Finite Group Databasebeta

← C3×C5×#2538 #14275572

Subgroups — C3×C5×#2538 #14275572

Order 1920.

1240 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C22⋊Q8C212yesmaximal, complemented
2C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
3C2×C3×C4×C5×D4C211yesmaximal, characteristic
4C3×C5×#396C214yesmaximal, complemented
5C3×C5×#390C212yesmaximal
6C3×C5×#389C212yesmaximal, complemented
7C3×C5×#384C211yesmaximal, characteristic, complemented
8C3×C5×#380C212yesmaximal, complemented
9C5×#2538C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2212yes—
13C3×C5×C22⋊Q8—28no—
14C3×C4×C5×Q8—24no—
15C3×C4×C5×D4—24no—
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C4⋊C4C2212yes—
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C4⋊C4C2214yes—
21C2×C3×C5×C4⋊C4C2212yes—
22C2×C3×C5×C4⋊C4C2212yes—
23C2×C3×C5×C22⋊C4C2212yes—
24C2×C3×C5×C22⋊C4C2214yes—
25C2×C3×C5×C22⋊C4C2214yes—
26C2×C3×C42×C5C2212yes—
27C2×C3×C42×C5C2211yescharacteristic
28C3×C5×#95C2214yes—
29C3×#2538C511yesmaximal, characteristic, direct product factor, Hall
30C2×C5×C22⋊Q8C2×C312yescomplemented
31C2×C4×C5×Q8C2×C311yescharacteristic, complemented
32C2×C4×C5×D4C2×C311yescharacteristic
33C5×#396C2×C314yescomplemented
34C5×#390C2×C312yes—
35C5×#389C2×C312yescomplemented
36C5×#384C2×C311yescharacteristic, complemented
37C5×#380C2×C312yescomplemented
38C24×C3×C5C2312yes—
39C2×C3×C5×Q8—22no—
40C2×C3×C5×Q8—22no—
41C2×C3×C5×Q8D414yescomplemented
42C2×C3×C5×D4—22no—
43C2×C3×C5×D4—22no—
44C2×C3×C5×D4Q814yescomplemented
45C22×C3×C4×C5—24no—
46C22×C3×C4×C5—24no—
47C22×C3×C4×C5—24no—
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2314yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C3×C5×C4⋊C4—22no—
56C3×C5×C4⋊C4—24no—
57C3×C5×C4⋊C4—22no—
58C3×C5×C4⋊C4—24no—
59C3×C5×C4⋊C4—28no—
60C3×C5×C4⋊C4—24no—
61C3×C5×C4⋊C4—42no—
62C3×C5×C22⋊C4—24no—
63C3×C5×C22⋊C4—28no—
64C3×C5×C22⋊C4—44no—
65C3×C42×C5—24no—
66C3×C42×C5D414yes—
67C2×C3×C22⋊Q8C2×C512yescomplemented
68C2×C3×C4×Q8C2×C511yescharacteristic, complemented
69C2×C3×C4×D4C2×C511yescharacteristic
70C3×#396C2×C514yescomplemented
71C3×#390C2×C512yes—
72C3×#389C2×C512yescomplemented
73C3×#384C2×C511yescharacteristic, complemented
74C3×#380C2×C512yescomplemented
75C22×C5×Q8C22×C311yescharacteristic
76C22×C5×D4C22×C311yescharacteristic
77C23×C4×C5C22×C312yes—
78C5×C22⋊Q8—28no—
79C4×C5×Q8—24no—
80C4×C5×D4—24no—
81C2×C5×C4⋊C4C22×C311yescharacteristic
82C2×C5×C4⋊C4C22×C312yes—
83C2×C5×C4⋊C4C22×C312yes—
84C2×C5×C4⋊C4C22×C311yescharacteristic
85C2×C5×C4⋊C4C22×C314yes—
86C2×C5×C4⋊C4C22×C312yes—
87C2×C5×C4⋊C4C22×C312yes—
