Finite Group Databasebeta

← C2×C3×C5×#387 #14275322

Subgroups — C2×C3×C5×#387 #14275322

Order 1920.

1424 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
1C22×C3×C5×C4⋊C4C211yesmaximal, characteristic
2C22×C3×C5×C4⋊C4C211yesmaximal, characteristic
3C22×C3×C42×C5C211yesmaximal, characteristic
4C3×C5×#387C218yesmaximal, direct product factor
5C2×C3×C5×#95C211yesmaximal, characteristic
6C2×C3×C5×#95C211yesmaximal, characteristic
7C2×C3×C5×#95C211yesmaximal, characteristic
8C2×C3×C5×#95C211yesmaximal, characteristic
9C2×C5×#387C311yesmaximal, characteristic, direct product factor, Hall
10C23×C3×C4×C5C2211yescharacteristic
11C23×C3×C4×C5C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C23×C3×C4×C5C2211yescharacteristic
16C23×C3×C4×C5C2211yescharacteristic
17C2×C3×C5×C4⋊C4—24no—
18C2×C3×C5×C4⋊C4C418yescomplemented
19C2×C3×C5×C4⋊C4C2214yes—
20C2×C3×C5×C4⋊C4C2214yes—
21C2×C3×C42×C5—24no—
22C2×C3×C42×C5C2214yes—
23C3×C5×#95C2214yes—
24C3×C5×#95C2214yes—
25C3×C5×#95C2214yes—
26C3×C5×#95C2214yes—
27C2×C3×#387C511yesmaximal, characteristic, direct product factor, Hall
28C22×C5×C4⋊C4C2×C311yescharacteristic
29C22×C5×C4⋊C4C2×C311yescharacteristic
30C22×C42×C5C2×C311yescharacteristic
31C5×#387C2×C318yesdirect product factor
32C2×C5×#95C2×C311yescharacteristic
33C2×C5×#95C2×C311yescharacteristic
34C2×C5×#95C2×C311yescharacteristic
35C2×C5×#95C2×C311yescharacteristic
36C24×C3×C5C2311yescharacteristic
37C22×C3×C4×C5—22no—
38C22×C3×C4×C5—22no—
39C22×C3×C4×C5—22no—
40C22×C3×C4×C5—22no—
41C22×C3×C4×C5—22no—
42C22×C3×C4×C5—22no—
43C22×C3×C4×C5—22no—
44C22×C3×C4×C5—22no—
45C22×C3×C4×C5—22no—
46C22×C3×C4×C5—22no—
47C22×C3×C4×C5—22no—
48C22×C3×C4×C5—22no—
49C22×C3×C4×C5—22no—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—22no—
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5C2×C414yes—
54C22×C3×C4×C5C2×C414yes—
55C22×C3×C4×C5C2×C414yes—
56C22×C3×C4×C5D414yes—
57C22×C3×C4×C5Q814yes—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2312yes—
61C22×C3×C4×C5C2312yes—
62C22×C3×C4×C5C2312yes—
63C22×C3×C4×C5C2312yes—
64C22×C3×C4×C5C2312yes—
65C3×C5×C4⋊C4—28no—
66C3×C5×C4⋊C4C2×C4116yescomplemented
67C3×C42×C5—28no—
68C22×C3×C4⋊C4C2×C511yescharacteristic
69C22×C3×C4⋊C4C2×C511yescharacteristic
70C22×C3×C42C2×C511yescharacteristic
71C3×#387C2×C518yesdirect product factor
72C2×C3×#95C2×C511yescharacteristic
73C2×C3×#95C2×C511yescharacteristic
74C2×C3×#95C2×C511yescharacteristic
75C2×C3×#95C2×C511yescharacteristic
76C23×C4×C5C22×C311yescharacteristic
77C23×C4×C5C22×C311yescharacteristic
