Finite Group Databasebeta

← C2×C3×#64748 #14280772

Subgroups — C2×C3×#64748 #14280772

Order 1920.

656 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×C4⋊C4C211yesmaximal, characteristic
2C2×C3×C22⋊#168C212yesmaximal
3C3×#64748C214yesmaximal, direct product factor
4C2×#64748C311yesmaximal, characteristic, direct product factor, Hall
5C23×C3×C4×C5C411yescharacteristic
6C23×C3×C4×C5C411yescharacteristic
7C23×C3×C4×C5C2211yescharacteristic
8C2×C3×C5×C4⋊C4—22no—
9C2×C3×C5×C4⋊C4—22no—
10C2×C3×C5×C4⋊C4C412yes—
11C2×C3×C5×C4⋊C4C2212yes—
12C22×C3×#168—22no—
13C3×C22⋊#168—22no—
14C3×C22⋊#168C2214yes—
15C2×C3×#324—51nomaximal, Hall
16C22×C5×C4⋊C4C2×C311yescharacteristic
17C2×C22⋊#168C2×C312yes—
18#64748C2×C314yesdirect product factor
19C24×C3×C5C2×C411yescharacteristic
20C22×C3×C4×C5—21no—
21C22×C3×C4×C5—21no—
22C22×C3×C4×C5—22no—
23C22×C3×C4×C5—21no—
24C22×C3×C4×C5—21no—
25C22×C3×C4×C5—42no—
26C22×C3×C4×C5—42no—
27C22×C3×C4×C5C814yescomplemented
28C22×C3×C4×C5C2×C412yes—
29C22×C3×C4×C5C2×C412yes—
30C22×C3×C4×C5C2×C411yescharacteristic
31C22×C3×C4×C5D412yes—
32C22×C3×C4×C5D412yes—
33C22×C3×C4×C5C2311yescharacteristic
34C3×C5×C4⋊C4—24no—
35C3×C5×C4⋊C4—24no—
36C2×C3×#168—24no—
37C2×C3×#168—44no—
38C22×C3×C4⋊C4D511yescharacteristic
39C2×C3×C22⋊C8—52no—
40C3×#324—54no—
41C23×C4×C5C3×C411yescharacteristic
42C23×C4×C5C3×C411yescharacteristic
43C23×C4×C5C22×C311yescharacteristic
44C2×C5×C4⋊C4—22no—
45C2×C5×C4⋊C4—22no—
46C2×C5×C4⋊C4C3×C412yes—
47C2×C5×C4⋊C4C22×C312yes—
48C22×#168—22no—
49C22⋊#168—22no—
50C22⋊#168C22×C314yes—
51C2×#324—51noSylow
52C23×C3×C5—22no—
53C23×C3×C5—22no—
54C23×C3×C5C22⋊C411yescharacteristic
55C23×C3×C5C22⋊C411yescharacteristic
56C23×C3×C5C2×C812yes—
57C23×C3×C5M1612yes—
58C23×C3×C5C22×C411yescharacteristic
59C2×C3×C4×C5—24no—
60C2×C3×C4×C5—24no—
61C2×C3×C4×C5—24no—
62C2×C3×C4×C5—22no—
63C2×C3×C4×C5—22no—
64C2×C3×C4×C5—22no—
65C2×C3×C4×C5—41no—
66C2×C3×C4×C5—42no—
67C2×C3×C4×C5—41no—
68C2×C3×C4×C5—42no—
69C2×C3×C4×C5—42no—
70C2×C3×C4×C5—42no—
71C2×C3×C4×C5C22⋊C412yes—
72C2×C3×C4×C5C2×C814yescomplemented
73C2×C3×C4×C5C2×D412yes—
74C3×#168—44no—
75C23×C3×C4Dic511yescharacteristic
76C23×C3×C4Dic511yescharacteristic
77C23×C3×C4C2×D511yescharacteristic
78C22×C3×C8—102no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4Dic512yes—
82C2×C3×C4⋊C4C2×D512yes—
83C3×C22⋊C8—54no—
84C3×C22⋊C8—102no—
85C24×C5C2×C3×C411yescharacteristic
86C22×C4×C5—21no—
87C22×C4×C5—21no—
88C22×C4×C5—22no—
89C22×C4×C5—21no—
90C22×C4×C5—21no—
91C22×C4×C5—42no—
92C22×C4×C5—42no—
93C22×C4×C5C3×C814yescomplemented
94C22×C4×C5C2×C3×C412yes—
95C22×C4×C5C2×C3×C412yes—
96C22×C4×C5C2×C3×C411yescharacteristic
97C22×C4×C5C3×D412yes—
98C22×C4×C5C3×D412yes—
99C22×C4×C5C23×C311yescharacteristic
100C5×C4⋊C4—24no—
101C5×C4⋊C4—24no—
102C2×#168—24no—
103C2×#168—44no—
