Finite Group Databasebeta

← C3×C5×#2120 #14275154

Subgroups — C3×C5×#2120 #14275154

Order 1920.

528 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#525C211yesmaximal, characteristic
2C3×C5×#413C211yesmaximal, characteristic, complemented
3C3×C5×#410C211yesmaximal, characteristic, complemented
4C3×C5×#352C212yesmaximal, complemented
5C3×C5×AΣL(1,𝔽16)C212yesmaximal, complemented
6C5×#2120C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41C2211yescharacteristic
8C23×C3×C4×C5C411yescharacteristic, complemented
9C2×C3×C5×M16—21no—
10C3×C5×#123C411yescharacteristic, complemented
11C3×C5×#123C2211yescharacteristic
12C3×C5×C22⋊Q8—21no—
13C3×C5×#121—21no—
14C3×C5×#120C411yescharacteristic, complemented
15C3×C5×#120C2211yescharacteristic
16C3×C4×C5×D4—21no—
17C3×C4×C5×D4—21no—
18C3×C5×#117—21no—
19C3×C5×#117C411yescharacteristic, complemented
20C3×C5×#101—21no—
21C3×C5×#100C2212yes—
22C3×C5×#99—21no—
23C3×C5×#99C2212yes—
24C3×#2120C511yesmaximal, characteristic, direct product factor, Hall
25C5×#525C2×C311yescharacteristic
26C5×#413C2×C311yescharacteristic, complemented
27C5×#410C2×C311yescharacteristic, complemented
28C5×#352C2×C312yescomplemented
29C5×AΣL(1,𝔽16)C2×C312yescomplemented
30C24×C3×C5C2×C411yescharacteristic
31C3×C5×#41—41no—
32C3×C5×#41—41no—
33C2×C3×C5×Q8D411yescharacteristic
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4C2×C411yescharacteristic
37C2×C3×C5×D4D411yescharacteristic
38C2×C3×C5×D4C2311yescharacteristic
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—41no—
42C22×C3×C4×C5—41no—
43C22×C3×C4×C5C2×C411yescharacteristic
44C22×C3×C4×C5C2×C411yescharacteristic
45C22×C3×C4×C5D411yescharacteristic
46C22×C3×C4×C5D411yescharacteristic
47C3×C5×M16—22no—
48C3×C5×M16—41no—
49C2×C3×C5×C8—41no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4—41no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C5×C22⋊C4—22no—
59C3×C5×C22⋊C4C2×C411yescharacteristic
60C3×C5×C22⋊C4C2×C411yescharacteristic
61C3×C42×C5—21no—
62C3×C42×C5—41no—
63C3×#525C2×C511yescharacteristic
64C3×#413C2×C511yescharacteristic, complemented
65C3×#410C2×C511yescharacteristic, complemented
66C3×#352C2×C512yescomplemented
67C3×AΣL(1,𝔽16)C2×C512yescomplemented
68C2×C5×#41C22×C311yescharacteristic
69C23×C4×C5C3×C411yescharacteristic, complemented
70C2×C5×M16—21no—
71C5×#123C3×C411yescharacteristic, complemented
72C5×#123C22×C311yescharacteristic
73C5×C22⋊Q8—21no—
74C5×#121—21no—
75C5×#120C3×C411yescharacteristic, complemented
76C5×#120C22×C311yescharacteristic
77C4×C5×D4—21no—
78C4×C5×D4—21no—
79C5×#117—21no—
80C5×#117C3×C411yescharacteristic, complemented
81C5×#101—21no—
82C5×#100C22×C312yes—
83C5×#99—21no—
84C5×#99C22×C312yes—
85#2120C3×C511yescharacteristic, direct product factor, Sylow
86C23×C3×C5—21no—
87C23×C3×C5—41no—
88C23×C3×C5—42no—
89C23×C3×C5C22⋊C411yescharacteristic
90C23×C3×C5C22×C411yescharacteristic
91C23×C3×C5C2×D411yescharacteristic