78C23×C4×C5C22×C311yescharacteristic
79C23×C4×C5C22×C311yescharacteristic
80C23×C4×C5C22×C311yescharacteristic
81C23×C4×C5C22×C311yescharacteristic
82C23×C4×C5C22×C311yescharacteristic
83C2×C5×C4⋊C4—24no—
84C2×C5×C4⋊C4C3×C418yescomplemented
85C2×C5×C4⋊C4C22×C314yes—
86C2×C5×C4⋊C4C22×C314yes—
87C2×C42×C5—24no—
88C2×C42×C5C22×C314yes—
89C5×#95C22×C314yes—
90C5×#95C22×C314yes—
91C5×#95C22×C314yes—
92C5×#95C22×C314yes—
93C2×#387C3×C511yescharacteristic, direct product factor, Sylow
94C23×C3×C5C22×C412yes—
95C23×C3×C5C2×D412yes—
96C23×C3×C5C2×Q812yes—
97C23×C3×C5#4112yes—
98C23×C3×C5#4112yes—
99C23×C3×C5#4112yes—
100C23×C3×C5#4112yes—
101C23×C3×C5C2411yescharacteristic
102C2×C3×C4×C5—28no—
103C2×C3×C4×C5—28no—
104C2×C3×C4×C5—28no—
105C2×C3×C4×C5—28no—
106C2×C3×C4×C5—28no—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5—22no—
110C2×C3×C4×C5—22no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—44no—
124C2×C3×C4×C5—44no—
125C2×C3×C4×C5C22×C414yes—
126C2×C3×C4×C5C22×C414yes—
127C2×C3×C4×C5C22×C414yes—
128C2×C3×C4×C5C2×D414yes—
129C2×C3×C4×C5C2×Q814yes—
130C23×C3×C4C22×C511yescharacteristic
131C23×C3×C4C22×C511yescharacteristic
132C23×C3×C4C22×C511yescharacteristic
133C23×C3×C4C22×C511yescharacteristic
134C23×C3×C4C22×C511yescharacteristic
135C23×C3×C4C22×C511yescharacteristic
136C23×C3×C4C22×C511yescharacteristic
137C2×C3×C4⋊C4—24no—
138C2×C3×C4⋊C4C4×C518yescomplemented
139C2×C3×C4⋊C4C22×C514yes—
140C2×C3×C4⋊C4C22×C514yes—
141C2×C3×C42—24no—
142C2×C3×C42C22×C514yes—
143C3×#95C22×C514yes—
144C3×#95C22×C514yes—
145C3×#95C22×C514yes—
146C3×#95C22×C514yes—
147C24×C5C23×C311yescharacteristic
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—22no—
151C22×C4×C5—22no—
152C22×C4×C5—22no—
153C22×C4×C5—22no—
154C22×C4×C5—22no—
155C22×C4×C5—22no—
156C22×C4×C5—22no—
157C22×C4×C5—22no—
158C22×C4×C5—22no—
159C22×C4×C5—22no—
160C22×C4×C5—22no—
161C22×C4×C5—22no—
162C22×C4×C5—22no—
163C22×C4×C5—22no—
164C22×C4×C5C2×C3×C414yes—
165C22×C4×C5C2×C3×C414yes—
166C22×C4×C5C2×C3×C414yes—
167C22×C4×C5C3×D414yes—
168C22×C4×C5C3×Q814yes—
169C22×C4×C5C23×C312yes—
170C22×C4×C5C23×C312yes—
171C22×C4×C5C23×C312yes—
172C22×C4×C5C23×C312yes—
173C22×C4×C5C23×C312yes—
174C22×C4×C5C23×C312yes—
175C22×C4×C5C23×C312yes—
176C5×C4⋊C4—28no—
177C5×C4⋊C4C2×C3×C4116yescomplemented
178C42×C5—28no—
179C22×C4⋊C4C2×C3×C511yescharacteristic
180C22×C4⋊C4C2×C3×C511yescharacteristic
181C22×C42C2×C3×C511yescharacteristic
182#387C2×C3×C518yesdirect product factor
183C2×#95C2×C3×C511yescharacteristic
184C2×#95C2×C3×C511yescharacteristic
