Finite Group Databasebeta

← C3×C4×C5×#126 #14275338

Subgroups — C3×C4×C5×#126 #14275338

Order 1920.

1032 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#126C211yesmaximal, characteristic
2C3×C5×#388C213yesmaximal
3C3×C5×#387C213yesmaximal, complemented
4C3×C5×#383C211yesmaximal, characteristic, complemented
5C3×C4×C5×C4⋊C4C213yesmaximal, complemented
6C3×C4×C5×C22⋊C4C213yesmaximal
7C3×C43×C5C211yesmaximal, characteristic, complemented
8C4×C5×#126C311yesmaximal, characteristic, direct product factor, Hall
9C23×C3×C4×C5C2211yescharacteristic
10C3×C5×#126C418yesdirect product factor
11C2×C3×C5×C4⋊C4C2213yes—
12C2×C3×C5×C4⋊C4C2213yes—
13C2×C3×C5×C22⋊C4C2213yes—
14C2×C3×C5×C22⋊C4C2213yes—
15C2×C3×C42×C5C2213yes—
16C2×C3×C42×C5C2213yes—
17C2×C3×C42×C5C2213yes—
18C2×C3×C42×C5C2211yescharacteristic
19C3×C5×#95C2213yes—
20C3×C5×#95C2213yes—
21C3×C5×#95C2213yes—
22C3×C5×#95C2213yes—
23C3×C4×#126C511yesmaximal, characteristic, direct product factor, Hall
24C2×C5×#126C2×C311yescharacteristic
25C5×#388C2×C313yes—
26C5×#387C2×C313yescomplemented
27C5×#383C2×C311yescharacteristic, complemented
28C4×C5×C4⋊C4C2×C313yescomplemented
29C4×C5×C22⋊C4C2×C313yes—
30C43×C5C2×C311yescharacteristic, complemented
31C24×C3×C5C2311yescharacteristic
32C22×C3×C4×C5—26no—
33C22×C3×C4×C5C2313yes—
34C22×C3×C4×C5C2311yescharacteristic
35C22×C3×C4×C5C2313yes—
36C22×C3×C4×C5C2313yes—
37C22×C3×C4×C5C2313yes—
38C22×C3×C4×C5C2311yescharacteristic
39C3×C5×C4⋊C4—26no—
40C3×C5×C4⋊C4C2×C4112yescomplemented
41C3×C5×C22⋊C4—26no—
42C3×C5×C22⋊C4C2×C4112yes—
43C3×C42×C5—26no—
44C3×C42×C5—26no—
45C3×C42×C5—26no—
46C3×C42×C5C2×C414yescomplemented
47C2×C3×#126C2×C511yescharacteristic
48C3×#388C2×C513yes—
49C3×#387C2×C513yescomplemented
50C3×#383C2×C511yescharacteristic, complemented
51C3×C4×C4⋊C4C2×C513yescomplemented
52C3×C4×C22⋊C4C2×C513yes—
53C3×C43C2×C511yescharacteristic, complemented
54C23×C4×C5C22×C311yescharacteristic
55C5×#126C3×C418yesdirect product factor
56C2×C5×C4⋊C4C22×C313yes—
57C2×C5×C4⋊C4C22×C313yes—
58C2×C5×C22⋊C4C22×C313yes—
59C2×C5×C22⋊C4C22×C313yes—
60C2×C42×C5C22×C313yes—
61C2×C42×C5C22×C313yes—
62C2×C42×C5C22×C313yes—
63C2×C42×C5C22×C311yescharacteristic
64C5×#95C22×C313yes—
65C5×#95C22×C313yes—
66C5×#95C22×C313yes—
67C5×#95C22×C313yes—
68C4×#126C3×C511yescharacteristic, direct product factor, Sylow