88C2×C5×C22⋊C4C22×C314yes—
89C2×C5×C22⋊C4C22×C312yes—
90C2×C5×C22⋊C4C22×C314yes—
91C2×C42×C5C22×C312yes—
92C2×C42×C5C22×C311yescharacteristic
93C5×#95C22×C314yes—
94#2538C3×C511yescharacteristic, direct product factor, Sylow
95C23×C3×C5—22no—
96C23×C3×C5—22no—
97C23×C3×C5—22no—
98C23×C3×C5—22no—
99C23×C3×C5—22no—
100C23×C3×C5—22no—
101C23×C3×C5C2×Q814yes—
102C23×C3×C5C2411yescharacteristic
103C3×C5×Q8—28no—
104C3×C5×D4—28no—
105C2×C3×C4×C5—22no—
106C2×C3×C4×C5—22no—
107C2×C3×C4×C5—21no—
108C2×C3×C4×C5—21no—
109C2×C3×C4×C5—22no—
110C2×C3×C4×C5—24no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—24no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—24no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—48no—
119C2×C3×C4×C5C2×D412yes—
120C2×C3×C4×C5C2×D414yes—
121C2×C3×C4×C5C2×D412yes—
122C2×C3×C4×C5C2×D412yes—
123C2×C3×C4×C5C2×D412yes—
124C2×C3×C4×C5C2×Q812yes—
125C2×C3×C4×C5#4112yes—
126C2×C3×C4×C5#4112yes—
127C22×C3×Q8C22×C511yescharacteristic
128C22×C3×D4C22×C511yescharacteristic
129C23×C3×C4C22×C512yes—
130C3×C22⋊Q8—28no—
131C3×C4×Q8—24no—
132C3×C4×D4—24no—
133C2×C3×C4⋊C4C22×C511yescharacteristic
134C2×C3×C4⋊C4C22×C512yes—
135C2×C3×C4⋊C4C22×C512yes—
136C2×C3×C4⋊C4C22×C511yescharacteristic
137C2×C3×C4⋊C4C22×C514yes—
138C2×C3×C4⋊C4C22×C512yes—
139C2×C3×C4⋊C4C22×C512yes—
140C2×C3×C22⋊C4C22×C514yes—
141C2×C3×C22⋊C4C22×C512yes—
142C2×C3×C22⋊C4C22×C514yes—
143C2×C3×C42C22×C512yes—
144C2×C3×C42C22×C511yescharacteristic
145C3×#95C22×C514yes—
146C24×C5C23×C312yes—
147C2×C5×Q8—22no—
148C2×C5×Q8—22no—
149C2×C5×Q8C3×D414yescomplemented
150C2×C5×D4—22no—
151C2×C5×D4—22no—
152C2×C5×D4C3×Q814yescomplemented
153C22×C4×C5—24no—
154C22×C4×C5—24no—
155C22×C4×C5—24no—
156C22×C4×C5C23×C312yes—
157C22×C4×C5C23×C314yes—
158C22×C4×C5C23×C312yes—
159C22×C4×C5C23×C311yescharacteristic
160C22×C4×C5C23×C312yes—
161C22×C4×C5C23×C311yescharacteristic
162C22×C4×C5C23×C311yescharacteristic
163C5×C4⋊C4—22no—
164C5×C4⋊C4—24no—
165C5×C4⋊C4—22no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—28no—
168C5×C4⋊C4—24no—
169C5×C4⋊C4—42no—
170C5×C22⋊C4—24no—
171C5×C22⋊C4—28no—
172C5×C22⋊C4—44no—
173C42×C5—24no—
174C42×C5C3×D414yes—
175C2×C22⋊Q8C2×C3×C512yescomplemented
176C2×C4×Q8C2×C3×C511yescharacteristic, complemented
177C2×C4×D4C2×C3×C511yescharacteristic
178#396C2×C3×C514yescomplemented
179#390C2×C3×C512yes—
180#389C2×C3×C512yescomplemented
181#384C2×C3×C511yescharacteristic, complemented
182#380C2×C3×C512yescomplemented
183C22×C3×C5—24no—
184C22×C3×C5—24no—
185C22×C3×C5—24no—
186C22×C3×C5—42no—
187C22×C3×C5—42no—
188C22×C3×C5—42no—
189C22×C3×C5—42no—
