Finite Group Databasebeta

← C3×C5×#2731 #14275765

Subgroups — C3×C5×#2731 #14275765

Order 1920.

984 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×C5×#124C211yesmaximal, characteristic
3C3×C5×#396C212yesmaximal, complemented
4C3×C5×#394C212yesmaximal
5C3×C5×#393C212yesmaximal, complemented
6C3×C5×#391C211yesmaximal, characteristic, complemented
7C3×C5×#388C212yesmaximal, complemented
8C3×C5×#388C212yesmaximal, complemented
9C3×C5×#387C212yesmaximal, complemented
10C5×#2731C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C3×C5×Q8C2211yescharacteristic
13C22×C3×C5×D4C2211yescharacteristic
14C3×C5×C4⋊Q8—24no—
15C3×C5×#124—24no—
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C5×C4⋊C4C2212yes—
19C2×C3×C5×C4⋊C4C2211yescharacteristic
20C2×C3×C5×C22⋊C4C2212yes—
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×C5C2211yescharacteristic
27C2×C3×C42×C5C2211yescharacteristic
28C2×C3×C42×C5C2211yescharacteristic
29C3×C5×#95C2212yes—
30C3×C5×#95C2212yes—
31C3×C5×#95C2212yes—
32C3×C5×#95C2212yes—
33C3×C5×#95C2212yes—
34C3×#2731C511yesmaximal, characteristic, direct product factor, Hall
35C2×C5×C4⋊Q8C2×C311yescharacteristic, complemented
36C2×C5×#124C2×C311yescharacteristic
37C5×#396C2×C312yescomplemented
38C5×#394C2×C312yes—
39C5×#393C2×C312yescomplemented
40C5×#391C2×C311yescharacteristic, complemented
41C5×#388C2×C312yescomplemented
42C5×#388C2×C312yescomplemented
43C5×#387C2×C312yescomplemented
44C24×C3×C5C2312yes—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8—41no—
47C2×C3×C5×Q8—41no—
48C2×C3×C5×Q8—41no—
49C2×C3×C5×Q8—41no—
50C2×C3×C5×Q8D414yes—
51C2×C3×C5×D4—22no—
52C2×C3×C5×D4—41no—
53C2×C3×C5×D4—41no—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C22×C3×C4×C5C2311yescharacteristic
61C22×C3×C4×C5C2312yes—
62C3×C5×C4⋊C4—24no—
63C3×C5×C4⋊C4—24no—
64C3×C5×C4⋊C4—24no—
65C3×C5×C4⋊C4—41no—
66C3×C5×C22⋊C4—24no—
67C3×C5×C22⋊C4—24no—
68C3×C5×C22⋊C4—24no—
69C3×C5×C22⋊C4—24no—
70C3×C5×C22⋊C4—42no—
71C3×C5×C22⋊C4—42no—
72C3×C42×C5—22no—
73C3×C42×C5—22no—
74C3×C42×C5—41no—
75C2×C3×C4⋊Q8C2×C511yescharacteristic, complemented
76C2×C3×#124C2×C511yescharacteristic
77C3×#396C2×C512yescomplemented
78C3×#394C2×C512yes—
79C3×#393C2×C512yescomplemented
80C3×#391C2×C511yescharacteristic, complemented
81C3×#388C2×C512yescomplemented
82C3×#388C2×C512yescomplemented
83C3×#387C2×C512yescomplemented
84C22×C5×Q8C22×C311yescharacteristic
85C22×C5×Q8C22×C311yescharacteristic
86C22×C5×D4C22×C311yescharacteristic
87C5×C4⋊Q8—24no—
88C5×#124—24no—
89C2×C5×C4⋊C4C22×C312yes—
90C2×C5×C4⋊C4C22×C312yes—
91C2×C5×C4⋊C4C22×C312yes—
92C2×C5×C4⋊C4C22×C311yescharacteristic
93C2×C5×C22⋊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×C312yes—
99C2×C42×C5C22×C311yescharacteristic
100C2×C42×C5C22×C311yescharacteristic
101C2×C42×C5C22×C311yescharacteristic
102C5×#95C22×C312yes—