92C3×C5×Q8—41no—
93C3×C5×D4—41no—
94C3×C5×D4—41no—
95C3×C5×D4—41no—
96C3×C5×D4—42no—
97C3×C5×D4—42no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—42no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5—42no—
112C2×C3×C4×C5C22⋊C411yescharacteristic
113C2×C3×C4×C5C22⋊C411yescharacteristic
114C2×C3×C4×C5C22⋊C411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C3×C5×C8—42no—
117C2×C3×#41C22×C511yescharacteristic
118C23×C3×C4C4×C511yescharacteristic, complemented
119C2×C3×M16—21no—
120C3×#123C4×C511yescharacteristic, complemented
121C3×#123C22×C511yescharacteristic
122C3×C22⋊Q8—21no—
123C3×#121—21no—
124C3×#120C4×C511yescharacteristic, complemented
125C3×#120C22×C511yescharacteristic
126C3×C4×D4—21no—
127C3×C4×D4—21no—
128C3×#117—21no—
129C3×#117C4×C511yescharacteristic, complemented
130C3×#101—21no—
131C3×#100C22×C512yes—
132C3×#99—21no—
133C3×#99C22×C512yes—
134C24×C5C2×C3×C411yescharacteristic
135C5×#41—41no—
136C5×#41—41no—
137C2×C5×Q8C3×D411yescharacteristic
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4C2×C3×C411yescharacteristic
141C2×C5×D4C3×D411yescharacteristic
142C2×C5×D4C23×C311yescharacteristic
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—41no—
146C22×C4×C5—41no—
147C22×C4×C5C2×C3×C411yescharacteristic
148C22×C4×C5C2×C3×C411yescharacteristic
149C22×C4×C5C3×D411yescharacteristic
150C22×C4×C5C3×D411yescharacteristic
151C5×M16—22no—
152C5×M16—41no—
153C2×C5×C8—41no—
154C5×C4⋊C4—21no—
155C5×C4⋊C4—21no—
156C5×C4⋊C4—21no—
157C5×C4⋊C4—41no—
158C5×C22⋊C4—21no—
159C5×C22⋊C4—21no—
160C5×C22⋊C4—21no—
161C5×C22⋊C4—21no—
162C5×C22⋊C4—22no—
163C5×C22⋊C4C2×C3×C411yescharacteristic
164C5×C22⋊C4C2×C3×C411yescharacteristic
165C42×C5—21no—
166C42×C5—41no—
167#525C2×C3×C511yescharacteristic
168#413C2×C3×C511yescharacteristic, complemented
169#410C2×C3×C511yescharacteristic, complemented
170#352C2×C3×C512yescomplemented
171AΣL(1,𝔽16)C2×C3×C512yescomplemented
172C22×C3×C5—22no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5—21no—
176C22×C3×C5—41no—
177C22×C3×C5—41no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5—42no—
181C22×C3×C5—42no—
182C22×C3×C5—41no—
183C22×C3×C5C2×C22⋊C411yescharacteristic
184C3×C4×C5—21no—
185C3×C4×C5—21no—
186C3×C4×C5—21no—
187C3×C4×C5—41no—
188C3×C4×C5—41no—
189C3×C4×C5—41no—
190C3×C4×C5—82no—
191C3×C4×C5#9911yescharacteristic
192C3×C4×C5#9911yescharacteristic
193C24×C3C2×C4×C511yescharacteristic
194C3×#41—41no—
195C3×#41—41no—
196C2×C3×Q8C5×D411yescharacteristic
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4C2×C4×C511yescharacteristic
200C2×C3×D4C5×D411yescharacteristic
201C2×C3×D4C23×C511yescharacteristic
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—41no—
205C22×C3×C4—41no—
206C22×C3×C4C2×C4×C511yescharacteristic
207C22×C3×C4C2×C4×C511yescharacteristic
208C22×C3×C4C5×D411yescharacteristic
209C22×C3×C4C5×D411yescharacteristic
210C3×M16—22no—
211C3×M16—41no—
