(5,𝔽2)11yescharacteristic
218C22×C3×C52−1+411yescharacteristic
219C3×C4×C5—22no—
220C3×C4×C5—22no—
221C3×C4×C5—24no—
222C3×C4×C5—42no—
223C3×C4×C5—42no—
224C3×C4×C5—42no—
225C3×C4×C5—42no—
226C24×C3C23×C511yescharacteristic
227C24×C3C23×C512yes—
228C2×C3×D4—42no—
229C2×C3×D4—42no—
230C2×C3×D4—42no—
231C2×C3×D4C2×C4×C514yes—
232C2×C3×D4C2×C4×C518yescomplemented
233C22×C3×C4—42no—
234C22×C3×C4—42no—
235C22×C3×C4—42no—
236C22×C3×C4—42no—
237C22×C3×C4C2×C4×C514yescomplemented
238C22×C3×C4C23×C512yes—
239C22×C3×C4C23×C512yes—
240C22×C3×C4C23×C512yes—
241C22×C3×C4C23×C512yes—
242C22×C3×C4C23×C511yescharacteristic
243C22×C3×C4C23×C512yes—
244C22×C3×C4C23×C511yescharacteristic
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4C2×C4×C514yescomplemented
248C3×C22⋊C4—42no—
249C3×C22⋊C4—41no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—41no—
254C3×C22⋊C4C2×C4×C518yescomplemented
255C3×C42—42no—
256C23×C5—21no—
257C23×C5—22no—
258C23×C5—22no—
259C23×C5—22no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5—22no—
263C23×C5—22no—
264C23×C5—21no—
265C23×C5—22no—
266C23×C5C22×C3×C414yes—
267C23×C5C22×C3×C412yes—
268C23×C5C24×C311yescharacteristic
269C5×D4—44no—
270C5×D4—48no—
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—22no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—44no—
288C2×C4×C5—48no—
289C2×C4×C5C22×C3×C412yes—
290C2×C4×C5C22×C3×C414yes—
291C2×C4×C5C22×C3×C412yes—
292C22×D4C22×C3×C511yescharacteristic
293C22×D4C22×C3×C512yes—
294C23×C4C22×C3×C512yes—
295C23×C4C22×C3×C511yescharacteristic
296#121C3×C4×C518yescomplemented
297C2×C4⋊C4C22×C3×C512yes—
298C2×C4⋊C4C22×C3×C512yes—
299C2×C4⋊C4C22×C3×C511yescharacteristic
300C2×C22⋊C4C22×C3×C511yescharacteristic
301C2×C22⋊C4C22×C3×C511yescharacteristic
302C2×C22⋊C4C22×C3×C512yes—
303C2×C22⋊C4C22×C3×C512yes—
304C2×C22⋊C4C22×C3×C512yes—
305C2×C22⋊C4C22×C3×C512yes—
306C2×C22⋊C4C22×C3×C512yes—
307C2×C42C22×C3×C512yes—
308#95C22×C3×C512yes—
309#95C22×C3×C511yescharacteristic
310#95C22×C3×C512yes—
311#95C22×C3×C511yescharacteristic
312#95C22×C3×C512yes—
313#95C22×C3×C512yes—
314C2×C3×C5—42no—
315C2×C3×C5—44no—
316C2×C3×C5#51811yescharacteristic
317C2×C3×C5#51811yescharacteristic
318C2×C3×C5#52011yescharacteristic
319C2×C3×C5#56012yes—
320C2×C3×C5#56212yes—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—21no—
330C23×C3—22no—
331C23×C3C22×C4×C514yes—
332C23×C3C22×C4×C512yes—
333C23×C3C24×C511yescharacteristic
334C3×D4—44no—
335C3×D4—48no—
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—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—44no—
353C2×C3×C4—48no—
354C2×C3×C4C22×C4×C512yes—
355C2×C3×C4C22×C4×C514yes—
356C2×C3×C4C22×C4×C512yes—
357C22×C5—22no—
358C22×C5—22no—
359C22×C5—24no—
360C22×C5—24no—
361C22×C5—22no—
362C22×C5—24no—
363C22×C5—42no—
364C22×C5—42no—
365C22×C5—42no—
366C22×C5—42no—
367C22×C5—42no—
368C22×C5—42no—
369C22×C5C23×C3×C411yescharacteristic
370C22×C5C3×H(5,𝔽2)11yescharacteristic
371C22×C5C3×H(5,𝔽2)12yes—
372C22×C5C3×H(5,𝔽2)12yes—
373C22×C5C3×2−1+411yescharacteristic
374C4×C5—22no—
375C4×C5—22no—
376C4×C5—24no—
377C4×C5—42no—
378C4×C5—42no—
379C4×C5—42no—
380C4×C5—42no—
381C24C23×C3×C511yescharacteristic
382C24C23×C3×C512yes—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4C2×C3×C4×C514yes—
387C2×D4C2×C3×C4×C518yescomplemented
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C3×C4×C514yescomplemented
393C22×C4C23×C3×C512yes—
394C22×C4C23×C3×C512yes—
395C22×C4C23×C3×C512yes—
396C22×C4C23×C3×C512yes—
397C22×C4C23×C3×C511yescharacteristic
398C22×C4C23×C3×C512yes—
399C22×C4C23×C3×C511yescharacteristic
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4C2×C3×C4×C514yescomplemented
403C22⋊C4—42no—
404C22⋊C4—41no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—41no—
409C22⋊C4C2×C3×C4×C518yescomplemented
410C42—42no—
411C3×C5#237711yescharacteristic, direct product factor, Hall