104C22×C4⋊C4C3×D511yescharacteristic
105C2×C22⋊C8—52no—
106#324—54no—
107C22×C3×C5—24no—
108C22×C3×C5—24no—
109C22×C3×C5—21no—
110C22×C3×C5—21no—
111C22×C3×C5—21no—
112C22×C3×C5—21no—
113C22×C3×C5C22⋊C812yes—
114C22×C3×C5C22⋊C812yes—
115C22×C3×C5#9912yes—
116C22×C3×C5#10112yes—
117C22×C3×C5C2×C22⋊C411yescharacteristic
118C22×C3×C5C22×C811yescharacteristic
119C22×C3×C5C2×M1611yescharacteristic
120C3×C4×C5—24no—
121C3×C4×C5—22no—
122C3×C4×C5—22no—
123C3×C4×C5—42no—
124C24×C3C2×Dic511yescharacteristic
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4#16814yescomplemented
133C22×C3×C4C2×Dic512yes—
134C22×C3×C4C2×Dic512yes—
135C22×C3×C4C2×Dic511yescharacteristic
136C22×C3×C4C22⋊D512yes—
137C22×C3×C4C22⋊D512yes—
138C22×C3×C4C22×D511yescharacteristic
139C2×C3×C8—104no—
140C2×C3×C8—204no—
141C3×C4⋊C4—24no—
142C3×C4⋊C4—24no—
143C23×C5—22no—
144C23×C5—22no—
145C23×C5C3×C22⋊C411yescharacteristic
146C23×C5C3×C22⋊C411yescharacteristic
147C23×C5C2×C3×C812yes—
148C23×C5C3×M1612yes—
149C23×C5C22×C3×C411yescharacteristic
150C2×C4×C5—24no—
151C2×C4×C5—24no—
152C2×C4×C5—24no—
153C2×C4×C5—22no—
154C2×C4×C5—22no—
155C2×C4×C5—22no—
156C2×C4×C5—41no—
157C2×C4×C5—42no—
158C2×C4×C5—41no—
159C2×C4×C5—42no—
160C2×C4×C5—42no—
161C2×C4×C5—42no—
162C2×C4×C5C3×C22⋊C412yes—
163C2×C4×C5C2×C3×C814yescomplemented
164C2×C4×C5C2×C3×D412yes—
165#168—44no—
166C23×C4C3×Dic511yescharacteristic
167C23×C4C3×Dic511yescharacteristic
168C23×C4C2×C3×D511yescharacteristic
169C22×C8—102no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C3×Dic512yes—
173C2×C4⋊C4C2×C3×D512yes—
174C22⋊C8—54no—
175C22⋊C8—102no—
176C2×C3×C5—22no—
177C2×C3×C5—21no—
178C2×C3×C5—21no—
179C2×C3×C5#32414yescomplemented
180C2×C3×C5C2×C22⋊C811yescharacteristic
181C2×C3×C5C2×#9911yescharacteristic
182C2×C3×C5C2×#10111yescharacteristic
183C23×C3—22no—
184C23×C3—22no—
185C23×C3C2×#16812yes—
186C23×C3#68112yes—
187C23×C3C22⋊Dic511yescharacteristic
188C23×C3C22⋊Dic511yescharacteristic
189C23×C3C22×Dic511yescharacteristic
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4C2×#16814yescomplemented
203C2×C3×C4C22⋊Dic512yes—
204C2×C3×C4C2×C22⋊D512yes—
205C3×C8—204no—
206C22×C5—21no—
207C22×C5—21no—
208C22×C5—21no—
209C22×C5—21no—
210C22×C5—24no—
211C22×C5—24no—
212C22×C5C3×C22⋊C812yes—
213C22×C5C3×C22⋊C812yes—
214C22×C5C3×#9912yes—
215C22×C5C3×#10112yes—
216C22×C5C2×C3×C22⋊C411yescharacteristic
217C22×C5C22×C3×C811yescharacteristic
218C22×C5C2×C3×M1611yescharacteristic
219C4×C5—24no—
220C4×C5—22no—
221C4×C5—22no—
222C4×C5—42no—
223C24C2×C3×Dic511yescharacteristic
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C3×#16814yescomplemented
232C22×C4C2×C3×Dic512yes—
233C22×C4C2×C3×Dic512yes—
234C22×C4C2×C3×Dic511yescharacteristic
235C22×C4C3×C22⋊D512yes—
236C22×C4C3×C22⋊D512yes—
237C22×C4C22×C3×D511yescharacteristic
238C2×C8—104no—
