Finite Group Databasebeta

← C2×C5×A4×M16 #14511363

Subgroups — C2×C5×A4×M16 #14511363

Order 1920.

800 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C5×A4C211yesmaximal, characteristic
2C5×A4×M16C214yesmaximal, direct product factor
3C2×C5×C8×A4C212yesmaximal, complemented
4C23×C5×M16C311yesmaximal, characteristic, complemented, Hall
5C23×C5×A4C411yescharacteristic
6C2×C4×C5×A4C412yes—
7C2×C4×C5×A4C411yescharacteristic
8C2×C4×C5×A4C2211yescharacteristic
9C2×C4×C5×A4C2212yes—
10C2×C3×C5×M16—41nomaximal
11C5×C8×A4C2214yescomplemented
12C2×A4×M16C511yesmaximal, characteristic, direct product factor, Hall
13C24×C4×C5C2×C311yescharacteristic
14C22×C5×M16—31no—
15C22×C5×M16—32no—
16C22×C5×M16—31no—
17C22×C5×M16—34no—
18C22×C5×M16C2×C314yescomplemented
19C23×C5×C8C2×C312yescomplemented
20C22×C5×A4—22no—
21C22×C5×A4C2×C412yes—
22C22×C5×A4C2×C411yescharacteristic
23C22×C3×C4×C5—41no—
24C4×C5×A4C2×C411yescharacteristic
25C4×C5×A4C2×C412yes—
26C4×C5×A4C2311yescharacteristic
27C3×C5×M16—44no—
28C2×C3×C5×C8—42no—
29C22×C4×A4C2×C511yescharacteristic
30A4×M16C2×C514yesdirect product factor
31C2×C8×A4C2×C512yescomplemented
32C25×C5C3×C411yescharacteristic
33C23×C4×C5—31no—
34C23×C4×C5—32no—
35C23×C4×C5—31no—
36C23×C4×C5—31no—
37C23×C4×C5—31no—
38C23×C4×C5—32no—
39C23×C4×C5C3×C412yes—
40C23×C4×C5C3×C411yescharacteristic
41C23×C4×C5C22×C311yescharacteristic
42C23×C4×C5C22×C312yes—
43C2×C5×M16—34no—
44C2×C5×M16—38no—
45C2×C5×M16—32no—
46C2×C5×M16—31no—
47C2×C5×M16—34no—
48C2×C5×M16—34no—
49C2×C5×M16—38no—
50C2×C5×M16—34no—
51C2×C5×M16—32no—
52C2×C5×M16A411yescharacteristic, direct product factor
53C22×C5×C8—32no—
54C22×C5×C8—32no—
55C22×C5×C8—34no—
56C22×C5×C8C22×C314yescomplemented
57C23×M16C3×C511yescharacteristic, complemented, Sylow
58C23×C3×C5—41no—
59C2×C5×A4—22no—
60C2×C5×A4M1612yesdirect product factor
61C2×C5×A4C22×C411yescharacteristic
62C2×C3×C4×C5—41no—
63C2×C3×C4×C5—42no—
64C2×C3×C4×C5—42no—
65C2×C3×C4×C5—41no—
66C3×C5×C8—44no—
67C23×A4C4×C511yescharacteristic
68C2×C4×A4C4×C512yes—
69C2×C4×A4C4×C511yescharacteristic
70C2×C4×A4C22×C511yescharacteristic
71C2×C4×A4C22×C512yes—
72C2×C3×M16—41no—
73C8×A4C22×C514yescomplemented
74C24×C5—22no—
75C24×C5—32no—
76C24×C5—31no—
77C24×C5—31no—
78C24×C5—62no—
79C24×C5C2×C3×C412yes—
80C24×C5C2×C3×C411yescharacteristic
81C22×C4×C5—32no—
82C22×C4×C5—34no—
83C22×C4×C5—32no—
84C22×C4×C5—32no—
85C22×C4×C5—34no—
86C22×C4×C5—31no—
87C22×C4×C5—32no—
88C22×C4×C5—31no—
89C22×C4×C5—31no—
90C22×C4×C5—32no—
91C22×C4×C5—32no—
92C22×C4×C5—32no—
93C22×C4×C5—31no—
94C22×C4×C5—31no—
95C22×C4×C5—31no—
96C22×C4×C5—32no—
97C22×C4×C5—31no—
98C22×C4×C5—32no—
99C22×C4×C5—32no—
100C22×C4×C5—32no—
101C22×C4×C5—32no—
102C22×C4×C5—31no—
103C22×C4×C5—32no—
104C22×C4×C5—31no—
105C22×C4×C5—32no—
106C22×C4×C5C2×C3×C411yescharacteristic
107C22×C4×C5C2×C3×C412yes—
108C22×C4×C5C2×A411yescharacteristic
109C22×C4×C5C23×C311yescharacteristic
110C5×M16—38no—
111C5×M16—34no—
112C5×M16—38no—
