Finite Group Databasebeta

← C2×C3×C5×#354 #14275159

Subgroups — C2×C3×C5×#354 #14275159

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#124C211yesmaximal, characteristic
2C2×C3×C5×#99C212yesmaximal
3C3×C5×#354C214yesmaximal, direct product factor
4C2×C5×#354C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5×Q8C411yescharacteristic, complemented
6C22×C3×C5×D4C2211yescharacteristic
7C3×C5×#124—22no—
8C3×C5×#124C412yescomplemented
9C3×C5×#124C2212yes—
10C2×C3×C5×C22⋊C4—22no—
11C2×C3×C5×C22⋊C4—22no—
12C2×C3×C42×C5C411yescharacteristic, complemented
13C3×C5×#99—22no—
14C3×C5×#99C2214yes—
15C2×C3×#354C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×#124C2×C311yescharacteristic
17C2×C5×#99C2×C312yes—
18C5×#354C2×C314yesdirect product factor
19C24×C3×C5D412yes—
20C2×C3×C5×Q8—21no—
21C2×C3×C5×Q8—41no—
22C2×C3×C5×Q8—41no—
23C2×C3×C5×Q8C2×C412yescomplemented
24C2×C3×C5×D4—41no—
25C2×C3×C5×D4—41no—
26C2×C3×C5×D4C2×C411yescharacteristic
27C2×C3×C5×D4D412yes—
28C2×C3×C5×D4C2311yescharacteristic
29C22×C3×C4×C5—21no—
30C22×C3×C4×C5—21no—
31C22×C3×C4×C5—42no—
32C22×C3×C4×C5C2×C411yescharacteristic
33C3×C5×C22⋊C4—24no—
34C3×C5×C22⋊C4—24no—
35C3×C5×C22⋊C4—24no—
36C3×C5×C22⋊C4—42no—
37C3×C42×C5—21no—
38C3×C42×C5C2×C412yescomplemented
39C2×C3×#124C2×C511yescharacteristic
40C2×C3×#99C2×C512yes—
41C3×#354C2×C514yesdirect product factor
42C22×C5×Q8C3×C411yescharacteristic, complemented
43C22×C5×D4C22×C311yescharacteristic
44C5×#124—22no—
45C5×#124C3×C412yescomplemented
46C5×#124C22×C312yes—
47C2×C5×C22⋊C4—22no—
48C2×C5×C22⋊C4—22no—
49C2×C42×C5C3×C411yescharacteristic, complemented
50C5×#99—22no—
51C5×#99C22×C314yes—
52C2×#354C3×C511yescharacteristic, direct product factor, Sylow
53C23×C3×C5—22no—
54C23×C3×C5—22no—
55C23×C3×C5—44no—
56C23×C3×C5C22⋊C412yes—
57C23×C3×C5C22⋊C411yescharacteristic
58C23×C3×C5C2×D412yes—
59C3×C5×Q8—42no—
60C3×C5×Q8—42no—
61C3×C5×D4—41no—
62C3×C5×D4—42no—
63C3×C5×D4—41no—
64C2×C3×C4×C5—21no—
65C2×C3×C4×C5—21no—
66C2×C3×C4×C5—22no—
67C2×C3×C4×C5—22no—
68C2×C3×C4×C5—42no—
69C2×C3×C4×C5—41no—
70C2×C3×C4×C5—41no—
71C2×C3×C4×C5—44no—
72C2×C3×C4×C5—84no—
73C2×C3×C4×C5C22⋊C411yescharacteristic
74C2×C3×C4×C5C22×C411yescharacteristic
75C22×C3×Q8C4×C511yescharacteristic, complemented
76C22×C3×D4C22×C511yescharacteristic
77C3×#124—22no—
78C3×#124C4×C512yescomplemented
79C3×#124C22×C512yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42C4×C511yescharacteristic, complemented
83C3×#99—22no—
84C3×#99C22×C514yes—
85C24×C5C3×D412yes—
86C2×C5×Q8—21no—
87C2×C5×Q8—41no—
88C2×C5×Q8—41no—
89C2×C5×Q8C2×C3×C412yescomplemented
90C2×C5×D4—41no—
91C2×C5×D4—41no—
92C2×C5×D4C2×C3×C411yescharacteristic
93C2×C5×D4C3×D412yes—
94C2×C5×D4C23×C311yescharacteristic
95C22×C4×C5—21no—
96C22×C4×C5—21no—
97C22×C4×C5—42no—
98C22×C4×C5C2×C3×C411yescharacteristic
99C5×C22⋊C4—24no—
100C5×C22⋊C4—24no—
101C5×C22⋊C4—24no—
102C5×C22⋊C4—42no—
103C42×C5—21no—
