Finite Group Databasebeta

← C3×C5×#2748 #14275782

Subgroups — C3×C5×#2748 #14275782

Order 1920.

928 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#399C212yesmaximal, complemented
2C3×C5×#398C212yesmaximal, complemented
3C3×C5×#396C212yesmaximal, complemented
4C3×C5×#390C211yesmaximal, characteristic
5C3×C5×#387C212yesmaximal, complemented
6C3×C5×#386C211yesmaximal, characteristic
7C3×C5×#386C211yesmaximal, characteristic
8C3×C5×#385C212yesmaximal, complemented
9C3×C5×#380C212yesmaximal, complemented
10C5×#2748C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2212yes—
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⋊C4C2212yes—
24C2×C3×C5×C22⋊C4C2211yescharacteristic
25C2×C3×C5×C22⋊C4C2211yescharacteristic
26C2×C3×C42×C5C2211yescharacteristic
27C3×C5×#95C2212yes—
28C3×C5×#95C2212yes—
29C3×C5×#95C2212yes—
30C3×C5×#95C2212yes—
31C3×C5×#95C2212yes—
32C3×#2748C511yesmaximal, characteristic, direct product factor, Hall
33C5×#399C2×C312yescomplemented
34C5×#398C2×C312yescomplemented
35C5×#396C2×C312yescomplemented
36C5×#390C2×C311yescharacteristic
37C5×#387C2×C312yescomplemented
38C5×#386C2×C311yescharacteristic
39C5×#386C2×C311yescharacteristic
40C5×#385C2×C312yescomplemented
41C5×#380C2×C312yescomplemented
42C24×C3×C5C2311yescharacteristic
43C24×C3×C5C2311yescharacteristic
44C2×C3×C5×D4—41no—
45C2×C3×C5×D4—41no—
46C2×C3×C5×D4Q814yes—
47C22×C3×C4×C5—24no—
48C22×C3×C4×C5—41no—
49C22×C3×C4×C5—42no—
50C22×C3×C4×C5—41no—
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2312yes—
58C3×C5×C4⋊C4—24no—
59C3×C5×C4⋊C4—42no—
60C3×C5×C4⋊C4—41no—
61C3×C5×C4⋊C4—42no—
62C3×C5×C4⋊C4—42no—
63C3×C5×C4⋊C4—42no—
64C3×C5×C22⋊C4—24no—
65C3×C5×C22⋊C4—42no—
66C3×C5×C22⋊C4—42no—
67C3×C5×C22⋊C4—41no—
68C3×C5×C22⋊C4—41no—
69C3×C5×C22⋊C4—42no—
70C3×C42×C5—41no—
71C3×#399C2×C512yescomplemented
72C3×#398C2×C512yescomplemented
73C3×#396C2×C512yescomplemented
74C3×#390C2×C511yescharacteristic
75C3×#387C2×C512yescomplemented
76C3×#386C2×C511yescharacteristic
77C3×#386C2×C511yescharacteristic
78C3×#385C2×C512yescomplemented
79C3×#380C2×C512yescomplemented
80C22×C5×D4C22×C311yescharacteristic
81C23×C4×C5C22×C311yescharacteristic
82C23×C4×C5C22×C311yescharacteristic
83C2×C5×C4⋊C4C22×C312yes—
84C2×C5×C4⋊C4C22×C312yes—
85C2×C5×C4⋊C4C22×C312yes—
86C2×C5×C4⋊C4C22×C312yes—
87C2×C5×C4⋊C4C22×C311yescharacteristic
88C2×C5×C4⋊C4C22×C312yes—
89C2×C5×C22⋊C4C22×C312yes—
90C2×C5×C22⋊C4C22×C312yes—
91C2×C5×C22⋊C4C22×C312yes—
92C2×C5×C22⋊C4C22×C312yes—
93C2×C5×C22⋊C4C22×C311yescharacteristic
94C2×C5×C22⋊C4C22×C311yescharacteristic
95C2×C42×C5C22×C311yescharacteristic
96C5×#95C22×C312yes—
97C5×#95C22×C312yes—
98C5×#95C22×C312yes—
99C5×#95C22×C312yes—
100C5×#95C22×C312yes—
101#2748C3×C511yescharacteristic, direct product factor, Sylow
102C23×C3×C5—22no—