69C23×C3×C5—23no—
70C23×C3×C5—23no—
71C23×C3×C5C22×C412yes—
72C23×C3×C5C2411yescharacteristic
73C2×C3×C4×C5—23no—
74C2×C3×C4×C5—23no—
75C2×C3×C4×C5—23no—
76C2×C3×C4×C5—23no—
77C2×C3×C4×C5—23no—
78C2×C3×C4×C5—23no—
79C2×C3×C4×C5—23no—
80C2×C3×C4×C5—23no—
81C2×C3×C4×C5—23no—
82C2×C3×C4×C5—23no—
83C2×C3×C4×C5—44no—
84C2×C3×C4×C5C22×C416yes—
85C2×C3×C4×C5C22×C416yes—
86C2×C3×C4×C5#4116yes—
87C2×C3×C4×C5#4116yes—
88C23×C3×C4C22×C511yescharacteristic
89C3×#126C4×C518yesdirect product factor
90C2×C3×C4⋊C4C22×C513yes—
91C2×C3×C4⋊C4C22×C513yes—
92C2×C3×C22⋊C4C22×C513yes—
93C2×C3×C22⋊C4C22×C513yes—
94C2×C3×C42C22×C513yes—
95C2×C3×C42C22×C513yes—
96C2×C3×C42C22×C513yes—
97C2×C3×C42C22×C511yescharacteristic
98C3×#95C22×C513yes—
99C3×#95C22×C513yes—
100C3×#95C22×C513yes—
101C3×#95C22×C513yes—
102C24×C5C23×C311yescharacteristic
103C22×C4×C5—26no—
104C22×C4×C5C23×C313yes—
105C22×C4×C5C23×C311yescharacteristic
106C22×C4×C5C23×C313yes—
107C22×C4×C5C23×C313yes—
108C22×C4×C5C23×C313yes—
109C22×C4×C5C23×C311yescharacteristic
110C5×C4⋊C4—26no—
111C5×C4⋊C4C2×C3×C4112yescomplemented
112C5×C22⋊C4—26no—
113C5×C22⋊C4C2×C3×C4112yes—
114C42×C5—26no—
115C42×C5—26no—
116C42×C5—26no—
117C42×C5C2×C3×C414yescomplemented
118C2×#126C2×C3×C511yescharacteristic
119#388C2×C3×C513yes—
120#387C2×C3×C513yescomplemented
121#383C2×C3×C511yescharacteristic, complemented
122C4×C4⋊C4C2×C3×C513yescomplemented
123C4×C22⋊C4C2×C3×C513yes—
124C43C2×C3×C511yescharacteristic, complemented
125C22×C3×C5—26no—
126C22×C3×C5—41no—
127C22×C3×C5—43no—
128C22×C3×C5C23×C411yescharacteristic
129C22×C3×C5C2×#4113yes—
130C22×C3×C5C2×#4113yes—
131C3×C4×C5—26no—
132C3×C4×C5—26no—
133C3×C4×C5—26no—
134C3×C4×C5—41no—
135C3×C4×C5—43no—
136C3×C4×C5#12614yesdirect product factor
137C24×C3C23×C511yescharacteristic
138C22×C3×C4—26no—
139C22×C3×C4C23×C511yescharacteristic
140C22×C3×C4C23×C513yes—
141C22×C3×C4C23×C513yes—
142C22×C3×C4C23×C513yes—
143C22×C3×C4C23×C513yes—
144C22×C3×C4C23×C511yescharacteristic
145C3×C4⋊C4—26no—
146C3×C4⋊C4C2×C4×C5112yescomplemented
147C3×C22⋊C4—26no—
148C3×C22⋊C4C2×C4×C5112yes—
149C3×C42—26no—
150C3×C42—26no—
151C3×C42—26no—
152C3×C42C2×C4×C514yescomplemented
153C23×C5—23no—
154C23×C5—23no—
155C23×C5C22×C3×C412yes—
156C23×C5C24×C311yescharacteristic
157C2×C4×C5—23no—
158C2×C4×C5—23no—