113C5×M16C2×A414yesdirect product factor
114C2×C5×C8—34no—
115C2×C5×C8—32no—
116C2×C5×C8—34no—
117C2×C5×C8—34no—
118C2×C5×C8—34no—
119C2×C5×C8C2×A412yescomplemented
120C24×C4C2×C3×C511yescharacteristic
121C22×M16—31no—
122C22×M16—32no—
123C22×M16—31no—
124C22×M16—34no—
125C22×M16C2×C3×C514yescomplemented
126C23×C8C2×C3×C512yescomplemented
127C22×C3×C5—41no—
128C22×C3×C5—42no—
129C22×C3×C5—82no—
130C5×A4C2×M1611yescharacteristic, direct product factor
131C3×C4×C5—42no—
132C3×C4×C5—41no—
133C3×C4×C5—41no—
134C22×A4—22no—
135C22×A4C2×C4×C512yes—
136C22×A4C2×C4×C511yescharacteristic
137C22×C3×C4—41no—
138C4×A4C2×C4×C511yescharacteristic
139C4×A4C2×C4×C512yes—
140C4×A4C23×C511yescharacteristic
141C3×M16—44no—
142C2×C3×C8—42no—
143C23×C5—22no—
144C23×C5—31no—
145C23×C5—32no—
146C23×C5—31no—
147C23×C5—32no—
148C23×C5—32no—
149C23×C5—32no—
150C23×C5—31no—
151C23×C5—32no—
152C23×C5—62no—
153C23×C5—62no—
154C23×C5—62no—
155C23×C5—64no—
156C23×C5—62no—
157C23×C5—62no—
158C23×C5—64no—
159C23×C5C3×M1612yescomplemented
160C23×C5C4×A411yescharacteristic
161C23×C5C22×C3×C411yescharacteristic
162C2×C4×C5—32no—
163C2×C4×C5—32no—
164C2×C4×C5—34no—
165C2×C4×C5—34no—
166C2×C4×C5—32no—
167C2×C4×C5—32no—
168C2×C4×C5—31no—
169C2×C4×C5—31no—
170C2×C4×C5—32no—
171C2×C4×C5—32no—
172C2×C4×C5—32no—
173C2×C4×C5—31no—
174C2×C4×C5—32no—
175C2×C4×C5—32no—
176C2×C4×C5—31no—
177C2×C4×C5—31no—
178C2×C4×C5—31no—
179C2×C4×C5—32no—
180C2×C4×C5—31no—
181C2×C4×C5—32no—
182C2×C4×C5—31no—
183C2×C4×C5C4×A411yescharacteristic
184C2×C4×C5C4×A412yes—
185C2×C4×C5C22×A412yes—
186C2×C4×C5C22×A411yescharacteristic
187C5×C8—34no—
188C5×C8C22×A414yescomplemented
189C25C3×C4×C511yescharacteristic
190C23×C4—31no—
191C23×C4—31no—
192C23×C4—32no—
193C23×C4—31no—
194C23×C4—31no—
195C23×C4—32no—
196C23×C4C3×C4×C512yes—
197C23×C4C3×C4×C511yescharacteristic
198C23×C4C22×C3×C511yescharacteristic
199C23×C4C22×C3×C512yes—
200C2×M16—34no—
201C2×M16—38no—
202C2×M16—32no—
203C2×M16—31no—
204C2×M16—34no—
205C2×M16—34no—
206C2×M16—38no—
207C2×M16—34no—
208C2×M16—32no—
209C2×M16C5×A411yescharacteristic, direct product factor
210C22×C8—32no—
211C22×C8—32no—
212C22×C8—34no—
213C22×C8C22×C3×C514yescomplemented
214C2×C3×C5—41no—
215C2×C3×C5—42no—
216C2×C3×C5—82no—
217C23×C3—41no—
218C2×A4—22no—
219C2×A4C5×M1612yesdirect product factor
220C2×A4C22×C4×C511yescharacteristic
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C3×C8—44no—
226C22×C5—22no—
227C22×C5—32no—
228C22×C5—31no—
229C22×C5—32no—
230C22×C5—31no—
231C22×C5—31no—
232C22×C5—32no—
233C22×C5—32no—
234C22×C5—32no—
235C22×C5—62no—
236C22×C5—64no—
237C22×C5—62no—
238C22×C5—62no—
239C22×C5—62no—
240C22×C5—64no—
241C22×C5—62no—
242C22×C5C2×C3×M1611yescharacteristic, complemented
243C22×C5C2×C4×A412yes—
244C22×C5C2×C4×A411yescharacteristic
245C4×C5—31no—
246C4×C5—31no—
247C4×C5—32no—
248C4×C5C2×C4×A412yes—
249C4×C5C2×C4×A411yescharacteristic
250C4×C5C23×A411yescharacteristic
251C24—22no—
252C24—32no—
253C24—31no—
254C24—31no—