103C5×#95C22×C312yes—
104C5×#95C22×C312yes—
105C5×#95C22×C312yes—
106C5×#95C22×C312yes—
107#2731C3×C511yescharacteristic, direct product factor, Sylow
108C23×C3×C5—22no—
109C23×C3×C5—22no—
110C23×C3×C5—22no—
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5—22no—
114C23×C3×C5—22no—
115C23×C3×C5C2411yescharacteristic
116C3×C5×Q8—44no—
117C3×C5×Q8—44no—
118C3×C5×D4—44no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—21no—
122C2×C3×C4×C5—21no—
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—21no—
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×C5C2×D412yes—
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5#4114yes—
143C22×C3×Q8C22×C511yescharacteristic
144C22×C3×Q8C22×C511yescharacteristic
145C22×C3×D4C22×C511yescharacteristic
146C3×C4⋊Q8—24no—
147C3×#124—24no—
148C2×C3×C4⋊C4C22×C512yes—
149C2×C3×C4⋊C4C22×C512yes—
150C2×C3×C4⋊C4C22×C512yes—
151C2×C3×C4⋊C4C22×C511yescharacteristic
152C2×C3×C22⋊C4C22×C512yes—
153C2×C3×C22⋊C4C22×C512yes—
154C2×C3×C22⋊C4C22×C512yes—
155C2×C3×C22⋊C4C22×C512yes—
156C2×C3×C22⋊C4C22×C512yes—
157C2×C3×C22⋊C4C22×C512yes—
158C2×C3×C42C22×C511yescharacteristic
159C2×C3×C42C22×C511yescharacteristic
160C2×C3×C42C22×C511yescharacteristic
161C3×#95C22×C512yes—
162C3×#95C22×C512yes—
163C3×#95C22×C512yes—
164C3×#95C22×C512yes—
165C3×#95C22×C512yes—
166C24×C5C23×C312yes—
167C2×C5×Q8—22no—
168C2×C5×Q8—41no—
169C2×C5×Q8—41no—
170C2×C5×Q8—41no—
171C2×C5×Q8—41no—
172C2×C5×Q8C3×D414yes—
173C2×C5×D4—22no—
174C2×C5×D4—41no—
175C2×C5×D4—41no—
176C22×C4×C5C23×C312yes—
177C22×C4×C5C23×C312yes—
178C22×C4×C5C23×C312yes—
179C22×C4×C5C23×C312yes—
180C22×C4×C5C23×C311yescharacteristic
181C22×C4×C5C23×C311yescharacteristic
182C22×C4×C5C23×C311yescharacteristic
183C22×C4×C5C23×C312yes—
184C5×C4⋊C4—24no—
185C5×C4⋊C4—24no—
186C5×C4⋊C4—24no—
187C5×C4⋊C4—41no—
188C5×C22⋊C4—24no—
189C5×C22⋊C4—24no—
190C5×C22⋊C4—24no—
191C5×C22⋊C4—24no—
192C5×C22⋊C4—42no—
193C5×C22⋊C4—42no—
194C42×C5—22no—
195C42×C5—22no—
196C42×C5—41no—
197C2×C4⋊Q8C2×C3×C511yescharacteristic, complemented
198C2×#124C2×C3×C511yescharacteristic
199#396C2×C3×C512yescomplemented
200#394C2×C3×C512yes—
201#393C2×C3×C512yescomplemented
202#391C2×C3×C511yescharacteristic, complemented
203#388C2×C3×C512yescomplemented
204#388C2×C3×C512yescomplemented
205#387C2×C3×C512yescomplemented
206C22×C3×C5—42no—
207C22×C3×C5—42no—
208C22×C3×C5—42no—
209C22×C3×C5—42no—
210C22×C3×C5—42no—
211C22×C3×C5—42no—
212C22×C3×C5—42no—
213C22×C3×C5C22×D411yescharacteristic
214C22×C3×C5C2×#4112yes—
215C22×C3×C5H(5,𝔽2)11yescharacteristic
216C22×C3×C5H(5,𝔽2)11yescharacteristic
217C22×C3×C52−1+412yes—
218C3×C4×C5—22no—
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×C512yes—
227C2×C3×Q8—22no—
228C2×C3×Q8—41no—
229C2×C3×Q8—41no—
230C2×C3×Q8—41no—
231C2×C3×Q8—41no—
232C2×C3×Q8C5×D414yes—
233C2×C3×D4—22no—
234C2×C3×D4—41no—