159C2×C4×C5—23no—
160C2×C4×C5—23no—
161C2×C4×C5—23no—
162C2×C4×C5—23no—
163C2×C4×C5—23no—
164C2×C4×C5—23no—
165C2×C4×C5—23no—
166C2×C4×C5—23no—
167C2×C4×C5—44no—
168C2×C4×C5C22×C3×C416yes—
169C2×C4×C5C22×C3×C416yes—
170C2×C4×C5C3×#4116yes—
171C2×C4×C5C3×#4116yes—
172C23×C4C22×C3×C511yescharacteristic
173#126C3×C4×C518yesdirect product factor
174C2×C4⋊C4C22×C3×C513yes—
175C2×C4⋊C4C22×C3×C513yes—
176C2×C22⋊C4C22×C3×C513yes—
177C2×C22⋊C4C22×C3×C513yes—
178C2×C42C22×C3×C513yes—
179C2×C42C22×C3×C513yes—
180C2×C42C22×C3×C513yes—
181C2×C42C22×C3×C511yescharacteristic
182#95C22×C3×C513yes—
183#95C22×C3×C513yes—
184#95C22×C3×C513yes—
185#95C22×C3×C513yes—
186C2×C3×C5—42no—
187C2×C3×C5C4×#4113yes—
188C2×C3×C5C2×#12611yescharacteristic
189C2×C3×C5#52913yes—
190C23×C3—23no—
191C23×C3—23no—
192C23×C3C22×C4×C512yes—
193C23×C3C24×C511yescharacteristic
194C2×C3×C4—23no—
195C2×C3×C4—23no—
196C2×C3×C4—23no—
197C2×C3×C4—23no—
198C2×C3×C4—23no—
199C2×C3×C4—23no—
200C2×C3×C4—23no—
201C2×C3×C4—23no—
202C2×C3×C4—23no—
203C2×C3×C4—23no—
204C2×C3×C4—44no—
205C2×C3×C4C22×C4×C516yes—
206C2×C3×C4C22×C4×C516yes—
207C2×C3×C4C5×#4116yes—
208C2×C3×C4C5×#4116yes—
209C22×C5—26no—
210C22×C5—41no—
211C22×C5—43no—
212C22×C5C23×C3×C411yescharacteristic
213C22×C5C2×C3×#4113yes—
214C22×C5C2×C3×#4113yes—
215C4×C5—26no—
216C4×C5—26no—
217C4×C5—26no—
218C4×C5—41no—
219C4×C5—43no—
220C4×C5C3×#12614yesdirect product factor
221C24C23×C3×C511yescharacteristic
222C22×C4—26no—
223C22×C4C23×C3×C511yescharacteristic
224C22×C4C23×C3×C513yes—
225C22×C4C23×C3×C513yes—
226C22×C4C23×C3×C513yes—
227C22×C4C23×C3×C513yes—
228C22×C4C23×C3×C511yescharacteristic
229C4⋊C4—26no—
230C4⋊C4C2×C3×C4×C5112yescomplemented
231C22⋊C4—26no—
232C22⋊C4C2×C3×C4×C5112yes—
233C42—26no—
234C42—26no—
235C42—26no—
236C42C2×C3×C4×C514yescomplemented
237C3×C5C4×#12611yescharacteristic, direct product factor, Hall
238C22×C3—26no—
239C22×C3—41no—
240C22×C3—43no—
241C22×C3C23×C4×C511yescharacteristic
242C22×C3C2×C5×#4113yes—
243C22×C3C2×C5×#4113yes—
244C3×C4—26no—
245C3×C4—26no—
246C3×C4—26no—
247C3×C4—41no—
248C3×C4—43no—
249C3×C4C5×#12614yesdirect product factor
250C2×C5—42no—
251C2×C5C3×C4×#4113yes—
252C2×C5C2×C3×#12611yescharacteristic
253C2×C5C3×#52913yes—
254C23—23no—
255C23—23no—
256C23C22×C3×C4×C512yes—
257C23C24×C3×C511yescharacteristic