212C2×C3×C8—41no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—41no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4C2×C4×C511yescharacteristic
223C3×C22⋊C4C2×C4×C511yescharacteristic
224C3×C42—21no—
225C3×C42—41no—
226C23×C5—21no—
227C23×C5—41no—
228C23×C5—42no—
229C23×C5C3×C22⋊C411yescharacteristic
230C23×C5C22×C3×C411yescharacteristic
231C23×C5C2×C3×D411yescharacteristic
232C5×Q8—41no—
233C5×D4—41no—
234C5×D4—41no—
235C5×D4—41no—
236C5×D4—42no—
237C5×D4—42no—
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—21no—
245C2×C4×C5—21no—
246C2×C4×C5—21no—
247C2×C4×C5—42no—
248C2×C4×C5—41no—
249C2×C4×C5—41no—
250C2×C4×C5—41no—
251C2×C4×C5—42no—
252C2×C4×C5C3×C22⋊C411yescharacteristic
253C2×C4×C5C3×C22⋊C411yescharacteristic
254C2×C4×C5C3×C22⋊C411yescharacteristic
255C2×C4×C5C2×C3×D411yescharacteristic
256C5×C8—42no—
257C2×#41C22×C3×C511yescharacteristic
258C23×C4C3×C4×C511yescharacteristic, complemented
259C2×M16—21no—
260#123C3×C4×C511yescharacteristic, complemented
261#123C22×C3×C511yescharacteristic
262C22⋊Q8—21no—
263#121—21no—
264#120C3×C4×C511yescharacteristic, complemented
265#120C22×C3×C511yescharacteristic
266C4×D4—21no—
267C4×D4—21no—
268#117—21no—
269#117C3×C4×C511yescharacteristic, complemented
270#101—21no—
271#100C22×C3×C512yes—
272#99—21no—
273#99C22×C3×C512yes—
274C2×C3×C5—21no—
275C2×C3×C5—41no—
276C2×C3×C5—41no—
277C2×C3×C5—42no—
278C2×C3×C5—41no—
279C2×C3×C5C2×#9911yescharacteristic
280C23×C3—21no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3C5×C22⋊C411yescharacteristic
284C23×C3C22×C4×C511yescharacteristic
285C23×C3C2×C5×D411yescharacteristic
286C3×Q8—41no—
287C3×D4—41no—
288C3×D4—41no—
289C3×D4—41no—
290C3×D4—42no—
291C3×D4—42no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4C5×C22⋊C411yescharacteristic
307C2×C3×C4C5×C22⋊C411yescharacteristic
308C2×C3×C4C5×C22⋊C411yescharacteristic
309C2×C3×C4C2×C5×D411yescharacteristic
310C3×C8—42no—
311C22×C5—22no—
312C22×C5—21no—
313C22×C5—21no—
314C22×C5—21no—
315C22×C5—41no—
316C22×C5—41no—
317C22×C5—41no—
318C22×C5—41no—
319C22×C5—42no—
320C22×C5—42no—
321C22×C5—41no—
322C22×C5C2×C3×C22⋊C411yescharacteristic
323C4×C5—21no—
324C4×C5—21no—
325C4×C5—21no—
326C4×C5—41no—
327C4×C5—41no—
328C4×C5—41no—
329C4×C5—82no—
330C4×C5C3×#9911yescharacteristic
331C4×C5C3×#9911yescharacteristic
332C24C2×C3×C4×C511yescharacteristic
333#41—41no—
334#41—41no—
335C2×Q8C3×C5×D411yescharacteristic
336C2×D4—21no—
337C2×D4—21no—
338C2×D4C2×C3×C4×C511yescharacteristic
339C2×D4C3×C5×D411yescharacteristic
340C2×D4C23×C3×C511yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4C2×C3×C4×C511yescharacteristic
346C22×C4C2×C3×C4×C511yescharacteristic
347C22×C4C3×C5×D411yescharacteristic
348C22×C4C3×C5×D411yescharacteristic
349M16—22no—
350M16—41no—
351C2×C8—41no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—21no—
360C22⋊C4—22no—
361C22⋊C4C2×C3×C4×C511yescharacteristic
362C22⋊C4C2×C3×C4×C511yescharacteristic