255C24—62no—
256C24C2×C3×C4×C512yes—
257C24C2×C3×C4×C511yescharacteristic
258C22×C4—32no—
259C22×C4—34no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—34no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—32no—
266C22×C4—32no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—32no—
270C22×C4—32no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—32no—
274C22×C4—31no—
275C22×C4—32no—
276C22×C4—32no—
277C22×C4—32no—
278C22×C4—32no—
279C22×C4—31no—
280C22×C4—32no—
281C22×C4—31no—
282C22×C4—32no—
283C22×C4C2×C3×C4×C511yescharacteristic
284C22×C4C2×C3×C4×C512yes—
285C22×C4C2×C5×A411yescharacteristic
286C22×C4C23×C3×C511yescharacteristic
287M16—38no—
288M16—34no—
289M16—38no—
290M16C2×C5×A414yesdirect product factor
291C2×C8—34no—
292C2×C8—32no—
293C2×C8—34no—
294C2×C8—34no—
295C2×C8—34no—
296C2×C8C2×C5×A412yescomplemented
297C3×C5—41noHall
298C22×C3—41no—
299C22×C3—42no—
300C22×C3—82no—
301A4C2×C5×M1611yescharacteristic, direct product factor
302C3×C4—41no—
303C3×C4—42no—
304C3×C4—41no—
305C2×C5—22no—
306C2×C5—32no—
307C2×C5—31no—
308C2×C5—31no—
309C2×C5—62no—
310C2×C5A4×M1612yesdirect product factor
311C2×C5C22×C4×A411yescharacteristic
312C23—22no—
313C23—31no—
314C23—32no—
315C23—31no—
316C23—32no—
317C23—32no—
318C23—32no—
319C23—32no—
320C23—31no—
321C23—62no—
322C23—62no—
323C23—62no—
324C23—64no—
325C23—62no—
326C23—62no—
327C23—64no—
328C23C3×C5×M1612yescomplemented
329C23C4×C5×A411yescharacteristic
330C23C22×C3×C4×C511yescharacteristic
331C2×C4—32no—
332C2×C4—32no—
333C2×C4—34no—
334C2×C4—34no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—32no—
341C2×C4—32no—
342C2×C4—31no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—32no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—32no—
349C2×C4—31no—
350C2×C4—32no—
351C2×C4—31no—
352C2×C4C4×C5×A412yes—
353C2×C4C4×C5×A411yescharacteristic
354C2×C4C22×C5×A411yescharacteristic
355C2×C4C22×C5×A412yes—
356C8—34no—
357C8C22×C5×A414yescomplemented
358C2×C3—41no—
359C2×C3—42no—
360C2×C3—82no—
361C5C2×A4×M1611yescharacteristic, direct product factor, Sylow
362C22—22no—
363C22—32no—
364C22—31no—
365C22—32no—
366C22—31no—
367C22—31no—
368C22—32no—
369C22—32no—
370C22—32no—
371C22—62no—
372C22—62no—
373C22—64no—
374C22—62no—
375C22—62no—
376C22—64no—
377C22—62no—
378C22C2×C3×C5×M1611yescharacteristic, complemented
379C22C2×C4×C5×A412yes—
380C22C2×C4×C5×A411yescharacteristic
381C4—31no—
382C4—32no—
383C4—31no—
384C4C2×C4×C5×A411yescharacteristic
385C4C2×C4×C5×A412yes—
386C4C23×C5×A411yescharacteristic
387C3—41noSylow
388C2—22no—
389C2—31no—
390C2—32no—
391C2—31no—
392C2—62no—
393C2C5×A4×M1612yesdirect product factor
394C2C22×C4×C5×A411yescharacteristic
subgroupG/H
centerC2×C4×C5C22×A4
commutatorC23C22×C3×C4×C5
FrattiniC4C23×C5×A4
FittingC23×C5×M16C3
socleC24×C5C2×C3×C4
prime radicalC23×C5×A4C4
cosocleC22×C4C23×C3×C5
hypercenterC2×C5×M16A4
hypocenterC22C2×C3×C5×M16
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C4×C5 ◁ C2×C5×M16