103C23×C3×C5—22no—
104C23×C3×C5—22no—
105C23×C3×C5—22no—
106C23×C3×C5—21no—
107C23×C3×C5—21no—
108C23×C3×C5—21no—
109C23×C3×C5—21no—
110C23×C3×C5C2×Q812yes—
111C23×C3×C5C2×Q812yes—
112C23×C3×C5C2411yescharacteristic
113C3×C5×D4—44no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—21no—
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—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—44no—
134C2×C3×C4×C5—44no—
135C2×C3×C4×C5C2×Q812yes—
136C2×C3×C4×C5#4114yes—
137C22×C3×D4C22×C511yescharacteristic
138C23×C3×C4C22×C511yescharacteristic
139C23×C3×C4C22×C511yescharacteristic
140C2×C3×C4⋊C4C22×C512yes—
141C2×C3×C4⋊C4C22×C512yes—
142C2×C3×C4⋊C4C22×C512yes—
143C2×C3×C4⋊C4C22×C512yes—
144C2×C3×C4⋊C4C22×C511yescharacteristic
145C2×C3×C4⋊C4C22×C512yes—
146C2×C3×C22⋊C4C22×C512yes—
147C2×C3×C22⋊C4C22×C512yes—
148C2×C3×C22⋊C4C22×C512yes—
149C2×C3×C22⋊C4C22×C512yes—
150C2×C3×C22⋊C4C22×C511yescharacteristic
151C2×C3×C22⋊C4C22×C511yescharacteristic
152C2×C3×C42C22×C511yescharacteristic
153C3×#95C22×C512yes—
154C3×#95C22×C512yes—
155C3×#95C22×C512yes—
156C3×#95C22×C512yes—
157C3×#95C22×C512yes—
158C24×C5C23×C311yescharacteristic
159C24×C5C23×C311yescharacteristic
160C2×C5×D4—41no—
161C2×C5×D4—41no—
162C2×C5×D4C3×Q814yes—
163C22×C4×C5—24no—
164C22×C4×C5—41no—
165C22×C4×C5—42no—
166C22×C4×C5—41no—
167C22×C4×C5C23×C312yes—
168C22×C4×C5C23×C312yes—
169C22×C4×C5C23×C312yes—
170C22×C4×C5C23×C312yes—
171C22×C4×C5C23×C312yes—
172C22×C4×C5C23×C311yescharacteristic
173C22×C4×C5C23×C312yes—
174C5×C4⋊C4—24no—
175C5×C4⋊C4—42no—
176C5×C4⋊C4—41no—
177C5×C4⋊C4—42no—
178C5×C4⋊C4—42no—
179C5×C4⋊C4—42no—
180C5×C22⋊C4—24no—
181C5×C22⋊C4—42no—
182C5×C22⋊C4—42no—
183C5×C22⋊C4—41no—
184C5×C22⋊C4—41no—
185C5×C22⋊C4—42no—
186C42×C5—41no—
187#399C2×C3×C512yescomplemented
188#398C2×C3×C512yescomplemented
189#396C2×C3×C512yescomplemented
190#390C2×C3×C511yescharacteristic
191#387C2×C3×C512yescomplemented
192#386C2×C3×C511yescharacteristic
193#386C2×C3×C511yescharacteristic
194#385C2×C3×C512yescomplemented
195#380C2×C3×C512yescomplemented
196C22×C3×C5—22no—
197C22×C3×C5—22no—
198C22×C3×C5—24no—
199C22×C3×C5—24no—
200C22×C3×C5—42no—
201C22×C3×C5—42no—
202C22×C3×C5—41no—
203C22×C3×C5—41no—
204C22×C3×C5—41no—
205C22×C3×C5—41no—
206C22×C3×C5C22×Q811yescharacteristic
207C22×C3×C5C2×#4112yes—
208C22×C3×C5H(5,𝔽2)11yescharacteristic
209C22×C3×C5H(5,𝔽2)12yes—
210C22×C3×C5H(5,𝔽2)11yescharacteristic
211C3×C4×C5—22no—
212C3×C4×C5—24no—
213C3×C4×C5—42no—
214C3×C4×C5—42no—
215C3×C4×C5—42no—
216C3×C4×C5—42no—
217C3×C4×C5—42no—
218C24×C3C23×C511yescharacteristic
219C24×C3C23×C511yescharacteristic
220C2×C3×D4—41no—
221C2×C3×D4—41no—
222C2×C3×D4C5×Q814yes—
223C22×C3×C4—24no—
224C22×C3×C4—41no—
225C22×C3×C4—42no—
226C22×C3×C4—41no—
227C22×C3×C4C23×C512yes—