363C42—21no—
364C42—41no—
365C3×C5#212011yescharacteristic, direct product factor, Hall
366C22×C3—22no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—41no—
377C22×C3C2×C5×C22⋊C411yescharacteristic
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—82no—
385C3×C4C5×#9911yescharacteristic
386C3×C4C5×#9911yescharacteristic
387C2×C5—21no—
388C2×C5—41no—
389C2×C5—42no—
390C2×C5—41no—
391C2×C5—41no—
392C2×C5C2×C3×#9911yescharacteristic
393C23—21no—
394C23—41no—
395C23—42no—
396C23C3×C5×C22⋊C411yescharacteristic
397C23C22×C3×C4×C511yescharacteristic
398C23C2×C3×C5×D411yescharacteristic
399Q8—41no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—42no—
404D4—42no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4C3×C5×C22⋊C411yescharacteristic
420C2×C4C3×C5×C22⋊C411yescharacteristic
421C2×C4C3×C5×C22⋊C411yescharacteristic
422C2×C4C2×C3×C5×D411yescharacteristic
423C8—42no—
424C2×C3—21no—
425C2×C3—41no—
426C2×C3—42no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3C2×C5×#9911yescharacteristic
430C5C3×#212011yescharacteristic, direct product factor, Sylow
431C22—22no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—42no—
440C22—42no—
441C22—41no—
442C22C2×C3×C5×C22⋊C411yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—41no—
447C4—41no—
448C4—41no—
449C4—82no—
450C4C3×C5×#9911yescharacteristic
451C4C3×C5×#9911yescharacteristic
452C3C5×#212011yescharacteristic, direct product factor, Sylow
453C2—21no—
454C2—41no—
455C2—42no—
456C2—41no—
457C2—41no—
458C2C2×C3×C5×#9911yescharacteristic
subgroupG/H
centerC3×C4×C5#99
commutatorC23C22×C3×C4×C5
Frattini, cosocleC2×D4C23×C3×C5
socleC2×C3×C5C2×#99
prime radicalC3×C5×#525C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C5×#413C2semidirect
3. C3×C5×#410C2semidirect
4. C3×C5×#352C2semidirect
5. C3×C5×AΣL(1,𝔽16)C2semidirect
6. C5×#2120C3direct (N × H)
8. C23×C3×C4×C5C4semidirect
10. C3×C5×#123C4semidirect
14. C3×C5×#120C4semidirect
19. C3×C5×#117C4semidirect
24. C3×#2120C5direct (N × H)
26. C5×#413C2×C3semidirect
27. C5×#410C2×C3semidirect
28. C5×#352C2×C3semidirect
29. C5×AΣL(1,𝔽16)C2×C3semidirect
64. C3×#413C2×C5semidirect
65. C3×#410C2×C5semidirect
66. C3×#352C2×C5semidirect
67. C3×AΣL(1,𝔽16)C2×C5semidirect
69. C23×C4×C5C3×C4semidirect
71. C5×#123C3×C4semidirect
75. C5×#120C3×C4semidirect
80. C5×#117C3×C4semidirect
85. #2120C3×C5direct (N × H)
118. C23×C3×C4C4×C5semidirect
120. C3×#123C4×C5semidirect
124. C3×#120C4×C5semidirect
129. C3×#117C4×C5semidirect
168. #413C2×C3×C5semidirect
169. #410C2×C3×C5semidirect
170. #352C2×C3×C5semidirect
171. AΣL(1,𝔽16)C2×C3×C5semidirect
258. C23×C4C3×C4×C5semidirect
260. #123C3×C4×C5semidirect
264. #120C3×C4×C5semidirect
269. #117C3×C4×C5semidirect
365. C3×C5#2120direct (N × H)
430. C5C3×#2120direct (N × H)
452. C3C5×#2120direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5252
C3×C5×#4132
C3×C5×#4102
C3×C5×#3522
C3×C5×AΣL(1,𝔽16)2
C5×#21203
C3×#21205