Complements

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

NH ≅ G/Ntype
2. C5×A4×M16C2direct (N × H)
3. C2×C5×C8×A4C2semidirect
4. C23×C5×M16C3semidirect
11. C5×C8×A4C22semidirect
12. C2×A4×M16C5direct (N × H)
18. C22×C5×M16C2×C3semidirect
19. C23×C5×C8C2×C3semidirect
30. A4×M16C2×C5direct (N × H)
31. C2×C8×A4C2×C5semidirect
52. C2×C5×M16A4direct (N × H)
56. C22×C5×C8C22×C3semidirect
57. C23×M16C3×C5semidirect
60. C2×C5×A4M16direct (N × H)
73. C8×A4C22×C5semidirect
113. C5×M16C2×A4direct (N × H)
119. C2×C5×C8C2×A4semidirect
125. C22×M16C2×C3×C5semidirect
126. C23×C8C2×C3×C5semidirect
130. C5×A4C2×M16direct (N × H)
159. C23×C5C3×M16semidirect
188. C5×C8C22×A4semidirect
209. C2×M16C5×A4direct (N × H)
213. C22×C8C22×C3×C5semidirect
219. C2×A4C5×M16direct (N × H)
242. C22×C5C2×C3×M16semidirect
290. M16C2×C5×A4direct (N × H)
296. C2×C8C2×C5×A4semidirect
301. A4C2×C5×M16direct (N × H)
310. C2×C5A4×M16direct (N × H)
328. C23C3×C5×M16semidirect
357. C8C22×C5×A4semidirect
361. C5C2×A4×M16direct (N × H)
378. C22C2×C3×C5×M16semidirect
393. C2C5×A4×M16direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C5×A42
C5×A4×M162
C2×C5×C8×A42
C23×C5×M163
C2×C3×C5×M164
C2×A4×M165