185C2×#95C2×C3×C511yescharacteristic
186C2×#95C2×C3×C511yescharacteristic
187C22×C3×C5#11714yes—
188C22×C3×C5C4×D414yes—
189C22×C3×C5C4×Q814yes—
190C22×C3×C5C22⋊Q814yes—
191C22×C3×C5#12314yes—
192C22×C3×C5#12514yes—
193C22×C3×C5#12614yes—
194C22×C3×C5C23×C411yescharacteristic
195C22×C3×C5C22×D411yescharacteristic
196C22×C3×C5C22×Q811yescharacteristic
197C22×C3×C5C2×#4111yescharacteristic
198C22×C3×C5C2×#4111yescharacteristic
199C22×C3×C5C2×#4111yescharacteristic
200C22×C3×C5C2×#4111yescharacteristic
201C3×C4×C5—24no—
202C3×C4×C5—24no—
203C3×C4×C5—24no—
204C3×C4×C5—24no—
205C3×C4×C5—24no—
206C3×C4×C5—42no—
207C3×C4×C5—42no—
208C24×C3C23×C511yescharacteristic
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—22no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4—22no—
224C22×C3×C4—22no—
225C22×C3×C4C2×C4×C514yes—
226C22×C3×C4C2×C4×C514yes—
227C22×C3×C4C2×C4×C514yes—
228C22×C3×C4C5×D414yes—
229C22×C3×C4C5×Q814yes—
230C22×C3×C4C23×C512yes—
231C22×C3×C4C23×C512yes—
232C22×C3×C4C23×C512yes—
233C22×C3×C4C23×C512yes—
234C22×C3×C4C23×C512yes—
235C22×C3×C4C23×C512yes—
236C22×C3×C4C23×C512yes—
237C3×C4⋊C4—28no—
238C3×C4⋊C4C2×C4×C5116yescomplemented
239C3×C42—28no—
240C23×C5C22×C3×C412yes—
241C23×C5C2×C3×D412yes—
242C23×C5C2×C3×Q812yes—
243C23×C5C3×#4112yes—
244C23×C5C3×#4112yes—
245C23×C5C3×#4112yes—
246C23×C5C3×#4112yes—
247C23×C5C24×C311yescharacteristic
248C2×C4×C5—28no—
249C2×C4×C5—28no—
250C2×C4×C5—28no—
251C2×C4×C5—28no—
252C2×C4×C5—28no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—44no—
270C2×C4×C5—44no—
271C2×C4×C5C22×C3×C414yes—
272C2×C4×C5C22×C3×C414yes—
273C2×C4×C5C22×C3×C414yes—
274C2×C4×C5C2×C3×D414yes—
275C2×C4×C5C2×C3×Q814yes—
276C23×C4C22×C3×C511yescharacteristic
277C23×C4C22×C3×C511yescharacteristic
278C23×C4C22×C3×C511yescharacteristic
279C23×C4C22×C3×C511yescharacteristic
280C23×C4C22×C3×C511yescharacteristic
281C23×C4C22×C3×C511yescharacteristic
282C23×C4C22×C3×C511yescharacteristic
283C2×C4⋊C4—24no—
284C2×C4⋊C4C3×C4×C518yescomplemented
285C2×C4⋊C4C22×C3×C514yes—
286C2×C4⋊C4C22×C3×C514yes—
287C2×C42—24no—
288C2×C42C22×C3×C514yes—
289#95C22×C3×C514yes—
290#95C22×C3×C514yes—
291#95C22×C3×C514yes—
292#95C22×C3×C514yes—
293C2×C3×C5#38718yesdirect product factor
294C2×C3×C5C2×#11711yescharacteristic
295C2×C3×C5C2×C4×D411yescharacteristic
296C2×C3×C5C2×C4×Q811yescharacteristic
297C2×C3×C5C2×C22⋊Q811yescharacteristic
298C2×C3×C5C2×#12311yescharacteristic
299C2×C3×C5C2×#12511yescharacteristic
300C2×C3×C5C2×#12611yescharacteristic
301C23×C3C22×C4×C512yes—
302C23×C3C2×C5×D412yes—