190C22×C3×C5C22×D411yescharacteristic
191C22×C3×C5C22×D411yescharacteristic
192C22×C3×C5C22×Q811yescharacteristic
193C22×C3×C5C2×#4111yescharacteristic
194C22×C3×C5C2×#4111yescharacteristic
195C22×C3×C5H(5,𝔽2)11yescharacteristic
196C22×C3×C52−1+411yescharacteristic
197C3×C4×C5—24no—
198C3×C4×C5—22no—
199C3×C4×C5—42no—
200C3×C4×C5—42no—
201C3×C4×C5—44no—
202C3×C4×C5#12114yes—
203C3×C4×C5C22⋊Q814yescomplemented
204C24×C3C23×C512yes—
205C2×C3×Q8—22no—
206C2×C3×Q8—22no—
207C2×C3×Q8C5×D414yescomplemented
208C2×C3×D4—22no—
209C2×C3×D4—22no—
210C2×C3×D4C5×Q814yescomplemented
211C22×C3×C4—24no—
212C22×C3×C4—24no—
213C22×C3×C4—24no—
214C22×C3×C4C23×C512yes—
215C22×C3×C4C23×C512yes—
216C22×C3×C4C23×C514yes—
217C22×C3×C4C23×C511yescharacteristic
218C22×C3×C4C23×C512yes—
219C22×C3×C4C23×C511yescharacteristic
220C22×C3×C4C23×C511yescharacteristic
221C3×C4⋊C4—22no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—28no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—42no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—44no—
231C3×C42—24no—
232C3×C42C5×D414yes—
233C23×C5—22no—
234C23×C5—22no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—22no—
239C23×C5C2×C3×Q814yes—
240C23×C5C24×C311yescharacteristic
241C5×Q8—28no—
242C5×D4—28no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—21no—
246C2×C4×C5—21no—
247C2×C4×C5—22no—
248C2×C4×C5—24no—
249C2×C4×C5—22no—
250C2×C4×C5—22no—
251C2×C4×C5—24no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—24no—
255C2×C4×C5—22no—
256C2×C4×C5—48no—
257C2×C4×C5C2×C3×D412yes—
258C2×C4×C5C2×C3×D414yes—
259C2×C4×C5C2×C3×D412yes—
260C2×C4×C5C2×C3×D412yes—
261C2×C4×C5C2×C3×D412yes—
262C2×C4×C5C2×C3×Q812yes—
263C2×C4×C5C3×#4112yes—
264C2×C4×C5C3×#4112yes—
265C22×Q8C22×C3×C511yescharacteristic
266C22×D4C22×C3×C511yescharacteristic
267C23×C4C22×C3×C512yes—
268C22⋊Q8—28no—
269C4×Q8—24no—
270C4×D4—24no—
271C2×C4⋊C4C22×C3×C511yescharacteristic
272C2×C4⋊C4C22×C3×C511yescharacteristic
273C2×C4⋊C4C22×C3×C512yes—
274C2×C4⋊C4C22×C3×C512yes—
275C2×C4⋊C4C22×C3×C514yes—
276C2×C4⋊C4C22×C3×C512yes—
277C2×C4⋊C4C22×C3×C512yes—
278C2×C22⋊C4C22×C3×C512yes—
279C2×C22⋊C4C22×C3×C514yes—
280C2×C22⋊C4C22×C3×C514yes—
281C2×C42C22×C3×C512yes—
282C2×C42C22×C3×C511yescharacteristic
283#95C22×C3×C514yes—
284C2×C3×C5—44no—
285C2×C3×C5C2×#12111yescharacteristic
286C2×C3×C5C2×C22⋊Q811yescharacteristic
287C2×C3×C5#53711yescharacteristic
288C2×C3×C5D4×Q811yescharacteristic
289C2×C3×C5#55011yescharacteristic
290C2×C3×C5#55411yescharacteristic
291C2×C3×C5#55611yescharacteristic
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3C2×C5×Q814yes—