239C2×C8—204no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C3×C5C2×#32411yescharacteristic, complemented, Hall
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—24no—
248C22×C3—24no—
249C22×C3C22⋊#16812yes—
250C22×C3C22⋊#16812yes—
251C22×C3#396612yes—
252C22×C3#396812yes—
253C22×C3C22×#16811yescharacteristic
254C22×C3C2×#68111yescharacteristic
255C22×C3C2×C22⋊Dic511yescharacteristic
256C3×C4—24no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—42no—
260C2×C5—22no—
261C2×C5—21no—
262C2×C5—21no—
263C2×C5C3×#32414yescomplemented
264C2×C5C2×C3×C22⋊C811yescharacteristic
265C2×C5C2×C3×#9911yescharacteristic
266C2×C5C2×C3×#10111yescharacteristic
267C23—22no—
268C23—22no—
269C23C2×C3×#16812yes—
270C23C3×#68112yes—
271C23C3×C22⋊Dic511yescharacteristic
272C23C3×C22⋊Dic511yescharacteristic
273C23C22×C3×Dic511yescharacteristic
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4C2×C3×#16814yescomplemented
287C2×C4C3×C22⋊Dic512yes—
288C2×C4C2×C3×C22⋊D512yes—
289C8—204no—
290C2×C3—22no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3#6474814yesdirect product factor
294C2×C3C2×C22⋊#16811yescharacteristic
295C2×C3C2×#396611yescharacteristic
296C2×C3C2×#396811yescharacteristic
297C5C2×C3×#32411yescharacteristic, complemented, Sylow
298C22—24no—
299C22—24no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22C3×C22⋊#16812yes—
305C22C3×C22⋊#16812yes—
306C22C3×#396612yes—
307C22C3×#396812yes—
308C22C22×C3×#16811yescharacteristic
309C22C2×C3×#68111yescharacteristic
310C22C2×C3×C22⋊Dic511yescharacteristic
311C4—24no—
312C4—22no—
313C4—22no—
314C4—42no—
315C3C2×#6474811yescharacteristic, direct product factor, Sylow
316C2—22no—
317C2—21no—
318C2—21no—
319C2C3×#6474814yesdirect product factor
320C2C2×C3×C22⋊#16811yescharacteristic
321C2C2×C3×#396611yescharacteristic
322C2C2×C3×#396811yescharacteristic
subgroupG/H
centerC23×C3C22⋊Dic5
commutatorC22×C5C22×C3×C8
FrattiniC22×C4C22×C3×D5
FittingC22×C3×C5×C4⋊C4C2
socleC23×C3×C5C22⋊C4
prime radicalC24×C3×C5C2×C4
cosocleC22×C4×C5C23×C3
hypercenterC22×C3×C4⋊C4D5
hypocenterC5C2×C3×#324
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C22×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#64748C2direct (N × H)
4. C2×#64748C3direct (N × H)
18. #64748C2×C3direct (N × H)
27. C22×C3×C4×C5C8semidirect
72. C2×C3×C4×C5C2×C8semidirect
93. C22×C4×C5C3×C8semidirect
132. C22×C3×C4#168semidirect
163. C2×C4×C5C2×C3×C8semidirect
179. C2×C3×C5#324semidirect
202. C2×C3×C4C2×#168semidirect
231. C22×C4C3×#168semidirect
242. C3×C5C2×#324semidirect
263. C2×C5C3×#324semidirect
286. C2×C4C2×C3×#168semidirect
293. C2×C3#64748direct (N × H)
297. C5C2×C3×#324semidirect
315. C3C2×#64748direct (N × H)
319. C2C3×#64748direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×C4⋊C42
C2×C3×C22⋊#1682
C3×#647482
C2×#647483
C2×C3×#3245