228C22×C3×C4C23×C512yes—
229C22×C3×C4C23×C512yes—
230C22×C3×C4C23×C512yes—
231C22×C3×C4C23×C512yes—
232C22×C3×C4C23×C511yescharacteristic
233C22×C3×C4C23×C512yes—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—41no—
244C3×C22⋊C4—41no—
245C3×C22⋊C4—42no—
246C3×C42—41no—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—22no—
250C23×C5—22no—
251C23×C5—21no—
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5C2×C3×Q812yes—
256C23×C5C2×C3×Q812yes—
257C23×C5C24×C311yescharacteristic
258C5×D4—44no—
259C2×C4×C5—22no—
260C2×C4×C5—21no—
261C2×C4×C5—22no—
262C2×C4×C5—21no—
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—22no—
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—44no—
279C2×C4×C5—44no—
280C2×C4×C5C2×C3×Q812yes—
281C2×C4×C5C3×#4114yes—
282C22×D4C22×C3×C511yescharacteristic
283C23×C4C22×C3×C511yescharacteristic
284C23×C4C22×C3×C511yescharacteristic
285C2×C4⋊C4C22×C3×C512yes—
286C2×C4⋊C4C22×C3×C512yes—
287C2×C4⋊C4C22×C3×C512yes—
288C2×C4⋊C4C22×C3×C512yes—
289C2×C4⋊C4C22×C3×C511yescharacteristic
290C2×C4⋊C4C22×C3×C512yes—
291C2×C22⋊C4C22×C3×C512yes—
292C2×C22⋊C4C22×C3×C512yes—
293C2×C22⋊C4C22×C3×C512yes—
294C2×C22⋊C4C22×C3×C512yes—
295C2×C22⋊C4C22×C3×C511yescharacteristic
296C2×C22⋊C4C22×C3×C511yescharacteristic
297C2×C42C22×C3×C511yescharacteristic
298#95C22×C3×C512yes—
299#95C22×C3×C512yes—
300#95C22×C3×C512yes—
301#95C22×C3×C512yes—
302#95C22×C3×C512yes—
303C2×C3×C5—42no—
304C2×C3×C5—42no—
305C2×C3×C5#54012yes—
306C2×C3×C5#54311yescharacteristic
307C2×C3×C5#55111yescharacteristic
308C2×C3×C5#55412yes—
309C2×C3×C5#56011yescharacteristic
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3C2×C5×Q812yes—
319C23×C3C2×C5×Q812yes—
320C23×C3C24×C511yescharacteristic
321C3×D4—44no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
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—44no—
342C2×C3×C4—44no—
343C2×C3×C4C2×C5×Q812yes—
344C2×C3×C4C5×#4114yes—
345C22×C5—22no—
346C22×C5—22no—
347C22×C5—24no—
348C22×C5—24no—
349C22×C5—41no—
350C22×C5—41no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—41no—
354C22×C5—41no—
355C22×C5C22×C3×Q811yescharacteristic
356C22×C5C2×C3×#4112yes—
357C22×C5C3×H(5,𝔽2)12yes—
358C22×C5C3×H(5,𝔽2)11yescharacteristic
359C22×C5C3×H(5,𝔽2)11yescharacteristic
360C4×C5—22no—
361C4×C5—24no—
362C4×C5—42no—
363C4×C5—42no—
364C4×C5—42no—
365C4×C5—42no—
366C4×C5—42no—
367C24C23×C3×C511yescharacteristic
368C24C23×C3×C511yescharacteristic
369C2×D4—41no—
370C2×D4—41no—
371C2×D4C3×C5×Q814yes—
372C22×C4—24no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4C23×C3×C512yes—
377C22×C4C23×C3×C512yes—
378C22×C4C23×C3×C512yes—
379C22×C4C23×C3×C512yes—
380C22×C4C23×C3×C512yes—
381C22×C4C23×C3×C511yescharacteristic
382C22×C4C23×C3×C512yes—
383C4⋊C4—24no—