303C23×C3C2×C5×Q812yes—
304C23×C3C5×#4112yes—
305C23×C3C5×#4112yes—
306C23×C3C5×#4112yes—
307C23×C3C5×#4112yes—
308C23×C3C24×C511yescharacteristic
309C2×C3×C4—28no—
310C2×C3×C4—28no—
311C2×C3×C4—28no—
312C2×C3×C4—28no—
313C2×C3×C4—28no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—44no—
331C2×C3×C4—44no—
332C2×C3×C4C22×C4×C514yes—
333C2×C3×C4C22×C4×C514yes—
334C2×C3×C4C22×C4×C514yes—
335C2×C3×C4C2×C5×D414yes—
336C2×C3×C4C2×C5×Q814yes—
337C22×C5C3×#11714yes—
338C22×C5C3×C4×D414yes—
339C22×C5C3×C4×Q814yes—
340C22×C5C3×C22⋊Q814yes—
341C22×C5C3×#12314yes—
342C22×C5C3×#12514yes—
343C22×C5C3×#12614yes—
344C22×C5C23×C3×C411yescharacteristic
345C22×C5C22×C3×D411yescharacteristic
346C22×C5C22×C3×Q811yescharacteristic
347C22×C5C2×C3×#4111yescharacteristic
348C22×C5C2×C3×#4111yescharacteristic
349C22×C5C2×C3×#4111yescharacteristic
350C22×C5C2×C3×#4111yescharacteristic
351C4×C5—24no—
352C4×C5—24no—
353C4×C5—24no—
354C4×C5—24no—
355C4×C5—24no—
356C4×C5—42no—
357C4×C5—42no—
358C24C23×C3×C511yescharacteristic
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4C2×C3×C4×C514yes—
376C22×C4C2×C3×C4×C514yes—
377C22×C4C2×C3×C4×C514yes—
378C22×C4C3×C5×D414yes—
379C22×C4C3×C5×Q814yes—
380C22×C4C23×C3×C512yes—
381C22×C4C23×C3×C512yes—
382C22×C4C23×C3×C512yes—
383C22×C4C23×C3×C512yes—
384C22×C4C23×C3×C512yes—
385C22×C4C23×C3×C512yes—
386C22×C4C23×C3×C512yes—
387C4⋊C4—28no—
388C4⋊C4C2×C3×C4×C5116yescomplemented
389C42—28no—
390C3×C5C2×#38711yescharacteristic, direct product factor, Hall
391C22×C3C5×#11714yes—
392C22×C3C4×C5×D414yes—
393C22×C3C4×C5×Q814yes—
394C22×C3C5×C22⋊Q814yes—
395C22×C3C5×#12314yes—
396C22×C3C5×#12514yes—
397C22×C3C5×#12614yes—
398C22×C3C23×C4×C511yescharacteristic
399C22×C3C22×C5×D411yescharacteristic
400C22×C3C22×C5×Q811yescharacteristic
401C22×C3C2×C5×#4111yescharacteristic
402C22×C3C2×C5×#4111yescharacteristic
403C22×C3C2×C5×#4111yescharacteristic
404C22×C3C2×C5×#4111yescharacteristic
405C3×C4—24no—
406C3×C4—24no—
407C3×C4—24no—
408C3×C4—24no—
409C3×C4—24no—
410C3×C4—42no—
411C3×C4—42no—
412C2×C5C3×#38718yesdirect product factor
413C2×C5C2×C3×#11711yescharacteristic
414C2×C5C2×C3×C4×D411yescharacteristic
415C2×C5C2×C3×C4×Q811yescharacteristic
416C2×C5C2×C3×C22⋊Q811yescharacteristic
417C2×C5C2×C3×#12311yescharacteristic
418C2×C5C2×C3×#12511yescharacteristic
419C2×C5C2×C3×#12611yescharacteristic
420C23C22×C3×C4×C512yes—
421C23C2×C3×C5×D412yes—
422C23C2×C3×C5×Q812yes—
423C23C3×C5×#4112yes—
424C23C3×C5×#4112yes—
425C23C3×C5×#4112yes—
426C23C3×C5×#4112yes—
427C23C24×C3×C511yescharacteristic