104C42×C5C2×C3×C412yescomplemented
105C2×#124C2×C3×C511yescharacteristic
106C2×#99C2×C3×C512yes—
107#354C2×C3×C514yesdirect product factor
108C22×C3×C5—22no—
109C22×C3×C5—22no—
110C22×C3×C5—22no—
111C22×C3×C5—42no—
112C22×C3×C5—42no—
113C22×C3×C5—44no—
114C22×C3×C5—82no—
115C22×C3×C5#9911yescharacteristic
116C22×C3×C5#9911yescharacteristic
117C22×C3×C5C2×C22⋊C411yescharacteristic
118C3×C4×C5—21no—
119C3×C4×C5—21no—
120C3×C4×C5—42no—
121C3×C4×C5—42no—
122C3×C4×C5—84no—
123C24×C3C5×D412yes—
124C2×C3×Q8—21no—
125C2×C3×Q8—41no—
126C2×C3×Q8—41no—
127C2×C3×Q8C2×C4×C512yescomplemented
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4C2×C4×C511yescharacteristic
131C2×C3×D4C5×D412yes—
132C2×C3×D4C23×C511yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—42no—
136C22×C3×C4C2×C4×C511yescharacteristic
137C3×C22⋊C4—24no—
138C3×C22⋊C4—24no—
139C3×C22⋊C4—24no—
140C3×C22⋊C4—42no—
141C3×C42—21no—
142C3×C42C2×C4×C512yescomplemented
143C23×C5—22no—
144C23×C5—22no—
145C23×C5—44no—
146C23×C5C3×C22⋊C411yescharacteristic
147C23×C5C3×C22⋊C412yes—
148C23×C5C2×C3×D412yes—
149C5×Q8—42no—
150C5×Q8—42no—
151C5×D4—41no—
152C5×D4—42no—
153C5×D4—41no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—22no—
157C2×C4×C5—22no—
158C2×C4×C5—42no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5—44no—
162C2×C4×C5—84no—
163C2×C4×C5C3×C22⋊C411yescharacteristic
164C2×C4×C5C22×C3×C411yescharacteristic
165C22×Q8C3×C4×C511yescharacteristic, complemented
166C22×D4C22×C3×C511yescharacteristic
167#124—22no—
168#124C3×C4×C512yescomplemented
169#124C22×C3×C512yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C42C3×C4×C511yescharacteristic, complemented
173#99—22no—
174#99C22×C3×C514yes—
175C2×C3×C5—21no—
176C2×C3×C5—21no—
177C2×C3×C5—42no—
178C2×C3×C5—42no—
179C2×C3×C5#35412yesdirect product factor
180C2×C3×C5C2×#9911yescharacteristic
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—44no—
184C23×C3C5×C22⋊C411yescharacteristic
185C23×C3C5×C22⋊C412yes—
186C23×C3C2×C5×D412yes—
187C3×Q8—42no—
188C3×Q8—42no—
189C3×D4—41no—
190C3×D4—42no—
191C3×D4—41no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—44no—
200C2×C3×C4—84no—
201C2×C3×C4C5×C22⋊C411yescharacteristic
202C2×C3×C4C22×C4×C511yescharacteristic
203C22×C5—22no—
204C22×C5—22no—
205C22×C5—22no—
206C22×C5—42no—
207C22×C5—42no—
208C22×C5—44no—
209C22×C5—82no—
210C22×C5C3×#9911yescharacteristic
211C22×C5C3×#9911yescharacteristic
212C22×C5C2×C3×C22⋊C411yescharacteristic
213C4×C5—21no—
214C4×C5—21no—
215C4×C5—42no—
216C4×C5—42no—
217C4×C5—84no—
218C24C3×C5×D412yes—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8C2×C3×C4×C512yescomplemented
223C2×D4—41no—
224C2×D4—41no—
225C2×D4C2×C3×C4×C511yescharacteristic
226C2×D4C3×C5×D412yes—
227C2×D4C23×C3×C511yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4C2×C3×C4×C511yescharacteristic
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—42no—
236C42—21no—