299C23×C3C24×C511yescharacteristic
300C3×Q8—28no—
301C3×D4—28no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—48no—
316C2×C3×C4C2×C5×D412yes—
317C2×C3×C4C2×C5×D414yes—
318C2×C3×C4C2×C5×D412yes—
319C2×C3×C4C2×C5×D412yes—
320C2×C3×C4C2×C5×D412yes—
321C2×C3×C4C2×C5×Q812yes—
322C2×C3×C4C5×#4112yes—
323C2×C3×C4C5×#4112yes—
324C22×C5—24no—
325C22×C5—24no—
326C22×C5—24no—
327C22×C5—42no—
328C22×C5—42no—
329C22×C5—42no—
330C22×C5—42no—
331C22×C5C22×C3×D411yescharacteristic
332C22×C5C22×C3×D411yescharacteristic
333C22×C5C22×C3×Q811yescharacteristic
334C22×C5C2×C3×#4111yescharacteristic
335C22×C5C2×C3×#4111yescharacteristic
336C22×C5C3×H(5,𝔽2)11yescharacteristic
337C22×C5C3×2−1+411yescharacteristic
338C4×C5—24no—
339C4×C5—22no—
340C4×C5—42no—
341C4×C5—42no—
342C4×C5—44no—
343C4×C5C3×#12114yes—
344C4×C5C3×C22⋊Q814yescomplemented
345C24C23×C3×C512yes—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8C3×C5×D414yescomplemented
349C2×D4—22no—
350C2×D4—22no—
351C2×D4C3×C5×Q814yescomplemented
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4C23×C3×C512yes—
356C22×C4C23×C3×C512yes—
357C22×C4C23×C3×C514yes—
358C22×C4C23×C3×C511yescharacteristic
359C22×C4C23×C3×C512yes—
360C22×C4C23×C3×C511yescharacteristic
361C22×C4C23×C3×C511yescharacteristic
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—42no—
369C22⋊C4—24no—
370C22⋊C4—28no—
371C22⋊C4—44no—
372C42—24no—
373C42C3×C5×D414yes—
374C3×C5#253811yescharacteristic, direct product factor, Hall
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3C22×C5×D411yescharacteristic
383C22×C3C22×C5×D411yescharacteristic
384C22×C3C22×C5×Q811yescharacteristic
385C22×C3C2×C5×#4111yescharacteristic
386C22×C3C2×C5×#4111yescharacteristic
387C22×C3C5×H(5,𝔽2)11yescharacteristic
388C22×C3C5×2−1+411yescharacteristic
389C3×C4—24no—
390C3×C4—22no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—44no—
394C3×C4C5×#12114yes—
395C3×C4C5×C22⋊Q814yescomplemented
396C2×C5—44no—
397C2×C5C2×C3×#12111yescharacteristic
398C2×C5C2×C3×C22⋊Q811yescharacteristic
399C2×C5C3×#53711yescharacteristic
400C2×C5C3×D4×Q811yescharacteristic
401C2×C5C3×#55011yescharacteristic
402C2×C5C3×#55411yescharacteristic
403C2×C5C3×#55611yescharacteristic
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23C2×C3×C5×Q814yes—
411C23C24×C3×C511yescharacteristic
412Q8—28no—
413D4—28no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—48no—
428C2×C4C2×C3×C5×D412yes—
429C2×C4C2×C3×C5×D414yes—
430C2×C4C2×C3×C5×D412yes—
431C2×C4C2×C3×C5×D412yes—
432C2×C4C2×C3×C5×D412yes—
433C2×C4C2×C3×C5×Q812yes—
434C2×C4C3×C5×#4112yes—