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—28no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—28no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4C22×C3×C4×C514yes—
452C2×C4C22×C3×C4×C514yes—
453C2×C4C22×C3×C4×C514yes—
454C2×C4C2×C3×C5×D414yes—
455C2×C4C2×C3×C5×Q814yes—
456C2×C3C5×#38718yesdirect product factor
457C2×C3C2×C5×#11711yescharacteristic
458C2×C3C2×C4×C5×D411yescharacteristic
459C2×C3C2×C4×C5×Q811yescharacteristic
460C2×C3C2×C5×C22⋊Q811yescharacteristic
461C2×C3C2×C5×#12311yescharacteristic
462C2×C3C2×C5×#12511yescharacteristic
463C2×C3C2×C5×#12611yescharacteristic
464C5C2×C3×#38711yescharacteristic, direct product factor, Sylow
465C22C3×C5×#11714yes—
466C22C3×C4×C5×D414yes—
467C22C3×C4×C5×Q814yes—
468C22C3×C5×C22⋊Q814yes—
469C22C3×C5×#12314yes—
470C22C3×C5×#12514yes—
471C22C3×C5×#12614yes—
472C22C23×C3×C4×C511yescharacteristic
473C22C22×C3×C5×D411yescharacteristic
474C22C22×C3×C5×Q811yescharacteristic
475C22C2×C3×C5×#4111yescharacteristic
476C22C2×C3×C5×#4111yescharacteristic
477C22C2×C3×C5×#4111yescharacteristic
478C22C2×C3×C5×#4111yescharacteristic
479C4—24no—
480C4—24no—
481C4—24no—
482C4—24no—
483C4—24no—
484C4—42no—
485C4—42no—
486C3C2×C5×#38711yescharacteristic, direct product factor, Sylow
487C2C3×C5×#38718yesdirect product factor
488C2C2×C3×C5×#11711yescharacteristic
489C2C2×C3×C4×C5×D411yescharacteristic
490C2C2×C3×C4×C5×Q811yescharacteristic
491C2C2×C3×C5×C22⋊Q811yescharacteristic
492C2C2×C3×C5×#12311yescharacteristic
493C2C2×C3×C5×#12511yescharacteristic
494C2C2×C3×C5×#12611yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3×C5C23
commutatorC22C23×C3×C4×C5
Frattini, cosocleC23C24×C3×C5
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×C5×#387C2direct (N × H)
9. C2×C5×#387C3direct (N × H)
18. C2×C3×C5×C4⋊C4C4semidirect
27. C2×C3×#387C5direct (N × H)
31. C5×#387C2×C3direct (N × H)
66. C3×C5×C4⋊C4C2×C4semidirect
71. C3×#387C2×C5direct (N × H)
84. C2×C5×C4⋊C4C3×C4semidirect
93. C2×#387C3×C5direct (N × H)
138. C2×C3×C4⋊C4C4×C5semidirect
177. C5×C4⋊C4C2×C3×C4semidirect
182. #387C2×C3×C5direct (N × H)
238. C3×C4⋊C4C2×C4×C5semidirect
284. C2×C4⋊C4C3×C4×C5semidirect
293. C2×C3×C5#387direct (N × H)
388. C4⋊C4C2×C3×C4×C5semidirect
390. C3×C5C2×#387direct (N × H)
412. C2×C5C3×#387direct (N × H)
456. C2×C3C5×#387direct (N × H)
464. C5C2×C3×#387direct (N × H)
486. C3C2×C5×#387direct (N × H)
487. C2C3×C5×#387direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×C5×C4⋊C42
C22×C3×C42×C52
C3×C5×#3872
C2×C3×C5×#952
C2×C5×#3873
C2×C3×#3875