384C4⋊C4—42no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C22⋊C4—24no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—42no—
395C42—41no—
396C3×C5#274811yescharacteristic, direct product factor, Hall
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3C22×C5×Q811yescharacteristic
408C22×C3C2×C5×#4112yes—
409C22×C3C5×H(5,𝔽2)11yescharacteristic
410C22×C3C5×H(5,𝔽2)12yes—
411C22×C3C5×H(5,𝔽2)11yescharacteristic
412C3×C4—22no—
413C3×C4—24no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—42no—
419C2×C5—42no—
420C2×C5—42no—
421C2×C5C3×#54012yes—
422C2×C5C3×#54311yescharacteristic
423C2×C5C3×#55111yescharacteristic
424C2×C5C3×#55412yes—
425C2×C5C3×#56011yescharacteristic
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23C2×C3×C5×Q812yes—
435C23C2×C3×C5×Q812yes—
436C23C24×C3×C511yescharacteristic
437D4—44no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4C2×C3×C5×Q812yes—
460C2×C4C3×C5×#4114yes—
461C2×C3—42no—
462C2×C3—42no—
463C2×C3C5×#54012yes—
464C2×C3C5×#54311yescharacteristic
465C2×C3C5×#55111yescharacteristic
466C2×C3C5×#55412yes—
467C2×C3C5×#56011yescharacteristic
468C5C3×#274811yescharacteristic, direct product factor, Sylow
469C22—22no—
470C22—22no—
471C22—24no—
472C22—24no—
473C22—42no—
474C22—42no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22C22×C3×C5×Q811yescharacteristic
480C22C2×C3×C5×#4112yes—
481C22C3×C5×H(5,𝔽2)11yescharacteristic
482C22C3×C5×H(5,𝔽2)12yes—
483C22C3×C5×H(5,𝔽2)11yescharacteristic
484C4—22no—
485C4—24no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C3C5×#274811yescharacteristic, direct product factor, Sylow
492C2—42no—
493C2—42no—
494C2C3×C5×#54012yes—
495C2C3×C5×#54311yescharacteristic
496C2C3×C5×#55111yescharacteristic
497C2C3×C5×#55412yes—
498C2C3×C5×#56011yescharacteristic
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. C3×C5×#399C2semidirect
2. C3×C5×#398C2semidirect
3. C3×C5×#396C2semidirect
5. C3×C5×#387C2semidirect
8. C3×C5×#385C2semidirect
9. C3×C5×#380C2semidirect
10. C5×#2748C3direct (N × H)
32. C3×#2748C5direct (N × H)
33. C5×#399C2×C3semidirect
34. C5×#398C2×C3semidirect
35. C5×#396C2×C3semidirect
37. C5×#387C2×C3semidirect
40. C5×#385C2×C3semidirect
41. C5×#380C2×C3semidirect
71. C3×#399C2×C5semidirect
72. C3×#398C2×C5semidirect
73. C3×#396C2×C5semidirect
75. C3×#387C2×C5semidirect
78. C3×#385C2×C5semidirect
79. C3×#380C2×C5semidirect
101. #2748C3×C5direct (N × H)
187. #399C2×C3×C5semidirect
188. #398C2×C3×C5semidirect
189. #396C2×C3×C5semidirect
191. #387C2×C3×C5semidirect
194. #385C2×C3×C5semidirect
195. #380C2×C3×C5semidirect
396. C3×C5#2748direct (N × H)
468. C5C3×#2748direct (N × H)
491. C3C5×#2748direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#3992
C3×C5×#3982
C3×C5×#3962
C3×C5×#3902
C3×C5×#3872
C3×C5×#3862
C3×C5×#3852
C3×C5×#3802
C5×#27483
C3×#27485