235C2×C3×D4—41no—
236C22×C3×C4C23×C512yes—
237C22×C3×C4C23×C512yes—
238C22×C3×C4C23×C512yes—
239C22×C3×C4C23×C512yes—
240C22×C3×C4C23×C511yescharacteristic
241C22×C3×C4C23×C511yescharacteristic
242C22×C3×C4C23×C511yescharacteristic
243C22×C3×C4C23×C512yes—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—41no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—42no—
254C3×C42—22no—
255C3×C42—22no—
256C3×C42—41no—
257C23×C5—22no—
258C23×C5—22no—
259C23×C5—22no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5—22no—
263C23×C5—22no—
264C23×C5C24×C311yescharacteristic
265C5×Q8—44no—
266C5×Q8—44no—
267C5×D4—44no—
268C2×C4×C5—22no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
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—22no—
288C2×C4×C5C2×C3×D412yes—
289C2×C4×C5C2×C3×D412yes—
290C2×C4×C5C2×C3×D412yes—
291C2×C4×C5C3×#4114yes—
292C22×Q8C22×C3×C511yescharacteristic
293C22×Q8C22×C3×C511yescharacteristic
294C22×D4C22×C3×C511yescharacteristic
295C4⋊Q8—24no—
296#124—24no—
297C2×C4⋊C4C22×C3×C512yes—
298C2×C4⋊C4C22×C3×C512yes—
299C2×C4⋊C4C22×C3×C512yes—
300C2×C4⋊C4C22×C3×C511yescharacteristic
301C2×C22⋊C4C22×C3×C512yes—
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×C511yescharacteristic
308C2×C42C22×C3×C511yescharacteristic
309C2×C42C22×C3×C511yescharacteristic
310#95C22×C3×C512yes—
311#95C22×C3×C512yes—
312#95C22×C3×C512yes—
313#95C22×C3×C512yes—
314#95C22×C3×C512yes—
315C2×C3×C5—82no—
316C2×C3×C5#53411yescharacteristic
317C2×C3×C5#54212yes—
318C2×C3×C5#54812yes—
319C2×C3×C5#55511yescharacteristic
320C2×C3×C5#56311yescharacteristic
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3C24×C511yescharacteristic
329C3×Q8—44no—
330C3×Q8—44no—
331C3×D4—44no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
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×C4C2×C5×D412yes—
353C2×C3×C4C2×C5×D412yes—
354C2×C3×C4C2×C5×D412yes—
355C2×C3×C4C5×#4114yes—
356C22×C5—42no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5—42no—
360C22×C5—42no—
361C22×C5—42no—
362C22×C5—42no—
363C22×C5C22×C3×D411yescharacteristic
364C22×C5C2×C3×#4112yes—
365C22×C5C3×H(5,𝔽2)11yescharacteristic
366C22×C5C3×H(5,𝔽2)11yescharacteristic
367C22×C5C3×2−1+412yes—
368C4×C5—22no—
369C4×C5—22no—
370C4×C5—22no—
371C4×C5—24no—
372C4×C5—42no—
373C4×C5—42no—
374C4×C5—42no—
375C4×C5—42no—
376C24C23×C3×C512yes—
377C2×Q8—22no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8C3×C5×D414yes—
383C2×D4—22no—
384C2×D4—41no—
385C2×D4—41no—
386C22×C4C23×C3×C512yes—
387C22×C4C23×C3×C512yes—
388C22×C4C23×C3×C512yes—
389C22×C4C23×C3×C512yes—
390C22×C4C23×C3×C511yescharacteristic
391C22×C4C23×C3×C511yescharacteristic
392C22×C4C23×C3×C511yescharacteristic
393C22×C4C23×C3×C512yes—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—41no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C42—22no—