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—22no—
417C22×C3—24no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3C23×C4×C511yescharacteristic
425C22×C3C5×H(5,𝔽2)12yes—
426C22×C3C5×H(5,𝔽2)12yes—
427C22×C3C5×H(5,𝔽2)11yescharacteristic
428C22×C3C5×2−1+411yescharacteristic
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—24no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C2×C5—42no—
437C2×C5—44no—
438C2×C5C3×#51811yescharacteristic
439C2×C5C3×#51811yescharacteristic
440C2×C5C3×#52011yescharacteristic
441C2×C5C3×#56012yes—
442C2×C5C3×#56212yes—
443C23—21no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—21no—
452C23—22no—
453C23C22×C3×C4×C514yes—
454C23C22×C3×C4×C512yes—
455C23C24×C3×C511yescharacteristic
456D4—44no—
457D4—48no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
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—44no—
475C2×C4—48no—
476C2×C4C22×C3×C4×C512yes—
477C2×C4C22×C3×C4×C514yes—
478C2×C4C22×C3×C4×C512yes—
479C2×C3—42no—
480C2×C3—44no—
481C2×C3C5×#51811yescharacteristic
482C2×C3C5×#51811yescharacteristic
483C2×C3C5×#52011yescharacteristic
484C2×C3C5×#56012yes—
485C2×C3C5×#56212yes—
486C5C3×#237711yescharacteristic, direct product factor, Sylow
487C22—22no—
488C22—22no—
489C22—24no—
490C22—24no—
491C22—22no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22C23×C3×C4×C511yescharacteristic
500C22C3×C5×H(5,𝔽2)12yes—
501C22C3×C5×H(5,𝔽2)12yes—
502C22C3×C5×H(5,𝔽2)11yescharacteristic
503C22C3×C5×2−1+411yescharacteristic
504C4—22no—
505C4—22no—
506C4—24no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C3C5×#237711yescharacteristic, direct product factor, Sylow
512C2—42no—
513C2—44no—
514C2C3×C5×#51811yescharacteristic
515C2C3×C5×#51811yescharacteristic
516C2C3×C5×#52011yescharacteristic
517C2C3×C5×#56012yes—
518C2C3×C5×#56212yes—
subgroupG/H
center, socleC23×C3×C5C24
commutatorC22C23×C3×C4×C5
Frattini, cosocleC23C24×C3×C5
prime radicalC2×C3×C5×#121C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C5×#390C2semidirect
3. C3×C5×#388C2semidirect
4. C3×C5×#387C2semidirect
5. C3×C5×#386C2semidirect
6. C3×C5×#386C2semidirect
7. C3×C5×#385C2semidirect
8. C3×C5×#381C2semidirect
9. C3×C5×#380C2semidirect
10. C5×#2377C3direct (N × H)
15. C3×C5×#121C4semidirect
33. C3×#2377C5direct (N × H)
35. C5×#390C2×C3semidirect
36. C5×#388C2×C3semidirect
37. C5×#387C2×C3semidirect
38. C5×#386C2×C3semidirect
39. C5×#386C2×C3semidirect
40. C5×#385C2×C3semidirect
41. C5×#381C2×C3semidirect
42. C5×#380C2×C3semidirect
49. C2×C3×C5×D4C2×C4semidirect
54. C22×C3×C4×C5C2×C4semidirect
64. C3×C5×C4⋊C4C2×C4semidirect
71. C3×C5×C22⋊C4C2×C4semidirect
74. C3×#390C2×C5semidirect
75. C3×#388C2×C5semidirect
76. C3×#387C2×C5semidirect
77. C3×#386C2×C5semidirect
78. C3×#386C2×C5semidirect
79. C3×#385C2×C5semidirect
80. C3×#381C2×C5semidirect
81. C3×#380C2×C5semidirect
86. C5×#121C3×C4semidirect
104. #2377C3×C5direct (N × H)
145. C3×#121C4×C5semidirect
169. C2×C5×D4C2×C3×C4semidirect
174. C22×C4×C5C2×C3×C4semidirect
184. C5×C4⋊C4C2×C3×C4semidirect
191. C5×C22⋊C4C2×C3×C4semidirect
194. #390C2×C3×C5semidirect
195. #388C2×C3×C5semidirect
196. #387C2×C3×C5semidirect
197. #386C2×C3×C5semidirect
198. #386C2×C3×C5semidirect
199. #385C2×C3×C5semidirect
200. #381C2×C3×C5semidirect
201. #380C2×C3×C5semidirect
232. C2×C3×D4C2×C4×C5semidirect
237. C22×C3×C4C2×C4×C5semidirect
247. C3×C4⋊C4C2×C4×C5semidirect
254. C3×C22⋊C4C2×C4×C5semidirect
296. #121C3×C4×C5semidirect
387. C2×D4C2×C3×C4×C5semidirect
392. C22×C4C2×C3×C4×C5semidirect
402. C4⋊C4C2×C3×C4×C5semidirect
409. C22⋊C4C2×C3×C4×C5semidirect
411. C3×C5#2377direct (N × H)
486. C5C3×#2377direct (N × H)
511. C3C5×#2377direct (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×#1212
C3×C5×#3902
C3×C5×#3882
C3×C5×#3872
C3×C5×#3862
C3×C5×#3852
C3×C5×#3812
C3×C5×#3802
C5×#23773
C3×#23775