258C2×C4—23no—
259C2×C4—23no—
260C2×C4—23no—
261C2×C4—23no—
262C2×C4—23no—
263C2×C4—23no—
264C2×C4—23no—
265C2×C4—23no—
266C2×C4—23no—
267C2×C4—23no—
268C2×C4—44no—
269C2×C4C22×C3×C4×C516yes—
270C2×C4C22×C3×C4×C516yes—
271C2×C4C3×C5×#4116yes—
272C2×C4C3×C5×#4116yes—
273C2×C3—42no—
274C2×C3C4×C5×#4113yes—
275C2×C3C2×C5×#12611yescharacteristic
276C2×C3C5×#52913yes—
277C5C3×C4×#12611yescharacteristic, direct product factor, Sylow
278C22—26no—
279C22—41no—
280C22—43no—
281C22C23×C3×C4×C511yescharacteristic
282C22C2×C3×C5×#4113yes—
283C22C2×C3×C5×#4113yes—
284C4—26no—
285C4—26no—
286C4—26no—
287C4—41no—
288C4—43no—
289C4C3×C5×#12614yesdirect product factor
290C3C4×C5×#12611yescharacteristic, direct product factor, Sylow
291C2—42no—
292C2C3×C4×C5×#4113yes—
293C2C2×C3×C5×#12611yescharacteristic
294C2C3×C5×#52913yes—
subgroupG/H
centerC22×C3×C4×C5C23
commutatorC22C23×C3×C4×C5
Frattini, cosocleC23C24×C3×C5
socleC23×C3×C5C24
prime radicalC24×C3×C5C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×C5×#387C2semidirect
4. C3×C5×#383C2semidirect
5. C3×C4×C5×C4⋊C4C2semidirect
7. C3×C43×C5C2semidirect
8. C4×C5×#126C3direct (N × H)
10. C3×C5×#126C4direct (N × H)
23. C3×C4×#126C5direct (N × H)
26. C5×#387C2×C3semidirect
27. C5×#383C2×C3semidirect
28. C4×C5×C4⋊C4C2×C3semidirect
30. C43×C5C2×C3semidirect
40. C3×C5×C4⋊C4C2×C4semidirect
46. C3×C42×C5C2×C4semidirect
49. C3×#387C2×C5semidirect
50. C3×#383C2×C5semidirect
51. C3×C4×C4⋊C4C2×C5semidirect
53. C3×C43C2×C5semidirect
55. C5×#126C3×C4direct (N × H)
68. C4×#126C3×C5direct (N × H)
89. C3×#126C4×C5direct (N × H)
111. C5×C4⋊C4C2×C3×C4semidirect
117. C42×C5C2×C3×C4semidirect
120. #387C2×C3×C5semidirect
121. #383C2×C3×C5semidirect
122. C4×C4⋊C4C2×C3×C5semidirect
124. C43C2×C3×C5semidirect
136. C3×C4×C5#126direct (N × H)
146. C3×C4⋊C4C2×C4×C5semidirect
152. C3×C42C2×C4×C5semidirect
173. #126C3×C4×C5direct (N × H)
220. C4×C5C3×#126direct (N × H)
230. C4⋊C4C2×C3×C4×C5semidirect
236. C42C2×C3×C4×C5semidirect
237. C3×C5C4×#126direct (N × H)
249. C3×C4C5×#126direct (N × H)
277. C5C3×C4×#126direct (N × H)
289. C4C3×C5×#126direct (N × H)
290. C3C4×C5×#126direct (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×#1262
C3×C5×#3882
C3×C5×#3872
C3×C5×#3832
C3×C4×C5×C4⋊C42
C3×C4×C5×C22⋊C42
C3×C43×C52
C4×C5×#1263
C3×C4×#1265