237C42C2×C3×C4×C512yescomplemented
238C3×C5C2×#35411yescharacteristic, direct product factor, Hall
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—42no—
243C22×C3—42no—
244C22×C3—44no—
245C22×C3—82no—
246C22×C3C5×#9911yescharacteristic
247C22×C3C5×#9911yescharacteristic
248C22×C3C2×C5×C22⋊C411yescharacteristic
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—42no—
252C3×C4—42no—
253C3×C4—84no—
254C2×C5—21no—
255C2×C5—21no—
256C2×C5—42no—
257C2×C5—42no—
258C2×C5C3×#35412yesdirect product factor
259C2×C5C2×C3×#9911yescharacteristic
260C23—22no—
261C23—22no—
262C23—44no—
263C23C3×C5×C22⋊C411yescharacteristic
264C23C3×C5×C22⋊C412yes—
265C23C2×C3×C5×D412yes—
266Q8—42no—
267Q8—42no—
268D4—41no—
269D4—42no—
270D4—41no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—84no—
280C2×C4C3×C5×C22⋊C411yescharacteristic
281C2×C4C22×C3×C4×C511yescharacteristic
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—42no—
285C2×C3—42no—
286C2×C3C5×#35412yesdirect product factor
287C2×C3C2×C5×#9911yescharacteristic
288C5C2×C3×#35411yescharacteristic, direct product factor, Sylow
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—44no—
295C22—82no—
296C22C3×C5×#9911yescharacteristic
297C22C3×C5×#9911yescharacteristic
298C22C2×C3×C5×C22⋊C411yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—84no—
304C3C2×C5×#35411yescharacteristic, direct product factor, Sylow
305C2—21no—
306C2—21no—
307C2—42no—
308C2—42no—
309C2C3×C5×#35412yesdirect product factor
310C2C2×C3×C5×#9911yescharacteristic
subgroupG/H
center, socleC22×C3×C5#99
commutatorC2×C4C22×C3×C4×C5
Frattini, cosocleC2×D4C23×C3×C5
prime radicalC22×C3×C5×D4C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C23×C3×C5 ◁ C22×C3×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×C5×#354C2direct (N × H)
4. C2×C5×#354C3direct (N × H)
5. C22×C3×C5×Q8C4semidirect
8. C3×C5×#124C4semidirect
12. C2×C3×C42×C5C4semidirect
15. C2×C3×#354C5direct (N × H)
18. C5×#354C2×C3direct (N × H)
23. C2×C3×C5×Q8C2×C4semidirect
38. C3×C42×C5C2×C4semidirect
41. C3×#354C2×C5direct (N × H)
42. C22×C5×Q8C3×C4semidirect
45. C5×#124C3×C4semidirect
49. C2×C42×C5C3×C4semidirect
52. C2×#354C3×C5direct (N × H)
75. C22×C3×Q8C4×C5semidirect
78. C3×#124C4×C5semidirect
82. C2×C3×C42C4×C5semidirect
89. C2×C5×Q8C2×C3×C4semidirect
104. C42×C5C2×C3×C4semidirect
107. #354C2×C3×C5direct (N × H)
127. C2×C3×Q8C2×C4×C5semidirect
142. C3×C42C2×C4×C5semidirect
165. C22×Q8C3×C4×C5semidirect
168. #124C3×C4×C5semidirect
172. C2×C42C3×C4×C5semidirect
179. C2×C3×C5#354direct (N × H)
222. C2×Q8C2×C3×C4×C5semidirect
237. C42C2×C3×C4×C5semidirect
238. C3×C5C2×#354direct (N × H)
258. C2×C5C3×#354direct (N × H)
286. C2×C3C5×#354direct (N × H)
288. C5C2×C3×#354direct (N × H)
304. C3C2×C5×#354direct (N × H)
309. C2C3×C5×#354direct (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×#1242
C2×C3×C5×#992
C3×C5×#3542
C2×C5×#3543
C2×C3×#3545