405C42—22no—
406C42—41no—
407C3×C5#273111yescharacteristic, direct product factor, Hall
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3C22×C5×D411yescharacteristic
416C22×C3C2×C5×#4112yes—
417C22×C3C5×H(5,𝔽2)11yescharacteristic
418C22×C3C5×H(5,𝔽2)11yescharacteristic
419C22×C3C5×2−1+412yes—
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—24no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—42no—
428C2×C5—82no—
429C2×C5C3×#53411yescharacteristic
430C2×C5C3×#54212yes—
431C2×C5C3×#54812yes—
432C2×C5C3×#55511yescharacteristic
433C2×C5C3×#56311yescharacteristic
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23C24×C3×C511yescharacteristic
442Q8—44no—
443Q8—44no—
444D4—44no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
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—22no—
465C2×C4C2×C3×C5×D412yes—
466C2×C4C2×C3×C5×D412yes—
467C2×C4C2×C3×C5×D412yes—
468C2×C4C3×C5×#4114yes—
469C2×C3—82no—
470C2×C3C5×#53411yescharacteristic
471C2×C3C5×#54212yes—
472C2×C3C5×#54812yes—
473C2×C3C5×#55511yescharacteristic
474C2×C3C5×#56311yescharacteristic
475C5C3×#273111yescharacteristic, direct product factor, Sylow
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22C22×C3×C5×D411yescharacteristic
484C22C2×C3×C5×#4112yes—
485C22C3×C5×H(5,𝔽2)11yescharacteristic
486C22C3×C5×H(5,𝔽2)11yescharacteristic
487C22C3×C5×2−1+412yes—
488C4—22no—
489C4—22no—
490C4—22no—
491C4—24no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C3C5×#273111yescharacteristic, direct product factor, Sylow
497C2—82no—
498C2C3×C5×#53411yescharacteristic
499C2C3×C5×#54212yes—
500C2C3×C5×#54812yes—
501C2C3×C5×#55511yescharacteristic
502C2C3×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
1. C2×C3×C5×C4⋊Q8C2semidirect
3. C3×C5×#396C2semidirect
5. C3×C5×#393C2semidirect
6. C3×C5×#391C2semidirect
7. C3×C5×#388C2semidirect
8. C3×C5×#388C2semidirect
9. C3×C5×#387C2semidirect
10. C5×#2731C3direct (N × H)
34. C3×#2731C5direct (N × H)
35. C2×C5×C4⋊Q8C2×C3semidirect
37. C5×#396C2×C3semidirect
39. C5×#393C2×C3semidirect
40. C5×#391C2×C3semidirect
41. C5×#388C2×C3semidirect
42. C5×#388C2×C3semidirect
43. C5×#387C2×C3semidirect
75. C2×C3×C4⋊Q8C2×C5semidirect
77. C3×#396C2×C5semidirect
79. C3×#393C2×C5semidirect
80. C3×#391C2×C5semidirect
81. C3×#388C2×C5semidirect
82. C3×#388C2×C5semidirect
83. C3×#387C2×C5semidirect
107. #2731C3×C5direct (N × H)
197. C2×C4⋊Q8C2×C3×C5semidirect
199. #396C2×C3×C5semidirect
201. #393C2×C3×C5semidirect
202. #391C2×C3×C5semidirect
203. #388C2×C3×C5semidirect
204. #388C2×C3×C5semidirect
205. #387C2×C3×C5semidirect
407. C3×C5#2731direct (N × H)
475. C5C3×#2731direct (N × H)
496. C3C5×#2731direct (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×C4⋊Q82
C2×C3×C5×#1242
C3×C5×#3962
C3×C5×#3942
C3×C5×#3932
C3×C5×#3912
C3×C5×#3882
C3×C5×#3872
C5×#27313
C3×#27315