435C2×C4C3×C5×#4112yes—
436C2×C3—44no—
437C2×C3C2×C5×#12111yescharacteristic
438C2×C3C2×C5×C22⋊Q811yescharacteristic
439C2×C3C5×#53711yescharacteristic
440C2×C3C5×D4×Q811yescharacteristic
441C2×C3C5×#55011yescharacteristic
442C2×C3C5×#55411yescharacteristic
443C2×C3C5×#55611yescharacteristic
444C5C3×#253811yescharacteristic, direct product factor, Sylow
445C22—24no—
446C22—24no—
447C22—24no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22C22×C3×C5×D411yescharacteristic
453C22C22×C3×C5×D411yescharacteristic
454C22C22×C3×C5×Q811yescharacteristic
455C22C2×C3×C5×#4111yescharacteristic
456C22C2×C3×C5×#4111yescharacteristic
457C22C3×C5×H(5,𝔽2)11yescharacteristic
458C22C3×C5×2−1+411yescharacteristic
459C4—24no—
460C4—22no—
461C4—42no—
462C4—42no—
463C4—44no—
464C4C3×C5×#12114yes—
465C4C3×C5×C22⋊Q814yescomplemented
466C3C5×#253811yescharacteristic, direct product factor, Sylow
467C2—44no—
468C2C2×C3×C5×#12111yescharacteristic
469C2C2×C3×C5×C22⋊Q811yescharacteristic
470C2C3×C5×#53711yescharacteristic
471C2C3×C5×D4×Q811yescharacteristic
472C2C3×C5×#55011yescharacteristic
473C2C3×C5×#55411yescharacteristic
474C2C3×C5×#55611yescharacteristic
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×C22⋊Q8C2semidirect
2. C2×C3×C4×C5×Q8C2semidirect
4. C3×C5×#396C2semidirect
6. C3×C5×#389C2semidirect
7. C3×C5×#384C2semidirect
8. C3×C5×#380C2semidirect
9. C5×#2538C3direct (N × H)
29. C3×#2538C5direct (N × H)
30. C2×C5×C22⋊Q8C2×C3semidirect
31. C2×C4×C5×Q8C2×C3semidirect
33. C5×#396C2×C3semidirect
35. C5×#389C2×C3semidirect
36. C5×#384C2×C3semidirect
37. C5×#380C2×C3semidirect
41. C2×C3×C5×Q8D4semidirect
44. C2×C3×C5×D4Q8semidirect
67. C2×C3×C22⋊Q8C2×C5semidirect
68. C2×C3×C4×Q8C2×C5semidirect
70. C3×#396C2×C5semidirect
72. C3×#389C2×C5semidirect
73. C3×#384C2×C5semidirect
74. C3×#380C2×C5semidirect
94. #2538C3×C5direct (N × H)
149. C2×C5×Q8C3×D4semidirect
152. C2×C5×D4C3×Q8semidirect
175. C2×C22⋊Q8C2×C3×C5semidirect
176. C2×C4×Q8C2×C3×C5semidirect
178. #396C2×C3×C5semidirect
180. #389C2×C3×C5semidirect
181. #384C2×C3×C5semidirect
182. #380C2×C3×C5semidirect
203. C3×C4×C5C22⋊Q8semidirect
207. C2×C3×Q8C5×D4semidirect
210. C2×C3×D4C5×Q8semidirect
344. C4×C5C3×C22⋊Q8semidirect
348. C2×Q8C3×C5×D4semidirect
351. C2×D4C3×C5×Q8semidirect
374. C3×C5#2538direct (N × H)
395. C3×C4C5×C22⋊Q8semidirect
444. C5C3×#2538direct (N × H)
465. C4C3×C5×C22⋊Q8semidirect
466. C3C5×#2538direct (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×C22⋊Q82
C2×C3×C4×C5×Q82
C2×C3×C4×C5×D42
C3×C5×#3962
C3×C5×#3902
C3×C5×#3892
C3×C5×#3842
C3×C5×#3802
C5×#25383
C3×#25385